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					Cardiovascular Systems
A 4-Volume Set
Editor: Cornelius T Leondes (University of California, Los Angeles, USA)

Computational Methods
ISBN-13 978-981-270-981-3
ISBN-10 981-270-981-9

Cardiovascular Systems
ISBN-13 978-981-270-982-0
ISBN-10 981-270-982-7

Muscular Skeletal Systems
ISBN-13 978-981-270-983-7
ISBN-10 981-270-983-5

General Anatomy
ISBN-13 978-981-270-984-4
ISBN-10 981-270-984-3
                                              A 4-Volume Set

    Cardiovascular Systems


       Cornelius T Leondes
             University of California, Los Angeles, USA

                                      World Scientific
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

A 4-Volume Set
Cardiovascular Systems
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
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ISBN-13 978-981-270-798-7 (Set)
ISBN-10 981-270-798-0     (Set)

ISBN-13 978-981-270-982-0
ISBN-10 981-270-982-7

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Printed in Singapore.

Because of rapid developments in computer technology and computational tech-
niques, advances in a wide spectrum of technologies, and other advances coupled
with cross-disciplinary pursuits between technology and its applications to human
body processes, the field of biomechanics continues to evolve. Many areas of sig-
nificant progress can be noted. These include dynamics of musculosketal systems,
mechanics of hard and soft tissues, mechanics of bone remodeling, mechanics of
implant-tissue interfaces, cardiovascular and respiratory biomechanics, mechanics
of blood and air flow, flow-prosthesis interfaces, mechanics of impact, dynamics of
man-machine interaction, and many more. This is the second of a set of four volumes
and it treats the area of Cardiovascular Systems in biomechanics.
    The four volumes constitute an integrated set. The titles for each of the volumes
•   Biomechanical   Systems   Technology:   Computational Methods
•   Biomechanical   Systems   Technology:   Cardiovascular Systems
•   Biomechanical   Systems   Technology:   Muscular Skeletal Systems
•   Biomechanical   Systems   Technology:   General Anatomy
    Collectively they constitute an MRW (Major Reference Work). An MRW is a
comprehensive treatment of a subject area requiring multiple authors and a number
of distinctly titled and well integrated volumes. Each volume treats a specific but
broad subject area of fundamental importance to biomechanical systems technology.
    Each volume is self-contained and stands alone for those interested in a spe-
cific volume. However, collectively, this 4-volume set evidently constitutes the first
comprehensive major reference work dedicated to the multi-discipline area of biome-
chanical systems technology.
    There are over 120 coauthors from 18 countries of this notable MRW. The chap-
ters are clearly written, self contained, readable and comprehensive with helpful
guides including introduction, summary, extensive figures and examples with com-
prehensive reference lists. Perhaps the most valuable feature of this work is the
breadth and depth of the topics covered by leading contributors on the interna-
tional scene.
    The contributors of this volume clearly reveal the effectiveness of the techniques
available and the essential role that they will play in the future. I hope that prac-
titioners, research workers, computer scientists, and students will find this set of
volumes to be a unique and significant reference source for years to come.

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Preface                                                               v

Chapter 1
A Simulation Study of Hemodynamic Benefits and Optimal Control
of Axial Flow Pump-Based Left Ventricular Assist Device               1
Huiting Qiao, Jing Bai and Ping He

Chapter 2
Techniques in Visualization and Evaluation of the
In Vivo Microcirculation                                              29
Shigeru Ichioka

Chapter 3
Impedance Cardiography: Development of the Stroke Volume
Equations and their Electrodynamic and Biophysical Foundations        49
Donald Philip Bernstein and Hendrikus J. M. Lemmens

Chapter 4
Indicator Dilution Techniques in Cardiovascular Quantification         89
Massimo Mischi, Zaccaria Del Prete and Hendrikus H. M. Korsten

Chapter 5
Analyzing Cardiac Biomechanics by Heart Sound                        157
Andreas Voss, R. Schroeder, A. Seeck and T. Huebner

Chapter 6
Methods in the Analysis of the Effects of Gravity and Wall
Properties in Blood Flow Through Vascular Systems                    207
S. J. Payne and S. Uzel

Chapter 7
Numerical and Experimental Techniques for the Study
of Biomechanics in the Arterial System                               233
Thomas P. O’Brien, Michael T. Walsh, Liam Morris, Pierce A. Grace,
Eamon G. Kavanagh and Tim M. McGloughlin

                                      CHAPTER 1


                               HUITING QIAO∗ and JING BAI
                   Institute of Biomedical Engineering, School of Medicine
                                 Beijing, 100084, P. R. China

                                           PING HE
            Department of Biomedical, Industrial and Human Factors Engineering
                                 Wright State University
                               Dayton, Ohio, 45435, USA

    This chapter gives an overview of various ventricular assist devices with a particular
    focus on axial flow pumps. Using the Hemopump as an example, we developed a model
    of a canine circulatory system assisted by an axial flow pump, and used computer simu-
    lation to predict the effects of the assist device under various hemodynamic conditions.
    In general, the results from the simulation are in good agreement with that observed in
    clinical and animal experiments. The same model is further used to explore the tech-
    niques and strategy for optimum control of the assist device by introducing an objective
    function, and choosing suitable membership functions with associated weighting factors,
    the model offers great flexibility in choosing the targeted hemodynamic variables, in
    specifying the particular way that each of these variables is to be optimized, and in
    assigning the relative importance of each targeted variable. The methods used and the
    results obtained in this study can be incorporated into the design of an advanced phys-
    iological controller for a long-term operation of the axial flow pump-based assist device
    as well as other types of continuous-flow LVAD.

    Keywords: Ventricular assist device; axial flow pump; model; simulation; optimum

1. Introduction
Heart failure is a major public health issue in the developed world. It is estimated
that in the United States alone, heart failure affects nearly 5 million patients.1
For patients with the most advanced heart failure, heart transplantation has been
the only means that prolongs and improves the quality of life. On the other hand,
because of the restrictive criteria for heart transplantation and the chronic shortage
of available donors, significant amount of research has been devoted to develop
mechanical cardiac assist devices that provide circulatory support to a failing heart
either as a destination therapy, or to serve as a bridge to heart transplantation or
    Many different types of ventricular assist devices have been developed and sig-
nificant amount of research has been devoted to improve the material, structure and

2                                H. Qiao, J. Bai & P. He

control strategy of these devices.4–8 Since many of these devices have been exten-
sively reviewed recently,3,9–12 in this chapter, we will just give a brief discussion
of several ventricular assist devices, and then focus on one particular type of left
ventricular assist devices that uses axial flow pump. We will use a computer model
to study the hemodynamic effects of the device in supporting a failing heart, and
to explore the techniques and strategy for optimum control of its operation.

2. A Brief Review of Ventricular Assist Devices
2.1. General objectives of ventricular assist devices
The heart serves as a mechanical pump of the circulatory system. In heart fail-
ure, its ability to pump the blood can be greatly diminished which can lead to
death. The best solution to a severely damaged heart is surgical replacement, or
heart transplantation. Despite the great success in cardiac transplantation, healthy
donor hearts are always in short supply. For example, in the United States, each
year approximately 40,000 healthy donor hearts are needed while only 2,300 are
available.13 The shortage of healthy heart donors and the restrictive criteria for
heart transplantation have prompted continuous research and development of vari-
ous mechanical cardiac assist devices.
    Ventricular assist device (VAD) is a kind of cardiac assist device that can par-
tially or totally replace the functions of the natural heart in pumping blood. The
general objectives of ventricular assist devices include improving the circulatory
state of the patient, maintaining adequate blood circulation throughout the body,
reducing the preload and afterload of the failing heart, decreasing myocardial oxy-
gen consumption, increasing oxygen supply to the heart by increasing coronary flow,
improving the contractibility of myocardium, strengthening the pump function of
the heart, and even replacing the failing heart temporarily before surgical heart
transplantation. According to the state of the cardiovascular system, ventricular
assist device may be used to provide left-ventricular assist, right-ventricular assist
and double-ventricular assist. Due to the high incidence of left-ventricle failure, the
great majority of ventricular assist devices are used for left-ventricular assist. For
simplicity, when ventricular assist is mentioned in this chapter, it generally refers to
left-ventricular assist, and the device is called the Left-Ventricular Assist Devices

2.2. Classification of ventricular assist devices
Ventricular assist devices are commonly composed of a pump, a control con-
sole, a power supply and cables. The pump is the central part of the ventric-
ular assist device. Ventricular assist devices are classified according to different
                      A Simulation Study of Hemodynamic Benefits                      3

2.2.1. Pulsatile and nonpulsatile assist devices
Based on the blood flow produced by the assist devices, ventricular assist devices
can be classified as pulsatile assist devices and nonpulsatile assist devices. Since the
flow pumped by a natural heart is pulsatile, it is generally considered that pulsatile
assist is preferred, but clinical evidences are not conclusive. For example, Nose14
reported that long-term nonpulsatile biventricular bypass did not produce harmful
physiological effects in animals, and nonpulsatile assist devices can function as well
as pulsatile assist devices if 20% higher blood flows are pumped. On the other hand,
Nishimura15 and Nishinaka16 suggested that prolonged nonpulsatile left ventricular
bypass may produce certain effects on the arterial structure and function, such as
on the wall thickness, smooth muscles and vasoconstrictive function of the vessels.
Structurally, while the devices that produce a pulsatile flow are relatively large, the
nonpulsatile ventricular assist devices can be much smaller and simpler, and have
a smaller thrombogenic foreign surface.

2.2.2. Parallel and sequential assist devices
Based on the relation between the blood flow in the assist device and the blood flow
in the ventricle, ventricular assist devices can also be classified as parallel assist
devices and sequential assist devices.
    When the pump of an assist device is connected directly between the left atrium
and the aorta, a portion of the total aortic flow is delivered by the assist device
and the remaining portion is pumped by the left ventricle. These devices are called
Parallel Left Ventricular Assist Device (PLVAD). An example is the atrio-aortic
LVAD (AA-LVAD), which withdraws blood from the left atrium and pumps it into
the ascending aorta.17 In the parallel assistant mode the assist device can take over
varying percentages of the pumping workload and blood flow. This kind of devices
aids the heart by pumping a portion of, or in some cases, all of aortic blood flow.
    Another kind of ventricular assist devices withdraws blood from the left ventricle
and delivers it to the aorta. Since the blood pumped by this kind of device first
passes through the left ventricle, such a device is called Sequential Left Ventricular
Assist Device (SLVAD). SLVAD can also be called as Serial Left Ventricular Assist
Device. Comparing with PLVAD, SLVAD is easier to implement.

2.2.3. Axial blood pump and other types of ventricular assist devices
Based on the structure of the pump, ventricular assist devices can be labeled as
the impeller type,18 the diaphragm type,19 the sac type,20 and the pusher-plate
type.21 The advantages of the impeller type pump include simple structure, small
volume, the ease of implant, low thrombogenic possibility and so on. The impeller
type pump can be further divided into centrifugal pump and axial flow pump.
   A centrifugal pump moves the blood by the centrifugal force generated by the
rotating impellers. In general, a higher aortic blood pressure can be produced when
4                                H. Qiao, J. Bai & P. He

the rotating speed of the impellers increases. In an axial flow pump, the rotating
impellers initially move the blood spirally. A diffuser is used to guide the blood to
finally move along the axis of the pump. Compared with the centrifugal pumps,
the axial flow pumps have smaller sizes, require less time to implant, consume less
power, and operate at much higher rotational speeds to deliver the desired blood
flow.13 Over the past 20 years, many intracorporeal axial flow pumps have been
developed and tested clinically that include the Hemopump, MicroMed DeBakey
VAD, Jarvik 2000, HeartMate II, and Berlin INCOR I.12,13,22,23 Combining with
other advantages, axial flow pumps have become the mainstream of ventricular
assist device used in clinic today.

2.3. Intra-aortic balloon pump, Hemopump and dynamic aortic
In the earlier years of the development of ventricular assist devices, great efforts
were concentrated on developing pulsatile assist devices. Prolonged clinical trials
however, also exposed some problems with these devices, such as the complicated
structure, the large volume, heavy weight and so on.24 The research efforts were
then turned to develop smaller and lighter ventricular assist devices with a simple
structure and low power-consumption. Since the ventricular assist devices are mostly
used as an implanted device, axial flow pumps have received much attention. The
Hemopump is an early type of LVAD that uses axial flow pump. Dynamic aortic
valve is a new type of axial flow pump-based LVAD that is still at the stage of
research and development.
    Though different ventricular assist devices are based on different principles and
are used for different cardiovascular situations, there are several common goals
of using these devices that include decrease of the workload of the left ventricle,
increase of the cardiac output, and increase of the oxygen supply to myocardial
tissues by increasing the blood flow to the coronary arteries. In the following para-
graphs, we will first give a brief description of three ventricular assist devices, Intra-
Aortic Balloon Pump, Hemopump and Dynamic Aortic Valve. We will then develop
a dynamic model of an axial flow pump and incorporate it into a canine circulatory
model to investigate the hemodynamic benefits and optimal control of the assist

2.3.1. Intra-aortic balloon pump
Intra-aortic balloon pump (IABP) has been widely used for cardiac assist since
Kantrowitz reported the first successful clinical trial in 1967.25 IABP played an
important role in the early development of cardiac assist devices, and it is the most
widely used cardiac assist device in clinic.
    The IABP consists of an inflatable balloon, a pressure source, and a control unit
that causes inflation and deflation of the balloon. The balloon is surgically inserted
into the femoral artery and advanced to the descending aorta, just distal to the
                        A Simulation Study of Hemodynamic Benefits                     5

             Fig. 1.   Illustration of the placement of intra-aortic balloon pump.

aortic arch, as shown in Fig. 1. The intra-aortic balloon pump is operated in a
counterpulsation mode, and its inflation and deflation are timed according to the
cardiac cycle: the balloon is deflated just before the opening of the aortic valve and is
inflated after aortic valve closure. Inflation of the balloon in early diastole displaces
a volume of blood back towards the aortic root, increasing the blood pressure there
and thus enhancing the coronary flow. Deflation of the balloon in end diastole or
early systole reduces aortic pressure and consequently reduces LV after-load for the
succeeding ejection.
    The most critical control of the operation of the IABP is the timings of bal-
loon inflation and deflation. For example, while active balloon deflation in early
systole greatly helps the left ventral to pump the blood, a delay of balloon deflation
into later systole would have much less beneficial effects. Similarly, while the balloon
inflation in diastole can greatly increase the coronary flow, the balloon inflation dur-
ing systole will cause a severe harm to the failing heart. To search for the optimal
control of IABP, various approaches have been used that include mathematical mod-
eling using lumped parameters,26 computer simulation with a multi-segment arterial
model,27 and direct experimental measurements using an performance index.28
    Though IABP is widely used to assist a failing heart during and after Car-
diotomy, the support provided by the IABP is sometime insufficient as it can only
increase cardiac output by 10% to 15%. More powerful assist devices are needed to
increase the survival rate of patients suffering from severe heart failure.
6                                 H. Qiao, J. Bai & P. He

2.3.2. Hemopump
The Hemopump was designed around 1982 while Dr. Wampler worked with the
Nimbus Corporation in California and a patent was filed in 1983.29 Clinical trials
of the Hemopump were conducted under an investigational device exemption (IDE)
approved by the FDA in 1988 and the preliminary results of the clinical trials were
encouraging.8,13,30 The Hemopump has been applied to the patients undergoing
typical cardiac surgery, minimally invasive coronary artery bypass graft,31 patients
suffering cardiogenic shock and in studies of percutaneous transluminal coronary
angioplasty.29 Though the clinical study with this device has been discontinued in
the United States, the experimental results with the Hemopump provided a valuable
basis for the design and development of other axial flow ventricular assist devices
that are undergoing clinical tests today.13
    The Hemopump consists of a disposable pump assembly, a high-speed motor
and a console, as shown in Fig. 2. The miniature axial flow pump is inserted into
the femoral artery and threaded up towards the aorta root. The inlet flow cannula
of the pump passes through the aortic valve with its tip situated inside the left

                        Fig. 2.   The composition of Hemopump.
                      A Simulation Study of Hemodynamic Benefits                       7

ventricle. Power to drive the pump is transmitted from an external electric motor
with a flexible cable threaded through a sheath. When the device is activated, the
pump impellers rotate and blood is drawn through the inlet cannula into the pump
and then discharged into the aorta. In this process the left ventricle is relieved of
its major workload. The capability of being inserted into the circulatory system
without major surgery makes the Hemopump as safe and convenient as IABP. On
the other hand, due to its ability to actively pump the blood and automatically
adjust the flow rate, the Hemopump is more effective than IABP in assisting left
    The rotating speed and pumping capacity of the Hemopump vary according to
the type of the device. For example, the Hemopump type 14F has a rotation speed
ranging from 27,600 rpm to 45,000 rpm, and the pump generates 1.3 L/min ∼ 2.3
L/min of blood flow. For type 24F (HP31), the rotation speed ranges from 17,000
rpm to 26,000 rpm, and the flow ranges from 3 L/min ∼ 5.1 L/min. The rotation
speed of the impeller is controlled by a control console. In general, a higher rotation
speed produces more blood flow. Yet a high speed is often associated with some
problems such as increased power consumption and the risk of blood cell damage.
As a result, optimum control of the Hemopump is an important area of research.

2.3.3. Dynamic aortic valve
As a new type of a LVAD, Dynamic aortic valve (DAV) was proposed by Li et al.
in 2001.32 Though the research of DAV is still in its early stage in laboratory, it has
caught much attention from researchers and surgeons due to its simple structure.
    The DAV consists of an external drive unit and an internal pump, which are
working together without any power cable connecting them. The pump is made up
of a support cage and a magnetic rotor-impeller, as shown in Fig. 3. The magnetic
rotor rotates in the presence of a revolving magnetic field generated by the external
drive unit. A typical DAV is 22 mm in diameter and 30 mm in length, and is inserted
to replace the natural aortic valve.
    The principle of the operation of DAV is to pump the blood out of the left
ventricle by rotating its impeller at a high speed as well as to replace the function of
the aortic valve. This latter function is achieved by maintaining a pressure difference

                     Fig. 3.   Schematic show of dynamic aortic valve.
8                                H. Qiao, J. Bai & P. He

between the left ventricle and aortic chamber all the time, and therefore preventing
blood regurgitation during diastole. Owning to its small volume, compact structure
and enough flux, DAV has a great potential to become an effective cardiac assist

3. Simulation Study of the Hemopump
Assessment of the performances and assistant effects of an axial flow pump-based
assist device under various hemodynamic conditions in a clinical setting is difficult,
while the method of modeling and simulation is more flexible and convenient. Using
an adequately developed model of the axial flow pump (in the following text, we
may simply call it the flow pump, or the pump) and the circulatory system, the
operation mode of the pump and the parameters of the circulatory system can be
adjusted easily, and the hemodynamic variables of the system at various locations
can be observed conveniently. Consequently, the performance and assistant effects
of the pump can be assessed under different conditions. Such a model can also be
used for the study of the optimal control of the pump.

3.1. Modeling the Hemopump
To evaluate the assist effects of the pump, a static model and a dynamic model of the
pump are established based on the experimental data obtained from a particular
kind of axial flow pump, the Hemopump type HP31, and the dynamic model is
incorporated into a multi-element cardiovascular model.33

3.1.1. Static model of the pump
Assuming that the inflow of the pump is not hampered, the flow produced by the
pump is a function of the rotation speed of the pump and the pressure difference
between the outflow and inflow of the pump.34 In a typical clinical application, the
pressure difference between the outflow and inflow of the pump is the aortic pressure
minus the left ventricular pressure.
   The idealized relation between the static pump flow, Qstat , and the pump pres-
sure head, ∆h, can be described by the following Euler equation35 :

                              ∆Pstat   u2  u cot β
                       ∆h =          =    − ·      Qstat ,                           (1)
                               ρg      g   g πd b

where ∆Pstat is the static pressure difference, ρ is the density of the fluid, g is the
gravity acceleration, u is the linear velocity of the liquid (e.g. blood), β is the angle
of the impeller outlet, d is the diameter at the impeller output, and b is the width
of the pump impeller.
                     A Simulation Study of Hemodynamic Benefits                      9

   If we express u as a function of the rotation speed n in rpm
                                       u=                                        (2)
   Equation (1) can be transformed into

                            Qstat = An − Bn · ∆Pstat ,                           (3)

                            nbπ 2 d2                   60b
                     An =               and Bn =              .                  (4)
                            60 cot β                 nρ cot β

    Equation (3) shows a linear relationship between the pump flow and the pres-
sure difference for a fixed rotation speed n. Given the values of An and Bn at a
certain rotation speed, the static pump flow under various pressure differences can
be determined.
    The Hemopump type HP31 can be operated at 7 different rotation speeds rang-
ing from 17,000 rpm (Speed 1) to 26,000 rpm (Speed 7) with an interval of 1,500
rpm. Meyns et al.34 performed in vitro measurements of the static flow–pressure
difference relations of HP31 Hemopump at the above 7 speeds up to a pressure dif-
ference of 230 mmHg. The data (Fig. 2, in Ref. 34) showed a nearly linear relation
between the flow and pressure difference in the region of low-pressure difference.
In the region of high-pressure difference, this linear relation broke down due to
flow separation within the rotor. To establish a static model of the Hemopump, an
up-limit of the pressure difference of 130 mmHg is used so that no flow separation
occurs. Consequently, the values of the two coefficients, An and Bn in Eq. (3), are
obtained by fitting a least squares line to the corresponding data in the range of
0–130 mmHg. Figure 4 shows the seven fitted straight lines and Table 1 lists the
values of An and Bn for each of the seven pump rotation speeds.
    A static model of the pump based on Eq. (3) is shown in Fig. 5, which consists
of a constant flow source Qn and a resistance Rn . The model gives the following
relation between the pump flow Qstat and the pressure difference ∆Pstat :
                            Qstat = Qn −        ∆Pstat .                         (5)
By comparing Eq. (5) with Eq. (3), we have:

                         Qn = An       and Rn = 1/Bn ,                           (6)

both depend upon the rotation speed of the Hemopump, as shown in Table 1.
   In the static model shown in Fig. 5, Qn represents the blood flow of the pump
with rotation speed n when pressure difference between inlet and outlet is zero, while
Rn represents the total resistance to the blood flow in the flow pump, including both
the effect of the impeller and blood viscosity.
10                                   H. Qiao, J. Bai & P. He

Fig. 4. Results of least squares fit to the reported data obtained from in vitro measurements of
Hemopump HP31 operated at seven rotation speeds.

         Table 1. Values of the two parameters of the linear model shown in Eq. (3)
         for the Hemopump HP31 operated at seven levels of rotation speed.

         Speed level      1         2        3         4        5           6     7
             An        56.744     64.056   69.655   75.118   80.928   85.105    87.598
             Bn        0.5381     0.5072   0.4851   0.4260   0.3726   0.3014    0.2274

                       Fig. 5.   The static model for the axial flow pump.
                     A Simulation Study of Hemodynamic Benefits                     11

3.1.2. Dynamic model of the axial flow pump
Equation (3) is for static flow under constant pressure difference. When the pump
works in a living circulatory system, the pressure at inlet (Pin = blood pressure in
left ventricle), the pressure at the outlet (Pout = blood pressure in aorta), and the
pump flow all change with time during a cardiac cycle, and the inertial property of
the liquid needs to be considered. To include the effects of the inertial property of
the blood in the pump, the static model of the pump is modified by including an
inductor H, as shown in Fig. 6. According to the mechanical property of the blood,
the inertial property of the blood manifests itself as a force that opposes the change
of the flow velocity36 :
                             d            ∂m    ∂v    dv
                     F =        (m · v) =    v+    m=    m,                       (7)
                             dt           ∂t    ∂t    dt
where F is the force, v and m are, respectively, the velocity and mass of the fluid
within the cannula of the pump. Equation (7) is applicable only for incompressible
    Equivalently, the inertial property of the blood can be modeled by a pressure
difference ∆P , which is induced by the change of the flow rate. Numerically, ∆P
is equal to the force F divided by the cross section area A of the cannula of the
                              F  1 dv       ρ dQH
                     ∆P =       = ·   ·m= L· ·    ,                               (8)
                              A  A dt       A  dt
where L is the length of the cannula; ρ is the fluid density; and QH is the volumetric
flow rate. Based on Eq. (8), the value of H in Fig. 6 can be determined as:
                                     H =L·       .                                (9)
   Using the data from Meyns et al.37 and Reul38 :

                                       L = 8.5 cm

                   Fig. 6.   The dynamic model for the axial flow pump.
12                               H. Qiao, J. Bai & P. He

                            A ≈ πd2 /4 = π · 8.12 /4 mm2
                                ρ = 1.06 · 103 kg/m3
we obtain:
                                H = 1.748 Pa · S2 /ml

3.1.3. Cardiovascular model including the axial flow pump
A canine cardiovascular model has been developed previously that can simulate
the dynamic relationship between the cardiac function and the vascular function,
as well as be used as a tool to investigate the mechanism of heart failure and the
effects of cardiac assist device.26,39,40 As shown in Fig. 7, the model consists of four
heart chambers, a pulmonary circulation, 11 aortic segments, and lumped venous
and peripheral vascular systems. The detailed structure and operation of each block
of the model can be found in Bai and Zhao40 and Zhou et al.39 Here we just give a
brief description of the model for the left ventricle with regional ischemia.
    The left ventricle is functionally divided into two non-physiological
compartments, one consisting of all the normal myocardium and the other consisting
of all the ischemic myocardium. The ratio of the ischemic myocardial mass to the
total myocardial mass, denoted by Rm , is set to be 30% in this study. During each
cardiac cycle, each compartment generates its own pressure-volume loop, which in
turn determines the pressure and volume of the entire left ventricle. The maximum
elasticity (which is related to myocardial contractility) of the normal compartment
(Ees1 ) is 5 mmHg/ml and the maximum elasticity of the ischemic compartment
(Ees2 ) is 1.5 mmHg/ml.
    The canine circulatory model shown in Fig. 7 also includes the dynamic model
of the pump shown in Fig. 6. To simulate the typical clinic setting,30 the inlet of the
pump in our model is inserted into the left ventricle where the pressure is denoted
as PLV , and the outflow of the pump is discharged into the descending aorta where
the pressure is denoted as Pa [4] (the number 4 refers to the 4th aortic segment). As
a result, Pin and Pout in Fig. 6 become PLV and Pa [4], respectively.

3.2. Computer simulations and results
Simulations were performed using the above model which represents a canine car-
diovascular system assisted by the axial flow pump. The heart rate was set to be 120
beats/min. The computer program for the simulation study was written in Delphi
language and run on a PC. The time interval for computation was 0.001 second.
During each time interval, the pressure, flow and volume in each block of the model
were computed and updated sequentially, starting from the left and right ventricles.
   Using the above model and computer simulation, the system parameters can be
changed easily and the hemodynamic variables in various parts of the circulatory
system can be observed conveniently. Consequently, the effects of the assist device
                       A Simulation Study of Hemodynamic Benefits                            13

Fig. 7. Block diagram of the model for the canine circulatory system including a axial flow pump

under different cardiovascular conditions can be assessed conveniently. In general,
the results obtained by computer simulation are consistent with the experimental
results reported previously.41,42

3.2.1. The pump flow and pressure difference during one cardiac cycle
The simulation results show that the change of flow (Q) lags behind that of pressure
difference (∆P ) by about 0.5 s at a heart rate of 120/s. The time course of Q versus
∆P during a cardiac cycle forms a loop which proceeds clockwise. This phenomenon,
which is due to the inertial property of the blood within the cannula and pump, was
also observed previously by Meyns et al.34 in their in vivo study of the Hemopump.
14                              H. Qiao, J. Bai & P. He

The hysteresis of the loop reduces as pump speed increases. This can be explained
by the fact that when the pump speed is increased, more blood is expelled by the
pump and the ventricular cavity is reduced. As a result, the contribution of the heart
to the pump flow is reduced, which is illustrated by the decrease in the upstroke in
flow during the ejection phase at high speeds.
    At low pump rotation speed, the flow shows significant pulsation. As the pump
speed rises, the pulsation of pump flow decreases.

3.2.2. The effects of the pump on various hemodynamic variables
Figure 8 shows the effects of the pump on various hemodynamic variables when the
pump speed is increased from 1 to 5. Figure 8(a) plots the aortic pressure during
a cardiac cycle. As the rotation speed increases, the mean aortic pressure increases
while the pulsation of the aortic pressure diminishes. The same trends have been
reported in clinical experiments.37,41,43
    Figure 8(b) shows the blood flow pumped out by the left ventricle. Only at
Speed 1, there is noticeable blood flow pumped out by the left ventricle during
early systole. As the pump rotation speed is further increased, the amount of blood
pumped out by the left ventricle becomes negligible. This observation is consistent
with the results reported by Peterzen et al.41 They noticed that the aortic valve
was almost always closed when the Hemopump was operating.
    The change in the volume of the left ventricle during each cardiac cycle is shown
in Fig. 8(c). The volume of the left ventricle decreases with the increase of the
rotation speed of the pump. At Speed 5, the volume of the left ventricle decreases
to a rather small value of 20 ml.
    The pressure-volume loop of the left ventricle is depicted in Fig. 8(d). As the
pump speed increases, the loop moves to the left (the mean volume decreases) and
the area of the loop decreases, indicating a decrease in the workload of the left
    Figure 8(e) shows the change of the left atrial volume during the cardiac cycle.
As the pump speed increases, the left atrial volume decreases, which may explain
the potential collapse of the left atrium at high speeds reported in literature.42

3.2.3. The beneficial effects of the axial flow pump on a failing heart
Figure 9(a) shows the simulation results of the effects of the pump on the stroke
volume which is the summation of the blood pumped out by the left ventricle and
the blood pumped out by the pump during a cardiac cycle. Since the contribution
of the left ventricle in pumping blood quickly becomes insignificant as the pump
rotation speed increases (see Fig. 8(b)), the increase in the stroke volume is almost
entirely due to the increase in the pump flow.
                        A Simulation Study of Hemodynamic Benefits                               15

Fig. 8. The effects of the pump on various hemodynamic variables when operated at five different
speeds. (a) aortic pressure (b) blood flow pumped out through the aortic valve by the left ventricle
(c) volume of left ventricle (d) pressure-volume relationship of left ventricle during the cardiac
cycle (e) volume of left atrium.
16                                H. Qiao, J. Bai & P. He

Fig. 9. The assistant effects of the pump at different rotation speeds: (a) stroke volume (b)
myocardial oxygen consumption (white bars) and oxygen supply (black bars).

    The total myocardial oxygen consumption, VO2 , of the left ventricle is considered
as the summation of two parts45 :
                              VO2 = VO2 (1) + VO2 (2)                                 (10)
          VO2 (1) = B · PVA(1) + C · Ees1 · (1 − Rm) + D · (1 − Rm)                   (11)
                 VO2 (2) = B · PVA(2) + C · Ees2 · Rm + D · Rm                        (12)
where VO2 (1) and VO2 (2) are the volume of oxygen consumed by the normal com-
partment and the ischemic compartment of the left ventricle, respectively; PVA(1)
and PVA(2) are the area of the individual pressure-volume loop of each compart-
ment, and Ees1 and Ees2 are the maximal elasticity of the normal region and the
ischemic region of the left ventricle, respectively. The values of the three coefficients
used in the simulation are obtained based on the work of Suga et al.46 :
                                B = 1.65 × 10−5 /mmHg
                                C = 0.0024 ml2 /mmHg
                                     D = 0.0177 m
   The volume of myocardial oxygen supply (VOS ) is calculated from the total
coronary flow during one cardiac cycle (FT C ):
                                   (A · T )O2
                              VOS =           · FT C                          (13)
where A is the volume percent of the oxygen content in the coronary arterial blood
and T is a variable that depicts the ability of the myocardial oxygen absorption.
The typical values of A and T are determined from the work of Walley et al.47 and
a final value of VOS = 0.0015 FT C is used.
   Figure 9(b) compares the myocardial oxygen consumption (VOC ) and the volume
of myocardial oxygen supply (VOS ) at each of the 5 pump speeds. The myocardial
                      A Simulation Study of Hemodynamic Benefits                    17

oxygen consumption (VOC ) of the left ventricle is directly related to the area of
pressure-volume loop. Figure 8(d) indicates when the pump speed increases, the area
of the pressure-volume loop of the left ventricle decreases. As a result, the oxygen
consumption decreases as the pump speed increases. Alternatively, the decrease
in oxygen consumption of the left ventricle can be explained by the reduction of
its preload that is related to the left atrium volume and the left ventricular end-
diastolic volume. With the increase of pump speed, both the left atrium volume
and the left ventricular end-diastolic volume decrease, as indicated in Figs. 8(e) and
8(c). Consequently, the oxygen consumption is decreased. As the pump rotation
speed increases, the aortic pressure increases (Fig. 8(a)). As a result, the coronary
flow increases, so does the oxygen supply.
    Figure 9 depicts the three most important beneficial effects of the pump to a
failing heart: an increase in the stroke volume, an increase in the oxygen supply,
and a reduction in the oxygen consumption. These benefits were also reported by
Peterzen et al.41 based on their clinic experiments. Figure 9 also shows that each of
the effects becomes more significant when the pump rotation speed increases.
    Finally, the results of the computer simulation regarding the beneficial effects
of pump in terms of the values of several relevant hemodynamic variables at five
pump speeds are summarized in Table 2.

               Table 2. Values of various hemodynamic variables when the
               pump is operated at five different rotation speeds.

               Pump Speed           1        2         3         4      5
               Qmpm (ml/s)       19.917    27.693    30.783   33.819   37.924
               Pmao (mmHg)       99.419    105.23    113.54   122.24   133.70
               Paosy (mmHg)      108.31    110.72    116.33   123.63   133.92
               Paodi (mmHg)      90.402    100.90    111.15   121.22   133.58
               Ppao (mmHg)       17.913    9.8177    5.1769   2.4047   0.3369
               Plvsy (mmHg)      108.33    99.500    68.586   31.273   4.4318
               Vlvsy (ml)        51.035    46.200    35.601   23.350   17.032
               Vladi (mmHg)      9.3042    8.1257    6.3562   4.7155   3.1842
               SV (ml/beat)      12.962    13.951    15.380   16.865   18.853
               VOC (ml/beat)     0.0683    0.0568    0.0421   0.0311   0.0274
               VOS (ml/beat)     0.0579    0.0673    0.0830   0.0998   0.1163

               Qmpm = mean pump flow
               Pmao = mean aortic pressure
               Paosy = aortic peak systolic pressure
               Paodi = aortic minimum diastolic pressure
               Ppao = pulsation of aortic pressure (Paosy – Paodi )
               Plvsy = left ventricle peak systolic pressure
               Vlvsy = left ventricle end systolic volume
               Vladi = left atrium end diastolic pressure
               SV = stroke volume
               VOC = volume of myocardial oxygen consumption
               VOS = volume of myocardial oxygen supply
18                               H. Qiao, J. Bai & P. He

4. Optimum Control of the Axial Flow Pump
The results of the simulation study indicate that the direct effects of the axial
flow pump include an increase in stroke volume, an increase in mean aortic pres-
sure and coronary blood flow, a decrease in the area of the pressure-volume loop
of the left ventricle and a decrease in left-atrial volume. The results also show
that all these changes are further enhanced when the rotation speed of the pump
impellers is increased. In practical applications however, it is not always benefi-
cial to operate the pump at the highest possible speed. For example, in animal
experiments with the Hemopump, Siess et al.36 found that at high pump speeds,
a total collapse of the left ventricle can occur that leads to a sudden decrease
in stroke volume. In addition, it was suggested that the shear forces inside the
high-speed pump can cause blood trauma and thrombosis.48 Finally, a high pump
speed requires greater power consumption. An important question is therefore
how to achieve the clinical objectives of the assist device with the lowest pump
    In the following sections, we describe a general framework for designing an opti-
mum control strategy for the axial flow pump and test the strategy using the sim-
ulation model described in the previous sections.49 An objective function is first
defined that includes four hamodynamic variables with suitable weighting factors:
stroke volume, mean left-atrial pressure, aortic diastolic pressure and mean pump
speed. The flow pump is then allowed to operate at either a constant speed or
two different speeds during a cardiac cycle. The goal is to maximize the objective
function by varying the magnitude and timing of the pump speed. The results of
simulation indicate that in general, different clinical objectives or different cardiac
conditions require different operation parameters. The results also suggest that it
is more beneficial to operate the pump at two different speeds than to maintain a
constant speed throughout the cardiac cycle.

4.1. The objective function for optimal control of the axial flow
A general approach in optimal control is to maximize the value of an objective
function (or a performance index) by adjusting a certain set of operation variables.50
The objective function (OF) has the following general form:

                OF = w1 · µ(v1 ) + w2 · µ(v2 ) + · · · + wn · µ(vn ),             (14)

where v1 , v2 , . . . , vn are the members of the objective function, µ(vi ) represents
the membership function of vi , and w1 , w2 , . . . , wn are the weighting factors. To
quantitatively evaluate the objective function, three questions need to be answered:
how to choose the members, how to construct the membership functions, and how
to determine the weighting factors.
                     A Simulation Study of Hemodynamic Benefits                       19

4.1.1. Determine the members of the objective function
The general clinical objectives of the flow pump-based assist device are to increase
the cardiac output, to increase the oxygen supply to the failing heart, and to decrease
the workload of the heart. The actual effects produced by the assist device can
be assessed by measuring the changes in the major hemodynamic variables from
their preoperative values. In choosing the members of the objective function for
optimal control of the pump, a logical approach is to include a minimum number of
hemodynamic variables that best represent the clinical objectives and are relatively
easy to measure clinically.
    The selected members of the objective function are: stroke volume (defined as
the combined volume of the blood pumped out by the left ventricle and the blood
pumped out by the pump during a cardiac cycle), mean left atrial pressure, mini-
mum aortic (diastolic) pressure, and mean pump speed. The first two variables are
directly related to the clinical objectives of the assist device. In fact, a low cardiac
index (< 2.0 L·min−1 · m−2 ) and a high pulmonary wedge pressure (> 20 mmHg),
which is directly related to mean left atrial pressure, are two important criteria
for selecting patients for receiving LVAD.51,52 After the assist device is implanted,
these two variables can readily be measured in vivo for hemodynamic assessment.22
The other two members, minimum aortic pressure and pump speed, are also eas-
ily measurable. The choice of minimum aortic pressure can be justified as follows.
As mentioned previously, increase of coronary blood flow is one of the important
objectives of the assist device. Coronary flow is mainly determined by mean aortic
diastolic pressure. An increase in minimum aortic pressure directly increases coro-
nary flow. On the other hand, an abnormally high aortic diastolic pressure increases
the afterload of the left ventricle, and can produce damaging effects to certain organs
such as eyes, brain and kidney. Consequently, there is a desired range of aortic dias-
tolic pressure for a healthy cardiovascular system, and this information can be used
to construct the membership function. Finally, the mean pump speed is included in
the objective function as a penalty term.

4.1.2. Establish the membership function
The four members of the objective function, stroke volume (SV ), mean left atrial
pressure (PMLA ), minimum aortic pressure (PMAO ) and mean pump speed (M P S)
have different desired ranges and different units. In order to have a uniform treat-
ment of their individual contributions to the objective function, the membership
function for each member is established. The value of each membership function
ranges from 0 to 1, with 1 represents the condition that the member reaches its
desired value. In general, the desired value is the one found in a healthy cardiovas-
cular system.
   The membership function for stroke volume (SV ) is chosen to be a sigmoid
                            µ(SV ) =                 ,                           (15)
                                      1 + e−a(SV −b)
20                                  H. Qiao, J. Bai & P. He

where a and b are two parameters which can be determined by two boundary values
of SV . Let SV2 represent the desired stroke volume (upper bound) that µ(SV2 ) =
1 − ε where ε is a small positive value (e.g. 0.01), and let SV1 represent the lower
bound that µ(SV1 ) = ε. Then
                         2 ln[(1 − ε)/ε]                SV1 + SV2
                    a=                      and b =               .                       (16)
                           SV2 − SV1                        2
    Based on out simulation results from a health canine heart, SV2 = 15 ml/beat,
SV1 = SV2 /3, and ε = 0.01. That makes a = 0.919, and b = 10. The actual curve
is shown in Fig. 10(a).
    The membership function for mean left atrial pressure (PMLA ) is defined as
                           1               PMLA ≤ 8 mmHg
                µ(PMLA ) =     −(PMLA −8)2                   ,               (17)
                           e      2σ2      PMLA > 8 mmHg

Fig. 10. Membership function of (a) stroke volume (SV ); (b) mean left atrial pressure (PMLA );
(c) minimum aortic pressure (PMAO ); (d) mean pump rotation speed (MPS ).
                     A Simulation Study of Hemodynamic Benefits                     21

where PMLA = 8 mmHg is considered as the normal value for a healthy canine
heart. The parameter σ determines the rate of decrease of the membership function
as PMLA increases. Based on the clinical criterion of patient selection52 as well as
simulation results, a σ = 7 mmHg is chosen. The actual curve is shown in Fig. 10(b).
    The membership function for minimum aortic pressure (PMAO ) is determined
based on the following considerations. The International Society of Hypertension,
World Health Organization defined the desirable range of PMAO in human as 70
mmHg – 80 mmHg.53 Considering that in this study the optimum control strategy
is tested with a canine model and the health canine model produces a PMAO of
93 mmHg, the desirable range of PMAO is shifted to 90 mmHg–100 mmHg. More
specifically, the membership function for PMAO have a value of 1 when PMAO falls
into this range, and have a descending branch when PMAO is either below 90 mmHg
or above 100 mmHg. Mathematically, the membership function for PMAO is defined
by the following expressions:
                                  2
                    e −(PM2σ2 −90)
                                     (σ = 20) 0 ≤ PMAO < 90 mmHg
      µ(PMAO ) = 1                             90 ≤ PMAO ≤ 100 mmHg             (18)
                    −(PM AO −100)2
                              2
                     e      2σ       (σ = 30) PMAO > 100 mmHg

The actual curve is shown in Fig. 10(c).
   Finally, a straight line is used to define the membership function of mean pump
speed (MPS). It is assigned that µ(MPS ) = 1 when MPS = 0, and µ(MPS ) = 0.5
when the pump is operated at the highest speed (26,000 rpm, Speed 7).34 The
expression for the membership function is therefore:

                         µ(MPS ) = 1 − MPS /52, 000,                             (19)

and the actual curve is shown in Fig. 10(d).

4.1.3. Determine the weighting factor for each membership
The weighting factors in Eq. (14) represent the relative importance of each member
in achieving the clinical objectives of the assist device. Depending upon the car-
diovascular conditions of an individual patient and the specific clinical objectives,
a different set of the weighting factors may be assigned. In this study, two cases
are demonstrated. In the first case, the main clinical objective is to increase cardiac
output, and the membership function µ(SV ) receives the largest weight. Without
quantitative justification, the following set of weighting factors is defined for the
purpose of demonstration, w(µ(SV )) = 1, w(µ(PMLA )) = 1/3, w(µ(PMAO )) = 1/3,
and w(µ(MPS )) = 1/5. This set of weighting factors means that to increase SV is 3
times more important than to increase PMAO or to lower PMLA , and 5 times more
important than to lower MPS. In the second case, the main clinical objective is to
increase coronary flow and the membership function µ(PMAO ) receives the largest
22                                H. Qiao, J. Bai & P. He

weight. In this case, the relative weights become [1/3 1/3 1 1/5]. After normal-
ization (the sum of the four weighting factors equals 1), the objective function in
each case can be expressed as:
    Case 1 (emphasizes SV ):

                    OF = 0.5357µ(SV ) + 0.1786µ(PMLA)
                           + 0.1786µ(PMAO ) + 0.1071µ(MPS ),                     (20)

     Case 2 (emphasizes PMAO ):

                    OF = 0.1786µ(SV ) + 0.1786µ(PMLA)
                           + 0.5357µ(PMAO ) + 0.1071µ(MPS ).                     (21)

4.2. The model used in the simulation
The canine cardiovascular model shown in Fig. 7 is used again in this part of the
study. The model includes a failing heart assisted by the axial flow pump. Three
conditions of the left ventricle are simulated in this study: normal (the ratio of the
ischemic myocardial mass to the total mass, denoted by Rm , is 0), minor ischemia
(Rm = 25%), and severe ischemia (Rm = 50%). When the model simulates a
normal left ventricle, it produces the following hemodynamic values: SV = 13.4
ml, PMLA = 10.8 mmHg, and PMAO = 93.1 mmHg. These values are used as the
baseline, preoperative hemodynamic values.

4.3. Optimum control of the axial flow pump using a single
     pump speed
Table 3 lists the values of SV, PMLA , and PMAO for a heart with minor ischemia
(Rm = 25%) when the Hemopump is off as well as when it is operated at Speed 1
to Speed 4. The last two rows of the table list the values of the objective function
calculated using the two sets of weighting factors defined by Eq. (20) and Eq. (21).
    As shown in Table 3, when the ischemic left ventricle is not assisted by the pump,
the SV is decreased by 10%, PMLA is increased by 19%, and PMAO is decreased by
11%, as compared with the baseline values of a normal heart. When the pump is
turned on, these variables move towards the normal values. The objective function
is maximized at Speed 2 under both sets of weighting factors.
    Table 4 lists the values of SV , PMLA , and PMAO for a heart with severe ischemia
(Rm = 50%) under various operating speeds of the pump. As compared with the
baseline values of a normal heart, the SV is decreased by 19%, PMLA is increased by
37%, and PMAO is decreased by 22% when the pump is turned off. When the pump
is operating, the objective function is maximized at Speed 3 when SV is emphasized
(Case 1), and maximized at Speed 2 when PMAO is emphasized (Case 2).
                      A Simulation Study of Hemodynamic Benefits                        23

        Table 3. Single pump speed and minor ischemia. SV, PMLA , and PMAO at
        different pump speed and corresponding values of the objective function (OF )
        using two sets of weighting factors (Case 1 and Case 2).

        Rm = 25%            Pump       Pump         Pump        Pump         Pump
                            is off     Speed 1      Speed 2     Speed 3      Speed 4
        SV (ml/beat)         12.0       13.5         14.3        15.7         17.2
        PMLA (mmHg)          12.8       10.7         9.68        8.07         6.61
        PMAO (mmHg)          82.4       91.5         101         112          123
        OF (Case 1)         0.877       0.932       0.946        0.942       0.910
        OF (Case 2)         0.901       0.946       0.953        0.915       0.819

        Table 4. Single pump speed and severe ischemia. SV, PMLA , and PMAO at
        different pump speed and corresponding values of the objective function (OF )
        using two sets of weighting factors (Case 1 and Case 2).

        Rm = 50%            Pump       Pump         Pump        Pump         Pump
                            is off     Speed 1      Speed 2     Speed 3      Speed 4
        SV (ml/beat)         10.8       12.4         13.7        15.3         17.0
        PMLA (mmHg)          14.8       12.3         10.6        8.68         6.89
        PMAO (mmHg)          72.9       87.5         98.7        110          122
        OF (Case 1)         0.707       0.880       0.932        0.943       0.912
        OF (Case 2)         0.712       0.913       0.944        0.925       0.827

4.4. Optimum control of the axial flow pump using two
     pump speeds
When the pump is operated at a constant speed throughout the cardiac cycle,
it is referred to the non-synchronous (with the heart) operation. Considering the
prospective long-term usage of the assist device, we cannot ignore the concern
regarding the long-term physiological effects of a nonpulsatile circulation. In the
synchronous pumping mode, the pump is operated at a lower speed during diastole
(to avoid backflow) and is operated at a higher speed during systole. Figure 11
depicts the general timing of the pump speed in one cardiac cycle which is 500 ms
in our simulation. Speed A represents the lower speed and Speed B represents the
higher speed; t1 is the time when the pump speed is switched from Speed A to Speed
B, and t2 is the time when the pump speed returns from Speed B to Speed A. For
each heart condition (Rm = 25% and Rm = 50%) and each weighting set (Case 1
and Case 2), all possible combination of two speeds and all possible timing of each
speed — from 0 to 500 ms with an increment of 20 ms, are systematically tested
and the corresponding value of the objective function is calculated.
    Table 5 gives the optimal pump speed and timing that maximize the value of
the objective function for the case of minor ischemia (Rm = 25%), and Table 6
gives the results for the case of severe ischemia (Rm = 50%). In both tables, Speed
A represents the lower speed and Speed B represents the higher speed; t1 is the
24                                    H. Qiao, J. Bai & P. He

Fig. 11. Schematic two speed control strategy during one cardiac cycle with the left ventricular
pressure (Plv ).

                Table 5. Optimal pump speeds and timings that maximize the
                objective function (OF ) in Case 1 and Case 2 for minor ischemic

                Rm = 25%      t1 (ms)    t2 (ms)   Speed A      Speed B     OF
                  Case 1         40        400         1           4       0.954
                  Case 2         40        440         1           3       0.956

                Table 6. Optimal pump speeds and timings that maximize the
                objective function (OF ) in Case 1 and Case 2 for severe ischemic

                Rm = 50%      t1 (ms)    t2 (ms)   Speed A      Speed B     OF
                  Case 1         40        380         1           5       0.953
                  Case 2         20        400         1           4       0.955

time when the pump speed is switched from Speed A to Speed B, and t2 is the time
when the pump speed returns from Speed B to Speed A.
    When the time course of the pump speed is plotted together with the time course
of the left ventricular pressure PLV during a cardiac cycle, it is found that in general,
optimal pump operation calls for a higher pump speed during systole and early
diastole and a lower pump speed during late diastole. In addition, a severely ischemic
left ventricle requires a higher pump speed during systole than a left ventricle with a
minor ischemia (Speed B = 5 & 4 versus 4 & 3). The results also indicate that if the
                     A Simulation Study of Hemodynamic Benefits                    25

increase of SV is more important (Case 1), the pump should be operated at a higher
Speed B but with a shorter duration (a more ‘pulsatile’ operation). On the other
hand, if the increase of PMAO is more important (Case 2), a smaller Speed B can
be used with a prolonged duration (a less ‘pulsatile’ operation). This observation
is consistent with the results reported by Klots et al.54 which showed that left
ventricular volume unloading is more pronounced in pulsatile LVAD as compared
with nonpulsatile LVAD. Finally, by comparing the values of the objective function
in Tables 5 and 6 with that in Tables 3 and 4, one can see that by allowing the
Hemopump to operate at two different speeds, a slightly larger value of the objective
function can be obtained.

5. Conclusion
Despite extensive research and great advancement in recent years, the search for
ideal mechanical cardiac assist device continuous. While the early left ventricular
assist device (LVAD) was mainly devised to serve as a bridge to heart transplanta-
tion, the recently completed REMATCH (the Randomized Evaluation of Mechan-
ical Assistance for the Treatment of Congestive Heart Failure) trail has clearly
demonstrated the feasibility of using LVAD as destination therapy. The REMATCH
trail however, also revealed many problems such as infection, bleeding and a high
probability (35%) of device failure.2 Another concern is the cost of these devices.
For example, cost data from REMATCH showed that the median overall cost was
about $250,000 as compared with the current US cost of $205,000 for each cardiac
transplantation.9 These problems have prompted increasing development of smaller,
simpler, and more reliable devices that are more affordable, easy to use and have
less risk for infection or hematologic aberrancies. The axial flow pump possesses
many promising advantages to meet these demands.
    The dynamic model for the axial flow pump described here was developed based
on a theoretical analysis as well as in vitro and in vivo experimental data. Incorpo-
rated in a canine circulatory system, the model has been used to predict the effects
of the pump on various hemodynamic variables that are in good agreement with
the reported results in clinical and animal experiments.
    After the above validation, the model was further used to study the optimal con-
trol of the assist device through the objective functions. By using the membership
functions and the associated weighting factors, the method offers great flexibility in
choosing the targeted hemodynamic variables, in specifying the particular way that
each of these variables is to be optimized, and in assigning the relative importance
of each targeted variable.
    Although the present study is carried out in the canine model, the general con-
clusions should be applicable to a human model. The methods used and the results
obtained in this study can be incorporated into the design of an advanced phys-
iological controller for a long-term operation of the axial flow pump-based assist
device as well as other types of continuous-flow LVAD.13,55
26                                H. Qiao, J. Bai & P. He

This work is partially supported by the National Nature Science Foundation of
China, the Tsinghua-Yue-Yuen Medical Science Foundation, the National Basic
Research Program of China, and the Special Research Fund for the Doctoral
Program of Higher Education of China.

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                      A Simulation Study of Hemodynamic Benefits                          27

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28                                 H. Qiao, J. Bai & P. He

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    Inflammatory response after implantation of a left ventricular assist device: compari-
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49. P. He, J. Bai and D. D. Xia, Optimum control of the Hemopump as a left-ventricular
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    unloading during pulsatile versus nonpulsatile left ventricular assist device support,
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    logical controller design for a left ventricular assist device to prevent left ventricular
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    Foreign Medical Biomedical Engineering Fascicle 28, 6 (2005) 366–369 (in Chinese).
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    valve. Progress in Natural Science (accepted).
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    pumps, 27th Annual International Conference of the IEEE Engineering in Medicine
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    pp. 234–244.
60. D. B. Olsen, The history of continuous-flow blood pumps, Artif. Organs. 24 (2000)
                                        CHAPTER 2


                                      SHIGERU ICHIOKA
        Department of Plastic and Reconstructive Surgery, Saitama Medical University
              38 Morohongo, Moroyama, Iruma-gun, Saitama, 350-0495, Japan

    Microcirculation plays a direct role in the accomplishment of the principal purpose of
    the circulation system. The functions of the microcirculation incorporates oxygen supply,
    transport, diffusion, and exchange of nutrients and metabolites between blood and tis-
    sue, maintenance of body temperature, regulation of blood pressure, tissue defense and
    repair. These complex functions are carried out in the microcirculation by a number of
    dynamic changes in the blood within the vessels, the vessels themselves or the tissues sur-
    rounding the vessels. Intravital microscopic approaches have greatly contributed to the
    advancement of research in the fields of microcirculation. The techniques represent the
    only method that allows direct visualization and quantitative analysis of the microcircu-
    lation. In this chapter in vivo microscopic techniques with useful experimental models,
    which provide significant information regarding the microcirculation, are introduced.

    Keywords: Microcirculation; intravital microscope; visualization; experimental model.

1. Introduction
Although there is no universally accepted definition of the microcirculation, it is
widely taken to encompass the flow of blood in the circulatory system of small
vessels less than 150 µm in diameter. It includes arterioles, capillaries, and venules.
In spite of their small size in diameter, the capillaries are enormous in total length,
which is about 90–110 thousand kilometers i.e. two and half times the perimeters
of the earth. The microcirculation constitutes a huge biological system essential for
maintenance of life.
    The microcirculation plays a direct role in the accomplishment of the principal
purpose of the circulation system. The functions of the microcirculation incorpo-
rates oxygen supply, transport, diffusion, and exchange of nutrients and metabolites
between blood and tissue, maintenance of body temperature, regulation of blood
pressure, tissue defense and repair. These complex functions are carried out in the
microcirculation by a number of dynamic changes in the blood within the vessels,
the vessels themselves or the tissues surrounding the vessels. Such changes include
vasospasm, vasomotion, blood flow, velocities, leukocyte-endothelium interaction,
macromolecular leakage, angiogenesis, microembolic and thrombotic events, and
functional capillary densities, and so forth.
    A number of expedient techniques have enriched the armamentarium for the
evaluation of the microcirculation. Most of them have used methods such as
microspheres,1 xenon washout,2 tissue oxygen levels,3 laser Doppler4,5 and dye

30                                         S. Ichioka

diffusion methods,6 but these methods characteristically measure indirect indica-
tors of blood perfusion, and do not allow distinct analysis of microcirculatory or
microhemodynamic mechanisms within individual segments of the microvascula-
ture. The use of these indirect techniques has been criticized for providing only
speculation to the microcirculatory status.
    In contrast, intravital microscopic approaches have overcome the limitation of
the indirect methods and have greatly contributed to the advancement of research in
the fields of microcirculation. The techniques represent the only method that allows
direct visualization and quantitative analysis of the microcirculation. In this chap-
ter in vivo microscopic techniques with useful experimental models, which provide
significant information regarding the microcirculation, are introduced.

2. Intravital Microscopic System
2.1. Microscope
In vivo microcirculation can be studied directly with an ordinary light, preferably
stereo, microscope with variable magnifications. The microscope used in our lab-
oratory is the stereo microscope (SZH 10, Olympus, Japan), with a useful zoom
function (Fig. 1).

2.2. Suspension of the microscope
Since small movements and vibrations of the microscope would interfere with the
observations, the microscope should be mounted on a firm stand. The stand should
include a movable arm so that the whole apparatus can be moved in any direction.

Fig. 1. Equipment for intravital microscopy. (a) Stereo microscope with variable magnifications
and a zoom function. (b) Halogen projection lamp for trans-illumination. (c) Mercury lamp for
epi-illumination. (d) Charged-coupled device (CCD) camera. (e) Hard disk video recorder. (f) TV
                Visualization and Evaluation of In Vivo Microcirculation          31

2.3. Illumination
A 150 W halogen projection lamp (Nikon) or a 100 W mercury lamp achieves suf-
ficient illumination as a light source. In vivo microscopy includes trans- and/or
epi-illumination techniques for the direct visualization of the microvasculature. The
trans-illumination technique requires thin, translucent preparations for the investi-
gation to guarantee exact imaging of microvessels and blood cell flow. Narrow band
filters (of a wave length between 400 and 440 nm) allow for contrast enhancement by
dark staining of erthrocytes, facilitating quantitative analysis of the microvascular
    Epi-illumination may be performed without contrast enhancement; however,
optimal images and analysis of the microcirculation are best achieved using flu-
orescent markers. By staining different components of the blood, such markers
permit visualization of plasma [fluorescein-isothiocyanate (FITC)-Dextran, FITC-
Albumin], red cells (FITC-labeled), and white cells (rhodamin 6G).

2.4. Imaging and recording
Microvascular appearances are imaged by charged-coupled device (CCD), inten-
sified charged-coupled device (ICCD) or silicon-intensified-target (SIT) cameras
connected to the stereo microscope. The images are transferred to video systems
and stored on videotapes or hard disk video recorders, permitting quantitative off-
line analysis of microcirculatory parameters by the use of computer-assisted image
analysis systems.

3. Tissue Preparation
Intravital microscopic techniques allow study in a variety of experimental models.
Microcirculation is visualized mostly by means of placing the surgically prepared
tissue of the experimental animals under the microscope.

3.1. Anesthetics
Several anesthetics commonly used to immobilize small rodents, and other small to
medium-sized mammals for surgical procedure as well as microvascular observations
include sodium pentobarbital, urethane, and inhalation anesthetics.
    Sodium pentobarbital is the most commonly used anesthetic. The animals are
anesthetized with an intraperitoneal injection of 50–75 mg/kg body weight or an
intramuscular injection of approximately 30 mg/kg body weight. It is the most
frequently used anesthetic during surgical intervention, but researchers should take
into consideration the fact that sodium pentobarbital generally depresses arterial
blood pressure and respiration below normal when it is used for hemodynamic
    Urethane is a favorite choice for acute experiments. It is injected intraperi-
toneally in a dose of 1 g/kg body weight. Urethane anesthesia has been verified to be
32                                     S. Ichioka

suitable for physiopharmacological investigation in cardiovascular system, because
it preserves a number of cardiovascular reflexes.7 But, we recommend it only for
acute studies and not for chronic experiments. Our experience is that repetitive
intraperitoneal administrations of urethane solution tend to induce inflammation of
the abdominal cavity or peritonitis.
    The commonly used inhalant anesthetics include isoflurane and sevoflurane.
Inhalation anesthesia offers the advantage of accurately controlling anesthetic depth
with the safety of being able to discontinue the administration of the inhalant anes-
thetic immediately if any problem should arise. The depth of anesthesia during
maintenance is easily controlled by adjusting the vaporizer output and the total flow
rate. Inhalant anesthetics can be eliminated quickly and rapid recovery is expected
when compared to the injectable agents. Inhalant anesthetics are mostly eliminated
through ventilation, whereas injectable anesthetics rely on the liver and kidney
for metabolism/elimination. For hemodynamic measurements care should be taken
because all inhalant anesthetics depress cardiopulmonary function a dose-dependent
manner as shown by the decreases in cardiac output, blood pressure, and respiratory

3.2. Experimental models
There are two types of experimental models; acute models and chronic models.
The former type is available for single use in an acute experiment principally based
on exteriorization and in situ visualization of inner organs and tissues. The latter
mainly utilizes chronically instrumented animals the aim of which is microcircula-
tory observation for prolonged periods of time.

3.2.1. Acute experimental models
(i) Mesentery
Rats are used for studies of the mesenteric microcirculation.8 The abdomen is
opened along the midline by using a radiocautery, and a loop of small intestine
is exposed. A section of the small intestine is carefully drawn through the incision
and positioned on a transparent observation stage (Fig. 2a). The mesentery is cov-
ered with a piece of film dressing (e.g. Saran wrap) to prevent drying of the tissue.
The model with transillumination provides best-contrasted microvascular images
among the various experimental models (Fig. 2b).
(ii) Cremaster muscle
Both rats and mice are suitable for studies of the cremaster muscle microcircula-
tion.9,10 The cremaster muscle surrounds the testes. It receives its vascular supply
from the pudic-epigastric truncus. Distal to the external iliac-femoral intersection,
the pudic-epigastric truncus lies in the abdominal muscles parallel to the inguinal
ligament. The trunk subtends many branches at the perineal region. One of these
terminal branches forms the cremaster muscle first order vessels.
                  Visualization and Evaluation of In Vivo Microcirculation                     33

Fig. 2. Observation of the rat mesentery. (a) The small intestine positioned on a transparent
observation stage. (b) Microvascular appearance of the rat mesentery providing excellent contrast.
Scale bar = 50 µm.

    In preparing this model, the scrotum is pulled away from the underlying cremas-
ter and testes while a longitudinal incision is made in the ventral scrotal sac. The
exposed cremaster muscle is continuously irrigated with suffusate, and is then freed
from the scrotal sac and surrounding fascia by blunt dissection. The thin fascia cov-
ering the cremaster muscle is carefully removed with forceps. A ligature is attached
to the extreme distal tip of the muscle. The testicle and cremaster are pushed into
the abdomen through the inguinal canal, which everts the cremaster muscle. An
abdominal incision is then made and the vessels connecting the cremaster to the epi-
didymis are cauterized. Using the distal ligature, the cremaster muscle is then exte-
riorized through the annulis inguinalis, leaving the freed testicle in the abdominal
cavity. The cremaster muscle is positioned over an optical port for transillumination
(Fig. 3).

(iii) Hamster cheekpouch
Golden hamsters or Syrian hamsters are used.11,12 The cheek pouch is gently everted
and pinned with 4–5 needles into a circular well filled with silicone rubber to provide
a flat bottom layer, thus avoiding stretching of the tissue, but preventing shrinkage.
In this position, the pouch is submerged in a superfusion solution that continuously
flushes the pool of the microscope stage. Before the pouch was pinned, large arte-
rioles and venules are located with the aid of an operating microscope. In order to
produce a single-layer preparation, an incision is made in the upper layer so that
atriangular flap can be displaced to one side. The exposed area is dissected under
the microscope, and the fibrous, almost avascular, connective tissue converging the
vessels is removed with ophthalmic or microsurgical instruments. The dissected part
of the pouch is 125 to 150 µm thick. Dissected pouches with petechial hemorrhages
and those without blood flow in all vessels are discarded. The preparation is placed
under an intravital microscope (Fig. 4).
34                                          S. Ichioka

     Fig. 3.   Skeletal muscle microcirculation of the mouse cremaster. Scale bar = 100 µm.

