ame 60676 biofluid & bioheat transfer 4. mathematical modeling

AME 60676Biofluid & Bioheat Transfer

4. Mathematical Modeling

Objectives

• Applications of hydrostatics and steady flow models to describe blood flow in arteries

• Unsteady effects:– pressure pulse propagation through arterial wall– Effects of inertial forces due to blood

acceleration/deceleration– Effects of artery distensibility on blood flow

Outline• Steady flow considerations and models:

– Hydrostatics in circulation– Rigid tube flow model– Application of Bernoulli equation

• Unsteady flow models:– Windkessel model for human circulation– Moens-Korteweg relationship (wave propagation, no viscous effects)– Womersley model for blood flow (wave propagation, viscous effects)– Wave propagation in elastic tube with viscous flow (wave

propagation, viscous effects)• Bioheat transfer models:

– Pennes equation– Damage modeling

Damage modelingPennes equationComplete model

1. Hydrostatics

Womersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

Hydrostatics in the Circulation

• Blood pressure in the “lying down” position– Arterial: 100 mmHg– Venous: 2 mmHg

• Distal pressure is lower

Hydrostatic pressure differences in the circulation

“lying down” position

Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics


• Blood pressure in the “standing up” position– Head artery: 50 mmHg– Leg artery: 180 mmHg– Head vein: -40 mmHg– Leg vein: 90 mmHg

• Pressure differences due to gravitational effects Hydrostatic pressure differences

in the circulation“standing up” position



• Bernoulli equation:

• Tube of constant cross section:

• Effects of pressure on vessels:– Arteries are stiff: pressure does

not affect volume– Veins are distensible: pressure

causes expansion

Hydrostatic pressure differences in the circulation

“standing up” position

2

2

V pz const

g g

p g z


2. Rigid Tube Flow Model

Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostaticsRigid tube flow model

Hagen Poiseuille Model

• Assumptions:– incompressible– steady– laminar– circular cross section

• From exact analysis:

2

2

4 4z

p dv r r

L

L

r

z

4

128

p dQ

L



• Assumptions:– incompressible– steady– laminar– circular cross section

• From control volume analysis:

L

r

z1P 2P

w

w

Control volume

4 wLp

d



• Validity considerations– Newtonian fluid: reasonable• Casson model: linear at large shear rate

– Laminar flow: reasonable• Average flow: Re=1500 (< Recr=2100)• Peak systole: Re = 5100

– Blood vessel : compliance– Flow measurements: no evidence of sustained turbulence

– No slip at vascular wall: reasonable• Endothelial cell lining



• Validity considerations– Steady flow: not valid for most of circulatory system

• Pulsatile in arteries– Cylindrical shape: not valid

• Elliptical shape (veins, pulmonary arteries)• Taper (most arteries)

– Rigid wall: not valid• Arterial wall distends with pulse pressure

– Fully developed flow: not valid• Finite length needed to attain fully developed flow• Branching, curved walls


Blood Vessel Resistance

• On time-average basis:

– p: time-averaged pressure drop (mmHg)– Q: time-averaged flow rate (cm3/s)– R: resistance to blood flow in segment (PRU,

peripheral resistance unit)

4

128

p LR

Q d


Blood Vessel Resistance

• Series connection:

• Parallel connection:

31 21 2 3

pp ppR R R R

Q Q Q Q

31 2

1 2 3

1 1 1 1QQ QQ

R p p p p R R R

R3R2R1

R1

R2

R3


Transition Flow in Pipes• Entrance

– V=0 at wall – Velocity gradient in radial direction

• Downstream– Fluid adjacent to wall is retarded – Core fluid accelerates

• Viscous effects diffuse further into center

U

region dominated by viscous effects

region dominated by inertial effects

parabolic velocity profile

Entrance region Fully developed flow region


BL Thickness and Entrance Length

• Balance inertial force and viscous force:

• Entrance length definition:

– Re > 50:

– Re 0:

1

2~

xx

U

L R ReL kD k = 0.06

0.65L D


3. Application of Bernoulli Equation

Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics Bernoulli applications

Bernoulli Equation

• Assumptions:– Steady– Inviscid– Incompressible– Along a streamline

2

2

V pz const

g g


Stenosis

• Narrowing of artery due to:– Fatty deposits– Atherosclerosis

• Effects of narrowing:– p1, V1, V2: known

a1

p1, V1

δ

p2, V2

a2

2 22 1 1 22

p p V V


Stenosis

• Arterial flutter– Low pressure at

contraction– Complete obstruction

of vessel under external pressure

a1

p1, V1

δ

p2, V2

a2

– Decrease in flow velocity– Increase in pressure– Vessel reopening (cycle)


