1. NBCR Summer Institute 2007: Multi-Scale Cardiac Modeling with Continuity 6.3 Friday: Cardiac Biomechanics Andrew McCulloch, Fred Lionetti and Roy Kerckhoffs

2. Friday: Cardiac Biomechanics
Modeling Ventricular Wall Mechanics
Constitutive Models of Cardiac Muscle
Galerkin FEM for ventricular stress analysis
Newton’s method
Ventricular-Vascular Coupling

3. Ventricular Wall Mechanics
Passive Stress (kPa)
Conservation of mass, momentum and energy
Geometry and Structure
3D geometry
Myofiber angle and sheet distributions
Myofiber angle dispersion, lattice structure
Material Properties
3-D resting mechanical properties
Residual stresses (Growth)
Myofilament active contractile mechanisms
Perfusion
Boundary conditions
Pressure boundary conditions from hemodynamic model
Displacement constraints

4. Myofilament Models
Niederer SA, Hunter PJ, Smith NP. A quantitative analysis of cardiac myocyte relaxation: a simulation study. Biophys J Mar 1;90(5):

5. Myofilament Activation/Crossbridge Cycling Kinetics
Roff
Ron A1 *
kon f g
Ca2+ kb kn
Non-permissive Tropomyosin
koff
koff
Permissive Tropomyosin f
Permissive Tropomyosin, 1-3 crossbridges attached (force generating states)
Ca2+
bound
to TnC
Ca2+
not
bound
to TnC
This scheme is used to find A(t), the time-course of attached crossbridges for a given input of [Ca2](t)

6. Myofilament Model Equations
Total force is the product of the total number of attached crossbridges, average crossbridge distortion, and crossbridge stiffness:
Average crossbridge distortion is obtained by the solution to the following differential equation:

7. Myofilament Model Equations
Governing ordinary differential equations (ODEs) come from mass conservation:
Because mass is conserved, one of the above equations can be replaced by the following algebraic expression:

8. 3-D Active Stress Cauchy stress tensor:
tensor Tactive is a function of peak intracellular calcium [Ca]i and sarcomere length.

9. Biomechanics Governing Equations
Kinematics
E = ½(FTF – I ) R i iR X x F =
Strain-displacement relation
Constitutive law
Second Piola-Kirchhoff Stress
Cauchy Stress
Total Stress
Equilibrium equations
Force balance equation
Moment balance
divT + rb = 0
T = T T

10. Cauchy Stress Tensor is Eulerian
Cauchy’s formula: t(n) = n•T
In index notation: e1 e2 e3
Tij = ti•ej
T11
T12
T22
T33
T13
T31
T23
T21
T32 R S n a
Tij is the component in the xj direction of the traction vector t(n) acting on the face normal to the xi axis in the deformed state of the body. The "true" stress.

11. Lagrangian Stress Tensors
The (half) Lagrangian Nominal stress tensor S
SRj is the component in the xj direction of the traction measured per unit reference area acting on the surface normal to the (undeformed) XR axis. Useful experimentally
S = detF.F-1.T  ST
The symmetric (fully) Lagrangian Second Piola-Kirchhoff stress tensor
Useful mathematically but no direct physical interpretation
For small strains differences between T, P, S disappear

12. Example: Uniaxial Stress
undeformed length = L
undeformed area = A
deformed length = l
deformed area = a L A a F l
Cauchy Stress
Nominal Stress
Add diagram – done Jan 2000
Second Piola-Kirchhoff Stress

13. 2-D Example: Exponential Strain-Energy Function
Stress components have interactions

14. 3-D Orthotropic Exponential Strain-Energy Function
From: Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng 1986;108:

15. Strain Energy Functions
Transversely Isotropic (Isotropic + Fiber) Exponential
Transversely Isotropic Exponential
Transversely Isotropic Polynomial
Orthotropic Power Law

16. Incompressible Materials
Stress is not completely determined by the strain because a hydrostatic pressure can be added to Tij without changing CRS. The extra condition is the kinematic incompressibility constraint
To avoid derivative of W tending to 
p is a Lagrange multiplier (a negative stress)

17. Minimizing Stress Gradients
Residual Stress
Fiber Angles
Torsion

18. 1. Formulate the weighted residual (weak) form
2. Divergence (Green-Gauss) Theorem
Note: Taking w=du*, we have the virtual work equation

19. Lagrangian Virtual Work Equations for Large Deformation Elasticity
Divergence Theorem

20. Lagrangian Finite Element Equations for Large Deformation Elasticity

21. Newton’s Method in n Dimensions
f’(x) is an n  n Jacobian matrix J
Gives us a linear system of equations for x(k+1)

22. 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

23. Strain Energy Function
Boundary Conditions
Fiber Coordinates P L V  X F C R e x t
= 0
epicardium
endocardium
(-37°)
(+83°)
Strain Energy Function

24. Numerical Convergence
9.0
9.5
10.0
10.5
11.0
Cubic Hermite
interpolation
3 elements
104 d.o.f.
14 sec/iteration
Total Strain Energy (Joules)
70 elements
340 d.o.f.
12 sec/iteration
Linear Lagrange
interpolation
600
500
400
300
200
100
Total Degrees of Freedom

25. Inflation of a High-order Passive Anisotropic Ellipsoidal Model of Canine LV

26. Coupling FE Models to the Circulation
Pulmonary
circulation
Atria
FE ventricles
Systemic
circulation

27. Methods: Coupling Estimate LV & RV cavity pressure FE model
Circulatory model
FE Cavity volumes
Circ Cavity volumes
Calculate difference R
R < criterion?
yes no
Update
Jacobian
Do not update
Jacobian
next timestep

28. Estimation 1: Estimate pressure from history
Methods: coupling
FE compliance matrix
Circ compliance matrix
Estimation 1: Estimate pressure from history
Estimation 2: Perturb LV pressure
Estimation 3: Perturb RV pressure
Estimations >3: Update pressures

29. Results normal beat followed by regional LV ischemia

30. Results normal beat followed by regional LV ischemia
stroke
volume
[ml]
Beat number