     Fig. 4.   Fluorescent micro-angiogram of the hamster cheek pouch. Scale bar = 100 µm.

3.2.2. Chronic experimental models
(i) Rabbit ear chamber
A transparent chamber is implanted in the surgically prepared rabbit ear.13–15 The
chamber is made of transparent acryl-resin, and is composed of a disk with a central
round table and three peripheral pillars, a cover plate, and a holder ring (Fig. 5).
    White rabbits weighing 3.0–3.5 kg are the appropriate size for most studies. The
ear is shaved, treated with a depilatory cream and sterilized with povidone-iodine.
Four holes are punched through the cartilage and skin of the distal portion of the
ear with a specially designed puncher (Fig. 6a). The size and position of the holes
are adjusted to match those of the round table and pillars of the disk. Care should
be taken to avoid large blood vessels. The epidermis on both sides of the ear around
the central hole is carefully retracted so as to leave the subcutaneous blood vessels
intact (Fig. 6b). Three peripheral pillars of the disk are inserted through the outer
                   Visualization and Evaluation of In Vivo Microcirculation                      35

Fig. 5. Complete rabbit ear chamber set. The chamber is composed of a disk with a
central round table and three peripheral pillars, a cover plate, and a holder ring. A
10×10 grid (grid constant 500 µm) is superimposed on the cover plate for microcirculatory analysis.

Fig. 6. Procedure for rabbit ear chamber installation. (a) Four holes are punched through the
cartilage and skin of the distal portion of the ear. (b) The epidermis on both sides of the ear
around the central hole is carefully retracted so as to leave the subcutaneous blood vessels intact.
(c) Installed rabbit ear chamber. (d) The disk is designed to leave a 50-µm-thick space into which
the new microvessels sprout.

small holes of the ear to position the chamber and the central round table plugges
the large puncture in the center. A cover plate is fixed on the pillars by the holder
ring (Fig. 6c). The disk is designed to leave a 50-µm-thick and 6-mm-diameter space
between the central round table and the cover plate. The new microvessels begin
to sprout into the space in several days after implantation (Fig. 6d). An intravi-
tal microscopic system under transillumination allows observations of angiogenesis
during wound healing as well as microcirculation after completion of repair process
(Fig. 7).
36                                            S. Ichioka

Fig. 7.   Gross appearance of microcirculation in the rabbit ear chamber 3 weeks after implantation.

Fig. 8. Aluminum chamber inserted into the dorsal skinfold. (A) Side view of the chamber frames.
The skinfold chamber consists of two frames held together with polyethylene tubes. (a) Opening
(7 mm on each side) for intravital microscopic investigation of underlying tissue; (b) holes drilled
for polyethylene tubes to allow fixation and adjustment of both chamber frames. (B) Transectional
view of the aluminum chamber. (c) Opening for intravital microscopic observation; (d) coverglass;
(e) polyethylene tube; (f) fluke for fixing the chamber to the observation stage.

(ii) Mouse skinfold chamber
The basic structure of a mouse chamber (Yasuhisa-koki, Ltd. Japan) is shown in
Fig. 8.
     The skinfold chamber is inserted using the following procedure (Fig. 9).16,17
The dorsal skin is pulled and fixed to a board with 26-gauge injection needles.
Four small holes are punched to penetrate the double-layered skinfold. In the center
of the four holes, a square area of one layer of skin (about 7 mm in one side) is
removed, and the remaining layer, consisting of epidermis, subcutaneous and thin
striated skin muscle, is covered with a coverglass incorporated in one of the frames
                   Visualization and Evaluation of In Vivo Microcirculation                      37

Fig. 9. Skinfold chamber installation. (a) The dorsal skin was drawn upward and fixed to the
board with 26-gauge injection needles. (b) A square area of one layer of skin was removed (about
7 mm on each side), maintaining the other layer intact. (c) The frames were implanted so as to
sandwich the stretched double layer of skin. The intact layer, consisting of epidermis, subcutaneous
tissue, and a thin layer of striated skin muscle, was covered with a coverglass incorporated in one
of the frames. (d) General appearance of a mouse with the skinfold chamber in place.

of the chamber. A polyethylene tube is passed through each small hole and melted
at both ends with heat to hold the two chamber halves together. The frames are
implanted so that they sandwich the extended double layer of skin perpendicular
to the animal’s back.
    A recovery period of 72 hours between chamber implantation and the micro-
scopic investigation is generally allotted to eliminate the effects of immediate sur-
gical trauma on the chamber tissue. The microcirculation of the skin as well
as the striated skin muscle can be inspected with an intravital microscope and
transillumination (Fig. 10). Similar skinfold chambers are applicable for rats and

Fig. 10. The microcirculation of the skin (a) as well as the striated skin muscle (b) is inspected
with the mouse skinfold chamber. Scale bar = 50 µm.
38                                          S. Ichioka

Fig. 11. (a) The hairless mouse ear spread on an observation stage. (b) Microvascular appearance
of the hairless mouse ear. Scale bar = 50 µm.

(iii) Hairless mouse ear
The homozygous (hr/hr) hairless mouse, although similar in appearance to the nude
mouse, is immunologically intact, with a normally functioning thymus and T-cells.
The use of the hairless mouse ear allows studies of intact skin microcirculation
without surgical intervention.22,23 The ear of the hairless mouse measures 10–13 mm
in width and length and comprises approximately 6% of the animal’s total body
surface area. It presents a central cartilaginous sheet (approximately 50 µm thick)
sandwiched between two full thickness dermal layers together, giving the ear a total
thickness of approximately 300 µm. The relatively large size and thin structure of
the hairless mouse ear make it easy to study the ear microcirculation using vital
     Its nourishment is supplied by three to four neurovascular bundles entering the
ear at its base and branching out periphery into descending orders of arterioles
that feed the capillaries. The capillaries form characteristic loops around empty
but otherwise normal hair follicles and drain into post-capillary venules and veins.
This entire vascular network can be visualized by transilluminating the spread ear
through the vital microscope (Fig. 11). Although the optical resolution is not so good
due to the overlying skin, the hairless mouse ear offers the advantage of a simple
and reproducible inspection of the microcirculation of intact mammalian skin.

4. Microcirculatory Measurement
Visualized microcirculatory images are analyzed using the following measurable

4.1. Vessel diameter
The measurement of the diameter of microvessels is a key feature in the classification
of the vessels and in the design of physiological theories that address homeostatic
                Visualization and Evaluation of In Vivo Microcirculation             39

phenomena such as blood flow distribution, the regulation of blood pressure, and
fluid exchange.
    Video-recorded microvascular images captured and stored as static digitalized
images onto a personal computer are analyzed using an image analysis program
(e.g. Scion Image for the Windows, NIH image for the Macintosh platform). After
setting the scale per pixel using a calibration function, measurement of the trans-
verse distance of the vascular regions of interest provides the vessel diameter. If the
boundary of the vessels is unclear in the original captured images, contrast modifi-
cation and an accurate trace of vascular regions with the pencil function of an image
processing software (e.g. Adobe Photoshop) helps detection of the vessel walls by
the analysis software.

4.2. Tissue vascularity
Capillary perfusion is a prerequisite for the transport of oxygen, nutrients and
metabolites to and from the tissue cells. Therefore, reliable quantification of capil-
lary perfusion represents an important goal of experimental and clinical research.
The nutritive perfusion of a tissue depends on the number and anatomical distri-
bution of capillaries and on the blood flow within these conduits. As a quantitative
parameter, functional capillary density (FCD) has been proposed as a measure of
capillary perfusion. FCD is defined as the total length of red cell-perfused capillaries
per unit area, and it has been used as an indicator of the quality of tissue vascularity
in various animal models.24
    In FCD measurement capillaries are conventionally distinguished from other
microvessels i.e. arterioles and venules. However, capillaries are not only sites that
supply oxygen to the tissue. Microvascular PO2 measurements have revealed the
longitudinal perivascular PO2 gradient along the arteriole and have suggested a sig-
nificant amount of O2 diffusion from the arteriolar network.25 In this sense, the total
length of all the microvessels including arterioles, capillaries and venules per unit
area or the functional microvascular density properly represents tissue vascularity.
The term FCD was recently used to indicate functional microvascular density.17
This interpretation is essential in the quantification of angiogenesis where the cap-
illaries can hardly be distinguished from other microvessels due to immaturation of
the microvasculature.26
    A microvessel is defined to be functional if passing RBCs are noted in the
dynamic microvascular image. FCD is assessed by an image processing software
(e.g. Adobe Photoshop CS) (Fig. 12).27 The Pencil tool is selected from the tool-
box and the capillaries (microvessels) depicted on the screen are manually redrawn
in a layer superimposed over the image, using a distinctive color. Because the
redrawn pencil line is automatically shown on the computer screen, it can be
controlled at every step and necessary corrective measures can be taken. The
color line is then selected with the Magic Wand tool and the Select Similar com-
mand in the Select menu, and the area covered by the pencil line is quantified
40                                         S. Ichioka

Fig. 12. (a) Representative image of fluorescently stained microvessels. (b) FCD is assessed by
Photoshop image analysis. Microvessels depicted are manually redrawn in a layer superimposed
over the image, using a distinctive color and the total length is calculated.

by using the Histogram command in the Window menu. On the basis of the res-
olution and magnification of the image and the predefined width of the pencil
line, the length of the pencil line is calculated and the length of the capillar-
ies (microvessels) is expressed relative to the area covered by the image (given
in cm/cm2 ).

4.3. Blood flow velocity
Recently developed image acquisition and analysis software (CapiScope: KK Tech-
nology, UK) provides a feasible measurement of red cell velocity using a spatial
correlation technique. Measurements can be made in real time, directly from the
subject. However, it is usually more convenient to record onto good quality video
tape or, for best quality, directly into the computer memory, or for longer sequences,
directly to a hard disk. The CapiScope also has functions of morphometric analysis
including vessel diameter and capillary density.
    Velocity is measured by using the mouse to draw a line along the vessel. The
gray level profile along the line is taken for each field. The gray level pattern along
each line is compared to the pattern from the next field (or several fields later for
very low velocities). The comparison is performed by calculating the correlation
coefficient for every possible shift of the previous gray level profile relative to the
new profile. The shift which produces the highest correlation, and which can be
seen as the peak in the correlation in the following figure, indicates the distance
that the pattern travelled between the two gray level profile measurements. Since
the time lapse between the two gray level profiles is known (i.e. 1/60th second for
NTSC based systems) the velocity can be calculated (Fig. 13).
                  Visualization and Evaluation of In Vivo Microcirculation                      41

Fig. 13. Blood flow velocity measurement using the Capiscope. (Above) The gray level pattern
along each line is measured. (Below) The pattern is compared to that from the next field (or
several fields later for very low velocities). Velocity can be calculated based on the distance that
the correlated pattern has travelled between the two gray level profile measurements.

4.4. Leukocyte behavior
Circulating leukocytes play a key role in the pathogenesis of microvascular injury
through a process first involving leukocyte rolling on the venular endothelium, which
is then followed by firm adherence of leukocytes to endothelial cells. Adherent leuko-
cytes may promote venular injury by releasing reactive oxidants and proteolytic
enzymes, which then further damage underlying endothelial cells.
    Rhodamine 6G is a cationic fluorescent dye that selectively accumulates in the
nuclei and mitochondria of living cells.28 Intravenous administration of rhodamine
6G (ICN Biomedicals Inc.) (0.5 mg/kg body weight) allows visualization of fluores-
cently labeled-leukocytes under intravital fluorescence microscopy (Fig. 14).29
    The straight portions of venules are usually selected as sites of interest. Leuko-
cyte behavior is classified as nonadherent, rolling, or sticking. Nonadherent leuko-
cytes are defined as cells passing the observed vessel segments without interacting
with the endothelial lining and are expressed as cells per unit time (i.e., 30 sec-
onds). Rolling leukocytes are defined as cells moving along the endothelial lining at
a velocity markedly less than that of the surrounding red cell column and are given
42                                        S. Ichioka

Fig. 14. Fluorescently visualized leukocytes labeled by rhodamine 6G under an intravital fluo-
rescence microscopy. Scale bar = 50 µm.

as the number of cells traversing an observed vessel segment within 30 seconds. The
rolling fraction (%) is calculated as the ratio of rolling cells/rolling cells plus non-
adherent cells times 100. Sticking leukocytes are defined as cells that do not detach
from the endothelial lining within 30 seconds of observation time and are expressed
as cells/mm2 .

4.5. Oxygen measurement
One of the most important molecular species which are subject to convective trans-
port within the macro- and microcirculatory system is oxygen. Several methods
have been utilized in efforts to measure tissue oxygen using microelectrodes, spec-
trophotometry, etc. Among these the O2 -dependent quenching of phosphorescence
technique,25 which allows regional determination of intravascular and interstitial
PO2 values, is herein explained.
    Pd-meso-tetra (4-carboxyphenyl) porphyrin (Pd-porphyrin, Porphyrin Prod-
ucts, Logan, UT) bound to bovine serum albumin is used as the phosphorescent
probe for the O2 -dependent quenching. The basis of this technique is a well-known
reaction in which Pd-porphyrin is excited to its triplet state by exposure to pulsed
light, after which phosphorescence intensity is reduced by emission and energy trans-
fer to O2 . The quenching of the phosphorescence by the O2 is diffusion limited. Thus,
if the PO2 distribution is homogeneous, it can be described as:

                I(t)/I0 = exp[−(1/τ0 + Kq × PO2 )t] = exp(−t/τ ),                        (1)

where I(t) is the light intensity at time t, and I0 is the initial value of light intensity
at t = 0. The Stern-Volmer equation can be written as

                              1/τ = 1/τ0 + Kq × PO2 ,                                    (2)

where τ0 and τ are the phosphorescence lifetimes in the absence of O2 and in the area
being measured, respectively, and Kq is the quenching constant. The PO2 values can
                Visualization and Evaluation of In Vivo Microcirculation             43

be obtained from the measurement of τ , because Kq and τ0 are substance specific.
However, Golub et al.30 pointed out that the rectangular distribution model should
be used to analyze the data when the PO2 distribution is heterogeneous in the
measuring area. The fitting equation they proposed is

            I(t)/I0 = exp[−(1/τ0 + Kq × PO2 )t] × [1 + (Kqσt)2 /2].                 (3)

By using this equation to calculate the PO2 values, the accuracy of the curve fitting
is improved compared with a conventional curve fitting, especially at the initial
segment of the decay curve.
    Figure 15 shows a schematic diagram of phosphorescence quenching laser micro-
scope system. The Pd-porphyrin phosphorescent probe is excited by epi-illumination
using a N2-dye pulse laser (LN120C, Laser Photonics). The area of the epi-
illuminated tissue is 10 µm in diameter on the surface. The phosphorescent emissions
from the tissue are captured by a photomultiplier (C6700, Hamamatsu Photonics,
Hamamatsu, Japan) through a longpass filter at 610 nm. Signals from the photomul-
tiplier are converted to 10-bit digital signals at intervals of 3 µs. The averaged 20–40
data are calculated by mathematically fitting the decay of the phosphorescence to
the rectangular model equation30 using online computer analysis. For regional anal-
ysis, intravascular PO2 measurements are carried out immediately, whereas intersti-
tial PO2 measurements are started 30 min after injection of the Pd-porphyrin solu-
tion (25 mg/kg body wt) usually into the cannulated jugular vein of the experimental

4.6. Permeability measurement
One of the functions of the microcirculatory vessel wall is to act as a molecular
sieve, so that many plasma constituents have limited access to the surrounding
tissue. This barrier function of the vessel wall is compromised in certain conditions
(e.g. acute inflammation), and the resulting leakage of fluid and protein into the
interstitium can result in morphological and functional disturbance such as edema
and exudates. Therefore, a frequent assay of microvascular integrity is the rate at
which protein leaks from plasma to tissue.31
    Intravascular injected fluorescein isothiocyanate (FITC)-labeled macromolecules
(e.g. FITC-Dextrans or FITC-Albumins) have been used to study vascular per-
meability under an intravital fluorescence microscope. The injected fluorescently
labeled molecules trace the movement of macromolecules across the vascular wall.
    Animals receive FITC-labeled macromolecules intravenously and fluorescein
angiograms are recorded and digitized. In general, the fields selected for investi-
gation are the areas around the postcapillary venules that are relatively free of
capillaries and other leaking postcapillary venules. Fluorescence intensities in the
intravenular and corresponding perivascular area are measured using a computer-
assisted digital image processing and analyzing software (Scion Image or Adobe
44                                           S. Ichioka

Fig. 15. Arrangement of the intravital laser microscope using O2 -dependent phosphorescence
quenching. A general observation of the microcirculation is performed using an intravital micro-
scope coupled with CCD camera. Pd-porphyrin is excited by epi-illumination with a N2 -dye pulse
laser with a 535-nm line at 20 Hz via the objective. The epi-illuminated tissue area is 10 µm in
diameter. Phosphorescence emission is captured by a photomultiplier through a long-pass filter at
610 nm, and signals from the photomultiplier are converted to 10-bit digital signals at intervals of
3 µs. ADC, analog-to-digital signal converter; PMT, photomultiplier tube.

    Leakage is quantified by measuring the gray levels in selected areas directly over
and immediately adjacent to the blood vessels. The areas are specified by the Select
Regions tool (Scion Image) or by the lasso tool from the tool box (Adobe Photo-
shop). Mean gray levels within the predefined area are assessed using the Measure
command in the Analyze menu (Scion Image) or the Histogram command in the
Window menu (Adobe Photoshop). The difference in the gray levels of areas overly-
ing and adjacent to the blood vessels is used as a measure of vascular permeability
(Fig. 16).
                  Visualization and Evaluation of In Vivo Microcirculation                     45

Fig. 16. Permeability measurement. (a) Microvascular image immediately after injection of FITC-
labeled macromolecular dextran. The fluorescent marker is retained within the vessel. (b) Microvas-
cular image 15 minutes after the marker injection. Leakage is quantified based on the difference
in the gray levels of areas overlying and adjacent to the blood vessels.

5. Application of the Microcirculatory Experimental Models
Visualization models of the microcirculation can be applied to the studies of diverse
pathophysiological phenomena. The current section introduces two examples of this
application; experimental models of angiogenesis and ischemia-reperfusion that have
been matters of concern in various fields of medicine.

5.1. Angiogenesis model
Angiogenesis is the process of new vessel formation. It is essential in all connective
tissue healing, as well as a wide variety of other physiologic and pathologic processes.
Application of the mouse skin fold chamber model and the hairless mouse ear model
allow direct visualization of angiogenesis during wound healing.

5.1.1. Angiogenesis model using mouse skinfold chamber
A wound is created in the mouse skin fold chamber model (Fig. 17).26 The area
selected for the wound on the intact side of the folded skin (opposite the coverglass)
is marked with a round-shaped ink stamp having a diameter of 1.6 mm. A circular
area of skin is then excised down to, but not including, the underlying striated
skin muscle layer using microforceps and scissors under an operating microscope.
The preparation provides a full-thickness dermal wound having a surface area of
approximately 2 mm2 and a depth of approximately 0.3 mm. The agent to be studied
and/or vehicle are applied topically in the wound defect if necessary. The wound
is covered with a dressing and the plastic cap is used to fix the dressing in place
throughout the experimental period.
    The model allows direct and repeated observation of microcirculatory changes
during the wound healing angiogenesis process in the same animal over an extended
period of time (Fig. 18).
46                                          S. Ichioka

Fig. 17. Wound creation procedure. Transectional schema (above) and appearance under an
operating microscope (below). A circular area of skin was excised with microforceps and scissors
down to, but not including, the underlying striated skin muscle layer. A dressing (a) covered the
wound, and the plastic cap (b) fixed the dressing.

Fig. 18. Changes in microvasculature during wound healing angiogenesis in the same animal on
day 3 (a), day 5 (b), day 7 (c), and day 10 (d) after wounding.

5.1.2. Angiogenesis model using hairless mouse ear
The hairless mouse ear also offers an angiogenesis model.32,33 Circular wounds are
created on the dorsal aspect of the ears, down to, but not including, the pineal
                Visualization and Evaluation of In Vivo Microcirculation          47

cartilage layer. The wounds are positioned between the anterior and middle principal
neurovascular bundles approximately 1.5 mm from the ear border. After wounding
and hemostasis, the agent to be studied and/or vehicle are applied topically in the
wound defect and the wound is covered with a thin sheet thus helping to maintain
a stable environment in the wound. Over the agent and the plastic covering, all
ears/wounds are covered in their entirely with a bioadhesive dressing.
   Wound epithelization and neovascularization are directly visualized and mea-
sured using an intravital microscopic system. The digitized images of the epithelial
and neovascular edges are traced and the area within the tracing is measured and
used to calculate the progression of the epithelial or neovascular edge at the time
the measurement is performed.

5.2. Ischemia-reperfusion
Ischemia-reperfusion injury has become of the major concerns in many fields of
medicine. A number of investigations have proved that impairment mechanisms of
various organs are quite attributable to the process of ischemia-reperfusion injury.
Ischemia-reperfusion injury has been defined as cellular injury resulting from the
reperfusion of blood to a previously ischemic tissue. When a tissue has been depleted
of its blood supply for a significant amount of time, the tissue may reduce its
metabolism to preserve function. The reperfusion of blood to the nutrient- and
oxygen-deprived tissue can result in a cascade of harmful events.
    Most of the microcirculatory models provide modes for studies of ischemia-
reperfusion. Clamping of the pedicled nutritional vessels or direct compression of
the observed microvessels induces ischemia, and release of the obstruction results in
reperfusion. Various microcirculatory measurements have been performed after the
ischemia-reperfusion procedure.

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 2. P. M. Hendel, D. L. Lilien and H. J. Buncke, Plast. Reconstr. Surg. 387 (1983).
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11. B. R. Duling, Microvasc. Res. 423 (1973).
12. T. J. Verbeuren, M. O. Vallez, G. Lavielle and E. Bouskela, Br. J. Pharmacol. 859
13. M. Asano, Jpn. J. Pharmacol. 225 (1964).
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15. S. Ichioka, M. Shibata, K. Kosaki, Y. Sato, K. Harii and A. Kamiya, Microvasc. Res.
    165 (1998).
16. S. Ichioka, T. C. Minh, M. Shibata, T. Nakatsuka, N. Sekiya, J. Ando and K. Harii,
    Microsurgery 304 (2002).
17. S. Tsuji, S. Ichioka, N. Sekiya and T. Nakatsuka, Wound Repair Regen. 209 (2005).
18. D. Nolte, M. D. Menger and K. Messmer, Int. J. Microcirc. Clin. Exp. 9 (1995).
19. S. Ichioka, M. Iwasaka, M. Shibata, K. Harii, A. Kamiya and S. Ueno, Med. Biol. Eng.
    Comput. 91 (1998).
20. S. Ichioka, M. Minegishi, M. Iwasaka, M. Shibata, T. Nakatsuka, K. Harii, A. Kamiya
    and S. Ueno, Bioelectromagnetics 183 (2000).
21. S. Ichioka, T. Nakatsuka, Y. Sato, M. Shibata, A. Kamiya and K. Harii, J. Surg. Res.
    42 (1998).
22. E. Eriksson, J. V. Boykin and R. N. Pittman, Microvasc. Res. 374 (1980).
23. J. H. Barker, F. Hammersen, I. Bondar, E. Uhl, T. J. Galla, M. D. Menger and K.
    Messmer, Plast. Reconstr. Surg. 948 (1989).
24. D. Nolte, H. Zeintl, M. Steinbauer, S. Pickelmann and K. Messmer, Int. J. Microcirc.
    Clin. Exp. 244 (1995).
25. M. Shibata, S. Ichioka, J. Ando and A. Kamiya, J. Appl. Physiol. 321 (2001).
26. S. Ichioka, S. Kouraba, N. Sekiya, N. Ohura and T. Nakatsuka, Br. J. Plast. Surg.
    1124 (2005).
27. J. Brunner, F. Krummenauer and H. A. Lehr, Microcirculation 103 (2000).
28. R. K. Saetzler, J. Jallo, H. A. Lehr, C. M. Philips, U. Vasthare, K. E. Arfors and R.
    F. Tuma, J. Histochem. Cytochem. 505 (1997).
29. H. A. Lehr, B. Vollmar, P. Vajkoczy and M. D. Menger, Meth. Enzymol. 462 (1999).
30. A. S. Golub, A. S. Popel, L. Zheng and R. N. Pittman, Biophys. J. 452 (1997).
31. N. R. Harris, S. P. Whitt, J. Zilberberg, J. S. Alexander and R. E. Rumbaut, Micro-
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    Exp. Med. (Berl.) 379 (1991).
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                                        CHAPTER 3


                                DONALD PHILIP BERNSTEIN
                              The Department of Anesthesia
                     Palomar Medical Center, Escondido, California, USA

                                HENDRIKUS J. M. LEMMENS
                                The Department of Anesthesia
                          The Stanford University School of Medicine
                                  Stanford, California, USA

    Impedance cardiography is a branch of bioimpedance primarily concerned with the deter-
    mination of left ventricular stroke volume. The technique involves applying a current field
    across a segment of the body, usually the thorax, by means of a constant magnitude,
    high frequency, low amplitude alternating current. By Ohm’s Law, the voltage measured
    within the current field is proportional to the electrical impedance Z. Upon ventricu-
    lar ejection, a characteristic pulsatile impedance change occurs ∆Z(t), which, in early
    systole, represents both the radially oriented volumetric expansion of and longitudinal
    forward blood flow within the great thoracic arteries. Past methods assumed a volumetric
    origin for the peak systolic upslope of ∆Z(t), that is, dZ/dtmax . A new method assumes
    the rapid ejection of forward flowing blood in early systole causes significant changes in
    blood resistivity, and it is the peak rate of change of blood resistivity that is the origin
    of dZ/dtmax . As proposed, dZ/dtmax is an ohmic mean acceleration equivalent, and,
    as necessary for stroke volume calculation, must undergo square root transformation to
    yield an ohmic mean velocity equivalent. Further changes over older methods include a
    variable magnitude volume conductor based on body mass, rather than on current field
    segment length or body height.

    Keywords: Impedance cardiography; stroke volume equations; signal processing;
    dZ/dtmax ; square root acceleration step-down transformation; volume conductors;
    intrathoracic blood volume; blood resistivity.

1. Introduction
Bioimpedance is a branch of biophysics concerned with the electrical hindrance
offered by biological tissues to the flow of low amplitude, high frequency alternating
current (AC). Bioimpedance encompasses a broad range of techniques, including
impedance spectroscopy, single and multi-frequency bioimpedance body composi-
tion analysis, impedance tomography, impedance plethysmography and impedance
cardiography (ICG). ICG is generally subdivided into the transthoracic and whole
body techniques.1 This chapter will concern itself with the transthoracic technique,
which is the one most often applied clinically and studied experimentally.

50                           D. P. Bernstein & H. J. M. Lemmens

    Impedance cardiography, also known as transthoracic electrical bioimpedance
cardiography (TEBC) is a clinically applicable, non-invasive, continuous, beat-
to-beat method for estimating left ventricular stroke volume (SV) and cardiac
output (CO).2–6 The method is attractive, because it does not require a pro-
tracted learning curve on the part of the user, or the necessity of a skilled
technician. Furthermore, data acquisition and waveform processing are not inor-
dinately time consuming, and with the availability of user-friendly, menu-driven,
programmable computers, data are displayed in a signal-averaged, real-time format.
As compared to its original operational implementation, using circumferential band-
electrodes, the advent of the tetra-polar eight spot-electrode patient interface
now renders the technique satisfactory for both patients and medical personnel
(Fig. 1).
    While it is generally acknowledged that directional changes in impedance-derived
SV follow changes of accepted standard reference methods, there is, unfortunately,
no consensus as to the validity of absolute measurements, especially in patients
with cardiopulmonary pathology.7,8 A review of the literature indicates that exist-
ing SV equations, originally modeled for simple extremity blood volume changes
(impedance plethysmography),9 inadequately account for the much more com-
plex and dynamic, vascular and hemorheologic interrelationships of pulsatile tho-
racic blood flow.10–13 Thus, a theoretically robust mathematical expression, linking
impedance-derived SV to other standard methods has been suggested as necessary.7
As a consequence of these important issues, impedance cardiography has not yet
found an established place in mainstream medicine for modulation of therapy in
critically ill humans.

Fig. 1. Ohm’s Law as applied to the human thorax. An AC (∼) field is applied to the thorax
generating a static voltage U0 and a cardiac-synchronous, time-dependent voltage drop ∆U (t).
By Ohm’s Law, the ratio of the measured voltage (U ) to the applied current (I) is equal to the
calculated impedance Z. Shown are the poorly conductive tissue impedances in the aggregate
Zt and the highly conductive blood impedance Zb . L is the distance between the voltage sens-
ing electrodes (Kubicek), approximated as 17% of overall body height for the Sramek–Bernstein
                               Impedance Cardiography                                51

2. Operational Implementation and Development of the Dual
   Compartment Parallel Conduction Model
Described most simply, the impedance technique involves applying a current field
across the thoracic volume over a defined segment L (cm). This is effected by means
of a constant magnitude, high frequency (50–100 kHz) alternating current (AC)
(I, mA) of small amplitude (1.0–4.0 mA rms). The AC is applied by means of a pair
of current injecting electrodes; one electrode (or set of two) placed at the base of the
neck, and the other electrode (or set of two) at the xiphisternal junction (Fig. 1).
A potential difference is thus established, wherefrom a baseline thoracic voltage
U0 (U, Volt) and a time-variable cardiovascular component ∆U (t) are obtained by
means of a pair of voltage sensing electrodes. As shown in Fig. 1, the voltage sensing
electrodes are placed proximal to the current injectors within the current field. The
static, relatively non-conductive thoracic tissues, including bone, muscle, connective
tissue, fat, vascular tissue, the thoracic gas volume, and the highly conductive fluids,
including the intra-thoracic blood volume (ITBV, mL) and interstitial fluids, act as
parallel impedances Z (Ω, Ohm) to AC flow. By virtue of Ohm’s Law, as applied to
AC (Z = U/I), the measured quasi-static voltage U0 , and the voltage change ∆U (t),
are proportional to the quasi-static impedance Z0 , and the cardiogenically-induced
pulsatile impedance change ∆Z(t), respectively. Disregarding the impedance change
caused by pulmonary ventilation ∆Zvent , the aforementioned relationships are best
described by the following modification of Ohm’s Law:

                         I · [Z0 + ∆Z(t)] = U0 + ∆U (t),                            (1)

where Z0 comprises a static, relatively non-conductive tissue impedance Zt (Ω), the
static component of the highly conductive blood resistance (impedance) Zb , and
the normally present, very highly conductive interstitial extra-vascular lung water
(EVLW) Ze (i.e. Z0 = Zt + Zb + Ze ) (Fig. 2). Since the combined tissue and pul-
satile impedances are exposed to a field of AC, they are not summed directly, but
rather, are added in parallel as their respective reciprocals (i.e. ). Thus, the fol-
lowing equation pertains to a two-compartment parallel conduction model in which
all current is assumed to traverse the static DC (Zb ) and dynamic AC (∆Zb (t))
components of the blood resistance:

                           I · [Z0 ∆Z(t)] = U0 ∆U (t).                              (2)

   Biophysically, the systolic portion of the cardiovascular component represents
the simultaneously generated, serial composite of both the instantaneous volumetric
changes of the great thoracic vessels ∆Zvol (t), principally from the aorta,14,15 and
the instantaneous velocity (v) changes ∆Zv (t) of axially directed forward flowing
blood contained therein.10–13,16 Thus, ∆Z(t) is a composite waveform, as depicted
in Fig. 3, and can be expressed as follows:

                           ∆Z(t) = ∆Zv (t) + ∆Zvol (t).                             (3)
52                            D. P. Bernstein & H. J. M. Lemmens

       Zt          Zb         Ze         Zo       ∆ Zb(t)     Z (t)        U

Fig. 2. Schematic of a multi-compartment parallel conduction model of the thorax. The
transthoracic electrical impedance Z(t) to an applied AC field represents the parallel
connection of a quasi-static base impedance Z0 and a dynamic, time-dependent compo-
nent of the blood impedance ∆Zb (t). Z0 represents the parallel connection of all static
tissue impedances Zt and the static component of the blood impedance Zb . In dis-
ease states characterized by excess EVLW, quasi-static Ze is added in parallel with Zt
and Zb . Voltmeter (U ) and AC generator (∼) are shown.

Fig. 3. The components of ∆Z(t). The thoracic cardiogenic impedance change ∆Z(t) is com-
posed of the blood velocity-induced impedance change ∆Zv (t), shown as v, and the pulsatile
vessel volume impedance change ∆Zvol (t) shown as ∆V (t). The ∆Z(t) shown is an improvisa-
tion, superimposed on a velocity and pressure waveform. It is assumed that ∆V (t) = ∆P (t) · C.
Velocity and pressure waveforms are from Van den Bos, G.C. et al. (1982) Circ. Res. 51, 479–485.
Reproduced with permission.
                                Impedance Cardiography                                 53

3. Hemodynamic Biomechanical Analogs and Bioelectric
   Components of the Transthoracic Cardiogenic
   Impedance Pulse Variation ∆Z(t)
3.1. The volumetric component: ∆Zvol (t)
Biomechanically, the net aortic volume change ∆V (t)net over an ejection interval is a
function of the aortic pulse pressure ∆P (t)(Psystolic −Pdiastolic ), modulated by vessel
compliance (∆V /∆P ), minus the volume lost to simultaneous outflow.17,18 Because
of the viscoelastic nature of the ascending aorta, in addition to distal vascular
hindrance, the proximal aorta expands in systole. These factors cause the volume
flow into an ascending aortic segment to be greater than the simultaneous outflow
(i.e. Vin > Vout ) during the early portion of the ejection interval. The systolic volume
lost to outflow is approximated as the quotient of the integral of the systolic portion
of the pressure curve P (t) in (mmHg · s) and systemic vascular resistance (SVR) Rs .
Rs is further defined as the mean systemic arterial blood pressure Pm divided by
mean flow Qm (Rs = Pm /Qm ). Thus, the equation defining the mechanics of the
net systolic volume change (mL) can be viewed most simply as conforming to that
of a two-element Windkessel model, comprising a compliance chamber and a distal
vascular resistance.17,18
                                            dP (t)       P (t)
                        ∆V (t)net =                C−           dt
                                      tavo     dt         Rs                           (4)
                                    Vin − Vout .
In practical operational terms equation (4) is unsolvable, because Rs and C, the
Windkessel parameters, are derived values, requiring a priori knowledge of both the
systemic mean flow and vessel compliance.
    Bioelectrically, the decrease in transthoracic impedance, corresponding to the
pulsatile systolic expansion of the aorta, is due to the volumetric addition of blood
of low static specific resistance (i.e. impedance) (ρb(stat) = 100− Ω cm in the phys-
iologic range),19 ejected as SV, into a relatively fixed-volume, gas-filled thorax of
much higher static specific impedance (ρ(gas) = 1020 Ω cm).16 The specific resistance
ρ of a cylindrical electrical conductor is equal to its measured impedance Z over a
defined segment, multiplied by its cross-sectional area (CSA, A, cm2 ) and divided
by its length (L, cm) (i.e. ρ = ZA/L). By virtue of the simultaneous displacement
of tracheobronchial gas by the ejected SV, there is also an absolute reduction of tho-
racic gas volume, provided that a positive pressure gradient does not exist between
the upper airway and tracheobronchial tree. As determined by pneumocardiography,
Wessale et al.20 showed that, although the displaced thoracic gas volume is several
magnitudes smaller than measured SV, it is highly correlated with respect to track-
ing directional changes. Similar to the mechanical principle underlying pneumocar-
diography, the impedance change caused by volume displacement of bronchoalveolar
gas by expanding vascular volumes, partially accounts for the original designation
of the technology, impedance plethysmography. Thus, the plethysmographic compo-
nent of the systolic time dependent decrease in transthoracic impedance ∆Zvol (t)
54                            D. P. Bernstein & H. J. M. Lemmens

is bioelectrically one of two components comprising the time dependent decrease
in transthoracic specific impedance ∆ρT (t). The volumetric component ∆ρT vol (t)
can be viewed as the oscillating, time variable ratio of the thoracic gas volume
∆Vg (t), simultaneously compressed and displaced by expanding vascular blood vol-
umes ∆Vb (t). By magnitude, this ratio is defined as the relative change in volume of
bronchoalveolar gas expelled from the thorax, [(Vg + Vb ) − ∆Vg (t)]/(Vg + Vb ), versus
the relative change of the volume of blood simultaneously added during systole,
[(Vg + Vb ) + ∆Vb (t)]/(Vg + Vb ). Simplifying and rearranging results in the following:

                              ∆ρT(vol) (t)      (Vg + Vb ) − ∆Vg (t)
               ∆Zvol (t) ∝                 ∝ 1−                      ,                  (5)
                                 ρT             (Vg + Vb ) + ∆Vb (t)
where ρT is the static transthoracic specific resistance and Vg and Vb are the thoracic
gas and blood volumes at end expiratory apnea and at end-diastole, respectively.

3.2. The blood velocity-induced “resistivity” component: ∆Zv (t)
The contemporaneous blood velocity-induced transthoracic impedance decrease
∆Zv (t) is caused by dynamic, time-dependent changes in the specific impedance
(“resistivity”) of flowing blood ∆ρb (t).10–13,16 These dynamic, temporally con-
cordant changes are generated by phasic alterations in erythrocyte orientation
and deformation during the cardiac cycle.6,10,21,22 At end-diastole, the state of
highest specific impedance, poorly conductive erythrocytes are oriented randomly,
causing the applied AC to take a circuitous route through the highly conduc-
tive plasma (ρ(plasma) = 60–70 Ω cm) (Fig. 4). By comparison, during the peak
acceleratory and rapid ejection phase of early systole, erythrocytes are oriented
with their long axes of symmetry parallel to the direction of flow. Parallel ori-
entation establishes clear current pathways through the plasma, with an atten-
dant steep decrease in transthoracic impedance.10,21 The impedance decrease, cor-
responding to the blood “resistivity” change, is known to be a function of the
shear rate profile. When forward flowing blood is interrogated by AC, longitudi-
nally, or parallel to the axis of flow, the shear rate profile is a function of the
reduced average velocity. The reduced average velocity is defined as the mean veloc-
ity divided by the vessel radius, namely, vmean /R.21,22 In vitro, the magnitude of
the relative resistivity change ∆ρb (t)/ρb(stat) is dependent on hematocrit and an
exponential value of the reduced average blood velocity, (vmean /R)n . Thus, the
dynamic, velocity-induced blood resistivity component of ∆ρT (t) (i.e. ∆ρT v (t)) can
be viewed as the time variable, oscillating, relative resistivity change of flowing blood
∆ρb (t)/ρb . Thus, in relative terms, the sum of the volumetric and blood resistiv-
ity changes comprises the transthoracic cardiogenically-induced impedance pulse.
           ∆ρT (Vel) (t) + ∆ρT (Vol ) (t)       ∆ρb (t)      (Vg + Vb ) − ∆Vg (t)
∆Z(t) ∝                                     ∝           + 1−                        .   (6)
                        ρT                       ρb          (Vg + Vb ) + ∆Vb (t)
                                  Impedance Cardiography                                     55

                                       | ∆ρ b(t)| >> 0
                                           v >> 0



                  v=0                                               v=0
                ∆ρb(t) = 0                                        ∆ρb(t) = 0
                                         Qin + Qout

                  End Diastole               Systole              Diastole

Fig. 4. Behavior of alternating current as applied to pulsatile blood flow — = AC flow;
v = velocity of red cells; ∆ρb (t) = changing specific resistance of flowing blood; Qin = flow into
aortic segment; Qout = simultaneous flow out of aortic segment; dA(t) = time-dependent change in
vessel CSA.

    The combined effects, corresponding to both vessel volume and blood veloc-
ity changes, contribute equivalently, producing a measured decrease in transtho-
racic impedance and transthoracic specific impedance to applied AC.10,13,21 Stated
differently, these contemporaneous hemorheologic and plethysmographic changes
produce enhanced electrical conductivity of the thorax. Despite the reported equiv-
alency of the velocity and volume contributions to the impedance pulse,10,13 it is
generally assumed that the peak magnitude of the first time-derivative of ∆Z(t),
namely d∆Z(t)/dtmax , or simply dZ/dtmax (historically and empirically given the
units Ω · s−1 ), represents the ohmic image of the peak rate of change of aortic vol-
ume or flow, namely dV /dtmax (Qmax , mL · s−1 ).1–6 Conceptually, this assumption
is embodied in the two most widely used impedance SV models: the Kubicek and
Sramek-Bernstein equations.23,24 For the remainder of the discussion, and the sake
of simplicity, the extreme negative absolute impedance change, |∆Z(t)|min , and
the extreme negative absolute value for its first time-derivative, |dZ/dt|min , will be
referred to as ∆Z(t)max and dZ/dtmax , respectively.

4. Development of the Kubicek and Sramek-Bernstein
   SV Equations
4.1. Nyboer equation: Foundation and rationale for the
     plethysmographic hypothesis in impedance cardiography
The Nyboer equation9 was originally proposed for determination of segmental blood
volume changes in the upper and lower extremities. It is based upon the assump-
tion that the arteries of the extremities are rigidly encased in muscle and connective
56                        D. P. Bernstein & H. J. M. Lemmens

tissue, and that both the arteries and surrounding tissue can be approximated as
cylindrical electrical conductors placed in parallel alignment. Nyboer found that,
by placing current injecting and voltage sensing electrodes on a limb segment, con-
figured analogously as in Fig. 1, impedance changes proportional to strain gauge
determined limb blood volume changes could be measured. He also determined that,
in order to compensate for simultaneous volume flow out of the limb segment during
arterial inflow, venous outflow occlusion of the limb was necessary. Using venous
outflow obstruction, absolute arterial volume changes during each pressure pulse
were accurately measured. Nyboer referred to this method, appropriately, as elec-
trical impedance plethysmography.9 The foundation of Nyboer’s method is based
on a companion equation to Ohm’s Law, relating to the electrical impedance offered
by inanimate cylindrical conductors of static composition to the flow of alternating
current. It is given as:
                                    Z=       ,                                    (7)
where Z = the impedance in Ohms (Ω), ρ = the static specific resistance of the
conductor (Ω cm), and in AC terminology, more correctly addressed as specific
impedance (ζ, zeta), L = the length of the conductor (cm), and A = the cross sec-
tional area (CSA) of the conductor (cm2 ). The numerically equivalent expression,
modified for use as the developmental platform in both impedance plethysmography
and impedance cardiography, is given as:
                                         ρb L 2
                                   Z=           ,                                 (8)
where Vb = blood vessel volume (cm3 , mL), and ρb = the hematocrit dependent,
static specific resistance of blood. If ρb and L are constants, and Vb is allowed
to vary with time t, then the change in impedance of a conductor over time t,
∆Z(t), varies inversely with its change in volume over the same time interval ∆Vb (t).
This is the basic premise in impedance cardiography, relating volumetric expansion
of the ascending aorta over a fixed thoracic length L to a decrease in transtho-
racic impedance during the systolic ejection period.23 This concept is embodied in
Sec. 3.1. Thus, rearranging Eq. (8) and solving for the time dependent net change
in aortic volume:
                                              ρb L 2
                               ∆Vb (t)net =          .                            (9)
Thus, according to Nyboer’s theory, Eq. (9) is the bioelectric equivalent of Eq. (4),
relating ∆V (t)net to purely mechanical events. If the vessel segment described
above is embedded in a larger thoracic encompassing cylinder of high static specific
impedance and equivalent length L, but of much larger CSA, then the total parallel
impedance of the thorax (Z(t)) can be expressed as:

                               Z(t) = Z0 ∆Z(t),                                  (10)
                               Impedance Cardiography                              57

where Z0 = the base, or static transthoracic impedance of the thorax (males: 22 Ω–
33 Ω, females: 28 Ω–45 Ω). By virtue of the magnitude of ∆Z(t) normally being only
0.3%–0.5% of total thoracic impedance (i.e. ∆Z(t)max = 0.1 Ω–0.2 Ω), Z(t) and Z0
are approximated as equivalents. Thus, solving Eq. (10) for ∆Z(t) by the reciprocal
rule for parallel impedances yields,
                                  Z0 · Z(t) ∼   2
                        ∆Z(t) =             =                                    (11)
                                  Z0 − Z(t)   −∆Z(t)
Substituting the parallel equivalence of ∆Z(t) from Eq. (11) into Eq. (9), and solving
for ∆V (t)net , yields a modification of the original Nyboer equation.9 Thus,
                                            ρb L 2
                           ∆Vb (t)net = −      2 ∆Z(t).                          (12)
    The Nyboer equation, though originally proposed for extremity blood vol-
ume changes, is the plethysmographic foundation for existing SV methods. As
directly applied to thoracic measurements, it implies that a measured change in
transthoracic impedance ∆Z(t) is directly proportional to the measured net SV
change ∆V (t)net . Clearly, Eq. (12) is also the bioimpedance equivalent of Eq. (4).
As discussed above, the condition under which the change in impedance is represen-
tative of the absolute change in vessel volume is when complete outflow obstruction
is imposed.25 This assumes that all other factors in Eq. (12) remain constant.
As applied to thoracic applications, Nyboer professed to have solved the outflow
problem by manually determining the maximum systolic down-slope of ∆Z(t) and
extrapolating it backwards to the beginning of ejection. The maximum impedance
change resulting from the backward extrapolation procedure was believed to be
equivalent to the maximum impedance change attained as if no arterial runoff
occurred during ejection (Fig. 5). Thus, the maximum volume change ∆Vmax , as
a result of ventricular ejection (SV), was thought to be proportional to Nyboer’s
∆Zmax .
                           ρb L 2
            SVNyboer = −      2 ∆Zmax (down-slope extrapolation).                (13)
Nyboer’s method was not widely accepted, because it was considered cumbersome,
and, because of the difficulty in determining the true down-slope, inaccurate.

4.2. Kubicek equation
In response to the objections of Nyboer’s down-slope extrapolation method, Kubicek
et al.23 made the assumption that, if the maximum systolic up-slope of ∆Z(t) (i.e.
∆Z = ∆Z · s−1 , Ω · s−1 ) is held constant throughout the ejection period, then com-
pensation for outflow, before and after attainment of peak flow results.23,26 The
theory underlying Kubicek’s forward extrapolation procedure assumes that little
peripheral arterial runoff occurs during the rapid ejection phase of systolic ejection.
While Nyboer’s ∆Zmax is directly measured, Kubicek’s method requires multiply-
ing ∆Z by left ventricular ejection time Tlve (i.e. ∆Z × Tlve = ∆Zmax ).6 In the
58                            D. P. Bernstein & H. J. M. Lemmens

Fig. 5. ECG, ∆Z(t) and dZ/dt waveforms from a human subject TRR = the R-R interval, or
the time for one cardiac cycle; Q = onset of ventricular depolarization; ——— = maximum systolic
upslope extrapolation of ∆Z(t); B = aortic valve opening; C = peak rate of change of the thoracic
cardiogenic impedance variation, dZ/dtmax ; X = aortic valve closure; Y = pulmonic valve closure;
O = rapid ventricular filling wave; Q-B interval = pre-ejection period, TP E ; B-C interval = time-
to-peak dZ/dt, TTP; B-X interval = left ventricular ejection period, Tlve . dZ/dt waveform to the
right shows dZ/dtmax remaining constant throughout the ejection interval, Tlve , which represents
outflow compensation. Nyboer’s down-slope backward extrapolation of ∆Z(t) is not shown.

original description of the technique, aortic valve opening and closure, and thus left
ventricular ejection time (Tlve , s), were determined by phonocardiography. Finally,
and without further explanation, this peak value, square wave extrapolation was
assumed to produce the equivalent impedance change proportional to the mean
flow velocity calculation required for SV determination. As opposed to manually
extrapolating the maximum forward slope of ∆Z(t), Kubicek et al.23 found that it
was obtained most accurately by electronically differentiating the ∆Z(t) waveform
with respect to time (Fig. 5). This was especially true in the presence of sponta-
neous ventilation, where the breathing artifact ∆Zvent caused a wandering baseline
and gross distortion of the cardiogenic component of Z(t), ∆Zcardiac (t). In modern
impedance devices, however, the respiratory artifact is suppressed, assuring a sta-
ble baseline for ∆Z(t) and dZ/dt. The first derivative signal clearly identifies the
maximum slope as a discreet landmark, point C (dZ/dtmax ), as well as defining Tlve
as the interval from point B to point X. Differentiating both sides of Eq. (12) with
respect to time purportedly yields the rate of change of volume, or flow (mL · s−1 ),
                               Impedance Cardiography                              59

its peak first time-derivative ostensibly related to its corresponding impedance ana-
log by the following:
                            dV (t)   ρb L2 dZ(t)
                                   =− 2          ,                               (14)
                            dtmax     Z0 dtmax
where dZ/dtmax is the maximal systolic upslope, peak first time-derivative, or for-
ward peak rate of change of the transthoracic cardiogenic impedance pulse varia-
tion d[∆Z(t)]/dt (Ω · .s−1 ). In an in vitro pulsatile open outflow system, using the
Kubicek equation for SV, Yamakoshi et al.25 demonstrated that the outflow extrap-
olation procedure yielded close approximations of corresponding electromagnetic
measurements of SV. As predicted, the Nyboer equation (Eq. 12), without out-
flow obstruction, systematically underestimated SV, thus tentatively validating the
outflow hypothesis. Despite the rather sweeping assumptions justifying the outflow
hypothesis, and its never having been directly confirmed experimentally, in vivo,
it is still considered, by most, a theoretically plausible solution. In terms of basic
hemodynamic theory, the outflow correction purportedly compensates for arterial
runoff as defined in the Windkessel model for flow Q, which is given as follows:17,18
                                         dV (t) P (t)
                              Qtotal =         +      ,
                                          dt     Rs                              (15)
                                          Qin + Qout
where dV (t)/dt (i.e. (dP (t)/dt) × C, Qin ) equals the flow of blood entering segment
L during ejection, and Qout , simultaneous aortic runoff (outflow). Thus, equating
the differentiated Nyboer equation, containing Kubicek’s outflow correction, with
the classical Windkessel expression for SV, found through integration of Eq. (15)
over the ejection interval, yields:
                           ρb L2 dZ(t)                  dV (t) P (t)
           SVKubicek = −      2        Tlve ≡                 +        dt ,
                            Z0 dtmax            tavo     dt     Rs               (16)
                           Vin + Vout                   Vin + Vout
where the left hand side is known as the Kubicek equation and t0 and t1 represent
aortic valve opening and closure, respectively. It is thus stipulated that impedance
derived SV, employing dZ/dtmax , is expressly dependent on dV /dt, the components
of which are the rate of change of ascending aortic pressure, Aortic dP/dt, and
aortic compliance C. If compliance C of a vessel is equal to a given change of
volume for a given change in pressure (pulse pressure) dV /dP , then dV /dt is equal
to the following:
                                 dV   dP dV
                                    =       ,                                    (17)
                                 dt   dt dP
where, for a cylindrical conduit of cross-sectional area A, (πr2 , cm2 ), and length
L (cm), the volume V (mL) is given as:

                                   V = πr2 L.                                    (18)
60                        D. P. Bernstein & H. J. M. Lemmens

Differentiating both sides of 18 with respect to the radius r yields,
                                      = 2πrdrL,                                  (19)
where dV /dr is the volume change of a cylinder with respect to the change in
radius dr, 2πr the circumference, and 2πrL the internal surface area A (cm2 ) of the
cylinder. It follows that the change in volume is directly proportional to the change
in vessel internal surface area, dV /dA. Thus,
                                    = dA × L.                                 (20)
Hence, the rate of change of dV /dr (or, analogously, dV /dA) with respect to time
is given as:
                        dV /dr    dV        dr      dA
                               =       = 2πr L ≡        L.                        (21)
                          dt       dt       dt       dt
Substitution of the right hand side of Eq. (21) into the integral containing dV /dt
in Eq. (16) shows that dZ/dtmax is explicitly dependent on the rate of change of
internal and cross-sectional aortic area dA/dt, as a function of the rate of change of
aortic radius, dr/dt. Equation (16) can therefore be rewritten as:
                   ρb L2 dZ(t)                         dr(t) P (t)
            SV =      2        Tlve =           2πrL        +        dt          (22)
                    Z0 dtmax            tavo            dt    Rs
Further inspection of Eqs. (16), (17) and (22) demonstrates that the magnitude of
impedance-derived SV is expressly dependent on aortic dP/dt and pulse pressure
dP , mean arterial blood pressure Pm , the static component of the ventricular after-
load, Rs (systemic vascular resistance, SVR), and, using a three-element Windkessel
model, the dynamic mechanical component of the ventricular afterload, character-
istic aortic impedance.27 The mechanical impedance, also called Z0 , is a frequency
dependent parameter, equal to the pulse pressure divided by peak aortic blood flow,
(Ps –Pd )/Qmax . Furthermore, since ICG-derived SV is hypothesized as conforming
to simple Windkessel theory as its foundation, it should also vary directly with C
and Aortic distensibility (AoDistens), which are both age-dependent.28–31 Aortic
cross-sectional distensibility is given in its various forms as:
                       C         (AoArea)max − (AoArea)min     2∆d
     AoDistens =               ≡                           ≡        ,
                   (AoArea)min       (AoArea)min × ∆P        d × ∆P
where (AoArea)min is the aortic CSA area at end-diastolic pressure, (AoArea)max
the aortic CSA at peak systolic pressure, ∆d the pulsatile change in aortic diameter
measured over the pulse pressure, d the end diastolic aortic diameter, and ∆P the
pulse pressure.30,31 However, studies have shown that, whereas reference-method
SV and stroke distance S (i.e. S = t01 v(t)dt, cm) decrease ∼ 20%–30% from age
20–60, or 5%–7% per decade, respectively,32,33 AoDistens and C decrease 80%–100%
from age 20–60, or 20%–25% per decade.30,31 Thus, it appears highly unlikely that
                               Impedance Cardiography                               61

the theory behind present ICG equations adequately accounts and compensates for
the mechanical changes in the aorta attendant with increasing age. Moreover, judg-
ing by the rather poor correlations between blood pressure levels and dZ/dtmax ,34
it is doubtful that ICG-derived SV is linearly correlated with changes in aortic
diameter and cross-sectional area over a wide spectrum of mean, end-diastolic, and
pulse pressures. Thus, while pulse pressure increases as C decreases with age, the
changes in aortic diameter and CSA do not parallel the actual magnitude of SV.
Given the complex relationship between Pm and Rs , it is rather doubtful that
Kubicek’s outflow correction can compensate for this discrepancy. As verification
of this assertion, Yamakoshi et al.35 demonstrated in an in vitro tube model that
ICG-derived SV, using dZ/dtmax , shows little change, if any, over a wide range of
vascular compliance C, systemic vascular resistance Rs , mean arterial pressure Pm ,
and vessel dP/dtmax . Clinically, Brown et al.36 showed that, despite progressive
stiffening of the aorta over the full spectrum of age (20–80 years), ICG-derived CO
is nonetheless highly correlated and in agreement with thermodilution CO. Upon
reassessment of equations (16) through (23), these findings are antithetical to the
very foundations upon which ICG SV is based. Specifically, a Windkessel model for
impedance-derived SV seems improbable. Thus,
                 ρb L2 dZ(t)                     dP (t)    P (t)
                    2        Tlve =                     C+          dt            (24)
                  Z0 dtmax               tavo     dt        Rs
Although the right hand sides of Eqs. (16) and (22) are considered a priori unsolv-
able, the bioimpedance expression for SV is assumed to produce a numeric out-
put equivalent to the simple Windkessel models delineated above, as well as to
those equations employed in other pulsatile methods; namely, electromagnetic and
Doppler flowmetry. While both latter methods are prone to methodological error,37
they each employ the universally accepted, theoretically assumption-free statements
defining SV, namely,
                     SV = πr2           v(t)dt ≡ πr2 vmean Tlve ,                 (25)

where πr2 is the aortic valve cross-sectional area and t01 v(t)dt the time-velocity
integral measured over the ejection interval (t0 → t1 ), which is equivalent to stroke
distance, S (cm). In practice, the normalized peak value of the first time-derivative
(dZ/dtmax )/Z0 is multiplied by the product of a static or quasi-static volumetric
constant V (ρb L2 /Z0 ), hereafter known as the volume conductor Vc , and left ventric-
ular ejection time Tlve to presumably yield SV. The Kubicek equation23 in modified
form is thus given as:
                                  ρb L2 dZ/dtmax
                           SV =                  Tlve ,                           (26)
                                   Z0      Z0
where ρb equals the static specific resistance of blood, usually fixed at 135–150 Ω
cm,2–5,23 L, the measured distance between the voltage sensing electrodes (cm)
62                        D. P. Bernstein & H. J. M. Lemmens

(Fig. 1), and Z0 , the quasi-static transthoracic base impedance (Ω) measured
between the voltage sensing electrodes.

4.3. Assumptions implicit to the accuracy of Nyboer-Kubicek
 (1) The transthoracic impedance is considered the parallel connection of an aggre-
     gate of static tissue impedances Zt , considered as one, and a dynamic blood
     resistance Rb , otherwise known as the blood impedance Zb .
 (2) The blood resistance is considered a homogeneously conducting tube of con-
     stant length L, or a parallel connection of an aggregate of tubes considered
     as one.
 (3) The current distribution in the blood resistance is uniform.
 (4) All current flows through the blood resistance.
 (5) The volume conductor Vc is homogeneously perfused with blood of specific
     resistance ρb .
 (6) The magnitude of SV is directly related to power functions of the measured
     distance L between the voltage sensing electrodes (Kubicek), or to height-
     based (cm) thoracic length equivalents (Sramek, Sramek-Bernstein) (Fig. 1).
 (7) All pulsatile impedance changes ∆Z(t) are due to vessel volume changes
     ∆V (t), and, in the context of assumption 2, ∆Z(t) is due exclusively to
     changes in vessel radius dr and CSA dA (Eqs. (9), (12) and (22) and Fig. 4).
 (8) In the context of assumption 7, dZ/dtmax is the bioelectric equivalent of
     dV /dtmax .
 (9) Outflow, or runoff, during ventricular ejection can be compensated for by
     extrapolating the peak rate of change of ∆Z(t) over the ejection interval (i.e.
     dZ/dtmax × Tlve ) (Eq. (16) and Fig. 5).
(10) The specific resistance (“resistivity”) of blood ρb is constant during ejection.
(11) The specific resistance of the thorax ρT is constant.23

4.4. Sramek–Bernstein equation
The Sramek–Bernstein24 equation and its assumptions are similar to Kubicek’s,
except for the physical definition and magnitude of the Vc . In Sramek’s interpreta-
tion of Kubicek’s Vc , ρ, and a Z0 variable are eliminated by mathematical substitu-
tion. This simplification assumes that ρ is constant, equal to ZA/L, thus rendering
the volume a constant for each individual. The resultant cylindrical model is con-
structed assuming the circumference of the thorax at the xiphoid process to be
three times the thoracic length L (i.e. C = 3L). Solving for the CSA at C, and
multiplying by L, yields a cylinder volume three times the magnitude required for
physiologic levels of SV. Taking one third of the resultant volume produces a best-fit
Vc . Sramek’s Vc is known as the volume of electrically participating thoracic tissue
VEP T and is modeled conceptually as a frustum, rather than as a cylinder in the
                              Impedance Cardiography                             63

Kubicek approach. At variance with assumption 6 of Kubicek’s method, this equa-
tion assumes the magnitude of SV to be proportional to the third power of a fixed
percentage of overall body height, and modified by Bernstein24 for deviation from
ideal body weight. The Sramek–Bernstein equation is given as:
                                   L3 dZ(t)/dtmax
                      SVS−B = δ                   Tlve ,                       (27)
                                  4.25    Z0
where δ = a dimensionless parameter which corrects for deviation from ideal body
weight at any given height, and further modified for the indexed blood volume at
that weight deviation, L = 17% of overall body height (cm), and 4.25 = a dimen-
sionless, empirically-derived constant of proportionality.
    Thus, the purported equivalency of Eqs. (26) and (27) with (25) can be expressed
as follows:
           SV = Area × vmean × Tlve = V olume ×                 × Tlve          (28)
where, (dZ/dtmax /Z0 ) × Tlve is assumed to be the dimensionless ohmic equivalent
of stroke distance, vmean × Tlve . Thus, SV equivalency is assumed on the basis of
dZ/dtmax being the ohmic analog of peak flow velocity (equivalent to dV /dtmax ),
and the correctness of the outflow extrapolation procedure. Therefore, as governed
by constraints of their ohmic origins, all other factors remaining constant, both
ICG equations rigidly ascribe the magnitude of SV solely to the phasic peak rate of
change of the transthoracic, volume-related, impedance variation; specifically, the
relationship expressed in proportionality 5 and Sec. 3.1.

