Session 9 Hemodynamics Doppler

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```					Hemodynamics
Hemodynamics
- the study of blood circulation and the
forces and motion of blood flow
   Flow - the ability to move from one point to
another when a force is applied
Matter is classified into 3 categories:

   Gas
   Liquid
   Solid
Fluid
 Substances that flow & assume the
shape of their container

Gases & liquids are fluids
Blood

A   liquid
 Supplies    nutrients & oxygen to the cells
 Removes      waste products
 Average     human has 5 liters of blood
Blood
Comprised of plasma, erythrocytes (red cells),
leukocytes (white cells) & platelets
   Plasma - approx. 90% H2O; remainder is proteins
   Blood cells - 40% of blood’s volume; known as
the hematocrit
   Erythrocytes - about 99% of all the blood cells
   Leukocytes - larger than erythrocytes & function
to protect the body against disease organisms
   Platelets - smaller than erythrocytes & are
important in blood clotting
Blood

-is a fluid because it is a liquid
- it follows the properties of a fluid
Fluid
Fluids have 2 important characteristics:

1. Density
2. Viscosity
Density

Mass per unit volume; grams/milliliter (g/mL)
Mass - a measure of an object’s inertia
(resistance to acceleration)
 Thegreater the mass - the greater the force
must be to accelerate it
Blood’s density (1.05 g/mL) > water’s (1
g/mL) because of the proteins & cells
Viscosity (‫)ת‬
   fluid’s ability to resist a change in flow
 Unit  - poise or kg/m-s
One poise is 1 g/cm-s

 Viscosity   varies with flow speed
Viscosity
Blood is a viscous fluid containing cells & plasma
   Blood viscosity - 0.035 poise at 370C - approx. 5X
that of water
   Blood viscosity varies from 0.02 (anemia) to 0.10
(polycythemia)
   Since erythrocytes are the major cellular component of
blood, if the number of RBCs increases, the viscosity of
the blood increases (directly related)
   Anemia (low # of RBCs) has low viscosity
   Polycythemia (high # of RBCs) has high viscosity
Friction
 A force that occurs when two bodies in
contact are in motion relative to one
another (rub together)
 Frictional force acts in a direction opposite
to the direction of motion
 One  cause of friction is the roughness of
the two surfaces in contact
Pressure

 force   per unit area
 is equally distributed throughout a static
fluid & exerts its force in all directions

A pressure difference is required for flow to occur
Fluids
   substances that flow & conform to the shape of
their containers, such as gases & liquids
   Pressure applied to a fluid acts different than
pressure applied to a solid.
   The difference is that not only is there pressure pushing
down at a given point, but there is also the same
pressure pushing up and to the sides.
   The pressure is the same in all directions in a fluid at a
given point. This is true because liquids and gases to
take the shape of their container.
PRESSURE’S EFFECT
ON FLUIDS
FORCE

When force is applied to a fluid,
pressure , even though the force
is directional, the pressure is
omnidirectional.
Fluid pressure at a given point is
the same in all directions
Pressure is applied equally
throughout a liquid
Pressure
   Equal pressures applied at both ends of a
liquid-filled tube results in no flow
   If the pressure is greater at one end, the
liquid will flow from the higher-pressure end
to the lower-pressure end
   This pressure difference can be generated by
a pump or by the force of gravity
   The greater the pressure difference, the
greater the flow rate will be
Pressure
The difference in pressure (between the
high pressure end & the low pressure
end) is called a pressure gradient or
Pressure gradient is calculated by taking
the pressure difference & dividing it by the
distance between the 2 pressure locations

P1-P2
Distance (length) between P1 and P2
Volume Flow Rate (Q)

= volume of blood passing a point per unit time
-   expressed in milliliters (mL)/minute or /second or
cc/second (this is not a measurement of speed)
5,000 mL/min (our total blood volume circulates in
-   Q in a long straight tube is determined by the
pressure difference & the resistance to flow
Volume Flow Rate (Q)

Volume flow rate (mL/s) =

∆P (dyne/cm2)
Flow resistance (g/cm4-s)
Pressure gradient & volume flow rate are
directly related (in a straight tube).
Therefore, the greater the pressure
difference, the greater the flow rate
Flow Resistance

