HemodynamicsHemodynamicsHemodynamicsHemodynamics

Michael G. Levitzky, Ph.D.Professor of Physiology

LSUHSCmlevit@lsuhsc.edu

(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.