HemodynamicsHemodynamicsHemodynamicsHemodynamics
Michael G. Levitzky, Ph.D.Professor of Physiology
(504)568-6184
FLUID DYNAMICSFLUID DYNAMICS
PRESSURE = FORCE / UNIT AREA = Dynes / cm 2PRESSURE = FORCE / UNIT AREA = Dynes / cm 2
FLOW = VOLUME / TIME = cm3 / secFLOW = VOLUME / TIME = cm3 / sec
RESISTANCE :RESISTANCE :
POISEUILLE’S LAW P1 - P2 = F x RPOISEUILLE’S LAW P1 - P2 = F x R
R =R =P1 - P2P1 - P2
FF
Dynes / cm 2Dynes / cm 2
cm3 / seccm3 / sec cm5cm5
Dyn secDyn sec== ==
POISEUILLE’S LAWPOISEUILLE’S LAW
AIR FLOW :P1 - P2 = V x R
.
BLOOD FLOW :P1 - P2 = Q x R
.
RESISTANCERESISTANCE
R = R = 8L8L
r 4 r 4
= viscosity of fluid = viscosity of fluid
L = Length of the tubeL = Length of the tube
rr = Radius of the tube = Radius of the tube
Constant flow
P1 P2
PL
(P1 – P2)r4
8LPoiseuille’s law: Q =
.
POISEUILLE’S LAW - ASSUMPTIONS:POISEUILLE’S LAW - ASSUMPTIONS:
1. Newtonian or ideal fluid - viscosity of fluid is independent of force and velocity gradient
1. Newtonian or ideal fluid - viscosity of fluid is independent of force and velocity gradient
2. Laminar flow2. Laminar flow
3. Lamina in contact with wall doesn’t slip3. Lamina in contact with wall doesn’t slip
4. Cylindrical vessels4. Cylindrical vessels
5. Rigid vessels5. Rigid vessels
6. Steady flow6. Steady flow
RESISTANCES IN SERIES :RESISTANCES IN SERIES :
RESISTANCES IN PARALLEL :RESISTANCES IN PARALLEL :
RT = R1 + R2 + R3 + ...RT = R1 + R2 + R3 + ...
RTRT R1 R1 R2R2 R3R3
11 11 11 11== ++ ++ +...+...
R1 R2 R3
RT = R1 + R2 + R3
R1
R2
R3
1/RT = 1/R1 + 1/R2 + 1/R3
x
Boundary layer edge
LAMINAR FLOWLAMINAR FLOW
P P Q x R Q x R..
TURBULENT FLOWTURBULENT FLOW
P P QQ2 2 x R x R..
ml /
sec
0
5
10
15
100 200 300 400 500
Pressure Gradient (cm water)
TURBULENCETURBULENCE
REYNOLD’S NUMBERREYNOLD’S NUMBER
==() (Ve) ( D)() (Ve) ( D)
= Density of the fluid = Density of the fluid
Ve = Linear velocity of the fluidVe = Linear velocity of the fluid
D = Diameter of the tubeD = Diameter of the tube
= Viscosity of the fluid = Viscosity of the fluid
HYDRAULIC ENERGYHYDRAULIC ENERGY
ENERGY = FORCE x DISTANCEunits = dyn cm
ENERGY = FORCE x DISTANCEunits = dyn cm
ENERGY = PRESSURE x VOLUME
ENERGY = (dyn / cm2 ) x cm3 = dyn cm
ENERGY = PRESSURE x VOLUME
ENERGY = (dyn / cm2 ) x cm3 = dyn cm
HYDRAULIC ENERGYHYDRAULIC ENERGY
THREE KINDS OF ENERGY ASSOCIATED WITH LIQUID FLOW:THREE KINDS OF ENERGY ASSOCIATED WITH LIQUID FLOW:
1. Pressure energy ( “lateral energy”)a. Gravitational pressure energyb. Pressure energy from conversion
of kinetic energyc. Viscous flow pressure
1. Pressure energy ( “lateral energy”)a. Gravitational pressure energyb. Pressure energy from conversion
of kinetic energyc. Viscous flow pressure
2. Gravitational potential energy2. Gravitational potential energy
3. Kinetic energy = 1/2 mv2 = 1/2 Vv23. Kinetic energy = 1/2 mv2 = 1/2 Vv2
TPir
T T
PoLaplace’s Law
Transmural pressure = Pi - Po
T = Pr
GRAVITATIONAL PRESSURE ENERGYGRAVITATIONAL PRESSURE ENERGY
PASCAL’S LAWPASCAL’S LAW
The pressure at the bottom of a column of liquid isequal to the density of the liquid times gravity timesthe height of the column.
