Post on 19-Jan-2016
description
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