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

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

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

Myofilament Models Niederer SA, Hunter PJ, Smith NP. A quantitative analysis of cardiac myocyte relaxation: a simulation study. Biophys J. 2006 Mar 1;90(5):1697-722.

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)

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:

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:

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

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

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.

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

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

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

3-D Orthotropic Exponential Strain-Energy Function From: Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng 1986;108:189-192

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

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)

Minimizing Stress Gradients Residual Stress Fiber Angles Torsion

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

Lagrangian Virtual Work Equations for Large Deformation Elasticity Divergence Theorem

Lagrangian Finite Element Equations for Large Deformation Elasticity

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)

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

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

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

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

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

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

Results normal beat followed by regional LV ischemia

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