The pressure at the bottom of a column of liquid isequal to the density of the liquid times gravity timesthe height of the column.
P = x g x h P = x g x h
GRAVITATIONAL PRESSURE ENERGY =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.
IN A CLOSED SYSTEM OF A LIQUID AT
CONSTANT TEMPERATURE THE TOTAL
OF GRAVITATIONAL PRESSURE ENERGY
AND GRAVITATIONAL POTENTIAL ENERGY
IS CONSTANT.
Referenceplane
E1
E2
Gravitational pressure E = 0 (atmospheric)
Gravitational potential E = X + gh·V
Thermal E = UV
h
Total E1 = X + gh·V + UV
Gravitational pressure E = gh·V
Gravitational potentialat reference plane E = X
Thermal E = UV
Total E2 = X + gh·V + UV
E = ( P + gh + 1/2 v2 ) V
GravitationalPotential
KineticEnergy
Gravitationaland ViscousFlow Pressures
TOTAL HYDRAULIC ENERGY(E)
TOTAL HYDRAULIC ENERGY(E)
(P1 + gh1 + 1/2 v12) V = (P2 + gh2 + 1/2 v2
2) V
BERNOULLI’S LAWBERNOULLI’S LAW
FOR A NONVISCOUS LIQUID IN STEADYLAMINAR FLOW, THE TOTAL ENERGY PERUNIT VOLUME IS CONSTANT.
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 v2
2 ] V
lateral pressure
kineticenergy
The Bernoulli Principle
PL
PL
PL
Constant flow(effects of resistance and viscosity omitted)
Increased velocityIncreased kinetic energyDecreased lateral pressure
LOSS OF ENERGY AS FRICTIONAL HEATLOSS OF ENERGY AS FRICTIONAL HEAT
U x V
E = (P•V) + (± gh •V) + ( 1/2 v2 •V) + (U •V)
TOTALENERGY PER UNIT VOLUME AT ANY POINT
PRESSUREENERGY
GRAVITATIONALPOTENTIAL ENERGY
KINETICENERGY
THERMALENERGY
VISCOUS FLOWPRESSURE
GRAVITATIONALPRESSURE
(± gh)(Q•R)•
TOTAL ENERGYTOTAL ENERGY
UV = Frictional heat ( internal energy)
½ v2·V = Kinetic energy
PV = Viscous flow pressure energy
E = Total energy
h
KE +UV
E1 E2 E3
Referenceplane
P1P2 P3
Referenceplane
E1 E2 E3
P1 P2 P3
KE + UV
Reference
plane
E1
E2
E3
E4
P1
P2
P3
P4
viscous
flow P
gravitational
energy
b
a
gh
Reference plane
P1 P2 P3 P4 P5
E1 E2 E3 E4 E5
KE + UV
Pressure equivalent of KE
Arteries Capillaries Veins
15
10
5
0
0
4
8
12
h(cm)
P’(mmHg)
4 4
12 12 12 12
Arteries Capillaries Veins
15
10
5
0
0
4
8
12
h(cm)
P’(mmHg)
1 -5
12 9 3
(12) (9) (3)
0
(0)
(9) (3)
Q = 1.0
Q = 1.0
15
10
5
0
0
4
8
12
h(cm)
P’(mmHg)
2.7-6.7
12 9 3
(12) (9) (3)
0
(0)
(10.7) (8.1)
Q = 0.43
0.1
Q = 1.43
Q = 1.0
12 10
a
4
b
Q
-6
c
d-8
15
10
5
0
0
4
8
12
h(cm)
P’(mmHg)
-5 16
-10 20
(Pa – Pv) (mmHg)
20 15 10 5 0
-5 0 5 10 150
100
Flow
(
ml/m
in)
Pv (mmHg)