Aneurysm

• Definition: Arterial wall bulge at weakening site, resulting in considerable increase in lumen cross-section

• Characteristics:– Elastase excess in blood– Decrease in flow velocity– Limited increase in pressure (<5 mmHg)– Significant increase in pressure under exercise– Increase in wall shear stress– Bursting of vessel


Heart Valve Stenoses

• Flow through a nozzle

– Flow separation recirculation region– Fluid in core region accelerates– Formation of a contracted cross section: vena

contracta

2

1 2 2 20

1

2 d

Qp p

A C

Cd: discharge coefficient (function of nozzle, tube, throat geometries)



• Effective orifice area:

• Gorlin equations (clinical criteria for surgery):

0 2d

QEOA A

C p

Q: mean flow rate (CO)

For aortic valve: Q: mean systolic flow rate

44.5

MSFAVA

p

AVA: aortic valve area (cm2)MVA: mitral valve area (cm2)MSF: mean systolic flow rate (cm3/s)MDF: mean diastolic flow rate (cm3/s)p: mean pressure drop across valve (mmHg)

31.0

MDFMVA

p



• Effects of flow unsteadiness and viscosity:

2dQp A BQ CQ

dt Young, 1979

p = temporal acceleration

convective acceleration

viscous dissipation+ +

pp

EOA KQp

rmsm

EOA KQp

based on mean values based on peak-systolic values


4. Windkessel Models for Human Circulation

Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics Windkessel model

Windkessel Theory

• Simplified model

• Arterial system modeled as elastic storage vessels

• Arteries = interconnected tubes with storage capacity

Unsteady flow due to pumping of heart

Steady flow in peripheral organsAttenuation of unsteady

effects due to vessel elasticity


Windkessel Theory

Definitions:• Inflow: fluid pumped intermittently by ventricular ejection• Outflow: calculated based on Poiseuille theory

inflow outflow

Q(t)p(t), V(t), Di

RS

Windkessel chamber

Variables Definition

p Windkessel chamber pressure

V Windkessel chamber volume

DiChamber distensibility

RSPeripheral resistance

Q Ventricular ejection flow rate

pVVenous pressure

pV


Windkessel Solution• Pressure pulse solution– Systole (0 < t < ts):

– Diastole (ts < t < T):

• Stroke volume

0 0 0S i

t

R DS Sp t R Q R Q p e

S i

T t

R DTp t p e

p0: pressure at t=0pT: pressure at t=T

Windkessel (left) vs. actual (right) pressure pulse

00

St

S SV Q t dt Q t Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics Windkessel model

Windkessel Theory Summary

• Advantages:– Simple model– Prediction of p(t) in arterial system

• Limitations:– Model assumes an instantaneous pressure pulse

propagation (time for wave transmission is neglected)

– Global model does not provide details on structures of flow field


5. Moens-Kortweg relationship

Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics Moens-Korteweg

Wave Propagation Characteristics

• Speed of transmission depends on wall elastic properties

• Pressure pulse:– depends on wall/blood interactions– Changes shape as it travels downstream due to

interactions between forward moving wave and waves reflected at discontinuities (branching, curvature sites)

Need for model of wave propagation speed


Moens-Korteweg Relationship

• Speed of pressure wave propagation through thin-walled elastic tube containing an incompressible, inviscid fluid

• Relationship accounts for:– Fluid motion– Vessel wall motion


Problem Statement

z

rR

h

Vz(r, z, t)

Vr(r, z, t)

flow

Infinitely long, thin-walled elastic tube of circular cross-section


Derivation Outline

Equations of fluid motion in infinitely long, thin-walled elastic

tube of circular cross section

Equations of vessel wall motion(inertial force neglected on wall)

Simplified Moens-Korteweg relationship

Equations of vessel wall motion(with inertial force on wall)

Moens-Korteweg relationship


Simplified Moens-Korteweg relationship• Reduced Navier-Stokes equations:

• Inviscid flow approximation:

2

2

2

2

10

1

1

r z

r r r rr z r

z z z z zr z

rv v

r r z

v v v v pv v rv

t r z r r r z r

v v v v v pv v r

t r z r r r z z

2

z

zr

v p

t z

vRv R

z



• Tube equation of motion:

• Coupling with fluid motion (without inertial effects):

2

2t

dhRd Rpd hd

dt

2 2

2 2 20

1p p

z c t

20 constant

2

hEc

R where



• Tube equation of motion:

• Coupling with fluid motion (with inertial effects):

2

2t

dhRd Rpd hd

dt

2 2

2 2 20

1p p

z c t

2 220 1

2t RhE

cR E

where


Experimental vs. Theoretical c0


6. Womersley model for blood flow

Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics Womersley model

Problem Statement

z

rR

h

Vz(r, z, t)

