Hemodynamics

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Hemodynamics
Michael G. Levitzky, Ph.D. Professor of
Physiology LSUHSC mlevit_at_lsuhsc.edu [phone]
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FLUID DYNAMICS
PRESSURE FORCE / UNIT AREA Dynes / cm 2
FLOW VOLUME / TIME cm3 / sec
RESISTANCE
POISEUILLES LAW P1 - P2 F x R
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POISEUILLES LAW
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RESISTANCE
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PL
P1
P2
Constant flow
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POISEUILLES LAW - ASSUMPTIONS
1. Newtonian or ideal fluid - viscosity of fluid
is independent of force and velocity gradient
2. Laminar flow
3. Lamina in contact with wall doesnt slip
4. Cylindrical vessels
5. Rigid vessels
6. Steady flow
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RESISTANCES IN SERIES
RT R1 R2 R3 ...
RESISTANCES IN PARALLEL
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R1
R2
R3
RT R1 R2 R3
R1
R2
1/RT 1/R1 1/R2 1/R3
R3
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?
x
Boundary layer edge
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LAMINAR FLOW ?P ? Q x R
.
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15
10
ml / sec
5
0
100
200
300
400
500
Pressure Gradient (cm water)
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TURBULENCE
(?) (Ve) ( D)
REYNOLDS NUMBER

?
? Density of the fluid
Ve Linear velocity of the fluid
D Diameter of the tube
? Viscosity of the fluid
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HYDRAULIC ENERGY
ENERGY FORCE x DISTANCE units dyn cm
ENERGY PRESSURE x VOLUME ENERGY (dyn
/ cm2 ) x cm3 dyn cm
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HYDRAULIC ENERGY
THREE KINDS OF ENERGY ASSOCIATED WITH LIQUID
FLOW
1. Pressure energy ( lateral energy) a.
Gravitational pressure energy b. Pressure
energy from conversion of kinetic
energy c. Viscous flow pressure
2. Gravitational potential energy
3. Kinetic energy 1/2 mv2 1/2 ? Vv2
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Laplaces Law
Po
r
Pi
T
T
T
T Pr
Transmural pressure Pi - Po
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GRAVITATIONAL PRESSURE ENERGY
PASCALS LAW
The pressure at the bottom of a column of liquid
is equal to the density of the liquid times
gravity times the height of the column.
P ? x g x h
GRAVITATIONAL PRESSURE ENERGY
? x g x h x V
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IN A CLOSED SYSTEM OF A LIQUID AT CONSTANT
TEMPERATURE THE TOTAL OF GRAVITATIONAL PRESSURE
ENERGY AND GRAVITATIONAL POTENTIAL ENERGY IS
CONSTANT.
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Gravitational pressure E 0 (atmospheric)
Gravitational potential E X ?ghV
Thermal E UV
Total E1 X ?ghV UV
h
Gravitational pressure E ?ghV
Gravitational potential at reference plane
E X
Thermal E UV
Total E2 X ?ghV UV
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TOTAL HYDRAULIC ENERGY (E)
E ( P ?gh 1/2 ?v2 ) V
Gravitational Potential
Kinetic Energy
Gravitational and Viscous Flow Pressures
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BERNOULLIS LAW
FOR A NONVISCOUS LIQUID IN STEADY LAMINAR FLOW,
THE TOTAL ENERGY PER UNIT VOLUME IS CONSTANT.
(P1 ?gh1 1/2 ?v12) V (P2 ?gh2 1/2
?v22) V
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Linear Velocity Flow / Cross-sectional area
cm/sec (cm3 / sec) / cm2
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Bernoullis Law of Gases (or liquids in
horizontal plane)
P1 ½ ?v12 V P2 ½ ?v22 V
lateral pressure
kinetic energy
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The Bernoulli Principle
PL
PL
PL
Increased velocity Increased kinetic
energy Decreased lateral pressure
Constant flow (effects of resistance and
viscosity omitted)
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LOSS OF ENERGY AS FRICTIONAL HEAT
U x V
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TOTAL ENERGY
TOTAL ENERGY PER UNIT VOLUME AT ANY POINT
PRESSURE ENERGY
GRAVITATIONAL POTENTIAL ENERGY
KINETIC ENERGY
THERMAL ENERGY
E (PV) ( ?gh V) ( 1/2 ?v2 V)
(U V)

( ?gh)
(?QR)
VISCOUS FLOW PRESSURE
GRAVITATIONAL PRESSURE
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?UV Frictional heat (? internal energy)
½ ?v2V Kinetic energy
PV Viscous flow pressure energy
E Total energy
E1
E2
E3
h
KE UV
Reference plane
P1
P2
P3
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E1
E2
E3
Reference plane
KE UV
P1
P2
P3
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E4
E3
E2
E1
P4
viscous flow P
P1
gravitational energy
Reference plane
P3
P2
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a
?gh
b
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E1
E2
E3
E4
E5
KE UV
Reference plane
P1
P2
P3
P4
P5
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Pressure equivalent of KE
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P (mm Hg)
h (cm)
15
0
4
4
4
10
8
5
0
12
12
12
12
12
Arteries
Capillaries
Veins
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P (mm Hg)
h (cm)
Q 1.0
15
0
(9)
(3)
1
-5
4
10
8
5
Q 1.0
0
12
3
0
12
9
(9)
(3)
(0)
(12)
Arteries
Capillaries
Veins
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P (mm Hg)
h (cm)
Q 0.43
15
0
(10.7)
(8.1)
-6.7
2.7
4
10
0.1
8
5
Q 1.43
0
12
3
0
12
9
(9)
(3)
(0)
(12)
Q 1.0
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Q
4
12
10
-6
c
a
b
-8
d
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(Pa Pv) (mmHg)
100
Flow (ml/min)
0
-5
0
5
10
15
Pv (mmHg)
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VISCOSITY
Internal friction between lamina of a fluid
STRESS (S) FORCE / UNIT AREA
S ?
Is called the rate of shear units are sec -1
The viscosity of most fluids increases as
temperature decreases
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v1
v2
A
dx

