Nonlinear biomechanics of tissue with stochastic microstructure
Biological tissue exhibits significant variability in mechanical properties that is usually attributed to random microscopic realizations. A computational approach to nonlinear mechanics of tissue with an underlying stochastic component is introduced. At the bulk continuum level, a hyperelastic formulation is combined with linear viscous models to provide the mean stress and strain fields for microscale constituent elements of the tissue. The hyperelastic behavior is expressed as a system of conservation laws, $\boldsymbol{q}_{, t} \nabla _{\boldsymbol{a}} \boldsymbol{f} (\boldsymbol{q}) =\boldsymbol{\psi} (\boldsymbol{q}, \boldsymbol{\xi})$, for the velocity and deformation gradient $\boldsymbol{q}= (\boldsymbol{v}, \mathbf{F})$ in a Lagrangian framework. The flux $\boldsymbol{f}= (-\mathbf{P} (\mathbf{F}), -\mathbf{Z} (\boldsymbol{v}))$ requires definition of the dependence of the first Piola-Kirchhoff tensor $\mathbf{P}$ in terms of the deformation gradient $\mathbf{F}$. Viscous effects are captured through the sink term $\boldsymbol{\psi}$, that incorporates history dependence through the auxiliary variables $\boldsymbol{\xi}$.
Constitutive laws of known analytical form utilize model parameters $\boldsymbol{y} (\boldsymbol{\omega})$ that are assumed here to be random functions of the microstate $\omega$, and can be determined by carrying out integration over all possible microstates $\mathbf{P} (\mathbf{F}) = \int_{\Omega} \mathbf{G} (\mathbf{F}, \omega) p (\omega) \mathrm \omega$, where $\mathbf{G}$ is the generating function for $\mathbf{P}$. The probability density function (PDF) $p (\omega)$ is extracted from microscopic simulations carried out under known mean fields; Brownian Dynamics and Markov Chain Monte Carlo methods are considered. Probability distribution functions characterizing the microscale configuration are organized as a statistical manifold $\mathcal{M}= { p (\omega ; \lambda) }$, where $\lambda$ is a parametrization of the manifold. Singular value decomposition of data from successive microscopic states defines the local curvature of $\mathcal{M}$, and computational geometry provides a local reconstruction of the manifold. Low-parameter PDFs are obtained by geodesic transport on $\mathcal{M}$ applying concepts of information geometry. The construction is updated during time evolution, forming a macro-microscale interaction loop. The procedure is applied to propagation of shear waves in the brain, using realistic data obtained from histology and compared to ultrasonic shear wave imaging.