A mathematical model for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding

Abstract

A continuum hypothesis-based model is presented for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding. Solely the dermal layer of the skin is modeled explicitly and it is modeled as a heterogeneous, isotropic and compressible neo-Hookean solid. With respect to the constituents of the dermal layer, the following components are selected as primary model components: fibroblasts, myofibroblasts, a generic signaling molecule and collagen molecules. A good match with respect to the evolution of the thickness of the dermal layer of scars between the outcomes of simulations and clinical measurements on hypertrophic scars at different time points after injury in human subjects is demonstrated. Interestingly, the comparison between the outcomes of the simulations and the clinical measurements demonstrates that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves more like normal scar tissue with respect to the evolution of the thickness of the tissue over time, while a relatively low apoptosis rate results in scar tissue that behaves like hypertrophic scar tissue with respect to the evolution of the thickness of the tissue over time. Our ultimate goal is to construct models with which the properties of newly generated tissues that form during wound healing can be predicted with a high degree of certainty. The development of the presented model is considered by us as a step toward their construction.

Biomedical implications from a morphoelastic continuum model for the simulation of contracture formation in skin grafts that cover excised burns

“…The mathematical descriptions for the movement of the cells, the biochemical kinetics associated with these cells, the dispersion of the generic signaling molecule, and the release, consumption, and removal of both the collagen molecules and the generic signaling molecule are nearly identical to the functional forms used previously (Koppenol et al. 2017b). …”

“…2017b). For the biochemical kinetics, the functional forms are whereThe parameter is the cell division rate, is the maximum factor with which the cell division rate can be enhanced due to the presence of the signaling molecule, is the concentration of the signaling molecule that causes the half-maximum enhancement of the cell division rate, represents the reduction in the cell division rate due to crowding, q is a fixed constant, is the signaling molecule-dependent cell differentiation rate of fibroblasts into myofibroblasts, is the apoptosis rate of fibroblasts, and is the apoptosis rate of myofibroblasts.…”

“…2017b): The parameter is the collagen molecule secretion rate, is the maximum factor with which the secretion rate can be enhanced due to the presence of the signaling molecule, is the concentration of the signaling molecule that causes the half-maximum enhancement of the secretion rate, and is the proteolytic breakdown rate of the collagen molecules.…”

“…(1986) Koppenol et al. (2017b) 1.02 Liang and Boppart (2010) TW TW Liang and Boppart (2010) − Liang and Boppart (2010) Maskarinec et al. (2009)&Wrobel et al.…”

Review on experiment-based two- and three-dimensional models for wound healing

“…This necessitates the use of either upwind-like techniques, such as streamline upwind Petrov-Galerkin (SUPG) methods, or the use of limiters, in order to preserve positivity and monotonicity of the numerical approximations; in any case, it becomes attractive to use a high mesh resolution in regions that are characterized by sharp transitions. SUPG techniques are used for instance in the works of Javierre et al [32] and Valero et al [34], whereas slope and flux-limiting strategies have been used in Koppenol et al [70]. Epshteyn [71] uses a discontinuous Galerkin (DG) method to solve the equations for chemotaxis.…”