4.5. Validity of the plethysmographic techniques
Despite the experimentally verified equivalence of impedance changes caused by
red blood cell velocity, as discussed earlier, both equations obligatorily ignore
this influence as a trivial contaminant. Indeed, some in vitro pulsatile and non-
pulsatile models have shown that the resistivity of flowing blood contributes
merely 5%–10%,38,39 or less,40 to the magnitude of ∆Z(t); they did not, how-
ever, study the effects on the peak first time-derivative, dZ/dtmax . While it is true
that many studies, especially those of clinical nature, have shown good to excel-
lent results using both the Kubicek and Sramek–Bernstein equations,1,2,7 there
are unanswered questions regarding the theoretical underpinnings of the method.
Although it has been proven that the parallel conduction model is valid for thoracic
applications,41,42 there remain some unresolved issues, a few of which comprise the
 (1) The proper value, physiologic definition, and theoretical basis for ρ: blood
     resistivity ρb (hematocrit dependent or constant) versus thoracic resistivity
     ρT (a constant).43,44
 (2) The proper methodology for determining thoracic length L in the Kubicek and
     Sramek–Bernstein equations and its physiologic relevance to SV.45
64                         D. P. Bernstein & H. J. M. Lemmens

 (3) The validity of the outflow extrapolation procedure.46
 (4) A coherent physiologic basis and correlate for the empirically-derived volume
     conductors defined above.
 (5) The relevance of δ in the Bernstein modification of the Sramek equation.45,47
 (6) Lack of consideration for the changing transthoracic specific impedance,48
     especially in critical illnesses characterized by increased thoracic liquids caus-
     ing aberrant electrical conduction.49,50
 (7) The effect of the blood velocity-induced change in transthoracic specific resis-
     tance ∆ρb (t).
 (8) Failure of the literature to validate a Windkessel model for ICG-derived SV.35

Therefore, it is the stated objective of this chapter to show how existing SV equa-
tions might be improved by exploring the following possibilities:

(1) The general assumption of ρb L2 /Z0 × (dZ/dtmax ) or VEP T /Z0 × (dZ/dtmax )

    being equivalent to dV /dtmax might be in error (Eq. (14)).
(2) dZ/dtmax × Tlve might not be the ohmic equivalent of stroke distance (Eq. (28)).
(3) The influence of the dynamic, blood velocity-induced resistivity change ∆ρb (t)
    is not trivial as concerns dZ/dtmax .
(4) The magnitude of, and theoretical basis for existing electrical volume conductors
    might be incorrect.
(5) Changing transthoracic specific impedance (“thoracic resistivity”) must be
    accounted for in states characterized by excess extra-vascular lung water.49
(6) Amendments to existing SV equations might be appropriate.3,4,7

5. Theory and Rationale for a New Stroke Volume Equation
5.1. Analytical methods
Based on a literature search relevant to impedance-derived SV, a critical analy-
sis of the biophysical and electrodynamic systolic origins of ∆Z(t) and dZ/dt was
performed and extrapolative comparisons made with their respective mechanically-
based hemodynamic counterparts. Equations descriptive of the reported phenomena
were then designed to elucidate the various factors responsible for the systolic com-
ponents of dZ/dt, and specifically the origin of dZ/dtmax . To this end, a differential
and time domain analysis was performed, and conclusions drawn with regard to the
physical time domain in which dZ/dtmax resides. An analytically-based mathemat-
ical transformation, derived and extrapolated from the relevant bioimpedance and
hemodynamic literature, was employed to find the ohmic mean velocity equivalent
necessary for SV calculation. A new volume conductor Vc is also proposed, based on
proven allometric relationships and accepted physiologic principles, as well as the
constraints imposed by the magnitude of the resultant transformation. Additionally,
a transthoracic electrical shunt index for aberrant conduction is introduced, render-
ing the magnitude of the volume conductor dynamic, rather than static, as in the
                                Impedance Cardiography                                65

Sramek–Bernstein approach. To test the above hypotheses, the results of a recently
published study are presented, wherein the Kubicek, Sramek, Sramek–Bernstein
equations were compared to the new equation, using thermodilution cardiac output
as a reference standard.51 As a consequence of the theoretical discussion, empirical
time domain analysis, equation derivations and results of the clinical study, a new
SV equation is proposed.

5.2. Origin of dZ/dtmax
5.2.1. Resolution of origin by anatomic domain
The systolic portion of ∆Z(t) has been shown to be comprised of elements from
both the pulmonary artery and aorta.14,15 By magnitude, Saito et al.14 and Ito
et al.15 demonstrated that about 30%–40% of the area beneath the ∆Z(t) wave-
form is generated by pulmonary blood flow and 60%–70% by aortic blood flow.
Early correlative phonocardiographic studies of the dZ(t)/dt waveform by Lababidi
et al.52 clearly show fiducial landmarks consistent with aortic valve opening and
aortic and pulmonary valve closure (Fig. 5). When electrocardiographic signs of
left bundle branch conduction delay are present, pulmonary valve opening can also
be readily distinguished.53 Likewise, with right bundle branch block, the delay in
pulmonary valve closure is also clearly apparent.53 As concerns the anatomic ori-
gin of dZ/dtmax , there is compelling evidence that it derives from the ascending
aorta.2–4,54 Similar to Doppler velocimetry, in which the aortic peak velocities are
proportional to the highest frequency shifts detected, the peak rate of change of aor-
tic volume (or peak flow) is assumed to be proportional to the peak rate of change of
impedance (Eq. (14)). In normal individuals, Gardin et al.55 reported mean values
of 92 cm · s−1 , and 63 cm · s−1 , respectively, for the peak velocities of the aorta and
pulmonary artery. Additionally, mean values for time to peak velocity (TTP), or
acceleration time, were reported as 98 ms and 159 ms for the ascending aorta and
pulmonary artery, respectively. Therefore, if one assumes a plethysmographic ori-
gin for dZ/dtmax , then it must be concluded that dZ/dtmax represents the highest
aortic velocities detected, and that, by timing, aortic peak velocity would precede
pulmonary artery peak velocity. Thus, the peak pulmonary artery component of
dZ(t)/dt (dZ[pulmonic]/dtmax ) would be obscured and delayed in timing within
the composite first derivative impedance envelope.
    Alternatively, if dZ/dtmax represents the peak time derivative of the resistivity-
based velocity component of ∆Z(t) (Eq. (3) and Sec. 3.2) dρb (t)/dtmax (Ω cm ·
s−2 )(i.e. d[∆ρb (t)]/dtmax or d[∆Zv (t)]/dtmax ), then the aortic component would still
predominate. Justification for this conclusion is based on the finding that average
values for aortic and pulmonary artery mean blood flow acceleration are reported as
940 cm · s−2 and 396 cm · s−2 , respectively.55 Thus, if dZ/dtmax represents the great-
est rate of change of the blood resistivity variation, which, in turn, is related to the
peak acceleration of red blood cells, then dZ/dtmax would peak with aortic dv/dtmax
and dQ/dtmax . In addition to purely mechanical considerations, dynamic changes
66                        D. P. Bernstein & H. J. M. Lemmens

in the specific resistance of flowing blood are characterized as being anisotropic.56
In concert with, and analogous to the magnitude of the ultrasonic frequency shift in
Doppler velocimetry, this means that, in addition to red cell velocity, the measured
magnitude of the resistivity change is related to the angle of interrogation between
the applied AC and the direction of the forward flowing erythrocytes. The greatest
negative resistivity changes of flowing blood occur when, from a state of random
orientation in end-diastole, the long axis of symmetry of the red cells become ori-
ented axially, aligned in parallel with the applied AC during systole (Fig. 4).21,22
This condition is fulfilled in the ascending and descending thoracic aorta, but not in
the pulmonary artery, because only a short segment of the pulmonary artery trunk
is aligned longitudinally, in parallel with the applied AC. Since the direction of the
vast majority of pulmonary blood flow is via the left and right pulmonary arteries,
and thus, perpendicular to the axis of the applied AC, the total pulmonary artery
resistivity contribution is considered trivial as compared to that of the aorta.16

5.2.2. Resolution of origin by differential time domain analysis
Since ∆Z(t) is a composite waveform, containing ohmic analogs of both velocity and
volume changes (proportionality 6, and Fig. 3), dZ/dt should reflect their respective
time-derivatives, linear acceleration and flow. Thus, by differentiating Eq. (3) with
respect to time:

                          dZ(t)   dZv (t) dZvol (t)
                                =        +          .                            (29)
                           dt       dt      dt

    If a segment of aorta is considered a cylindrical, thin-walled blood-filled struc-
ture at end-diastole, its impedance Z to an applied AC field, across the measured
segment, can be defined as given in Eq. (8). In this example, ρb is the static specific
resistance of blood (Ω cm), L is the segment of aorta under electrical interrogation
(cm), and Vb is the volume of blood (mL) in the aorta over segment L in end-
diastole. If upon ventricular ejection, all variables in Eq. (8) become continuously
differentiable functions of time, then, using the product and quotient rules, the
rate of change of the cardiogenically-induced impedance variation dZ/dt is given as

                 dZ(t)   ρb 2L dL(t) L2 dρb (t) ρb L2 dVb (t)
                       =            +          −              .                  (30)
                  dt     Vb 1 dt      Vb dt      Vb2    dt
                             (1)         (2)           (3)

If dL/dt → 0, then dZ/dtmax peaks in the time domain of either derivative 2 or 3.
Conceptually, SV obtained by the Kubicek, Sramek and Sramek–Bernstein equa-
tions implement the peak first time-derivative of derivative 3, relating dZ/dtmax to
                               Impedance Cardiography                               67

the peak rate of change of the aortic volume variation, dV /dtmax (mL · s−1 ), the
genesis of which has already been discussed. Thus,
                       dZvol (t)   ρb L2 dVb (t)
                                 =− 2            (Ω · s−1 ).                      (31)
                        dtmax       Vb dtmax
Alternatively, dZ/dtmax might possibly peak with derivative 2, relating dZ/dtmax
to the peak first time-derivative of the velocity-induced blood resistivity variation,
d[∆ρb (t)]/dtmax , corresponding biophysically to, and in the time domain with peak
aortic blood acceleration dv/dtmax . Therefore,

                          dZv (t)   L2 dρb (t)
                                  =            (Ω · s−2 ).                        (32)
                          dtmax     Vb dtmax
Analysis of Eqs. (31) and (32) demonstrate the two possible origins of dZ/dtmax , but
unfortunately, this theoretical conundrum cannot be resolved by simple differential
analysis. Although a temporal difference in time-to-peak (TTP) magnitude of the
derivatives in Eqs. (31) and (32) is implied, there is insufficient information in either
equation to establish a definitive biophysical origin for dZ/dtmax in the time domain.

5.2.3. Resolution of origin by comparative time domain analysis:
       time-to-peak (TTP) of dZ/dtmax
While the blood resistivity change is necessarily deemed a trivial contaminant, using
the plethysmographic approach, it has been demonstrated that the changing “resis-
tivity” (specific resistance) of flowing blood contributes approximately 50% to the
amplitude of ∆Z(t).10,13,57 Others11,12,16 have observed lesser, but still substan-
tial contributions of the blood velocity induced resistivity change to the magnitude
of ∆Z(t). Therefore, since dZ/dt is the first time-derivative of ∆Z(t) and accel-
eration is the derivative of velocity, it is suggested that dZ/dt might peak in the
time domain of acceleration (dv/dt, dQ/dt), rather than flow (Q, dV /dt). Proof that
dZ/dtmax peaks synchronously with dv/dtmax , is found to reside in its empirical
correlation with known TTP durations (ms) of the time-derivatives of pressure and
flow in the mammalian cardiovascular system. Physical measurements of the TTP
from opening of the aortic valve to dZ/dtmax is reported to be 60 ± 20 ms in nor-
mal individuals at rest (Fig. 6),58–60 whereas peak velocity (vmax ), and in the time
domain, equivalently maximum flow (Qmax , dV /dtmax ), peak at a mean of 98 ms, or
at the end of the first third of systole.55 Concordant with the TTP of dZ/dtmax , it
has been demonstrated that the TTP of peak aortic blood acceleration (dv/dtmax ,
dQ/dtmax ) occurs equivalently at about 50 ± 20 ms from valve opening, or in the
first 10%–20% of systole.37,61,62 In Fig. 6 it is to be noted that, for an ejection time
of ∼ 360 ms, dZ/dt peaks in ∼ 60 ms, or in the first 10%–20% (16.7%) of systole.
Similarly, graphic evidence provided by Sheps et al.63 shows that, for an ejection
time of approximately 250 ms, TTP for dZ/dtmax occupies 20% of Tlve , or about
50 ms. Corroboratively, close examination of the fiducial landmarks and time lines
68                           D. P. Bernstein & H. J. M. Lemmens

Fig. 6. dZ/dt peaks in the time domain of acceleration TTP = time-to-peak; LVET = left ventric-
ular ejection time; % of systole = % LVET; (TTP/LVET) × 100 = % of systole.

supplied by Lababidi et al.52 show that dZ/dtmax peaks in the first 15–20% of
the ejection phase of systole, as does peak acceleration.37,61,62 Matsuda et al.60
have shown that, while peaking out of phase, the maximum slope and TTP of
LV dP/dtmax and dZ/dtmax are identical. It can also be demonstrated from their
data that the time difference between the onset of the Q wave of the ECG to
aortic valve opening, (i.e. Q wave to dP/dtmax ), and the time of onset from Q
wave to dZ/dtmax , is approximately 60 ± 15 ms (i.e. [Q → dZ/dtmax (ms)] – Q →
LV dP/dtmax (ms)] = 60±15 ms). Corroborative data from Lozano et al.,106 obtained
by subtracting the R(ECG)–B(dZ/dt) interval (mean 68 ms, range 47–84 ms) from
the R–dZ/dtmax interval (mean 120 ms, range 80–160 ms), shows that the TTP
of dZ/dtmax from point B (rise time) is ∼ 52 ± 20 ms (i.e. [R–dZ/dtmax (ms)]–
[R–B (ms)] = 52 ± 20 ms). In normal children, Barbacki et al.64 showed that the
TTP for dZ/dtmax was appreciably less than for Doppler-derived peak ascending
aortic blood flow. Figure 7 graphically demonstrates that dZ/dtmax (waveform G)
precedes peak ascending aortic blood flow (waveform D), monotonically, during the
period of peak acceleration. Kubicek54 and Mohapatra and Hill65 demonstrated that
dZ/dtmax corresponds precisely in time with the “I” wave of the acceleration ballis-
tocardiogram (BCG). The “I” wave of the acceleration BCG is known to represent
the peak acceleratory component of the initial ventricular impulse during the iner-
tial phase of very early systole.61 Kubicek et al.26 also showed that, when measured
simultaneously, dZ/dtmax peaks substantially before ascending aortic Qmax . In this
study,26 correct interpretation of both waveforms’ maxima requires measuring left
                                   Impedance Cardiography                                    69

Fig. 7. Relationship between esophageal ECG (A), aortic pressure (B), aortic expansion (C), aor-
tic blood flow (D), pulmonic expansion (E), surface ∆Z(t) (F) and dZ/dt (G) waveforms. Calibra-
tion signals: aortic and pulmonic expansion (1 mm), aortic blood flow (10 L · min−1 ), ∆Z(t) (0.1
Ω), dZ/dt (1 Ω · s−1 ). Note that dZ/dtmax precedes peak aortic blood flow (Qmax ). Figure 7 with
permission from John K. Hayes, PhD. Influence of simulated gastric reflux, body posture, aortic
and pulmonic vascular expansion and erythrocytes on the impedance cardiogram recorded from an
esophageal impedance probe. Doctoral Dissertation, University of Texas at Austin, p. 186, 1994.

ventricular ejection time on the dZ/dt waveform from point B, corresponding to
aortic valve opening, which is above the baseline, to point X, corresponding to
aortic valve closure on the ascending aortic flow tracing. Confirming this observa-
tion, image enlargement of the work of Ehlert and Schmidt66 subtly, but convinc-
ingly demonstrates that, over a wide range of SV, dZ/dtmax peaks prior to Qmax .
Rubal et al.67 also showed that, when measured simultaneously, dZ/dtmax peaks
appreciably before Qmax . Finally, Welham et al.68 showed that, over a full range
of myocardial depression with halothane anesthesia, dZ/dt peaked simultaneously
with dv/dtmax (acceleration), and very subtly before peak flow (Qmax ). As a result
of this evidence-based, empirical TTP analysis, it is suggested that dZ/dtmax is
the ohmic impedance analog of dv/dtmax , and not dV /dtmax , as generally believed.
Based on this discussion, and the conclusions drawn therefrom, it is suggested that
dZ/dtmax is best described by differential Eq. (32), relating dZ/dtmax to the peak
rate of change of the erythrocyte velocity-induced resistivity change, dρb (t)/dtmax ;
that is, peak aortic red blood cell acceleration. This implies that the requirements
of Eq. (25) are unmet, insofar as the Kubicek and Sramek Bernstein equations are
concerned, and Eqs. (14), (16), (22), and (26–28) do not represent equality. Clearly,
SV is obtained through a mean flow (velocity) calculation, precluding the use of
acceleration as a surrogate: Specifically,
                                dv                    dZ(t)/dtmax
        SV = Area ×                dt ⇒ SV = Volume ×             × Tlve .                 (33)
                          t0    dt                        Z0
70                          D. P. Bernstein & H. J. M. Lemmens

5.2.4. Hemodynamic and biomechanical origins of ∆Z(t) and dZ/dtmax :
       Analysis by means of a differentiable cylindrical model of ascending
       aortic blood flow
Consider a cylinder of length S, and radius r, and assume that the cylinder at its
maximum volume V is equal to SV . Hemodynamically, let the time-velocity inte-
gral, also known as stroke distance, or the displacement of blood over one ejection
interval, equal S (cm). Also, assume that the mean radius of the aortic root over
the ejection interval is a constant, equal to r (cm), as well as the aortic wall being
visco-elastic and thin. Equation (25) then takes the form of,

                                 SV = V = πr2 S.                                     (34)

If r and S are allowed to vary with time, t, then V is a function of time.

                                V (t) = πr(t)2 S(t).                                 (35)

If r(t) and S(t) are continuously differentiable functions, then,
                        dV (t)           dr(t) ∗      dS(t)
                               = Q = 2πr      S + πr2       ,                        (36)
                         dt               dt           dt
                                            (1)            (2)

where, dV (t)/dt equals the rate of change of volume, or flow Q, and dr/dt is the
time rate of change of the aortic radius (wall velocity, cm · s−1 ) from its end-diastolic
nadir r. In this context, derivative 1 corresponds to (dA/dt) · S ∗ ,25 which is a func-
tion of, and equivalent to (dP/dt) · C (Eqs. (17) and (21)). For derivative 1, let S ∗
be a fixed-length aortic segment over which dr/dt is generated and analogous to
thoracic segment length L over which ∆Z(t) and dZ/dtmax are measured. It is also
assumed that vascular segment S ∗ in derivative 1 provides for simultaneous inflow
and outflow during systole. In the currently used plethysmographic approaches,
dS/dt (cm · s−1 ) in derivative 2, corresponding to axial blood velocity v in the New-
tonian domain, and to ∆ρb (t) in the ohmic domain, is obligatorily considered trivial
(vide supra). Thus, in the context of its composite architecture, ∆Z(t) contains the
volume change from Eq. (35), S remaining constant, plus the velocity component
of derivative 2 in Eq. (36), dS/dt, r (i.e. area) remaining constant (see Figs. 3
and 4). It is generally believed that dZ/dtmax peaks in the time domain of Eq. (36)
and that the differential component best describing this biophysical phenomenon in
impedance cardiography corresponds to derivative 1. Derivative 1 is in agreement
with Kubicek assumption 7, which states that all volume changes are due only to
changes in CSA. By contrast, it should be noted that in Doppler and electromag-
netic flowmetry (EMF), both techniques implement derivative 2, where r(t) and
dr/dt are assumed to be negligibly small (i.e. r(t) → 0, dr/dt → 0) as compared
to dS/dt.37 License for this convention is considered valid, because, attendant with
systole, the aortic annulus and ascending aortic radius in humans change by only
± 3.3% from their end-diastolic nadir (i.e. ∆r/r = ±3.3%).69 The small change in
                                     Impedance Cardiography                            71

radius generates a maximum change in CSA of about 6–7%. Clearly, integrating dif-
ferential 1 of Eq. (36) over the ejection interval is equivalent to dV /dr in Eq. (19).
Thus, for an aortic segment length of 30 cm with open outflow, segment mean end-
diastolic radius of 1.5 cm and a 3.3% change in radius over the pulse pressure (i.e.
dr = 0.0495 cm), the net change in volume (i.e. net SV) would be approximately
14 mL. This value is approximately five to six times smaller than physiologic (i.e.
15%–20%), and thus, similar in magnitude to the expelled gas volume obtained and
predicted from pneumocardiography (vide supra).20 As discussed earlier, this vol-
ume represents net volume inflow into the aortic segment, with the remainder of SV
(i.e. 80%–85%) being simultaneously dissipated to the periphery as axially directed
outflow. For this simple model of pulsatile blood flow, in contrast to classic Wind-
kessel approaches, it thus follows that the outflow is represented by derivative 2 of
Eq. (36). To compensate for radial displacement and volume inflow, Doppler-derived
SV implements the measurement of aortic diameter at peak systolic displacement.
Thus, by magnitude,
                                 t1                         t1
                                      dS(t)                      dr(t)
                        ACSA                dt      AISA               dt,
                                t0     dt                  t0     dt                 (37)
                                            SVout     SVin ,

where ISA is the vessel internal surface area over segment S and CSA is the vessel
cross-sectional area at the point of measurement of dS/dt.
   Differentiating flow (Q) with respect to time, yields radial wall and axial aortic
blood acceleration dQ/dt (i.e. d2 V /dt2 ). Hence,
    dQ          dr(t)                  d2 r(t)         dr(t) dS(t)       d2 S(t)
       = 2π                  S + 2πr       2
                                               S + 4πr             + πr2         .   (38)
    dt           dt                     dt              dt    dt           dt2
                  (1)                    (2)                    (3)          (4)

As concerns ventricular-vascular coupling, derivatives 1–3 are considered compar-
atively trivial at dQ/dtmax , because aortic wall velocity dr/dt, wall acceleration,
d2 r/dt2 , and axial dS/dt approximate zero by comparison with d2 S/dt2 (cm · s−2 )
during the inertial phase of ventricular ejection. Thus, dZ/dt is a composite wave-
form, comprising ohmic analogs of derivative 1 of Eq. (36), and derivative 4 of
Eq. (38). Therefore, based on this and preceding discussions, it is the thesis of
this chapter that dZ/dtmax peaks in the time domain of derivative 4, the second
derivative of stroke distance; that is, linear peak aortic blood acceleration, dv/dtmax
(d2 S/dt2 ). Clearly, this conclusion is consistent with the theories of origin pro-
posed in Eqs. (29)–(32) and the discussion in Sec. 5.2.3. Thus, in order to compute
SV , the ohmic equivalent of derivative 4 in Eq. (38) (i.e. Eq. (32)) must be trans-
formed into the ohmic equivalent of derivative 2 in Eq. (36). Ordinarily, velocity
is obtained from acceleration through simple integration. Unfortunately, however,
ohmic mean velocity cannot be directly extracted from dZ/dt using this method.
72                         D. P. Bernstein & H. J. M. Lemmens

6. Rationale for Application of the Square Root Acceleration
   Step-down Transformation in Impedance Cardiography:
   Conversion of dZ/dtmax Normalized by its Base Impedance
   Z0 (dZ/dtmax /Z0 ) to Normalized Ohmic Mean Velocity
   ∆Zv (t)max /Z0
Hypothetically assuming dZ/dt to be a mono-component, blood resistivity-based
acceleration waveform (i.e. dZv (t)/dt), and conforming to derivative 2 in Eq. (30),
simple integration would yield ∆Zv (t), its sole composition being the velocity-
induced blood resistivity change ∆ρb (t). However, as discussed, ∆Z(t) is a com-
posite waveform, its peak magnitude ∆Z(t)max probably representing a point
temporally approximating peak pressure, which physically corresponds to the net
volume change of the aorta in mid systole (Fig. 3, Eqs. (4) and (12)).35 This
effectively voids integration as a solution for finding ∆ρb (t)max , and thus, ohmic
mean velocity. Since it is suggested that dZ/dtmax possesses the dimensions of
Ω · s−2 (Eq. (32)), and normalized by its base impedance Z0 to a dimensionless
ohmic acceleration equivalent, s−2 , a square root transformation would seem math-
ematically plausible and intuitively appealing. When equation (32) is normalized
to relative ohmic acceleration by Z0 , a dimensionless ohmic equivalent results,
namely, [(L2 /Vb )dρb (t)/dtmax ]/Z0 , which reduces to s−2 . Operationally, the numer-
ator would possess dimensionless magnitude            0 during early systolic ejection.
Thus, it is suggested that dZ/dtmax /Z0 is related to the normalized differential in
Eq. (32), as follows:
                      dZv (t)/dtmax   L2 dρb (t) 1
                                    =              = s−2 .                        (39)
                            Z0        Vb dtmax Z0
Since both sides of Eq. (39) are suggested to be dimensionless ohmic equivalents of
acceleration, the square root of both sides should yield dimensionless ohmic equiv-
alents of velocity. That is:

            dZv (t)/dtmax         L2 dρb (t) 1        ∆Zv (t)max
                          =                       ≡              = s−1 .          (40)
                  Z0              Vb dtmax Z0            Z0

                      dZv (t)/dtmax
                                    = ohmic mean velocity,                        (41)
where Eq. (41) is to be known as the square root acceleration step-down transfor-
mation, or simply, ICG acceleration step-down transformation.

6.1. Peak aortic reduced average blood acceleration (PARABA):
     Hemodynamic analog of dZ/dtmax /Z0
The rationale for, and the correctness of the square root transformation is to be
found in a modified version of theory originally propounded by Visser.21 Consider
                                   Impedance Cardiography                                                  73

the following relationship for blood accelerating axisymmetrically through a circular
aorta with radius R (cm):
                            dQ              dv
                               =               2πrdr = mL · s−2 ,                                        (42)
                            dt      0       dt
where dQ/dt is the instantaneous acceleration. Integrating Eq. (42) by parts, and
assuming the blood does not slip at the vessel wall, the following is ultimately
                        R                                         1                    2
        dQ                  (dv/dt) 2                                 (dv/dt)      r            r
           = −π                    r dr = −πR3                                             d      .      (43)
        dt          0         dr                              0         dr         R            R
                                                1                          2
             dQ/dt   dv/dtmean                          (dv/dt)        r           r
                   =           =                    −                          d     = s−2 .             (44)
              πR         R                  0             dr           R           R
The results obtained from Eq. (44) demonstrate that the blood flow acceleration
resulting from integration over the vessel radius R is the mean acceleration divided
by the aortic radius. At its aortic maximum, this is to be referred to as peak aortic
reduced average blood acceleration (PARABA), [(dv/dtmean )/R]max , and is solely a
function of the maximum shear acceleration, [−(dv/dt)/dr]max , and the ratio of r/R.
Since r/R is dimensionless, the shear acceleration is purely a function of the reduced
average acceleration. Thus, rearranging Eq. (44) and solving for volumetric mean
                   dQ                               dv/dtmean
                                   = πR3                                = mL · s−2 .                     (45)
                   dt       mean                        R
Concordant with this analysis, Visser21 showed that, in vitro, the peak relative
resistivity change ∆ρ(t)max /ρstat is a function of the maximum reduced aver-
age blood velocity [(vmean /R)max ]exp . In vivo, peak aortic reduced average blood
velocity (PARABV) is determined by peak aortic blood velocity, vmax . Analo-
gously, an equivalent relationship exists between dZ/dtmax /Z0 , PARABA, and
dv/dtmax . Thus, to obtain ohmic mean velocity by means of impedance cardiog-
raphy, the square root of the impedance analog of PARABA must be taken, which
is (dZ/dtmax )/Z0 . It can thus be stated that mean flow Q (mL · s−1 ) is equivalent
                               dZv (t)/dtmax                               dv/dtmean
             Qmean = K1                      ≡ K2
                                     Z0                                        R               max
                               vmean                    m             ∆Zv (t)max
                   ≡ K2                                     ≡ K1                 ,                       (46)
                                 R          max                          Z0
where K1 and K2 , are volumetric constants of proportionality, and n and m are
exponents equating the bioelectric and hemodynamic domains. Clearly, the rela-
tionship between dZv /dtmax /Z0 and ∆Zv (t)max /Z0 is parabolic in nature (y = x2 ),
74                             D. P. Bernstein & H. J. M. Lemmens



                            dZ/dtmax/Zo   0.08


                                                 0.0   0.1   0.2    0.3      0.4

Fig. 8. Square root acceleration step-down transformation. Relationship between the bioelectric
equivalent of peak aortic reduced average blood acceleration (PARABA), which is dZ/dtmax /Z0 (y-
axis), and the bioelectric equivalent of peak aortic reduced average blood velocity (dZ/dtmax /Z0 )

as graphically demonstrated in Fig. 8. Shown on the x-axis is the linear extrapo-
lation of the square root transformation of dZ/dtmax /Z0 (x = y), which yields
dimensionless ohmic equivalents of peak aortic reduced average blood velocity. Thus,
it is suggested that Eqs. (16) and (26–28) do not represent equality, the suggested
correct relationship between impedance-derived SV and classical expressions being,
                       SV = Area × vmean × Tlve = πr2                          v(t)dt

                                            dZv (t)/dtmax
                           = Volume                       Tlve .                              (47)

    Despite Visser’s experimental in vitro finding that the measured relative blood
resistivity change is proportional to hematocrit,21,22 in vivo SV calculation using
dZ/dtmax is known to be minimally affected (± 5%) over the full range of physiologi-
cally permissible hematocrit levels.44,70 Corroborative in vitro evidence reported by
Visser et al.13 clearly shows that, while the measured peak magnitude of ∆Z(t) (i.e.
∆Z(t)max ) is dependent upon hematocrit, its maximal upslope, that is, dZ/dtmax
is not (see Fig. 3 in Ref. 13). In as much as the ohmic mean velocity calculation is
derived from the square root transformation of dZ/dtmax , and not from an in vivo
measurement, impedance-derived SV , using the transformation, is predicted to be
unaffected by hematocrit as well. By being mathematically and not physiologically
coupled, this is surely the case. Thus,
                                                 ∆Zv (t)max
                          SV = Volume ×                     × Tlve ,                          (48)
where ∆Zv (t)max /Z0 is the calculated ohmic equivalent of (vmean /R)max .21
                               Impedance Cardiography                               75

    Thus, it is suggested that the mean value square root transformation renders,
moot, the theoretical justification for the peak value outflow correction extrapola-
tion procedure, and consequently, assumption 9 of Kubicek’s hypothesis. Evidence
suggesting that the theory behind the maximal forward systolic upslope extrapola-
tion of ∆Z(t) is flawed resides in the sweeping assumption that little outflow exists
during the rapid ejection phase of systole. Johnson et al.71 showed that, by the end
of the rapid ejection phase, or first third of systole, about 45% of total SV is already
ejected in normal individuals. Thus, the extrapolation procedure would have to com-
pensate for the roughly 50% of SV ejected before peak flow velocity occurs, plus
the 50% of SV ejected in the final two-thirds of systole. The end of the first third of
systole is precisely in time where dV /dtmax occurs in normal individuals and other
primates,37,58,72 and where Kubicek54 and Kubicek et al.26 believed dZ/dt peaked.
As per the discussion in 5.2.3., dZ/dtmax peaks in the very early, inertial phase of
systolic ejection (i.e. peak acceleration), where only 5%–10% of SV is ejected.61,73
This, however, is irrelevant, insofar as Kubicek’s hypothesis is concerned, because
dZ/dtmax is a priori a mean acceleration variable. Thus, while the outflow theory is
flawed, the square wave integration is operationally correct, providing acceleration
surrogates of mean flow over a narrow range of values (Fig. 8).
    Further evidence showing that dZ/dtmax is a mean value equivalent is demon-
strated by the observation that, over a full range of Vc and dZ/dtmax /Z0 , both the
Kubicek and the Sramek–Bernstein equations yield numeric outputs roughly 50%
of those actually predicted for peak flow.72 Therefore, the direct proportionality
of dV /dtmax and dZ/dtmax expressed in Eq. (14), the equivalencies expressed in
Eqs. (16–22) justifying the plethysmographic hypothesis, and assumptions 7 and 8
of Kubicek’s method appear invalid.
    Interestingly, as compared to standard reference methods, the correlation of
impedance derived “SV ”, using mean acceleration equivalents, is considered good
(r2 = 0.67, range = 0.52–0.81 and, r = 0.82, range = 0.70–0.90).7 The closeness
of association can be fully explained by the observation that both mean and peak
acceleration, per se, are both highly correlated with peak velocity and SV (r = 0.75–
0.775),74,75 as well as with the systolic velocity integral (r = 0.75).76 Undoubtedly,
the close association between acceleration with both SV and stroke distance, across
a full range of values, has fortuitously kept the technology alive since its clini-
cal introduction forty years ago.23 Thus, in conjunction with a Vc of appropriate
magnitude, the “extrapolation procedure” mathematically forces the ohmic analog
of PARABA through systole (Tlve ) producing a sometimes close facsimile of SV .
Therefore, the well known overestimations of SV reported in both young normal
individuals,77 and animals,78–80 using both the Kubicek and Sramek–Bernstein SV
equations, can be fully explained on the basis of PARABA equivalents, proportion-
ately disparate with their reduced average blood velocity counterparts. By inspec-
tion of Fig. 8, it becomes clear that there will be overestimations of mean velocity
at higher levels of PARABA and underestimations at lower levels of PARABA.
Indeed, there is evidence based data provided by Yamakoshi et al.35 showing that
76                        D. P. Bernstein & H. J. M. Lemmens

impedance-derived SV overestimates its EMF counterpart in healthy canines by
as much as 70%, and correspondingly underestimates its EMF counterpart by as
much as 25% when myocardial failure is induced. These findings are consistent with
those of Ehlert and Schmidt,66 who reported that a linear relationship could not be
found between impedance and EMF-derived SV over a wide range of hemodynamic
perturbations. As a result of this discussion, it seems improbable that assumptions
7 and 8 of Kubicek’s hypothesis can be supported.
    Evidence suggesting that dZ/dtmax varies inversely with aortic valve CSA, or
radius, as a function of PARABA, is inferentially demonstrated through the work
of Sageman.81 He clearly showed that an inversely proportional and highly nega-
tively correlated (r = −0.75) relationship exists between dZ/dtmax /Z0 and body
mass in otherwise healthy individuals. The literature also shows that aortic valve
CSA is highly and directly correlated with both body mass (kg) and body surface
area (BSA).82 This would corroborate the experimental results and hypotheses of
Visser,21 and explain the findings of Sageman,81 showing progressively diminished
values of dZ/dtmax /Z0 as weight, aortic valve radius, and CSA increase. This is in
stark contrast to Doppler or electromagnetically derived peak velocities and systolic
velocity integrals, which are totally independent of body mass.83 Thus, peak New-
tonian velocities of equal magnitude, measured between age-matched normal indi-
viduals of different body mass and aortic valve CSAs, will produce correspondingly
disparate values of dZ/dtmax /Z0 . However, while there is no clear direct proportion-
ality between hemodynamically-based, and impedance-derived systolic velocity inte-
grals within or between individuals (i.e. vmean × Tlve versus dZ/dtmax /Z0 × Tlve ),
a linear equivalence does exist between Newtonian-based mean flow and impedance-
derived mean flow. Specifically,

                  Area × vmean = Volume ×                         .              (49)
Taken to its logical conclusion, using the relationships established in Eqs. (42–47),
the right hand side of Eq. (49) can also be equated with the hemodynamic domain
through a convoluted abstraction of the equation of continuity:
                   Qmean = πR3
                                 πr2                         dZ/dtmax
                          =                    v(t)dt = Vc            ,          (50)
                              t 1 − t0   t0                     Z0
where exponent n has been experimentally determined to be between 1.20–
1.25 (authors’ unpublished results). Because of the absolute dependency of
(dZ/dtmax )/Z0 upon PARABA, as established earlier, this means that, for any given
value of mean acceleration, the magnitude of dZ/dtmax is related to, and explicitly
dependent on aortic root CSA by its dependency on R. Since aortic root CSA is
a function of body mass,82 age,33 and gender, dZ/dtmax will also vary accordingly
                              Impedance Cardiography                              77

with these parameters.84 Thus, the magnitude of dZ/dtmax is multi-factorial and
not explicitly dependent on the respective levels of myocardial contractility or Z0
alone.35,68,85 Specifically, as a first-order approximation, the input variables deter-
mining the magnitude of dZ/dtmax are given as follows:
                                                       n 2
            dZ(t)      πR3          dv/dtmean
                  = Z0                                       (Ω · s−2 ).        (51)
            dtmax       Vc              R        max

7. Stroke Volume Equation Implementing the Square Root
   Acceleration Step-down Transformation
7.1. The volume conductor
As a consequence of the square root acceleration step-down transformation of
(dZ/dtmax )/Z0 , the resultant dimensionless values are numerically greater by sev-
eral magnitudes (Fig. 8). Thus, the volume conductors of the Kubicek and Sramek–
Bernstein methods are approximately four to five magnitudes larger than required,
and therefore, as utilized with the square root transformation, inappropriate. As
an alternative to present approaches, it is suggested that the thoracic Vc , or VEP T
would then best approximate the intra-thoracic blood volume (ITBV). The ratio-
nale for the new hypothesis derives from studies correlating both body mass and
ITBV with SV. It is well known that TBV is a major input for venous return and
ITBV, which, in turn, are major input variables for ventricular preload, and thus,
SV.86 Furthermore, SV is highly correlated with (r = 0.97), and virtually a lin-
ear allometric function of body mass (0.66 W(kg)1.05 ).87 It is also known that body
mass is linearly and highly correlated (r2 = 0.997) with TBV by the allometric rela-
tionship, 73 W(kg)1.02 .88 The allometric relationships linking body mass to both SV
and TBV89 extend to ITBV and global end-diastolic volume (GEDV).90,91 These
indices are now recognized as the measurements most closely correlated with left
ventricular preload, and, as such, the major determinants of SV, CO, and changes
thereof.92–96 As reported by Feldschuh and Enson,97 indexing TBV for body height
shows poor correlation (r = 0.35–0.37) as compared to BSA (r = 0.80–0.87), weight
(0.77–0.88), and deviation from ideal body weight (r = 0.90). The relatively poor
correlation of overall body height with indexed values of blood volume is concordant
with its poor correlation with SV.98 Indeed, allometrically, SV is most highly cor-
related with body mass, and fat-free body mass by univariate statistical analysis.98
Thus, the high correlation of body mass with ITBV, and both body mass and
ITBV with SV, suggests that the most appropriate input variable for the Vc would
be body mass (kg), allometrically related to the ITBV. Therefore, in the other-
wise normal individual, and as a first order approximation, Vc is suggested to be
equivalent in magnitude to the ITBV (VITBV ), which approximates 17–20 mL/kg,91
or about 25% of TBV. For purposes of SV determination, VITBV is approximated
as 16 W(kg)1.02 (or 0.25 × TBV). Thus, in otherwise healthy individuals of ideal
body mass, the Vc in impedance cardiography is linked with the aortic radius and
78                        D. P. Bernstein & H. J. M. Lemmens

CSA by the relationship established in Eq. (50). In defense of the new hypothe-
sis, an alternative explanation supporting the biophysical appropriateness of the
thoracic length or height-based volume conductors has never been proposed, much
less proven.99 Thus, in retrospect, given the range of magnitude of dZ/dtmax /Z0 ,
it is suggested that the empirically derived “best-fit” volume conductors of both
the Kubicek and Sramek–Bernstein methods were unknowing searches for close
approximations of TBV. Supporting this contention, consider the following argu-
ment: For the Kubicek equation, using 17% of overall height as a surrogate45 for
measured L in healthy young persons of 150–180 cm, where ρ = 135 Ω cm, and
Z0 = 24–30 Ω, produces volume conductors in the range of 3,500–5,500 mL. Corre-
spondingly, indexing blood volume at 65–75 mL/kg in humans of ideal body mass
index (BMI(ideal) = 22 kg/m2 ) from 50–80 kg, produces TBV closely approximating
the volume conductors constructed above. As concerns the appropriate value for
ρ, Quail et al.44 rearranged the Kubicek equation, solved for ρ as the dependent
variable, and measured SV by EMF. By means of normovolemic exchange transfu-
sion, they showed that over a wide range of hematocrit from 26%–66%, the value
of ρ remained virtually constant about a mean of approximately 135 Ω cm. Accord-
ingly, they changed the designation of the hematocrit dependent specific resistance
of blood ρb in the Kubicek equation to the assumed constant specific resistance of
the thorax ρT . These findings were seminal in generating the Sramek equation,24
wherein he substituted ZA/L for ρT in the Kubicek equation. Meijer et al.,48 how-
ever, measured the transthoracic specific impedance and found the mean value for
ρ approximated 330 Ω cm ±40 Ω cm, which is about 2.5 times greater than the value
predicted by Quail et al.44 They also found that ρT varied with age, the extremes
being 290 Ω cm at age ten to 430 Ω cm at age fifty, probably reflecting the increased
ratio of thoracic gas volume to ITBV. At variance with Quail et al.,44 they also
demonstrated that ρT was non-constant intra-individually and varied directionally
with Z0 . As predicted by Eq. (7), and solving for ρ, this would be the expected
result with varying levels of hydration, intravascular blood volume and interstitial
extra-vascular lung water (EVLW). Therefore, as suggested, it is not ρT that is
constant, but rather, the monotonic constancy of the ratio ρT /Z0 . Thus, the non-
constant magnitude of Kubicek’s volume conductor, based on changing values of Z0
and a constant value for ρT , appears somewhat untenable. Sramek’s conductor is
somewhat more compatible with the findings of Meijer et al.,48 because it is oper-
ationally more consistent with the concept that ρT varies directly with Z0 ; that is,
VEP T is constant over a full range of ρT and Z0 .
    The concept of the ITBV being the appropriate Vc is consistent with Kubicek
assumption 4, stipulating that all electrical conduction flow through the blood
resistance. As implicitly mandated and proposed herein, the ITBV is, in fact, the
equivalent physical embodiment of the ohmic blood resistance Rb . Furthermore,
except for errors imposed by its allometrically defined magnitude, ITBV is anatom-
ically and biophysically assumption-free, inherently unambiguous in meaning, and
intuitively understood. By comparison, existing volume conductors are modeled as
                                   Impedance Cardiography                           79

simple geometric abstractions, which are firmly rooted, uniquely, in basic electrical
theory.99 By virtue of their “best-fit” mathematical construction, they bare little
relevance to, and have virtually no association with other commonly acknowledged
physiologic, anatomic or hemodynamic parameters.

7.2. Index of transthoracic aberrant conduction ζ (zeta): Genesis
     of the three-compartment parallel conduction model
Analysis of the appropriate volume conductor does not address the issue of changing
thoracic resistivity in pathologic states, especially those characterized by excess
intra-thoracic liquids. Using the Sramek–Bernstein method, it is well known that
SV is systematically underestimated in sepsis and other conditions characterized
by increased thoracic liquids, such as pulmonary edema.49,100–102 Because quasi-
static Z0 appears as a squared function in the Kubicek equation, with a constant
value for ρ (Eqs. (16) and (26)), the underestimation of SV is less marked.102 The
underestimation is most likely due to a bypassing or shunting of AC, circumventing
the VITBV to the more highly conductive edema fluid.49,50,103 In its extreme form,
a virtual short circuit is produced. Since the specific resistance of plasma is about
60–70 Ω cm,104 and that of blood 100–180 Ω cm, within the physiologic range,19,44 it
is likely that the AC takes the path of least resistance (i.e. lowest impedance), which
is the plasma-like edema fluid. Thus, the magnitude of the Vc increases as a function
of an enlarging edema volume Ve , causing pathologic parallel conduction between
the blood and edema fluid compartments. In this scenario, Vc = VIT BV + Ve . Thus,
in effect, a three-compartment parallel conduction model is generated. According
to Ohm’s Law, this produces an appropriate, but pathologic transthoracic voltage
drop, and a progressive decrease in AC flow through the dynamic VIT BV as Ve
increases. Consequently, a smaller than predicted pulsatile change in voltage occurs,
rendering it no longer proportional to its mechanical hemodynamic counterpart.35 If
Eqs. (1) and (2) represent normal conduction, wherein all current flows through the
static Zb and dynamic blood resistance ∆Zb (t), then the following interpretation of
Ohm’s Law characterizes the discussion above, if Z is defined in terms of Eq. (8).
              ρt L 2   ρb L 2   ρe L 2   ∆ρb (t)L2     ρb L 2
       I·                                          +            = U0 ∆U (t) ,     (52)
               Vt       Vb       Ve        Vb         ∆Vb (t)
              (1)       (2)       (3)              (4)

where subscripts t, b, and e represent thorax, blood, and edema, respectively (Fig. 2).
Compartment 1 represents the non-conductive static thoracic tissue impedance Zt ;
compartment 2, the highly conductive, static component of the blood resistance,
or impedance Zb ; compartment 3, the most highly conductive, variable magnitude
static edema impedance Ze ; and compartment 4, the pulsatile, dynamic components
of the blood resistance, comprising ∆ρb (t) and ∆Vb (t). It should be noted that
compartments 1 through 3 represent the quasi-static base impedance Z0 . Clearly,
as compartment 3 increases in volume, expanding within a fixed volume thoracic
80                         D. P. Bernstein & H. J. M. Lemmens

cage, the total transthoracic impedance Z(t) decreases as a function of displace-
ment, compression and reduction of the thoracic gas volume. The result of this is
an appropriate, but pathologic drop in the measured static transthoracic voltage
U0 . Moreover, since a progressively increasing proportion of AC flows through com-
partment 3, drawing current at the expense of compartments 2 and 4, an inap-
propriate electrical damping of the peak magnitude of ∆Z(t) (∆Z(t)max ), and
reduction of its maximum systolic upslope dZ/dtmax ensue.35,50 Specifically, this
sequence of events causes spuriously low values of dZ/dtmax /Z0 , which, by absolute
magnitude, no longer parallel PARABA or proportional changes thereof.49,50 The
diminishing magnitude of dZ/dtmax , as the volume conductor increases in magni-
tude, PARABA remaining constant, is predicted by Eq. (51). Thus, the equality
established in Eq. (50) no longer exists, the final result being underestimation of
ICG-derived SV . In conjunction with reduced levels of myocardial contractility,
as in congestive cardiomyopathy, it is suggested that this is the major cause of
underestimation of SV and CO in sepsis and other pulmonary edematous states
using present impedance techniques. While it is beyond the scope of this chapter
to fully elucidate the bioelectric mechanisms operative in states of excess thoracic
liquids, it has been found that the magnitude of the volume conductor must be
corrected for by a dimensionless AC shunt index. This parameter is to be known as
the Index of Transthoracic Aberrant Conduction, ζ (zeta), and becomes operative
when pathologic parallel conduction evolves. ζ is mathematically equivalent to:

                         Zc − Z0 · Zc + K
                ζ=                                (Dimensionless),                 (53)
                     2Zc + Z0 − 3Zc · Z0 + K
                       2     2

where Zc = critical level of base impedance = 20 Ω, Z0 = measured transthoracic base
impedance ≤Zc , and K = a trivial constant → 0.
    It has been empirically established that pulmonary edema with pathologic con-
duction supervenes at critical values of Z0 (Zc ) between 20 Ω ± 5 Ω.49,103 The pre-
cise value for Zc depends on premorbid Z0 , age, gender, body habitus, and the
frequency of the applied AC.105 Until being further quantifiable, Zc is ascribed a
nominal default value of 20 Ω at 2.5 mA (rms) and 70 kHz, using a tetra-polar spot-
electrode array. Operationally, ζ is a dimensionless index, mirroring the magnitude
of the AC shunt, and decreases progressively from a value of unity at levels below
Zc (i.e. If Z0 < 20 Ω, 0 < ζ < 1.0). Consistent with a cylindrical model of constant
length, but expanding CSA, ζ is implemented as an exponential function, reflecting
a change in edema volume. At all levels of Z0 ≥ 20 Ω, ζ = 1.0. As the denominator
of the VIT BV , diminishing values of ζ produce progressively larger values of Vc , thus
accounting for AC lost from normal conduction through the VITBV to pathologic
conduction through the Ve . Thus, a new SV equation is proposed51,107 :

                               VITBV    dZ(t)/dtmax
                        SV =                        Tlve ,                         (54)
                                 ζ2         Z0
                                             Impedance Cardiography                                        81


(1) VITBV = K · 16 W 1.02
(2) ζ = index of transthoracic aberrant conduction, (0 < ζ ≤ 1.0)
    ζ = (Zc − Zc Z0 + K)/(2Zc + Z0 − 3Zc Z0 + K), where Zc is the critical level of
            2                   2     2

    Z0 , nominally ascribed a default value = 20 Ω, and K is a trivial constant → 0
(3) dZ/dtmax = the peak rate of change of the transthoracic cardiogenic impedance
    pulse variation, (Ω · s−2 )
(4) Z0 = the transthoracic quasi-static base impedance (Ω)
(5) Tlve = left ventricular ejection time (s)

8. Proof of Hypothesis: The New Equation versus the Kubicek,
   Sramek, and Sramek–Bernstein Equations using
   Thermodilution as the Standard Reference Technique
To show the superior behavior of Eq. (54) over the Kubicek, Sramek, and Sramek–
Bernstein equations, 106 adult cardiac surgery patients were studied within 24
hours of cardiopulmonary bypass.51 Thermodilution cardiac output (TDCO) was

                                 Kubicek                                          Sramek
                           Rs = 0.46 (P<0.0001)                             Rs = 0.51 (P<0.0001)
          4000 8000

                                                            4000 8000
         K (mL/min)

                                                           S (mL/min)


                       0   2000     6000          10000                 0   2000     6000          10000
                               TD (mL/min)                                      TD (mL/min)

                             Sramek-Bernstein                                  New Equation
                            Rs = 0.6 (P<0.0001)                             Rs = 0.78 (P<0.0001)
          4000 8000

                                                            4000 8000
         SB (mL/min)

                                                           N (mL/min)


                       0   2000     6000          10000                 0   2000     6000          10000
                               TD (mL/min)                                      TD (mL/min)

Fig. 9. Scatterplots of cardiac output (CO) obtained by the Kubicek (K), Sramek (S), Sramek–
Bernstein (S–B), and the new equation (N) versus thermodilution CO (TDCO) The solid line is
the line of identity. The dotted line is the smooth through the data obtained by robust locally
weighted regression. Rs is the non-parametric Spearman correlation coefficient. CO is given in
82                                                         D. P. Bernstein & H. J. M. Lemmens

performed on all patients and represented the mean value of at least 5 consecu-
tively obtained acceptable measurements. Simultaneous with the thermodilution
measurements, a commercially available ICG device was used for raw data acquisi-
tion, which comprised dZ/dtmax and Z0 . Left ventricular ejection time was obtained
from radial intra-arterial pressure pulse tracings. The volume conductors for the
four equations were calculated and the raw data substituted within each equa-
tion to obtain SV. Thereafter, CO was calculated as the product of SV and heart
rate. CO calculated using the new equation (6060 ± 1484 mL/min) was not dif-
ferent from TDCO (5966 ± 1411 mL/min). By contrast, CO calculated using the
equation of Kubicek (3700 ± 1531 mL/min), Sramek (4161 ± 1829 mL/min) and
Sramek–Bernstein (4368 ± 1818 mL/min) were significantly less than cardiac out-
put obtained by thermodilution and the new equation. Scatterplots of TDCO and
CO determined with each of the ICG equations are shown in Fig. 9. For the range of
CO measurements in this study, the relationship between TDCO and CO obtained
by the new equation was linear. For CO obtained by the Kubicek, Sramek, and
Sramek–Bernstein equations, the relation with TDCO was non-linear with most of

                                                     Kubicek                                                             Sramek

     (TD-K) / ((TD+K)/2) (%)

                                                                                 (TD-S) / ((TD+S)/2) (%)

                                         +2SD                                                                +2SD


                                                                                          0 50




                                            2000       6000         10000                                       2000       6000       10000
                                        Average CO by two methods (mL/min)                                  Average CO by two methods (mL/min)

                                                 Sramek-Bernstein                                                      New Equation

     (TD-SB) / ((TD+SB)/2) (%)

                                                                                 (TD-N) / ((TD+N)/2) (%)


                                                                                          0 50

                                          mean                                                               +2SD

                                          -2SD                                                                -2SD


                                            2000       6000         10000                                       2000       6000       10000
                                        Average CO by two methods (mL/min)                                  Average CO by two methods (mL/min)

Fig. 10. Bland and Altman analysis. Comparison of cardiac output (CO, mL/min) determinations
using the bland and Altman analysis shows poor agreement between thermodilution (TD) and the
Kubicek (K), Sramek (S), and Sramek–Bernstein equations, but not with the new equation (N).
The difference (bias) between CO methods on the y-axis is expressed as a percentage of the mean
CO (i.e. [(TD − ICG)]/[(TD + ICG)/2] × 100 = %).
                                  Impedance Cardiography                               83

                               Table 1.   Bland-Altman analysis.

       TDCO vs ICG*              Bias (%)    Precision (%)   95% Limits of Agreement

                                                             Upper (%)    Lower(%)
       K                             51            32              113      −12
       S                             41            34              108      −26
       SB                            36            33              100      −28
       N (All patients)              −1            16               30      −32
       N (Z0 ≥ 20 Ω, n = 78)         −2            13               24      −27
       N (Z0 < 20 Ω, n = 28)          0            22               42      −43
  ∗ Difference   as % of mean: (TDCO − ICG)/(TDCO + ICG)/2)(%)

the data points below the line of identity. In the correlation analysis the correlation
coefficient was highest for the new equation. Bland and Altman analysis showed
poor agreement between TDCO and CO obtained by the Kubicek, Sramek, and
Sramek–Bernstein equations but not between TDCO and CO obtained by the new
equation (Fig. 10). Bias, precision, and limits of agreement are shown in Table 1.
The difference (bias) between CO methods is presented as a percentage of the mean
CO (i.e. [(TD − ICG)]/[(TD + ICG)/2] × 100). When CO obtained by the new equa-
tion was stratified by Z0 , patients in whom Z0 ≥ 20 Ω, showed better agreement
with TDCO than in patients in which Z0 < 20 Ω.
    In assessing the accuracy of one CO method over another, it should be clearly
understood that there is no true “gold-standard” of blood flow in clinical medicine.
TDCO, for example, is known to possess many sources of error, routinely causing
a 15–20% deviation from actual flow in the critically-ill, ventilated patient. Based
upon the reproducibility of TDCO, it is generally accepted that limits of agreement
up to ±30% between blood flow methods are acceptable. The CO values that are
obtained by the new equation fall just within these parameters when compared
with TDCO. Other recent studies show that, when implemented by a state-of-the-
art bioimpedance cardiac output computer, Eq. (54) provides equivalent cardiac
output determinations versus both transesophageal Doppler echocardiography and
pulmonary artery thermodilution cardiac output.108,109

9. Conclusions
As a basic premise in impedance cardiography, it is suggested that the commonly
held belief of a plethysmographic origin for dZ/dtmax is incorrect. In defense of
this hypothesis, there is compelling evidence indicating that dZ/dtmax is caused
by the maximum acceleration of red blood cells, dv/dtmax . Bioelectrically, this is
a function of the peak rate of change of the velocity-induced pulsatile resistivity
variation of forward flowing blood when exposed to a field of AC. From a state of
random orientation of red cells in end-diastole, the resistivity change ∆ρb (t), and
its rate of change, dρb (t)/dt, are caused by a well-defined state of parallel orien-
tation of the erythrocytes along their long axes of symmetry during early systole.
84                         D. P. Bernstein & H. J. M. Lemmens

Based on the data and theory put forth, it is concluded that dZ/dtmax /Z0 is not the
ohmic analog of dV /dtmax , but rather that of a more obscure parameter, peak aor-
tic reduced average blood acceleration, [(dv/dtmean )/R]max (PARABA). Thus, the
differentiated peak, dZ/dtmax , cannot be implemented in its raw normalized form
to directly predict SV. The appropriate parameter for SV calculation in impedance
cardiography is the ohmic counterpart of peak aortic reduced average blood veloc-
ity, ∆Zv (t)max /Z0 , which is a function of ∆ρb (t)max . This parameter is derived by
means of the square root acceleration step-down transformation of dZ/dtmax /Z0 .
As a consequence of the magnitude of the resultant transformation, and with appro-
priate physiologic license, the volume conductor Vc in impedance cardiography is
designated dimensionally as the ITBV. The magnitude of the volume conductor
is modified in states of excess thoracic liquids by a dimensionless parameter, the
index of transthoracic aberrant conduction ζ. Based on the biophysical and elec-
trodynamic origins of dZ/dtmax , it is suggested that SV computed by means of
the transthoracic impedance technique be known as impedance cardiovelocimetry

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                                       CHAPTER 4


      ∗,† Department of Electrical Engineering, Eindhoven University of Technology

                  P.O. Box 513, 5600 MB Eindhoven, The Netherlands
                  ∗∗ Department  of Mechanical and Aeronautical Engineering
                    University of Rome “La Sapienza”, 00184, Rome, Italy
                      † Department  of Anesthesiology, Catharina Hospital
                                  Eindhoven, The Netherlands

    The injection of indicators in the blood stream is used for several diagnostic measure-
    ments of the cardiovascular system functionality. An indicator is a non-toxic substance
    that can be detected by a specific sensor. The detection often implies a contact between
    indicator and sensor. As a result, the measurement is invasive, i.e. a catheter must be
    inserted through the blood vessels in the measurement site. The invasiveness issue can
    be overcome by means of contrast imaging techniques. Contrast agents are substances
    that enhance the signal detected by specific imaging modalities. The measurement of
    the contrast (or indicator) concentration versus time results in an indicator dilution
    curve. Several hemodynamic models can be adopted for the curve interpretation and
    for the estimation of the cardiovascular parameters of interest. This chapter provides an
    overview on the indicator dilution theory and the clinical techniques that can be adopted
    for the measurement of dilution curves. Both invasive and minimally invasive techniques
    are discussed. Particular attention is dedicated to the use of medical imaging techniques
    such as ultrasound and magnetic resonance imaging. These methods promise to open
    new possibilities for minimally invasive cardiovascular diagnostics based on indicator

    Keywords: Indicator dilution technique; cardiovascular quantification.

1. Introduction
An indicator is a non-toxic substance that can be injected into the blood stream
and detected by specific sensors, which depend on the characteristics of the indi-
cator. The injection and consequent detection of an indicator in specific sites of
the cardiovascular system allows the assessment of several cardiovascular parame-
ters for diagnostics and follow up of cardiovascular dysfunctions. This chapter aims
at providing a broad overview on the major indicator dilution techniques that are
currently used (or emerging) in the field of cardiovascular quantification.
    Section 2 gives a brief introduction on the anatomy and physiology of the car-
diovascular system. Section 3 describes the theoretical background of the indicator
dilution theory. In Section 4, the application of the indicator dilution theory for
the assessment of cardiac output (CO), blood volumes, ventricular ejection fraction
(EF), and myocardial perfusion is discussed. Section 5 presents the major indicator

90                     M. Mischi, Z. Del Prete & H. H. M. Korsten

dilution techniques that are used for cardiovascular quantification. A distinction is
made between invasive techniques (Sec. 5.1), where the contrast detection needs
a contact between the sensor and the indicator, and minimally-invasive medical
imaging techniques (Sec. 5.2), where the detection is performed by external imag-
ing of the inner body. Although an indicator injection is still needed, these methods
are referred to as minimally invasive. Eventually, in Sec. 6 a comparative evalua-
tion of the presented techniques is provided together with a discussion on emerging

2. The Cardiovascular System
The heart is basically a pump, which makes the blood flow through the circula-
tory system. The blood carries oxygen and nutrients to and waste materials away
from all body tissues. The overall efficiency of the heart and the circulatory sys-
tem is usually characterized by a few important parameters. They are the Pulse
Rate (PR, number of cardiac cycles per minute), the Cardiac Output (CO, blood
flow through the heart), and the Ejection Fraction (EF, percentage of volumetric
variation of the ventricles). As the PR is accurately measured by pressure varia-
tions or electrical cardiac activity, an accurate assessment of CO and EF is more
    A local indicator of the cardiac condition is the level of perfusion. Like all the
human tissues, also the muscle that contracts the ventricles, i.e. the myocardium,
needs blood perfusion. This is provided by the coronary arteries. When a perfusion
defect is present, the viability of the myocardial tissue may be reduced. As a result,
the contraction becomes less efficient and asynchronous, and the ventricle may be
dilated to provide sufficient CO. Eventually, if necrosis of myocardial tissue occurs,
that results in a myocardial infarction.
    Another indicator of the cardiovascular system condition is the amount of blood
that circulates through the pulmonary and systemic circulation. Due to the compli-
ance of the cardiovascular system, volume variations are related to pressure varia-
tions, and the pulmonary blood volume (PBV) may be a good indicator for cardiac
preload, i.e. the blood pressure in the left atrium (LA).

3. Indicator Dilution Modeling
In this section the theory of the indicator dilution analysis is discussed. We assume
that an indicator bolus is injected and subsequently detected in the blood stream.
The measurement of the concentration-versus-time curve in a specific site of the
circulatory system is referred to as Indicator Dilution Curve (IDC). The IDC can
be represented and interpreted by several hemodynamic models in order to derive
the cardiovascular parameters of interest.
               Indicator Dilution Techniques in Cardiovascular Quantification      91

3.1. Overview of the indicator dilution models
Before discussing the IDC models, it is important to state that all the indicator
dilution methods are based on the following assumptions, which are also cause of
measurement errors1,2 :

•   The   blood flow is constant during the measurement (about one minute).
•   The   indicator shows an instantaneous and uniform mixing.
•   The   injection is sufficiently fast to be modelled by a dirac impulse.
•   The   loss of indicator is either absent or known.

The first IDC model was developed by Hamilton.3–5 He noticed that the IDC is
mainly composed by a sharp rise followed by a slower descent, which resembles an
exponential function. Thus, he modeled this descent by an exponential decay with
time-constant τ as
                                 C(t) = C(t0 )e−     τ    ,                      (1)

where C(t) is the IDC (i.e. the concentration-time curve) and t0 is the injection
time. This model, which represents the wash-out curve in a mono-compartment, also
allows the estimation of the EF (see Sec. 4.3). The IDC is interpolated along a short
segment in the down-slope of the curve before the rise given by the recirculation of
the contrast (see Fig. 1), resulting in estimates that are strongly dependent on the
noise level.6
    The interpretation of (1) can be related to an exponential distribution or the
impulse response of a mono-compartment model. We consider a chamber (compart-
ment) of volume V with one input and one output where a fluid (blood) is flowing.
If the chamber is not elastic and the fluid incompressible, the input and output flow
Φ are equal. For the mass conservation principle, the variation of indicator mass
in the chamber (i.e. V dC(t)) equals the mass of indicator that leaves the chamber
(i.e. C(t)dV ). Therefore, since C(t)dV = C(t)Φdt, the system can be described by
the differential equation

                                 V dC(t) = −C(t)Φdt.                             (2)

(2) can be written as given in (3) and solved in the Laplace domain as given in (4),
where C0 represents the initial condition at time t0 :
                                 V dC(t)
                                         + C(t) = 0,                             (3)
                                 Φ dt
                                    C(s)    1
                                         =    Φ
                                                .                                (4)
                                     C0    s+ V

   If an indicator-bolus is rapidly injected in the chamber at time t0 and the mix-
ing is perfect, then C0 = m/V (with m equal to the injected mass of indicator).
The result of the anti-transformation of (4) in the time domain is given as in (1),
92                         M. Mischi, Z. Del Prete & H. H. M. Korsten

Fig. 1. In (a) the IDC (i.e. C(t)) with recirculation is shown. In (b) the logarithmic plot of the
IDC as well as the short segment that is suitable for the exponential fitting are shown.

with τ = V /Φ and C (t0 ) = C0 = m/V . In fact, all the compartmental models
assume implicitly an instantaneous and complete mixing of the indicator in each
    A model that is often adopted in the IDC theory is the two-compartment model,
which is obtained by adding a second equation (representing the second compart-
ment) to (3). It is related to the cardiac functionality, which can be represented
by two double-compartment pumps. The resulting differential system is given in
(5), where C1 (t) and C2 (t) are the contrast concentrations in the two chambers of
volume V1 and V2 , respectively (see Fig. 2):

                            V1 dC1 (t) + C1 (t) = 0,
                             Φ dt                                                             (5)
                            V2 dC2 (t)
                                       + C2 (t) − C1 (t) = 0.
                             Φ dt
              Indicator Dilution Techniques in Cardiovascular Quantification             93

Fig. 2. Scheme of a two-compartment model. Vi and Ci (i = 1, 2) represent the compartment
volume and concentration, respectively. hi (t) is the impulse response of each compartment.