Vr(r, z, t)

flow

Infinitely long, thin-walled elastic tube of circular cross-section


Problem Assumptions

• Flow assumptions:– 2D– Axisymmetric– No body force– Local acceleration >> convective acceleration

• Tube assumptions:– Rigid tube– No radial wall motion ( no radial fluid velocity)


Equation of Motion

• Pressure gradient:

• Axial flow velocity:

• Axial flow velocity magnitude:

i tpt Ae

z

, i tzv r t w r e

2 22

2

1d w dw ARi w

dr r dr

where: R

rr

R

Womersley number


Examples of Womersley Number


Flow Solution

• Flow solution:

20

02

410

102

sin

sin

z

A R Mv t

A R MQ t

0

0

3 20

0 3 20

3 21

0 3 2 3 20

1

21

i

i

J i rM e

J i

J iM e

i J i

where:

kJ x : Bessel function of order k


Flow Solution


Application• Flow rate calculation for complex (non-

sinusoidal) pulsatile pressure gradients


Flow Solution• Time history of the

axial velocity profile (first 4 harmonics)

• Characteristics:– Hagen-Poiseuille parabolic

profile never obtained during the cardiac cycle

– Presence of viscous effects near the wall makes the flow reverse more easily than in the core region

– Main velocity variations along the tube cross section are produced by the low-frequency harmonics

– High-frequency harmonics produce a nearly flat profile due to absence of viscous diffusion


7. Complete model:Wave propagation in elastic tube with viscous

flow

Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics Complete model

Elastic Tube Equations of Motion

Stresses on a tube element

2

2

2

2

t rr

zt rz

hh S

t R

Sh h

t z


Elastic Tube and Fluid Stresses

• Tube stresses (from Hooke’s law)

• Fluid stresses (cylindrical coordinates)

2

2

2

2

2

1

1

1

z

z

dR dzS E

ddz RS E

d ddS dz R dzEdz

2 rrr

z rrz

vp

rv v

r z


Elastic Tube Equations of Motion

• Equations of motion for tube and flow must be solved simultaneously to obtain solutions for:

2

2 2 2

2 2

2 2 2

21

1

rt

r R

z rt

r R

v hEh p

t r R R z

v v hEh

t r z z R z

Governing equations of motion for elastic tube

, , , ,z rv v p


Flow Equations

• Non-linear inertial terms can be neglected (see order-of-magnitude study performed in Moens-Korteweg derivation)

• Along with the 2 tube equations, we obtain a set of 5 equations with 5 unknowns

2 2

2 2 2

2 2

2 2

1

1

10

r r r r r

z z z z

r z

v v v v vp

t r r r r z r

v v v vp

t z r r r z

rv v

r r z

Governing flow equations


Flow Equations

• Boundary conditions at fluid-tube interface:

r r Rv

t

z r Rv

t

No penetration No slip

2 2

2 2 2

2 2

2 2

1

1

10

r r r r r

z z z z

r z

v v v v vp

t r r r r z r

v v v vp

t z r r r z

rv v

r r z

Governing flow equations


Solutions• All variables are, at least, functions of z and t Seek for solutions that vary as:

where:k1 : wave number (= 1/)k2 : damping constant (decay along z)

• Solutions:

i kz te

1 2k k ik

1

1

i kz t

i kz t

i kz t

e

e

p Pe

1

1

i kz tr r

i kz tz z

v v r e

v v r e


Solutions• After performing an order-of-magnitude study,

the problem statement reduces to:

11 1

22

112

21 1

1 2

11

1

1

10

z

r R

z

r R

z zz

rz

viEkR rhk

vEh iP

R kR r

v vi v ikP

r r r

rvikv

r r

1 1

1 1

z

r

v R i

v R i

BCs:

Equation for 1

Equation for 1

NS / z

continuity


Solutions• By combining the 4 equations and applying the

BCs, the problem can be expressed as a system of 2 equations with 2 unknowns:

1 2 1 22

1 1 11 2 1 22

1 2

1 2 1 2

1 0 122

2

12

2

1

k Eh k i k iA J R J R

R i iR

ii

kEh i iA J R J R

EkR R ihk R

1 2

11 2

iJ R


Solutions• Non trivial solutions if and only if determinant

of the system = 0• If (k/ω) = φ is the root of the determinant:

1 2

1

2

Re

Im

k ik

k

k


ame 60676 biofluid & bioheat transfer 4. mathematical modeling

Documents

blood flow slide

effects of pressure

circulation blood pressure

blood flow wave propagation

steady flow considerations

viscous flow wave propagation

mmhg pressure differences

reasonable average flow