Flow resistance (in a long, straight tube)
depends on:
 Fluid’s viscosity
Flow Resistance
Flow Resistance =

8 X length x viscosity (‫)ת‬
Poiseuille’s Equation

Substituting the flow resistance equation
into the flow-rate equation & using tube
Poiseuille’s equation for volume flow rate
Poiseuille’s Equation
volume flow in long straight tubes

Thus, it serves only as a rough
approximation to the conditions in
blood circulation
Flow Resistance = 8 X length X viscosity (‫)ת‬

Volume flow rate (Q) =       ∆P
Flow resistance
Q=         ∆P
Flow resistance

Q=           ∆P
8 X length x viscosity

Q = ∆P X  X radius4
8 X length X ‫ת‬
Poiseuille’s Equation

Q = ∆P  r4
8‫ת‬L

Q = ∆P  d4
128 ‫ ת‬L
Extra Credit – 10 points

Prove to me on paper why both of
the equations given are equal.

Due next class meeting
Using this formula, what
relationships can be seen?

Q = ∆P  r4
8‫ת‬L

Q = ∆P  d4
128 ‫ ת‬L
Imagine you are drinking through a straw &
you want to get more flow through the straw:

1.  pressure gradient (suck harder)

2.  radius of the vessel (use a bigger straw)

3.  viscosity of the fluid (dilute or heat it)

4.  length of the vessel (cut the straw in ½)
Which factor affects flow, resistance &
velocity the most?

The one raised to the fourth power!!!

A small change in vessel diameter will
create a large difference.

Resistance to flow depends on viscosity of
blood, the radius of the blood vessel’s
lumen & the length of the vessel.
Increases in:   Causes increased   Causes decreased

Flow                          Volume flow rate
resistance
Pressure     Volume flow rate
difference
Length       Flow resistance    Volume flow rate

Radius       Volume flow rate   Flow resistance

Diameter      Volume flow rate   Flow resistance

Viscosity     Flow resistance    Volume flow rate
Types of Flow Profiles

Flow is divided into 5 spatial categories:
1) plug
2) laminar
3) parabolic
4) disturbed
5) turbulent
Flow Profiles
   Velocity (speed & direction RBCs are traveling) is
not constant or uniform across the vessel lumen
   Various flow profiles are seen in normal flow in:
   various vessels
   at different points in the cardiac cycle

   Velocity profile across a vessel depends on:
   curvature of a vessel
   branching to a smaller vessel
   obstruction in a vessel
   diverging cross section
Plug Flow

- is constant flow velocity
across the vessel

It occurs in large vessels
such as the aorta
Laminar Flow

- is flow that occurs when straight, parallel
layers of fluid slide over each other
 In a normal vessel, friction produces lowest
velocities along the vessel wall & highest
velocities occur in the center of the vessel
 Laminar flow is commonly seen; its
absence often indicates abnormal flow
conditions at a site where there is vascular
or cardiac-valvular disease
Laminar Flow

 flow that occurs
when straight,
parallel layers of
fluid slide over
each other
Parabolic Flow

- flow is steady laminar flow whose varying
flow speeds across the tube are described
by a parabola
- ave. flow speed = ½ max. flow speed
(at the center)
Parabolic Flow

- not commonly
seen in blood
circulation because
the vessels
generally are not
long & straight
Disturbed Flow

- a form of laminar flow but the parallel
streamlines are altered from their
straight form

Disturbed flow fluid flows
in a forward direction
Disturbed Flow

This occurs in the
region of stenosis or
at a bifurcation (the
point at which a
vessel splits into two)
Turbulent Flow
- (turbulence) is nonlaminar flow with random &
chaotic speeds
   Turbulence with multiple velocity components is
known as chaotic flow
   Eddies currents (flow particles moving in circles)
occur creating regions of reverse flow
   As flow speed , turbulence will ultimately occur
Turbulent Flow
Turbulent Flow
 Flowspeed for turbulent flow depends
on the fluid’s density & viscosity and the
vessel’s diameter
 Reynolds  number predicts the onset of
turbulent flow.
   If the Reynolds number (Re) exceeds
about 2000 to 2500 (depending on tube
geometry), flow becomes turbulent
   This is called the critical Reynolds number
Reynolds number =
Average flow speed x tube diameter X density
Viscosity
If there is an increase in:

