Nonlinear soft-tissue elasticity, remodeling, and degradation described by an extended Finsler geometry

Abstract

A continuum mechanical theory incorporating an extension of Finsler geometry is formulated for fibrous soft solids. Especially if of biologic origin, such solids are nonlinear elastic with evolving microstructures. For example, elongated cells or collagen fibers can stretch and rotate independently of motions of their embedding matrix. Here, a director vector or internal state vector, not always of unit length, in generalized Finsler space relates to a physical mechanism, with possible preferred direction and intensity, in the microstructure. Classical Finsler geometry is extended to accommodate multiple director vectors (i.e., multiple fibers in both a differential-geometric and physical sense) at each point on the base manifold. A metric tensor can depend on the ensemble of director vector fields. Residual or remnant strains from biologic growth, remodeling, and degradation manifest as non-affine fiber and matrix stretches. These remnant stretch fields are quantified by internal state vectors and a corresponding, generally non-Euclidean, metric tensor. Euler-Lagrange equations derived from a variational principle yield equilibrium configurations satisfying balances of forces from elastic energy, remodeling and cohesive energies, and external chemical-biological interactions. Given certain assumptions, the model can reduce to a representation in Riemannian geometry. Residual stresses that emerge from a non-Euclidean material metric in the Riemannian setting are implicitly included in the Finslerian setting. The theory is used to study stress and damage in the ventricle (heart muscle) expanding or contracting under internal and external pressure. Remnant strains from remodeling can reduce stress concentrations and mitigate tissue damage under severe loading.