A solution of (5), assuming the injection time is equal to zero, gives C1 (t) and C2 (t)
as in (6), where τ1 = V1 /Φ and τ2 = V2 /Φ:
                                   m − τt
                          C1 (t) =    e 1,
                                   V1                                                  (6)
                                      m         t       t
                          C2 (t) =         (e− τ1 − e− τ2 ).
                                   V1 − V2
    To make the model more general, only a fraction of the first compartment out-
flow can be considered as the in-flow of the second compartment (the rest of the flow
is lost). As a consequence, different flows (e.g. Φ1 and Φ2 , Φ1 ≥ Φ2 ) can be used to
model the output vessels of the compartments.7 However, the only difference with
respect to (6) concerns the coefficient in front of the exponentials that represent C2 .
The new coefficient equals m/(V1 − V2 Φ2 /Φ1 ).
    Also the IDC in a cascade of n equal chambers is an interesting model.8–10 The
nth equation of the system is given as
                         V dCn (t)
                                   + Cn (t) − Cn−1 (t) = 0.                            (7)
                         Φ dt
For each chamber (n ≥ 2) the initial condition is C(0) = 0. Thus, in the Laplace
domain the iterative Eq. (7) is given as
                                     τ                             V
                        Cn (s) =      1
                                   s+ τ
                                          Cn−1 (s) , with τ =      Φ.                  (8)

    If the input to the first chamber (n = 1) is an impulse (Dirac distribution) at
time t0 = 0 of area m/Φ (i.e., (m/Φ)u0 (t)), then, for n ≥ 1, Cn (s) can be expressed
as given in (9) (Laplace domain) and (10) (time domain):
                                                    1      n
                                         m          τ
                                Cn (s) =             1         ,                       (9)
                                         Φ        s+ τ
                                                 n−1 − τ
                                              mt     e
                                Cn (t) =                   .                          (10)
                                             Φτ n (n − 1)!
Cn (t) in (10) also represents a χ2 -distribution with 2n degrees of freedom. There-
fore, the n-compartment model (with equal volume of the chambers) is basically
a distribution with three independent parameters, like the standard distributed
94                     M. Mischi, Z. Del Prete & H. H. M. Korsten

    If the volumes of the chambers are different, a different model, which is basi-
cally a multi-exponential model, is obtained. It is the summation of a number of
exponential functions equal to the number of compartments. The number of degrees
of freedom increases as well as the complexity of the fitting algorithm. In general,
developing such models is an effort that does not lead to better results in terms of
representation and interpolation of a real dilution system.9
    Rather than focusing only on the impulse response, also the system response to
different inputs can be considered and the input function integrated in the model.
A common approach replaces the standard Dirac input with a rectangular input.8
A valid alternative for modelling the input function is represented by a Gaussian
function. In Ref. 11, for instance, a Gaussian input is convoluted with a mono-
compartment exponential model in order to fit the measured IDC.
    The impulse response of the compartment could be chosen also according to
different criteria. Often the IDC is interpreted by means of distributed models,
which are usually chosen among statistical distributions. Distributions that are com-
monly adopted to fit the IDC are the lognormal distribution and the gamma-variate
distribution.8,12–14 Even without an hemodynamic interpretation, these models
provide accurate IDC interpolations. The formulation of the lognormal and the
gamma-variate model, as typically used for IDC fitting, is given as in (11) and (12),
                                0                            t≤0
                       C(t) =    A      (ln(t)−µ)2                  ,           (11)
                              √      e− 2σ2                  t>0
                               0        t≤0
                       C(t) =    α −tβ          .                               (12)
                               At e     t>0

   The parameter A is the scale factor of the model. The integrals of (11) and
(12), which can be used for flow measurements, equal A and Aβ −(1+α) Γ(1 + α),
respectively. Γ represents the Gamma integral
                           Γ(n) =            x(n−1) e−x dx,                     (13)

whose fundamental property is that Γ(n + 1) = nΓ(n), with n ∈ IN. The integral
of (12) can be directly derived by means of this property after the substitution
α = n − 1 and x = tβ. Also the first statistical moments of the distributions (11)
and (12) are important, as they are used for the estimation of the indicator mean
transit time (MTT) for volume measurements (see Sec. 4.2). The first statistical
moments (defined as the integral over time between 0 and ∞ of tC(t) normalized
by the integral of C(t)) of the lognormal and the gamma variate distributions equal
eµ+ 2 and (1 + α)β −1 , respectively. In the lognormal distribution, σ is related to
the skewness of the curve while µ is related to the peak-concentration time.
               Indicator Dilution Techniques in Cardiovascular Quantification                95

Remark. Notice that any such statistical model has to obey the mass conservation
law as given in (14), where m is the injected mass (dose) :
                                              C(t)dt =     .                              (14)
                                      0                  Φ
    Two distributed models that provide an interesting interpretation of the indica-
tor dispersion process are the Local Density Random Walk (LDRW) and the First
Passage Time (FPT) models.6,7,9,10,15 –17 These models, which are based on the
assumption of random walk of the indicator particles, are described more in details
in the next sections.
    The IDC fitting is usually executed by means of nonlinear regression techniques,
such as Levemberg-Marquardt or Gauss-Newton algorithms.11,18 The application of
a linear regression technique in a nonlinear domain is also possible. For instance,
linear regression in the logarithmic domain has been successfully proposed for the
LDRW model.19 Also some geometric techniques have been developed. They give
an estimate of the area below the curve in order to measure the CO. The inflection
triangle technique, for instance, is a technique that has been applied to all the
distributions.4,13,16 The area estimate is based on the triangle that is tangent to
the two inflections (second derivative equal to zero) of the IDC.

3.2. The local density random walk model
The Local Density Random Walk (LDRW) model is a mono-dimensional character-
ization of the dilution process (see Fig. 3). It describes the injection of an indicator
into a straight infinitely-long tube where a fluid (carrier) flows with constant velocity
u. This model was introduced by Sheppard and Savage in 1951.10,15 The assump-
tions of the model are a fast injection (modeled as a Dirac impulse) and a Brownian
motion of the indicator, whose particles interact by pure elastic collisions. Without
any loss of generality, we consider the injection time t0 and the injection position
x(t0 ) equal to zero. If we focus on the discrete motion of a single particle, its position
X(nT ) at time nT can be described by the stochastic process given in (15), where
S is a random variable that represents the distance covered by the particle in the
time interval T (single step):
                                   X(nT ) =            S(iT ).                            (15)

   No assumptions are made on the probability density function of the random
variable S. As a consequence of the Brownian motion hypothesis, each step S(iT ) is
independent from the previous ones and X(nT ) is a Markov processa .20 Therefore,

a A Markov process is a stochastic process where the past has no influence on the future, if its

present is specified. In mathematical terms,
                    P {x(tn ) = xn |x(t), t ≤ tn−1 } = P {x(tn ) = xn |x(tn−1 )},
where P{·} represents the probability function.
96                       M. Mischi, Z. Del Prete & H. H. M. Korsten

for increasing n (or decreasing T ) the Central Limit Theorem is applicable to the
process X (nT ). If µ and σ are respectively the mean and the standard deviation
of S, then the probability density function of the random variable X at time nT is
described by the process
                                             1           (x−nµ)2
                          W (x, nT ) = √            e−     2nσ2    .                     (16)
                                           2πnσ 2
   In terms of continuous time t = nT (with T infinitely small), (16) can be
expressed by the Wiener process
                                          1     (x−tu)2
                             W (x, t) = √     e− 2tα ,                                   (17)
where α = σ 2 /T and u = µ/T .21
   The concentration of the indicator C(x, t) is determined by (m/A) W (x, t), where
m is the mass of injected indicator and A is the section of the tube. Thus, as shown in
Fig. 3, C(x, t) is described by a normal distribution that moves along the tube with
the same velocity as the carrier (mean equal to tu) and spreads with a variance that

Fig. 3. LDRW experimental model. The lower curves represent the probability distribution func-
tion of the indicator for increasing time.
             Indicator Dilution Techniques in Cardiovascular Quantification        97

is a linear function of time (variance equal to tα). If we consider α = 2D (with D
representing the diffusion coefficient), C(x, t) is a solution of the mono-dimensional
diffusion with drift equation, which is given as in (18) with boundary conditions
given as in (19) and (20)9 :

                      ∂C(x, t)    ∂ 2 C(x, t)    ∂C(x, t)
                               =D       2
                                              −u          ,                     (18)
                        ∂t            ∂x           ∂x
                                  C(x, 0) =       u0 (x),                       (19)
                                          C(x, t)dx =     .                     (20)
                                  0                     A
    The conditions stated in (19) and (20) express the fast injection hypothesis
and the mass conservation law, respectively. (18) represents the link between the
statistical and the physical interpretation of the dilution process.
    The derivation of a model for the IDC interpretation requires to focus on a
fixed section of the tube (detection section) where the concentration of the indicator
is evaluated versus time (see Fig. 3). The distance between the injection point and
the detection section is determined by x = x0 = uµ. Therefore, µ is the MTT of the
indicator from the injection to the detection site. Wise and Bogaard formalized the
concentration time curve evaluated at distance x0 as

                                  m λ        λµ − λ ( µ + µ )
                         C(t) =      e           e 2      t ,                   (21)
                                  µΦ         2πt
where Φ = uA is the flow of the carrier and λ = µu2 /2D = µΦ2 /2DA2 is a
parameter related to the skewness of the curve.6,16,17 √
    The maximum of C(t) is reached for t = µ · (2λ)−1 ( 1 + 4λ2 − 1) < µ. Notice
that max[C(t)] is given when t = µ only for λ → ∞. This can be explained by the
physics of the dilution process. If we consider L = x0 as the characteristic length
of the LDRW model, it follows that 2λ equals the Peclet number, which is defined
as uL/D and is the hydrodynamic parameter used to quantify the ratio between
convection and diffusion in a dilution process.17,22 The limit λ → ∞ can be inter-
preted as an infinitely small contribution of diffusion in comparison to convection.
As a consequence, all the particles reach the detection section at the same time
µ = x0 /u. Already for λ > 10, as shown in Fig. 4, the curve is almost symmetric,
while for λ < 2 the curve is very skew.
    The parameter λ (and therefore also the Peclet number divided by two) can be
also viewed as the ratio between the diffusive time τD (defined as τD = x2 /(2D))
and the convective time x0 /u = µ.17 The diffusive time τD can be interpreted as
the time taken by the standard deviation 2Dt, which describes the pure diffusion
process as given in (17) for u = 0, to reach the distance x0 .
    In conclusion, it is evident that the LDRW model is related to the physical
interpretation of the dilution process as described by classic fluid mechanics. Some
98                       M. Mischi, Z. Del Prete & H. H. M. Korsten

       Fig. 4.   LDRW model for λ equal to 2, 5, and 10. µ is fixed and equal to 100 s.

important properties of C(t) are listed below and are used for the assessment of
flow and volume in the Sec. 4:
                                            C(t) =     ,                                 (22)
                                    0                Φ
                                     tC(t)dt               1
                                   ∞            = µ 1+         .                         (23)
                                     C(t)dt                λ
The flow Φ can be directly calculated by (22) once the injected dose m is known. (23)
is the first moment of the LDRW model and it is referred to as Mean Residence Time
(MRT) of the indicator between the injection and detection sites.17 A discussion on
the MRT interpretation and the difference between MRT and MTT is postponed
to the Sec. 4.2. An analytical derivation of (23) is proposed Ref. 23.

3.3. The First Passage Time model
Like the LDRW model (see Fig. 3), also the FPT model represents the IDC as the
result of the passage of an injected indicator bolus (fast injection described by a
Dirac function), which flows in a fluid-dynamic system (an infinitely-long tube),
through a detection section.
   The only difference with respect to the LDRW model concerns the indicator
passage through the detection site. The FPT model hypothesis allows only a single
passage of the indicator particles while the LDRW model hypothesis contemplates
               Indicator Dilution Techniques in Cardiovascular Quantification                99

multiple passages. It is like if an absorber layer captured the particles right after the
detection site. Therefore, the FPT model is derived from the LDRW model when
only the first passage through the detection site is considered and its formulation
is given as6,7,17 :

                                     m λ       λµ − λ ( µ + µ )
                            C(t) =     e          3
                                                    e 2     t ,                           (24)
                                     Φ        2πt

where the meaning of the symbols is the same as for (21). Figure 5 shows three
different FPT curves for different values of λ.
   The derivation of the FPT model is rather complicated. However, it can be
simplified by exploiting the LDRW model definition. The probability p(x, t) that an
indicator particle moves from a distance d = 0 to d = x in time t is described by
the unbiased (diffusion without drift) LDRW model as

                                            1       x2
                                p(x, t) = √     e− 4Dt ,                                  (25)

where D is the diffusion constant.
   If the particle reaches the distance x for the first time at time t − τ (τ ∈ [0, t]),
the probability of finding the particle at distance x at time t can be divided into two
contributions: l(x, t − τ ) and p(0, τ ). l(x, t − τ ) is the probability of a first passage

         Fig. 5.   FPT model for λ equal to 2, 5, and 10. µ is fixed and equal to 100 s.
100                       M. Mischi, Z. Del Prete & H. H. M. Korsten

in x at time t − τ and p(0, τ ) is the probability of a subsequent passage in x at time
τ after the first passage. Thus, p(x, t) is given by the convolution operation
                          p(x, t) =           l(x, t − τ )p(0, τ )dτ.             (26)
      In the Laplace domain (26) can be expressed as
                             P (x, s) = L(x, s) · P (0, s).                       (27)
      As a consequence, L(x, s) is given as
                                                   P (x, s)
                                 L(x, s) =                  ,                     (28)
                                                   P (0, s)
and p(x, t) is expressed in the Laplace domain as
                                           1                           x2 s
                              P (x, s) = √     e−                       D     .   (29)
      Therefore, L(x, s) is given as
                                                            x2 s
                                 L(x, s) = e−                D     .              (30)
    The FPT process is the anti-transformation in the time domain of L(x, s), which
is given as
                                          x         x2
                             l(x, t) = √        e− 4Dt .                         (31)
                                         4πDt 3

    l(x, t) represents the probability density function that the particle reaches the
distance x for the first time at time t.
    The same procedure is applicable in case of drifting diffusion. With reference
to (17), p(x, t) is now defined as given in (32), where u is the carrier fluid velocity:
                                                1     (x−ut)2
                            p(x, t) = √             e− 4Dt .                      (32)
    If the detection distance is fixed to x = x0 = uµ, and D/u2 = K 2 , (32) can be
rewritten as
                                        1       (µ−t)2
                          p(µ, t) = √         e− 4K 2 t .                        (33)
                                    u 4πK 2 t
    The model is valid for u > 0. For u = 0 the diffusion without drift model must
be used. As for the diffusion without drift case, (33) is substituted into (26), which
is now expressed as
                          p(µ, t) =           l(µ, t − τ )p(0, τ )dτ.             (34)
   Once again, (34) can be solved in the Laplace domain. The resulting L(µ, s)
represents the FPT model in the Laplace domain when drift is included. L(µ, s) is
expressed as given in (35):
                                                         1−4sK 2 )
                                               − µ(1−
                             L(µ, s) = e                2K 2              .       (35)
             Indicator Dilution Techniques in Cardiovascular Quantification           101

   Taking the inverse Laplace transform, (35) is expressed in the time domain as
                                          µ         (µ−t)2
                            l(µ, t) = √           e− 4K 2 t .                     (36)
                                        4πK  2 t3

    l(µ, t) represents the FPT model with drift in the time domain. Since L(µ, 0) = 1,
the integral of l(µ, t) in the interval (0, ∞) equals 1 and l(µ, t) may be considered
as a statistical distribution. With the substitution λ = (u2 µ)/(2D), it follows that
K 2 = µ/2λ and (36) is expressed as given in (24), which is the FPT model definition
as reported by Bogaard, Reth, and Wise.6,7,17 The first moment of the model, as
proven by Sheppard, is equal to µ.10

4. Cardiovascular Quantification by Indicator Dilution Principles
The application of the indicator dilution principles for the measurement of several
cardiovascular parameters is explored. The link with specific medical technologies
is limited because it is the object of the next section.

4.1. Cardiac output
Several different techniques are clinically used for Cardiac Output (CO) measure-
ments. They are based on different principles with different reproducibility and
accuracy.24 In general, the CO is defined as the volume of blood that is ejected by
the left ventricle (LV) into the aorta in one minute, and it is expressed in liters per
    Due to the cyclic contraction-expansion of the ventricles, the CO, as well as the
blood pressure, is a periodic function of time. The first harmonic of these functions
is the Pulse Rate (PR), which represents the number of ventricular systoles (or
diastoles) per minute. Despite the fact that the CO is a periodic function, it is
usually represented by a single value, which is a measure of the average flow in
liters per minute. The CO is related to the ventricular volume variations during one
cardiac cycle. If Ved is the end diastolic volume (maximum volume) and Ves is the
end systolic volume (minimum volume), then the Stroke Volume (SV), which is the
volume of blood ejected from the ventricle to the artery in one cardiac cycle, equals
the difference Ved − Ves . Therefore, the CO can be defined as given in (37):
                                  CO = SV · PR.                                     (37)
The CO definition in (37) is correct only if no regurgitation (valve insufficiency) is
present, otherwise SV = Ved − Ves .
    The CO for an average size adult (70 kg) at rest is about 5 L/min. During severe
exercise it can increase to over 30 L/min. Miguel Indurain (who won the Tour de
France in five successive years) had a resting PR of 28 beats per minute and could
increase his CO to 50 L/min and his PR to 220 beats per minute.
    The CO is often divided by the Body Surface Area (BSA) to normalize the value
with respect to the size of the subject. In this case, it is referred to as Cardiac Index
102                        M. Mischi, Z. Del Prete & H. H. M. Korsten

(CI). Several formulas can be adopted to determine the BSA based on the weight and
the height of the patient. One of the most common is the Dubois and Dubois formula
(1917), which estimates the BSA as
                     BSA(cm2 ) = 71.84 · weight0.425 · height0.725 ,                        (38)
where weight and height are given in kg and cm, respectively.
    The indicator dilution theory is based on the following concept: If the concen-
tration of an indicator that is uniformly dispersed in an unknown volume V is
determined, and the volume of the indicator (dose) is known, then the unknown
volume can be determined too. Since Φ(t) = dV (t)/dt (Φ(t) and V (t) are respec-
tively the instantaneous flow and the volume of the carrier) and C(t) = dm/dV (m
and C(t) are respectively the mass of the tracer and its concentration at time t),
the following differential equation can be derived:
                                           dV (t)    1 dm
                                  Φ(t) =          =         .                               (39)
                                            dt      C(t) dt
    A typical approach for CO measurements makes use of a rapid injection of
an indicator dose. This permits to relax the constraints about the nature of the
indicator, since only a small bolus is injected. However, also methods based on
continuous indicator infusion are used (see Secs. 5.1.1 and 5.1.2). In rapid injection
methods, C(t) in (39) is not constant. In practice, as shown in Fig. 6, the indicator
is rapidly injected into a fluid dynamic system where a carrier fluid (in our specific
case blood) is flowing, and the indicator concentration C(t) is measured versus time
for the registration of an IDC. The IDC contains all the information to estimate
the flow, whose value is derived from (39) by an integration over time as shown
in (40). The flow Φ is assumed to be constant, so that it can be moved out of the
integration. The resulting formula is the Stewart-Hamilton equation,
            ∞                     ∞                  ∞
                                                         dm                      m
                ΦC(t)dt = Φ           C(t)dt =               dt = m =⇒ Φ =   ∞          ,
        0                     0                  0        dt                 0
which provides the measurement of the mean flow Φ.25,26 Therefore, the injection
and subsequent detection of an indicator allows the measurement of the mean flow.
    The calculation of the integral in (40) is not trivial. Since the circulatory system
is a closed system, the recirculation of the indicator produces rises of the concen-
tration that mask the tail of the IDC related to the fist passage of the contrast
(see Fig. 7). In addition, the IDC is often very noisy. Therefore, the estimation of
the integral of the first passage IDC in (40) requires the employment of one of the
models discussed in Sec 3.

4.2. Mean transit time and blood volumes
Blood volume measurements provide valuable information on the circulatory sys-
tem functionality. In particular, the Pulmonary Blood Volume (PBV, blood volume
              Indicator Dilution Techniques in Cardiovascular Quantification               103

Fig. 6. Measurement of the indicator concentration versus time (IDC) in an infinite tube model.
The indicator is injected at distance 0 and detected at distance x0 from the injection site.

between the pulmonary artery and the LA), the Central Blood Volume (CBV, blood
volume between the pulmonary artery and the LV) and the Intra-Thoracic Blood
Volume (ITBV, blood volume between the right atrium and the LV) are important
parameters in anaesthesiology, intensive care, and cardiology to evaluate the cardiac
preload and the symmetry of the cardiac efficiency. For instance, LV EF and SV
are closely related to PBV and CBV.27–29
    More in general, asymmetries in the cardiac efficiency due to LV malfunction
or heart failure lead to increased intrapulmonary pressures (cardiac preload) and,
therefore, volume.27,29–33 The persistency of such condition results in the increase of
pulmonary interstitial fluid and, eventually, in the formation of pulmonary edema,
e.g. extravasations of fluid into the alveoli.34 The interstitial fluid is often referred
to as Extra-Vascular Lung Water (EVLW). Since cardiogenic pulmonary edema
is usually combined with increased cardiac preload, invasive pulmonary pressure
measurements can serve as useful diagnostic means. Nevertheless, non-cardiogenic
pulmonary edema can also occur, due for instance to infections, kidney failure,
104                        M. Mischi, Z. Del Prete & H. H. M. Korsten

Fig. 7. The dots represent a measured IDC while the continuous line shows the real (theoretical)
first passage IDC. The second rise due to the recirculation and the noise due to the measurement
system are evident.

Fig. 8. Scheme of the blood volume measurements: Pulmonary Blood Volume (PBV), Central
Blood Volume (CBV), and Intra-Thoracic Blood Volume (ITBV). The volumes of the compart-
ments that are included in the measurement are filled with gray color.

and injuries. The fluid accumulated in the alveoli may become a barrier to normal
oxygen exchange, resulting in dangerous hypoxia.
    A scheme of the standard blood volume measurements is shown in Fig. 8. In
order to make the volumes independent on the size of the subject, they are usually
normalized (indexed) as already explained for the CI (Sec. 4.1). PBV measurements
are based on trans-pulmonary indicator dilution techniques, which make use of the
             Indicator Dilution Techniques in Cardiovascular Quantification         105

injection of an indicator bolus before the lungs and its subsequent detection after the
lungs.27,28,35 The measured IDC is analyzed for the estimation of the MTT that the
indicator takes to cover the distance between the injection site (pulmonary artery
or right ventricle) and the detection site (ascending aorta or LV). The estimated
MTT is then multiplied times the CO as given in (41) for the measurement of the
blood volume (V ) between injection and detection site:
                                 V = MTT · CO.                                    (41)
Nowadays, trans-pulmonary indicator dilution techniques are very invasive due to
the need for a double catheterization. In fact, a catheter for thermodilution (or dye
dilution) must be inserted through the femoral artery up to the aorta, where the
IDC is measured. Moreover, since the indicator must be injected into a central vein,
the insertion of a second catheter is necessary to reach the injection site. Common
applications make use of a Pulmonary Artery Catheter (PAC) in order to inject
the contrast in the pulmonary artery and measure the CBV.36 The measurement
of the MTT between the first and the second passage of the indicator in the same
site allows the assessment of the Total Circulating Blood Volume (TCBV).
    In this context, it is important to make the distinction between intravascular
and extravascular indicators. Quantification of EVLW is possible if an intravas-
cular indicator (e.g. a dye bolus) and an extravascular indicator (e.g. a cold saline
bolus) are injected. This technique is referred to as double indicator transpulmonary
dilution.37,38 Two IDCs, one for each indicator, are then registered and analyzed
to assess the MTT of the indicator between the injection and detection site. The
intravascular PBV can be estimated from the MTT of the intravascular indicator,
while the total (intravascular and extravascular) fluid can be estimated by the MTT
of the extravascular indicator, under the assumption of rapid extravascular diffu-
sion. The difference between the two estimated MTTs permits the quantification of
the EVLW.
    Based on a number of assumptions, the estimation of the EVLW can also be
derived by the injection of a single extravascular indicator.38,39 The time constant
of the exponential decay of the IDC is adopted as the estimate for the MTT of the
largest compartment along the transpulmonary circulation, i.e. the total PBV.40
The difference between the ITBV and the total PBV provides the global end-
diastolic volume (GEDV). In Ref. 38 the GEDV is reported to be linearly correlated
to the intravascular PBV (measured by an intravascular dye). In particular,
                              PBV = 1.25 · GEDV.                                  (42)
Therefore, the EVLW can be derived as the difference between the total PBV,
derived from the time constant of the IDC exponential decay, and the intravascular
PBV, measured by (42).
    Here below a more accurate derivation of the volume estimates based on the
dilution of an indicator bolus is provided. The obtained general results are discussed
with respect to the FPT and the LDRW models.
106                          M. Mischi, Z. Del Prete & H. H. M. Korsten

    The infinite tube model in Fig. 6 is adopted to derive a formula for the volume
measurement. A carrier fluid flows through the tube with a steady flow Φ. An
indicator bolus is injected (fast injection) at time t = 0 into the tube. The volume
to measure is defined as the tube segment between the indicator injection and
detection sections.
    We define f (t)dt as the fraction of injected indicator that leaves the tube segment
in the time interval [t, t + dt]. It is assumed that the indicator may not pass more
than once through the detection section (FPT model hypothesis).
    Due to the single passage hypothesis, the fraction of leaving particles corresponds
to the fraction of particles that appear at the detection section. Therefore, f (t)
equals the normalized indicator concentration that is measured at the detection
site and it is given as
                                     f (t) =          ∞          .                                         (43)
                                                         C(τ )dτ
    The fraction of indicator that has left the segment by time t is determined by
F (t) as
                                     F (t) =                 f (τ )dτ.                                     (44)
The volume of fluid that enters the tube segment in the time interval [0, dt] is Φ · dt
and the fraction that leaves the segment by time t is Φ · dt · F (t). Therefore, the
volume of fluid that enters and leaves the tube segment in the time interval [0, t] is
given as
                                         Φ           F (τ )dτ.                                             (45)
    The difference between the entering and the leaving fluid volume in the time
interval [0, t] is then given as
                                     Φt − Φ                  F (τ )dτ.                                     (46)
    Therefore, (46) expresses the volume of fluid that enters the segment in the time
interval [0, t] and is still in the segment at time t. For time t → ∞ all the fluid in the
segment is replaced by fluid that has entered for t ≥ 0. Therefore, the total volume
V of the segment is given as
                              V = lim Φ t −                           F (τ )dτ   .                         (47)
                                    t→∞                       0

      (47) can also be formulated asb
                                                         t                               ∞
            V = lim Φ t − [τ F (τ )]t +
                                    0                        τ f (τ )dτ = Φ                  τ f (τ )dτ.   (48)
                  t→∞                                0                               0

                                  R                         R                            R
b The integration per parts of t τ f (τ )dτ allows replacing t F (τ )dτ in (47). In fact, t τ f (τ )dτ =
Rt                          Rt     0                          0                           0
 0 τ dF (τ ) = [τ F (τ )]0 − 0 F (τ )dτ .
             Indicator Dilution Techniques in Cardiovascular Quantification          107

    Due to the definition of f (t) and the FPT hypothesis, the right term of (48)
represents the multiplication of the flow Φ times the MTT of the indicator, i.e.
the average time that the indicator takes to cover the distance between injection
and detection section. Moreover, due to the FPT hypothesis and (43), f (t) may be
represented by (24) (except for the coefficient m/Φ), which proves the convergence
of the integral in (48).
    In conclusion, the volume is given as in (41), where the MTT of the indicator,
which equals the first moment of the IDC, is given as
                                    ∞                   ∞
                                                       0 τ C(τ )dτ
                     MTT =              τ f (τ )dτ =    ∞            .             (49)
                                0                       0
                                                          C(τ )dτ
    The definition of the indicator MTT as the first moment of the IDC is appro-
priate only under single passage hypothesis. In this case, the MTT corresponds
to the MRT of the indicator in the defined segment and equals µ in (24).17 In
fact, the indicator appearance time at the detection section also corresponds to the
disappearance time from the segment.
    The LDRW model is more general and does not assume a single passage of the
indicator. As a consequence, the first moment of the model, which still represents
the MRT of the indicator in the tube segment, differs from the MTT.9,16 The MTT,
which is defined as the average time that the indicator takes to go from the injection
to the detection site, is by definition equal to µ in (21). In fact, in the LDRW model
µ equals the time that elapses to cover the distance between the injection and
detection site at the carrier fluid velocity, i.e. the MTT of the indicator. Instead, the
first moment of the model, which corresponds to the MRT, equals µ(1+1/λ) (see also
(23)).23 Therefore, the MRT exceeds the MTT by the term µ/λ=2D/u2, i.e. twice
the ratio between indicator diffusion constant D and squared velocity of the carrier
fluid u2 . Large diffusion constants lead to increased numbers of indicator particle
passages through the detection site and, therefore, to large differences between MRT
and MTT.
    In conclusion, when either the LDRW model or the FPT model is adopted to
interpolate the IDC, (41) corresponds to

                                        V = Φ · µ.                                 (50)

4.3. Ejection fraction
The measurement of the Ejection Fraction (EF) is a common clinical practice to
evaluate the cardiac condition. In particular, it is a measure of the efficiency of
the myocardial contraction. If the end-diastolic (Ved ) and the end-systolic (Ves )
ventricular volumes are measured, the percent EF is defined as given in (51):
                                          Ved − Ves
                             EF% =                  · 100.                         (51)
108                      M. Mischi, Z. Del Prete & H. H. M. Korsten

    The EF can be estimated by any clinical imaging technique that allows a geo-
metric estimation of Ved and Ves , such as magnetic resonance (MR), ultrasound,
X-ray, and nuclear imaging techniques.24,41–44 Three-dimensional techniques, such
as MR or three-dimensional ultrasound imaging, are particularly attractive for these
applications. The acquired images are analyzed by manual or automatic segmenta-
tion. Due to the complex geometry of the right ventricle (RV), the estimation of
the RV EF is not commonly performed in clinical practice.
    In general, geometric EF measurements are time consuming due to the need
for segmentation, which in clinical practice is usually performed manually. In fact,
the reliability of automatic border detection algorithms is sometimes very limited
and cardiologists prefer a manual delineation of the endocardial contours. As a con-
sequence, the EF assessment not only slows down the clinical practice, but also
requires the employment of specialized personnel, such as radiologists or cardiol-
ogists. Also magnetic resonance imaging (MRI), despite the better image quality,
requires a long procedure both for patient scanning and for data analysis. Moreover,
patients that are claustrophobic or have an implanted pace-maker cannot undergo
an MRI scan.
    The EF assessment can be also performed by means of indicator dilution tech-
niques. A cold saline (thermodilution) or a dye (dye dilution) bolus is injected for
the measurement. The injected indicator bolus is detected either in the LV or in the
ascending aorta. A mathematical interpretation of the measured IDC allows the EF
    A ventricle can be modeled as a mono-compartment system, whose volume
changes as a quasi-periodic function of time. This can be represented by a cylinder-
piston system as shown in Fig. 9. The system is filled with an incompressible fluid.
Two valves are used for the fluid input and output and are driven by pressure
    If an indicator bolus is rapidly injected within a diastolic phase and the mix-
ing is complete, then the contrast concentration at the nth end-diastolic phase
is given by Cn and it equals the indicator mass m in the ventricle divided by
Ved . During the following systole, part of the contrast mass (∆m) is ejected out

Fig. 9. Mono-compartment cylinder-piston LV model. An input and an output valve are included
and represent the mitral and aortic valve, respectively.
             Indicator Dilution Techniques in Cardiovascular Quantification         109

of the cylinder. The concentration Cn+1 at the subsequent end-diastole is given
as in (52):
                            m − ∆m            Ved − Ves
                  Cn+1 =           = Cn · 1 −                        .            (52)
                              Ved                Ved
  Combining (51) and (52), the percent EF can be expressed in terms of Cn and
Cn+1 as given in (53):
                           EF% =      1−                  · 100.                  (53)
    The sampling timing of Cn and Cn+1 is usually controlled by an electrocardio-
graphic (ECG) trigger, so that the sampling rate is based on the cardiac electrical
activity. In order to ensure that the indicator recirculation (see Fig. 7) does not
influence the measurement, the samples Cn and Cn+1 should be measured before
the recirculation time.
    The EF estimate in (53) only considers the fluid that is ejected through the
output valve (aortic valve in the LV). If the input valve is insufficient, i.e. it leaks,
part of the contrast is ejected back through the input valve (mitral valve in the
LV). However, this fraction of contrast comes again into the ventricle during the
subsequent diastole and, therefore, does not contribute to ∆m. As a consequence,
the EF definition in (53) is better referred to as Forward EF (FEF). A similar
reasoning may be also applied to aortic valve insufficiency.
    If the SNR of the measured IDC is low, the definition of the right samples to
estimate Cn and Cn+1 is difficult, resulting in very inaccurate measurements. A
better approach makes use of an IDC model interpolation. The LV is well approx-
imated by a simple mono-compartment model, whose impulse response equals an
exponential function (see Sec. 3). If C0 is the concentration after a sudden indicator
injection in the compartment at time t = 0, then the IDC C(t) is represented as
                                 C(t) = C0 · e   τ    ,                           (54)

where τ is the time constant.3
    The exponential function in (54) can be used to fit the IDC as measured from
the model that is shown in Fig. 9. The ripple due to the pulsatile flow does not
disturb the measurement because it is averaged over a large number of cycles. Once
the exponential model is fitted to the IDC down-slope, the FEF is measured by (53)
as given in (55), where ∆t is the cardiac period:
                           Cn+1     e τ                            −∆t
                 FEF = 1 −      =1−   −t                    =1−e    τ    .        (55)
                            Cn       eτ
The FEF measurement by (55) is valid only when the contrast bolus is rapidly
injected into the ventricle within a diastolic phase, which is the reason why a cor-
rect FEF estimation requires a ventricular injection (Holt method).45 In fact, the
measurement must be performed in the LV (or ascending aorta) during contrast
110                     M. Mischi, Z. Del Prete & H. H. M. Korsten

wash-out with no incoming contrast.3 Therefore, catheterization is needed and the
clinical application of the method is very limited due to its invasiveness.
    An invasiveness reduction is accomplished by use of radio-opaque contrast or
radionuclides for X-ray or nuclear angiography (see Secs. 5.2.2 and 5.2.3), which
allow a non-invasive detection of the indicator. In fact, the FEF can be assessed
by videodensitometry of cine-loops.46 –48 However, despite a non-invasive contrast
detection, contrast injection still needs cardiac catheterization (invasiveness issue)
and, as discussed in Secs 5.2.2 and 5.2.3, the use of X-rays or radionuclides is not
recommended in several cases.
    In healthy people the EF is usually bounded between 50% and 85%. However,
in heart failure patients the EF can even drop down to 10%. In order to keep a
sufficient value of CO, the decrease of EF is compensated by an increase of heart
rate and both Ved and Ves (dilated ventricle), so that the SV can still guarantee a
sufficient CO.

4.4. Myocardial perfusion
The perfusion of the myocardium is an important indicator for the presence of coro-
nary artery diseases (CAD) and ischemia. Ischemia is an inadequate blood supply
to the myocardium, usually due to stenoses or blocked arteries (clots), which often
results in hypokinesis (reduction of wall motion and thickening) and asynchrony of
the cardiac wall motion. The tests to evaluate the myocardial perfusion are usually
qualitative, and an absolute quantification of the perfusion defects is not feasible.
    By X-ray angiography the perfusion of single arteries can be analyzed and evalu-
ated. This technique, however, requires the insertion of a catheter through a periph-
eral artery (e.g. the femoral artery) up to the LV coronary arteries for the indicator
injection. The level of perfusion in the myocardium can be also evaluated by looking
at the contrast enhancement of the myocardial region by nuclear imaging and, more
recently, also by contrast ultrasound and MR imaging. These techniques, however,
are limited in time (MRI) or spatial (ultrasound and nuclear imaging) resolution
with respect to X-ray angiography.
    A common test that is performed to have a better assessment of the blood supply
to the myocardium is referred to as stress test. A stress test is performed during or
immediately after exercise. An indicator is then injected and detected under stress
condition. The stress test is commonly performed by means of ultrasound or nuclear
imaging investigations.49 In patients that are unable to exercise, the stress test is
performed by the injection of vasodilator drugs, such as adenosine or dipyridamole,
which increase the level of perfusion. Other drugs that induce heart rate increase,
such as dobutamine, may also be used. However, these drugs are preferably adopted
for wall motion analysis.
    Recently, also semi-quantitative information on myocardial perfusion can be
obtained. The myocardium is considered as a mono-compartment dilution system
whose input is represented by the indicator concentration in the ascending aorta. If
             Indicator Dilution Techniques in Cardiovascular Quantification        111

perfusion intensity curves are measured by imaging techniques in several locations
in the myocardium, then the mono-compartment system between the aorta and
the measured output curve can be identified. Usually the input function can be
assumed to be well represented by a step function C0 u1 (t − t0 ), where u1 (t − t0 )
is the unitary step function at time t0 and C0 is the concentration after the step
rise. The system response for t ≥ t0 (causal system) is given by the convolution
between the step function (C0 u1 (t−t0 )) and the mono-compartment system impulse
response (τ −1 e− τ ) as
                            C(t) = C0 1 − e−      τ     .                        (56)

The time constant τ of the compartment as well as the steady state signal (con-
centration C0 ) are then associated to the level of perfusion and flow. An interesting
application of this method is the replenishment technique, which can be performed
by means of contrast ultrasound measurements as discussed in the end of Sec. 5.2.1.
An important parameter that can be estimated by means of the same methods is
the coronary flow reserve, i.e. the increase of myocardial blood flow in stress condi-
tion. It corresponds to the ability of the cardiovascular system to increase coronary
blood flow in response to vasoactive mechanisms.
    Perfusion techniques in the context of cardiovascular applications have recently
extended to novel applications. Of particular interest is the perfusion of vasa vaso-
rum and plaques in the carotid arteries.50 The presence of angiogenesis in plaques
might be very interesting to distinguish between stable and unstable plaques.51 A
plaque is unstable when the risk that it detaches and causes a stroke (in the brain as
well as in the myocardium) is high. Up until now, a qualitative evaluation of plaque
perfusion is mainly performed by contrast ultrasound or MRI.52 Semi-quantitative
analysis may be expected in the future. For this studies, the use of intravascular
ultrasound probes can be very useful.53

5. Indicator Dilution Techniques for Cardiovascular Quantification
The most common methods for cardiovascular quantification by means of indicator
dilution are discussed. A distinction is made between invasive techniques (Sec. 5.1
and Refs. 54, 55, 56), where the indicator detection requires contact between sensor
and indicator, and indicator imaging (minimally invasive) techniques (Sec. 5.2),
where the injected contrast is detected by means of non-invasive medical imaging
techniques. Particular attention is dedicated to echocardiographic and MR methods
(Secs. 5.2.1 and 5.2.4).

5.1. Invasive techniques
5.1.1. Fick method
These techniques uses a continuous indicator infusion. Due to the injection of a large
amount of contrast, the adopted contrast must be absolutely inert, harmless, and
112                         M. Mischi, Z. Del Prete & H. H. M. Korsten

                 Fig. 10.    Scheme of the Fick method for CO measurement.

non-toxic. There are at least two tracers that fulfill these requirements: Oxygen,
which is used in the Fick technique (Fick, 1870), and heat, which is used in the
continuous thermodilution technique (Fegler, 1954).57
   In the Fick method two sites for the indicator concentration measurement (see
Fig. 10) are fixed: the first site a is located before the injection point while the second
one b is located after it. Thus, assuming constant concentrations of the tracer Ca
and Cb , steady flow Φa = Φb = dV /dt, and using (39), (57) can be derived:
                                    dmb         dma       d(mb −ma )
                  Cb − Ca =          dt
                                          −      dt
                                                      =      dt
                                                                       .            (57)
                                     dt          dt
   Since d(mb −ma )/dt represents the tracer injection dm/dt between the sampling
points, (57) can be expressed as given in (58), which is the basic equation of the
Fick method:
                                     Φ=        .                                 (58)
                                      Cb − Ca
    The “trick” of the method is that the injection is naturally made by the lungs,
and Ca and Cb are respectively the venous and the arterial concentration of oxygen
(O2 ). Since the concentration of O2 is different in the different venous returns,
the sampling point a is placed in the pulmonary artery, after the venous blood has
been mixed by the RV. The blood samples of the so called “mixed venous blood” are
drawn by a catheter inserted through the jugular vein (or the subclavian vein) across
the right atrium (RA) and RV up to the pulmonary artery. Then the blood samples
are analyzed by a gas analyzer device for the measurement of the O2 concentration.
The choice for the arterial sampling site is not critical, since the blood from the
lung capillaries is well mixed. An arm or a leg artery is usually used.
    The measurement of the injected tracer (i.e. the inhaled oxygen) is performed
by making the patient breath pure oxygen from a spirometer.54 The exhaled CO2
is absorbed by a soda-lime absorber, so that the oxygen injection rate dm/dt (or
consumption) can be directly measured by the net gas-flow. This method had been
considered as the standard technique until the thermodilution replaced it.

5.1.2. Thermodilution
The thermodilution technique was first introduced by Fegler in 1954.57 This tech-
nique can be performed in a continuous mode, in which the indicator consists of the
               Indicator Dilution Techniques in Cardiovascular Quantification               113

heat diffused through a resistor (warm thermodilution), or in a single mode, in which
the indicator consists of a cold saline bolus (cold thermodilution). In both cases a
Swan-Ganz catheter is inserted via a central vein (usually the internal jugular or
subclavian) through the RA and RV, so that its tip lies in the pulmonary artery.58
This catheter is carried in the correct position by the dragging force of the flowing
blood thanks to a doughnut-shaped air-filled balloon on the tip of the catheter.
    In the warm thermodilution,59 the catheter (described in Fig. 11) includes the
circuitry for the thermistor that measures the temperature in the pulmonary artery
and the wires for the heating coil (resistor), which lies in the RA. A thermistor is
a semiconductor thermometer, which uses the relation between temperature and
material resistivity for temperature measurement.54 Like oxygen, heat is non-toxic
and naturally cleared. Therefore, it is a perfect indicator to perform a continuous
infusion technique.
    Heat may also be dissipated through the walls of the blood vessels (extravascu-
lar indicator) between the injection and the sampling site, therefore, this distance
should be as short as possible. Unfortunately, in order to have an adequate mixing
this distance should be long. The compromise that is usually adopted consists of
the indicator infusion in the RA and its sampling in the pulmonary artery.
    The warm thermodilution is still based on (58). The term dm/dt is given by
the heat derivative q (expressed in watt), and the term Cb − Ca is given by the
temperature difference Tb − Ta (expressed in kelvin) times the specific heat of the
blood cb (expressed in J · kg −1 · K −1 ) times the density of the blood ρb (expressed
in kg · m−3 ). In conclusion, the formulation of (58) for the thermodilution technique
is given as
                                  Φ=                     .                                (59)
                                        cb ρb (Tb − Ta )

    The mass of the tracer is now expressed by the amount of supplied energy q (in
joule) and its concentration is given in J · m−3 by the term cb ρb T . The thermistors
are usually placed in a Wheatstone configuration.54 This technique has become
very common in the clinical practice and everything is integrated in one single
catheter. The advantage of these continuous methods consists in the possibility of
a continuous CO monitoring.

Fig. 11.   Swan-Ganz catheter for warm thermodilution. Tb represents the reference temperature.
114                      M. Mischi, Z. Del Prete & H. H. M. Korsten

Fig. 12. Swan-Ganz catheter for cold thermodilution. The IDC is measured by the temperature
fall ∆T .

    A different application of (39), i.e. the cold thermodilution, makes use of an
injection of a cold indicator bolus, which can be either cold dextrose or saline (NaCl
blood-isotonic solution). In this case, the resistor is replaced by a port for the bolus
injection as shown in Fig. 12, which is performed by a syringe. The cold solution
mixes with blood in the RA and RV before passing into the pulmonary artery, where
the temperature fall is sensed by a thermistor on the side of the catheter. The CO
is then calculated from the temperature-time curve. With the same interpretation
of heat and temperature in (59), (40) can be written as
                             Φ=           ∞             ,                             (60)
                                   cb ρb t0   −∆T (t)dt
where q is the total injected heat, ∆T is the temperature fall, and cb and ρb are
the same as in (59). This method does not allow continuous CO monitoring.
    Apart for the PAC, sometimes a transpulmonary thermodilution is performed.
In this case, the bolus is injected in the RA or pulmonary artery and detected
in the aorta by a second catheter usually inserted through the femoral artery. The
transpulmonary thermodilution between RA and aorta can be performed in children
to assess the CO when cardiac catheterization is complicated by the small size of
the heart. This application is also necessary for the measurement of the ITBV or
the CBV based on the methods explained in Sec. 4.2.
    Cold saline is an extravascular indicator, since the heat is lost (diffused) through
the vessel walls. Therefore, these methods also allow the estimation of the EVLW
based on the models reported in Sec. 4.2. The same measurement can be performed
when an intravascular indicator, such as a dye (following section), is injected and
detected together with the cold saline bolus (double indicator method). In fact, the
MTT difference between the indicators is related to the EVLW.

5.1.3. Dye dilution
Also a colored dye such as indocyanine green,c usually referred to as cardiogreen,
meets all the necessary requirements of a “good” indicator: It is inert, non-toxic,
measurable, and even economic. Using the principle of absorption photometry, the

c Also other dye indicators are used, such as Evans Blue (absorption peak at 640 nm and 50%

clearance in 5 days) and Coomassie Blue (absorption peak around 590 nm and 50% clearance in
15–20 minutes).
            Indicator Dilution Techniques in Cardiovascular Quantification        115

concentration of cardiogreen, which is usually injected into the pulmonary artery,
can be detected by the light absorption peak at the wave-length of 805 nm.
    In the past, blood samples had to be drawn by a catheter placed in the femoral
or brachial artery and analyzed by an external photometry device. Nowadays, the
use of optical fibers allows in situ measurements.
    About 50% of the dye is cleared by the kidneys in the first 10 minutes, so that
repeated measurements are possible too. Once the system is calibrated, meaning
that the peak absorption is related to the concentration C(t) of the dye, the flow is
directly given by (40).

5.1.4. Lithium dilution
Lithium is a fluid that can be injected and detected by a lithium-selective electrode
in a flow-through cell.60–63 Two Ag-AgCl electrodes measure the potential across
a lithium selective membrane. According to the Nernst equation (more details are
reported in Ref. 54), the electric potential E (volt) across a membrane is given as

                                       RT ln(Cext )
                                 E=                 ,                          (61)
                                       nF ln(Cint )

where Cext and Cint are the external and the internal (with respect to the membrane)
lithium activities, which correspond to the lithium ionic concentrations, R is the
gas constant (8.314 J · mol−1 · K−1 ), T is the absolute temperature (kelvin), F is
the Faraday constant (96485 C · mol−1 ), and n is the valence of the ions, which
for lithium ions (Li+ ) is 1. As a result, the transducer measures a voltage that is
logarithmically related to the lithium concentration. The sensor is connected to a
three-way tap on the arterial line and a small peristaltic pump draws blood with a
flow of few mL per minute.
    After calibration, the lithium concentration is determined and a lithium IDC
generated. The time integral of the measured IDC is used as given in (40) for the CO
assessment. Moreover, based on the IDC CO measurement and the establishment
of the pressure-volume relation (compliance), also a continuous CO monitoring can
be performed.

5.2. Indicator imaging techniques for cardiovascular quantification
5.2.1. Echocardiography
Ultrasounds are elastic waves characterized by a frequency f that is higher than the
audibility threshold (20 kHz). Assuming a non-viscous medium, ultrasound waves
are longitudinal and are described by the wave equation. The real part of the wave
equation solution,

                          Re[A(t, z)] = A0 cos (k (vt − z)) ,                  (62)
116                                  M. Mischi, Z. Del Prete & H. H. M. Korsten

is commonly used for the wave representation: A(t, z) represents the medium dis-
placement as a function of time (t ) and position (z ), A0 is the maximum displace-
ment, k is the wave number, which is equal to 2πλ−1 (λ is the wave length and it
is equal to v · f −1 ), and v is the ultrasound velocity.64
    The ultrasound propagation velocity v is 343 ms−1 in air and 1480 ms−1 in water
at 20◦ C. The velocity of the sound in the soft biological tissues is about 1540 ms−1 .
It is similar to the velocity in water, since soft biological tissues are made by 80%
of water. The ultrasound propagation velocity is equal to Bρ−1 , where ρ is the
density of the medium (kg · m−3 ) and B is the bulk modulus, which is measured in
pascal (1 Pa = 1 N · m−2 ) and it is the measure of the stiffness of the material. The
energy carried by an acoustic wave is defined by its intensity I, which represents
the power across an unitary surface (watt · m−2 ). It is given as
                                                 I=      Z(2πf A)2 ,                                             (63)
where Z is the acoustic impedance, which for a planar wave equals P/A = ρv (P˙
is the ultrasound pressure and A   ˙ the oscillation velocity).64 The physical principle
of echography is the reflection of the ultrasound waves at the discontinuities of the
medium. The discontinuity is described in terms of acoustic impedance Z, which is
usually expressed in rayls: 1 rayl = 1 N · s · m−3 . For water Z = 1.49 M rayls. For
larger discontinuities a larger fraction of the energy is reflected. The amplitude P of
the pressure wave is given as P = 2ZI (i.e. P = 2πf AZ), therefore, the reflection
can be described in terms of pressure, intensity, or displacement.64,65
    An important aspect for a complete characterization of sound propagation is the
acoustic attenuation. The intensity of the sound wave decays with an exponential
law asd
                                                    I = Ii e−2az ,                                               (64)
where Ii is the initial intensity, z is the covered distance, and a is the absorption
coefficient in neper per cm.64,65
    The coefficient a depends on the material as well as on the frequency of the
ultrasound wave. It increases as the frequency increases. The relation between a
(usually given in cm−1 · MHz−1 ) and frequency for soft tissues is nearly linear. The
absorbtion coefficient for soft tissues is a linear function of f b , where b is slightly
larger than 1 (b 1.15) and f ∈ [0.1 MHz, 50 MHz].64
    Very often the attenuation a is measured in dB · cm−1 · MHz−1 . As a conse-
quence, (64) is given ase
                                                 I = Ii e−0.23adB zf .                                           (65)

d Since  the intensity is related to the squared value of the pressure amplitude, for pressure (64)
becomes P = Pi e−az .
e It is common to find a expressed in dB, i.e. a
                                                    dB = 20 log10 (P/Pi ). As a consequence, P =
         adB f z                    adB f z
Pi 10−     20      = Pi e− ln(10)     20      − Pi e−0.115adB f z . The conversion is then a =   ln 10
                                                                                                       adB   = 0.115 ·
adB .
               Indicator Dilution Techniques in Cardiovascular Quantification                     117

    Ultrasound transducers are made of piezoelectric crystals that are placed on
the tip of a probe. The crystals are the vibrating material that converts electrical
energy into acoustic energy and vice versa. Each crystal makes an approximately
linear conversion between mechanical pressure and electrical voltage and shows a
specific resonance frequency that maximizes the energy transfer.55,65
    The acoustic adaptation between crystals and skin tissue in order to transmit
with the maximum bandwidth and the minimum loss is made by a matching layer
designed with a thickness of λ/4 (λ is the length of the acoustic wave in the layer
at the resonance frequency of the transducer) and an acoustic impedance between
those of the crystal and the external medium (the acoustic impedance of the human
    Echography, as the name suggests, is based on the analysis of the echoes: when a
pulse (usually few ultrasonic cycles) is transmitted inside the body and the result-
ing echoes (reflections) received back, it is possible to derive the position of the
intercepted discontinuities that generated the echoes. In fact, since the propagation
velocity of ultrasonic waves through biological tissues is known, the delays of the
received echoes can be interpreted in terms of distance. The distance d between the
transducer and the discontinuity is given as
                                          d=       ,                                            (66)
where v is the ultrasound velocity in tissue and ∆t is the time interval between the
transmission and reception of the ultrasonic pulse. An example is given in Fig. 13.
   This is the basic Mode of an ultrasound scanner, and it is referred to as A-Mode
(“A” stands for “Amplitude”). A pulse is transmitted and received back in order to
reconstruct the discontinuities along a line. The received signal is demodulated in

Fig. 13. Distance estimation by means of echography. The discontinuity is represented by a plastic
layer inserted in a water-filled basin. Notice that the border of the basin represents a discontinuity
too, and that the resulting echo is more attenuated due to the longer path.
118                     M. Mischi, Z. Del Prete & H. H. M. Korsten

order to suppress the frequency (usually between 2.5 MHz and 7 MHz for clinical
applications) of the ultrasound pulse. This signal is referred to as A-line (Amplitude
line). The A-line that is obtained after demodulation represents the amplitude of
the ultrasound echoes as a function of the ultrasound depth, i.e. the profile of the
acoustic impedance discontinuities.
    An important characteristic of the system is the resolution. Two different reso-
lutions can be distinguished: the axial resolution and the lateral resolution.64 The
axial resolution depends on the length of the pulse envelope (e.g. 5 cycles at 5 MHz
result in a resolution distance in tissue d      0.8 mm). The lateral resolution is a
measure or the narrowness of the ultrasound beam. It can be derived from the Huy-
gens principle.55,64,66,67 For a circular transducer of radius r the lateral resolution
is maximal (approximately equal to r) for a distance corresponding to r2 /λ, i.e. the
transition between Fresnel and Fraunhofer zone. In the Fraunhofer zone the main
lobe of the beam can be described by the opening angle θ0 as
                             θ0 = arcsin            .                             (67)
Two points covered by the main lobe cannot be distinguished, therefore, θ0 describes
the angular lateral resolution of the system, which decreases as the distance of the
discontinuities from the transducer increases. Although the beam pattern depends
on the shape of the transducer, in every transducer the lateral resolution is propor-
tional to the ultrasound frequency and crystal size.
    The same principles that are used to reconstruct the distribution of the acous-
tic impedance discontinuities along one line (A-line) can be used to obtain the
discontinuity distribution on one plane.55,68 In fact, a plane can be spanned by
translating the transducer in one direction and taking measurements for a series of
lines. The voltage of the multiple A-lines (dynamic range) is mapped into gray-level
by using a logarithmic compression, or, more in general, a non-linear compression.
The result is a bi-dimensional image that represents a slice of the body. The bi-
dimensional application of echography is referred to as B-Mode, where “B” stands
for “Brightness”.
    The first simple B-Mode implementation was based on the mechanical trans-
lation of the transducer. Current solutions are mainly based on electronic array
processing, which is implemented in the array transducers and allows dynamic focal-
ization and steering of the ultrasound beam to different directions (beam steering).
An array transducer is made of an array of crystals. Each crystal can be activated
separately with a specific delay. By choosing the appropriate delays it is possible
either to focus the ultrasound beam (like an optical lens) or change (steer) its orien-
tation to span a wide angle. The crystals are therefore “phased” along one direction.
    The use of linear phased arrays is especially desirable in cardiology for transtho-
racic imaging, since the transducer must be sufficiently small to fit between the ribs
and, at the same time, it must be able to span a large plane inside the body to
image the heart. Linear phased arrays can be made in 1.5 dimensions, i.e. the array
             Indicator Dilution Techniques in Cardiovascular Quantification             119

also contains a few elements (3 to 9) along the second dimension (elevation plane),
which are used to improve the resolution along this second dimension (elevation
focus). Obviously, this solution can be extended to a fully bi-dimensional array,
which allows spanning complete volumes (3D imaging) inside the body.69
    The most common echocardiographic test is referred to as transthoracic echocar-
diography (TTE). The name derives from the fact that the ultrasound transducer
is positioned on the chest (thorax) of the patient. From that position, several win-
dows can be found between the ribs in order to obtain different views (slices) of the
    Miniaturized transducers are also available. They can be introduced through the
mouth into the esophagus or the stomach as shown in Fig. 14. Since the heart is
analyzed from the esophagus, this technique is known as transesophageal echocar-
diography (TEE) and has a significant advantage over classical TTE. Using TTE
the ultrasound beam has to pass through the ribs and the lungs, resulting in lower
quality images. The lungs may cause problems especially in patients with lung
emphysema. Furthermore, added difficulties arise from the female breast and the fat
tissue when dealing with obese patients. Instead, the esophagus is directly behind
the heart, separated only by a thin layer of tissue. That is the reason why TEE
produces higher Signal-to-Noise Ratio (SNR) images, which are also suitable for
contrast detection and quantification.71 However, a disadvantage of this technique
consists of its application procedure, which is unpleasant for the patient due to the
probe insertion through the mouth. Nevertheless, for applications during surgery
or in intensive care unit, TEE is becoming widely used. Related complications,
such as gastrointestinal tissue damage, are very low, but they are not completely
negligible.72 Figure 14 shows a TEE four chamber view, which permits the simul-
taneous observation of all the four cardiac chambers.

        Fig. 14.   Transesophageal echocardiography: example of a four-chamber view.
120                     M. Mischi, Z. Del Prete & H. H. M. Korsten

Ultrasound contrast agents
Ultrasound Contrast Agents (UCAs) are made of a solution of micro-bubbles (diam-
eter from 1 µm to 10 µm). They are smaller than red blood cells (diameter from 6 µm
to 8 µm) and, therefore, suitable for passage through the capillaries and, in particu-
lar, the transpulmonary circulation. As for all indicators, ultrasound contrast agents
have to be inert and non-toxic. Commercial agents can be distinguished in three
generations.73 The first generation includes the early air micro-bubbles that were
not stabilized by a shell, such as Echovist (Schering, Berlin, Germany), which
was approved in Europe in the early 90s. Second generation agents were introduced
in the mid-90s and included the first encapsuled air bubbles, such as Albunex
(Molecular Biosystems, San Diego, CA, USA) and Levovist (Schering), whose
shell was made of albumin and galactose, respectively.
    The latest third generation micro-bubbles for ultrasound detection are composed
of air, SF6 , C3 F8 , or other perfluorocarbons encapsuled in a phospholipid, albumin,
or polymer shell.74–77 The optimized use of a shell creates a strain that opposes to
the Laplace pressure and stabilizes bubbles against dissolution. In fact, according
to the Laplace law, the pressure inside a gas sphere, referred to as Laplace pressure,
equals 2σ/R, where σ and R are the surface tension (N · m−1 ) and the sphere
radius, respectively.78 The high internal pressure due to small radius is held by
the strain that is provided by the shell. Examples of third-generation agents that
are currently used in clinical practice are SonoVue (Bracco, Italy) and Definity
(Bristol-Myers Squibb, USA).
    Once injected into blood, the effect of the bubbles (see Fig. 15) is a significant
increase of the ultrasonic energy backscatter, which at first analysis could be simply
explained by the reflection across the blood-air acoustic-impedance discontinuity.
    Due to the natural oscillations (contraction-expansion) of the bubbles when
invested by a pressure input, the interaction between contrast agents and ultrasound
is a nonlinear process, which adds several harmonics to the backscattered ultra-
sound. The oscillations of a single bubble are commonly characterized by the model
developed by Rayleigh and Plesset to describe the motion of vibrating spheres.78,79
The bubble is represented as a sphere of radius R and its motion is considered spher-
ically symmetric as shown in Fig. 16. Therefore, bubble oscillations are described by
radius variations. The surrounding fluid is assumed to be Newtonian (incompressible
with constant viscosity).
    The equation that is generally used to describe the relation between velocity and
pressure in a fluid-dynamic system is the Navier-Stokes equation.80 If the steady
external forces (e.g. the gravitational force) are not considered, the Navier-Stokes
equation can be written as given in (68), where ρ and µf are the density and the
viscosity of the fluid, respectively:

                ∂v     v2                      4
            ρ      +        + µf ( ×     × v) − µf ( · v) = − P.                 (68)
                ∂t     2                       3
                 Indicator Dilution Techniques in Cardiovascular Quantification                  121

Fig. 15. TTE using SonoVue contrast agent. The contrast is recognizable in the left side of
the heart. Courtesy of the department of cardiology of the Catharina Hospital in Eindhoven, The

      Fig. 16.    Schematic drawing of an oscillating bubble in an ultrasound pressure field.