• Flow speed
• Diameter
• Density
Reynolds number 

If viscosity , Reynolds number 
With the exception of the heart and
proximal aorta, turbulent flow does not
typically occur in normal circulation

Turbulent flow most commonly occurs
beyond an obstruction, such as a
stenosis, particularly in systole but will
also occur in abnormal arterial geometry
(kinked, bent, or tortuous vessels)
- is nonpulsatile hemodynamics that
occurs in the venous system because the
veins offer little resistance to flow and can
accommodates a large change in volume
with little change in pressure
 volumeflow rate is simply related to
pressure difference & flow resistance
Venous pressure & flow are affected by:

1.   Respiration
2.   Hydrostatic pressure (p)
Respiration
Inspiration: diaphragm moves inferiorly,
causing an increase in abdominal
pressure, that slows down the flow of
blood in the lower extremities.
   This causes a decrease in the thoracic
pressure increasing the blood flow in the
veins of the thorax & upper extremities
   The opposite happens on expiration
This is called phasicity (waxes and wanes)
Hydrostatic Pressure (p)
- the gravitational weight of a column of blood
that extends from the heart to a level where
blood pressure is determined
   The higher the column of blood in the body, the
greater the pressure
   Hydrostatic pressure has greater effect on the
venous system than on the arterial system
Hydrostatic Pressure (p)
   can be measured by taking a blood
pressure, with measurement units of
millimeters of mercury (mm Hg).

Right atrium is considered the reference
point & has zero hydrostatic pressure
Pulsatile Flow
increasing & decreasing pressure and flow
speed (cardiac cycle & arterial circulation
The relationship between the varying
pressure & flow rate depends on the:
 flow impedance (resistance)

 inertia of the fluid as it accelerates

& decelerates
 compliance (expansion & contraction)

of the vessel walls
Major Characteristics of
Pulsatile Flow are:
1.   windkessel effect
2.   flow reversal
Windkessel Effect
- occurs when the pressure forces a fluid
into a compliant vessel, (aorta) expanding
& increasing the volume in it
When the pressure is reduced, the vessel
contracts producing extended flow later in
the pressure cycle.
Windkessel Effect

In the aorta, it
results in continued
flow in the forward
direction, because
aortic valve closure
prevents flow back
into the heart.
Flow Reversal
- is the reversal of flow in diastole as the
pressure decreases & the distended
vessels contract in the distal circulation
where there are no valves to prevent the
back flow
Flow Reversal
 inthe distal circulation
 reversal of flow in diastole
Continuity Rule
- states that the volume flow rate is equal to the
average flow speed across the vessel multiplied by
the cross-sectional area of vessel

A stenosis (narrowing of the lumen of a vessel)
produces disturbed & possibly turbulent flow

The average flow speed in the stenosis must be 
flow speed proximal & distal to it, so that the
volume flow rate is constant throughout the vessel
Continuity Rule
Volume flow rate must be constant
for the 3 regions of the stenosis:
1. proximal to
2. at
3. distal to
But…
Flow speed increases at a stenosis
& turbulence can occur distal to it
Poiseuille’s law converted to average flow
speed, rather than volume flow rate, is:

Average Flow Speed =

Pressure difference X diameter2
32 X length X viscosity
Poiseuille’s Equation states:
Volume Flow Rate (Q) = ∆P X  X radius4
8 X length X ‫ת‬

Continuity Rule states:
Average Flow Speed =      ∆P X diameter2
32 X length X ‫ת‬
Continuity Rule
If a stenosis has an area measuring ½ that of
the proximal & distal vessel, the average flow
speed in the stenosis is 2X that proximal &
distal to it.

Average Flow Speed =
Pressure difference X diameter2
32 X length X viscosity
Continuity Rule - E.C.

