Hemodynamics
(fluid mechanics)
A.
Flow: some basic definitions and relationships
Flow: volume that crosses a plane per unit of time (ml/min)
Perfusion: Flow per unit of tissue mass (ml/min*100g))
Flow_velocity
(cm/sec):
Q
v
A
v = velocity,
Q = flow rate
A = cross sectional area
Ohm’s Law for fluids: Flow is driven by a pressure gradient
Q
DP
R
DP = pressure gradient ,
R = resistance
cardiac output:
Q=
( MAP - MVP )
(total peripheral Resistance).
(note about pressure units: 1 mm Hg = 1.36 cm H2O = 1330 dynes/cm2 ,
1 Newton = 105 dynes = 0.22 lb)
B. Elastic Properties of Vessels.
1.
a.
Elasticity. – the vessel walls are elastic and deform if there is a
pressure gradient across them.
Hooke’s Law . As you apply force, the vessel deforms, storing
energy like a spring.
F  kx
b.
F = force,
x = displacement
Young’s elastic modulus: consider a rod with a specific cross
sectional area. The Y.M. is the specific stress (Force/Area) needed
to double the initial length of the rod. In the case of the vessels,
we look at the increase in radius.
Rubber
Young’s Elastic Mod.
dynes/cm2
4x107
Steel
2x1012
VSM
106
Elastin
6x106
Collagen
109
Material
2.
Compliance : How much the vessel’s volume changes as the
intraluminal pressure changes (at equilibrium).
C
3.
DV
DP
C = compliance,
DV = change in blood volume due to …
DP = change in blood pressure.
Distensibility: compliance relative to some initial state (at
equilibrium).
D
DV
DPVi
Vi = initial blood volume
D = distensibility,
4.
Windkessel Effect. The previous relationships are true for
equilibrium conditions. However, the vessels take some time to
distend. Relationship between the rate of pressure build/up and the
concomitant rate of volume change.
dV
dP
C
dt
dt
simple example: aortic pressure during diastole:
dV
 Qin (t )  Qout (t )
dt
P
dP
 Qin (t )   C
R
dt
dP
1

P (t )
dt
CR
behaves like a discharging capacitor!
Note the analogy between fluid mechanics and circuits:
Q= flow
DP= Pressure Drop
C=compliance
V= volume
R = resistance





I = current
DV = voltage drop
C = capacitance
Q = charge
R = resistance
You can use the same math techniques on both!
C. Blood’s viscosity and flow : Poiseuille ‘s equation
1.
Viscosity: mechanical property of fluids that slows down their flow
due to internal forces . Newton’s definition:

shear stress

F/A


shear rate du / dy U / Y
“non-Newtonian fluid” is one that doesn’t behave like this
(ie - non-constant relationship between shear stress and shear rate)
2. Poiseuille’s Equation: determines the resistance to flow of a vessel given
the viscoelastic properties of the fluid under the following assumptions:
-Laminar flow
-Newtonian fluid
-Straight, rigid pipe
-Constant flow
R
8L
r 4
R = resistance,
 = viscosity (function of hematocrit primarily)
L = length (won’t usually change)
r = radius : this is the most critical. Arterioles can essentially
shunt flow because of this property.
and therefore,
DP DPr 4
Q

R
8L
DP = pressure drop through a segment of length L
3. Considerations:
a. Combined resistance : this works just
like circuits do
i. Series
ii. Parallel
Considerations ….(cont’d)
b.
The real world: Non-Newtonian Behavior (??)
i. Plug flow happens near the inlet of a tube, before laminar flow is
fully developed. Capillaries can also show plug flow because of their
size relative to RBCs.
ii. Distortion of erythrocytes. Greater hematocrit  greater
viscosity
“shear thinning” At higher flows, the
RBC tend to travel through the center
of the tube
More Considerations ….
c.
Different types of flow exist:
i.
Plug flow: all molecules move at the same speed. Happens only at very small
diameters, and slow flows.
ii.
Laminar Flow. Due to friction against vessel walls, the blood near the center of
the tube flows faster than that on the periphery. Infinitesimally thin concentric
cylinders sliding past each other. The velocity profile is shaped like a parabola.
iii.
Turbulent flow. Chaotic, “random”. Occurs when the Reynolds number for a
fluid is exceeded .
Re 
2rv

Even more Considerations ….
d. shear stress (force/area) : the viscous drag of the blood creates a shear
force on the intraluminal side of the vessel walls. Using Poiseuille’s
eq.
w 
F D Pr 4Q

 3
A 2L
r
w = wall shear stress
this can cause tears inside the lumen (dissecting aneurysm). High
velocity in the aorta  more likely place to happen : bad news!
Pressure inside capillaries: Law of LaPlace
Sources of pressure:
a.
Hydrostatic pressure: pressure due to gravity  function of body part, height, position,
….etc.
Phs =  h g
 = fluid density ,
h =vertical distance to a reference (“phlebostatic”)level
g = gravitational force constant
b.
Static (intraluminal or transluminal) pressure : Pressure in the vessels without the
hydrostatic pressure. I.e. – measured at the reference level: patient is supine and all
organs are at the same level as the heart.
Law of Laplace:
T  rP
T = tension in vessel wall,
P = intraluminal pressure
r = radius of vessel
Implication  thin walled capillaries can stand high internal pressures,
because of their small lumen
Stress : force per unit area on the vessel wall. Strain is the resultant
deformation. Stress in vessel wall
.
rP
s
w
s = vessel wall stress
w = wall thickness
BUT: As the vessel gets stretched out, the wall gets thinner, more fragile (ie
greater stress with the same pressure), less compliant.
(notice table: the capillaries and the
aorta withstand “similar” (ratio ~ 10)
pressure, but there is a lot less tension
in capillaries (ratio~109). This radius
dependence keeps the capillaries from
rupturing.
•
Bernoulli’s Relationships
Under the following conditions:
i.Constant flow
ii.Non-viscous fluid
iii.Incompressible fluid
…the total pressure in a section of a vessel is constant and can be
divided into a static and a dynamic component.
.
Bernoulli’s Law:
v2
P
 constant
2
v2
Pd  
2
Pd = dynamic component to pressure
 = density of the fluid
v = velocity of flow
(analogous to conservation of energy: P.E. + K.E. = constant)
P=static pressure
Pd = 0.5  v2
-Flow must be the same in the
whole tube (conservation of
mass!) and v = Q/A
-The Total pressure must not
increase in theat segment.
Consequences:
1. Faster flow , grater dynamic
pressure (kinetic energy).
2. Smaller static pressure
Total pressure
Static radial pressure
Velocities through the different vessels vary because of different cross sectional areas at isobaric regions:
ie – more cross sectional area at sum of all capillaries : slower flow.
Physiological Examples of Bernoulli’s principle:
Stenosis, Aneurysm: Consider a long continuous tube. Flow must be the same
throughout the whole length (conservation of mass).
If we reduce the cross-sectional area of a segment (stenosis), then the flow velocity
must increase proportionally to maintain flow constant. The static pressure is
reduced. (more velocity  more shear stress )
An aneurysm is exactly the opposite effect.