VISCOSITYVISCOSITY
Internal friction between lamina of a fluidSTRESS (S) = FORCE / UNIT AREA
S = dvdx
= Sdvdx
dvdx
Is called the rate of shear;units are sec -1
The viscosity of most fluids increases as temperature decreases
A
v1 v2
===dx
VISCOSITY OF BLOODVISCOSITY OF BLOOD
1. Viscosity increases with hematocrit.
2. Viscosity of blood is relatively constant at high shearrates 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 theirdeformability.
4. Viscosity decreases at high shear rates in vessels < 1mmdiameter (FAHRAEUS-LINDQUIST EFFECT). This is because of “plasma skimming” of blood from outer lamina.
Non-Newtonian behavior of normal human bloodA
pp
are
nt
Vis
cosi
ty
(p
ois
e)
Rate of Shear (sec-1)
0
0.1
0.2
0.3
100 200
Hematocrit
Rela
tive V
isco
sity
Effects of Hematocrit on Human Blood Viscosity
0
2
4
8
0.8
6
0.60.40.2
52 / sec
212 / sec
PULSATILE FLOWPULSATILE FLOW1. 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 Definitions (Mostly from Milnor)Milnor)Definitions (Mostly from Definitions (Mostly from Milnor)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 = Stress ΔL/Lo Strain
Elastance: Inverse of compliance.
Distensibility: Virtually synonymous with compliance, but used more broadly.
Stiffness: Virtually synonymous with elastance. ΔF/ΔL
Distance from the Arch
m /
sec
15
10
5
20 10 0 10 3020 40 50 60 70 80
Carotid
Arch
ThoracicAorta
Diaphragm
Inguinalligament
Knee
Tibial
Femoral
Illiac
Abd
omin
al A
orta
Asc
endi
ng A
orta
Bifid
2.5
Progressive increase in wave front velocity of the pressure wave with increasing distance from the heart. Mean pressures were 97 – 120 mmHg.
(Average of 3 dogs)
-20
20
60
100
140
V
(cm
/sec)
P
(mm
Hg)
60
80
100
Ascending Thoracic Abdominal Femoral SaphenousAorta
1. Ascending aorta
2. Aortic arch
3. Descending thoracic aorta
4. Abdominal aorta
5. Abdominal aorta
6. Femoral Artery
7. Saphenous artery
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.
70
90
10065
0
Flow
(m
l /
sec)
Pre
ssu
re(m
mH
g)
Pressure
Flow
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.
Pulmonary Arteryflow
PulmonaryArtery PressurekPa / mmHg
Aortic PressurekPa / mmHg
2.5
kPa
20m
mH
g5 k
Pa
40m
mH
g
100
mls
-1100
mls
-1
Aortic flow
CAPACITANCECAPACITANCE
(COMPLIANCE)(COMPLIANCE)
Ca =Ca = VV
PP
During pulsatile flow, additional energy is needed toovercome 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 :
E = (P•V) + (± gh •V) + ( 1/2 v •V2) + (U •V)
TOTALENERGY
PRESSUREENERGY
GRAVITATIONALPOTENTIAL ENERGY
KINETICENERGY
THERMALENERGY
VISCOUS FLOWPRESSURE
GRAVITATIONALPRESSURE
STEADY FLOWCOMPONENT
PULSATILE FLOWCOMPONENT
STEADY FLOWCOMPONENT
(± gh)
PULSATILE FLOWCOMPONENT
“MEANVELOCITY”
“INSTANTANEOUSVELOCITY”
*(V/C)(Q•R)•
(POTENTIAL ENERGYIN WALLS OF VESSELS)
( 1/2v2 •V) ( 1/2v2 •V)
ReferencesReferencesReferencesReferences
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.