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.
Highlights
Burn wounds are a global problem and are the fifth most common cause of non-fatal childhood injuries
Dc is the Fickian diffusion coefficient of the signaling molecules, kc is the maximum net secretion rate of the signaling molecules, ηI is the ratio of myofibroblasts to fibroblasts in the maximum secretion rate of the signaling molecules, acI is the concentration of the signaling molecules that causes the half-maximum net secretion rate of the signaling molecules, δc is the proteolytic breakdown rate parameter of the signaling molecules, ηI I is the ratio of myofibroblasts to fibroblasts in the secretion rate of the Matrix Metallo Proteins (MMPs) and 1/[1 + acI I c] represents the inhibition of the secretion of the MMPs
We report the densities of the variables and the relative surface area of the wound (RSAW)
Summary
Keywords Burns · Wound contraction · Stability · Morphoelasticity · Moving-grid finite-element. Mathematics Subject Classification 35B20 · 35B35 · 35G20 · 35L65 · 35M10 · 35Q74 · 35Q80 · 35Q92 · 35R37 · 65C20 · 65M12 · 65M60 · 65N12 · 65N30 · 74-10 · 74L15 · 92-10 · 92C10 · 92C17 · 92C45 · 93B18
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