If a stenosis has a diameter that is ½ that
adjacent to it, the area at the stenosis is ¼ that
adjacent to it, so the average flow speed in the

Show how this mathematically logical
using the continuity rule.
Does Poiseuille’s law and the continuity rule
contradict each other in regards to a stenosis?

Poiseuille’s law states that flow speed  with
smaller diameters and the continuity rule states
that flow speed  with smaller diameters

Recall that Poiseuille’s law deals with a long
straight vessel (no stenosis), the diameter
refers to that of the entire vessel. If the
diameter of the entire vessel is reduced (as in
vasoconstriction), flow speed is reduced.
Bernoulli's Principle
 As the speed of a moving fluid increases,
the pressure within the fluid decreases
 As fluid passes through a tube that
narrows or widens, the velocity and
pressure of the fluid vary.
   If the tube narrows - fluid flows more quickly
Surprisingly, Bernoulli's Principle tells us that
as the fluid flows more quickly through the
narrow sections, the pressure actually
decreases rather than increases!
ri.html#animation
 http://library.thinkquest.org/27948/bernoulli
.html
In the continuity rule, the diameter is referred
to is for a short portion of a vessel (the
stenosis). If the diameter of only a short
segment of a vessel is reduced (stenosis),
the flow speed in the vessel is only affected
at the stenosis, where it is increased. The
stenosis has little effect on the flow
resistance of the entire vessel if the stenosis
length is small compared to the vessel length.

Therefore, the two situations are different.
Bernoulli Effect

   Explains why there is a drop in pressure
associated with high flow speed at a stenosis

   At a stenosis, the pressure is < the pressures
proximal & distal to it. This allows the fluid to
accelerate into the stenosis & decelerate out

   Energy balance is maintained because pressure
energy is converted to flow energy upon entry,
and then vice versa upon exit
Bernoulli’s equation states that as flow
energy  , pressure energy .

The magnitude of the decrease in
pressure (from the increasing flow speed)
at the stenosis can be found from a
modified form of Bernoulli’s equation.

pressure drop (∆P) = ½ density x (flow speed)2
pressure drop = ½ density x (flow speed)2

In this modified equation, the flow speed
proximal to the stenosis is assumed to be
small enough, compared to the flow speed
in the stenosis, to be ignored.

Simply stated:
when flow speed  ; pressure 
Pressure drop from a stenotic heart
valve can reduce cardiac output

The following form of the equation is
used in Doppler echocardiography for
calculating pressure drop across a
stenotic valve:

∆P = 4 (V2)
∆P = 4 (V2)
V = flow speed (m/s) in the jet
∆P (mm Hg) = pressure drop across the valve

If the flow speed in the jet is 5 m/s, the
pressure drop is 100 mm Hg
Pressure drop can be calculated from a
measurement of flow speed at the stenotic
valve using Doppler ultrasound
The  flow speed within a stenosis can
cause turbulence distal to it.
Sounds produced by turbulence are called
bruits & can be heard with a stethoscope
The ultimate stenosis is called an occlusion,
(the vessel is blocked & there is no flow)
Doppler
Doppler Effect or Shift
   the change in frequency (& wavelength) due
to motion of a sound source, receiver, or
reflector
   the difference between the received &
transmitted frequencies measured in Hz
   used to determine the velocity of the moving
reflectors (Note: velocity is speed & direction)
Doppler Effect or Shift

If the source is moving toward the receiver,
or the receiver is moving toward the
source, or the reflector is moving toward
has a higher frequency than would be
experienced without the motion.
A moving source
(red blood cells)
approaching a
(transducer), the
cycles are
compressed in
front of the source
as it moves into its
own wave.
The source motion shortens the

 wavelength =  frequency
as observed by a stationary receiver
in front of the approaching source
Conversely, if the source
wave has a lower frequency
&  wavelength
Only the moving reflector is of interest
for diagnostic Doppler ultrasound

The amount of  or  in the frequency
depends on the:
   speed of motion
 angle between the wave propagation
direction & the motion direction
 frequency of the source’s wave
Doppler equation
-   relates the detected Doppler shift (∆F) to
those factors that affect it

∆F = 2 x F0 x V x COS 
C
∆F = 2 x F0 x V x COS 
C

F0 is the central sound frequency (aka -
transmitted, emitted, incident, or
ongoing frequency) transmitted by the
transducer.