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VISCOSITY OF BLOOD
1. Viscosity increases with hematocrit.
2. Viscosity of blood is relatively constant at
high shear rates in vessels gt 1mm diameter
(APPARENT VISCOSITY)
3. At low shear rates apparent viscosity
increases (ANOMALOUS VISCOSITY) because
erythrocytes tend to form rouleaux at low
velocities and because of their deformability.
4. Viscosity decreases at high shear rates in
vessels lt 1mm diameter (FAHRAEUS-LINDQUIST
EFFECT). This is because of plasma skimming
of blood from outer lamina.
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Non-Newtonian behavior of normal human blood
0.3
0.2
Apparent Viscosity ? (poise)
0.1
0
100
200
Rate of Shear (sec-1)
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Effects of Hematocrit on Human Blood Viscosity
52 / sec
Relative Viscosity
212 / sec
Hematocrit
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PULSATILE FLOW
1. The less distensible the vessel wall, the
greater the pressure and flow wave velocities,
and the smaller the differential pressure.
2. The smaller the differential pressure in a
given vessel, the smaller the flow pulsations.
3. Larger arteries are generally more distensible
than smaller ones. A. More distal vessels are
less distensible. B. Pulse wave velocity
increases as waves move more distally.
4. As pulse waves move through the cardiovascular
system they are modified by viscous energy
losses and reflected waves.
5. Most reflections occur at branch points and at
arterioles.
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Definitions (Mostly from Milnor)
Elasticity Can be elongated or deformed by
stress and completely recovers original
dimensions when stress is removed. Strain
Degree of deformation. Change in length/Original
length. ?L/Lo Extensibility ?L/Stress (
Compliance ?V/?P) Viscoelastic Strain
changes with time. Elasticity Expressed by
Youngs Modulus. E ?F/A Stress
?L/Lo Strain Elastance Inverse of
compliance. Distensibility Virtually synonymous
with compliance, but used more broadly. Stiffness
Virtually synonymous with elastance. ?F/?L
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Progressive increase in wave front velocity of
the pressure wave with increasing distance from
the heart. Mean pressures were 97 120 mmHg.
Inguinal ligament
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Arch
Diaphragm
Knee
Bifid
Thoracic Aorta
Carotid
Tibial
Illiac
Ascending Aorta
Abdominal Aorta
10
Femoral
m / sec
5
2.5
20
10
0
10
30
20
40
50
60
70
80
Distance from the Arch
(Average of 3 dogs)
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1. Ascending aorta
2. Aortic arch
Pressure waves recorded at various points in the
aorta and arteries of the dog, showing the change
in shape and time delay as the wave is propagated.
3. Descending thoracic aorta
4. Abdominal aorta
5. Abdominal aorta
6. Femoral Artery
7. Saphenous artery
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65
100
Pressure (mmHg)
90
Flow (ml / sec)
Pressure
0
Flow
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Experimental records of pressure and flow in the
canine ascending aorta, scaled so that the
heights of the curves are approximately the same.
If no reflected waves are present, the pressure
wave would follow the contour of the flow wave,
as indicated by the dotted line. Sustained
pressure during ejection and diastole are
presumably due to reflected waves returning from
the periphery. Sloping dashed line is an
estimate of flow out of the ascending aorta
during the same period of time.
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Pulmonary Artery Pressure kPa / mmHg
Pulmonary Artery flow
100 mls-1
2.5 kPa 20mmHg
Aortic Pressure kPa / mmHg
Aortic flow
5 kPa 40mmHg
100 mls-1
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CAPACITANCE (COMPLIANCE)
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During pulsatile flow, additional energy is
needed to overcome the elastic recoil of the
larger arteries, wave reflections, and the
inertia of the blood. The total energy per unit
volume at any point equals
TOTAL ENERGY
PRESSURE ENERGY
GRAVITATIONAL POTENTIAL ENERGY
KINETIC ENERGY
THERMAL ENERGY
E (PV) ( ?gh V) ( 1/2 ?v V2)
(U V)
VISCOUS FLOW PRESSURE
GRAVITATIONAL PRESSURE
PULSATILE FLOW COMPONENT
STEADY FLOW COMPONENT
( ?gh)
( 1/2?v2 V)
( 1/2?v2 V)
MEAN VELOCITY
INSTANTANEOUS VELOCITY
PULSATILE FLOW COMPONENT

(?V/C)
(?QR)
STEADY FLOW COMPONENT
(POTENTIAL ENERGY IN WALLS OF VESSELS)
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References
Badeer, Henry.S., Elementary Hemodynamic
Principles Based on Modified Bernoullis
Equation. The Physiologist, Vol 28, No. 1,
1985. Milnor, W.R., Hemodynamics Williams and
Wilkins, 1982.