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):

v = velocity, Q = flow rate A = cross sectional area

 Ohm’s Law for fluids: Flow is driven by a pressure gradient

P = 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)

A

Qv

R

PQ

B. Elastic Properties of Vessels. 1.      Elasticity. – the vessel walls are elastic and deform if there is a

pressure gradient across them.a.       Hooke’s Law . As you apply force, the vessel deforms, storing

energy like a spring.

F = force, x = displacement

 b.      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.

 

kxF

Material Young’s Elastic Mod.dynes/cm2

Rubber 4x107

Steel 2x1012

VSM 106

Elastin 6x106

Collagen 109

 

 2.      Compliance : How much the vessel’s volume changes as the

intraluminal pressure changes (at equilibrium).

C = compliance, V = change in blood volume due to …P = change in blood pressure.

3. Distensibility: compliance relative to some initial state (at equilibrium).

Vi = initial blood volumeD = distensibility,

P

VC

iPV

VD

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.

simple example: aortic pressure during diastole:

dt

dPC

dt

dV

)(1

)(

)()(

tPCRdt

dPdt

dPC

R

PtQ

tQtQdt

dV

in

outin

behaves like a discharging capacitor!

Note the analogy between fluid mechanics and circuits:

Q= flow I = currentP= Pressure Drop V = voltage dropC=compliance C = capacitanceV= volume Q = chargeR = resistance 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:

 YU

AF

dydurateshear

stressshear

/

/

/

“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 = 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,

P = pressure drop through a segment of length L

4

8

r

LR

L

rP

R

PQ

8

4

3. Considerations:

a. Combined resistance : this works just like circuits do

i.     Seriesii.     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 .

rv2

Re

 

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 = wall shear stress

 this can cause tears inside the lumen (dissecting aneurysm). High velocity

in the aorta more likely place to happen : bad news!

3

4

2

Pr

r

Q

LA

Fw

 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 = tension in vessel wall, P = intraluminal pressurer = 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

. = vessel wall stressw = wall thickness

 BUT: As the vessel gets stretched out, the wall gets thinner, more fragile (ie greater stress with the same pressure), less compliant.

rPT

w

rP

(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:

Pd = dynamic component to pressure

= density of the fluidv = velocity of flow

(analogous to conservation of energy: P.E. + K.E. = constant)

constant2

2

vP

2

2vPd

P=static pressurePd = 0.5 v

-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 iso-baric 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.