Biomechanics. Soft Tissue Biomechanics

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Sharyl Shepherd
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Biomechanics cross-bridges 3-D myocardium ventricles circulation Image

Ca 2+ R0 on Ca 2+ g Ca 2+ A* 1 A0 1 Ca 2+ Myofilament kinetic model

stress Boundary conditions: circulatory systems model Soft Tissue

(geometric nonlinearity) Nonlinear, anisotropic stress-strain relations

pressure) Dynamic active systolic tension development as a function of

Myofiber angle dispersion and transverse active stress Dynamic impedance

1 Biomechanics cross-bridges 3-D myocardium ventricles circulation Image Research Machines plc R* off k n k b Ca 2+ 0 R off Ca 2+ * k on R* on g f Ca 2+ R0 on Ca 2+ g Ca 2+ A* 1 A0 1 Ca 2+ Myofilament kinetic model Hyperelastic constitutive equation 3-D finite element model of ventricular stress Boundary conditions: circulatory systems model 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 1.1 Extension Ratio 1.2 1

2 Nonlinear Biomechanics: Governing Equations 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 Conservation of Mass Lagrangian form (ρ is mass density) E = ½(F T F I ) x i F ir = F = Grad(x) X R = 1 W W P + RS ( ) 2 E RS E SR 1 T = F P F T det F Div(P F T ) + ρb = 0 P = P T ρ = ρ 0 detf 1. Formulate the weighted residual (weak) form 2. Divergence (Green-Gauss) Theorem Note: Taking w= u *, we have the virtual work equation 2

3 Lagrangian Virtual Work Equations for Large Deformation Mechanics Virtual Work Divergence Theorem Lagrangian FE Formulation 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 3

4 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 4

5 Fiber Material Coordinates η endocardium (+83 ) epicardium (-37 ) X C X F X R P L V P e x t = 0 Orthotropic Strain Energy Function! Boundary Conditions node 4 node 2 θ ξ 2 Y 2 λ ξ 1 d=3.7 cm ξ 3 P L V node 1 node 3 0 µ P e x t = 0 Y 1 Fixed Node λ µ θ 1 s(2), s(2)s(3) value, s(2), s(2)s(3) 2 value, s(2), s(2)s(3) 3 s(2), s(2)s(3) value, s(2), s(2)s(3) 4 value value, s(2), s(2)s(3) value 5

5 Cubic Hermite interpolation 3 elements 104 d.o.f.

6 Inflation of a High-order Passive Anisotropic Ellipsoidal Model of Canine LV Numerical Convergence Total Strain Energy (Joules) Cubic Hermite interpolation 3 elements 104 d.o.f. Linear Lagrange interpolation 70 elements 340 d.o.f Total Degrees of Freedom

7 Ca 2+ -Contraction Coupling: Unloaded Shortening FREE CYTOSOLIC Ca 2+ TnC THIN FILAMENT ACTIVATION XB DISTORTION XB KINETICS LENGTH MYOFILAMENT OVERLAP MYOFILAMENT FORCE PASSIVE CELL MECHANICS EXTERNAL LOADING (Campbell et al., Phil Trans R Soc A, 2008) Active Cardiac Muscle Contraction Rice et al (2008) Biophysical Journal 95(5):

8 Myofilament Activation and Interactions Rice et al (2008) Biophysical Journal 95(5): Pressure serves as hemodynamic boundary condition Cavity pressure Cavity pressure Flow Q FE Cavity volume Cavity volume from circulatory model 8

material: transversely isotropic, exponential Active material: time-, sarcomere length-,

9 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 ischemic Electrical stimulation: synchronous 48 tricubic elements 1968 DOFs timestep 4 ms 9

10 Circulation Compliance arteries/capill. Resistance arteries/capill. Compliance veins Resistance veins Pulmonary circulation Atria FE ventricles Compliance arteries/capill. Resistance veins Compliance veins Resistance arteries/capill. Systemic circulation Protocol / performance 12 normal beats 18 beats with ischemia Number of estimations Time [sec] Average: Computation time needed for first estimation: Each additional estimation: 5.1 estimations ~ 2 minutes ~ 5 seconds 10

11 Results normal heart followed by LV ischemic region Pressure [kpa] ischemia stroke volume [ml] ischemia Volume [ml] Beat number Results normal heart followed by LV ischemic region Normal 1 st beat with ischemia Last beat with ischemia Endsystolic endocardial myofiber strains Reference: end-diastole 11

12 Coupled Ventricular Electromechanics (Campbell et al., Exp Physiol, 2009) Algorithm for Fully Coupled Electromechanics Coupling algorithm 12

Lectures on. Constitutive Modelling of Arteries. Ray Ogden

Lectures on. Constitutive Modelling of Arteries. Ray Ogden Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics