New non-invasive method of determination of wave speed and by olliegoblue28


									    New non-invasive method of determination of wave speed and
                   attenuation in arterial system
            K.B. Abdessalem a, b, W. Sahtoutb, P. Flaud a, M.H. Gazzah c, Z. Fakhfakh b
                      Laboratoire Matière et Systèmes Complexes, , CNRS URA 343,
                       Université Paris VII, 2, Place Jussieu, 75005 Paris, France
                      Département de Physique, Faculté des sciences de Sfax, Tunisia
                  Département de Physique, Faculté des sciences de Monastir, Tunisia
                Khaled Ben Abedessalem : Faculté des sciences de Sfax, 3018, Tunisia.
                             E-mail address:


The propagation coefficient of pulse wave’s harmonics in an artery can be determined by measuring pulsating blood
velocity and radius at each of two points along the vessel. In this work an expression of propagation coefficient in
pulsatile flow through a viscoelastic vessel is derived and used to study the effects produced by changing in physical
tube parameters. We consider in particular, the effects of increasing the reflection coefficient, on the determination
of the true propagation coefficient. The method developed here is based on the knowledge of instantaneous velocity
and radius values at only two sites. It takes into account, a reflection site of unknown reflection coefficient, localised
in the distal end of the vessel. The results of wave speed and attenuation obtained from simulation of this method are
in good agreement with theoretical values.

Keywords: Pulse wave propagation, Wave reflection, phase velocity, attenuation.

1. Introduction:
The determination of propagation coefficient is of a             obtained by some authors [5] using the three points
great importance. To describe the contribution of                method, measuring pressures at three sites, is not in
arterial properties on pressure, velocities and radius           good agreement with the theory. Phase velocity found
wave propagation, the propagation coefficient should             by these investigators was 20% lower as the attempted
be calculated. Attenuation, a, is related to the energy          value, and the attenuation coefficient was about eight
dissipation due to the blood viscosity, viscoelasticity of       times higher. In other researches, an agreement was
the wall, reflection in bifurcations and disease sites as        found between theoretical and experimental
atherosclerosis or occlusion [1]. The imaginary part of          propagation coefficient values, for uniform tube [6].
propagation coefficient is inversely proportional to the             A second approach in case of unknown reflection
phase velocity that represents a parameter of clinical           coefficient, using four transducers, requiring a pair of
interest [2]. The determination of the propagation               pressure and flow measurement at two sections of the
coefficient in arterial system has been the subject of           vessel investigated, has been described by Milnor and
many researches in the last few decades. In absence of           Nichols [7], and used by Milnor and Bertram [8] to
significant reflection or when the reflection coefficient        measure the propagation coefficient in the femoral and
of the distal site is known (equal to unity), a two-point        carotid artery. Bertram et al. [9] described, an iterative
method based on the measurement of pressure or flow              general method, for calculating propagation coefficient
in two sites [3] can be used on the derivation of the true       using two pressure measurements, one flow rate and
propagation coefficient. Investigators using two-point           one vessel diameter.
method [4] found values in agreement with the                        All the methods found in literature, used for
Womersley theory. The accuracy of this two-point                 derivation of propagation coefficient are based on
method is acceptable, but it has a limited application :         invasive measurement. We solve this problem by using
It assumes a known reflection coefficient produced by            a non-invasive method based on ultrasound
a total occlusion, which is not the case in most                 measurements of velocities and radius at two sites
hemodynamic conditions, when the lumen of artery is              separated with a known distance (d). This method
partially or not occluded.                                       doesn't require the knowledge of the reflection
     Without any restrictive assumption about the                coefficient as it is generally the case in major literature
reflections from local vascular sites, Taylor developed          works using two-point method. We will critically re-
a method known as three-point method. It is based on             examine the exactitude of the method of wave speed
the measurements of pressure or flow in three                    and     attenuation     determination,     for    different
equidistant sites of the vessel of interest. The results         hemodynamic conditions, by using numerical

simulations. In particularly we will study, the effects of             At point x (2)     the expression of forward velocity is
increasing the reflection coefficient.
                                                                                 1           V      
                                                                       V2fn =     V2 n + R2 n 1an
                                                                                                    