    For a Newtonian incompressible fluid       · v = 0. If the vorticity is negligi-
ble, the velocity field is irrotational ( × v = 0) and the viscosity terms in (68)
disappears.78,80 Therefore, we can define a potential Ψ so that v = Ψ and replace
(68) by the Bernoulli equation as given in (69)78,80 :

                                       ∂Ψ v 2
                                 ρ        +         = − P.                                     (69)
                                       ∂t   2
122                     M. Mischi, Z. Del Prete & H. H. M. Korsten

    Due to the radial symmetry hypothesis, the fluid velocity field has a radial
symmetry and can be described as a function v(r, t) of the radial distance r from
the bubble center and the time t. As a direct consequence, v(R, t) = R, where R
is the bubble radius. Moreover, Ψ = 0 for r → ∞. Another condition concerns
the pressure field for r → ∞, which is defined as P∞ and equals the sum of two
contributes: the hydrostatic pressure P0 and the ultrasound driving pressure P (t).
Using these conditions, the integration of (69) over r allows defining the relation
between the pressure field P (r, t) and the bubble radius R(t), which for r = R is
given as

                                         ¨ 3 ˙
                          P (R) − P∞ = ρRR + ρR2 .                                   (70)
   P (R) can be related to the internal pressure Pi of the gas bubble as given in (71),
where Pv is the vapor pressure and σ is the surface tension coefficient (N · m−1 )79 :
                               P (R) = Pi −     + Pv .                        (71)
Since the gas expansion and contraction can be considered respectively isothermal
and adiabatic, the whole process is described by a polytropic transformation with
exponent k, i.e. P · V k is constant.81
    For a driving pressure that does not cause bubble collapse we may assume the
internal gas to be ideal. In this case Pi is given as
                                            2σ        R0
                        Pi =    P0 − Pv +                       ,                    (72)
                                            R0        R
where R0 is the bubble radius at the equilibrium condition.
    Combining (72) and (71) with (70), we obtain (73), which describes the non-
linear motion of an ideal bubble and is referred to as Rayleigh-Plesset equation 78 :
             ¨ 3 ˙
           ρRR + ρR2 =
                                  + P0 − Pv
                                                                − 1 − P (t).         (73)
                2              R0                     R

    A modified expression of the Rayleigh-Plesset equation also adds to the second
member of (73) the term 4µf RR−1 , which defines the pressure drop that is caused by
the viscous damping of the bubble-fluid system and is related to the fluid viscosity
µf (see Ref. 78 pp. 189–190 and Ref. 82 pp. 68–70). The resulting equation is given
as in (74). It is referred to as the modified Rayleigh-Plesset equation,83,84 which is
the result of the work of Noltingk, Neppiras, and Poritsky:
                                                      3k                ˙
        ¨ 3 ˙
      ρRR + ρR2 =
                            + P0 − Pv
                                                           −1 +
                                                                    4µf R
                                                                          − P (t).   (74)
           2             R0                      R                   R

    Since modern contrast agents (second and third generation) are made of encap-
suled bubbles, the shell properties must be included in the bubble motion equation.
Starting from the Rayleigh-Plesset equation, several authors have made modifica-
tions and added different terms according to different shell characterizations. The
              Indicator Dilution Techniques in Cardiovascular Quantification                  123

major contributions were proposed by de Jong (1993), Church (1995), Frinking-de
Jong (1998), and Hoff (2000).73,79,82,85–88
    Also other models have been derived to remove some of the assumptions of the
Rayleigh-Plesset equation. They are based on the Herring (1941) and Gilmore (1952)
equations,89,90 which consider the enthalpy energy and a compressible medium (non
Newtonian) in order to give a better prediction of large oscillations.78,91,92 Based
on these equations, Flynn (1975) developed a model that also included the thermal
effects inside the bubble.93
    Still based on the Herring equation, the Trilling (1952) and the Keller-Kolodner
(1952) models were derived.82 The Trilling model was then extended to a popu-
lation of bubbles by Chin and Burns (1997),94 while the Keller-Kolodner model
was modified first by Keller-Miksis (1980) to introduce the driving pressure field
and then by Prosperetti (1988).95,96 Another model — still based on a modified
Herring equation — was also introduced by Morgan and Hallen (2000).97
    A complete overview of all the proposed bubble dynamics models is beyond the
purpose of this chapter. Due to its simplicity, we limit our discussion to the model
proposed by de Jong in 1993.79
    With respect to (74), the model does not consider only the damping due to
the fluid viscosity µf , but also other contributes related to re-radiation and heat
conduction, together with the contributes to pressure due to the shell-properties.
The pressure that is related to the presence of a shell is mainly caused by the
shell viscous damping and elasticity. After collecting all these terms together, the
resulting equation is given as

            ¨ 3 ˙
          ρRR + ρR2 =
                                + P0 − Pv
                                                             − 1 − Sp
               2             R0                    R                      R0   R
                          − 2πf δt ρRR − P (t) .
                                     ˙                                                      (75)

    The pressure due to the shell elasticity (restoring force) is defined by the shell
elasticity parameter Sp , which is measured in N · m−1 and can be derived from
the application of the Hooke law (first order approximation) under hypothesis of
homogeneous, thin, and perfectly elastic shell.79,98 Notice that in this model the
shell thickness is assumed to be constant during the oscillations.f
    The term δt in (75) is the total damping factorg and it is the sum of four terms
as given in (76):

                               δt = δrad + δvis + δth + δf .                                (76)

f Other  models, such as the Hoff model, assume a constant shell volume and, therefore, the shell
thickness becomes a function of the bubble radius R.
g In a linear system of the second order, such as a mass-spring damped system, the damping factor

(non-dimensional) is defined as the inverse of the quality factor and equals b/2πfn m, where b is
the damping coefficient, fn is the natural frequency of the system, and m is the mass.
124                     M. Mischi, Z. Del Prete & H. H. M. Korsten

The term δrad represents the re-radiation damping, δvis the fluid viscosity damp-
ing (already included in (74)), δth the damping due to thermal losses, and δf the
damping due to the shell friction.78,82,99
    The expression for the four dimensionless terms, usually considered at the natu-
ral frequency fn (a derivation of fn for a linearized system is given in (80)), is given
below, where µs is the shell viscosity, v is the ultrasound velocity in the medium,
and the expression for B(f, R) can be found in Ref. 99 (pp. 296–299):
                                  δrad =       R0 ,                                 (77)
                                  δvis =        2,
                                         πfn ρR0
                                   δth = B(f, R0 ) n ,
                                          6µs Ts
                                    δf =        3.
                                         πfn ρR0
    All the damping factors are function of the frequency f and the bubble radius
R0 . For instance, the thermal damping δth is only effective for intermediate fre-
quencies (near the natural frequency fn ). In fact, for high frequencies the process is
adiabatic (no heat transfer due to the short period of the oscillations), while for low
frequencies the process is isothermal (the oscillations are sufficiently slow to allow
the heat transfer to keep the bubble temperature constant). Instead, for intermedi-
ate frequencies, like for instance around the natural frequency fn , the temperature
oscillates in the bubble and causes pressure variations that are out of phase with
respect to the driving pressure.
    The bubble dynamic model is used to study and predict the response of the
bubble system to a pressure input (driving pressure). The major interest is the
definition of a frequency range that maximizes the bubble oscillations and, therefore,
the backscattered ultrasound signal. In order to derive the frequency response of
the bubble, the system is analyzed for small oscillations and a Taylor first order
approximation of (75) is considered. The resulting system is a typical second order
linear system (like a damped mass-spring system, see Ref. 82, p. 109) as given in
(78), where represents the small radius oscillation R − R0 for a driving pressure
P (t) (see Fig. 16):
               m¨ + b ˙ + s = −4πR0 P (t),
               m = 4πρR0 ,
               b = 2πf mδt ,                                                       (78)
                                         2σ        2σ        2Sp
               s = 4πR0 3k P0 − Pv +             −    + pv +     .
                                        R0         R0        R0
   The frequency response | (f )| · |P (f )|−1 of the system (see also Refs. 82, 78) is
given as
                    | (f )|                   1
                            =                                  ,                  (79)
                   |P (f )|   R0 ρ (fn  2 − f 2 )2 + (f f δ )2
                                                         n t
               Indicator Dilution Techniques in Cardiovascular Quantification          125

where fn is the natural frequency of the bubble without damping and is given as

             1    s      1                             2σ          2σ        2Sp
     fn =           =   √           3k P0 − Pv +               −      + Pv +     .   (80)
            2π    m   2π ρR0                           R0          R0        R0

    Based on (79) and (80), the peak frequency response (resonance frequency f0 )
of the damped bubble is given as in (81):

                                f0 = fn    1−              .                         (81)

    As a consequence, the maximum oscillation, when for instance the viscous damp-
ing is included, is generated for a driving frequency f0 given as in (82) (see Ref. 78,
pp. 305–306):

                    1                        2σ        2σ        2Sp   4µ2
        f0 =       √       3k P0 − Pv +            −      + Pv +     −   f
                                                                         2.          (82)
                 2π ρR0                      R0        R0        R0    ρR0

    As already mentioned, the de Jong model, as well as all the models in literature,
are based on a series of assumptions that are not always realistic. In particular, in
a real clinical application a large number of bubbles may interact with each other
with multiple thermal and radiation energy exchanges. The interaction forces are
not considered in the presented model. Moreover, as recently observed by de Jong
and Versluis using an ultra-fast camera, the assumption of spherical symmetry of
the bubble oscillations is not realistic.100,101
    Bubbles oscillate using several different geometrical modes, so that more sophis-
ticated models are needed. Also a large variability of the maximum bubble expan-
sion for same driving pressures and same bubble original diameters is recognizable.
This phenomenon is complex to explain and might be the result of different elastic
properties of the bubble shell.102 In this case, the elasticity parameter Sp could be
substituted with a statistical distribution.
    However, especially for small pressure and low concentrations (negligible interac-
tion between bubbles), the available models can already predict with sufficient accu-
racy the interaction between ultrasound and bubbles and they are widely adopted
for contrast quantification.8,73,103–108 In particular, contrast quantification involves
the estimation of the agent echogenecity, which is commonly determined by the
measurement of the ultrasound backscatter.
    The ultrasound backscatter is defined by the backscatter coefficient β, which is
the scattering cross-section (cm2 ) per unit volume (cm3 ) and per scattering angle
(sr). The scattering cross-section of a bubble is the ratio between the power scattered
in all directions and the incident acoustic intensity. The scattered power equals the
energy that is dissipated because of the radiation damping.
126                              M. Mischi, Z. Del Prete & H. H. M. Korsten

    In general, if the damping coefficient equals b, then the force F that is necessary
to compensate for it equals ˙b. For (t) = 0 cos(2πf t), the average power W (energy
per cycle) that is dissipated is given as in (83), where T = f −1 :
                           T                             T
                 1                              1                                  1 2 2 2
          W =                  (F · ˙(t))dt =                ( ˙(t)b · ˙(t))dt =     4π f 0 b.              (83)
                 T     0                        T    0                             2
    The re-radiation average power is given as in (83) for radiation damping coeffi-
cient brad = 2πf mδrad = 16π 2 ρR0 f 2 v −1 . Combining (83) with (79), the scattering

cross-section Σ for a single bubble, which is defined as the ratio between average
scattered power and ultrasound intensity, is a function of the radius R of the bubble
and the ultrasound frequency f as given in (84), where W is the average scattered
power, Ii is the amplitude of the incident intensity (Ii = Pi2 /2Z), and fn is the
natural frequency of the bubble:78,79,82,88,108,109
                       W      4π 2 f 2 2 b
        Σ(R0 , f ) =      = 22             =                                                            .   (84)
                       Ii   Pi (2Z)−1                                    2
                                                         (fn (R0 )/f ) − 1             + δt (R0 , f )

    The term δt (R0 , f ) summarizes all the damping factors. Since the adopted model
represents a second order system, the scattering cross-section shows a resonance
frequency where the system gives the strongest response in terms of scattered power.
For f     fn ⇒ Σ (R0 , f ) 4πR0 , which is the physical cross-section, i.e. the bubble
surface.        The resonance frequency is inversely proportional to the radius R0 of
the bubbles at the equilibrium. Therefore, the total scattering cross-section Σtot (f )
depends on the normalized radius distribution n(R) of the bubbles as given in
(85)110,111 :
                                Σtot (f ) =              n(R)Σ(R, f )dR.                                    (85)

   n(R) is a characteristic of the specific contrast. Assuming an isotropic scatter-
ing and a low concentration of bubbles, the backscatter coefficient β (expressed in
cm−1 · sr−1 ) is given as in (86), where ρn is the number of bubbles per unit volume
(concentration)110 :
                                      ρn Σtot (f )
                                        β(f ) =    .                           (86)
    Therefore, the backscatter coefficient is a linear function of the UCA concentra-
tion and the backscattered acoustic intensity I that is measured by the transducer
can be approximated by8,105,106,110 –113
                             dV            dV ρn Σtot (f )
                                 β(f )Ii = 2
                                I=                         Ii ,                   (87)
                              z2           z      4π
where Ii is the acoustic intensity insonating the contrast, dV = dz ·dA is the volume
of the insonated contrast, and z is the distance between dV and the transducer.h

h An homogeneous scattering with a spherical symmetry of the backscattered pressure is assumed
(see Ref. 114, pp. 357–361).
             Indicator Dilution Techniques in Cardiovascular Quantification            127

When β(f ) is averaged over the frequency spectrum of the ultrasound transducer
it is referred to as Integrated Backscatter Index (IBI).
     The interaction between ultrasound and UCA is not only described by the
backscatter coefficient, but also by the attenuation coefficient, which represents
the loss of acoustic pressure in neper per cm. It occurs along the distance that
the ultrasound beam covers through the contrast solution, and it is related to the
micro-bubble decay (ad ) due to both chemical decay (dissolution) and dispersion
through the capillaries, the scattering of the acoustic energy in multiple directions
(as ), and the viscous, thermal, and friction damping (δvis , δth , and δf in (76)) of the
ultrasound waves (aδ ).
     The total increase of attenuation ∆a is given as in (88), where ad is proportional
to the chemical constituents of the contrast while (as + aδ ) is proportional to the
concentration of the contrast:

                                ∆a = ad + as + aδ ,                                  (88)

    The attenuation that is expressed by the term as +aδ is referred to as extinction.
It is possible to define an extinction cross section Σe as the sum of the backscatter
cross section Σ given in (84) and the absorption cross section Σa . Similarly to (84),
the absorption cross section is defined as the ratio between the loss of power and the
incident intensity Wloss /Ii . The power can be always expressed as a linear function
of the damping coefficient b as given in (83). Since b = 2πδmfn by definition,
the ratio between the loss of power caused by damping and the acoustic scattering
Wloss /W equals the ratio between the respective damping factors. This concept can
be formulated as given in (89) (see Ref. 82, pp. 25–28 and Ref. 99, pp. 302–304):
                                     δvis + δth + δf
                             Σa =                          Σ.                        (89)
Since Σe = Σa + Σ, Σe is also given as
                                  Σe =             Σ.                                (90)
    The extinction coefficient ae = as + aδ can be derived from the extinction cross
section. In fact, from the definition of extinction cross section, the differential loss
of power dWloss can be written as given in (91), where dV = dA · dz is a volume
sample as shown in Fig. 17:

                     dWloss = Iρn Σetot dV = Iρn Σetot dAdz.                         (91)

   Therefore, the loss of intensity dI is described by the differential

                                 dI = Iρn Σetot dz,                                  (92)

whose integral is given as

                                I(z) = Ii e−ρn Σetot z .                             (93)
128                         M. Mischi, Z. Del Prete & H. H. M. Korsten

Fig. 17. Schematic passage of an ultrasound beam of intensity I through a volume sample dV
of bubbles. The volume length is dz while the cross-section area is dA. The intensity loss through
the sample equals dI.

   The total extinction cross-section Σetot is derived from Σe as given in (85) for
the scattering cross-section Σtot :
                            Σetot =           n(R)Σe (R, f )dR.                              (94)

      As a consequence, the extinction coefficient ae is given as in (95):
                            ρn      ρn
                     ae =      Σe =                   n(R)Σe (R, f )dR.                      (95)
                            2 tot   2         Rmax

The division by a factor 2 allows deriving (64) from (93), and it is due to the fact
that the attenuation coefficient is usually defined in terms of pressure loss, which is
related to the intensity by the quadratic relation I = P 2 (2Z)−1 .
    For shell encapsuled bubbles, the first term ad in (88) can be neglected if the mea-
surement is executed in a short time (i.e. few minutes). When only the extinction is
considered, ae in (95) corresponds to the attenuation coefficient a introduced in (64).
As a result, the attenuation is linearly proportional to the concentration of the con-
trast ρn , as also shown in (86) for the backscatter coefficient β.79,105,110,111,115,116
    Because of the linear relation between total attenuation and contrast
concentration, some authors have considered the opportunity of measuring the
attenuation-time curve rather than the backscatter-time curve for the estimation of
fluid-dynamic parameters.116
    According to (64), (87), (88), and (95), the received echo intensity from a con-
trast perfused organ can be rewritten as
                       I=   (β + ∆β)Ii e−4a(z0 +∆z)−4∆a∆z ,                    (96)
where β is the backscatter coefficient in the examined organ (without contrast), a
is the attenuation coefficient of tissue, z0 is the depth of the organ, and ∆z is the
depth of the insonated volume sample dV within the contrast perfused organ. ∆β
             Indicator Dilution Techniques in Cardiovascular Quantification         129

and ∆a are the backscatter and attenuation increase due to the presence of contrast
in the investigated organ. The return distance that is covered by ultrasonic waves
(after reflection) is taken into account by the factor 4 at the exponent (instead of 2).
    This is the model that is usually adopted to interpret the backscattered intensity.
In fact, both ∆β and ∆a are linearly proportional to the contrast concentration.
A good measure of the efficacy of UCAs in terms of ultrasound detection, which
considers both backscattering and attenuation, is represented by the Scattering-To-
Attenuation Ratio (STAR) Σetot = δrad = 2π∆β .73,110,111
                                       δt    ∆ae

    This characterization of UCA is largely based on the assumption of small oscilla-
tions, i.e. small ultrasonic driving pressures. For higher pressures the bubble cannot
hold the inner gas and the bubble collapses. The violent collapse of bubbles is
referred to as cavitation. The standard indicator to determine the mechanical inter-
action or impact of ultrasonic waves on bubbles is the mechanical index (MI), which
is defined as
                                         P [MPa]
                                 MI =             ,                               (97)
                                          f [MHz]
where P is the peak rarefactive pressure measured in MPa and f is the central
frequency of the ultrasound pulses expressed in MHz.
    For MI > 0.6, a large fraction of the bubbles collapses. It is worth mentioning
that bubble cavitation phenomena find important applications in the field of drug
and genes delivery.75,117–120 In these applications, bubbles are loaded with either
drugs or genes. A targeting ligand in the bubble shell makes the injected bubbles
settle and attach to specific sites.121,122 Once the bubble is located where the drug is
intended to be delivered, high MI ultrasound pulses make the bubbles collapse and
release the drug. This technique allows both targeting the delivery and enhancing
the drug uptake through the increased porosity of the cell membranes due to bub-
ble cavitation. In cardiology, this technique is very promising for adenosine release
during cardiac reflow therapy (i.e. when the flow through the coronary arteries is
reestablished). Also applications for lysis of clots in the coronary arteries are very
interesting and promising.123
    The nonlinear behavior of bubble oscillations has given the opportunity to
develop specific imaging modes to preferentially detect the echoes from the agent
while suppressing those from other linear structures, such as tissue.76,77,105,124–126
The nonlinear behavior of the bubbles is mainly evident for driving pressures above
50 kPa.104 Above this threshold, the backscatter cross-section Σ differs from the
physical value of 4πR0 according to (84).
    We refer to as fundamental mode when the ultrasound scanner is set to transmit
and receive at the same frequency (fundamental frequency). Nowadays, also contrast
detection modes, referred to as harmonic modes, are widely available in ultrasound
scanners. The objective of these harmonic imaging techniques is to maximize the
Contrast-to-Tissue Ratio (CTR).127–129 In harmonic mode, the system transmits at
one frequency (resonance frequency of the transducer), but is tuned to receive echoes
130                      M. Mischi, Z. Del Prete & H. H. M. Korsten

                   Fig. 18.   Frequency response spectrum of SonoVue .

preferentially at a different frequency. Therefore, the bandwidth of the transducer
must be broad.
    As shown in Fig. 18, the frequencies that are commonly used for contrast detec-
tion are distinguished in sub-harmonic (about half of the fundamental harmonic),
ultra-harmonic (between the fundamental and the second harmonic), second and
higher harmonics, and super-harmonic (between higher harmonics).76,77,128,130,131
For these frequencies, the signal coming from contrast shows larger amplitudes with
respect to that coming from tissue. In fact, due to the linear response of tissue, tissue
echoes contain approximately the same frequency components as the pressure waves
that are generated by the transducer (fundamental harmonic). Figure 18 shows an
example of frequency spectrum of UCA backscatter.
    The implementation of filters in order to detect specific frequencies is not the
only solution to enhance the UCA detection. Alternative techniques involve the
modulation of amplitude and phase of the transmitted ultrasound. They are referred
to as power modulation imaging and phase modulation imaging, respectively. These
techniques are economically more convenient, since they do not require the employ-
ment of expensive broad-band transducers.
    The most common phase modulation technique is referred to as pulse inversion.
Two pulses p1 (t) and p2 (t) = −p1 (t) are transmitted in sequence into the tissue.
The sum of the received echoes results in the cancellation of the echoes from linear
structures (i.e. tissues), while the echoes from the bubbles do not cancel, resulting
in a selective detection of UCAs.
    Amplitude modulation (or power modulation) techniques quantify directly the
nonlinearity of the ultrasound transmission-detection channel (i.e. the medium),
which is related to the presence of bubbles. If h(P ) represents the channel for pres-
sure P and the channel is nonlinear, then 2h(P ) = h(2P ). Several implementations
combine Power Modulation and Pulse Inversion (PMPI) with a number of different
pulse sequence schemes.
    The use of high MI (MI > 1) produces a high-rate bubble disruption.104,120,132
Some techniques exploit the bubble disruption for contrast imaging.133,134 Two
             Indicator Dilution Techniques in Cardiovascular Quantification       131

pulses are transmitted in fast sequence. The first one is reflected and destroys the
bubble. After the bubble destruction, the second pulse is not reflected. By taking
the difference between the reflections from the two pulses it is possible to distin-
guish bubbles from tissue, unless the fast motion of tissue is confused with the
bubble destruction (clutter noise). More sophisticated implementations of the same
concepts are possible and are referred to as release-burst imaging.76 ,126,134

Cardiovascular quantification
Nowadays, the main application of UCA dilution in cardiology aims at the improve-
ment of the endocardial and epicardial visualization and segmentation.43 In partic-
ular, an accurate segmentation of the endocardium is fundamental for accurate
assessments of LV end-diastolic and end-systolic volumes, which also permit the EF
derivation. These applications are referred to as LV opacification (LVO) and require
UCA infusion rather than the injection of a single bolus. The use of LVO also allows
the detection of ventricular obstructions that are difficult to recognize otherwise.
    Apart from LVO, which is not directly a quantitative application of UCA dilu-
tion, a technique that is implemented in all recent ultrasound scanners is referred
to as replenishment. It consists of the destruction of micro-bubbles followed by a
low MI detection phase. This technique is used for myocardial flow quantification.
During perfusion, the contrast in the myocardium is destroyed by a high MI ultra-
sound burst (MI ≥ 1). The following replenishment curve is detected by record-
ing the acoustic intensity-versus-time curves in defined regions of interest of the
myocardium. The interpolation of these curves by a specific model allows the quan-
tification of the myocardial flow.135,136 Also this application requires UCA infusion
rather than a bolus injection.
    In practice, a simple mono-compartment model is adopted to represent the dilu-
tion system of the myocardium as shown in Fig. 19. Therefore, the replenishment
curve C(t) is modelled as the step response of a mono-compartment model as given
in (56). The flow is proportional to C0 · τ −1 and an image of the myocardium can
be generated such that the gray level (or color coding) corresponds to the perfusion
level.49,135,136 As discussed in Sec. 4.4, the myocardial flow can be estimated before
and after inducing stress (stress test) in order to assess the coronary flow reserve.
    An interesting emerging application of UCA dilution concerns the measurement
of the PBV. A region of interest (ROI) is placed on the RV and a second ROI is
placed on the LA after the injection of a small bolus of UCA. Two IDCs can be
measured (one for each ROI) and interpolated by the distributed models presented
in Sec. 4.2, so that the MTT of the transpulmonary circulation can be estimated
and multiplied times the CO for the assessment of the PBV.137
    Another novel technique consists of the EF assessment by analysis of two IDCs
in the LA and LV after a peripheral bolus injection. A deconvolution technique is
adopted to estimate the impulse response of the LV. Due to low SNR, the decon-
volution can be implemented by means of squared error minimization techniques,
such as for instance a Wiener filter.21 The LV is modelled by a mono-compartment
132                          M. Mischi, Z. Del Prete & H. H. M. Korsten

                                                           −1                        −1
      Fig. 19.   Replenishment curves for lower (Φ ∝ C1 · τ1 ) and higher (Φ ∝ C2 · τ2 ) flow.

model as described in Sec. 4.3, so that the EF can be measured by the Holt method
described in (55) in a minimally invasive manner.138

5.2.2. X-ray angiography
X-ray imaging is based on the detection of X-rays that pass through the body. The
different absorbtion of X-rays associated to different tissues permits to use the X-ray
projection to reconstruct a medical image of the inner body. The X-ray angiography
is the standard technique to diagnose blood-vessel stenosis and aneurysm, but it
can be also adopted to estimate CO and MTT, especially in the myocardium.139
Stenosis is an abnormal narrowing of an artery or vein (it can be also referred to
the cardiac valves) while aneurysm is an abnormal widening of it.
     Cardiac catheterization is a combined hemodynamic and angiographic proce-
dure, which can be done for diagnostic and therapeutic purposes. It is an invasive
procedure, therefore, the decision to perform cardiac catheterization must be based
on a balance of the risk of the procedure against the possible benefit to the patient.
Catheterization is performed mainly by percutaneous approach, through femoral or
radial arteries, transseptal catheterization, or apical left ventricular puncture. While
it is technically possible to perform a catheterization in a freestanding facility, it is
still strongly advised to have it performed in the hospital. For the operating person-
nel, cardiac catheterization requires a thorough training, which includes study of
cardiac anatomy, physiology, and instruction about radiologic equipment, radiation
safety, and the proper use of contrast agents.
     During angiography, the X-ray scanner is used in fluoroscopy-mode so that a
continuous monitoring can be performed.140 This is realized by means of a BI
              Indicator Dilution Techniques in Cardiovascular Quantification               133

(Brightness Intensifier ) or flat panels made of TFTs (Thin-Film Transistors) for
real-time imaging. To permanently acquire a movie of the catheterized anatomic
districts, 35 mm cine-cameras have been utilized so far in every cardiac catheteriza-
tion facility although, more recently, these devices have been substituted by digital
encoders which operate with the “filmless” technology. Because of historical reasons,
this operating mode is still referred to as cineangiography. During a high resolution
image recording performed with the cineangiographic technique, the X-ray beam
intensity can be 100 times higher than it is during fluoroscopy, when the images are
acquired by a standard television chain; this has to be taken into high consideration
for the calculation of the X-ray dose absorbed by the patient and by the operator.
    A standard angiographic procedure requires the insertion of a catheter through
the femoral artery up to the heart. The injection of a radiopaque medium is per-
formed through this catheter. A radiopaque medium is a substance that absorbs
X-radiation (e.g. iodine). As a consequence, the blood mixed with the radiopaque
absorbs the X-rays and the blood vessels can be detected by the X-ray scanner.
Figure 20 shows an example of angiographic image.
    Early generations of intravascular contrast agents contained heavy metals (e.g.
bismuth and barium). However, all modern contrast agents are based only on iodine,
which has a high atomic number and, therefore, it is proved to be a very good
intravascular opacification agent. Inorganic iodine, however, cause toxic reactions.
Ratio 1.5 ionic compounds with a sodium concentration around that of blood, a
pH between 6.0 and 7.0, and a low concentration of calcium disodium ethylenedi-
aminetetraacetate (i.e. chelated with ethylenediaminetetraacetic acid) are the tra-
ditional high osmolar ionic contrast media (HOCM): Renografin (Bracco, Milan,
Italy), Angiovist (Berlex-Schering), Hypaque (Nycomed). Ratio 3 or low osmo-
larity ionic and non-ionic contrast media (LOCM), which reduce hypertonicity side
effects, have been introduced in the end of the 1980s. Hexabrix (Mallinckrodt,
Hazelwood, MO) is the only ionic LOCM while several examples of non-ionic LOCM
are available on the market. Both HOCM and LOCM have advantages and draw-
backs, however, LOCM have not yet completely substituted HOCM only because of
the cost of LOCM, still 2 to 3 times higher than that of HOCM.140 A new class of

Fig. 20. Example of X-ray coronary angiography with evident stenosis of the middle right coro-
nary artery (indicated by an arrow).
134                         M. Mischi, Z. Del Prete & H. H. M. Korsten

isomolar contrast medium (IOCM), which is a ratio 6 non-ionic dimmer, has been
recently made available on the market: Visipaque (Nycomed).
    The accuracy of X-ray analysis is improved by the employment of three-
dimensional imaging techniques. The technology that allows a three-dimensional
image reconstruction is the Computerized Axial Tomography (CAT). It is based on
the Radon transform (Johann Radon, 1917) of multiple X-ray projections.141,142
    Since an angiographic procedure takes about 30 minutes, the dose of X-rays
absorbed by the patient should be seriously considered. This dose is defined as
the energy that is absorbed per unit mass of material invested by X-rays. In the
International System it is measured in gray [Gy], 1 Gy = 1 Jkg−1 . In case of biological
tissues it is measured by the equivalent dose, expressed in sievert [Sv]. The equivalent
dose takes into account many factors that determine the biological interaction. The
limit dose for common people is 1.7 mSv/year, while the dose given by a fluoroscopy
is about 1 cGy 1 mSv.

5.2.3. Nuclear imaging
In the radionuclide angiography a small amount of radioisotopes, normally indi-
cated as radiopharmaceuticals, is peripherally injected. The radioactive decay of
radioisotopes leads to the emission of α and β particles as well as γ- and x-radiation.
Hence, the detection of the indicator is performed by radioactivity measurements. It
is performed by a scintillation camera (Anger, 1959), usually referred to as gamma-
camera. The gamma-camera is a photon counter (minimal photon energy 50 keV).i
It consists of a scintillator crystal (NaI or NaTl) that covers a matrix of photomul-
tiplier tubes. Only γ- or x-radiation can be detected with detectors that are outside
the body. The information detected and recorded by these scanners is analyzed and
processed to generate images of the target anatomical structures.
    The emission tomography systems are divided in two main groups, depending
on the type of radiation emitted by the adopted radiopharmaceutical.

• The Single Photon Emission Computed Tomography (SPECT) system makes use
  of routine single photon gamma emitters such as 99m Tc, 131 I, 123 I, 67 Ga, and
      Tl. It is generally designed to collect data from different angles.
• The Positron Emission Tomography (PET) system detects annihilation radiation
  from positron emitters such as 11 C, 13 N, 15 O, 18 F, and 68 Ga. It consists of two
  or more opposed detectors that permit the detection of the two 511 keV gamma
  photons that are emitted simultaneously in opposite directions by the annihilation
  process. Therefore, the line that intercepts the emission point can be determined.

i 1 eVis the energy acquired by an electron (1.602 × 10−19 C) when inserted in an electrical field of
1 V. Hence, 1 eV = 1.602×10−19 J. For instance, in case of X-radiation, we can calculate the energy
by the formula E = λ , with c equal to the light speed, λ equal to the radiation wave-length, and
  equal to the Planck’s constant (6.626 × 10−34 Js). The average wave-length of the X-radiation
                                          6.626·10−34 Js·3·108 ms−1
is 10−1 nm, resulting in an energy E =   10−10 m·1.602·10−19 eV−1 J
                                                                      = 12.5 keV.
             Indicator Dilution Techniques in Cardiovascular Quantification         135

    The most advanced radionuclide technique for dynamic image analysis is the
Multi-Gate imaging (MUGA, also known as ventriculogram), where the gamma-
camera takes more images triggered by the ECG signal. This technique permits
the geometrical estimation of EF as well as the application of the indicator dilution
theory for blood flow, perfusion, and volume measurements according to the models
described in Sec. 3.29,33,143–145 In fact, since the images represent the concentration
of the radiopharmaceuticals, it is always possible to place a ROI and record an IDC
based, for instance, on the average video-intensity in the ROI.
    Since radionuclides cannot be considered as inert, patients in special condition,
such as for instance women who are pregnant or breast-feeding, are not allowed to
undergo nuclear imaging.

5.2.4. Magnetic resonance imaging
Principles of magnetic resonance imaging

Magnetic Resonance Imaging (MRI) is an imaging technique used to produce high
quality images of the inside of the human body. To this end, the MRI scanner uses
high magnetic fields (several tesla). A detailed discussion of the MRI physics and
technology is beyond the purpose of this chapter. Here we provide a brief description
of the basic principles. A complete description of MRI can be found in Refs. 142,
    The MRI technique is based on the principles of Nuclear Magnetic Resonance
(NMR), a spectroscopic technique that allows the distinction between different
materials depending on the magnetic properties of their atoms. In particular, bio-
logical tissues can be analyzed depending on the hydrogen concentration.
    Hydrogen-1 (mass number = 1) is the most abundant element in the human
body in its association in water molecules. Due to an uneven atomic number, each
hydrogen nucleus (proton) shows an angular momentum (spin), which is due to
the nucleus rotation around its own axis. Since the proton is electrically charged
and rotates, it generates a magnetic field, which can be represented by a magnetic
dipole moment. Therefore, each proton can be considered as a small magnet with
separate poles as shown in Fig. 21. Its magnetic moment is parallel and proportional
to the spin by a physical constant, the gyromagnetic ratio γ, which is a property
characterized by quantum mechanics, and it is equal to 2.685 × 108 rad s−1 T−1 .
Thus, the relationship between the spin (angular momentum J ) and the magnetic
moment − is given as

                                     − = γ− .
                                     µ    J                                       (98)

    In an external magnetic field, the spinning protons tend to align themselves
in the direction of field. According to the quantum theory, only two orientations
of the nuclear magnetic moment relative to the field direction are possible. Both
136                     M. Mischi, Z. Del Prete & H. H. M. Korsten

                                Fig. 21.   Spinning nuclei.

orientations are at an angle to the external field and are labeled parallel and antipar-
allel. These two states correspond to two allowed energy levels. At equilibrium, the
number of spins at the lower energy level exceeds that at the higher level.
    According to classical mechanics, the magnetic moment experiences a torque
→ from the external magnetic field − 0 , which is equal to the rate of change of
−τ                                      B
its angular momentum J , and it is given by the equation of motion for isolated
                                           → →
                              − = ∂ J = − × − 0.
                              →τ            µ    B                                 (99)
Combining (98) and (99), we can derive
                               ∂−µ    → − →
                                   = γ− × B 0.
                                      µ                                          (100)
As a result, the magnetic moment precesses around the axis of the external magnetic
field (Fig. 22) at a particular frequency that is referred to as Larmor frequency and
is given as
                                    fL =      B0 .                               (101)

                         Fig. 22.   Magnetic moment precession.
             Indicator Dilution Techniques in Cardiovascular Quantification          137

This is the frequency at which the nuclei can receive the RF energy to change their
states and exhibit nuclear magnetic resonance.
    In a macroscopic sense, the sum of the spin vectors leads to a net magnetization
in the direction of the field, which is the vectorial sum of all the magnetic moments
of the nuclei in the considered object, and it is referred to as longitudinal mag-
netization. If an additional alternate field B 1 perpendicular to the z-axis with a
frequency equal to the Larmor frequency is applied, then the spin system absorbs
energy. The population at the upper energy level increases while that at the lower
level decreases. The alternate field B 1 is referred to as RF pulse and the following
process is referred to as resonant process. As a result, the net magnetization vector
is no longer in the direction of the external magnetic field. It moves away from
the z-direction and it is flipped of an angle — referred to as flip angle — equal to
θ = γB1 ∆t, where ∆t is the duration of the RF pulse.
    Once the RF pulse is finished, the net magnetization vector returns to its original
equilibrium (steady) state through a process that is referred to as relaxation. The
previously absorbed energy is emitted at the Larmor frequency and it is detected by
receiving coils perpendicular to the z axis (see Fig. 22). The MR signal (transient
response) after a RF pulse excitation is referred to as free induction decay (FID)
signal. The induced voltage V (t) in the receiving coil can be derived as follows,
applying the principle of reciprocity (102) and the Faraday law (103):
                                    − −
                                    →→           →
                                                 − →
                   Φ(t) =                                  →
                                    B ( r , t) · M (− , t)d− ,
                                                    r      r                      (102)

                              ∂Φ    ∂                →→
                                                     − −          →
                                                                  − →
                  V (t) = −      =−                                         →
                                                     B ( r , t) · M (− , t)d− .
                                                                     r      r     (103)
                              ∂t    ∂t         obj
    →                                                        →
    B is the magnetic field generated by the coil at location − and Φ is the magnetic
flux through the coil. The longitudinal (z direction) component varies much slower
than the transversal (xy-plane) component and can be neglected. As a result, the
voltage detected by the receiving coil can be approximated as
                                     → −
                                     −            ∂− → →
                  V (t) ∼ −
                        =                                       →
                                     B xy (→, t) · M xy (− , t)d− .
                                           r             r      r                 (104)
                               obj                ∂t
Magnetic relaxation
The relaxation expresses the recovery towards equilibrium of nuclear dipoles that
have been perturbed by RF excitations. The time-dependent behavior of the net
                      −                                             −
magnetization vector M in the presence of an applied magnetic field B 1 is described
by the Bloch equation as given in (105),
               ∂M       → −
                        −    → Mx i + My j       (Mz − M0 )k
                     = γM × B −                −              ,               (105)
                ∂t                     T2             T1
where M0 is the equilibrium Mz magnetization in the presence of B 0 only. T1 and
T2 are, respectively, the longitudinal and transverse relaxation times. There are
relaxation parameters unique for each tissue. The solution of (105) is given as
                              Mx (t) = M0 e− T2 cos(γB0 t),                       (106)
138                     M. Mischi, Z. Del Prete & H. H. M. Korsten

                            My (t) = M0 e− T2 sin(γB0 t),
                            Mz (t) = M0 1 − e      1   ,
which represents the time dependent behavior of the magnetization vector.
   Representing the xy-plane by means of the complex plane with i = −1, the
transverse magnetization can be expressed as
       −                                           −t          t
       M xy (t) = Mxy (cos(γB0 t) − i sin(γB0 t))e T2 = M0 e− T2 e−i(γB0 t) .   (107)
From the definition of Mz in (106) it is clear that the longitudinal or spin-lattice
relaxation time T1 refers to the time that is necessary for the tissue magnetization
to return to its steady state, i.e. parallel to the external magnetic field, after the
RF pulse.
    Since an exchange of energy between protons and environment takes place, the T1
relaxation time depends on the nature of the surrounding molecules. For instance,
the magnetization associated with lipids relaxes faster than that associated with
much smaller, such as water, or much larger molecules, such as proteins.
    The second relaxivity property of tissue is the transverse or spin-spin relaxation,
referred to as T2 relaxation. In this relaxation process the magnetic moments of
the spins turn out of phase as a result of their mutual interaction. Each of the
spins experiences a slightly different magnetic field and rotates at its own Larmor
frequency, leading to loss of spin phase coherence (dephasing). As a result, the
transverse magnetization Mxy decays. The longer the elapsed time, the larger the
phase difference and the transverse magnetization decay.
    Unlike the T1 relaxation, no energy is transferred from the nuclei to the environ-
ment during the T2 relaxation. Each tissue has a characteristic T2 relaxation time
that is referred to as the decay of the transverse magnetization. However, the actual
rate of signal decay is faster than the prediction based on T2 . In fact, the actual
observed transverse relaxation time, referred to as T2 *, is affected by local magnetic
field inhomogeneities, which cause the precessional rates of the individual spins to
differ from each other and dephase.
    An important parameter in MRI is the spin density ρ. The spin density is pro-
portional to the effective number of hydrogen nuclei per unit volume contributing
to the MR signal. Thus, the amplitude of M0 in (106) is proportional to ρ. The
density ρ can be used together with T1 and T2 to distinguish different tissues.
    In general, the T2 * relaxation does not provide reliable information on the spe-
cific tissue. In order to obtain information related to the T2 relaxation time of tissue,
a 180o RF pulse is used to invert the dephasing process and rephase the spins at
a later moment as an echo. When the MR signal decay is evaluated for refocused
spins, then the signal depends on T2 rather than T2 *. An example of FID refocusing
is shown in Fig. 23. This is a typical Spin-Echo (SE) sequence, which begins with a
90o RF pulse that initiates the relaxation precession and is followed after TE/2 (TE
is the echo time) by a rephasing pulse. This sequence can be repeated several times
in order to obtain all the information that is necessary to allow reconstruction of
             Indicator Dilution Techniques in Cardiovascular Quantification       139

                        Fig. 23.   T2 relaxation and echo formation.

an image from a slice of tissue inside the body. Spatial localization is fundamental
for the reconstruction of an image.
Spatial localization
The spatial localization of the signal contribution per voxel is implemented by means
                        − →                                               −
of a spatial gradients G (− ). These are added to the magnetic field B 0 . After a
number of approximations, the expression of the MR signal S(t) detected by the
receiving coil from an object with spin density ρ(− ) can be derived from (104) and
(107) as given in (108) :
                                                     − −
                          S(t) ∝             →                  →
                                           ρ(− )e−iγ( G · r )t d− .
                                             r                  r              (108)

After demodulation (i.e. removal of the carrier signal e−iγB0 t by a mixer) (108) can
be written as:
                                                   − −
                        S(t) ∝             →                  →
                                         ρ(− )e−iγ( G · r )t d− .
                                           r                  r                (109)
If we define a real vector k as
                                     → γt→
                                     −    −
                                     k =    ,                                  (110)
then (109) can be written as
                          −                         − −
                        S( k ) ∝            →               →
                                          ρ(− )e−i2π K · r d− .
                                            r               r                  (111)

     Expression (111) corresponds to a three-dimensional Fourier transformation of
the space − into the space k , which is commonly defined as k-space. From (111)
it is clear that the reconstruction of an image ρ(− ) consists of the Fourier inverse
transformation of S( k ) (see Fig. 24). Notice that both the k-space and the output
image are discrete.
     Since the origins of MRI, extensive developments have been implemented to
reduce the time required to fill the samples of the k-space. The selection of a slice
140                        M. Mischi, Z. Del Prete & H. H. M. Korsten

Fig. 24. LV short axis view (courtesy of dr. vd Bosch, Catharina Hospital Eindhoven, The Nether-
lands) in k-space and in real space after Fourier inverse transformation.

perpendicular to the z direction is usually performed by choosing the frequency
of the RF pulse. Only the slice of which the Larmor frequency γ(B0 + Gz z)/2π
corresponds to the frequency of the RF pulse is excited. Once a slice is selected, the
space (kx , ky ) must be filled.
    Typically, the bi-dimensional k-space is filled line by line. This is accomplished
by the combination of phase and frequency encoding. A phase encoding gradient
∆Gy is activated for a time Tpe before the signal readout in order to select one line
of the k-space, while a frequency gradient Gx is activated during the readout to span
the selected line of the k-space. For ∆t equal to the time sampling period during
readout, the discrete k-space in Fig. 24 shows the following sampling distances:
                                  ∆kx =      Gx ∆t,
                                         γ                                      (112)
                                ∆ky =       ∆Gy Tpe .
    As for most discrete systems, the Nyquist threshold must be fulfilled in order
to avoid aliasing. For an image size (Wx × Wy ), the Nyquist theorem imposes the
following limits:
                                      ∆t =        ,
                                           γGx Wx
                                     ∆Gy =          .
    Apart from the Cartesian structure of the k-space given in Fig. 24, alternative
strategies to fill the k-space exist. For instance, one can make use of polar coor-
dinates, resulting in a polar sampling of the k-space made of concentric circles or
a spiral trajectory. When polar coordinates are used, the image reconstruction is
based on the Inverse Radon Transform.141,142
    As an example, the description of a SE sequence is provided (see Fig. 25). A
SE sequence starts with the slice selection gradient, which spreads out the Larmor
frequency over a broad range, so that the frequencies contained in the RF pulse affect
only one slice perpendicular to the z direction. Then the phase-encoding gradient
             Indicator Dilution Techniques in Cardiovascular Quantification        141

                           Fig. 25.   Spin echo timing sequence.

Gy is pulsed briefly (time Tpe) after the first RF excitation pulse has been turned
off, making the precession frequency depend on spatial position along the phase-
encoding direction (y). Simultaneously, a frequency encoding gradient Gx is turned
on to prepare the initial position to span the k-space line. After time TE/2 a 180◦
pulse is transmitted, so that an echo centered at time TE is generated and recorded
during the readout time with the frequency encoding gradient Gx turned on.
   The following pulse is sent after time TR (repetition time) since the initial 90◦
RF pulse. Each sequence permits the registration of one line of the k-space. The
MR signal SSE measured by a SE sequence depends on the settings TR and TE as
well as on the tissue properties T1 , T2 , and ρ as given in (114):
                                      ( 1 T E−T R)        −T R       −T E
                  SSE = ρ 1 − 2e            T1
                                                     +e    T1
                                                                 e    T2
                                                                            .   (114)

     Expression (114) can be derived from (105) in case of a SE sequence. It is clear
that the dependency of the image contrast on T1 or T2 can be varied by choice of
the scan parameters TE and TR. In that way, the contrast in the resulting image
can be made depend strongly on either T1 or T2 , and the image contrast is referred
to as T1 or T2 weighted, respectively.
     The SE sequence is one of the most common sequences that have been introduced
in MRI. Another common technique is to leave out the RF refocusing pulse. This
method is referred to as Gradient Echo (GE) or Field Echo (FE).146,148–150 An
image is still generated by a number of excitation pulses that is equal to the number
of lines. Parameters like the TR and TE can be used to control the image contrast.151
142                      M. Mischi, Z. Del Prete & H. H. M. Korsten

The characteristics of the image and its relation with different tissues can also be
controlled by use of pre-pulses (magnetic preparation), like for instance in Inversion
Recovery (IR) pulse sequences. In general, even though the use of smaller flip angles
(< 90◦ ) can increase the time resolution of GE scans, these techniques are too slow
for real-time imaging of moving structures as well as for perfusion imaging.
    Especially for rapid processes, the need for a fast scanning technique becomes a
fundamental requirement. Recent technical developments have increased the time
resolution, which is now approaching that of X-ray CT scan and ultrasound echog-
raphy. These improvements mainly consist of advanced strategies to fill the k-space
with a reduced number of excitation pulses. Developments in this direction of SE and
GE techniques result in the Fast or Turbo SE (FSE or TSE) and in the Echo Pla-
nar Imaging (EPI) techniques.146,150 By EPI, a single excitation pulse can lead to
the reconstruction of the entire image (single shot technique). Combinations of EPI
and FSE are also used and referred to as Gradient Echo And Spin Echo (GRASE)
or Turbo Gradient Spin Echo (TGSE).150,152,153 Especially for GE techniques, the
scan time can be reduced by using short flip angles and TR. These techniques are
often referred to as Fast Field Echo (FFE).146,150,153 In this family of techniques,
a surprisingly large number of members exists and the associated names differ per
    The signal is usually read during magnetic steady state, i.e. when the magne-
tization has reached a stable value depending on the flip angle and the sequence
parameters TR and TE. Sometimes the readout is performed prior steady state to
reduce the scanning time. This technique is referred to as Turbo Field Echo (TFE).
In this case, the degree of T1 -weighing can be increased by use of inversion or sat-
uration pre-pulses. A survey of MRI nomenclature for different manufacturers is
presented in Ref. 154.
    Alternative techniques to boost the time resolution of MRI based on smart
k-space construction strategies consist of exploiting the symmetry of the Fourier
transform of a real signal.150,153 An example is the Half Fourier Single-Shot Turbo
Spin Echo (HASTE) pulse sequence.155 As already mentioned, the k-space can be
also filled by a radial or spiral trajectory.150,153 This permits to fill the center of the
k-space in the initial part of the readout, resulting in better contrast and reduced
sensitivity to motion. In fact, the external part of the Fourier spectrum contains
limited information, and it could be filled by zeros (zero padding) to further decrease
the scan time.
    Recent developments to increase speed of image production consist of the com-
bined use of arrays of receiving coils and frequency-temporal interpolation of the
k-space. These go by the names of SiMultaneous Acquisition of Spatial Harmonics
(SMASH) or SENSitivity Encoding (SENSE) method.156,157 In sequential scan-
ning of the dynamics of events, in addition to all techniques already mentioned,
spatio-temporal k-space interpolation can often be used. A modern version of this
technique is the k-t BLAST (or k-t SENSE if combined with the SENSE technique)
             Indicator Dilution Techniques in Cardiovascular Quantification           143

method, where a least squares approach is used to treat aliasing problems due to
simultaneous down-sampling in frequency and time.158
MRI contrast agents
MRI contrast agents often are molecular constructs containing paramagnetic atoms
such as gadolinium (Gd3+ ), iron (Fe2+ , Fe3+ ), and manganese (Mn2+ ) ions. Their
influence on the MR signal is caused by the induced change in the relaxation times
of nearby protons. By proper choice of the molecular structure, this effect is maxi-
mized. Since the contrast-agent enhancement is based on the alteration of the two
relaxivity parameters, it can be categorized according to the shortening it produces
on either T1 or T2 . Shortening of T1 is the result of dipole-dipole interactions, which
occur when the frequency of time-varying magnetic fields produced by contrast
agents (as a result of molecular rotation and tumbling) is similar to the resonance
(Larmor) frequency of hydrogen nuclei. Increases in local magnetic field inhomo-
geneities enhance dephasing of non-stationary nuclei, resulting in decreased T2 or
T2 *. A contrast agent that predominantly affects T1 relaxation is referred to as a
positive relaxation agent because the enhanced shortening of T1 relaxation results
in increased signal intensity on a T1 -weighted image. Instead, a contrast agent that
predominantly affects T2 or T2 * relaxation is referred to as a negative relaxation
agent because the reduced T2 * results in decreased signal intensity on a T2 -weighted
    The influence on the T1 and T2 * values of tissue can be formalized as given in
(115) and (116), respectively146,160–163 :
                             1                  1
                                (observed) =       + R1 C,                         (115)
                             T1                 T1
                             1                  1
                                (observed) =       + R2 C.                         (116)
                             T2                 T2
Here R is the relaxivity of the contrast agent and C is its concentration.
    At low concentration, the dominant effect often is the reduction of T1 so that
T1 -weighted imaging is preferred in most contrast enhanced MRI applications. Espe-
cially gadolinium is currently used in several MRI studies. Its toxicity is reduced by
chelation to molecules such as diethylene triamine pentaacetic acid (DPTA).

Cardiovascular quantification
Applications of contrast MRI in cardiovascular diagnostics are mainly qualitative.
Contrast agent infusion is performed either to delineate the cardiovascular sys-
tem (major arteries and cardiac lumina) or to assess the level of perfusion of the
myocardium. The use of contrast agents improves the detection of stenoses and
aneurisms in the vessels as well as abnormal structures in the cardiac cavities. The
assessment of regional perfusion in the myocardium is an important procedure for
the detection of infarcted areas, i.e. areas in which the tissue viability is reduced due
to lack of oxygenation.164,165 In general, perfusion defects are related to stenosis
144                       M. Mischi, Z. Del Prete & H. H. M. Korsten

or strokes in the coronary arteries, i.e. the myocardial perfusion arteries. Figure 26
shows a cardiac opacification by gadolinium infusion.
    As a consequence of (115) and (116), several recent studies confirm that an
approximately linear relationship between MR signal and gadolinium concentration
can be found for low concentrations.161,166,167 This finding has opened the way
to novel perfusion imaging applications. Therefore, although the use of contrast
MRI mainly aims at qualitative diagnostics, some studies are available that provide
more quantitative measurements of blood perfusion and volumes. Perfusion studies
in the myocardium, brain, kidney, and liver have been reported.167–170 Despite
a linear relationship between contrast concentration and MR signal, an absolute
relationship is difficult to establish because it is flow dependent.171
    Due to their small size, gadolinium molecules can diffuse through the capillary
wall into the interstitial space. Several multi-compartment models (see also Sec. 3.1)
have been tested to describe this diffusion process.167,168,172–174 For some cardio-
vascular studies, the use of blood pool contrast is more suitable. Novel opportunities
for perfusion can be provided by novel blood pool agents under development, such
as Gd labeled albumin (e.g. MS-325 , Epix Pharmaceutical, Cambridge, MA) and
Gd-DPTA labelled dextran (e.g., Gadomer , Schering, Berlin, Germany), whose
characteristic is a longer intravascular life time (slow clearance).167,175–177 Extrava-
sations are avoided by albumin binding or larger molecular size for the first and the
second class of contrasts, respectively.
    An interesting novel application of quantitative perfusion imaging relates to
the assessment of the blood flow through the myocardium. To this end, a bolus
of contrast agent is injected in a peripheral vein. The MR signal due to the bolus
passage is registered versus time in the LV or the ascending aorta (input IDC) and
in several regions of the myocardium (output IDC). The impulse response of the
dilution system between the input and output IDC is estimated by a deconvolu-
tion technique (see also the end of Sec. 5.2.1).178,179 The dilution system is usu-
ally assumed to be represented by a mono-compartment model and the estimated
impulse response is interpolated by an exponential decay (see Sec. 3.1). The time

Fig. 26. Four chamber view MRI before (left side) and after (right side) gadolinium infusion.
Courtesy of dr. vd Bosch, Catharina Hospital Eindhoven, The Netherlands.
             Indicator Dilution Techniques in Cardiovascular Quantification        145

constant of the fitted model, which corresponds to the MTT of the contrast between
the detection sites, is used to estimate flow and perfusion. If calculated pixel by
pixel, a new image that corresponds to perfusion timing (and therefore flow) can be

6. Conclusions and Outlook
This chapter concerns the quantification of cardiovascular parameters by means of
indicator dilution methods. The interpretation of an indicator dilution curve permits
to characterize the dilution system where the indicator has passed. The mathemat-
ical background that is necessary for the interpretation of an indicator dilution
curve is provided and tailored to the estimation of the cardiovascular parameters of
interest. The main focus is on the quantification of cardiac output, blood volumes
in the pulmonary circulation, ejection fraction of the ventricles, and perfusion of
the myocardium. The clinical methods for the measurement of the indicator and
the assessment of the clinical parameters of interest are introduced. A distinction is
made between invasive methods that require catheterization and minimally invasive
methods that use medical imaging techniques.
    The invasive methods are usually employed in anesthesiology, both in the oper-
ating rooms and in the intensive care units. The need for catheterization does not
represent an obstacle for these applications and the derived parameters are consid-
ered important for monitoring the condition of the patient. Instead, the indicator
imaging methods, with the exception of the X-ray angiography, are minimally inva-
sive as only the injection of the indicator in a peripheral vein is necessary.
    Large part of the chapter is dedicated to contrast agent imaging techniques and,
in particular, ultrasound and magnetic resonance imaging. The use of these mini-
mally invasive methods for the measurement of indicator dilutions have two major
advantages. The first one concerns the applicability of these methods without the
need for hospitalization. This makes the methods available to cardiologists, adding
diagnostic possibilities to standard cardiologic investigations made for instance by
ultrasound or magnetic resonance imaging. The second one, still under development,
is related to the possibility of measuring multiple indicator dilution curves, so that
several central dilution systems can be characterized by means of system identifica-
tion methods (deconvolution) without catheterization and central injection of the
indicator. This adds diagnostic potential to these methods and represents a major
advantage over invasive methods.
    Indicator dilution methods are under continuous development due to the intro-
duction of new types of indicators. Novel technologies that extend the use of indi-
cators from purely diagnostic applications to therapeutic ones are also growing.
A clear example is given by the use of ultrasound contrast agents for myocardial
perfusion. Apart from the assessment of perfusion and viability of the myocardial
tissue, contrast agents can also be loaded with adenosine and used to locally release
it for cardiac reflow therapy.
146                      M. Mischi, Z. Del Prete & H. H. M. Korsten

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                                       CHAPTER 5

                         HEART SOUND

                         ∗,†,‡ University of Applied Sciences Jena,

       Department of Medical Engineering and Biotechnology, Carl-Zeiss-Promenade 2,
                                  Jena D-07745, Germany

                           ‡,§ Medtrans GmbH, Tatzendpromenade 2
                                    Jena D-07745, Germany

    Diseases of the heart have become the Number One cause of death in the industrialized
    nations of the world. Every heart disease affects the biomechanics of the heart in a direct
    or indirect way. These effects can primarily be analyzed by the signals of the heart sounds
    and cardiac murmurs using techniques such as auscultation and phonocardiography. But
    these methods are very sophisticated and require a high degree of specialization. During
    the last years electronic stethoscopes and commercial PC techniques have improved
    essentially so an automatic analysis of heart sound has become a potential supporting
    tool for physicians in particular as a screening method for heart diseases by the general
    practitioner. This paper introduces a new automatic system to diagnose heart valve
    diseases based on time and frequency analyzing methods and feature extraction. It also
    describes an multivariate approach for an enhanced risk stratification in patients with
    heart failure considering the time interval between the ECG signal, especially the R-wave,
    and the first heart sound.
         In conclusion we could demonstrate, that analysis of the heart sound is a suitable
    method to evaluate the state of the heart and to detect changes on the biomechanics at
    an early stage.

    Keywords: Heart sound; heart valves; heart failure; frequency analysis; signal processing.

1. Introduction
The incidence of heart diseases has increased dramatically during the last few
decades, especially in the industrialized nations of the world. According to mor-
tality statistics, heart diseases have become the Number One cause of death in
these countries. For example, 27.3% of all deaths in the USA in 2004 were due to
heart diseases (NCHS).1 The American Heart Association estimates for 2006 that
the direct and indirect cost of cardiovascular heart diseases amount to more than
$142.5 billion a year in the US alone.2
    Every heart disease affects the biomechanics of the heart. Some of these diseases
affect the biomechanical system directly (heart defects, diseases of the myocardium,
heart valve disease and destruction of heart muscular cells by myocardial infarction),
others have an indirect effect on it because of insufficient supply of the myocardial
tissue or pathological changes of the generation or conduction of electrical stimuli
(coronary heart disease, cardiac arrhythmia).

158                                   A. Voss et al.

     Heart valve diseases, congenital heart diseases and myocardial diseases cause
typical cardiac murmurs. Listening to the sound of the murmur and its different
characteristics, the trained medical doctor is able to identify not only the type of
a heart defect but often also the severity. Such characteristics include, for example,
the frequency of the murmur (high or low frequency) or the way in which a murmur
develops in a course of a heart beat, e.g. whether it starts faintly and becomes louder
or it starts loud and then decreases. The relation of the cardiac murmur with the
preceding and the following heart sounds is also important.
     Conventional stethoscopes in the market transmit the heart sound and murmur
to the human ear via a diaphragm and airborne sound conduction.
     Auscultation is a purely manual method without any possibility of obtaining an
objective measurement and analysis. This method depends entirely on the routine
and experience of the examiner. Not infrequently, physicians of advanced age suffer
from impaired hearing or are even hard of hearing. They can no longer sufficiently
detect the very faint heart sounds and even more faint cardiac murmurs.
     For this reasons, efforts to develop electronic stethoscopes have been reinforced
especially during the last five years. These stethoscopes record sound signals, for
example, by small encapsulated microphones or vibration sensors and amplify them.
Then the signals are output by small encapsulated headphones in different filter
steps. In most cases, these devices do not improve the quality of the signals signifi-
cantly, they only amplify them (and along with the signals also the noise).
     At present, there are only a few electronic stethoscopes which also permit record-
ing the heart sound data in a PC or similar device. By recording the heart sound
in a PC, phonocardiography is generally possible and the heart sounds and car-
diac murmurs can be visualized. With the patient in supine position, the physician
places a stethoscope (formerly a heart sound microphone, today an electronic stetho-
scope) at several points of the thorax (similar to established auscultation sites). The
microphone/stethoscope is connected to the phonocardiograph (formerly) or the
PC (today) which records the heart sound and displays it graphically. Because the
microphone records every sound, including noise, the examination must take place
in a low-noise environment. Besides, there should be neither breathing nor speaking
for the time of the examination. The traditional phonocardiography is extremely
demanding and as a rule is or was applied only by specialized cardiologists.
     Because echocardiography, as an imaging method, is a more reliable method
for identifying congenital or acquired heart defects, phonocardiography is rarely
used today. As in the industrialized countries of the world, ultrasonic examination
equipment for echocardiographic examination has become standard for cardiologists
in hospitals as well as in private practices, these phonocardiographic methods have
been replaced, by and large. Phonocardiography was a purely visualizing method for
manual diagnosis, without any automatic measurement or interpretation. In view
of the fact that phonocardiography was pushed in the background, there is very
little recent research in the area of heart sound analysis anywhere in the world.
Automatic interpretation algorithms (similar to automatic ECG interpretation) are
not known in daily medicine.
                    Analyzing Cardiac Biomechanics by Heart Sound                    159

    The approaches discussed below are not intended to revive phonocardiography
for the cardiologist; their objective is to transfer a modified form of phonocardiogra-
phy as a generalized method of examination to the level of the general practitioner
(GP). The interpretation of difficult to understand heart sound tracings can be
automated and assist the family doctor in diagnosing heart diseases.
    Heart sounds are no longer recorded by a phonocardiographic sensor but a newly
developed electronic stethoscope with low-noise high amplification of the signal and
minimization of noise signals and external disturbances e.g. contact pressure, trem-
ble or temperature influences. Electrodes integrated in the head of the stethoscope
provide an ECG recording at the same time. The standardized digitization of ECG
and heart sound, the control of electronic auscultation program and PC control and
data transmission are handled by a small manual control.
    The newly developed methods of heart sound analysis in time and frequency
domains will be discussed in detail. The gold standard for the evaluation and classi-
fication, especially of subjects and patients with heart valve disease, were parame-
ters from echocardiographic reference examinations. It is a main objective to obtain
information via heart sound analysis comparable to that obtained by echocardio-
graphic examination for early detection of pathological changes of biomechanical
processes as heart valve diseases.
    To reduce the number of surgeries on heart valves it is necessary to identify
the defect as early as possible. If a correct diagnosis is available in time, surgical
operation could be avoided by the reduction of risk factors by the patient and/or
treatment with drugs which might still be effective at that stage or at least could
be delayed significantly. There may be cases in which a heart valve can be corrected
surgically instead of replacing it, which would avoid other consequential factors
significantly impairing the quality of a patient’s life, such as life-long anticoagulation
in case of implanted artificial mechanical heart valve.
    A patient with complaints will at first consult his family doctor or nearest GP.
The further efficient medical treatment depends on the family doctor’s or GP’s abil-
ity of assessing the patient’s complaints and refer him or her to the right specialist
in time. Hence, the GP or family doctor should already be in a position to diag-
nose and interpret pathological biomechanical changes, e.g. heart valve defects, by
a simple and reliable method.

2. Background
2.1. The biomechanics of the healthy heart
The primary function of the heart is to pump blood through blood vessels to the
body’s cells. To make the supply as efficient and effective as possible, high-precision
valve systems are needed. This function is served in the body especially by the heart
valves. In fact, the heart valves control the blood flow and its direction in the heart
and are opened and closed by the pressures generated in the heart.3
160                                       A. Voss et al.

Fig. 1. The heart and the heart valves (T Tricuspidal valve, P Pulmonary valve, M Mitral valve,
A Aortic valve).43

     The heart consists of four chambers, the right and left atrium and the right and
left ventricle. The atrium and ventricle of each side of the heart are separated by
heart valves (atrioventricular valves). These are the tricuspid valve in the right half
of the heart and the mitral valve in the left half (Fig. 1). Two other valves located
at the two points where blood exits the heart: these are the pulmonary valve (at
the exit of the right ventricle into the pulmonary artery) and the aortic valve (exit
of the left ventricle into the aorta). Both these valves are also known as semilunar
valves (from Latin semi = half, luna = moon).3,4
     A cardiac cycle is divided in two stages, the systole and the diastole. The systole
is the contraction phase of the heart, the diastole is the phase during which the
heart relaxes.
     The systole is again divided into two phases. During the tension phase (approx.
50 ms), the contraction of the myocardium causes the intraventricular pressure to
rise quickly and the atrioventricular valves (AV valves) to close. At this time, the
pulmonary and aortic valves are also closed because the pressure in the ventricle is
still below the pressure in the outgoing blood vessels (Fig. 2; phase I). The tension
of the ventricular muscles around the incompressible blood volume now leads to a
pressure increase. If the pressure in the ventricle exceeds the pressure in the vessels,
the semilunar valves open and the ejection phase (approx. 210 ms) begins (Fig. 2;
Phase II). The pulmonary valve opens at a pressure of about 10 mmHg, whereas a
much higher pressure of about 80 mmHg is needed for opening the aortic valve. The
reason why the pressure in the right ventricle is much lower than in the left is the
                    Analyzing Cardiac Biomechanics by Heart Sound                    161

very much weaker resistance of the vessels in the pulmonary circuit. The ejected
blood volume is the same and is about 70 ml at rest. This volume is referred to as
ejection fraction and amounts to approx. 50% of the blood volume in the ventricle.
In this phase, the atria also fill with new blood from the large body veins.
    Following the systole with tension and ejection phase is the diastole, which is
also divided in two phases. The relaxation phase (approx. 60 ms) begins when the
ventricle pressure drops below the arterial pressure. This causes the pulmonary and
aortic valves to close so that all heart valves are closed at that time (Fig. 2; phase
III). The pressure in the ventricle drops very quickly to below the pressure level in
the atria. The AV valves open and the blood volume in the atria can flow in the
ventricles. This phase is the filling phase (approx. 500 ms) (Fig. 2; phase IV). If the
needed blood volume has flown in the ventricles, the next tension phase begins.4,5
    Figure 2 illustrates the time relationship between ECG, heart sound and the
corresponding mechanical action of the heart during the four phases of cardiac

2.2. Heart sounds and cardiac murmur
Heart sounds and cardiac murmurs6,7 can be identified during a heart action (Fig. 3).
Heart sounds include vibrations up to 0.1 s caused by the vibration of structures
capable of oscillation. Cardiac murmurs are vibrations of longer than 0.1 s duration
and are generated by turbulent flow of blood within the heart. Murmurs are also
transmitted to structures capable of vibration.

