NBCR Summer Institute 2006:Multi-Scale Cardiac Modeling with

Continuity 6.3

Friday:Cardiac Biomechanics

Andrew McCulloch, Fred Lionetti and Stuart Campbell

Friday: Cardiac Mechanics and Electromechanics

Modeling Ventricular Wall Mechanics

Analysis of ventricular wall stress

Galerkin FEM for ventricular stress analysis

Newton’s method

Examples

homogeneous cube

prolate spheroid

Ventricular-Vascular Coupling

Soft Tissue Biomechanics

• Conservation of mass, momentum and energy for finite elasticity

• 3-D geometry and tissue structure• Boundary conditions: displacement

pressure, impedance, isovolumic• Nonlinear, anisotropic stress-strain

relations• Active systolic stress development as a

function of time, intracellular calcium and sarcomere length history

• Myofiber angle dispersion and transverse active stress

• Residual strain• Growth• Perfusion 0

30

60

90

120

150

180

1.6 1.8 2 2.2

Sarcomere Length (m)

Act

ive

Str

ess

(kP

a)

1.21.11.00

10

20

30 TffTcc

Extension Ratio

Str

ess

(KP

a)

Pas

sive

Str

ess

(kP

a)

kinematics

Strain-displacement relation

constitutive law

Stress-strain relation

equilibrium

Force balance equationMoment balance

1

2

W

E RS

W

E SR

+=PRS ( )

E = ½(FTF – I )

R

iiR X

xF

divT + b = 0T = T T

Nonlinear Elasticity: Governing Equations

jSRSiRij det1

FPFTF

1. Formulate the weighted residual (weak) form

0d)(

0d)(

w

xwfLuR

bT

2. Divergence (Green-Gauss) Theorem

ddd

0ddd

)( www

www

ntbT

TnbT

Note: Taking w=u*, we have the virtual work equation

Lagrangian Virtual Work Equations for Large Deformation Elasticity

2A A

A

V

V

T

T T

T

• dA • dA

dA

dV

- dV

s u t u

N P F u

P F u+ P F u

P F u b u

0 (Equilibrium) TP F b

Divergence Theorem

2V V A

-1 T Twhere = = det =

T

iiJ J

dV dV dA

W x; ; ;

X

sP F u b u u

P T F F P F C F F FC

Virtual Work

Newton’s Method in n Dimensions

f’(x) is an n n Jacobian matrix J

njixf

fJj

iijij

,1),()(' xx

Gives us a linear system of equations for x(k+1)

0xfxxxf )())((' )()()1()( kkkk

2)()()()( ))((')()( kkkk O xxxxxfxfxf

Newton’s Method

Each step in Newton’s method requires the solution of the linear system At each step the n2 entries of Jij have to be computed In elasticity, the method of incremental loading is often useful It might be preferable to reevaluate Jij only occasionally (Modified

Newton’s Method) Matrix-updating schemes: In each iteration a new approximation to the

Jacobian is obtained by adding a rank-one matrix to the previous approximation

Often the derivatives in J are evaluated by finite differences

Strain Energy Function

2 2 2 2 21 2

2 2 2 23

2 ln 1compr

FF CC RR CR RC

FC CF FR RF

c QW e C J J J

Q b E b E E E E

b E E E E

Boundary ConditionsFiber Coordinates

P LV

XFXCXR

Pext= 0

epicardiumendocardium

(-37°)(+83°)

Numerical Convergence

60050040030020010009.0

9.5

10.0

10.5

11.0

Total Degrees of Freedom

To

tal S

tra

in E

ner

gy

(Jo

ule

s)

Linear Lagrangeinterpolation

Cubic Hermiteinterpolation

70 elements340 d.o.f.12 sec/iteration

3 elements104 d.o.f.14 sec/iteration

Inflation of a High-order Passive Anisotropic

Ellipsoidal Model of Canine LV

Coupling FE Models to the Circulation

R1p

right atriumleft atrium

FE rightventricle

FE leftventricle

R2p

R1sR2s

C2s

C2pC1p

C1s

Pulmonarycirculation

Systemiccirculation

Atria

FE ventricles

• Pressure protocol in finite element (FE) model

• Maximum and minimum elastances

• Time-varying elastance (VE) model

• Run VE model coupled to circulation

• Run fully coupled FE – circulation model

– Test case: normal heart followed by LV ischemia

Methods: Ventricular-Vascular Coupling

Elastances

Methods: Coupling

FE model Circulatory model

Estimate LV & RV cavity pressure

FE Cavity volumes Circ Cavity volumes

Calculate difference R

R < criterion?

no

RP

RPP

oldp

oldnew

1

yes

next timestep

R

L

P

PP

FER

FELFE

V

VV

circR

circLcirc

V

VV

circFE VVR

Do not updateJacobian

UpdateJacobian

R

circR

L

circR

R

circL

L

circL

R

FER

L

FER

R

FEL

L

FEL

P

circ

P

FE

P

P

V

P

VP

V

P

V

P

V

P

VP

V

P

V

P

V

P

V

P

R

iii

Estimation 1: Estimate pressure from history

Estimation 2: Perturb LV pressure

Estimation 3: Perturb RV pressure

Estimations >3: Update pressures

FE compliance matrix Circ compliance matrix

Methods: coupling

Resultsnormal beat followed by regional LV ischemia

ischemia

ischemia

Resultsnormal beat followed by regional LV ischemia

strokevolume

[ml]

Beat number