A one-dimensional morphoelastic model for burn injuries: stability analysis, numerical validation and biological interpretation

TL;DR

This paper develops and analyzes a one-dimensional morphoelastic model for scar formation post-wound healing, incorporating mechanical and biological factors, and validates stability constraints through numerical simulations with biological insights.

Contribution

It introduces stability analysis for a coupled mechanical-biological morphoelastic model and validates these constraints numerically, linking stability to biological parameters.

Findings

Eigenvalue truncation error is of order h^2.

Mechanical components reach equilibrium monotonically or non-monotonically based on viscosity.

Chemical stability depends on signaling molecule decay rate.

Abstract

To deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order $\mathcal{O}(h^2)$. Next, we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results.