2.2.1. Heart sounds
During a cardiac cycle, a healthy heart produces two heart sounds which are rather
intensive compared to the other heart sounds. These heart sounds differ in amplitude
and frequency. The first heart sound is associated with the closing of the AV valves,
i.e. the tricuspidal and mitral valves, at the beginning of ventricular contraction, or
systole and ejection of the ventricle volume. The origin of the second heart sound is
the closing of the semilunar valves, i.e. the aortic and pulmonary valves, at the end
of ventricular systole. As the right/left ventricle empties, its pressure falls below the
pressure in the aorta/pulmonary artery, and the respective valve closes. The first heart sound
The first heart sound begins about 0.01 to 0.03 s after the Q wave of the elec-
trocardiogram (electromechanical latency). It can be divided into three sections
with different frequencies and physiological origins. The pre-segment with lower
frequencies of up to about 50 Hz is caused by the first movements of the ventri-
cles. The main-segment starts approximately 0.06 s after the beginning of the QRS
complex and comprises frequencies between 60 and 100 Hz. It is produced by the
sudden tension of the cardiac wall after closure of the AV valves, when the ventricle
162                                     A. Voss et al.

Fig. 2. ECG, heart sound and heart mechanics during a cardiac cycle. Pressures and volumes
and valve states during the different action phases of the heart.

contracts around the non-compressible blood volume and due to this starts oscillat-
ing, including the valve mechanism. The post-segment with frequencies up to about
30 Hz already marks the beginning of the ejection phase. The semilunar valves have
opened under the ventricular pressures and the ventricle volume flows into the aorta
                       Analyzing Cardiac Biomechanics by Heart Sound                      163

                         Fig. 3.   Heart sound, heart murmur and ECG [5].

and the pulmonary artery, respectively, which also start oscillating due to the pres-
sure rise in the vessels. The second heart sound
The second heart sound comprises higher frequencies of up to 150 Hz. It is of shorter
duration than the first heart sound and of higher pitch. It is generated by pressure
changes and vibration of valves and contiguous structures induced by motion of the
aortic and pulmonic leaflets toward their respective ventricles. The second heart
sound can be divided into an aortal and a pulmonary section, with the aortal section
generally preceding the pulmonary section. The third and fourth heart sounds
Occasionally, a third heart sound occurs during diastolic rapid inflow of the blood
into the ventricle. This can be heard only in children or young adults due to their
more favorable conditions of sound conduction. The third heart sound is low in
frequency and intensity. A fourth heart sound can occasionally be noted between
the P wave and the Q wave in the ECG. This heart sound is caused by the ejection
phase of the atrium just before the first heart sound. Under normal auscultation
conditions,a this sound is not audible.
    In addition to the third and fourth heart sounds, further extra sounds, such as
the opening sound of the mitral valve or tricuspidal valve, may occur. Extra sounds
are also caused by artificial heart valves.

a Auscultation   — Listening, mostly with a stethoscope, to the murmurs and sounds produced by
the body.8
164                                   A. Voss et al.

   The intensity of the heart sounds depends on a multitude of cardiac and extrac-
ardiac factors, such as:
• the intensity of the heart action or the force and speed of contraction of the heart
• the oscillation capability of the heart valve apparatus, which can be impaired by
  calcification or scarring,
• the ventricular mass; the larger the ventricular mass, the lower the frequency and
  amplitudes of the oscillations,
• the blood viscosity.

2.2.2. Cardiac murmurs
Cardiac murmurs are caused most frequently by blood flow turbulences. In cases
where the flow velocity of fluid within a pipe exceeds a certain value, turbulence
develops and energy is dissipated generating audible vibrations. The determining
factors in this process are the viscosity of the fluid, the lumen and the flow rate.
    Practically, the sound level of the murmur depends on the rate of the blood
flow, the viscosity of the blood, the oscillation capability of the cardiac wall and
the valve apparatus and the big vessels leading away from the heart, the diameter
of the opened valve and the width of the vessels downstream the valves. Thus, the
sound level of the murmur is not necessarily an indicator of the severity of a valve

2.3. The pathological heart
Basically, there are five different groups of heart diseases:
(1)   Arrhythmia
(2)   Coronary heart diseases
(3)   Myocardial diseases
(4)   Heart defects
(5)   Valve diseases
These five groups of heart diseases can have significant effects on the biomechanical
function of the heart and can be diagnosed by medical specialists/cardiologists using
the usual cardiological diagnostic methods, such as heart catheter (HC), echocar-
diography (Echo), 12-channel electrocardiography (ECG), ergometry (ERGO) and
long term Holter ECG (LECG). Opposed to these is the very exact (but expensive)
method of computer tomography (CT). Phonocardiography (Phono) can take over
part of the diagnostic action (Table 1).

2.3.1. Cardiac arrhythmias
Cardiac arrhythmias are abnormalities in rate, regularity or sequence of cardiac
activation. In a healthy person, the cardiac rhythm is generated by the sinus node
                   Analyzing Cardiac Biomechanics by Heart Sound                      165

                     Table 1.   Methods for diagnosing heart diseases.

                                    ECG        ECG        Echo    HC     CT   Phono
                                   +LECG      +ERGO
       Arrhythmia                     x           x
       (blocks, tachycardia,
       Coronary heart disease                     x                x     x
       (ischemia, infarction)
       Myocardial diseases                                  x      x            x
       Heart defect                                         x      x            x
       Atrial and ventricular
       septal disease
       Valve diseases                                       x      x            x
       (insufficiency, stenosis)

generating electrical impulses (about 60–100 beats per minute) which are conducted
via the AV node and along specific pathways to the ventricles, where they cause
ventricular contraction. Arrhythmias are caused by malfunctions of the heart beat
generation centers or the electrical conduction system.
   There are different types of arrhythmias:

• Tachycardia: heart rate that is faster than the normal range (from 100 up to 200
  heartbeats per minute). This reduces the pumping efficiency and increases the
  oxygen consumption of the heart.
• Bradycardia: heart rate that is lower than the normal range (less than 60 heart-
  beats per minute) at rest (threshold). As a consequence of this, the body is not
  supplied with enough oxygen and nutrients. The threshold heartbeat is variable;
  for example, the heart rate at rest can be substantially lower in trained sportsmen.
• Atrial fibrillation or atrial flutter: the electrical signals that coordinate the muscle
  of the upper atria of the heart become rapid and disorganized, typically causing
  the atria to beat faster than 300 beats per minute. The myocardium has insuffi-
  cient time for coordinated blood filling of the ventricles. This impairs the pumping
  performance and leads to a reduced blood flow to the body.
• Ventricular fibrillation: The frequency of the ventricle contractions rises up to 300
  beats per minute. This prevents further pumping of the blood. This state is also
  known as functional cardiac arrest.
• Heart block: This is a delay or inhibition of the pathway of the stimulus. The
  following blocks are known: sinoatrial (SA) block (block between sinus node and
  atrium), atrioventricular (AV) block (block between atrium and ventricle), and
  bundle-branch block (block in ventricular pathway).
• Premature beats: Sudden heartbeats outside the normal baseline rhythm (if orig-
  inating in the ventricle: ventricular premature beats; if originating in the atrium:
  premature atrial contractions).
166                                   A. Voss et al.

2.3.2. Coronary heart disease
A coronary heart disease (CHD) is a constriction or occlusion (narrowing) of one or
several coronary arteries. As a consequence the blood supply of the heart muscle cells
is reduced. The most frequent cause of CHD is an arteriosclerosis of the coronary
arteries. Fat deposits on the walls of the vessels, which together with calcium form
so called plaques. The vessel loses the ability to expand and constricts gradually. If
a vessel is occluded, the person suffers a myocardial infarction. The risk of coronary
heart disease is increased by having a family history of coronary heart disease before
age 50, older age, stress, smoking, high blood pressure, high cholesterol, diabetes,
lack of exercise, and overweight. Treatment includes the elimination of the risk
factors and a number of methods of vessel expansion and drug therapy. Constricted
vessels can be bridged by implants inserted in a by-pass operation.

2.3.3. Myocardial diseases
A myocardial disease is a structural change of the myocardium caused by the
changes or degeneration of individual muscle cells. Myocardial inflammation is a
frequent cause, others are drug or alcohol abuse or hereditary disposition. In all
such cases, the heart becomes weak, and its capacity to pump is diminished. Symp-
toms of cardiac insufficiency (heart failure) start to develop. This change is seen as
a thickening, enlargement or stiffening of the ventricular muscles.
    There are three different types:

• Dilatative cardiomyopathy: This disease implies enlargement of the left or right
  or of both ventricles with restriction of the force of contraction. The ventricular
  systolic pump function of the heart is impaired, leading to progressive cardiac
  enlargement and hypertrophy (remodeling). Possible consequences are arrhyth-
  mia, cardiac insufficiency or sudden cardiac death.
• Hypertrophic cardiomyopathy: Typical of this disease is a massive muscle thick-
  ening and the ventricular cavities are small without an obvious cause. In addition,
  microscopic examination of the heart muscle in HCM shows an abnormal align-
  ment of muscle cells. The walls are stiff, and relax poorly during diastole, leading
  to increased end-diastolic pressure and resulting in pulmonary congestion and
• Restrictive cardiomyopathy: While the heart rhythm and the pumping action
  seem to be healthy, the stiff walls of the heart chambers keep them from filling
  normally. So blood flow is reduced, and blood that would normally enter the heart
  is backed up in the circulatory system.

2.3.4. Heart defect
A heart defect is a congenital or acquired specific structural phenomenon of the
heart or the adjacent vessels impairing the function of the cardiovascular system.
                   Analyzing Cardiac Biomechanics by Heart Sound                   167

Most frequent are atrial or ventricular septum diseases and valve anomalies (e.g.
Ebstein anomaly, aortal stenosis, and mitral valve prolaps).

2.3.5. Heart valve diseases
Most heart valve diseases are acquired defects. Only about 1% of heart valve diseases
are congenital. Acquired heart valve diseases can have several causes.
    About one in three of all acquired heart valve diseases is due to calcification
and degeneration. These diseases are often due to the typical ways of life in the
industrialized nations. Unhealthy food, overweight and lack of physical exercise
mostly lead to arteriosclerosis and high blood pressure. As people become older,
calcific degeneration of the valves increases and the valves lose mobility and become
stiff. As a consequence of this, heart valves fail to open and close properly, possibly
hampering the heart’s ability to pump blood adequately through the body. Due to
the high mechanical load of the aortic valve, this is affected most often. The most
frequent cause of aortic valve stenosis is arteriosclerosis. Compared with other valve
diseases, aortic valve diseases have become most frequent.8
    Another cause of heart valve disease is endocarditis, an inflammation of the
endocardium. It can be caused by bacteria or after an infection with rheumatic
fever. The valve cusps and flaps are part of the endocardium. During endocarditis,
they are also affected by bacteria. This leads to swelling and conglutination of the
cusp or flaps leading to valvular stenosis and/or insufficiency.
    Until 1970, mitral stenosis was the most frequently diagnosed heart valve disease.
The combined mitral valve disease (stenosis and insufficiency) occurred second most
frequently. Mitral valve diseases almost always occur in connection with rheumatic
fever. Advances in the treatment with antibiotics have dramatically reduced the
incidence of these diseases. Today, aortic valve diseases (aortic valve stenosis, in
particular), account for 65% of all heart valve diseases (Table 2) whereas the mitral
valve defects has dropped down to about 30%.9
    Acquired heart valve diseases can also occur after a preceding myocardial infarc-
tion. The AV valves are most at risk because a myocardial infarction usually dam-
ages the muscles which prevent disruption of the cusps when the ventricle becomes
tense. These muscles develop small fissures and become necrotic with time causing
valve insufficiency. Often, mitral valve insufficiency is a consequence.
    Right-side heart valve disease is very rare and mostly congenital or the conse-
quence of severe left-sided insufficiency of the heart.

                   Table 2.     Frequency of acquired heart valve diseases.

               Valves disease                                          Frequency
             left-sided    Aortic valve disease                           65%
                           Mitral valve disease                           30%
             right-sided   Pulmonary and tricuspid valve disease          5%
168                                    A. Voss et al.

      Heart valves can have one of two malfunctions leading to cardiac murmur:

• Regurgitation
  The valve does not close completely, causing the blood to flow backward instead
  of forward through the valve.
• Stenosis
  The valve opening becomes narrowed or does not fit properly, inhibiting the ability
  of the heart to pump blood to the body due to the increased force required to
  pump blood through the stiff (stenotic) valve.

(Besides, shunts, can cause turbulence in the blood accompanied by cardiac murmur.
As they are rare, they will not be considered in the following.)
   Stenosis or insufficiency can occur alone or in combination with each other at
each of the four heart valves. It is important for diagnosis, in which phase of the
heart action the murmur occurs:

• if the AV valves are insufficient, blood flows back into the atria during contraction
  of the ventricles. Therefore, murmur is in the systole of the heart action, directly
  after the first heart sound.
• if the AV valves are stenotic, the murmurs are caused by the flow of blood from
  the atrium in the ventricle, i.e., during the diastole after the second heart sound.
• if the semilunar valves are insufficient, the blood, after ejection from the heart,
  flows back from the vessels into the ventricles. The murmur occurs during the
  diastole, immediately after the second heart sound.
• if the semilunar valves are stenotic, turbulences occur during ejection of the blood
  from the ventricles into the large vessels during the systole after the first heart

    In simplified terms: At the AV valves, stenoses cause diastolic murmur, insuffi-
ciencies cause systolic murmurs. At the valves to the large vessels, it is vice versa.6,7
In this way, typical heart sound curves are obtained for each heart valve disease;
which are illustrated in Fig. 4.
    In the following, the examined heart valve diseases will be described in greater
detail as to their origin and specific change of the heart sound. Aortic valve stenosis
Normally, the opening area of the aortic valve is about 2.5 to 3 cm2 . Blood flows
through it with a velocity of up to 0.2 l/s during the systole. If the aortic valve is
stenotic (narrowing greater 50%), the pressure in the left ventricle increases (up to
300 mmHg) and a pressure difference occurs between the left ventricle and the aorta.
To compensate for the increasing resistance at the aortic valve, the muscles of the
left ventricle thicken to maintain pump function and appropriate cardiac output.
This muscle thickening causes a stiffer heart muscle which requires higher pressures
in the left atrium and the blood vessels of the lungs to fill the left ventricle.
                      Analyzing Cardiac Biomechanics by Heart Sound                      169

           Fig. 4.   Typical heart sound characteristics of different heart diseases.10

    Even though these patients may be able to maintain adequate and normal car-
diac output at rest, the ability of the heart to increase output with exercise is
limited by these high pressures. This constantly higher pressure causes hypertro-
phy (pathological enlargement of an organ in response to the extra load) of the
left heart, which again results in reduced coronary blood circulation and increased
oxygen demand of the heart. Affected patients complain of pain in the chest, loss of
efficiency, respiratory distress and sudden unconsciousness, mainly under physical
    The changes of the heart sound curve are typical and, as a rule, easy to find
by auscultation in patients with higher severity of the disease. Typical of it is a
spindle-like murmur during the ejection phase, i.e. between the first and second
heart sounds. This murmur reaches its maximum the later the stronger the stenosis.
Its intensity increases if the stroke volume increases, e.g. under physical strain.
Another change of the heart sound curve occurs at the heart tones. As the mass
of the left ventricle increases, the oscillation frequency is lower and the attenuation
stronger, letting the first heart sound appear weaker. In cases of very severe aortic
valve stenosis, the sequence of the aortic and the pulmonary valve portions can
occur inversely at the second heart sound because the systole at the aortic valve
is extended. The hardening of the aortic valve and the related decrease in the
oscillation capability also make the second heart sound weaker.6,7,12 Aortic valve insufficiency
After ejection of the blood from the left ventricle, the pressure in the ventricle drops
below the pressure in the aorta and the aortic valve closes. This causes a pressure
gradient towards the ventricle to be established. If the aortic valve is insufficient, i.e.
leaks, blood flows back into the heart. Aortic valve insufficiency can be congenital
170                                    A. Voss et al.

or caused by inflammatory changes of the valves (endocarditis after rheumatic fever
or bacterial infection).
    The consequences of aortic valve insufficiency depend on the amount of back-
flow (regurgitant blood flow) which, in turn, is determined by the size of the leaky
opening area and the pressure difference and the duration of the diastole. To ensure
that a sufficient amount of blood reaches the periphery, the stroke volume must
be increased. Constant pumping of a higher blood volume finally causes dilata-
tion and hypertrophy of the left ventricle. Unlike the aortic valve stenosis, patients
are not necessarily incapacitated by high physical strain because when the heart
rate is higher, less time is available for the diastole, during which blood can flow
back. Dizziness, respiratory distress and physical capability at rest and during light
physical activity are observed.7,11
    The main indication for aortic valve insufficiency in the heart sound curve is a
decrescendo murmur which directly follows the second heart sound. The duration
of the murmur depends on the severity of the insufficiency. In severe cases, it can
extend into the presystole. Another typical indication is a fainter second heart sound
in severe cases of the disease. The reason for this is the reduced oscillation capability
of the aortic root and the aortic valve.6,12 Mitral valve insufficiency
The closing ability of the mitral valve depends on the function of the cusps and
the mitral ring, also on the papillary muscles and tendons which tension the cusps.
If the valve cusps do not close completely, blood flows back from the left ventricle
into the atrium during the systole. Hence, an additional regurgitant blood flow
between the atrium and the ventricle must be managed without being available for
    In the long term, this extra muscle work also causes hypertrophy of the left
heart. Patients suffering from mitral valve insufficiency cannot perform normally
and fatigue more easily due to insufficient supply of blood. The number of mitral
valve insufficiency after myocardial infarction has increased especially in recent
years. But also endocarditis or a Marfan’s syndrome (hereditary disease of the con-
nective tissue) are potential triggers of the disease. Endocarditis causes the cusps
and tendinous cords to shrink, become thick or stiff, which impairs valve closure.
Another frequent cause of mitral valve insufficiency is the Barlow’s syndrome (also
mitral valve prolaps syndrome), which is characterized by too long tendinous cords.
The cusps bulge into the atrium far enough as to open again. This effect is also
caused by a necrosis of the papillary muscles (tensor muscles) after myocardial
    By auscultation, mitral valve insufficiency is identified as a murmur directly after
the first heart sound. This murmur is heard loudest at the apex of the heart. The
murmur decrescends towards the second heart sound but ends before the second
heart sound starts because the contraction of the ventricle causes the leaky spot to
become smaller. Only in very severe cases will the murmur persist at the same level
                   Analyzing Cardiac Biomechanics by Heart Sound                   171

until the second heart sound. The difference to aortic valve stenosis is in the onset
of the murmur. Whereas the murmur of mitral valve insufficiency follows the first
heart sound directly, a pause until the actual expulsion phase can be considered in
the case of aortic valve stenosis.
    Another typical feature of the heart sound curve in the presence of mitral valve
insufficiency is the fainter first heart sound. Changes of the valve (e.g. due to cal-
cification) impair the latter’s capacity for vibration. Besides, less tension energy is
available. In most cases, the third heart sound is also more pronounced. As blood
flows back into the atrium, the latter is filled strongly at the end of a systole. This
large volume very quickly flows into the ventricle as a wave at the beginning of the
diastole. In addition, a click murmur can occur as third heart sound in patients with
Barlow’s syndrome; it is caused by the reversal of the cusps in the atrium due to
overlength of the tendinous cords.6,12 Tricuspid valve insufficiency
Typical of a tricuspid valve insufficiency is the backflow of blood from the right
ventricle in the right atrium. The most frequent cause is preceding rheumatic fever,
which is often accompanied by mitral valve disease. In this case also, the cusps and
the tendinous chords shrink, deform or become thicker.
    As with mitral valve insufficiency, auscultation will note a systolic decrescendo
murmur after the first heart sound. Frequently, the level of the murmur is constant.
Compared with murmurs of the left heart, the loudness of the murmurs with tricus-
pid valve insufficiency increase during inspiration. Occasionally, there is an inverse
splitting of the second heart sound due to the shorter systole in the right half of the
heart. Often, the third heart sound is more pronounced because of the large volume
flowing into the right ventricle.6,7 Pulmonary valve insufficiency
Pulmonary valve insufficiency occurs if the blood flows back from the pulmonary
artery into the right ventricle during the diastole. Pulmonary valve insufficiency
alone is a very rare disease. Frequently the causes are dilatation of the valve ring in
case of pulmonary hypertension, but also connective tissue disorders, for example,
the Marfan’s syndrome. Inflammatory processes or malformation are involved rarely.
Pulmonary valve insufficiency leads to extra volume strain on the right ventricle.
Because it usually occurs together with pulmonary hypertension, the consequences
for the patient are normally determined by the hypertension.
    As with aortic valve insufficiency, the principal acoustic feature is a diastolic
decrescendo murmur. The difference is in the beginning of the murmur. The earliest
possible onset is after the pulmonary section of the second heart sound, but mostly
after a short pause. The frequency and duration of the murmur depend on the
pressure in the pulmonary artery.6,7
    Table 3 summarizes the important heart sound indicators of heart valve diseases.
172                                        A. Voss et al.

          Table 3.   Typical changes of the heart sound caused by heart valve diseases.

       Heart valve disease          Heart sounds                  Cardiac murmurs
      Aortic valve            First and second heart        Spindle murmur between first
        stenosis                sounds fainter                and second heart sound
      Aortic valve            Second heart sound            Decrescendo murmur directly
        insufficiency             fainter                       after the second heart sound
      Mitral valve            First heart sound fainter     Decrescendo murmur after the
        insufficiency                                           first heart sound
      Tricuspid valve         Frequent third heart          Decrescendo murmur after the
        insufficiency             sound                       first heart sound
      Pulmonary valve         No change                     Decrescendo murmur after a
        insufficiency                                           short pause following the sec-
                                                            ond heart sound

2.3.6. Biomechanical replacement systems Replacement of the heart valves
The target of heart valve surgery is to recover the impaired pumping capacity of the
heart caused by pathological changes of the valves. Because natural human heart
valves are unique in life and functionality, the first method of choice is to repair the
diseased heart valve whenever possible (valve reconstruction) and spare the patient
possible disadvantages of an artificial heart valve. The repair option of the heart
valves is limited by marked calcification and severe changes of the valve cusp tissue.
In these cases, replacement of the heart valve is the only possibility. The ideal heart
valve replacement should meet the following requirements:

• unlimited durability
• normal blood flow conditions in the vessel (there should be no pressure difference
  before or after the implantation of the heart valve replacement and the blood flow
  through the heart valve replacement should be normal and without obstruction)
• no complications due to the heart valve replacement, e.g. increased thrombogenic-
  ity (clot formation), susceptibility to endocarditis (inflammation of the endo-
• no complications due to the heart valve replacement, e.g. cradle fracture
• simple to implant
• low-murmur and comfortable
• constant availability

   None of the present valves meets all these criteria. Generally, there are two
different valve types: biological heart valves, which are divided further into biological
prostheses (xerografts) and human valves (homografts), and mechanical artificial
heart valves.
   All types have specific advantages and disadvantages. Which valve type is used
depends on age, medication, other diseases and the personal conditions of the
                   Analyzing Cardiac Biomechanics by Heart Sound                 173

Bioprostheses (xerografts)

Bioprostheses are heart valves made of animal material. Some prostheses are made
from pig heart valves (porcine prostheses) and some prostheses from bovine peri-
cardial tissue (bovine prostheses). The prostheses are prepared by special chemical
and physical methods. Their function and appearance is very similar to those of
human heart valves. Some tissue heart valves are sewn onto a plastic valve frame
to which an external ring of dacron or teflon tissue is applied, which ensures easier
implantation. Other tissue heart valves have no frame.
    The advantage of biological heart valves prostheses is that their hemodynamic
behavior is very similar to that of the natural human heart valve. Besides, anticoag-
ulation treatment is not normally required. The essential disadvantage of biopros-
theses is their limited durability; about 5% of all modern bioprostheses in 65-year
old patients show marked changes after about 10 years. Degenerative changes, calci-
fication, in particular, occur the earlier the younger the patients. Another problem
is sudden avulsion or tearing of the cusp, which can lead to massive cardiac insuf-
ficiency within very short time.

Human valve replacement (homografts)

These valve replacements are heart valves obtained either during a post mortem or
a heart transplantation. They are prepared and preserved for storage in an organ
bank. Generally, the advantages and drawbacks are the same as those of animal
heart valves. The durability here is between 15 and 20 years.

Mechanical heart valves prostheses

Mechanical heart valves prostheses are heart valves made of extremely durable mate-
rials, such as metal or plastic. They have an outer ring of synthetic tissue (dacron
or teflon). The purpose of the ring is to sew the heart valve in the tissue of the
patient’s heart. The large number of different mechanical heart valve prostheses at
present in clinical use indicates that the ideal prosthesis has still not been found.
The big advantage of mechanical heart valves in comparison with bioprostheses is
in their virtual unlimited durability and constant availability. However, extensive
anticoagulation therapy is required, in spite of which thrombo-embolic complica-
tions are frequent. The risk of endocarditis is also higher in case of a mechanical
heart valve replacement. Replacement systems for cardiac arrhythmia
If cardiac arrhythmia cannot be controlled by drug therapy, a replacement system
may have to be implanted. There are the following systems:

• Cardiac pacemakers, which are implanted in cases of low heart frequency, sinus
  node disorder or a heart block. If the heart frequency is low, the pacemaker
  sends electrical signals to the heart through electrodes which cause the heart to
174                                  A. Voss et al.

• Implantable cardioverter/defibrillators (ICD), which are implanted preferably in
  patients with ventricular tachycardia. If a life-threatening arrhythmia (tachy-
  cardia) occurs, the device releases an electric shock which normalizes the heart
  frequency. Cardiac assist device and artificial heart
A cardiac assist device is usually used in addition to the natural heart in case of
heart insufficiency, to bridge the time until a donor organ is available. Another use
is as temporary relief of the heart if the heart muscles can regenerate, e.g. after a
severe myocarditis.
    The term artificial heart describes an electromechanical pumping system within
(intracorporeal) or outside (extracorporeal) the body which unlike the cardiac assist
device takes over the complete function of the natural heart.

3. Methods of Analyzing Cardiac Biomechanics
3.1. Computer tomography (CT) and magnetic resonance
     tomography (MRT)
Until a few years ago, computer tomography and magnetic resonance tomogra-
phy were only used as imaging methods for the anatomy and morphology of the
heart. The main uses were and still are congenital heart diseases, heart tumors
and the diseases of the thoracic aorta. Rapid progress in science and technology in
the last few years has opened new applications for both methods in hospital rou-
tine examinations of heart function. For example, functional MRT can be used to
assess function, perfusion and vitality by a single examination. MRT is an imaging
process for anatomy and morphology, malformation of coronary arteries, quantifi-
cation of valvular defects and shunt defects, the examination of myocardial diseases
and the assessment of the metabolism of the heart. The focus of CT is on the
examination of the coronary arteries, in particular, the quantification of coronary

3.2. Ultrasound — echocardiography
In addition to electrocardiography (ECG), echocardiography is one of the most
important non-invasive examination methods of the heart and by now an indis-
pensable tool in cardiological diagnostics. It allows to assess the function of the
ventricles, the atria and the heart valves. Quantification of defects by measuring
certain parameters is also possible. Congenital heart defects can also be diagnosed
reliably. In particular, the examiner obtains information on the structure of car-
diac walls and heart valves as well as their movements, the wall thickness of heart
atria and ventricles, the size of heart chambers and the ejection capacity of he
                   Analyzing Cardiac Biomechanics by Heart Sound                  175

3.3. Heart sound — auscultation and phonocardiography
Many heart diseases, most of all the heart valve diseases and the congenital heart
defects, produce typical cardiac murmurs due to a sometimes massive change of the
biomechanics of the heart (see Chapter 2).
    Mostly, cardiac murmurs are the first symptoms of a pathological process at the
heart valves. For a long time, auscultation was the only method by which heart
valve diseases could be detected.14 Auscultation is listening with a stethoscope
to sounds produced by the heart, lungs, and blood. The cardiac murmurs which
the physician hears are diagnosed directly. Phonocardiography was introduced by
EINTHOVEN as late as in 1907.15 Phonocardiography makes the murmurs visible as
a curve and records them. From these curves, several parameters can be selected for
diagnosis. Mostly, the curves are referenced to the heart curves of healthy or other
pathological heart curves. Despite its high diagnostic potential, phonocardiography
never prevailed in everyday hospital work. One reason for this is the sensitivity of
this method and, in relation with it, the difficulty in obtaining noise-free records
whose quality is sufficient for exploitation.16,17 Likewise, the interpretation of heart
sound curves requires high medical qualification and clinical experience on the part
of the physician.6,7 An important step ahead in phonocardiography was made by
MAASS and WEBER in 1952. They introduced a standardized recording technology
(amplifiers, filters, etc.) allowing more exact comparisons between different heart
sound curves.18
    The introduction of ultrasonic diagnostics and especially of echocardiography
(first examinations were made by KEIDEL around 1950) almost completely dis-
placed phonocardiography. This imaging method is a safe and exact diagnostic tool
but requires substantial technical equipment and related high costs and is, as a
rule, used only by cardiologists and internal specialists but not by the general prac-
titioner.19 Only the development of electronic stethoscopes gives diagnosis by means
of heart sounds a new lease of life.

4. Heart Sound Evaluation for Analyzing Cardiac Biomechanics
4.1. Stethoscopes and heart sound sensors
Whereas, in earlier times, a difference used to be made between purely acoustic
stethoscopes and sensors for phonocardiography, today these functions mostly merge
to a single product, the electronic stethoscope, it “hears”and “records” at the same

4.1.1. State of the art
The present stethoscopes and sensors can generally be grouped in three categories
(Fig. 5):
• conventional, purely acoustic stethoscopes,
176                                       A. Voss et al.

                  Fig. 5.   Overview of stethoscopes and sensor principles.

• electronic stethoscopes, airborne sound microphone principle,
• electronic stethoscopes, structure-borne sound or vibration sensor principle.

    Conventional stethoscopes with and/or without diaphragm are for the purely
acoustic hearing of heart sound phenomena and subjective diagnostic findings, they
are not able to record heart sound signals. They are comparatively low-cost and still
regarded as “status quo” in the hospital and doctor’s practice. At least, these stetho-
scopes support two mechanical frequency ranges (heart sound and lung sound).
    Electronic stethoscopes on the principle of airborne sound microphones can
essentially be compared with conventional stethoscopes with the difference that
an encapsulated microphone for recording the sound is integrated in the acoustic
path, and an electronic amplification and filter circuit with output via loudspeaker
or encapsulated headphone is available. The two frequency ranges for heart sound
and lung sound are controlled electronically. The almost only advantage of this first
generation of electronic stethoscopes is the better amplification of the sound events,
in particular for users whose hearing ability has suffered with age.
    The younger generation of electronic stethoscopes no longer records the
(airborne) sound of the heart or lung activities but the vibrations these activi-
ties produce at the surface of the thorax are recorded by a structure-borne sound or
vibration sensor. At present, stethoscopes of this type supply the best signal quality
                    Analyzing Cardiac Biomechanics by Heart Sound                  177

and signal-noise ratio. Today, vibration sensors can be contact film sensors based
on a piezo-polymer film.
   The common features of electronic stethoscopes in comparison with conventional
acoustic microphones are the following:
• Signal amplification (e.g. for hearing impaired due to age) or volume control,
• Electronic filtering, selectable frequency ranges for heart and lung sounds,
• Recording the signals in the stethoscope or the PC, with the possibility of play-
  back and analysis.

4.1.2. Design of a new electronic stethoscope
The “Audira” system (Fig. 6) developed by our group is an electronic stethoscope
of the latest generation with all above-named features and standard functionalities.
Essential component is a newly developed, high-sensitivity vibration sensor based
on a bimorph rod.
    The design provides a very clear and highly amplified signal with attenuation of
noise and jitter on the basis of floating and pressure-balancing sensors. This design
ensures essentially that the device is independent of the contact pressure as well as
skin and ambient temperatures. The Audira stethoscope simultaneously records a
single-channel ECG by electrodes integrated in the head of the stethoscope. A hand-
held module digitizes the ECG and the sound in a standard format, electrically
isolates and transmits them to the PC via USB or Bluetooth for further processing.
The selection of the point of auscultation and the entire program and process control
of electronic auscultation are also made with the manual control that the examining
physician is not required to operate the PC.

4.2. Heart sound analysis for detection of biomechanical disorders
     of the heart valves
A main objective of this work was to develop an automatic heart sound analysis
system for diagnosing acquired heart valve diseases which with simple handling and

          Fig. 6.   “AUDIRA” stethoscope, inclusive of hand-held control module.
178                                   A. Voss et al.

at reasonable cost, can be used by the general practitioner and the family doctor.
The algorithms for the classification of the different diseases of heart valves are
based on the feature extraction from heart sound curves.

4.2.1. State of the art
The last few years have originated several research works focusing on the use
of neuronal networks for the automatic diagnosis of heart disease. The differ-
entiation of functional and pathological heart murmurs in children for diagnos-
ing congenital heart defects, in particular, has archived high levels of sensitivity
and specificity. However, no commercial analytical systems for the diagnosis of all
heart valve diseases based on neuronal networks are available in the market so
    Another approach is that of correlation analyses, which bases on the similarity
of patterns of heart sound curves of identical diseases. For this, a new signal is
set in correlation with all diagnosed signals, for example, in a database. Signals of
high similarity to each other have a high correlation coefficient. The disease with
which the signal has the highest average correlation coefficient is diagnosed. For
example, if a signal has a higher average correlation coefficient to signals of the
group of aortic stenoses than to signals of the tricuspid valve insufficiency group
it is more similar to the aortic stenoses than the tricuspidal insufficiencies), the
patient suffers from aortic valve stenosis with higher probability than from tricus-
pid valve insufficiency. First internal studies showed 23 that the correlation analysis
for the detection of moderate and severe aortic valve stenosis achieves good results
(sensitivity 91%, specificity 94% in comparison with a healthy reference group; sen-
sitivity 94%, specificity 97% in comparison with a group with other heart valve
    Both approaches, i.e. neuronal networks and correlation analysis, require a
lot more computer hardware and computing time than parameter extraction
    For some time, several studies have applied different methods of signal analysis
for the interpretation of heart sounds. For example, high sensitivity and speci-
ficity levels were achieved in the detection of degenerated biological replacement
valves (aortic and mitral valves). For this, different methods of frequency analysis
were combined with traditional classification methods (Bayes classificator, nearest
neighbor) and neuronal networks.24–26
    Much more recently, a device for home monitoring of mechanical replacement
valves based on frequency analysis (Fast Fourier Transform) of the heart sounds is
available in the market. The purpose of the device is the early detection of disorders
of the mechanical heart valve to enable timely intervention.27
    Pavlopoulos et al.28 presented a method of differential diagnosis of aortic valve
stenosis and mitral valve insufficiency based on parameter extraction followed by
a decision-tree method. This method although of high accuracy is unsuitable as
                   Analyzing Cardiac Biomechanics by Heart Sound                   179

screening method on GP level because a valve disease must already have been
    Another study tried, by different methods of frequency analysis (CWT, FFT),
to determine the dominating frequency from the heart sound signal and use it as a
parameter for the degree of severity of aortic valve stenosis.29
    Kim et al. proved in a study that the duration of certain frequency ranges in
the spectrum correlate with the pressure gradient derived from the Doppler signal
and that these are useful indicators of an aortic valve stenosis.30
    Most recent studies also prove that the Wavelet Transformation, which we also
apply, is especially useful for a detailed analysis of the heart sound components.31
    Important research work on automatic interpretation using parameters of the
heart sound has been done by Barschdorff.17,46 However, the signal analytic methods
by which signals are analyzed (envelope formation and pattern recognition) and
parameters are defined (e.g. mean value calculation, variance, envelope function
form factor, flattening coefficient in different segments), are completely different,
classification is also based on neuronal networks.32,33
    So far, there have been no algorithms for the automatic interpretation of heart
sounds based on feature extraction using methods of the time and frequency domain
which are also applicable as screening method for all heart valve diseases.

4.2.2. Heart sound analysis based on feature extraction Signal recording and preliminary processing
For this study, the heart sound signals of patients were recorded using an electronic
stethoscope on five different auscultation areas for 15 seconds each. Auscultation
areas 1 to 5 were those which the physician normally checks with the acoustic
stethoscope. Their locations can be seen in Fig. 7. Simultaneous with the heart
sound, a single-channel ECG (Einthoven II lead) was recorded to obtain a time
relation of the heart sound signal.
    The cardiologist examined every patient’s heart extensively by echocardiogra-
phy. For this, a special appraisal form was developed which the cardiologist had to

R wave detection and noise eliminating
With reference to the detected R waves in the ECG, the heart sound was divided
into single heart periods. Each of these heart periods was replayed acoustically and
inspected visually for noise. Heart periods in which noise was detected were excluded
from the following examinations.

Wavelet decomposition
The heart sound signals were filtered by multi-scale analysis using wavelets. It is very
suitable for discriminating background noise from biosignals.34,35 The method which
180                                        A. Voss et al.

                           Fig. 7.   Auscultation areas AP1 to AP5.

       Fig. 8.   Scheme of stepwise decomposition of the signal by multi-scale analysis.

we applied splits a signal successively into a low-frequency segment and a high-
frequency segment. The low-frequency segment is split again for the next step.36
    In Fig. 8, c0 is the input signal. Signal c1 is the first low-frequency band generated
by low pass H and d1 the first high-frequency band generated by high pass G. The
filtering continued up to m = 10 so that totally 20 frequency bands (10 low-frequency
bands, 10 high-frequency bands) were obtained.
                   Analyzing Cardiac Biomechanics by Heart Sound                  181

Representative heart period
To reduce the time and computing effort, an attempt was made to carry out the
examinations only with one heart period for each auscultation area. For this, the
period in one recording was determined automatically which was most similar to
all other heart periods of this recording. For the purposes of the study, this heart
period was declared to be the representative heart period whose sum of correlation
coefficients was the highest in relation to all other heart periods.
    The calculations were not made with the original signals because this would have
consumed too much time due to the high sample frequency of 44100 Hz. Instead, this
analytical step was carried out in the wavelets within the frequency band from 0 to
2756 Hz, which saved computing time. The resulting information loss was tolerable
because all examinations were limited to the frequency bands of up to 2756 Hz
because only relevant information to the analyzed diseases were expected in the
range of up to approximately 2000 Hz.

Heart sound detection
The temporal occurrence of the heart sound phenomena is essential for the assign-
ment of the affected heart valve. Therefore, it is necessary to divide the heart period
in functional sections according to the mechanical phases of the heart action. The
maximums of the first and second heart sounds can serve as triggers. They were
detected by an algorithm and thus the heart period was divided. For detection, two
independent methods were developed and their results were compared with each
other to improve the degree of accuracy.
    At first, the time progression of the Shannon energy of the wavelet filtered
signal in the range from 0 to 172 Hz was calculated. The analysis of the energy
characteristic served as reference method. For this, the frequency-time spectrum was
determined by Short Time Fourier Transform (STFT) and all coefficients within a
time window were summated.
    In the two obtained time series, the two highest peaks within defined time win-
dows in which the first and second heart sounds were expected, were detected (first
heart sound — 0 to 150 ms, second heart sound — 350 to 600 ms). If the difference
of the detected positions in both methods did not exceed 70 ms, the mean of the
two methods was calculated and the maxima declared to be the first and second
heart sound.
    Figure 9 illustrates the method. The first graph shows the original heart sound
signal. The graphs shown are those of the STFT, the energy progression obtained
from it and the progression of the Shannon energy. The maximum values found in
graph three and four are back-projected onto the original signal. Finally, they are
compared and declared to be valid if the difference is sufficiently small after the
mean has been calculated.
    The problem of this method also becomes apparent with patients of strong heart
murmurs and faint heart sounds, e.g. in presence of severe aortic valve stenosis. In
182                                       A. Voss et al.

Fig. 9. Heart sound detection in a patient with healthy valves on the left and a patient with
severe aortic valve stenosis on the right (in each case original signal, STFT spectrum, squared
sum of frequency components from STFT and Shannon energy; maxima are marked, amplitudes
normalized, frequency in Hz, time in samples and window steps).

both detection methods, the heart murmurs have higher maximum peaks than the
heart sounds. For this reason, the maximum values were sought in limited time
windows in which the heart sounds with the utmost probability are situated. These
time windows were defined empirically.

Signal segmentation
Starting from the positions of the first and second heart sound, the signal was split
into 12 segments, S1 to S12 (Fig. 10). For this, the distance between the heart
sounds was determined and divided by five. This value, x, described the length of
segments S2 to S11, beginning from the first heart sound minus x/2. Segment S1
consists of all sample points from the R wave until that time. The last segment,
S12, consisted of all sample points after S11 until the end of the heart period. In

Fig. 10. Segmentation of a heart sound signal from segment S1 to S12 (amplitude normalized,
time in samples).
                   Analyzing Cardiac Biomechanics by Heart Sound                 183

this way a dynamic segmentation was obtained which guarantees that functionally
equivalent segments during a heart cycle among patients were studied independently
of the length of a heart period or the position of the heart sounds. Parameter extraction
Global parameters from the Fast Fourier Transform
The Fast Fourier Transform (FFT) splits a measured signal into its sinusoidal and
cosinusoidal frequency parts and displays them as a spectrum. This provides infor-
mation on the frequencies in the signal and their respective parts, but not on the
time of their occurrence.
    Stenosis or insufficiencies at a heart valve cause turbulences in the blood flow.
These flow murmurs lead to higher frequency portions in the heart sound signal.
Theoretically, sound signals from persons without heart diseases only contain fre-
quencies in the range of the heart sounds up to approximately 150 Hz. If a heart
valve disease exists, much higher frequencies (partly over 1000 Hz) can be contained
in the spectrum.
    For the parameter extraction, the amplitude spectrum of the heart sound signals
was calculated. For this, the original signals were weighted with the Blackman-Harris
window function to avoid the occurrence of high frequencies due to amplitude jumps
at the edges of the window. The window length was 4096 sample points. This yielded
a spectral resolution of 10 Hz.
    Now the frequency parts of different bands between 15 and 700 Hz were summed
as parameters. For this, the frequency range was divided in bands of 25 Hz width
each and the sum of all parts in each band was calculated. Each of these values
was one parameter which was checked for significance using the Mann-Whitney U
test, e.g. the parameter sum of all frequency components between 15 and 40 Hz, the
parameter sum of all frequency components between 40 and 65 Hz, etc. After this,
the same method was applied to frequency bands of 50 Hz, 75 Hz, 100 Hz, 125 Hz,
150 Hz, 175 Hz, and 200 Hz width.
    Figure 11 illustrates the method by the example of the 50 Hz frequency band
width. Graph A is the original heart period, graph B the associated FFT spectrum
from 15 to 700 Hz and its division into sections of 50 Hz width each. The sum of all
frequency components in each section is then calculated.

Global parameters from the time relation between ECG and heart sounds
Changes at the heart valves can shift the time of occurrence of the heart sounds
relative to the R-peak in the ECG (Fig. 12) because, for example, the pressure
conditions among the vessels change.4 Therefore, the following four parameters were

(1) Time between the R-peak and the first heart sound,
(2) Time between the first and second heart sound,
184                                       A. Voss et al.

Fig. 11. Example of FFT parameter extraction. (A) Original signal with heart sounds; amplitude
normalized, time in samples, (B) Fourier spectrum up to 700 Hz, divided into sections of 50 Hz,
each, in each of which the area was determined as parameter; amplitude normalized, frequency
in Hz.

(3) Time between the R-peak and the second heart sound,
(4) Time between the second heart sound and the end of the heart period.

    These parameters were also analyzed regarding to their significance for each
disease. In Fig. 12 an example is given for this parameter extraction method.
Graph A shows the ECG (Einthoven II lead), Graph B the associated heart sound

Segmental parameters from time and frequency domain analysis
For diagnosing a heart valve disease, it is extremely important to know in which
section of the heart period the heart sound changes. In most cases, this is the
only way to find the origin of the defect. Therefore, it was necessary, in addition
to the information on the frequency content, to analyze the temporal behavior of
the signal. For this, analytical methods from the time and frequency domain were
    In all the representations and curves of the signal derived from these methods,
the following parameter classes were extracted from each of the 12 segments:

   A total area of the segment
   B proportion of the area of the segment to the total area
                     Analyzing Cardiac Biomechanics by Heart Sound                       185

Fig. 12. Parameter extraction from the time relationship between ECG and heart sound. (A) ECG
with R-peak marked, (B) original heart sound with marked heart tones; amplitudes normalized;
time in samples.

    C sum of differences between two points
    D sum of the absolute values of the differences between two points.

Each parameter is composed of its class, the segment and the analytical presentation
of the signal.

Segmental parameters from time-domain analysis
The analysis in the time-domain was not carried out with the original signal but
with the wavelet-filtered signals. The frequency components in the range from 0 to
2756 Hz were included in the analysis. The following parameters were extracted:

•   Intensity (absolute value)
•   Energy
•   Shannon energy
•   Shannon entropy

Figure 13 illustrates this method by an example. Graph A is a wavelet-filtered heart
period at level 6.0 (0 to 172 Hz), B represents the intensity, C the energy, D the
Shannon energy, and E the Shannon entropy of the signal.

Segmental parameters from time/frequency methods
Heart murmurs are classified by high frequencies within the sound signal. Therefore,
the examination of the frequencies in connection with their time-related occurrence
186                                       A. Voss et al.

Fig. 13. Example of parameter extraction from time domain. (A) wavelet-filtered heart period 0
to 172 Hz, (B) Intensity, (C) Energy, (D) Shannon energy, (E) Shannon entropy; extracted from
each curve were in each segment the parameters: area, area proportion, sum total of differences
between two points and sum of absolute values of the differences between two points; the cursors
denote the heart sounds; amplitudes normalized, time in samples.

promised to yield the most significant information. Only the time/frequency meth-
ods can reveal information which frequencies occur and with what intensity at what
time during a heart cycle.

Parameters from STFT

The STFT was calculated for each heart period with the following parameters:

(1) Window length of 4096 samples; this yields a frequency resolution of 10 Hz and
    a time resolution of 100 ms,
                      Analyzing Cardiac Biomechanics by Heart Sound              187

(2) Shift of the window by 100 samples each; this somewhat impairs     the time res-
    olution but this had to be tolerated to reduce computing time      and memory
(3) Windowing of the signal by Blackman-Harris window function          to avoid the
    occurrence of high frequencies due to amplitude jumps at the        edges of the

From the calculated 3-dimensional transformations, new signal descriptions were
generated by calculating different sums of all frequency sections in each time

(1)   sum of    all frequency components
(2)   sum of    all frequency components and squaring of sum,
(3)   sum of    all logarithmized frequency components,
(4)   sum of    all logarithmized frequency components and squaring of sum,
(5)   scaling   of 3. between 0 and 1,
(6)   scaling   of 4. between 0 and 1.

    The method is illustrated by an example in Fig. 14. Graph A is the original
heart sound signal, graph B the associated STFT spectrum. Graph C to H show
the generated signal presentations in the above-named order. Extracted from each
curve in each segment were the four parameters absolute area, area proportion of
the total signal, sum of differences between two points and sum of absolute values
of the differences between two points.

Parameters from Wavelet Transform

The Wavelet Transform (WT) also supplies information on the time and frequency
dependence. Instead of a window of fixed width, such as for STFT, a changing win-
dow of a constant number of oscillations is slided over the signal and the similarity
with the current signal section is calculated. The variation of the analytical func-
tion, the so called wavelet, is obtained from the compression or elongation defined
by the so called scaling function. The basic form of the wavelet does not change.
    For parameter extraction, a WT with the following parameters was calculated
from every representative heart period:

 • “Mexican hat” mother wavelet because this shape resembles that of the heart
 • elongation of the wavelet from 20 to 4410 sample points in 100 steps after
   a cubic root function; in this way, the lower frequency ranges up to 500 Hz
   were analyzed in narrower steps than were the higher frequencies from 500 to
   2000 Hz, because more information was expected to be obtained from the lower
 • stepwise shifting of the wavelet by 10 sample points (to save computing time
   and memory capacity).
188                                        A. Voss et al.

Fig. 14. Example of parameter extraction from STFT. (A) Original signal, (B) STFT,
(C to H) from STFT generated different signal representations (see text); extracted from each
curve were in each segment the parameters: area, area proportion, sum total of differences between
two points and sum of absolute values of the differences between two points; the cursors denote
the heart sounds; amplitudes normalized, frequency in Hz, time in samples and window steps.
                       Analyzing Cardiac Biomechanics by Heart Sound                        189

From the calculated transformations, new signal descriptions were generated by
summation of the frequency sections of each time window similar to the approach
with STFT.
(1)   sum of    all frequency components,
(2)   sum of    all frequency components and squaring of sum,
(3)   sum of    all logarithmized frequency components,
(4)   sum of    all logarithmized frequency components and squaring of sum,
(5)   scaling   of 3. between 0 and 1,
(6)   scaling   of 4. between 0 and 1,
(7)   sum of    absolute values of all frequency components,
(8)   sum of    absolute values of all frequency components and squaring of sum.
Figure 15 illustrates this method graphically. The figure starts with the presentation
of the original signal (A) and the associated WT spectrum (B). Graph (C) to (J)
show the generated signal presentations in the above-named order. Extracted from
each curve in each segment were the parameters absolute area, area proportion of
the total signal, sum of differences between two points and sum of absolute values
of the differences between two points.
    All segmental parameters were now tested for significance to the tested item by
means of the Mann-Whitney U Test. Using the methods for analysis of heart sounds in patients with heart valve

Fig. 15. Example of parameter extraction from WT. (A) Original heart sound, (B) WT spectrum,
(C to J) signals derived from WT (see text); extracted from each curve were in each segment the
parameters: area, area proportion, sum of differences between two points and sum of absolute
values of the differences between two points; the two cursors denote the heart sounds; amplitudes
normalized, frequency in Hz, time in samples and window steps.
190                                        A. Voss et al.

Data of 402 patients were included in this retrospective randomized study. Not
included were patients with at least one non-native cardiac valve, i.e. a replacement
valve or a surgically corrected valve. Another condition for the usability of the
examination was that at least one heart period includes noise-free sound signals
from all auscultation areas.
    These 402 patients were divided in two groups, one training group (TrG) for the
determination of significant parameters and the resulting discriminant functions,
and one test group (TeG) for verification of the classification methods. Patients were
classified into these groups by a random procedure. The training group comprised
of 206 patients, the test group of 196 patients.
    Table 4 shows the group arrangement concerning sex, age and body weight. A
classification was made in patients with healthy heart valves (REF) and patients
with heart valve disease (VD).
    The parameter sets and the discriminant functions were determined from the
training group by Mann-Whitney U test and discriminant function analysis. The
obtained test methods were validated on the test group.
    The higher severity levels of the analyzed diseases were partly underrepresented
so that groups were summarized (Table 5).
Classification methods
Classification method 1
The simplest possibility was to separate the patients with the analyzed disease
(without considering the severity level) from those patients who were not affected
by that disease. This required only one parameter set with one discriminant function
(Fig. 16).
                     CS 1 : (ASl + ASms) versus (REF + OV D)

Classification method 2
This method considers the possibility that mild and higher severity levels of a disease
do not necessarily have the same significant parameters and as such are not, of

                      Table 4.    Patient data (BMI-Body mass index).

             Group                   Sex                     Age        BMI
                           Male            Female
             REF             44               33            58 ± 16     27 ± 4
             VD              50               79            66 ± 14     26 ± 4

             REF             51               25            57 ± 15     27 ± 5
             VD              58               62            66 ± 13     26 ± 4
                    Analyzing Cardiac Biomechanics by Heart Sound                191

               Table 5. Arrangement of the patient groups for the analysis
               of valve diseases (AS — Aortic valve stenosis, AI — Aor-
               tic valve insufficiency, MI — Mitral valve insufficiency, TI —
               Tricuspid valve insufficiency, PI — Pulmonary valve insuffi-

                    Heart valve disease           TrG       TeG        Total
               REF                                 77        76         153
               AS           mild                   20        12          32
                            moderate–severe        27        30          57
               AI           mild                   42        39          81
                            moderate–severe        20        21          41
               MI           mild                   53        50         103
                            moderate–severe        23        16          39
               TI           mild                   40        45          85
                            moderate–severe        23        16          39
               PI           mild–severe            22        17          39

                 Fig. 16.   Classification method 1 — aortic valve stenosis.

necessity, placed in the same group. In the first step of this method, only higher-
level patients are identified and in the second step, patients with mild levels of the
disease are identified (Fig. 17).

    For every classification step, the respective significance of each parameters was
to be determined in terms of a possible separation of the currently examined groups.
Due to the large number of parameters, the level of significance had to be reduced
192                                        A. Voss et al.

                   Fig. 17.   Classification method 2 — aortic valve stenosis.

according to the Bonferroni criterion:

                                                  global significancelevel α
                 Local significancelevel α =                                 .
                                                  N umber testparameters

So the significance level for each parameter was α < 3∗ 10−6 . In addition, it was
arranged that parameters of not more than three auscultation areas in each case
were included in the discriminant analysis. A linear stepwise discriminant analysis
with maximum four of the significant parameters was carried out.


Example: Aortic valve stenosis
For examination of the aortic valve stenosis, the patients were divided into four

      (I)     REF –  patients without valve disease
      (II)    OVD –  patients with a valve disease other then
                     aortic valve stenosis
      (III)   ASl – patients with mild aortic valve stenosis
                     (and possibly other valve diseases)
      (IV)    ASms – patients with moderate or severe aortic valve
                      stenosis (and possibly other valve diseases)
                   Analyzing Cardiac Biomechanics by Heart Sound                  193

    In Table 6 the following group arrangement was obtained:
    The following concrete test situations and results were obtained for the two
classification methods:
(1) CS 1: Discriminant function for separation of all patients with aortic valve
    stenosis from patients without Aortic valve stenosis (Table 7)
    (REF + OVD) versus (ASl + ASms)
(2) CS 1: Discriminant function for separation of all patients with moderate or
    severe aortic valve stenosis from all other patients
    (REF + OVD + ASl) versus (ASms)
    CS 2: Discriminant function for separation of the patients with mild aortic valve
    stenosis from patients without aortic valve stenosis (Table 8)
    (REF + OVD) versus (ASl)
   The primary aim was to achieve high sensitivity and specificity not less than
80%. For this reason, test method 2 was selected for aortic valve stenosis. This
method will be explained in detail.
Selected classification method for aortic valve stenosis
The purpose of the first stage of the test was to separate all patients with moderate
or severe aortic valve stenosis from all other patients. The results in Table 9 were
obtained for the training group with the developed parameter set PAS CS1.

                        Table 6.   Composition of patient groups.

                  Group arrangement         TrG        TeG          Total
                          REF                77         76           153
                          OVD                82         78           160
                           ASl               20         12           32
                          ASms               27         30           57

                  Table 7. Results of classification method 1 — Aortic
                  valve stenosis.

                                          TrG         TeG           Total
                  Sensitivity            85.1%       71.4%          78.7%
                  Specificity             93.1%       89.6%          91.4%
                  Correct diagnosis      91.3%       85.7%          88.6%

                  Table 8. Results of classification method 2 — Aortic
                  valve stenosis.

                                          TrG         TeG           Total
                  Sensitivity            89.4%        81%           85.4%
                  Specificity             93.7%       85.1%          89.5%
                  Correct diagnosis      92.7%       84.2%          88.6%
194                                             A. Voss et al.

                      Table 9.    Results of CSI — aortic valve stenosis (TrG).

                      AS CS 1             REF + OVD + ASl                  ASms
                      Total                     77 + 82 + 20                 27
                      Correct                   76 + 78 + 14                 26
                      In %                         93.9%                   96.3%

                      Table 10.   Result of CS2 — aortic valve stenosis (TrG).

                      AS CS 2                   REF + OVD                    ASl
                      Total                        77 + 82                   20
                      Correct                      74 + 77                   15
                      In %                          95%                     75%

    Parameter set PAS CS2 was to find patients with mild aortic valve stenosis
(Table 10).
    For the evaluation of the total result, the two test stages had to be considered
one after the other. For this, every patient who is not identified in CS1 as suffering
from moderate or severe aortic valve stenosis, is again examined at CS2 (which
includes undetected wrongly negative patients). This method was also evaluated
with the test group (Table 11).
    As expected, the test group performed less good than the training group. Alto-
gether, 94.7% of moderate or severe aortic stenoses and 68.8% of mild aortic stenoses
were detected. Total sensitivity for detection of aortic valve stenosis is 85.4%. Of
this, 4 of the detected 54 moderate or severe aortic valve stenosis cases (7%) were
only detected in the second test step and were therefore diagnosed as mild aortic
stenosis. In contrast 12 of 22 detected mild cases of aortic valve stenosis (54.6%) were
already detected at the first test step and therefore diagnosed as being moderate or
severe aortic valve stenosis than they actually are.
    The specificity for patients without valve disease is 93.5%, for patients with valve
diseases other than aortic valve stenosis it is 85.6%. Thus, a total specificity of
89.5% is obtained.
    Finally, 88.6% of all patients are classified correctly concerning the question
whether an aortic valve stenosis does or does not exist. The used parameters are
listed in Table 12.

            Table 11.      Overall result of the classification of aortic valve stenosis.

                                  TrG                        TeG                   Total
                           ASms          ASl        ASms            ASl     ASms            ASl
      True positive       27/27         15/20       27/30           7/12    54/57           22/32
      Sensitivity         100%           75%         90%           58.3%    94.7%           68.8%
                           REF           OVD         REF            OVD      REF             OVD
      True negative       74/77         75/82       69/76          62/78   143/153         137/160
      Specificity          96.1%         91.5%       90.8%          79.5%    93.5%           85.6%
                         Analyzing Cardiac Biomechanics by Heart Sound                                        195

               Table 12.      Explanation of parameter sets PAS CS1 and PAS CS2.

Parameter                                               Description

              AA        Parameter extrac-          Frequency          Time series     Extracted parameter
                        tion method                range (Hz)
PCS 1       REF + OVD + AVS l vs. AVS ms
Par1 1       3   STFT             15–172                              power-time-     area S4
Par1 2         2        WT                         172–689            Shannon         ratio   of area of
                          decomposition                                 entropy         S5    to total area
Par1 3         3        STFT                       15–689             power-time-     ratio   of area of
                                                                        STFT            S5    to total area
Par1 4         1        STFT                       172–345            power-time-     ratio   of area of
                                                                        STFT            S4    to total area
PCS 2       REF + OVD vs. AVS l
Par2 1       1   WT                                639–2205           power-time-     ratio of area of
                                                                        WT              S2 to total area
Par2 2         1        WT                         79–269             power-time-     ratio of area
                                                                        WT              of S4 to total area
Par2 3         2        FFT                        115–140            spectrum        total power
Par2 4         2        FFT                        115–165            spectrum        total power

   The associated discriminant functions are:
   CS1:      d = −2.827 + 0.073*Par1 1 + 7.273*Par1 2 + 9.975*Par1 3
             + 4.190*Par1 4.
   CS2:      d = −1.730 − 2.283*Par1 1 +2 4.818*Par1 2 − 0.008*Par1 3
             + 0.006*Par1 4.

Now, this method was also applied to the other four heart valve diseases: aor-
tic valve insufficiency, mitral valve insufficiency, tricuspid valve insufficiency and
pulmonary valve insufficiency. In contrast to aortic valve stenosis, classification
method 1 supplied the best results in each of these cases. The overall results of
the training and test groups analyses are presented in the following tables (Tables

Aortic valve insufficiency
            Table 13.     Overall results of the classification of aortic valve insufficiency.

                                       TrG                      TeG                 Total
                               AImms         AIg       AImms          AIg      AImms          AIg
          True positive        15/20      27/42        14/21          23/39    29/41        50/81
          Sensitivity          75%        64.3%        66.7%          59%      70.7%        61.7%
                               REF           OVD        REF           OVD       REF           OVD
          True negative        70/77      43/67        59/76      40/60       129/153       83/127
          Specificity           90.9%      64.2%        77.6%      66.7%        84.3%        65.4%
196                                              A. Voss et al.

Mitral valve insufficiency
          Table 14.     Overall results of the classification of mitral valve insufficiency.

                                    TrG                      TeG                 Total
                            MImms          MIg       MImms          MIg     MImms             MIg
         True positive      19/23        36/53        10/16         34/50    29/39          70/103
         Sensitivity        82.6%        67.9%        62.5%         68%      74.4%           68%
                             REF           OVD         REF          OVD       REF           OVD
         True negative      64/77        30/53        58/76        16/54    122/153         46/107
         Specificity         83.1%        56.6%        76.3%        29.6%     79.7%           43%

Tricuspid valve insufficiency

         Table 15.     Overall results of the classification of tricuspid valve insufficiency.

                                     TrG                     TeG                Total
                              TIms         TIg        TIms          TIg      TIms           TIg
         True positive       17/23         24/40      7/16        34/59     24/39        49/85
         Sensitivity         73.9%         60%       43.8%        57.6%     61.5%        57.7%
                              REF          OVD        REF          OVD       REF            OVD
         True negative       67/77       47/66       64/76        34/59     131/153      81/125
         Specificity          87%         71.2%       84.2%        57.6%      85.6%       64.8%

Pulmonary valve insufficiency

       Table 16.      Overall results of the classification of pulmonary valve insufficiency.