∆F and F0 are directly related;
if F0 is doubled, then ∆F is doubled
∆F = 2 x F0 x V x COS 
C
V - the velocity of the moving reflector
(i.e. red blood cells)

∆F and V are directly related;
if V is doubled, then ∆F is doubled
∆F = 2 x F0 x V x COS 
C

COS  is the cosine of the angle
between the direction of blood flow &
the axis of the beam (the Doppler
angle).

If the cosine is doubled,
the Doppler shift doubles.
sin & cos (in brief)
 A is the starting point
(in our case, the blood
cell), the side opposite
of angle A is called ‘a’
 C is the right angle; the
hypotenuse (the side
of the triangle opposite
of angle C) is called ‘c’
 B is the remaining
angle; the side of the
triangle opposite angle
B is called ‘b’
As  A , side ‘a’ also ;
eventually ‘a’ will equal ‘c’
As  A approaches 90°,
ratio a/c becomes closer to 1
When  A equals 90°,
a/c = 1

 sin A = 1
when  A = 90°
(sin 90° = 1)
∆F = 2 x F0 x V x COS 
C
 Cosine  values range from 0 to 1
 Is an inverse relationship.

As the angle , the cosine 

Example
cosine of 0° = 1
cosine of 90° = 0
∆F = 2 x F0 x V x COS 
C

C - speed of sound in the medium
The speed of sound is approximately
1540 m/s or 1.54 mm/s

If the speed of sound in the medium ;
the detected Doppler shift 
C is inversely related to ∆F
Detection of Doppler Shift

Doppler ultrasound units do not use the
Doppler equation to calculate the Doppler shift

The ultrasound unit compares the
frequency of the received echo (fr) to the
frequency of the transmitted pulse (ft)
Doppler Frequency (fd) =
Reflected Frequency (fr) - Transmitted Frequency (ft)

When:
   received frequency = transmitted frequency
there is no Doppler shift

   received frequency > transmitted frequency
Doppler shift is positive

   received frequency  transmitted frequency
Doppler shift is negative
Factors influencing the magnitude of
the Doppler shift frequency

1.   Doppler shift frequency occurs in
the audible range
Example: A probe emits 5.0 MHz ultrasound
beam striking the red blood cells traveling toward
the transducer. The unit detects the reflected
frequency to be 5.007 MHz. The transmitted &
received frequencies are in the ultrasonic range,
but the Doppler shift frequency is .007 MHz (7,000
Hz) (audible range).
Typical Doppler shifts range: –10 kHz to +10 kHz
Factors influencing Doppler shift frequency

2. As the angle between the transducer & flow
 (COS ), the Doppler shift 
3. If the RBCs are moving toward the
transducer, the received frequency is higher
than the transmitted frequency (shown as a
positive Doppler shift)
4. If the RBCs are moving away from the
transducer, the received frequency is lower
than the transmitted frequency (shown as a
negative Doppler shift)
Factors influencing Doppler shift frequency
5. If there is no motion of the RBCs, the reflected
frequency = transmitted frequency and the
Doppler shift is zero.
6. The faster the flow velocity, the higher the
Doppler shift. Flow speed & Doppler shift,
 with vessel diameter2.
7. If there is a  in concentration of RBCs or if
you are performing the Doppler exam of the
vessel off to one side, there may be a  in
intensity (hard to hear & see on spectral
display)
Factors influencing Doppler shift frequency
8. Since the RBCs are much smaller than
the wavelength of the sound beam,
Rayleigh scattering occurs
9.  the transducer frequency will 
scattering &  the Doppler shift
10. Although higher frequencies produce
more scatter, they are attenuated more
rapidly
Factors influencing Doppler shift frequency

11. Since these reflectors are very weak in
intensity, lower frequency transducers
may be needed to obtain Doppler
information at deeper depths
12. Modern transducer technology takes
advantage of the wide bandwidth emitted
by the transducer by allowing imaging at
high frequencies and then downshifting
into lower frequencies to acquire the
Doppler information

```
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