Abstract
We consider a mathematical model for skin contraction, which is based on solving a momentum balance under the assumptions of isotropy, homogeneity, Hooke’s Law, infinitesimal strain theory and point forces exerted by cells. However, point forces, described by Dirac Delta distributions lead to a singular solution, which in many cases may cause trouble to finite element methods due to a low degree of regularity. Hence, we consider several alternatives to address point forces, that is, whether to treat the region covered by the cells that exert forces as part of the computational domain or as ‘holes’ in the computational domain. The formalisms develop into the immersed boundary approach and the ‘hole’ approach, respectively. Consistency between these approaches is verified in a theoretical setting, but also confirmed computationally. However, the ‘hole’ approach is much more expensive and complicated for its need of mesh adaptation in the case of migrating cells while it increases the numerical accuracy, which makes it hard to adapt to the multi-cell model. Therefore, for multiple cells, we consider the polygon that is used to approximate the boundary of cells that exert contractile forces. It is found that a low degree of polygons, in particular triangular or square shaped cell boundaries, already give acceptable results in engineering precision, so that it is suitable for the situation with a large amount of cells in the computational domain.
Highlights
In this manuscript, we consider skin contraction after skin injury
Skin contraction takes place as a result of mechanical, pulling forces that are exerted by the cells that are responsible for the regeneration of collagen [2]
In Proposition 2, we have proved the equivalence between the finite element solutions to the adjusted immersed boundary approach and the ‘hole’ approach for E = 0
Summary
We consider skin contraction after skin injury. Since severe (burn) injuries involve a considerable loss of soft tissue, secondary healing takes place. Skin contraction takes place as a result of mechanical, pulling forces that are exerted by the cells (i.e. mainly fibroblasts and myofibroblasts) that are responsible for the regeneration of collagen [2]. Due to being triggered by the high concentration of TGF-beta, fibroblasts differentiate to myofibroblasts, which are known to exert even larger forces than fibroblasts These larger pulling forces result into the contraction of the tissue around the injury towards the wound centre [4,5,6]. The current manuscript focusses on hybrid models for simulating wound contraction in a small scale, where we consider cells as individual entities. One may include the region covered by the cell as part of computational domain This idea develops into the immersed boundary approach.
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