                                  TrG                        TeG                      Total
                                  PI                         PI                        PI
      True positive              18/22                      12/17                   30/39
      Sensitivity               81.8%                     70.6%                     76.9%
                           REF           OVD          REF           OVD       REF              OVD
      True negative       72/77         90/107       68/76         89/103    140/153          179/210
      Specificity          93.5%         84.1%        89.5%         86.4%      91.5%            85.2% Discussion
Very high values of sensitivity and specificity for the classification of aortic valve
stenosis were achieved. In particular, the detection of moderate and severe grades of
aortic valve stenosis is nearly 100% and even about 70% for mild forms. Besides, this
classification method provides information as to the severity of the disease. On the
other hand, most cases of mild aortic valve stenosis are classified higher whereas the
                   Analyzing Cardiac Biomechanics by Heart Sound                  197

classification of medium and severe levels of aortic valve stenosis is correct but for
a few exceptions. Most of the wrongly positive findings are not concerning patients
with healthy valves but patients suffering from valve diseases other than aortic valve
stenosis. In view of this, the result of specificity should also be rated highly. The
parameters Par 1 1 to Par 1 4 of the parameter set PAS CS1 concern all segments
4 and 5, i.e. the segments between the first and second heart sound. This is the
area at which the heart murmur of aortic valve stenosis typically occurs. Parameter
Par 2 1 of the parameter set PAS CS2 describes a change of the second segment
in which the first heart sound is located. This reflects the established diagnostic
criteria of the fainter heart sound. The changes in the frequency domain caused by
the occurrence of the heart murmur also appear, in particular, in parameters Par
2 3 and Par 2 4.
    The values for sensitivity and specificity of the test for aortic valve insufficiency
do not meet the expected target values. For the test, all aortic insufficiencies had
to be combined in one group. Mild, moderate or severe aortic insufficiencies are
detected at 70.7% with the result for the test group being about 8% below that
of the training group. The aortic insufficiencies of more severe aortic valve insuffi-
ciency only represent for about 30% of the total number so that their impact on
parameter optimization is not strong enough. Two-thirds of these patients suffer
only from a mild form whose characteristic changes apparently are not important
enough for better detection. Possibly these changes are masked by additional other
valve diseases. Thus, the sensitivity for mild aortic insufficiencies was only 61.7%.
The specificity also does not confirm the characteristic changes accompanying aor-
tic valve insufficiency or a mingling with other valve diseases. The specificity in
comparison with other valve diseases is only 65.4%, whereas it is a sufficient 84.3%
in comparison with patients with healthy valves.
    The classification of mild, moderate and severe mitral insufficiencies was suc-
cessful with a sensitivity of totally 74.4%. In this, the test group performed poorly
of 20% than the training group. The patient population in that group was rather
low giving the wrong classification of a single patient a massive impact on the per-
centage result. The detection of mild mitral insufficiencies reaches 68% sensitivity.
This result is satisfactory considering the mild grade of disease. The biggest problem
with the method chosen for mitral valve insufficiency is the very low specificity in
connection with patients with valve diseases other than mitral valve insufficiency,
in particular, in the test group, where it is below 50%. This means that more than
half of all patients who are suffering from a valve disease that is not mitral valve
insufficiency are diagnosed as being mitral insufficient. Also the number of patients
with wrongly positive detection who actually have healthy valves is fairly high when
compared with the other tests. One reason for this is the very high proportion of
mild mitral insufficiencies (over 70%). Less than 30% of all mitral insufficiencies are
of higher severity as a result of which the clear symptoms of this disease cannot
sufficiently be detected.
198                                    A. Voss et al.

    Typically, the tricuspid insufficiencies as diseases of the right heart very com-
monly occur together with left-side valve diseases. For example, 90% of patients
with higher-degree tricuspid valve insufficiency suffer from at least one other valve
disease, 80% even from at least 2 additional valve diseases. The detection of the
mild tricuspid valve insufficiency was not as high as expected. One should consider
that the pressure conditions in the right half of the heart are very much lower than
on the left side which explains that changes at the valves will only be noted at very
minor degree.
    The results of the pulmonary valve insufficiency test method are very good.
Sensitivity, however, is just short of 80% but it should be considered that with the
exception of one patient all patients have only mild valve insufficiencies at the right
half of the heart.
    The generated algorithms achieved a correct detection of patients with heart
valve disease of 76.3%, i.e. three of four patients are diagnosed correctly. This result
meets the expected target values, in particular, as most of the analyzed diseases are
at the initial stage. The specificities are a problem of the developed test method. If
a patient suffers from a cardiovascular disease other than a heart valve defect, there
is a high probability for that patient still to be diagnosed among the patients with
heart valve disease. In this case, it does make sense that the GP refers that patient
to the specialist despite the initially wrong suspected diagnosis.
    For a final validation of the method as a whole, a prospective study with entirely
new patient data will be carried out.
    At the next step, it will be necessary to optimize the preprocessing procedures,
for one, to make this process fully automatic (e.g. develop a system of automatic
noise detection) and, for another, to further reduce computing time (e.g. by calcu-
lating the representative heart period in a narrower frequency band).
    Another aim is the improvement of the classification results. For this, several
approaches can be combined. For example, one possibility is to include the corre-
lation coefficient as an additional parameter. Unfortunately, this would make the
time for computing even longer and necessary to reduce signal preprocessing which
might lead to worse quality of other parameters. Besides, the developed parameters
could be used as inputs for a neuronal network which can use non-linear relations
among them for classification. Finally, other approaches can be applied, such as the
development of a classification method on fuzzy basis.
    Furthermore, it is planned to apply these principles to classify other heart dis-
eases. These will mainly be congenital heart defects (e.g. septum diseases) and
myocardial diseases (cardiomyopathies) as well as vascular diseases as these lead to
changes in the heart and vessel sounds.
    Another area of application where the developed methods and principles could be
used is in follow-ups after heart valve operation or other cardiological interventions,
e.g., after implantation of a stent.
    Summarizing, it can be said that the developed algorithms are suitable for an
early detection of patients with heart valve disease. They require inexpensive tech-
nical equipment (commercial medical PC).
                       Analyzing Cardiac Biomechanics by Heart Sound                               199

4.3. Heart sound analysis for risk stratification in patients with
     heart failure
Heart failure is one of the most common diseases worldwide and has a high preva-
lence especially in the industrialized countries. In America, about 5 million people
suffer from heart failure37 and approximately 10 million in Europe.38 Every year,
further 550000 patients contract the disease, both in America and in Europe (inci-
dence). The direct and indirect costs for the treatment of heart failure are estimated
to amount to about USD 29.6 billion in America. According to the National Center
for Health Statistics (NCHS)1 57218 patients died of the direct consequences of
heart failure in America in 2003, the 1-year mortality rate is about 30%.39
    The term “heart failure” denotes the weakening of the right and/or left
myocardium, which changes the biomechanics and, in particular, restricts the ejec-
tion fraction of the heart. As a result of this, the organism is not supplied with
enough blood and nutrients. The development of heart failure can be triggered by
a large number of adverse factors and can occur quickly or take longer to develop
(months/years). The most frequent causes of heart failure are40 :

• Excessive pressure load on the heart, e.g. due to heart valve diseases (aortic valve
  stenoses) or high blood pressure causes myocardial hypertrophy (Fig. 18),
• Excessive volume load on the heart, e.g. due to heart valve diseases (aortic or
  mitral valve insufficiency) leads to a dilatation of the left and/or right ventricle
  (Fig. 18),
• Disturbed filling of the left ventricle, e.g. due to mitral stenosis, restrictive car-
• Diminished pumping efficiency due to primary myocardial disease, e.g. cardiomy-
  opathy, coronary heart disease,
• Indirect diseases, such as diabetes, thyroid disease.

   At the initial stage of heart failure (compensated heart failure), the organism can
compensate the reduced cardiac output by appropriate compensation mechanisms so

Fig. 18. Schemes of healthy heart, hypertrophic heart (myocardium is hypertrophied, increased
wall thickness especially of the left ventricle) and a dilated heart (enlarged heart, dilatation of the
left and/or right ventricles, reduced wall thickness).
200                                   A. Voss et al.

that none or only minor discomfort appears during the patients normal course of life.
On a continuing basis, compensation mechanisms damage the circulatory system,
the organs and also the tissue and lead to a progressive weakening of the heart. The
progressive heart failure cannot constantly be compensated (decompensated heart
failure) and causes symptoms such as shortness of breath, fatigue, fluid retention
(edema) in the lung, tissue and organs, and discomfort during mild physical activity.
    The diagnosis of heart failure can be based on resting or stress ECG, echocar-
diography, X-ray examination, heart catheter and laboratory values. The severity of
heart failure is assessed by means of the subjective NYHA (New York Heart Associa-
tion) index which classifies the functional capacity (NYHA I–IV) of a patient. Heart
failure can be treated as follows: By wholesome diet, adequate physical activity, med-
ication (ACE-blocker, beta-blocker, digitalis, diuretics) and when in advanced stage
by implantation of a cardiac pacemaker or a defibrillator or heart transplantation.41
    Until now an early detection of high-risk patients with heart failure has not
been solved satisfactorily. In a first pilot study we investigated how far different
methods for the analysis of cardiovascular variability and heart sound provide addi-
tional diagnostic information for a prognostic individual risk stratification of heart
    Under standardized resting conditions, 30-minutes of ECG and non-invasive con-
tinuous blood pressure (Portapres M2, TNO-TPD, Amsterdam, Netherlands) were
recorded from 43 patients with heart failure (NYHA ≥ 2). Afterwards, heart sound
on 9 defined auscultation areas (Welch Allyn r Master Elite Plus stethoscope) and
synchronized ECG and blood pressure were acquired over five heartbeats each. To
eliminate respiratory sounds, the acoustic signals were recorded during a short res-
piratory pause. Use of an USB sound card (Maya EX, Audiotrak, sample frequency:
44.1 kHz, resolution: 16 bit) allowed digitization of the heart sounds (Fig. 19).
    During data preprocessing the tachograms from ECG (time distance between
successive R waves) as well as systograms and diastograms (succession of systolic
and diastolic blood pressure values) from the blood pressure were extracted. Follow-
ing, artifacts and extrasystolic beats were replaced by interpolated beats. Heart rate
variability was quantified by defined parameters from time and frequency domain
according to the Task Force of the European Society for Cardiology.43
    For the classification of the dynamic, biomechanical interactions between ECG
and blood pressure, additional nonlinear parameters of the Joint Symbolic Dynamics
(JSD) were calculated.44 The analysis of interactions by means of JSD is based on
the transformation of the tachograms and systograms in symbol sequences consisting
of “0” (heart frequency and blood pressure decrease) and “1” (heart frequency and
blood pressure increase). Thereafter, words are formed from the symbol sequences.
The maximum word length is determined by a statistically required probability of
the occurrence of the words and thus the length of the symbol sequences. If a uniform
distribution and a required minimum frequency of occurrence of 30 events per bin
are assumed, a word length of three is obtained for the 30-minute standardized
recordings made as part of the analysis at an estimated mean heart frequency of
                   Analyzing Cardiac Biomechanics by Heart Sound                201

                            Fig. 19.   Measuring equipment.

75 beats per minute. Consequently, a 3 × 3 matrix W is spanned from the word type
[000,000] to the word type [111,111]. To ensure the comparability of data records of
different length the total occurrence is normalized to 1. Due to this simplification
of time series, some detailed information of the time series is lost whereas robust,
invariate information will be stayed.
    In addition to the analysis of variability within a time series and the interac-
tion analysis between the recorded signals, the examination of the blood pressure
morphology supplies important information.45 For this, parameters which describe
the amplitudes and the temporal characterization (intervals, slopes) of the blood
pressure morphology were estimated (Fig. 20).
    The analysis of heart sound as an important indirect method to determine the
biomechanical characteristics of the heart is based on the determination of param-
eters describing the first heart sound (frequency range: 5–100 Hz) and the second
heart sound (100–150 Hz) and the temporal coupling with ECG and blood pressure.
The heart sounds were detected by a wavelet method.23 At first, RR interval related
heart sound sections were generated for every auscultation point and decomposed
in the frequency range 0–172 Hz by wavelet decomposition.36 The time points of
the first and second heart sounds were detected applying the methods of Shannon
energy and STFT.46 In case of the Shannon energy method the considered filtered
heart sound interval was divided into 40 segments. From the mean values of the
202                                      A. Voss et al.

Fig. 20. Extracted parameters from blood pressure morphology. SYS — end-systolic blood pres-
sure; DIA — end-diastolic blood pressure; BPA — blood pressure amplitude; SSS — maximum
systolic slope; MSS — mean systolic slope; SDS — maximum diastolic slope; IC — incisure; DP —
dicrotic peak; T1-T4 — time intervals; A1-A6 — areas below the blood pressure curve.

Shannon energy of each segment, the envelope of the heart sound interval was con-
structed. Within this envelope, the maxima for the first and second heart sounds
were searched within certain time ranges. The time window for the first heart sound
is 0–200 ms (= 1st to 7th sample point of the envelope) after the R wave and for the
second heart sound it is 250–700 ms (= 15th to 25th sample points of the envelope)
after the R wave. In addition, the short-time Fourier transform (Blackman-Harris
window, window length = 4096 samples, time shift = 100 samples) was calculated
for the analyzed heart sound interval. Within the transform, the maximum of the
first heart sound was also determined in the range of 0–200 ms (STFT section from
0 to 60) and of the second heart sound in the range of 250–700 ms (STFT section
from 145 to 250) after the R wave. As a next step, the difference of the two time
points (from both methods) for the first heart sound was calculated. If the difference
was less than 3000 sample points, the mean of both time points was determined and
used as position of the first heart sound, otherwise the time point of the first heart
sound was discarded. The position of the second heart sound was defined in the
same way.
    During a 6-month follow-up, 7 of the total of 43 patients died due to a cardiac
event, in 15 patients the state of health was deteriorated in comparison with the
time of data acquisition (group of high-risk patients). The state of the remain-
ing 21 patients did not deteriorate (group of low-risk patients). The statistical
evaluation for determination of univariate differences between high-risk and low-
risk patients was based on the Mann-Whitney U test. Considering the Bonferroni
                   Analyzing Cardiac Biomechanics by Heart Sound                   203

criterion for multiple tests, the significance level of univariate statistics was cor-
rected. An optimized parameter set for risk stratification of heart failure patients
was determined applying the Cox regression model.
    The statistical analysis of the calculated parameters of different domains showed
that the standardized parameters of heart rate variability did not contribute to
the assessment of the individual risk in case of heart failure. On the other hand,
parameters of blood pressure morphology and non-linear JSD as well as from the
heart sound analysis showed significant differences between high-risk and low-risk
patients. Using one high-significant parameter from each of the three analysis meth-
ods, an optimized parameter set for risk stratification of heart failure was developed.
By application of this parameter set, 93.3% of survivors (specificity) and 91.7% of
high-risk patients (sensitivity) were classed correctly. The parameter HS1 in the
set corresponds with the time shift from the R wave to the first heart sound and
is increased in high-risk patients. The time delay between the QRS complex and
the first heart sound is based on the stimulus conduction in the ventricles. After
electrical excitation of the septum (Q wave), the excitation reaches the apex of the
heart (R wave) and causes increased tension of the ventricles (beginning of the first
heart sound). The maximum of the first heart sound is reached when the electrical
excitation has been conducted through the ventricular walls causing contraction of
the entire ventricular myocardium. The parameter HS1 describes the time delay
between R wave and maximum of the first heart sound. Another parameter BBI011
of JSD describes the probability of occurrence of the symbol sequence “011” within
the RR interval time series and is decreased in high-risk patients. In addition, a less
steep negative slope was found between the dicrotic wave and the diastolic minima
in high-risk patients (parameter SDS of the blood pressure morphology analysis).
    Several studies proved the occurrence of a third heart sound47,48 in patients
with heart failure. In patients with progressive heart failure, the third heart sound
occurs almost obligatory in 96% of all patients.49 The third heart sound indicates
a reduced left ventricular function and is generated by the transmitral inflow as an
oscillation signal of the intracavitary blood volume, the ventricular walls and the
surrounding structures.50 Patients with limited ejection fraction of less than 50%
were identified by the third heart sound with a sensitivity of 51% and a specificity
of 90%.51

5. Discussion
In the industrialized nations, the incidence of cardiac diseases with partly massive
impairment of the biomechanics of the heart is constantly increasing and is now
the Number One in the mortal statistics in many of these countries. This points to
the need of detecting cardiac diseases as early as possible to start treatment with
promise of success. Because a patient usually consults the family doctor first, it is
very important that the latter can rely on a simple-to-use, non-invasive and low-cost
system for screening.
204                                    A. Voss et al.

    The heart allows blood to flow from the venous to the arterial vascular system
and in this way supplies all organs of the body with sufficient amounts of oxygen.
This requires an intricate and accurate interplay of cardiac ventricles and muscles
and the heart valves. If this system is disordered, a heart disease develops as a result
of which certain parts of the body receive less oxygen than they need. The disease
or disorder can involve the biomechanical system as such (heart defect, myocardial
diseases, heart valve disease) or other areas of the heart as drivers or suppliers of
the system (coronary heart disease, arrhythmia).
    The biomechanics of the heart can primarily by analyzed by the signals of the
heart sound. The analysis of the heart sounds and cardiac murmurs can provide
information on the state of the heart. The position of cardiac murmurs in relation
to the heart sounds is a decisive criterion for the diagnosis of a cardiac disease. Heart
defects, myocardial diseases and heart valve diseases can be detected by the heart
sounds, in particular, because the disordering of the biomechanical system causes
typical changes. These changes can be analyzed and diagnosed by auscultation
and phonocardiography. Both methods are very sophisticated and require a high
degree of specialization if a disease is to be detected early. Another problem is the
susceptibility of these methods to patient movements and murmurs, e.g. breathing,
action of the bowels, and external noise.
    In the last few years modern imaging methods, such as computer tomography
and magnetic resonance tomography have been making inroads in the clinical arena.
These methods are not available to the general practitioner or the family doctor
due to the high level of technical equipment and costs.
    The purpose of these studies was, on the one hand, the development of a fully
automatic system for diagnosing heart valves diseases which relies on an electronic
stethoscope and a commercial PC. This should assist the GP in a first finding
after which the patient can be referred to the required specialist, in particular, a
cardiologist or internist. For this, the heart sound should be studied by methods of
time and frequency analysis. The parameters extracted from these methods were
statistically evaluated in a real patient population. On the other hand, the suitability
of heart sound analysis for characterizing basic cardiac diseases (no heart valve
diseases) was to be established. In this respect, it was shown that the time delay
between the R wave and the maximum of the first heart sound is an important
measure for the risk assessment in patients with heart failure.
    Proceeding from these very encouraging results from the author’s work and the
recent results of other groups, the analysis of the biomechanics of the heart based on
the heart sound analysis can be regarded as a very promising method, in particular,
for screening of heart diseases by the general practitioner.

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                                        CHAPTER 6

                 VASCULAR SYSTEMS

                                  S. J. PAYNE∗ and S. UZEL
                   Department of Engineering Science, University of Oxford
                            Parks Road, Oxford OX1 3PJ, UK

    Simple one-dimensional models of blood flow are widely used in simulating the transport
    of blood around the human vasculature. However, the effects of gravity have only been
    previously examined briefly and the effects of changes in wall properties and their interac-
    tion with gravitational forces have not been investigated. Here the effects of both gravita-
    tional forces and local changes in wall stiffness on the one-dimensional flow through axi-
    symmetric vessels are studied. The governing fluid dynamic equations are derived from
    the Navier-Stokes equations for an incompressible fluid and linked to a simple model of
    the vessel wall, derived here from an exponential stress-strain relationship. A closed form
    of the hyperbolic partial differential equations is found. The flow behavior is examined
    in both the steady state and for wave reflection at a junction between two sections of dif-
    ferent wall stiffness. A significant change in wave reflection coefficient is found under the
    influence of gravity, particularly at low values of baseline non-dimensional wall stiffness.

    Keywords: Gravity; wall properties; blood flow; vascular systems.

1. Introduction
The flow of blood around the circulatory system is the fundamental basis of the
human anatomy. The correct understanding of the behavior of blood flow in the
vasculature is thus crucial if any realistic physiological model of the human anatomy
is to be accurate and representative. The difficulties involved in modeling the flow
of blood in the vessels are well-known: There is considerable interaction between the
fluid flow and the vessel wall, which makes modeling this flow a non-trivial matter.
    The governing fluid dynamics equations, the Navier-Stokes equations, are known,
but to solve them the coupling between the fluid and vessel wall behaviors must be
known. Since the vessel wall has a very complex structure, comprising three layers of
different structure and function, it is frequently characterized with a simple pressure-
area relationship, although more advanced mechanical models have also been used,
as outlined below. This enables the fluid-structure equations to be reduced to two
area-averaged equations, as shown below.
    Under certain assumptions, these equations can be approximated by a set of
equations that is of identical form to an equivalent electrical circuit: The use of this
analogy is extremely widespread and forms the basis for most models of the whole
vasculature, although the precise nature of the equivalent circuit varies widely from
author to author. Units, each comprising resistors, inductors and capacitors, are

208                                S. J. Payne & S. Uzel

generally joined together to form a complete circuit. Essentially resistance, induc-
tance and capacitance are included to model friction, inertia and wall compliance
    With the advent of powerful computers and efficient Computational Fluid
Dynamics (CFD) solvers, many problems that were not previously amenable to
analysis can now be solved numerically. However, one of the major drawbacks of
using detailed numerical solvers is that only small regions of the vasculature can be
examined at any given time. The setting of suitable boundary conditions is crucial
and lumped parameter models are still frequently used.
    However, one-dimensional models of blood flow generally assume that the wall
properties are invariant with distance along the vessel. It is well known that vessel
walls exhibit local regions of increased stiffness, for example stenoses, or decreased
stiffness, for example aneurysms. CFD solvers have been widely used to examine the
details of the flow fields around such local variations, but the equations governing
one-dimensional flow around such features have not received equivalent attention.
Clearly, an assumption has to be made that the wall properties are invariant with
circumferential position, but this should still provide a good first approximation to
realistic situations. In addition, the effects of gravity on such features have not been
    In this chapter, the equations governing flow in an axi-symmetric tube with an
elastic wall are thus derived, including gravitational forces and variable wall stiffness.
The theory is presented and some numerical simulations are shown to illustrate the
effects of gravitational forces and variable wall stiffness on the flow field.

2. Theory
2.1. Fluid dynamic equations
The approach set out in Canic and Kim1 is followed here, starting with the
incompressible axi-symmetric form of the Navier-Stokes equations in cylindrical
co-ordinates (x, r, θ). The momentum equations are:
       ∂ux      ∂ux      ∂ux   1 ∂p    ∂ 2 ux   1 ∂ux ∂ 2 ux
           + ur     + ux     +      =ν        +      +       − gS,                   (1)
        ∂t       ∂r      ∂x    ρ ∂x     ∂r2     r ∂r   ∂x2

       ∂ur      ∂ur      ∂ur   1 ∂p    ∂ 2 ur   1 ∂ur  ur   ∂ 2 ur
           + ur     + ux     +      =ν        +       − 2 +        ,                 (2)
        ∂t      ∂r       ∂x    ρ ∂r     ∂r2     r ∂r   r    ∂x2
and the continuity equation is:
                                ∂ux   1 ∂(rur )
                                    +           = 0,                                 (3)
                                ∂x    r ∂r
where the velocity components are given by (ux , ur , uθ ) and the slope of the vessel
is denoted by S. The value of slope varies between 1 and −1 for vessels inclined
vertically upwards and downwards respectively in the direction of flow. It is equal
                 Effects of Gravity and Wall Properties in Blood Flow              209

to the sine of the angle of the vessel axis with the horizontal. The pressure is p and
the fluid has density ρ and viscosity ν. It is assumed that the vessel radius is small
in comparison with its length, so the gravitational force component in the radial
direction can be neglected.

2.1.1. One-dimensional flow
Using order of magnitude arguments, the penultimate term in Eq. (1) is neglected
and the radial pressure gradient can be assumed small. Equations (1) and (3) are
then integrated over the cross-sectional area to give:
                               ∂(R2 ) ∂(R2 U )
                                     +         = 0,                               (4)
                                 ∂t     ∂x
           ∂(R2 U ) ∂(αR2 U 2 ) R2 ∂p              ∂ux
                   +           +      + gSR2 = 2νR                           ,    (5)
             ∂t        ∂x        ρ ∂x               ∂r                 r=R

where the following non-dimensional parameters are used:
                               U=                  2rux dr,                       (6)
                                     R2      r=0
                             α=                     2u2 rdr,                      (7)
                                    R2 U 2    r=0

where U is the area-averaged axial velocity and α is the correction term, also known
as the Coriolis coefficient, which compensates for the fact that the final equation will
be based on conservation of the area-averaged momentum, rather than the actual
momentum. The wall inner radius is R, which varies with both longitudinal distance
and time.
    To calculate the viscous term on the right-hand side of Eq. (5), the velocity
profile is often assumed to be of the form of Hagen-Poiseuille flow:
                                    γ+2      r            γ
                           ux =         U 1−                   ,                  (8)
                                     γ       R
where γ is a constant that defines the shape of the velocity profile: A value of 2
corresponds to a Newtonian fluid, although much larger values have been reported
to be a good compromise fit to experimental data, reflecting the fact that blood is
not quite a Newtonian fluid.2 Equations (4) and (5) are then expressed in terms of
flow rate, Q, and area, A:
                                    ∂A ∂Q
                                       +    = 0,                                  (9)
                                    ∂t   ∂x
               ∂Q   ∂          Q2        A ∂p            παν Q
                  +        α         +        + gSA = −2       .                 (10)
               ∂t   ∂x         A         ρ ∂x            α−1A
   Although area and flow rate are used as the primary variables of interest,
these equations can be reformulated in terms of the area and area-averaged axial
210                                S. J. Payne & S. Uzel

velocity3 :
                            ∂A    ∂U    ∂A
                               +A    +U    = 0,                                   (11)
                            ∂t    ∂x    ∂x

        ∂U           U 2 ∂A             ∂U   1 ∂p           παν U
           + (α − 1)        + (2α − 1)U    +      + gS = −2       .               (12)
        ∂t           A ∂x               ∂x   ρ ∂x           α−1 A
The viscosity of the fluid is well known to vary across the vessel cross-sectional
area. Near to the wall, the fluid comprises a plasma layer, the red blood cells being
confined to a core layer in the center of the vessel. Although two-phase models of
the flow have been used,4 for a one-dimensional model of fluid flow only the cor-
rection factor and wall shear stress are required, so such models are not widely
used here. The velocity profile, although well fitted by a Hagen-Poiseuille profile
with a high value of γ, thus still has a low shear stress, since the plasma viscos-
ity is much lower than that of the core flow. The right-hand side of equations 10
and 12 is thus often neglected, even though α is close to 1, due to the high value
of γ.

2.1.2. Two-dimensional flow
Under certain conditions, the flow cannot be assumed to be one-dimensional, for
example where there are local changes in wall stiffness. If the two-dimensional flow
field is to be solved, assuming again that the flow field is axi-symmetric, the govern-
ing equations can be written in the vorticity/stream-function form,5 for example.
Since this approach is much less amenable to simple analysis, it is not examined
further here. In addition, the two-phase nature of the fluid means that, except in
larger blood vessels, the viscosity is not constant over the cross-sectional area and
some means must be employed to determine the boundary between the flow layers.

2.2. Vessel wall equations
Since the fluid dynamic equations are expressed in terms of three variables, but
there are only two equations, the vessel wall equations play a very important role in
determining the fluid behavior. This fluid-structure interaction is of great interest
for a number of reasons, including the fact that wall shear stress is implicated in
the behavior of the vessel wall. In particular, regions of low shear stress are thought
to be implicated in the onset of atherosclerosis,6 although the precise mechanisms
for this remain contested.7 The development of a physiologically realistic model for
the vessel wall is complicated by the fact that the vessel wall comprises three layers,
each of which has a different structure and function and the relative fractions of
which vary widely around the vasculature.
    However, it is generally assumed that the wall has a small thickness and thus
has negligible inertia in comparison to the fluid. The equilibrium equations are thus
very widely used. The stress field will be assumed uniform over the wall thickness
                  Effects of Gravity and Wall Properties in Blood Flow            211

here. We briefly examine a number of different models, considering first models that
consider both axial and radial deformations before examining those that include only
radial deformation. Since such models can be expressed in terms of a pressure-area
relationship, such relationships are finally reviewed.

2.2.1. Two-dimensional deformation
The general equations for an axi-symmetric membrane with finite deformation can
be expressed in terms of the principal deformation ratios8 :
                                              2             2
                                        ∂R             ∂S
                       λ1 =       1+                            ,               (13)
                                        ∂x             ∂x

                                     λ2 =      ,                                (14)
where R, S and Ru are the wall radius, axial position and undeformed radius respec-
tively. The equilibrium equations in the tangential and normal directions are:
                 ∂R                ∂T1                      ∂R
                    (T1 − T2 ) + R     = τR         1+                  ,       (15)
                 ∂x                ∂x                       ∂x

                ∂ 2R        T1                   T2
            −                          +                     = p,               (16)
                ∂x2 (1 + (∂R/∂x)2 )3/2   R(1 + (∂R/∂x)2 )1/2
in terms of the applied wall shear stress, τ , and pressure. T1 and T2 refer to wall
tensions in the longitudinal and circumferential directions respectively. Since the
membrane is assumed thin, it is frequently assumed to have negligible bending
stiffness, as in the analysis presented here.
    The relationship between the deformation ratios and the wall tensions is the final
link in the model. A linear visco-elastic model3,5,9 can be used, which character-
izes the wall behavior by a non-dimensional elasticity coefficient, K, and viscosity
coefficient, C:
                                 λ2   3            ∂λ1   1 ∂λ2
                 T1 = K λ1 +        −        +C        +                ,       (17)
                                 2    2             ∂t   2 ∂t
                                 λ1   3            ∂λ2   1 ∂λ1
                 T2 = K λ2 +        −        +C        +                .       (18)
                                 2    2             ∂t   2 ∂t
Although K can be determined in terms of the Young’s modulus and wall thickness,
the value of C is much less well characterized. Its value, however, has a significant
effect on the fluid flow patterns.9
   The linearized version of these equations3 can be written in the form:
                                        3 Eh ∆R
                                 ∆p =           ,                               (19)
                                        4 Ro Ro
where Ro is the characteristic value of radius.
212                                  S. J. Payne & S. Uzel

    The inclusion of axial deformation makes the analysis highly complex: however,
since the axial displacements are some orders of magnitude smaller than the radial
displacements,3 these are frequently neglected. The vessel deformation is thus con-
sidered purely in radial terms, which means that the model reduces to a simple
pressure-area relationship. This approach is thus now examined in detail as it forms
the basis for a considerable proportion of the fluid-structure interaction models.

2.2.2. One-dimensional deformation
The model for one-dimensional deformation is often referred to as the ‘independent
ring model’. It has been shown that this gives the leading-order term in the approx-
imation of the pressure in terms of the displacement.10 The equilibrium equation
reduces to11 :

                              pR = pext (R + h) + σθ h,                           (20)

where the vessel wall has thickness h, the circumferential stress component is σθ and
the external pressure is pext . Note that the external pressure is frequently neglected
or it is assumed that pressure refers directly to trans-mural pressure, which is only
the case for negligible wall thickness. This equation can be re-arranged into the
                                                h           h
                             p = pext 1 +            + σθ     .                   (21)
                                                R           R
      If it is assumed that the vessel wall is incompressible:
                         (R + h)2 − R2 = (Ro + ho )2 − Ro ,                       (22)

neglecting any longitudinal strain, the wall thickness to inner radius is:
                      h                    ho Ao           ho         Ao
                        = −1 +      1+2          +                       ,        (23)
                      R                    Ro A            Ro         A
which, for thin-walled vessels, is approximately:
                                     h   ho Ao
                                       =       .                                  (24)
                                     R   Ro A
      The circumferential strain can be directly related to the change in area:
                                             1 dA
                                     dεθ =        ,                               (25)
                                             2 A
which gives:
                                         1      A
                                  εθ =     ln          .                          (26)
                                         2      Ao
The relationship between circumferential stress and strain is thus the final step in
determining the pressure-area relationship, linking Eqs. (21) to (26). The simplest
                 Effects of Gravity and Wall Properties in Blood Flow               213

relationship is that which assumes a linear isotropic material with constant Young’s
modulus and Poisson’s ratio, i.e. Hooke’s law:
                                εθ =     (σθ − νσz ),                             (27)
radial stress being much smaller than circumferential or longitudinal stresses for thin
walled vessels. However, since the vessel wall increases in stiffness with increased
strain, a nonlinear relationship is more accurate. One such possible relationship is
of the form:
                                1       k
                         εθ =     ln 1 + (σθ − νσz ) ,                            (28)
                                k       E
where k is a constant that quantifies the non-linearity of the stress-strain relation-
ship, which assumes that the stress is related to the exponential of the strain. As
k tends to zero, Eq. (28) tends towards Eq. (27). Another, similar, relationship
between stress and strain that can be used is derived from the strain energy rela-
tionship proposed by Zhou and Fung12 :
                                  K          2
                            W =     (exp(a1 Eθ ) − 1),                            (29)
where K is a measure of the stiffness and a1 a measure of the non-linearity of
the material, W being the strain energy. Longitudinal strain is assumed zero here,
hence this expression is a simplified version of that given by Zhou and Fung.12
Circumferential stress is then:
                       σθ = Ka1 Eθ (1 + 2Eθ ) exp(a1 Eθ ),                        (30)

where the strain component is related to the change in cross-sectional area by:
                                        1 A
                                 Eθ =        −1 .                                 (31)
                                        2 Ao
    The relationship between pressure and area can be directly calculated using
either Eq. (28) (Eq. (27) being taken as a special case of Eq. (28), where k tends to
zero) or Eq. (30), using the equilibrium conditions for circumferential stress (Eq. 21)
and longitudinal stress:
                                       h              h        h
                     p = pext 1 +              + σz       2+       .              (32)
                                       R              R        R
   The pressure-area relationship for the logarithmic model (Eq. 28) is thus:
                          ho 1         E     1     ho                   1
           p = pext 1 +           +                        a(k/2)−1 −       ,     (33)
                          Ro a         k (1 − ν/2) Ro                   a
                                       a=         .                               (34)
214                                     S. J. Payne & S. Uzel

Since the wall thickness is assumed small, only the leading terms have been consid-
ered. In the limit as k → 0, Eq. (33) tends towards:
                                      ho 1            E      ho ln (a)
                    p = pext 1 +              +                        ,                       (35)
                                      Ro a        2(1 − ν/2) Ro a
which is the response for a linear elastic material. The corresponding relationship
for the strain energy model (Eq. 30) is:
                        ho 1        Ka1 ho    1                 a1
        p = pext 1 +            +                   (a − 1) exp    (a − 1)2 .                  (36)
                        Ro a         2 Ro (1 − ν/2)             4
      For convenience, we define the baseline stiffness of the vessel as:
                                      Go =               ,                                     (37)
                                              ∂a   a=1

which gives the logarithmic and strain energy pressure-area relationships as:
                                     ho 1           2                     1
                    p = pext 1 +             + Go            a(k/2)−1 −       ,                (38)
                                     Ro a           k                     a
                                 ho 1                             a1
                p = pext 1 +             + Go (a − 1) exp            (a − 1)2 .                (39)
                                 Ro a                             4
   The resulting pressure-area characteristics are shown in non-dimensional form
in Figs. 1 and 2 for varying values of k and a1 . External pressure is taken to be
zero here. Large values of k are required to give a significant non-linear effect in the

 Fig. 1.   Pressure-area relationship for varying values of k for logarithmic and linear function.
                    Effects of Gravity and Wall Properties in Blood Flow                           215

     Fig. 2.   Pressure-area relationship for varying values of a1 for strain energy function.

logarithmic model, since a linear material automatically gives a reduction in stiffness
with pressure. This indicates that the vessel wall material is very strongly nonlinear.
However, the stiffness increases with raised pressure for any positive value of a1 ,
although it should be noted that the value of a1 quoted by Zhou and Fung12 is 0.264,
for which the area increases very rapidly with pressure, which is not physiologically
    It is also convenient to consider the compliance of the vessel, since this is known
to exhibit certain characteristics. This is defined here in non-dimensional terms as:
                                        C = Go       ,                                           (40)
and for the logarithmic and strain energy relationships is, respectively:
                                    k            a2
                              C=                                ,                                (41)
                                    2     k
                                            − 1 ak/2 + 1
                                   1           a1
                     C=         a1        exp − (a − 1)2 .                                       (42)
                                        2      4
                             1 + (a − 1)
    The resulting compliance against area ratio for both relationships is shown in
Figs. 3 and 4. Note that for the definition of compliance used here, the value is
always equal to one at zero pressure. Both Eqs. (41) and (42) show a peak in com-
pliance, which is exhibited by experimental data.13 However, this peak is found at
216                                  S. J. Payne & S. Uzel

      Fig. 3.   Compliance for varying values of k for logarithmic and linear function.

          Fig. 4.   Compliance for varying values of a1 for strain energy function.
                    Effects of Gravity and Wall Properties in Blood Flow                         217

an area ratio that is dependent upon the non-linearity parameter for the logarithmic

                               apc =                           ,                            (43)
                                        (k − 2)(k − 4)

whereas it is always found at baseline area for the strain energy model. The former
is more physiologically realistic.13 For the logarithmic model, the area ratio at peak
compliance is a monotonically decreasing function for medium values of k, as shown
in Fig. 5. Note that a maximum is only found in the range k ≥ 4. As the value of
k increases past 6, the peak compliance is found below an area fraction of 1. As k
tends towards a very large value, the area at peak compliance tends back towards 1,
having a minimum value of 0.679 at a value of k of 13.4.
    The logarithmic model thus has two advantages: Peak compliance that occurs
at different area ratios, although, since there is only one degree of freedom in the
model, this is wholly dependent upon the nonlinearity parameter, and algebraic
simplicity in comparison with the strain energy relationship, which, although based
on experimental data, results in a complex pressure-area relationship that is not
amenable to inclusion within the one-dimensional flow equations. Before considering
this coupling of the fluid and vessel equations, we briefly survey some of the existing
pressure-area relationships available in the literature.

    Fig. 5.   Area at peak compliance as a function of k for logarithmic and linear function.
218                                S. J. Payne & S. Uzel

2.2.3. Existing pressure-area relationships
Probably the most popular form of the pressure-area relationship used in the fluid-
structure interaction equations is the power law relationship2 :
                        p − pext = Go                  −1 ,                      (44)

which is a generalized version of the expression quoted by Formaggia,14 where Go
is given by:
                                        ho    E
                               Go =                .                             (45)
                                        Ro (2 − ν)
This power law relationship is clearly very similar to that derived above, based on
the logarithmic model for stress-strain (Eq. (38)), although Eq. (38) includes a finite
wall thickness. Equation (44) also has a monotonically decreasing compliance with
area, which is not found in Eq. (38) and which is found in the experimental data.
Olufsen15 quotes a similar relationship:
                                   4 Eho          A
                      p − pext =            1−                     ,             (46)
                                   3 Ro           Ao
                               = k1 exp(k2 Ro ) + k3 ,                           (47)
based on the experimental results of several studies.16–18
   A quadratic relationship was proposed by Stergiopoulos17 :
                 A = Aref (1 + Co (p − pref ) + C1 (p − pref )2 ),               (48)
an exponential relationship by Hayashi19:
                         p = po exp β            −1        ,                     (49)

and a tangential relationship by Langewouters13:
                                                A   1
                       p = po + p1 tan π          −            ,                 (50)
                                               Am   2
where po is the pressure at maximum compliance and p1 the ‘half-width’ pressure,
i.e. the change in pressure from the pressure at maximum compliance required to
halve the compliance. Am is the maximum area of the vessel at high pressure,
beyond which the pressure cannot increase. All other parameters in Eqs. (48)–(50)
are constants. The data presented by Langewouters13 is of particular interest as the
compliance shows a clear peak at low area ratios.
     From the above survey, it is clear that there is no generally agreed form of
the pressure-area relationship for use in the fluid-structure interaction equations.
Although they all exhibit an increasing stiffness with pressure, they differ in their
algebraic complexity and in their ability to model the peak in compliance found
                 Effects of Gravity and Wall Properties in Blood Flow                          219

in experimental data. The optimal relationship, however, should be of a form that
it models the variation in compliance accurately and that it can be incorporated
within the fluid equations simply. The only such relationship is the one that we have
derived here (Eq. (38)). This is directly based on the material properties, assuming
small, but not negligible wall thickness, and predicts a peak in compliance, whilst
being algebraically simple. It will thus be incorporated within the fluid-structure
equations, as now outlined below.

2.3. Coupled form of equations
The pressure-area relationship in Eq. (38) is substituted into Eq. (10). The vessel
wall properties are assumed all to be invariant with time, although the wall stiffness
is allowed to vary with axial distance. This latter addition is used to model the fact
that there may be local variations in the wall properties, for example at an aneurysm
or a stent.
    The external pressure is determined by static equilibrium:
                              pext = pext,o − ρgSx,                                       (51)
relative to a reference pressure, defined at x = 0, which is taken to be the vessel
inlet. The external pressure is thus only constant for a vessel inclined along the
horizontal. Substitution of Eqs. (38) and (51) into Eq. (10) gives:
    ∂Q   ∂        Q2      2Go      k          A                  Ao    ∂A
       +        α       +            −1                      +
    ∂t   ∂x       A        kρ      2          Ao                 A     ∂x
             ho Ao pext ∂A        ho    παν Q 2Ao                     A               dGo
         −                 = gSAo    −2      −                                   −1       ,
             Ro A ρ ∂x            Ro    α−1A   kρ                     Ao               dx
where variations in wall stiffness only are assumed, the non-linearity parameter
being taken as constant.
    The non-zero wall thickness, which leads to a further term in the equilibrium
equation, thus contributes two terms in Eq. (52), one related to external pressure
and one to gravitational forces. The relative magnitude of these terms, even for
small values of the wall thickness to radius ratio, has been investigated in detail.20
The term relating directly to the external pressure on the left-hand side of Eq. (52)
is generally negligible, except for walls with very low stiffness, and will thus be
neglected here.
    However, the term due to the gravitational force (the first term on the right-hand
side of Eq. (52)) has been shown to be significant, even for thin-walled vessels, when
the vessel is not horizontal.20 As a fraction of the viscous term, the gravitational
term is:
                                      Sα gho Ro
                              ψ=                   ,                              (53)
                                   2(α − 1) νU
220                                    S. J. Payne & S. Uzel

where U is the flow velocity. The non-dimensional parameter ψ can be used to
determine the importance of the gravitational forces relative to the viscous forces
in a similar way to the Reynolds number being used to determine the relative
importance of momentum and viscous terms. For a Newtonian fluid with viscosity
ν = 3.2 × 10−6 m2 /s and a flow velocity of 0.5 m/s in a vessel of radius 5 mm and
wall thickness 0.5 mm, this ratio is approximately equal to twice the slope of the
vessel and is thus large even for a thin-walled vessel.
    A second non-dimensional parameter can be defined that compares the gravita-
tional term to the momentum term:
                                  ψ       Sα gho
                            φ=       =               .                         (54)
                                 Re     2(α − 1) U 2
This is similar in form to the Froude number and interestingly is primarily depen-
dent upon only the wall thickness and the flow velocity.
    It has been shown that gravitational forces are highly significant in comparison
with either the viscous forces or the momentum forces for every type of vessel.20 ψ is
large in the larger vessels since both the wall thickness and inner radius become small
in the microvasculature, whereas φ is larger in the smaller vessels, since the veloc-
ity becomes very small. It is suggested that, for any vessel, both non-dimensional
parameters are calculated to ascertain the importance of gravitational forces, since
gravitational forces should not in general be neglected unless the vessel under con-
sideration has low values of both ψ and φ.
    Before considering the solutions to the governing equations, we re-express them
in non-dimensional form. The following non-dimensional parameters are used: a =
A/Ao , q = Q/Uo Ao , go = Go /ρUo , z = x/L, τ = t/T , where Uo , L and T denote
characteristic velocity, length and time scales respectively. The resulting continuity
and momentum equations are thus:
                                        ∂a ∂q
                                   St      +    = 0,                              (55)
                                        ∂τ   ∂z
                    ∂q   ∂        q2         2go    k                1 ∂a
               St      +      α          +            − 1 a(k/2)−1 +
                    ∂τ   ∂x       a           k     2                a ∂z
                                    α    1 q  2           dgo
                     = gr S − 2              − [ak/2 − 1]     ,                   (56)
                                  α − 1 εRe a k           dz
                                        St =       ,                              (57)
                                             Uo T
                                            ho gL
                                       gr =       2
                                                     ,                            (58)
                                            Ro Uo
                                       Ro Uo Ro
                                   εRe =        .                              (59)
                                       L ν
In non-dimensional form, the equations are thus dependent upon four non-
dimensional parameters (St, gr , εRe, α) together with the wall stiffness parameters
                 Effects of Gravity and Wall Properties in Blood Flow               221

(the variation of stiffness along the vessel wall and the degree of non-linearity). The
first parameter determines the relative importance of the time-varying behavior,
the second the importance of gravitational forces, the third the viscous forces and
the last characterizes the velocity profile.
   In matrix form the equations become:
                                ∂θ        ∂θ
                                   + H(θ)      = B(θ),                             (60)
                                ∂τ        ∂z
                                      θ=       ,                                   (61)
                                                                      
                                             0                     1
                    1                                                 
          H (θ) =            2
                                                                     q ,          (62)
                   St −α q + 2go          k
                                             −1 a  (k/2)−1
                            a2      k     2                  a       a
                                                                  
                      1                                           
            B (θ) =                                           dgo  .             (63)
                      St g S − 2 α          1 q
                                                   2 k/2
                                                       a   −1
                                    α − 1 εRe a k              dz
   The eigenvalues of the matrix H(θ) are:

              q           q    2                2go    k                1
      λ1,2 = α ±      α            (α − 1) +             − 1 a(k/2)−1 +     .     (64)
              a           a                      k     2                a
Since the parameter α is approximately one, they are thus:
                           2go           k                1      √
             λ1,2 ≈ ±                      − 1 a(k/2)−1 +     ≈ ± go .            (65)
                            k            2                a
The orientation of the vessel has no effect on the eigenvalues; we have neglected
the external pressure, which will tend to reduce their magnitude, but this is a very
small effect.
    Having presented the governing equations, the solutions are now examined, both
the steady state solution and the wave behavior where there are changes in wall
stiffness under the influence of gravitational forces.

3. Model Predictions
3.1. Steady state behavior
In the steady state the flow rate is invariant with axial position (Eq. (55)) and the
steady state area varies according to:
                                         α    1 q   2 k/2    dgo
                        gr S − 2                  −   a   −1
             ∂a                        α − 1 εRe a k         dz
                =                  2
                                                                       ,          (66)
             ∂z            q             2go   k        (k/2)−1   1
                     −α                +         − 1 (a)        +
                           a              k    2                  a
222                                S. J. Payne & S. Uzel

where the over-bar denotes time-averaged values. This result is a more general
version of that presented previously.2,20
    Although Eq. (66) is somewhat complicated, some simplifications can be made
to interpret its behavior. Firstly, it can be noted that the numerator consists of three
separate terms, each of which contributes separately to changes in cross-sectional
area: The first two correspond to gravitational and frictional forces respectively,
whereas the third relates to the local change of wall stiffness. The importance of
this term will be examined in more detail below. Secondly, it can be shown that
the momentum term in the denominator of Eq. (66) is much smaller than the wall
stiffness term under most physiologically realistic conditions and that the local area
is normally close to the unstressed area.20 Equation (66) thus reduces, when wall
stiffness is constant, approximately to:
                         ∂a   1             α    1
                            =    gr S − 2           q .                            (67)
                         ∂z   go          α − 1 εRe
    For short lengths, the area variation with distance is approximately linear. A
positive change in area is induced by a positive slope and a reduction in area is
caused by a negative slope and by frictional forces. Changes in the wall stiffness
adjust the rate of change of area with axial distance.
    The area change due to frictional forces is relatively small in physiologically
realistic vessels.20 However, that due to gravitational forces alone is somewhat more
                              ∂a  gr  3ρgL
                                 = S=      S,                                      (68)
                              ∂z  go   2E
where Poisson’s ratio is assumed to be 0.5 for an incompressible material. For a
Young’s modulus E = 300 kN/m2 and blood density ρ = 1050 kg/m3 , this is equal
to 5.15% per meter per unit slope. Note that it is not dependent upon either the flow
field or the cross-sectional geometry of the vessel. In the longer blood vessels, the
gravitational force will thus have a significant effect on the flow field, even though
the change in cross-sectional area is only a few percent: it has been shown elsewhere1
that a small tapering in the vessel cross-sectional area has a significant effect on the
flow field, particularly the shock formation behavior. The wave behavior will thus
now be examined in more detail, in particular the response to local changes in wall
stiffness under gravitational forces.

3.2. Wave behavior
Since the changes in wall stiffness are likely to occur over short sections, such that
the distance over which the change occurs is of comparable length to the vessel
radius and less than the wavelength of pulsatile flow (of order 0.1–1 m), any local
change in wall stiffness can be considered as a first approximation as a step change.
The behavior of waves approaching such sharp changes in wall stiffness is thus
investigated here.
                      Effects of Gravity and Wall Properties in Blood Flow                 223

   To consider such behavior, the governing equations are linearized about the
baseline values:
                                           a=a+a,                                        (69)
                                           q =q+q ,                                      (70)
where the overbar is used to denote steady state conditions, as determined by
Eq. (67) for area and shown to be constant for flow rate, and the prime is used
to denote the fluctuating component.
   Substitution of these into Eqs. (55) and (56) and subtraction of the steady state
condition gives:
                                           ∂a   ∂q
                                      St      +    = 0,                                  (71)
                                           ∂τ   ∂z
           ∂q    ∂            q   a         2go    k                        1 ∂a a
      St      +α          2     −      +             −1        a(k/2)−1 −
           ∂z    ∂z           q   a          k     2                        a ∂z a

                2go     k                1 ∂a        α    1                  q   a
            +             − 1 a(k/2)−1 +      = −2                             −     ,   (72)
                 k      2                a ∂z      α − 1 εRe                 q   a
where the rate of change of stiffness term is now neglected, since it is assumed that
there is a step change in wall stiffness.
    It was previously shown that the rate of change of area in the steady state is
small under most conditions of physiological interest. It can thus be assumed that
both the steady state flow and area are approximately one. Substitution of Eq. (67)
into Eq. (72) thus gives:
                       ∂q       ∂q            ∂a        α    1
                  St       + 2α    + (go − α)     +2            q
                       ∂τ       ∂z            ∂z      α − 1 εRe
                              k                α    1     k
                         +      − 2 gr S − 2                − 1 a = 0.                   (73)
                              2              α − 1 εRe 2
   To analyze the behavior of this system, we consider a vessel with two sections:
the first section with wall stiffness goL and the second with wall stiffness goR . In
the first section, there are forward and backward waves, but there is only a forward
wave in the second section:
                                            z                                z
           aL = CaL exp jω τ −                    + DaL exp jω τ +               ,       (74)
                                           cL                               cL
                                            z                                z
           qL = CqL exp jω τ −                    + DqL exp jω τ +               ,       (75)
                                           cL                               cL
                              aR = CaR exp jω τ −                  ,                     (76)
                              qR = CqR exp jω τ −                 ,                      (77)
224                                 S. J. Payne & S. Uzel

where the coefficients C and D determine the relative magnitudes of the forward
and backward waves respectively and cL and cR are the wave speeds in the two
    Substitution of Eqs. (74)–(77) into the linearised continuity and momentum
equations gives the following equation for wave speed in either section:
           c2 (jω(St)2 + Kq St) + c(Ka − 2αjωSt) − jω(go − α) = 0,                   (78)
giving the following solution:

                         1                      (go − α)
                    c=      α +       α2+                       ,                    (79)
                         St                  (1 + Kq /jωSt)
                                      α − Ka /2jωSt
                               α =                  ,                                (80)
                                       1 + Kq /jωSt
                           k                α    1          k
                  Ka =       − 2 gr S − 2                     −1 ,                   (81)
                           2              α − 1 εRe         2
                                        α    1
                                 Kq = 2         .                               (82)
                                      α − 1 εRe
    To calculate the reflection coefficient, continuity of area and its first differential
at the boundary between the two sections is used to relate the wave coefficients:
                              CaL = DaL + CaR ,                                      (83)
                           ω        ω          ω
                         −j CaL = j DaL + j CaR ,                                    (84)
                           cL       cL        cR
which yields a reflection coefficient:
                                          c L − cR
                                  ρR =             .                                 (85)
                                          c L + cR
This reflection coefficient is a function of the two values of wall stiffness, the wall
nonlinearity parameter and the four non-dimensional parameters (St, gr , εRe, α)
that govern the flow field.
    In the limiting case where frictional and gravitational forces are negligible (i.e.
gr is small and εRe is large) and α is approximately one, the wave speed is directly
proportional to the square root of the wall stiffness. The reflection coefficient thus
tends towards:
                                      1−     goR /goL
                               ρR =                     ,                            (86)
                                      1+     goR /goL
which is only dependent upon the ratio of the wall stiffness values in the two sections.
This is shown in Fig. 6 for a range of values of this ratio: there is significant reflection
at both low and high values of this ratio. However, since gravitational forces have
been shown previously to have a significant impact on the flow behavior, the effects
of non-zero gravitational forces will now be examined.
                   Effects of Gravity and Wall Properties in Blood Flow                    225

Fig. 6. Reflection coefficient for changes in wall stiffness with stiff walls and zero friction and
gravitational forces.

            Table 1. Typical values of physiological parameters for human cardio-
            vascular system.21

                          Wall thickness   Inner radius   Length     Number of
                            ho (mm)          ro (mm)      L (mm)      vessels
            Aorta              2.5               10        0.5                1
            Arteries           0.5                1        0.01            3450
            Arterioles         0.025           0.050       0.001       18600000
            Capillaries        0.001           0.005       0.001    16100000000
            Venules            0.010           0.050       0.0005     160000000
            Veins              0.5                2        0.01             512
            Vena cavae         1.5               30        0.3                1

   To estimate typical values of the non-dimensional parameters, characteristic
vessel dimensions and numbers are used, Table 1. The wall is taken to have Young’s
modulus E = 300 kN/m2 .2 Blood is assumed to have density ρ = 1050 kg/m3
and viscosity ν = 3.2 × 10−6 m2 /s. The resulting non-dimensional parameters are
shown in Table 2, using a characteristic time period of 1 second. The characteristic
velocity is calculated using an assumed average flow velocity in the aorta of 0.5
m/s: The flow in the remaining vessels is calculated by dividing the corresponding
flow rate amongst the vessels equally. There is a very wide range of values for most
parameters, over a number of orders of magnitude, apart from St, which ranges
only between 0.4 and 8 over the entire cardiovascular system. The gravitational
parameter is largest in the smallest vessels, as is the frictional term, since εRe is
226                                    S. J. Payne & S. Uzel

                        Table 2. Typical values of non-dimensional
                        parameters for human cardiovascular system.

                                       St       gr        ε.Re       go
                        Aorta           1       4.9        31.3     381
                        Arteries      0.69       233      0.453    9.07e5
                        Arterioles    0.93      4240     8.40e-4   1.65e8
                        Capillaries   8.05    127,000    9.70e-7   4.94e9
                        Venules         4      62,800    1.95e-4   4.88e9
                        Veins         0.41      41.1       3.05    1.60e5
                        Vena cavae    5.4       47.7       52.1     6170

very small. The wall stiffness term is large in all vessels, but largest in the smallest
vessels: this is due to the flow velocity being very small. The value for k can be
approximately derived by comparing Eqs. (38) and (44) and using estimated values
for β from the literature: values quoted for the arterial network are in the range 3.23
to 3.53 and 2.15 for the venous network.2 This gives values of k in the approximate
range 4 to 9.
    The resulting variation in reflection coefficient with wall stiffness ratio (taken
throughout as the value in the second section as a fraction of the value in the first
section) is shown in Fig. 7 for the parameters for flow in the arteries. Throughout,
we plot the magnitude of the reflection coefficient, the phase being of less interest.
Although the reflection coefficient is predominantly determined by the wall stiffness
ratio, there is also an influence due to the gravitational force, with the reflection

Fig. 7. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters
from Table 2.
                  Effects of Gravity and Wall Properties in Blood Flow                  227

coefficient having a larger magnitude for positive slopes and a smaller magnitude
for negative slopes. This effect is, however, not symmetrical, as the effects of gravity
are much more significant at wall stiffness ratios less than one than at values greater
than one. The solution for zero slope is almost identical to the ‘ideal’ solution shown
in Fig. 6, showing that frictional forces have a negligible effect on the wave behavior,
as expected.
    Since the wave speed derived in Eq. (79) is complex, it should strictly be split
into real and imaginary components: the real part corresponding to the actual wave
speed and the imaginary part relating to the wave attenuation:

                                   1  1   1
                                     = +j .                                           (87)
                                   c  c  c

In the absence of frictional and gravitational forces, the attenuation is zero. The
wave speed and attenuation coefficient for flow in the arteries is shown in Fig. 8.
There is only a small change in either parameter as the slope varies, with the
wave speed increasing as slope increases (either positively or negatively), and the
attenuation coefficient, which is very small, increasing with slope in a near linear
manner. For small values of the attenuation coefficient, this coefficient is equivalent
to the fractional reduction in the wave amplitude over the length of the vessel in
which the wave is traveling. The reflection coefficient as a fraction of the value at
zero slope is shown in Fig. 9. There is a significant change in reflection coefficient

Fig. 8. Non-dimensional wave speed and attenuation coefficient for flow in the arteries, using
parameters from Table 2.
228                                    S. J. Payne & S. Uzel

Fig. 9. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters
from Table 2, as a fraction of the values at zero slope.

at low wall stiffness ratios with slope: even at a ratio of 0.1, there is a change of ±
4% with slope changing from −1 to +1.
    However, if the wall stiffness reduces, thus reducing the corresponding non-
dimensional parameter, the effect of gravitational forces becomes progressively more
significant. The reflection coefficient is shown in Fig. 10 for a baseline wall stiffness
reduced by a factor of 10 and the corresponding wave speed and attenuation coeffi-
cient in Fig. 11. The reflection coefficient as a fraction of the value at zero slope is
shown in Fig. 12. There is a more significant change in reflection coefficient at low
wall stiffness ratios with slope: Even at a ratio of 0.1, there is a change of ± 4%
with slope changing from −1 to +1. At ratios higher than one, the change is much
smaller: Reflection coefficient is thus predominantly affected by gravitational forces
when incurred by a reduction, rather than an increase, in wall stiffness.
    For small changes in reflection coefficient, the percentage change with slope
is proportional to the gravitational parameter gr . This scaling drops off for high
values, as the reflection coefficient saturates at one. The gravitational parameter is
dependent upon the vessel wall thickness to inner radius ratio and length and the
flow velocity: In thicker walled vessels, the gravitational parameter thus becomes
more significant. This also increases the non-dimensional wall thickness, the ratio
of the two being independent of both the wall thickness to inner radius ratio and
the flow velocity.
                    Effects of Gravity and Wall Properties in Blood Flow                        229

Fig. 10. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters
from Table 2, but with baseline wall stiffness reduced by a factor of 10.

Fig. 11. Non-dimensional wave speed and attenuation coefficient for flow in the arteries, using
parameters from Table 2, but with baseline wall stiffness reduced by a factor of 10.
230                                     S. J. Payne & S. Uzel

Fig. 12. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters
from Table 2, but with baseline wall stiffness reduced by a factor of 10, as a fraction of the values
at zero slope.

    From the numerical simulations, it is found that, at a fixed value of wall stiffness
ratio, the reflection coefficient is scaled under gravitational forces in an approxi-
mately linear manner with the gravitational parameter and the square root of the
baseline wall stiffness.

4. Discussion
It should be noted that the analysis presented here only considers a single boundary
between two regions of uniform stiffness. In reality, there are likely to be more than
one change in wall stiffness, as there may be regions of decreased stiffness (for
example an aneurysm) or increased stiffness (for example a stent). There will then
be multiple reflections, which will complicate the flow field, although the analysis
here could easily be extended to such situations.
    In earlier work,20 velocity waveforms propagating along the vessel were found to
exhibit similar behavior to that shown previously1: The waveform appears to ‘pile
up’ with the front edge becomes steeper. These solutions were obtained using the
Lax-Wendroff numerical solver.22 This ‘piling up’ behavior has not been examined
here as the focus has been on the behavior of waveforms at sharp discontinuities in
wall stiffness under the influence of gravity.
    The waveform amplitude increases under positive slope and vice versa, relative
to the behavior at zero slope. The amplitude increase with distance down the vessel
is almost linear and is strongly dependent on the slope. For a vertically inclined
                 Effects of Gravity and Wall Properties in Blood Flow              231

vessel, the amplitude of the pulse increases by approximately 4% per meter for flow
downwards and decreases by approximately 5% per meter for flow upwards.
   The front of the waveform is also steeper for positive slopes, growing more rapidly
and thus promoting shock formation. It was shown using numerical simulations20
that the rate of shock formation is increased for downwards flow and decreased for
upwards flow by a factor that is approximately 1.2 times the wall thickness to inner
radius ratio times the slope, i.e.:
                                xs (S)        ho
                                         ≈ 1.2 S.                                (88)
                              xs (S = 0)      Ro
Thus even for thin-walled vessels, a slope causes a significant change in the rate of
shock formation. A linear tapering in area, simulated by allowing the baseline area
to decrease linearly with distance, was found to postpone shock formation.1 Both
of these effects thus have an effect on the shock behavior, it being postponed by
both a linear tapering in baseline area and an upwards slope.

5. Conclusions
We have presented here an investigation into the effects of gravity on the flow field
in compliant axi-symmetric vessels. The particular focus has been on the cardiovas-
cular system, although the results here are applicable to any similar fluid-structure
interaction system. We have outlined the theory governing this flow in detail, cover-
ing the mechanics of both the fluid and the vessel wall and illustrating the complex
nature of this fluid-structure interaction.
    After presenting a survey of models for the vessel wall, we have presented a
new model that captures the characteristics of the vessel wall behavior well. This
has been coupled to the fluid dynamics equations and the governing equations thus
derived. The steady state and wave behavior have then been examined to investigate
the difference in behavior of the flow field under the influence of gravitational forces.
    It is found that gravitational forces induce significant changes in the steady
state area variation of such vessels: Even though this change is only a few percent,
this is known to have a significant effect on the flow field, particularly the shock
wave formation. The wave behavior at the junction between two sections of different
wall stiffness is also found to be affected by gravitational forces, particularly where
the wave encounters a region of reduced wall stiffness. The reflection coefficient
is increased in magnitude for positive slopes and decreased for negative slopes,
whereas the wave speed is increased slightly for either positive or negative slopes
and the attenuation coefficient increases near linearly with slope. This effect is most
significant when the non-dimensional gravitational parameter is large and the non-
dimensional stiffness parameter is small.
    The analysis presented here, however, is only a part of the answer: There remains
much work to be done to improve our understanding of the fluid-structure inter-
action found in the human vasculature. At each stage of modeling, rigorous math-
ematical analysis of the impact of different effects is vital. Although a number of
232                                 S. J. Payne & S. Uzel

assumptions that are widely made have been relaxed in this paper, there are still
several issues to be considered: In particular, the models of the wall behavior are
very simplistic, neglecting any damping or inertial effects. For better understand-
ing of the flow field around changes in wall stiffness, more detailed models will be
required, probably using two-dimensional models, where the radial velocity is also
considered and the velocity profile is not assumed to be of a fixed shape. This is the
subject of current research and will be very important if the wall shear stresses are
to be predicted accurately: this will be crucial in improving our understanding of
how the flow field influences the wall behavior in both normal and diseased states.

I would like to thank Professor Paul Taylor and members of my research group for
helpful comments.

