Biomechanics cross-bridges ventriclescirculation3-D myocardium Myofilament kinetic model Hyperelastic constitutive equation 3-D finite element model of ventricular stress Boundary conditions: circulatory systems model R off R on A1A1 A1A * * * k on f gg Ca 2+ * kbkb knkn Image © Research Machines plc

Soft Tissue Biomechanics Conservation of mass, momentum and energy Large deformations (geometric nonlinearity) Nonlinear, anisotropic stress-strain relations Boundary conditions: displacement and traction (e.g. pressure) Dynamic active systolic tension development as a function of intracellular Ca, sarcomere length, shortening rate Viscoelastic properties Myofiber angle dispersion and transverse active stress Dynamic impedance boundary conditions Coupled problems: growth, electromechanics, FSI Passive Stress (kPa) Tff Tcc Extension Ratio

Kinematics Strain-displacement relation Deformation gradient tensor Constitutive law Hyperelastic relation for Lagrangian 2 nd Piola-Kirchoff stress (W is the strain energy function) Eulerian Cauchy stress Conservation of Momentum Force balance Moment balance 1 2 ∂W∂W ∂E∂E RS ∂W∂W ∂E∂E SR + = P RS ( ) E = ½ (F T F – I ) R i iR X x F ∂ ∂ = Div(PF T ) +  b = 0 P = P T Nonlinear Biomechanics: Governing Equations det 1 T F = FPF T F = Grad(x) Conservation of Mass Lagrangian form (  is mass density)  =   detF

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

Lagrangian Virtual Work Equations for Large Deformation Mechanics Divergence Theorem Virtual Work

Galerkin is equivalent to a Virtual Work Formulation Rayleigh-Ritz gives rise to a Potential Energy Minimization The dependent variable are the deformed coordinates of the nodes While we can use linear interpolation of deformed coordinates (or displacements) this gives rise to discontinuous strains and hence stresses. Cubic Hermite interpolation allows continuous stress and strain solutions The essential boundary conditions are displacement constraints The natural boundary conditions are traction (stress) boundary conditions. No nodal boundary condition is equivalent to traction-free For incompressible materials, we can use a constrained (augmented Lagrangian) formulation, which introduces a hydrostatic pressure Lagrange multiplier as a dependent variable, or we can use a Penalty Formulation by setting the bulk modulus to be high. The strain-displacement and stress strain relationships are nonlinear, so the stiffness matrices are functions of the dependent variables Hence, we need a nonlinear solution scheme Lagrangian FE Formulation

Newton’s Method in n Dimensions f’(x) is an nxn 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 n 2 entries of J ij have to be computed  In elasticity, the method of incremental loading is often useful  It might be preferable to reevaluate J ij 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

Orthotropic Strain Energy Function Fiber Material Coordinates P LV  X F X C X R P e xt = 0 epicardium endocardium (-37°) (+83°)

Boundary Conditions P LV P e xt = 0 Fixed Node  1s(2), s(2)s(3)value, s(2), s(2)s(3) 2 3s(2), s(2)s(3)value, s(2), s(2)s(3) 4valuevalue, s(2), s(2)s(3)value       node 1 node 2 node 3 node 4 d=3.7 cm 0 Y1Y1 Y2Y2

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

Numerical Convergence Total Degrees of Freedom Total Strain Energy (Joules) Linear Lagrange interpolation Cubic Hermite interpolation 70 elements 340 d.o.f. 3 elements 104 d.o.f.

Ca 2+ -Contraction Coupling: Unloaded Shortening FREE CYTOSOLIC Ca 2+ THIN FILAMENT ACTIVATION XB KINETICS PASSIVE CELL MECHANICS LENGTH MYOFILAMENT FORCE XB DISTORTION EXTERNAL LOADING TnC MYOFILAMENT OVERLAP (Campbell et al., Phil Trans R Soc A, 2008)

Rice et al (2008) Biophysical Journal 95(5): Active Cardiac Muscle Contraction

Rice et al (2008) Biophysical Journal 95(5): Myofilament Activation and Interactions

Pressure serves as hemodynamic boundary condition Cavity pressure Flow Q FE Cavity volume Cavity pressure Cavity volume from circulatory model

Kerckhoffs RCP, Neal M, Gu Q, Bassingthwaighte JBB, Omens JH, McCulloch AD. Ann Biomed Eng 2007;35(1):1-18

Canine heart Canine geometry & fiber angles Passive material: transversely isotropic, exponential Active material: time-, sarcomere length-, and Calcium-dependent Electrical stimulation: synchronous 48 tricubic elements 1968 DOFs timestep 4 ms ischemic

Circulation Pulmonary circulation Systemic circulation Atria FE ventricles Resistance arteries/capill. Resistance arteries/capill. Resistance veins Resistance veins Compliance veins Compliance arteries/capill. Compliance arteries/capill. Compliance veins

Protocol / performance Average: Computation time needed for first estimation: Each additional estimation: 5.1 estimations ~ 2 minutes ~ 5 seconds Time [sec] Number of estimations 12 normal beats18 beats with ischemia

Results normal heart followed by LV ischemic region ischemia stroke volume [ml] Beat number Volume [ml] Pressure [kPa]

Results normal heart followed by LV ischemic region Endsystolic endocardial myofiber strains Reference: end-diastole Normal1 st beat with ischemiaLast beat with ischemia

Coupled Ventricular Electromechanics (Campbell et al., Exp Physiol, 2009)

Algorithm for Fully Coupled Electromechanics Coupling algorithm