BE 276 Biomechanics part 1 Roy Kerckhoffs

Cardiac Modeling Circulation Ventricles Cross-bridges 3-D myocardium Image © Research Machines plc Ventricles Cross-bridges 3-D myocardium Myofilament kinetic model Roff Ron A1 * kon f g Ca2+ kb kn Boundary conditions: circulatory systems model Hyperelastic constitutive equation 3-D finite-element ventricular 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) Passive Stress (kPa) 1.2 1.1 1.0 10 20 30 Tff Tcc Extension Ratio

Deformation F ∂xi FiR = ∂XR F = Grad(x) F = RU Deformation gradient tensor F: ∂xi FiR = ∂XR F = Grad(x) F describes rotation and deformation F = RU O

Deformation Lagrangian strain tensor describes deformation only E = ½(FTF – I ) E = ½((RU)T(RU) – I ) E = ½(UTRTRU – I ) E = ½(UT I U – I ) E = ½(UT U – I )

Equilibrium of a body All stress components acting in a 3D continuum y Tyy All stress components acting in a 3D continuum Tyx Txy Tyz Tzy Txx x Txz Tzz Tzx z

Equilibrium of a body Cauchy Stress tensor T y Tyy Tyx Txy Tyz Tzy Txx Txz Tzz Tzx The stress tensor defines a relation between the stress vector t which acts on a plane and the normal n of that plane in the deformed state of the body. z

Equilibrium of a body Conservation of momentum: Txy=Tyx Txz=Tzx Tzy=Tyz T=TT Hence there are only 6 independent stress components y Tyy Tyx Txy Tyz Tzy Txx x Txz Tzz Tzx The stress tensor defines a relation between the stress vector t which acts on a plane and the normal n of that plane in the deformed state of the body. z

Equilibrium and virtual work S ρ is mass density is gravity S is surface of body is stress vector

Equilibrium and virtual work Apply Green-Gauss or divergence theorem on first term S Physical interpretation on divergence theorem for fluid flow is that a volume integral of the divergence of the flow field equals the net flow over the boundary

Equilibrium and virtual work Because this must hold for each subpart of the body, we can remove the integrals S

Equilibrium and virtual work Or, in components: This is Newton’s second law, written in the strong form.

Equilibrium and virtual work To obtain weak form, we apply the notion of virtual work Now apply integration by parts and Green-Gauss theorem again on first term: Hence we multiply the strong form equation with a virtual displacement field and integrate over the volume V

Equilibrium and virtual work This leads to the weak form: Internal virtual work This is the principle of virtual work, expressed in the current (deformed) configuration Virtual work by gravity Virtual work by boundary tractions

Equilibrium and virtual work In the weak form we impose lower order differentiability requirements on the solution Here, no assumptions have been made yet regarding material properties magnitude of strains Therefore, this equation is valid for both linear and finite elasticity mechanics. This is the principle of virtual work, expressed in the current (deformed) configuration

Equilibrium and virtual work In a deforming body, volumes, surfaces, mass density, stresses and strains continuously change At t+Δt the state of the body is unknown To solve this, refer stresses and strains to a known configuration Refer to known configuration in finite elasticity, typically the unloaded state. For linear elasticity, one assumes that volumes and surfaces don’t change, so the integration is always performed over the original volume.

Lagrangian stress tensors The (half) Lagrangian Nominal stress tensor or First Piola-Kirchhoff stress tensor S S = detF.F-1.T  ST S provides forces in the deformed state measured per unit reference (unloaded) area. Useful experimentally.

Lagrangian stress tensors The symmetric (fully) Lagrangian Second Piola-Kirchhoff stress tensor P Useful mathematically but no direct physical interpretation For small strains differences between T, P, S disappear To get T from P: Just like the Lagrangian strain E, the 2nd PK stress tensor P components are invariant to rigid body rotations This represents the Cauchy stress tensor transformed back to the original geometric configuration (by inverse of F). Perhaps one can think of this as stresses being stretched and rotated, just like the body material.

Large Deformation Mechanics

Example: uniaxial stress undeformed length = L undeformed area = A mass density is ρ0 volume is V0=LA deformed length = l deformed area = a mass density is ρ volume is V=la L A a F l Stretch Cauchy Stress

Example: uniaxial stress undeformed length = L undeformed area = A mass density is ρ0 volume is V0=LA deformed length = l deformed area = a mass density is ρ volume is V=la L A a F l Nominal Stress Second Piola-Kirchhoff Stress

Need relation between stress and strain: Constitutive Models Equate stress in a body to the strain Metals are elastic for small strains Hyperelastic Elastic Tissues are classified as a hyper-elastic material Plastic Exponential relationship used to model soft tissue Young’s Modulus The derivative In large deformation finite elasticity, relate stress and strain through a strain energy function W=W(E):

2-D Example: Exponential Strain-Energy Function

2-D Example: Exponential Strain-Energy Function If b1=b2: tissue is isotropic If b1≠b2: tissue is anisotropic