                                                                                                           (3)
2. Mathematical model                                                            2           R1an   
      Consider pulsate, laminar flow through a
uniform, viscoelastic and impermeable vessels of
                                                                       Where     Vin    the running velocity waves at location
instantaneous radius R ( x, t ) , of length L, and                     i, Vin    the forward running velocity waves at location
terminated by an equivalent site of reflection, with                   i,
                                                                            Rin the running radius waves at location i, Rin the
reflection coefficient K.
     The fluid is assumed to be Newtonian, and                         forward running radius waves at location i,     V1an and
incompressible; the viscosity is taken to be constant,
and the effect of gravity is negligible. The velocity is               R1an are respectively the component of running axial
denoted by V = [u ( r , x, t ), v ( r , x, t )] , where r is the       velocity and radius waves function of Womersley
radial coordinate, x is the position along the vessel, t is            parameter, at location 1, U 1an component of running
time, u the radial velocity and v the axial velocity. We               radial velocity waves, dependant of variable r, at
assume that parietal deformations are small; the                       location 1.
behaviour of the system is linear, which is coherent                   Since the forward wave at site (1) can been expressed
with a fluid velocity small compared to wave speed.                    as a function of the forward wave at site (2), we can
The signal is assumed to be periodic.                                  write for sites (1-2):
                                                                             V2 fn = V1n .e −γ n .d (4)
    Let us consider a vessel, with a distal reflection site.                               f
We will take two measurements of radius and
velocities at two sites separated by the distance (d). At              With a development limited, of order 3, in the vicinity
each point of the vessel, the velocity or radius which is                           − γ .d
associated to the nth harmonic is the sum of two                       of zero of e        and using equation (2) and (3), we
components, a forward and backward travelling wave.                    can obtain an equation of fourth degree with complex
The forward component for each harmonic at point x                     coefficients: the unknown complex coefficient is
can be expressed as a function of total radius and                     propagation coefficient, and H is a function of the
velocity using an expression derived by Rogova [10].                   Womersley parameter α=R(ω/ν)1/2.
At point x (1) the expression of forward velocity is
         1          V                                                Expressions (4), shows that the determination of
V1nf =     V1n + R1n 1an
                               
                                        (2)                           propagation coefficient is possible, starting from the
         2          R1an                                             measurement of instantaneous radius and velocities in
                                                                       two sections of an arterial tree. The diameter of vessel
                                                                       can be obtained by echo-tracking; blood velocity can
                                                                       be also measured by ultrasound Doppler techniques.

     d 3V2n     3 d ²V   d 3 HR2n       2         d ² HR2n 
            + γ n ( 2n +          ) + γ n  dV2n +           + γ n (V2n − V1n + R2n dH ) + (R2n − R1n )H = 0 (4)
        6            2       6                        2 

3. Results:

                 Fig. 1: examples of signals used in simulation and there discrete spectrum at x=5cm of the origin
                                             of the tube (non noisy signals).

The number of harmonic of velocity and radius signal                method are in good agreement with theoretical values
is n=15, the theoretical wave speed is C=8.78m/s, the               for low and high reflection coefficient
tube length is L=44.3cm and the attenuation a=0.4m-1.               Fig. 4a shows three dimensional frequency pattern of
Figure 2 shows the frequency pattern of normalized                  wave speed, between 1 and 10Hz for different values of
phase velocity between 1 and 15Hz, in a tube of length              reflection coefficient, which increases from 0.06 to
L, for low reflection coefficient (top panels) and high             0.96. The phase velocity values computed by RV-two
reflection coefficient (bottom panel), obtained using a             point method are in close agreement with theoretical
small distance (d=3cm, left) or a large                             values over the investigated range of frequency. The
distance(d=8cm,) between transducers based on                       augmentation of reflection coefficient has no effect on
sections 1-2. In each panel we plotted the apparent                 the determination of the true phase velocity.
phase velocity derived from the following
equation γ n   = (1 / d ) ln( φ1n − φ 2 n ) where φ in is the       Figure 4.b shows three dimensional frequency pattern
                                                                    of Normalised attenuation, between 1 and 10Hz for
phase of the nth harmonic. The phase velocity                       different reflection coefficient values, which increase
computed by our method (RV-two point method), for                   from 0.06 to 0.96. Attenuation computed on all
the same physical and geometrical state in the case of              sections, is also in a good agreement with theoretical
non noisy signals.                                                  values. Increase of reflection coefficient has no effect
Figure 3 shows the results of Normalized attenuation                on the determination of wave speed computed by RV-
over the range of frequency in a similar manner as in               method. The deviations of attenuation values computed
fig.2. Figures 2 and 3 show a good agreement between                by RV-method over the range of frequency are very
simulated and theoretical wave speed and attenuation.               small; it varies between 1.001 and 0.999 times of
Values obtained are close to the theoretical one at low             theoretical values for all values of reflection coefficient
and high values of the reflection coefficient. While                used over the investigate range of frequency. The
apparent wave speed and attenuation show an                         deviations of phase velocity and attenuation, are small
oscillatory behaviour over the range of frequency                   compared to theoretical values, the maximum of wave
investigated. Moreover the amplitude of apparent wave               speed obtained by RV-method over the range of
speed and attenuation increases with the augmentation               frequency investigated is 1.001 times the theoretical
of reflection coefficient, as shown by fig. 2 (bottom).             values; the minimum simulated wave speed was about
The phase’s velocity and attenuation, derived from our              0.999 times the theoretical phase velocity.

     Fig. 2: Normalized phase velocity RV-method: without noise (∆), normalized apparent phase velocity
     (o), for uniform viscoelastic tube, of length L=43.3cm at low reflection coefficient K=0.36 (top panel)
     and high reflection coefficient K=0.86 (bottom panel). The panel in the left column show data obtained
              for (d=3cm), the panel in the right column show data obtained for (d=8cm).C=8.67m/s

   Fig. 3: Normalized attenuation RV-two point method: without noise (∆), normalized apparent attenuation
      (o), for uniform viscoelastic tube, of length L=43.3cm. The legend panels are the same as in (fig.2).

         Fig. 4: three dimensional view of frequency pattern of normalized phase velocity (left) and normalised
                   attenuation (right) for different reflection coefficient values (K=0.06: step=0.1:0.96)

    5.    Conclusions:                                           tube wall behaviour can be solved in the frequency
The aim of our study is to present a theoretical linear          The simulation of the mathematical formula of this
method, for estimating propagation coefficient in                method can be summarized as follows:
arterial vessels. We have validated the method by                In all case studied, both attenuation and phase velocity
simulation and investigated the effect produced by the           computed by our method are in good agreement with
increase of reflection coefficient on the determination          theoretical values. Small deviations are shown at high
of propagation coefficient. Our method based on four             frequency when increasing reflection coefficient. The
arterial wave form measurements, (radius and centre              accuracy of the method increases when decreasing
line velocities at two arterial sites) can be applied non-       frequency. The discrepancies between theoretical and
invasively using Doppler techniques. Our investigation           computed values of the attenuation are larger than
based on Womersley and transmission lines theories,              those of the phase velocity for the same physical and
takes into account the viscoelastic behaviour of wall            geometrical conditions, but still very small. The effect
vessel. Linear analysis, has the advantage that the              of noise can be studded in a next time to evaluate the
system of equations taking into account the viscoelastic         applicability of the method in vivo.

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