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

TL;DR

The paper develops a generalized Finsler-geometric continuum theory for nonlinear, fibrous soft tissues with remodeling and degradation, introducing multiple internal state vectors to capture evolving microstructure. Through a variational principle, it derives coupled Euler–Lagrange equations for linear momentum and internal-state equilibrium, allowing mass exchange with the environment and phase-field-like damage and remodeling energetics. The framework can recover osculating Riemannian limits and links residual stresses to Cartan tensors and gradients of internal state, while enabling non-affine microstructure evolution and multiple fiber families. The left ventricle application demonstrates that remnant strains from remodeling can reduce stress concentrations and tissue damage under severe loading, illustrating the practical impact for modeling remodeling and damage in cardiac tissue and informing potential extensions to anisotropic, large-deformation regimes.

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.

Figures (8)

• Figure 1: Elastic constant fits for cases C0, C1, C3, and C5 (Table \ref{['table1']}, uniform remnant stretch $\Lambda_\Theta$, cohesive fracture energy $\hat{e}_c$) to ventricular pressure-volume data spotnitz1966, inner radius $r_i$, internal pressure $p_i$, and external pressure $p_o$ (1 atm in experiments): (a) pressure range close to experiments (b) higher net pressures
• Figure 2: Stress versus inner radius $r_i$ for cases C1, C2, C3, and C4 (Table \ref{['table1']}, uniform remnant stretch $\Lambda_\Theta$): (a) transverse $\sigma^\phi_\phi = \sigma^\theta_\theta$ at null applied pressure (b) transverse at $\approx 1$ kPa pressure (c) radial $\sigma^r_r$ at $\approx 1$ kPa pressure
• Figure 3: Stress and degradation versus inner radius $r_i$ for cases C1, C2, C3, and C4 (Table \ref{['table1']}, uniform remnant stretch $\Lambda_\Theta$): (a) transverse stress $\sigma^\phi_\phi = \sigma^\theta_\theta$ at $\approx 4$ kPa net applied pressure (b) fracture order parameter $\eta$ at $\approx 4$ kPa pressure
• Figure 4: Stress and degradation versus inner radius $r_i$ for cases C1, C2, C3, and C4 (Table \ref{['table1']}, uniform remnant stretch $\Lambda_\Theta$): (a) transverse stress $\sigma^\phi_\phi = \sigma^\theta_\theta$ at $\gtrsim 7$ kPa net applied pressure (b) fracture order parameter $\eta$ at $\gtrsim 7$ kPa pressure
• Figure 5: Stress and internal state components versus inner radius $r_i$ for case C1 (Table \ref{['table1']}, $E_R = 0$, $\hat{n} = 2$): (a) transverse stress $\sigma^\phi_\phi = \sigma^\theta_\theta$ (b) radial stress $\sigma^r_r$ (c) fracture variable $\eta$ (d) remodeling variable $D_\Theta$