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

( ) Summary of Equations + = Kinematics Constitutive law Deformation gradient tensor Strain-displacement relation Constitutive law Hyperelastic relation Lagrangian 2nd Piola-Kirchoff stress W is the strain energy function Eulerian Cauchy stress Conservation of Momentum Force balance Moment balance Conservation of Mass Lagrangian form r: mass density = F = Grad(x) ∂XR ∂xi FiR E = ½(FTF – I ) 1 2 ∂W ∂ERS ∂ESR + = PRS ( ) det 1 T F = F•P•FT Div(P•FT) + rb = 0 P = PT r = r0detF

Lagrangian FE Formulation Dependent Variable Deformed coordinates of the nodes Choice of basis functions: Linear Interpolation Cubic Hermite interpolation Boundary Conditions Essential boundary conditions Displacement constraints Natural boundary conditions Traction (stress) boundary conditions No nodal boundary condition is equivalent to traction-free

Solution Scheme Non-linear strain-displacement and stress strain relationships Stiffness matrices are functions of the dependent variables Need a nonlinear solution scheme Newton Method f’(x) is an nxn Jacobian matrix J Gives us a linear system of equations for x(k+1)

Accelerating Newton’s Method Each step Requires the solution of the linear system n2 entries of Jij have to be computed Modified Newton’s Method Incremental loading in elasticity Re-evaluate Jij only occasionally Evaluate derivatives of J using finite differences Determine components of J over several processors (parallel solving)

Newton’s method in 1D force true mechanical response (nonlinear) force = K(u)u displacement

Newton’s method in 1D l Apply an external force F, find displacement

Newton’s method in 1D Determine derivative f’ (stiffness K(0)) at unloaded state force F F l K displacement

Newton’s method in 1D l Calculate displacement u1 = F/K(0) force F F K

Newton’s method in 1D Evaluating force at u1 leads to F1=K(u1)u1 ≠ F and we have an imbalance of F-F1 force F F1 F l K u1 displacement

Newton’s method in 1D Use stiffness at u1 and calculate next incremental displacement u2 force F K F l u1 u2 displacement

Newton’s method in 1D l Again, re-evaluating the force leads to F2 displacement

Newton’s method in 1D Repeating these steps leads to a small enough difference between F and Fn leading to a total final displacement of u=u1+u2+u3 force F Fn K F l u1 u2 u3 displacement

Newton’s method in 1D Therefore, stop Newton’s loop when either force imbalance is small enough, or the incremental displacement is small enough force F Fn K F l u1 u2 u3 displacement

Modified Newton’s method in 1D force etcetera F Fn F l Use same K as in unloaded state u1 u2 …. lots of un’s displacement

Myocardial Infarction

Myocardial Infarction FE mesh of left ventricle 32 elements with tricubic basis functions 80 nodes

Myocardial Infarction fibers endo>0 epi<0

Myocardial Infarction fibers Endocardium with fiber direction Epicardium with fiber direction

Myocardial Infarction Constitutive law C=0.45 kPa for healthy myocardium C=4.5 kPa for infarcted myocardium

Myocardial Infarction Location of infarction Healthy Infarct

Myocardial Infarction Displacement boundary conditions coor2 s(2) coor3 s(1) coor1 s(3) Fixed Node coordinate 1 coordinate 2 coordinate 3 BASE value, s(1), s(3), s(1)s(3) YCONS ZCONS APEX s(1), s(2), s(1)s(2), s(2)s(3), s(1)s(3), s(1)s(2)s(3) s(1), s(3)

Myocardial Infarction Displacement boundary conditions coor2 s(2) coor3 s(1) coor1 s(3) Fixed Node coordinate 1 coordinate 2 coordinate 3 BASE value, s(1), s(3), s(1)s(3) YCONS ZCONS APEX s(1), s(2), s(1)s(2), s(2)s(3), s(1)s(3), s(1)s(2)s(3) s(1), s(3)

Myocardial Infarction Displacement boundary conditions coor2 s(3) coor3 coor1 Fixed Node coordinate 1 coordinate 2 coordinate 3 BASE value, s(1), s(3), s(1)s(3) YCONS ZCONS APEX s(1), s(2), s(1)s(2), s(2)s(3), s(1)s(3), s(1)s(2)s(3) s(1), s(3)

Myocardial Infarction Displacement boundary conditions coor3 coor2 s(2) coor1 s(1) s(3) Fixed Node coordinate 1 coordinate 2 coordinate 3 BASE value, s(1), s(3), s(1)s(3) YCONS ZCONS APEX s(1), s(2), s(1)s(2), s(2)s(3), s(1)s(3), s(1)s(2)s(3) s(1), s(3)

Myocardial Infarction Pressure boundary conditions coor2 PLV Pext=0 coor3 coor1 Passive filling: Increase pressure incrementally on endocardial elements Pressure turned into force boundary conditions parallel to normal of element faces