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Hemodynamics
Michael G. Levitzky, Ph.D.
Professor of Physiology
LSUHSC
[email protected]
(504)568-6184
FLUID DYNAMICS
PRESSURE = FORCE / UNIT AREA = Dynes / cm 2
FLOW = VOLUME / TIME = cm3 / sec
RESISTANCE :
POISEUILLE’S LAW
R =
P1 - P2
F
=
P1 - P2 = F x R
Dynes / cm 2
cm3 / sec
=
Dyn sec
cm5
POISEUILLE’S LAW
AIR FLOW :
.
P1 - P2 = V x R
BLOOD FLOW :
.
P1 - P2 = Q x R
RESISTANCE
8L
R =
r4

=
viscosity of fluid
L
=
Length of the tube
r =
Radius of the tube
.
Poiseuille’s law: Q =
(P1 – P2)r4
8L
PL
P1
P2
Constant flow
POISEUILLE’S 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 doesn’t slip
4.
Cylindrical vessels
5.
Rigid vessels
6.
Steady flow
RESISTANCES IN SERIES :
RT = R1 + R2 + R3 + ...
RESISTANCES IN PARALLEL :
1
RT
=
1
R1
+
1
R2
+
1
R3
+...
R1
R2
R3
RT = R1 + R2 + R3
R1
R2
R3
1/RT = 1/R1 + 1/R2 + 1/R3

x
Boundary layer edge
LAMINAR FLOW
.
P  Q x R
TURBULENT FLOW
.
 P  Q2 x R
ml / sec
15
10
5
0
100
200
300
400
Pressure Gradient (cm water)
500
TURBULENCE
REYNOLD’S =
NUMBER
() (Ve) ( D)


=
Density of the fluid
Ve
=
Linear velocity of the fluid
D
=
Diameter of the tube

=
Viscosity of the fluid
HYDRAULIC ENERGY
ENERGY = FORCE x DISTANCE
units = dyn cm
ENERGY = PRESSURE x VOLUME
ENERGY = (dyn / cm2 ) x cm3
= dyn cm
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
Laplace’s Law
Po
Pi
r
T
T
T
T = Pr
Transmural pressure = Pi - Po
GRAVITATIONAL PRESSURE ENERGY
PASCAL’S 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
=
IN A CLOSED SYSTEM OF A LIQUID AT
CONSTANT TEMPERATURE THE TOTAL
OF GRAVITATIONAL PRESSURE ENERGY
AND GRAVITATIONAL POTENTIAL ENERGY
IS CONSTANT.
Gravitational pressure E = 0 (atmospheric)
Gravitational potential E = X + gh·V
E1
Thermal E = UV
Total E1 = X + gh·V + UV
h
Gravitational pressure E = gh·V
Reference
plane
E2
Gravitational potential
at reference plane
E = X
Thermal E = UV
Total E2 = X + gh·V + UV
TOTAL HYDRAULIC ENERGY
(E)
E = ( P + gh + 1/2 v2 ) V
Gravitational
and Viscous
Flow Pressures
Gravitational
Potential
Kinetic
Energy
BERNOULLI’S 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
Linear Velocity = Flow / Cross-sectional area
cm/sec = (cm3 / sec) / cm2
Bernoulli’s Law of Gases
(or liquids in horizontal plane)
[ P1 + ½ v12 ] V = [ P2 + ½ v22 ] V
lateral
pressure
kinetic
energy
The Bernoulli Principle
PL
PL
PL
Constant flow
(effects of resistance
and viscosity omitted)
Increased
velocity
Increased kinetic energy
Decreased lateral pressure
LOSS OF ENERGY AS
FRICTIONAL HEAT
U x V
TOTAL ENERGY
TOTAL
ENERGY
PER UNIT
VOLUME
AT ANY
POINT
PRESSURE
ENERGY
GRAVITATIONAL
POTENTIAL ENERGY
KINETIC
ENERGY
THERMAL
ENERGY
E = (P•V) + (± gh •V) + ( 1/2 v2 •V) + (U •V)
•
(Q•R)
VISCOUS FLOW
PRESSURE
(± gh)
GRAVITATIONAL
PRESSURE
UV = Frictional heat ( internal energy)
½ v2·V = Kinetic energy
PV = Viscous flow pressure energy
E = Total energy
E1
E2
E3
h
Reference
plane
KE +UV
P1
P2
P3
E1
E2
E3
Reference
plane
KE + UV
P1
P2
P3
E4
E3
E2
E1
P4
P1
P3
P2
a
gh
b
E1
E2
E3
E4
E5
KE + UV
Reference
plane
P1
P2
P3
P4
P5
Pressure equivalent
of KE
P’
h
(cm) (mm
Hg)
15
0
10
4
5
8
0
12
4
12
12
Arteries
4
12
Capillaries
12
Veins
P’
h
(cm) (mm
Hg)
15
Q = 1.0
0
(9)
10
4
5
8
1
(3)
-5
Q = 1.0
0
12
12
9
3
0
(12)
(9)
(3)
(0)
Arteries
Capillaries
Veins
P’
h
(cm) (mm
Hg)
15
Q = 0.43
0
(10.7) (8.1)
10
-6.7
2.7
4
0.1
5
8
Q = 1.43
0
12
12
9
3
0
(12)
(9)
(3)
(0)
Q = 1.0
h
(cm)
Q
P’
(mm
Hg)
15
0
10
4
5
8
0
12
-5
16
-10
20
4
12
a
10
-6
b
c
d
-8
(Pa – Pv)
Flow (ml/min)
20
15
(mmHg)
10
5
0
100
0
-5
0
Pv
5
(mmHg)
10
15
VISCOSITY
Internal friction between lamina of a fluid
STRESS (S) = FORCE / UNIT AREA
S =  dv
dx
dv
dx
 =
S
dv
dx
Is called the rate of shear;
units are sec -1
The viscosity of most fluids increases as
temperature decreases
v1
v2
A
dx
===
VISCOSITY OF BLOOD
1.
Viscosity increases with hematocrit.
2.
Viscosity of blood is relatively constant at high shear
rates in vessels > 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 < 1mm
diameter (FAHRAEUS-LINDQUIST EFFECT). This is
because of “plasma skimming” of blood from outer lamina.
Apparent Viscosity  (poise)
Non-Newtonian behavior of normal human blood
0.3
0.2
0.1
0
100
Rate of Shear (sec-1)
200
Effects of Hematocrit on Human Blood Viscosity
Relative Viscosity
52 / sec
8
6
212 / sec
4
2
0
0.2
0.4
Hematocrit
0.6
0.8
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.
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. ΔL/Lo
Extensibility: ΔL/Stress (≈ Compliance = ΔV/ΔP)
Viscoelastic: Strain changes with time.
Elasticity: Expressed by Young’s Modulus.
E = ΔF/A
ΔL/Lo
= Stress
Strain
Elastance: Inverse of compliance.
Distensibility: Virtually synonymous with compliance, but used more
broadly.
Stiffness: Virtually synonymous with elastance. ΔF/ΔL
Progressive increase in wave front velocity of the pressure wave with
increasing distance from the heart. Mean pressures were 97 – 120 mmHg.
Carotid
m / sec
10
Ascending Aorta
Arch
Diaphragm
Thoracic
Aorta
Abdominal Aorta
15
Inguinal
ligament
Bifid
Illiac
30
40
Knee
Tibial
Femoral
5
2.5
20
10
0
10
20
50
Distance from the Arch
60
70
80
(Average of 3 dogs)
P (mmHg)
100
80
60
Aorta
Ascending Thoracic Abdominal Femoral
V (cm/sec)
140
100
60
20
-20
Saphenous
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
65
Pressure
(mmHg)
Flow (ml / sec)
100
90
Pressure
0
Flow
70
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.
100 mls-1
2.5 kPa
20mmHg
5 kPa
40mmHg
Aortic flow
100 mls-1
Pulmonary
Artery
flow
Pulmonary
Artery
Pressure
kPa / mmHg
Aortic
Pressure
kPa / mmHg
CAPACITANCE
(COMPLIANCE)
Ca =
V
P
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 = (P•V) + (± gh •V) + ( 1/2 v •V2) + (U •V)
VISCOUS FLOW
PRESSURE
GRAVITATIONAL
PRESSURE
(± gh)
•
(Q•R)
STEADY FLOW
COMPONENT
PULSATILE FLOW
COMPONENT
STEADY FLOW
COMPONENT
PULSATILE FLOW
COMPONENT
( 1/2v2 •V)
( 1/2v2 •V)
“MEAN
VELOCITY”
*(V/C)
(POTENTIAL ENERGY
IN WALLS OF VESSELS)
“INSTANTANEOUS
VELOCITY”
References
Badeer, Henry.S., Elementary Hemodynamic Principles Based on
Modified Bernoulli’s Equation. The Physiologist, Vol 28, No. 1,
1985.
Milnor, W.R., Hemodynamics Williams and Wilkins, 1982.