Friday: Cardiac Mechanics and Electromechanics

1. NBCR Summer Institute 2006:Multi-Scale Cardiac Modeling with Continuity 6.3Friday:Cardiac BiomechanicsAndrew McCulloch, Fred Lionetti and Stuart Campbell

2. 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

3. Passive Stress (kPa) 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

4. E = ½(FTF– I ) ¶ x i = F iR ¶ X R W W 1 = ( ) P + RS E E 2 RS SR Nonlinear Elasticity: Governing Equations kinematics Strain-displacement relation constitutive law Stress-strain relation equilibrium divT + rb= 0 T=T T Force balance equation Moment balance

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

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

7. 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)

8. 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

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

10. 11.0 10.5 10.0 9.5 9.0 0 100 200 300 400 500 600 Numerical Convergence 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 Total Degrees of Freedom

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

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

13. Methods: Ventricular-Vascular Coupling • 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

14. Elastances

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

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

17. ischemia ischemia Resultsnormal beat followed by regional LV ischemia

18. Resultsnormal beat followed by regional LV ischemia stroke volume [ml] Beat number