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 3.   G. Pontrelli and E. Rossoni, Int. J. Num. Methods Fluids 43 (2003) 651–671.
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      E. Wofowitz, Prog. Biophys. Mol. Biol. 81 (2003) 177–199.
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10.   S. Canic and A. Mikelic, Comptes Rendus de l’Academie des Sciences, Serie I (2002)
11.   M. Ursino and C. A. Lodi, Am. J. Physiol. 274 (1998) H1715–H1728.
12.   J. Zhou and Y. C. Fung, Proc. Natl. Acad. Sci. 94 (1997) 14255–14260.
13.   G. Langewouters, K. Wesseling and W. Goedhard, J. Biomech. 17 (1984) 425–435.
14.   L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Comput. Visual. Sci. 2
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16.   N. Westerhof, F. Bosman, C. Vries and A. Noordergraf, J. Biomech. 2 (1969) 121–143.
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18.   P. Segers, F. Dubois, D. De Wachter and P. Verdonck, J. Cardiovasc. Eng. 3 (1998)
19.   K. Hayashi, K. Handa, S. Nagasawa and A. Okumura, J. Biomech. 13 (1980) 175–184.
20.   S. J. Payne, Med. Biol. Eng. Comp. 42 (2004) 799–806.
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22.   R. J. Leveque, Finite Volume Methods for Hyperbolic Problems (Cambridge University
      Press, 2002).
                                        CHAPTER 7


                   EAMON G. KAVANAGH and TIM M. McGLOUGHLIN∗

                     Centre for Applied Biomedical Engineering Research
                      MSSI, University of Limerick, Limerick, Ireland

    Cardiovascular disease is a major cause of death in Western society. Complications
    arise throughout the cardiovascular system with the arterial sub-system being prone
    to atherosclerosis and aneurysm formation. Atherosclerosis is a disease characterized by
    the deposition of lipoproteins in the arterial wall while an aneurysm is characterized by
    an abnormal swelling of the wall itself. Medical therapy, comprising life-style changes and
    drug administration is the mainstay of treatment for the majority of people with arterial
    vascular disease. However, a variety of surgical interventions are available to treat dis-
    eases associated with the arterial system and the primary objective of such interventions
    is to restore normal arterial function. These treatments may be performed using open
    surgery or minimally invasive techniques and are biomechanical in nature.
         A range of numerical and experimental techniques has been developed to quan-
    tify and qualify disease-influencing biomechanical factors in the arterial system. These
    techniques may be used to perform basic research on disease forming mechanisms, to
    diagnose disease in vivo and to assess or develop new biomedical treatments. The tech-
    niques presented in this chapter are applied for various purposes at different locations in
    the arterial system using an integrated research and development approach. The chapter
    concludes with a discussion on new medical devices currently under development in order
    to demonstrate how numerical and experimental methods may be applied in the search
    for new and improved treatments for arterial disease.

    Keywords: Cardiovascular; artery; hemodynamics; biomechanical modeling; computa-
    tional fluid dynamics.

1. Introduction
Numerical and experimental techniques have been widely used for the study of
biomechanics in the human vascular system.1–4 At the macroscopic or systemic
level, application of these techniques have yielded extensive knowledge on how blood
interacts with the blood vessels through which it flows, and, if complications in the
form of disease or injury arise, how the mechanical function of the vascular system
may be restored.5,6 At present, numerical and experimental techniques are being
used to qualify and quantify the biomechanical environment and to hypothesize,
design and develop new biomedical treatments for vascular disease.7–9
    The function of the cardiovascular system is to transport nutrients and oxygen
around the body and to remove waste products. In mechanical terms, it comprises
a pump, in the form of the heart, and a distribution network, consisting of arteries,

234                                T. P. O’Brien et al.

capillaries and veins.10 While complications can occur throughout the vasculature,
a major focus of research to date has been investigating the principal diseases asso-
ciated with the arterial system.11 The arterial system consists of major arteries such
as the aorta and the iliac arteries, and minor arteries such as the femoral, carotid
and coronary arteries. The major disease associated with the arterial system is
atherosclerosis characterized by the deposition of low-density lipoproteins onto the
endothelial cell layer that lines the lumen of the artery.12 These depositions may
accumulate over time causing narrowing of the artery lumen, which can result in
hypertension, stroke, limb ischemia or heart attack, some of the major causes of
death in western society today.13
    In a simple analysis, a segment of the arterial system may be modeled as a
Newtonian fluid moving through a constant diameter rigid-walled pipe at constant
velocity.14 A constant pressure gradient acts between the inlet and outlet of the
pipe and the blood develops fully according to Poiseuille’s flow. This simplification
has been widely used in biomechanical models to estimate the mean shear stresses
acting on the endothelial cells, often cited as being a major disease-influencing
factor.15–18 Previous work has assumed that the Newtonian assumption is valid
in larger diameter arteries such as the aorta and the iliac arteries.19 However, in
smaller diameter arteries, such as the coronary and femoral arteries, blood is more
accurately modeled as a non-Newtonian fluid, with a viscosity/shear characteristic
that may be determined experimentally.10 At the microscopic level, blood is com-
prised of red blood cells, white blood cells, platelets and plasma, but it is the red
blood cells which coalesce to form Rouleux chains, which result in non-Newtonian
characteristics in smaller diameter arteries. A number of non-Newtonian models
have been reported previously.19
    A further assumption, which limits the applicability of this simple model, is
that of the blood velocity boundary condition. The heart does not provide steady
flow of blood through the vascular system; rather, it is an oscillatory pump which
pulsates blood through the vasculature.20,21 The resulting transient flow rates vary
throughout the system depending on location, with the large diameter aortic flow
rates being far higher compared to those associated with the small diameter coro-
nary and femoral arteries.22 This creates a need to model arterial biomechanics
with three-dimensional transient blood flow according to the continuity and Navier-
Stokes equations.23 Transient blood flow differs significantly from steady flow and
is characterized by temporally varying velocity profiles, wall shears and flow sepa-
ration/recirculation regions.24
    The model may be improved by modeling the compliant nature of the artery wall.
Arteries vary in mechanical stiffness; as a person ages, their arteries become less
compliant.25 However, the elasticity is such that, in younger people, it can affect the
velocity profiles through the vasculature. In addition, the elasticity is responsible,
in part, for the reversal in direction of blood flow over the pulsatile cycle, caused
by vessel contraction and wave reflections.26 Though arteries are heterogenous and
                    Study of Biomechanics in the Arterial System                  235

anisotropic, their non-linear elastic behaviour may be modeled to an acceptable
degree by assuming linear elasticity.26,27
    The original simplified model has thus evolved to represent non-Newtonian, pul-
satile flow through a constant diameter complaint pipe. At this point, the model
captures the main biomechanical characteristics of an idealized model of an artery.
However, the foremost variable of interest throughout this chapter is that of geome-
try. Geometrical attributes have been cited as being the one of the most impor-
tant biomechanical variables contributing to the initiation and development of
atherosclerosis.28–30 In addition, for the treatment of disease, geometry is often
the single most influential factor in designing surgical interventions.31,32
    This chapter presents a range of numerical and experimental techniques used in
the study of the biomechanics of the arterial system. Medical imaging techniques
may be used to qualify and quantify biomechanical phenomena in the arterial sys-
tem and to measure quantities suitable for analysis using in vitro biomechanical
methods. Data acquired from medical images may be used to create models for
use in numerical and experimental studies. Digital image processing techniques
are discussed together with computer-aided methods to design and manufacture
test models. Numerical methods, which are used to investigate the biomechanics
of arteries and associated biomedical devices, are described with particular atten-
tion to the techniques of computational fluid dynamics and finite element analysis.
Next, experimental validation and testing techniques are described. Flow modeling
methods and measurement systems are presented with emphasis on laser Doppler
anemometry, video extensometry and photoelasticity. Finally, an application of the
aforementioned techniques is used to demonstrate the hypothesis, design and devel-
opment process used to create novel medical devices. These techniques may be
combined to describe an integrated investigative approach for arterial biomechanics
(Fig. 1).

2. Medical Imaging Techniques
Imaging methodologies are widely applied in the clinical environment for the diag-
nosis of disease and for follow-up evaluation of surgically implanted treatments.33,34
Methods such as magnetic resonance imaging (MRI) and computerized tomogra-
phy (CT) have been used to assess the geometrical nature of aneurysms, to locate
lesions and occlusions and to follow-up the incorporation and performance of vascu-
lar bypass grafts.35–37 What makes modern medical imaging techniques so powerful
is the portability of the resulting data. While traditional X-ray systems were lim-
ited in the portability of the photographic plate, modern MRI and CT systems
store their image data digitally, enabling rapid data transportation and processing.
Additional techniques, such as Doppler ultrasound, enable researchers to obtain
qualitative data as to the patency of bypass grafts and arteries and quantitative
data in the form of flow rates through blood vessels.38 Other techniques such as
236                                       T. P. O’Brien et al.

Fig. 1.   The integrated approach to performing biomechanical investigations of the arterial system.

positron emission tomography (PET) and electrical impedence tomography (EIT)
are less widely used but provide additional capabilities and applications.

2.1. Magnetic resonance imaging
Magnetic resonance imaging has attained widespread use in recent times for clinical
imaging and diagnostics. It is a non-invasive imaging technique posing minimal risk
to the patient due to its non-ionizing radiation. It is capable of measuring blood
velocity and acquiring geometrical information and is widely used for investigative
biomechanics studies.39–42

2.1.1. Operating principles
The core operating principle underlying MRI is that certain atomic nuclei, when
placed into a magnetic field, undergo realignment as a result of the effect that the
                     Study of Biomechanics in the Arterial System                   237

magnetic field has on their spin.43 The realigned nuclei, mainly hydrogen protons
in the case of MRI, may then be periodically ‘flipped’ from the alignment or lon-
gitudinal plane towards the direction of the transverse plane by targeting a radio
frequency (RF) pulse at them. Net magnetization is then in the transverse plane.
This results in a voltage being induced in a coil placed parallel to the transverse
plane thereby enabling data acquisition. Following the RF pulse, the protons realign
with the longitudinal magnetic field. The time it takes for realignment is termed
the relaxation time (T1). Different tissues have different relaxation times, resulting
in a difference between the transverse magnetization being received for different
tissues.37 These different tissues may then be distinguished on the basis of the
induced voltage in the receiver coil.
    Different RF signals have been developed for imaging different tissues. The sim-
plest is termed the saturation recovery (SR) signal. This involves a single excitation
pulse with a fixed time between each pulse, termed the repetition time (TR). The
relaxation time must be different from the repetition time; otherwise there will be
artifacts present in the images. This method is not capable of varying the contrast
between different tissues.
    A more widely used method incorporates an echo pulse following the excitation
pulse, termed the spin echo (SE) method. The echo pulse is used to gradually restore
phase coherency of the protons, which gradually decays after the RF pulse, as the
gradual loss of phase results in a gradual loss of signal. The echo time (TE) is the
time between the RF pulse and the echo pulse. TE may be varied in order to give
different contrasts between different tissues. Multi-spin echo imaging used several
echo pulses, following the initial RF pulse, to give images that have different tissue
contrasts. Other variations on the echo pulse include turbo spin echo (TSE)44 and
gradient echo (GE), a rapid acquisition method, which is used for imaging blood.
    There are a number of other parameters associated with the MR imaging of
tissue. The field of view (FOV) is the anatomical area being imaged and the matrix
(m × n) contains the 2D pixel information.29
    MRI scanners consist of a number of basic elements. There is a magnet, which
produces the main magnetic field, shim coils, which reduce inhomogenities in the
main field; gradient coils, which produce linear gradient magnetic fields along three
orthogonal planes to enable spatial location of the MR image; RF transmitter and
receiver coils; an MRI computer and surface coils.
    Image quality is quantified by the signal-to-noise ratio (SNR) and the contrast-
to-noise ratio (CNR). It is desirable to increase these ratios when possible for a given
image. There are a number of factors that affect these, the main ones being voxel
size, slice thickness, matrix resolution, field-of-view, number of signal averages, pulse
sequence, interslice gap, magnetic field strength and contrast media. Contrast agents
include gadolinium-based compounds. These agents help to reduce the relaxation
time of certain tissue thereby reducing TR, TE and the scan time.
    Image artifacts are prevalent in MRI arising from a number of different sources.45
Magnetic field artifacts are due to inhomogenities in the magnetic field and
238                                T. P. O’Brien et al.

distortion due to nearby magnetic materials. Hardware related artifacts are caused
by misalignment of gradient coils, receiver coils, image resolution and computer-
generated artifacts. Motion related artifacts are critical in imaging blood flow and
arterial walls and are due to the pulsatile nature of blood flow. Since MRI scans
take a certain length of time, there may be a blurring of boundaries which means
synchronizing the scan with the low velocity region following diastole.
    Magnetic resonance angiography (MRA) is the imaging of the cardiovascular sys-
tems using an MR machine.29,43 MRA may be used to determine both the velocity
of the blood as well as the cardiovascular geometry. There are a variety of differ-
ent methods used to acquire this information. These methods have several common
features including small flip angles, and short TR and TE.29,39,46,47
    Since blood flows, it is possible to capture the wall image by having such a time
lag between the RF pulse and the signal acquisition that the blood has already
left the imaging plane and so appears as a void on the image yielding detailed
information of the wall. This is termed time-of-flight angiography. In addition, blood
velocity may be determined by saturating the surrounding tissues with the RF pulse.
Since the blood is flowing through the image plane, the net magnetization shows
the unsaturated blood surrounded by saturated tissue.

2.1.2. Application
This example illustrates how MRI may be used to screen for disease. The veloc-
ity and geometry image files of the healthy human abdominal aorta from above the
renal arteries to below the aortic bifurcations were acquired using a 1.5T MR imager
(Philips Electronics N.V.). Time-of-flight angiography was used to acquire the geo-
metrical information. No contrast agent was used. A total of 50 geometry slices of
spacing 4 mm were acquired. Repetition time was 16.52 ms, echo time was 6.9 ms, a
70◦ flip angle was used and the field of view was 300 mm. The image information was
recorded into a 256×256 matrix. The velocity conditions at the inlet of the descend-
ing aorta, at the outlets of each renal artery, and, at the outlets below the aortic
bifurcations were determined in the temporal and spatial domain for a total of 16
time-intervals per pulse. The total time for scanner configuration and data acquisi-
tion was approximately 1 hour. The 178 files were saved in DICOM format and split
into separate header and raw image files and were made available for processing.
    An example of an MRI scan is shown in Fig. 2 (inset) together with models
resulting from MRI and CT scans. While the resolution is low, the aorta (circular
white region) is clearly evident in the image. The scans were each examined in order
to reveal that no obvious disease was present and that the aorta appeared to be
functioning as normal. Figure 2 also shows the inlet velocity profile at a certain
time in the pulse and illustrates the nature of the skewed velocity profile.

2.2. Computerized tomography
Computerized tomography (CT) is a non-invasive imaging technique that uti-
lizes non-ionizing radiation to generate a two-dimensional image of the tissue.
                       Study of Biomechanics in the Arterial System                          239

Fig. 2. Models of healthy and diseased abdominal aortas generated from MRI and CT data. The
inset shows the peak velocity profile occurring at the inlet to the healthy aorta as determined
using MRA together with an MRI scan. It is possible to apply this velocity profile to the inlet of
the model using CFD.

It offers high-resolution images and is capable of differentiating between a wide
range of different tissues. It may be used dynamically enabling surgeons to per-
form minimally invasive procedures using CT to guide instruments and catheters
around the patient’s internal organs. CT is widely used for imaging of the arterial

2.2.1. Operating principles
A CT scanner utilizes non-ionizing radiation to capture two-dimensional images of
internal organs.51 The patient may receive a dose of contrast agent such as iodine
or barium sulphate. The contrast agent absorbs some of the incident radiation
enhancing the contrast on the resulting image. The radiation source is mounted
on a gantry, which rotates about the patient, in the case of spiral CT, acquiring
a single image for each rotation. The gantry is then indexed a certain distance
in the axial direction, and further gantry rotation produces the next image. The
process is repeated until a series of contiguous images is acquired which may then
be processed to produce a 3D image of the internal organ under investigation. Each
image is separated by a certain distance, and each image has associated with it a
certain thickness. The image will be stored in digital format and the resolution, a
240                                 T. P. O’Brien et al.

measure of the size of each pixel in the digital image, may be determined by dividing
the field-of-view by the dimensions of the m×n pixel array.

2.2.2. Application
CT is generally used in disease diagnosis and follow-up. The CT scan shown in Fig. 2
illustrates a complication associated with the descending abdominal aorta. Abdom-
inal aortic aneurysm (AAA) is a potentially fatal swelling of the aortic wall and is
attributed to hereditary and lifestyle factors.52 A localized weakening in the artery
wall causes the wall to expand. As it expands, the thickness of the wall decreases
and an increase in wall stress results, leading to further expansion. Eventually, the
stress reaches a critical level causing the wall to rupture. The model shown was gen-
erated from a set of contiguous CT data acquired using a spiral CT imager (Siemens
A. G.). A spiral acquisition was performed with an interslice spacing of 4 mm. One
hundred slices in total were taken with a 422 × 422 mm2 FOV stored in a 512 × 512
pixel array. The model was generated using an image processing software (Scion
Image, Scion Corp.).

2.3. Ultrasound
Ultrasound is a flexible medical imaging technique capable of acquiring approximate
geometrical and flow-rate data rapidly for use in disease diagnosis and post-operative
follow-up.53,54 While it does not produce accurate quantitative data, the qualitative
data is useful as it is easily acquired and poses no risk to the patient. Ultrasound and
its variant, Doppler ultrasound, offer effective techniques in obtaining approximate
in vivo flow-rates and geometrical information.

2.3.1. Operating principle
When high frequency sound waves are emitted from one medium into another, some
of the soundwaves are reflected at the interface. It is possible to measure the time
delay from the emission of the waves to the detection of the reflected waves, thus
providing a measure of the depth of the interface from the emitter. Ultrasound is
a based on this concept, in that the geometrical nature of certain tissues can be
determined by measuring their depth in a body. 2D and 3D ultrasound machines are
available, the latter of which is essentially an extension of the 2D machine, capable
of creating approximate 3D models of tissue. Ultrasound is suitable for obtaining a
snapshot of a particular tissue and is widely used in gynecology for identifying the
development of the foetus.
    Doppler ultrasound operates on the principle that there is a change in the fre-
quency of a wave reflected from a moving body. In the case of angiography, the
moving body is the red blood cell, and the change in the frequency of the emitted
and received sound waves is dependant on the insonating frequency, the velocity of
the moving blood and the angle incident between the emitted wave and direction
                        Study of Biomechanics in the Arterial System                            241

of blood flow. There are several different methods of representing the velocity for
the end-user, the principal ones being colour, pulsed and power Doppler ultrasound.
Colour Doppler uses a colour code to identify regions of different mean flow velocity
while pulsed Doppler represents a spectrum or graph of temporally varying flow
velocity at a specific location. Power Doppler is a more sensitive technique than
colour Doppler.
    Several significant artifacts may occur with Doppler ultrasound. The first,
termed aliasing, happens when the frequency shift to be measured is over twice
the pulse repetition frequency. The result of this is a wraparound of the Doppler
spectrum in pulsed or colour Doppler ultrasound. On occasion, aliasing may indi-
cate high velocity flow resulting from stenoses. The second artifact, inherent to
colour Doppler, is termed bleeding. This occurs when the colour gain is set too high
and blood regions may appear larger than they actually are. This may veil certain
attributes of the flow volume, including aneurysms and stenoses.

2.3.2. Application
Doppler ultrasound may be used to acquire a snapshot of the patency of blood ves-
sels. The model (Fig. 3) illustrates a carotid bifurcation showing a patent junction.

2.4. Angiography
Angiography is an imaging technique, which operates using similar principles to
CT. It is used exclusively for imaging the vascular system. A contrast medium is
injected into the bloodstream proximal to the artery under investigation and the
blood vessel shows up as a white image when using standard angiography techniques
and a dark image when using digital subtraction angiography.55
    As an example, coronary angiography was used to image the left coronary artery
during a screening process for coronary artery disease (Fig. 4). Stenoses and plaques
may be located using angiography and regions necessary for stenting may be iden-
tified. As it is a dynamic process, angiography is used by cardiologists during the
implantation of coronary stents.

Fig. 3.   Doppler ultrasound of a carotid artery bifurcation. The sinus of the carotid is visible.
242                                   T. P. O’Brien et al.

               Fig. 4.   A left coronary artery generated from an angiogram.

3. Computer Aided Design and Manufacture
Concurrent engineering principles may be applied in the biomechanical study of the
arterial system by performing numerical and experimental investigations simulta-
neously. While numerical techniques are particularly useful in performing design,
research and optimization studies, there remains the need for experimental valida-
tion of numerically derived results and techniques.56,57 Advances in computer tech-
nology means that models designed and developed using computer aided design
(CAD) systems for use in computational simulations can be manufactured using
computer aided manufacturing (CAM) systems. Therefore, as research and opti-
mization studies are being performed using numerical methods, experimental vali-
dations of these studies can be carried out concurrently.
    While there is a wide range of different CAM techniques available and under
development, two techniques of particular interest in the study of arterial biome-
chanics are discussed here.

3.1. Digital image processing
The process of converting the two-dimensional digital medical images into three-
dimensional solid models for use in numerical and experimental investigations is
termed digital image processing (DIP). It encompasses a broad field of mathemat-
ical processes, which convert the unsmoothed, pixel images into smoothed vector
models. The essence of DIP applied to medical imaging involves acquiring geomet-
rical co-ordinates of an ambiguous object from a pixel array and assembling these
co-ordinates into a well defined 2 or 3 dimensional model.58
    Images acquired using CT or MRI are represented in digital form. Each image
scan results in the creation of one file that contains the information associated
with that particular image. The general file format used in medical imaging is the
digital imaging and communications in medicine (DICOM) format. A single DICOM
file consists of the image header, which contains information on the scan process
                     Study of Biomechanics in the Arterial System                 243

parameters, positioning and patient data, as well as the ASCII representations of the
m × n pixel array. Each pixel is represented by a certain number of bits depending
on how many levels of gray or colour was used. Typically a digital medical image is
represented by either 8 or 16 bit pixels. A 16 bit pixel results in 256 levels of gray
together with addressing and is suitable for most medical applications.
    When opening DICOM image data it is necessary to know a certain number
of parameters. The dimensions of the pixel matrix, m × n, must be known and
from this the total number of pixels may be calculated. Typical pixel arrays are
256 × 256 or 512 × 512. The pixel size may be determined by dividing the x and y
FOV lengths by their corresponding m and n pixel array values. The gray-scale or
colour levels and distance between slices must also be known. This information is
stored in the DICOM header. Once the files have been opened and all parameters
correctly specified, processing may occur.
    The quality of the images is determined by the resolution of the scanner. Res-
olution refers to the size of an individual pixel in the image. All digital images
have a certain resolution and it is this property that limits their accuracy. Tissues
with a relatively large cross-sectional area will be represented by a large number of
pixels, resulting in clear boundary definition. However, in small arteries, low reso-
lution means that a circular cross-section may be represented by a cluster of 4 or 5
pixels. Therefore, as the cross-sectional area of the tissue being scanned decreases,
so too does the accuracy. Recent developments in scanner technology means that
high-resolution images for major and minor arteries are more readily obtained.

3.1.1. Process application
This example is concerned with the human abdominal aorta screened for disease
(Sec. 2.1.2). This data was acquired using time of flight angiography. 178 DICOM
data files were available for processing.
     There are a number of different methods available for processing data of the
type described above. Commercially available software, such as Mimics (Materialise
Inc.), provides an efficient and comprehensive method for processing data. Mimics
is capable of importing and exporting a wide variety of different file formats. Image
files may be thresholded, edited and assembled into 3D models suitable for export
for numerical study or manufacture. Mimics forms an integral part of computer
aided engineering in the medical domain. Curve processing and smoothing is per-
formed using automatic curve and surface processing algorithms. A solid model of
the abdominal aorta was generated from the DICOM image data using Mimics and
is shown in Fig. 2.
     A second method involves using image processing software such as Scion Image
(Scion Corp.). This software is capable of reading raw image data and DICOM for-
mat, and provides a range of common image processing and analysis tools. However,
it is capable of 2D image processing only, and does not support 3D model generation.
Therefore, once the 2D boundary configuration is determined, further processing is
244                                      T. P. O’Brien et al.

needed to create a 3D model. In addition, controlled curve processing and surface
smoothing must be performed manually, according to a defined algorithm. There
are numerous curve-smoothing algorithms available and new algorithms may be
written and implemented using a high-level language. Algorithms may record the
spatial co-ordinates of pixels located at the boundary and fit polynomial curves of a
certain order to these spatial coordinates. Axial smoothing, or smoothing between
adjacent slices, is also necessary and may be accomplished using different mathe-
matical techniques. The main objective of curve and surface smoothing applied to
arterial biomechanics is to create an artery lumen wall, which approximates the wall
of the actual patient (Fig. 5).

3.1.2. Velocity profile processing
MRA can be used to determine the velocity in an arterial system.29 In this example,
the velocity and phase information for the various inlet and outlets were recorded
at 16 times in the cardiac cycle for the abdominal aorta scan. The volunteer’s heart
rate was approximately 60 beats per minute. The velocity data was represented as
a series of images, the gray-level of the blood pixels being proportional to the blood
velocity at that point. Depending on the cross-section of interest, the velocity was
encoded from 0 to 4095 (grayscale level). In the case of the inlet of the aorta, the
phase velocity encoding spanned –100 cm/s to +100 cm/s. Therefore a pixel value of
0 (black) corresponded to a velocity of –100 cm/s and a pixel value of 4095 (white)
corresponded to a velocity of 100 cm/s. By reading the coordinates of the pixels,

Fig. 5. The stages involved in generating a solid model from a set of CT scans. The set of scans is
thresholded individually (a), and the boundaries of the arteries are identified according to a certain
image processing procedure. Assembly of these boundaries results in the wire-frame geometry (b).
Further curve smoothing and surface fitting results in a surface model (c) which can then be
processed further to produce a solid model suitable for CFD or FEA investigations (d).
                        Study of Biomechanics in the Arterial System                           245

together with their gray-level value, a 3D velocity profile could be established. This
procedure was tested for the maximum velocity step at the inlet slice. Positions and
levels were recorded for all blood pixels and a 3D surface was created resulting in
the profile (Fig. 2).

3.1.3. Vector transformations
A pair of images acquired using angiography can also be used to create a 3D model
of an artery using vector transformation provided that the angle and scale between
the two images is known.59 Figure 6 illustrates the vector transformation process
applied to a right coronary artery. Two images were acquired and were used to
construct a 3D model.

3.2. Rapid prototyping
Rapid prototyping (RP) is a CAM technique that manufactures a solid homoge-
nous prototype directly from a CAD model. There is a range of different techniques
available capable of producing prototypes using a variety of different manufactur-
ing processes and materials. Stereolithography, selective laser sintering, 3D printing

Fig. 6. Ortohogonal angiograms of the right coronary artery. These images were processed using
vector transformations to produce a realistic representation of the geometrical nature of the right
coronary artery.
246                                    T. P. O’Brien et al.

and fused deposition modeling are all popular RP processes enabling fast and effi-
cient manufacture of prototypes of computer solid models. While RP has not been
widely used to date in investigating arterial biomechanics, it has been used exten-
sively for orthopaedic applications. Studies60,61 have manufactured components for
reconstructive surgery, while rapid prototyping has also been used to develop custom

3.2.1. Application
In this application of RP, a model of a rigid graft was manufactured from fused
deposition modeling (FDM). The dimensions of a commercially available abdominal
aortic graft was measured and used to generate a 3D computer model of the part.
This part was then split so the two halves modeled could be clamped together
(Fig. 7). As the halves were symmetrical one model was exported as an STL file to
the FDM processing software Insight (Stratasys Inc.). This software mathematically
slices and orientates the part. Fused deposition modeling operates by extruding a
filament of semi-molten material from an extrusion head in a prescribed pattern
onto a platform. This generates a layer or slice of the geometry. The material layer
then cures, the platform drops in the z direction according to the slice thickness,
and the next layer is generated. Successive layers of the part are then built up on
top of each other until the part is completed. In addition to the design of the part,
the user must specify the material choice, the tool path, the layer thickness and part
orientation. At present, FDM uses only thermoplastic materials. For the graft model
discussed here, acrylonitrile butadiene styrene (ABS) was used. Due to the nature
of the manufacturing process, accuracy and surface finish are the main limitations
of FDM.

Fig. 7. Abdominal aortic graft lumen model generated using fused deposition modelling. The
lumen computer model was split in two through its centreplane, a wall thickness was declared and
a clamping flange and holes were added to the model.
                     Study of Biomechanics in the Arterial System                   247

3.3. Casting
Casting is an additional method available to create both rigid and complaint models
of elements of the arterial system.63 There are a variety of different techniques
available, but common to all, is the need to manufacture a solid part, the outer
surface of which corresponds to the artery lumen wall. This male lumen cast defines
the geometry of the artery inner wall and so accurate casting of this is critical to
capturing the internal geometry of the artery for biofluid mechanics investigations.
This cast may then be placed into a female mould which is used to define the
outer wall of the artery, and pouring or injecting a curable material into the void
between the male cast and the female mould results in a solid model of the artery
with a certain wall thickness.20,64 Casts of the arterial system may be used in
flow visualization studies, wall deformation analyses, medical device implantation
tests and photoelasticity investigations.65–67 A general technique is described in the
following section for creating compliant casts of human arteries.

3.3.1. Application
A solid model is generated from medical images using the techniques described in
Sec. 3.1. In this example, a solid model of an aortic arch is developed using a CAD
package. A rendered model of the arch is shown in Fig. 8. This model is then split
along its centerline into two halves, each one corresponding to one mould half. Each
half is then defined by a curved surface representing the lumen wall, bounded by
two edges defining the split lines. These split lines are swept orthogonally away from
the model thereby defining two additional surfaces. A similar process is performed
for the other mould half. These two mould halves may then be brought together
to ensure that the swept surfaces are concurrent and that no interference or gaps
occur. The halves are then processed for machining into two blocks of material
such as aluminium or acrylic. Solid modeling packages are capable of defining the
computer numerical control (CNC) machining codes required for machining surfaces
defined in a computer solid model onto a workpiece.
    The workpiece is then mounted onto the table of a CNC machining center, a
datum is set, and the surfaces are machined. The two mould halves are manufactured
using this technique. Gates, vents, supports and guide-pins may be added during the
solid modeling procedure or after the surfaces have been machined. The completed
mould may then be prepared for casting.
    Prior to casting, a light coat of mould release agent is applied to the walls of the
lumen mould. The halves are then joined together. A low viscosity casting wax is
then melted and poured into the mould via the gate. Upon solidification of the wax,
the male component is then removed from the mould. A lumen cast, representing
the blood volume whose surface corresponds to the lumen wall, has been created
(Fig. 9).
    The principal difference between the female mould and the male mould is that
the female mould has a wall thickness associated with the artery wall included into
248                                     T. P. O’Brien et al.

Fig. 8. A computer solid model of the aorta generated from a set of CT images. Two sample
images and their location are also shown. A cast of the aortic arch section was manufactured using
an injection moulding procedure.

Fig. 9. A female wall mould manufactured using a CNC machine, a lumen cast which is inserted
into the mould and the resulting silicone rubber cast of the aortic arch.
                    Study of Biomechanics in the Arterial System                 249

the diameter of the solid model derived from the medical images. In addition to
this, supports must be added to the female mould to support the male cast rigidly
in place during material injection. The process for manufacturing the female mould
is similar to that of the male mould.
    The female mould is coated in mould release agent and the male cast is placed on
supports in the female mould. The two mould halves are then joined together, and
casting material is injected into the injection port, filling the gap between the male
lumen cast and the female mould wall. The mould is filled once the casting material
begins escaping from the vents. Different types of material may be injected depend-
ing on the application. Latex and silicone rubber have been used to manufacture
rigid and compliant casts of the arterial system.68,69

4. Numerical Investigations
Numerical studies are arguably the most powerful tools available to researchers
seeking to study the biomechanics of the arterial system. As was discussed in ear-
lier sections, the main areas of biomechanical study in the human artery involve
investigations of wall stresses in the artery and blood flow dynamics.70–73 Compu-
tational fluid dynamics (CFD) has been used to investigate artery hemodynamics,
mass transfer and disease formation mechanisms while finite element analysis has
been widely used in studying artery wall stresses, deformations and load responses.

4.1. Computational fluid dynamics
Computational Fluid Dynamics is a numerical technique capable of modeling the
hemodynamics of the arterial system.74 The technique involves taking a geometrical
description of the volume the fluid occupies and dividing or discretizing this into
a finite number of volumes. There are a number of different numerical algorithms
available for CFD and their operation depends on the discretization scheme used.
The classical method for CFD is called the finite volume method, whereby a fluid
volume is discretized into a finite number of control volumes over which the gov-
erning equations are imposed.75,76 The finite element method, originally developed
for structural analyses, may also be implemented in fluid analyses, and it has the
added advantage that it enables both fluid-structure investigations to be performed
using one integrated package.77,78 Additional methods applicable include the finite
difference method and the boundary element method.
    CFD operates on the principle that it is possible to take a complex fluid contin-
uum and divide it into a finite number of simple fluid elements. The intial process
in CFD involves creating a geometrical model of the fluid volume using a solid mod-
eler or a purpose build CFD pre-processor. Once the edges, surfaces and volumes
defining the fluid volume have been created, it is possible to subdivide or discretize
the model into a finite number of elements. During this pre-processing stage, the
250                                T. P. O’Brien et al.

names of the boundary entities are defined and edges and surfaces of interest in
analysis may also be defined.
    The next stage in CFD is to define the material properties, boundary condi-
tions and solution method and procedure. This is done using a specialist software
termed the ‘processor’. Numerous commercially available processors are available
and some specialized purpose built codes have been developed by research institu-
tions and organizations.70,75,78–81 The processor imports the discretized geometry
and applies the governing equations across it. By defining boundary conditions, such
as flow-rates and pressure waveforms, at the inlets or outlets of arterial systems, the
processor is able to calculate the entire flow-field. During processing, the material
properties, boundary conditions, species concentrations and temperatures may be
defined while additional sub-routine libraries are able to model many other flow
    The final stage of CFD involves post-processing the results. This stage involves
selecting areas of interest in the mesh, recording the associated data, and arranging
it to provide meaningful graphical methods of interpreting the flow.

4.1.1. Application
The following simulation was performed in order to investigate the nature of the
blood flow around the aortic arch and down into the abdominal aorta. The model
was generated from a set of CT images acquired using a spiral CT imager (Siemens
A. G.). A total of 198 images were acquired with an interslice spacing of 2 mm.
The model was processed according to a smoothing process outlined previously.52
The model was then exported to the CFD pre-processor Gambit (Fluent Europe)
where a paved quadrilateral mesh was specified. A total of 250000 first order ele-
ments were used to mesh the model. The resulting mesh was then exported to the
finite volume processor Fluent (Fluent Europe) for solution. Models were solved
on a Dell Dimension 4100 1Ghz personal computer with 512 Mb RAM. A laminar
flow, segregated solver incorporating the QUICK momentum scheme, and, PISO
pressure-velocity coupling was used to perform the analysis. A fully developed pro-
file inlet boundary condition used was described elsewhere.82 The convergence cri-
terion was set to 0.001 for the residuals of the continuity equation and of the X,
Y and Z momentum equations. The model was run, found to converge and the
peak velocity flowfield was recorded. Velocity vector and contour plots are shown
in Fig. 10. It is evident from the results that significant skewness of the flow occurs
upon passing around the arch. Slices 3 and 4 illustrate how significant that skew-
ness is.
    A second model illustrates how CFD can be used to measure how fluids interact
with artery walls. In this study, a model of a right coronary artery was generated
from a cine-angiogram. Using a similar numerical procedure outlined in the previous
model, the wall shear stresses of the coronary artery were recorded at a number of
critical locations in the geometry during the peak flow phase. Figure 11 illustrates
                       Study of Biomechanics in the Arterial System                        251

Fig. 10. Hemodynamic flow patterns through a rigid model of an aortic arch. The model was
generated from a CT image set. Pulsatile, Newtonian flow was applied at the inlet (the ascending
aorta). Note the skewed flow through the arch which develops again down through the descending

the nature of the flow through the geometry by means of velocity contours and also
how different levels of wall shear stress are evidenced at different locations.

4.2. Finite element analysis
Finite element analysis is a powerful numerical technique developed initially to solve
complex structural engineering problems, which could not be solved empirically or
analytically. From the geometrical definition of a physical body it is possible to
sub-divide or discretize the body into a finite number of individual geometrical
elements.83–85 The constitutive laws or equations governing the system are then
applied to individual geometrical entities.86 By assembling a global stiffness matrix
for the governing equations, creating geometrical relationships between each element
and its neighbour, it is possible to define a single matrix representing the entire
system. This may then be solved to yield an approximate solution for the boundary
value problem.
    A particular advantage of the finite element method over the finite volume
method is its capability in modeling both fluids and structures. While efficient
solvers that can capture physiologically realistic fluid-structure interactions are cur-
rently available and under continuous development this section will concentrate on
252                                    T. P. O’Brien et al.

Fig. 11. Hemodynamic flow patterns through a rigid model of a coronary artery. The model was
generated from cine-angiography. Pulsatile, Newtonian flow was applied at the inlet. The flow
contour plots show the nature of the flow through the artery at different cross-sections. Results
were then taken from the CFD for each time-step and plots show WSS occurring at inner and
outer locations at bends in the geometry were created.

the application of the finite element method in the investigation of physiologically
realistic artery wall stresses and deformations.
    Finite element solvers have a wider variety of element types within their libraries,
from simple first order 2D quadrilateral elements to higher order 3D tetrahedral
elements. The choice of element is dependant on the geometry of the model to be
investigated and also on the required accuracy and the nature of the deformations.
    As with CFD, FEA consist of three main stages.87 The first, pre-processing,
involves creating a geometrical representation of a body using a 2D or 3D CAD
system. This model may then be imported into a mesh generation software pack-
age, where the geometrical model is discretized into a finite number of individual
elements. Concentrations of elements are placed at regions of interest in the struc-
ture and where stresses are likely to be most complex. During the pre-processing
stage, names tags are attached to edges or surfaces for constraining or loading, and
to planes or volumes for defining continuums. Also, edges and surfaces of interest
may be pre-defined during pre-processing to aid post-processing of results.
    The second stage involves exporting the element mesh to a solver where the nec-
essary material properties, constraints and loads are defined. During this stage, the
solution method and convergence criteria are also set for the numerical procedure.
The model is run until the solution converges to a required limit, otherwise, the
model must be redefined or remeshed.
                        Study of Biomechanics in the Arterial System                           253

    The final stage in FEA is to take the solution and to analyse the results using a
post-processor. Here, geometrical definitions enable specific surfaces, edges and vol-
umes to be selected and various quantities, such as Von Mises stress and directional
strain, may be recorded and plotted.

4.2.1. Application
Solid models of two bypass graft anastomoses were generated using ProEngineer
Wildfire (Parametric Technology Corp.). One model consisted of a 6 mm diameter
artery with a 1 mm thick wall attached to a 6 mm diameter graft with a 1 mm
thick wall. The second model was that of a 6 mm diameter graft attached to a
6 mm diameter artery at an anastomotic angle of 45◦ . The models were exported
to ProMechanica (Parametric Technology Corp.) were automeshing was performed
using 9 node quadrilateral elements. A multipass solution process was performed
until the wall stresses converged to within 2% of their limiting value. The Von Mises
Stress along the suture lines (Fig. 12) shows how the wall stress characteristics of
the end-to-end anastomosis are superior to those of the end-to-side anastomosis.
This observation supports previously published work.86

5. Experimental Investigations
In vitro modeling of arterial systems using experimental techniques enables
researchers to validate numerical results and also to investigate phenomena that are
beyond the capability of numerical solvers. While numerical simulation enables quick
and efficient investigation, complex phenomena such as particulate flows frequently
need to be studied using experimental methods. In addition, numerical simulations

Fig. 12. A comparison between the stress distributions between a Dacron graft and a compliant
artery for an end-to-end anastomosis and an end-to-side anastomosis in femoral bypass procedures.
A mean pressure of 10 kPa was applied to the internal walls of the models. The peak stress measured
in the ESA was 56 kPa while in the EEA it was 28 kPa. For a normal healthy artery under similar
loads it is 18 kPa. It is evident from this study that the ESA produces wall stresses that are
significantly higher than those in the EEA and over three times those present in a healthy artery.
From the stress distributions, it is also evident that significant gradients in the wall exist.
254                                T. P. O’Brien et al.

need to be validated using experimental and theoretical models in order to establish
the accuracy and reliability of a modeling technique.88–90

5.1. Physiological flow modeling
The basic element required for enabling experimental investigations of the arterial
system is the flow circuit. A wide range of different flow circuits are possible each one
being tailored to fulfill a particular requirement.91–94 A basic flow circuit requires
a pressure head to drive fluid through the system under investigation, usually a
pump or reservoir. The fluid then passes through the test section, which may be a
geometrical model of an artery lumen or experimental medical device. In general,
steady flow conditions are modeled using a constant pressure head.93,95 Pressure
within the system may be set by controlling the head height or pressure within
the reservoir while a valve controls the flow rate. Steady flow models have been
used to investigate flow fields, wall stresses and device performances in the arterial
system.27 However, steady flow modeling is limited in that it fails to capture the
main attribute of the cardiovascular system, that of its pulsatile nature.
    Pulsating flow circuits are required to accurately replicate the in vivo flow condi-
tions. Numerous studies have used pulsatile flow circuits in order to create flow-rate
waveforms similar to those found in the body.20,91,92,94,96 A basic method to gener-
ate pulsatile flowrates involves using a piston pump with one-way inlet and outlet
valves.29 A motor is used to drive the piston and the resulting fluid displacement
through the test section is proportional to the displacement of the piston. Complex
waveforms may be modeled by using computer controlled stepper or servo-motors
to drive the piston. Fluid is drawn into the piston from a reservoir and is pumped
through a valve to the test section. The motion of the piston may be related to the
motion of the fluid by means of the continuity equation. Therefore, the temporal
integral of the desired mean velocity in the test section is proportional to the dis-
placement of the piston. While a piston pump is capable of producing flow-rates
similar to those found in the arterial system, the modeling of physiologically real-
istic pressure waveforms is more complex. While addition of a valve downstream
from the test section can serve to increase pressures acting within the test section
to realistic levels, accurate reproducibility of both pressure and velocity is particu-
larly difficult due to the fact that the arterial system is complex, and that pressure
waveforms are composed of numerous harmonics. These harmonics may result from
the pumping nature of the heart, from the elasticity of the arterial walls or from
the resistance of the peripheral system. In addition, wave reflections from branches
and bifurcations also play a major role in influencing the nature of the pressure
waveform. In a flow circuit which is predominately rigid, and the test section is
compliant, it may be necessary to incorporate a Windkessel, or pressure reservoir,
to mimic the contraction of the arteries located elsewhere in the arterial system.
The compliance inherent in the Windkessel serves to increase the steady system
                        Study of Biomechanics in the Arterial System                           255

Fig. 13. A computer controlled piston pump drives fluid through a test section at physiologi-
cally realistic flow-rates. Pressures within the test section are controlled by valves located down-
stream from the test section. Transient flow-rates and pressures are recorded and a laser Doppler
anemometer is used to determine the flow velocity at specific locations in the test section.

pressure to more physiologically realistic levels after the transient pressure pulse
has passed through.
    A schematic diagram of a pulsing flow circuit is shown below (Fig. 13). In this
example, pulsatile pressure and flow waveforms were passed through compliant mod-
els of an end-to-side bypass graft. The pressure transducer was used to record the
pressure within the model while laser Doppler anemometry was used to investigate
the nature of the flow through the junction.

5.2. Laser Doppler anemometry
Laser Doppler anemometry (LDA) is a non-contact optical measurement system
which offers excellent directional response and high temporal and spatial resolution.
It determines the average velocity of particles through a small finite volume of fluid
and, depending on the system, is capable of acquiring measurements in all three
Cartesian co-ordinates.
    The principle behind LDA is that of the Doppler shift; light reflecting from a
body experiences a shift in frequency. In operation, two laser beams intersect at a
point, with one of the beams at a slightly different frequency than the other. As
micro-particles pass through the intersection (measurement volume), light fluctuat-
ing in intensity is reflected off them and into a photo-detector. The frequency of this
light is equivalent to the Doppler shift between the incident and scattered light and
is proportional to the component of particle velocity which lies in the plane of the
256                                T. P. O’Brien et al.

two beams and perpendicular to their bisector. All 3 components of velocity can be
measured by using additional lasers set in the plane of each of the 3 Cartesian axes.
    LDA is often termed as being a ‘single point measurement’ system. However,
LDA relies on particles passing through a small finite volume. The velocity is sam-
pled each time a particle passes through the finite volume. The main limitations of
LDA are experienced due to the nature of the small finite volume. If the measure-
ment volume is large with respect to the flow volume being studied, a significant
velocity gradient may exist across the measurement volume. Also, if the velocity
increases, so too does the number of particles passing through the finite volume,
thereby affecting the accuracy of the system. The operation of an LDA system
requires that particles be suspended in the fluid. Therefore, the medium through
which the laser beams pass must be transparent. While LDA offers fast and accu-
rate single point measurement, the set-up and alignment of the laser beams can be
difficult. The beams must be aligned around the axis of rotation of the optics set
and after passing through a convex lens, must intersect at the focal point of the
lens. Each beam must have the same cross-sectional area at intersection.
    LDA has been used in a variety of different applications in arterial
biomechanics.4,67,97 Studies have been performed investigating the nature of pul-
satile flow94 and on the specifics of flow through arterial models such as the carotid
artery,67 the aorta,20 aneurysms98 and femoral bypasses.27 Rigid and compliant
walled artery models have been investigated, and Newtonian and non-Newtonian
flows have also been studied. Complex three dimensional pulsatile flow analy-
ses have been investigated in addition to simpler one-dimensional steady flow
    With the advent of CFD, complex and expensive LDA systems are presently
primarily used for validation and for performing investigations on models unsuit-
able for CFD. Recent studies have investigated arterial biomechanics using both
LDA and CFD as a means of validation and also to investigate the effectiveness of
experimental and numerical modeling techniques.88,99
    The following example illustrates the use of LDA in CFD validation. A computer
controlled piston pump was used to drive a Newtonian 42%/58% water/glycerine
fluid (by weight) through an idealised model of a femoral bypass graft.88 A 55x
series, one-dimensional LDA system (Dantec Dynamics Ltd.) using a 20 mW He-
Ne laser operating in forward scatter configuration was used to record the flow at
various points through the aneurysm. The focal length of the lens was 160 mm with
a beam separation of 76 mm. The refractive index of the fluid was 1.41 producing
a laser wavelength of 448.8 nm in the fluid. A 1.95 beam expander was used and
the beam waist was calculated after expansion to be 2.15 mm with a diameter after
focussing to be 0.042 mm. The resulting beam intersection volume dimensions were
calculated to be 0.043 mm × 0.043 mm × 0.255 mm giving a volume of 0.23 pm.3 45
fringes with a spacing of 2.72 µm were determined through the volume. Polystyrene
latex particles of 0.5 µm diameter were used to seed the fluid.
                      Study of Biomechanics in the Arterial System                       257

Fig. 14. Comparison between various investigative techniques applied to bypass junctions. MRI
and finite volume CFD are compared (top) while 1D LDA is used to validate finite element and
finite volume flow profiles (bottom).

    The resulting profiles at this time are shown (Fig. 14) together with those
acquired from corresponding CFD studies. It is evident from the results that there
is excellent agreement between the experimental and numerical results. Results
are also shown for steady flow through an idealised model of an abdominal aortic
aneurysm (Fig. 15).

5.3. Particle image velocimetry
While LDA is a point measurement technique, particle image velocimetry (PIV) is a
whole field method providing instantaneous velocity measurement in a cross-section
of flow.100,101 Two velocity components are measured while a stereo arrangement
enables measurement of all three components. The use of high-speed charge-coupled
device (CCD) cameras and digital processing enables computation of real-time
velocity maps. As with LDA, lasers are used in PIV. A pulsed laser light sheet is
targeted at the cross section of interest while velocity motion of particles suspended
in the flow is tracked using CCD cameras.
    Once a sequence of two light pulses is recorded, the images are divided into sub-
sections called interrogation areas. The interrogation areas for each image are then
cross-correlated with each other, pixel-by-pixel. The correlation produces a signal
peak, identifying the common particle displacement, and dividing by time yields the
speed of the particle. The direction is determined from the initial location of the
258                                    T. P. O’Brien et al.

Fig. 15. Validation of numerical results through an idealised model of an abdominal aortic
aneurysm using 1D LDA. Excellent agreement between the LDA and the CFD is evident. Steady
flow through a rigid model of the abdominal aorta was investigated.

Fig. 16. A comparison between a flowfield determined using PIV and one using CFD for a bypass
graft junction model. A pulsatile flow circuit was used and the velocities are shown for the peak
time in the inlet pulse.

particle to the final location by means of Cartesian co-ordinate system applied to the
field. A velocity vector map is then created over the whole area by cross-correlation
of all interrogation areas.
    PIV has been used to validate global flow fields in arterial biomechanics.
Figure 16 illustrates a comparison between flow through and end-to-side anasto-
mosis visualised using PIV and calculated using CFD.
                       Study of Biomechanics in the Arterial System                           259

5.4. Video extensometry
CCD cameras have been used in the past for a variety of different biomedical investi-
gations from monitoring cellular fluorescence to recording anatomical changes.102,103
Video extensometry is a technique, which uses CCD cameras to record the deforma-
tion of materials over time. It has been used in biomechanics to record the deforma-
tions of compliant walled arterial models.20,104 A major advantage of the technique
is its portability, enabling researchers to investigate model wall deformations as well
as arterial deformations during surgery.
     A video extensometer consists of a camera, which captures real-time images of
the deformation, and software, which processes and records the images. The system
must be calibrated using known dimensions of the material under investigation and
then it may determine the strain in the material over the course of the deformation.
     The strains in compliant models of femoral bypass grafts may be determined
in such a manner. As an example, a video extensometer (Messphysik Laborgeraete
GmbH.) was used to record the deformation of a rigid graft/compliant artery model.
The system was pressurized to physiologically realistic pressures and the resulting
deformations were recoded using the video extensometer. Figure 17 illustrates the
geometrical nature of the model together with the deformation that occurs due to
physiologically realistic pressures.

Fig. 17. Video extensometry. The bypass junction model is pressurised from the unloaded state to
the loaded state and deformations are calculated using associated software. Bed rise and side-wall
bulging is evident from the images.
260                                T. P. O’Brien et al.

5.5. Pressure and flow-rate measurement
The measurement of pressure and flow-rates in in vitro flow studies is possible
through the use of medical grade products such as pressure catheters and ultra-
sonic flowmeters. While traditional measurement systems such as elastic pressure
transducers, hot-wire anemometry, pitot tubes and turbine meters can play a role
in arterial biomechanics, piezoelectric, electromagnetic and ultrasonic transducers
offer minimal flow disturbance and high frequency operation. Medical grade pressure
catheter and ultrasonic systems enable researchers to perform both in vivo and in
vitro experiments. A metal diaphragm with directly deposited resistive strain gauges
(Gaeltec Ltd.) was used to measure the pressure in the aortic aneurysm model below
(Fig. 19). An ultrasonic flowmeter (Transonic Systems Inc.) incorporating a transit-
time flowsensor was used to measure corresponding flowrates.

5.6. Photoelasticity
A variety of different non-contact experimental strain measurement techniques have
been used to investigate compliant arterial models. Video extensometry,21,104,105
photonic sensor,106 photocell combined with light-emitting diode or scanning
laser107,108 have previously been used to quantify arterial wall strains. These meth-
ods enable measurement of surface stresses, however, they do not provide data on
the whole stress field unless numerous measurements are taken. However, the reflec-
tion photoelastic method enables viewing of the whole stress field on a loaded part.
Photoelastic fringe analysis is a widely used method for experimental visualizing
of stresses during static and dynamic loading. The photoelastic method has been
applied to the stress analysis of biomechanical applications.
    Photoelasticity works on the principle of birefringence, which is exhibited by
certain transparent materials. A ray of light passing through a birefringent material
experiences two refractive indices. The light is resolved along two principal stress
lines and each component experiences two different refractive indices. The difference
between the refractive indices produces phase retardation between the two compo-
nent waves proportional to the two principal stresses. As the level of birefringence is
proportional to the stress, differently stressed areas will produce different colors. In
photoelasticity, the birefringent property is expressed only when a stress is applied
to the material. Models are made from a birefringent material such as PL-3 epoxy
    The photoelastic technique may be used to investigate the stress distributions
in an aortic aneurysm.109 In this example, a photoelastic model of an idealized
abdominal aortic aneurysm is manufactured using an injection moulding technique
(Fig. 18). The model is manufactured using PL-3 epoxy resin and its interior surface
is coated in PC-11 reflective adhesive.
    A computer controlled piston pump is used to provide a physiologically realistic
pressure waveform to the interior of the model. Internal pressure is monitored using
a pressure catheter. The model was illuminated by plane polarized light using a
                        Study of Biomechanics in the Arterial System                            261

Fig. 18. Photographs of an aneurysm model used for photoelastic experiments. (a) An injection
moulded model manufactured from PL-3 epoxy resin. The model is removed from the mould, the
male cast is removed by cutting the model open which is then coated in PC-11 reflective adhesive
and glued back to its original shape using an elastic glue. The result is a photoelastic model shown
in (b) which may be inserted into a flow loop.

Fig. 19. A photograph of the photoelastic model under load. The photo is taken through the
optical head of the polariscope. The fringes corresponding to different stress levels are clearly
evident with maximum stress evident at the proximal and distal aneurysm ends.

030 series reflection polariscope (Measurement Group Inc.). A 12 V 100W tungsten-
halogen lamp was used. A model 232 null-balance compensator was used to measure
the fringe orders with a resolution of 0.01 fringe. In the unloaded state, there was
no fringe pattern, while under load, a fringe pattern was visible (Fig. 19). As the
262                                    T. P. O’Brien et al.

fringe orders observed in the material were proportional to the difference between
the principal strains in the material, it is possible to determine the principal stresses
from the elastic modulus and Poisson’s ratio of the epoxy resin. This method found
that hypertensive patients may experience wall stresses up to 35% greater than
normal. These findings also correlated with those from previous studies which show
that most aneurysm ruptures occur at the postero-lateral wall.110,111

6. Medical Device Research and Development
The techniques discussed in previous sections may be used for the study of arterial
biomechanics with a view to better understanding the mechanical environment and
to aid in the development of medical devices to treat various medical conditions.
This section is concerned with illustrating how a medical device may be developed
to address a certain condition.

6.1. Vascular grafts for abdominal aortic aneurysms
The problem under examination is that of the increase in cardiac load which results
following the implantation of an abdominal aortic aneurysm graft. In order to study
this problem, a sample of a conventional aortic aneurysm graft for open surgery
was sourced from a commercial manufacturer. The dimensions of the graft were
measured and a solid model of the graft was generated using a 3D solid modeler.
The resulting conventional graft geometry is shown in Fig. 20. The graft is a woven
Dacron graft consisting of an aortic section of uniform diameter, two iliac legs of
uniform diameter, and a bifurcation region where the three components are woven
together. Comparison of the cross-sectional area of the aortic graft section to the
sum of the cross-sectional areas of the iliac graft sections shows that the area ratio
between the aorta to the iliac arteries is approximately 2.

Fig. 20. An illustration of the abdominal aortic aneurysm, conventional graft and tapered graft
computer solid models. Two views of the tapered graft is shown clearly illustrating the gradual
change from the aorta inlet to the iliac outlets.
                     Study of Biomechanics in the Arterial System                   263

    An MRI scan of a healthy aorta was then used to determine the physiologically
realistic area ratio (Fig. 2). The scans showed that the aorta and iliac arteries
are not of constant cross sectional area, rather, the cross-sectional area reduces
along the axial distance from the heart. Therefore, the area ratio was determined
averaging the diameter of each vessel a short distance (∼ 10 mm from the apex of
the bifurcation. The resulting area ratio between the aorta and the iliac arteries
in the healthy case was found to be 1.3. Previous work has cited an area ratio of
approximately 1.27.112
    It was then hypothesized that the uniform cross-sectional areas and non-
physiologically realistic area ratio associated with the aortic graft may have a role
to play in augmenting the aortic pressure. There was a suggestion that pressure
wave reflections from the iliac aortic bifurcation may result in increases in aortic
pressure.113 Therefore, a study was performed in order to redesign the vascular graft
to a more physiologically realistic specification, and then to assess this new design
using in vitro experimental techniques.
    The initial step involved redesigning the graft. It was specified that the new
graft would have to have the same aortic inlet and iliac outlet diameters as those
of the original graft. In addition, the graft would have the same axial length and
be manufactured from the same materials. The only variable would be that of the
cross-sectional area, which would no longer be uniform in each vessel, rather, it
would be tapered gradually along the length of the graft. In addition, the area ratio
between the aortic and iliac segments would be less than that determined from the
physiologically realistic MRI scan, approximately 1.1.
    A novel tapered cross-section was defined which involved sweeping a variable
cross-sectional area aorta into two variable cross-sectioned iliacs (Fig. 20). The graft
was then subjected to in vitro tests in order to assess the effects of the geometrical
design on the system fluid dynamics.
    The first test involved using CFD on the new graft. Two models were investi-
gated, namely, the original commercial graft and the new tapered graft. The models
were meshed using a similar meshing technique in a CFD pre-processor and then
exported for solution in a finite volume processor. Here, a transient, pulsatile, New-
tonian analysis was performed and the fluid velocity profiles at various locations
through the geometries were compared. It was found that the tapered design pro-
duced less skewing of the flow in the iliac legs and that the fluid acceleration from
the aorta into the iliac arteries was less pronounced than in the commercial graft.
    The second test involved creating experimental models of the commercial and
new graft and subjecting them to pressure tests using an in vitro experimental flow
circuit. Aluminium moulds were manufactured corresponding to the graft lumen
dimensions using a CNC machining center. The resulting moulds were then used to
produce wax casts of the lumen. These were then placed in a box and surrounded
by a two-part liquid polyurethane that was allowed to cure. Once cured, the wax
lumens were melted out and the resulting lumen models were inserted into the flow
264                                T. P. O’Brien et al.

    A computer controlled piston pump was used to produce physiologically realistic
flow and pressure waveforms acquired from the literature.114,115 Water was pumped
at pressure through the models in turn. Each model was mounted into the flow
circuit and a series of pressure measurements were taken using a 1 mm diameter
pressure catheter located 20 mm upstream from the aortic bifurcation. These sets
of results were then compared to each other and it was determined that the tapered
design resulting in a reduction in aortic pressure of approximately 9 mmHg.
    The second test proved that the in vitro aortic pressure could be reduced by
tapering the graft along its length and designing the bifurcation so that a sudden
change in area ratio did not occur. The results of these in vitro investigations have
since been used in the development of a new medical device incorporating the design
attributes described here. The final stage in the process, performing in vivo tests and
establishing the efficacy of the proposed device, involves significant clinical input
and lies beyond the scope of this chapter. It should be noted, however, that though
the in vitro results appear promising, full in vivo studies are required to capture
phenomena assumed negligible in the in vitro flow circuit.

6.2. Vascular grafts for femoral bypass procedures
Atherosclerosis is a disease characterized by the deposition of low-density lipopro-
teins onto the intima of an artery. These depositions gradually increase in size until
they lead to the development of plaques, stenoses or occlusions. When an artery
becomes stenosed or occluded, blood supply to a location is reduced or stopped,
and this may result in tissue necrosis. In the case of the femoral artery, stenosis
leads to a reduction in blood supply to the leg, while complete occlusion of the
artery can lead ultimately to amputation of the affected limb. A variety of dif-
ferent treatments are available to treat atherosclerosis in the femoral artery, with
the endovascular methods of angioplasty and stenting and the open surgical tech-
nique of femoro-popliteal bypass being the most commonly used. The open surgery
technique involves bypassing the diseased portion of an artery with a new conduit,
typically made from autologous vein or from a synthetic material such as ePTFE
or Dacron. However, the long-term patency rates of prosthetic grafts remain low.
There have been two fundamental disease formation mechanisms suggested which
contribute to the poor long-term patency of femoral artery bypasses. Disease has
been attributed to the abnormal hemodynamic patterns associated with the geo-
metrical configuration of the end-to-side anastomosis.116,117 In addition, evidence
is available which suggests disease forms due to wall stress gradients resulting from
the material mismatch between the stiff graft and the compliant host artery.1 Two
different configurations may be used to create the distal anastomosis in a femoro-
popliteal bypass, end-to-end (ETE) anastomosis and end-to-side (ETS) anastomo-
sis. The latter is more commonly used and has a number of different geometrical
configurations, including the Miller cuff and Taylor patch, which may be used when
anastomosing a synthetic graft to arteries below the knee.79 Cuffs and patches have
                     Study of Biomechanics in the Arterial System               265

been developed in an attempt to buffer the material mismatch between the syn-
thetic graft and the artery by incorporating a section of vein between them. Also,
increasing the volume of the graft/artery junction can lead to a reduction in the
fluid forces impinging on the host artery wall. A recent review of these approaches
concluded that an optimum geometry for the ETS anastomosis may not exist.82
    Previous work has shown that the wall stresses and hemodynamic patterns asso-
ciated with the ETE anastomosis are less physiologically abnormal than those asso-
ciated with the ETS anastomosis. Therefore, a potential treatment was hypothesized
which seeks to improve the wall stress and hemodynamic characteristics of the ETS
anastomosis by means of the bifurcated graft design shown in Fig. 21. The bifur-
cated graft means that two ETE anastomoses are used to attach the graft to the host
artery rather than the traditional single ETS anastomosis. It is hypothesized that
this device design may increase the long-term patency of femoral bypass procedures,
however, to date, only in vitro evidence supporting this claim has been produced.27
Due to its bifurcated nature, flow through the graft into the host arteries is more
streamlined than in the traditional ETS anastomosis.
    A previous in vitro study used CFD to illustrate how this streamlined flow serves
to reduce abnormal wall shear stress magnitudes and gradients near the graft/artery
junction. A comparative CFD study of idealized models of a traditional ETS bypass
and a bifurcated ETE bypass graft was performed and velocity vector plots and WSS
contour plots showed significant advantage of streamlined flow over the disturbed
flow associated with the traditional ETS anastomosis. In addition to this study,
FEA has been used to study the wall stresses at the ETE anastomosis and the
ETS anastomosis (Fig. 13). Experimental investigations on the two anastomoses
have also been carried out using steady flow idealized models. LDA has been used
to investigate the flow profiles associated with the ETE and the ETS anastomses
while video extensometry was used concurrently to quantify the wall deformations.
Evidence from these studies also suggest that improved flow dynamics and wall
strain characteristics result from using two ETE anastomoses rather than one ETS

               Fig. 21.   A hypothetical femoral artery bypass graft design.
266                                 T. P. O’Brien et al.

7. Conclusions and Future Directions
Numerical and experimental techniques offer researchers powerful tools for the study
of arterial biomechanics, disease and potential treatment. Incorporating a variety
of multi-disciplinary techniques and methodologies is key to successful implementa-
tion of a biomechanics research program. It should be noted, however, that arterial
biomechanics is just one aspect of the greater study of the cardiovascular sys-
tem. While biomechanics concerns itself with flow and structural phenomena, ever
increasing effort is being placed on arterial biochemistry, pharmacology and genet-
ics. As arterial biomechanics progresses into the future, it is expected that the
technologies and techniques described here will be used to further the development
of machines and systems capable of investigating, diagnosing and treating disease
at the microscopic level. Indeed, it is this fusion of newly developed biomechani-
cal investigative techniques and biochemical and genetic advances which offer the
greatest potential in furthering future understanding of the artery, its complications
and its treatment.

  1. F. Loth, S. A. Jones, C. K. Zarins, D. P. Giddens, R. F. Nassar, S. Glagov and H. S.
     Bassiouny, J. Biomech. Eng. 124 (2002) 44–51.
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