Hemodynamics Bioengineering 6000 CV Physiology

Hemodynamics

Bioengineering 6000 CV PhysiologyHemodynamics

Overview and Terminology

Arterioles

• Parameters– Pressure– Velocity– Flow

• laminar vs. turbulent

– Resistance– Viscosity – Energy– Area– Volume

Bioengineering 6000 CV PhysiologyHemodynamics

Vessels of the Circulatory System

Endothelium

Elastic tissue

Smooth Muscle

Fibrous Tissue

Arteriole30 µm

Capillary8 µm

Venule20 µm

6 µm 0.5 µm 1 µm

Diameter

Wall thickness

Aorta25 mm

Artery4 mm

Vein5 mm

Vena Cava30 mm

2 mm 1 mm 0.5 mm 1.5 mm

Bioengineering 6000 CV PhysiologyHemodynamics

Blood Distribution• Velocities/Flows

– Aorta: 30-50 cm/s– Capillaries: 0-1 cm/s

or 5.5 hours/mm3 • Blood mass: 8% of

body mass• Volumes (percent

of total blood volume)– Systemic: 83%

• Arteries: 11%• Capillaries: 5%• Veins: 67%

– Pulmonary: 12%– Heart: 5%

Aorta Arteries Arterioles Capillaries Venules Veins Venae Cava

VelocityCross-sectional area

Pressure

Percent of blood volume

Velocity

Cross-sectional area

Pressure

Percent of blood volume

Bioengineering 6000 CV PhysiologyHemodynamics

Hemodynamics Basics

• "The problem of treating the pulsatile flow of blood through the cardiovascular system in precise mathematical terms is insuperable" (Berne and Levy) – Blood is not Newtonian (viscosity is not constant)– Flow is not steady but pulsatile– Vessels are elastic, multibranched conduits of constantly

changing diameter and shape.– Use equations qualitatively

• Local control of blood flow

R =�P

�Q̇

C =�V

�P

L =�P

�Q̇/�t

Q̇ = (P1 � P2)⇡r4

8⌘l

R =8⌘l

⇡r4(I =

V

R)

Bioengineering 6000 CV PhysiologyHemodynamics

Hemodynamic Parameters

• Resistance

• Compliance

• Inertance

• Poiseuille's Law– laminar flow– Newtonian fluid– rigid tube– works for small

arteries and veins

Q = flow.

P = pressure η = viscosityV = volume

Bioengineering 6000 CV PhysiologyHemodynamics

Resistance and Compliance

• Veins vs. arteries– have 24 times the

compliance of arteries– carry 65% of the blood– have even higher blood

storage capacity• Autonomic control

– alters resistance but not compliance (slopes of curves)

– acts to shift blood volume

Sympathetic inhibition

0 100 200Arterial Pressure [mm Hg]

Blo

od F

low

Norm

al

Sympathetic stimulation

0 70 140

Pressure [mm Hg]

Blo

od V

olum

e

Arterial

Venous C =�V

�P

R =�P

�Q̇

⌘ =⌧

du/dy=

F/A

U/Y=

Shear stressShear rate

Bioengineering 6000 CV PhysiologyHemodynamics

ViscosityF

U

uy Y

A

Definition for homogenous Newtonian fluid

• Viscosity increases with– increased hematocrit– constrictions in vessels

• Viscosity decreases with– increased flow velocity– vessel diameter below 300 µm

Poor formula for viscosity in small vessels

Hematocrit [%]

Visc

osity

(wat

er=1

)

10

2

20 30 40 50 60 70

4

6

8

10

Water

Plasma

Blood

Bioengineering 6000 CV PhysiologyHemodynamics

Factors that Affect Viscosity• Flow rate: as flow decreases, viscosity increases up to

10-fold. Mechanism: RBCs adhering to each other, and the vessel walls.

• RBCs stick at constrictions, increase viscosity.• Concentration, distribution, shape, and rigidity of the

suspended particles (e.g., RBCs drift to the center so velocity profile flattens from ideal parabolic)

• Fahraeus-Lindqvist effect: reduced η when RBCs line up in small vessels (< 300 µm).

• In very small vessels (< 20 µm), η increases as RBCs fill the capillaries, “tractor tread” motion

• Temperature, blood pressure, presence of anticoagulants,• Measurement conditions: higher in vitro than in vivo.• History (pulsatile flow)

Q̇ = Q̇o

v = vo

Ptot

= Po

= Pdo

+ Plo

Pd

=12⇢v2

o

= Pdo

Pl

= Plo

Bioengineering 6000 CV PhysiologyHemodynamics

Velocity and Pressure

• Example: Aortic stenosis– increased velocity– decreased lateral

pressure– reduced coronary

flow– coronary ischemia

Q̇ = Q̇o

v = vo

Ptot

= Po

Pd

= Pdo

Pl

= Plo

Q̇ = Q̇o

v = kvo

Ptot

= Po

= Pd

+ Pl

Pd

=12⇢(kv

o

)2

= k2Pdo

Pl

= Ptot

� Pd

< Plo

Bioengineering 6000 CV PhysiologyHemodynamics

Aortic Stenosis

Left ventricle

Coronary artery

Coronary artery

Pv

Pavo

Pao (lower than normal)

• Pressure losses– kinetic energy

conversion– energy loss (friction)

Bioengineering 6000 CV PhysiologyHemodynamics

Resistance of the Circulatory System

• Resistance high where pressure drops

• Arterioles have highest resistance

• Paradox?– arterioles have more total

area than arteries– vessels with larger area

have smaller resistance– but arterioles have larger

resistance than arteries?Aorta Arteries Arterioles Capillaries Venules Veins Venae

Cava

VelocityCross-sectional areaPressurePercent of blood volume

VelocityCross-sectional area

Pressure

An, rn, Rn

Rt

Rt = 4Rw!

Aw = 4An

rw = 2rn

1Rt

=4X

1=1

1Rn

=4

Rn

Rw =k

r4w

=k

(2rn)4=

k

16r4n

=14⇤ k

4r4n

=Rt

4

Bioengineering 6000 CV PhysiologyHemodynamics

Resistance-Area Paradox

• Net flow must be constant

• One vessel splitting to four increases total resistance!

Aw, rw, Rw

R =8⌘l

⇡r4=

k

r4

Rt =Rn

4=

k

4r4n

Rw = Rt

At = 16An = 4Aw

Bioengineering 6000 CV PhysiologyHemodynamics

Resistance Break Even Point

• Break-even point at 16 to 1 (for Rw=Rt).

• Capillaries have more than 16:1 ratio

Aw, rw, Rw

An, rn, Rn

Rt

Bioengineering 6000 CV PhysiologyHemodynamics

Laminar Flow and Turbulence

• Laminar flow– Parabolic profile

• Pulsatile laminar flow– Velocity changes– May reverse direction

• Turbulent flow– Nonaligned movement– Noisy (BP cuff)– Reynolds number

• > 1000 = turbulence• > 200 = eddies possible

– Rarely occurs in healthy vesselsVe

loci

ty P

rofil

es