Numerical Methods to Compute Stresses and Displacements from Cellular Forces: Application to the Contraction of Tissue

TL;DR

This paper compares numerical methods for modeling tissue contraction by cellular forces, focusing on handling point forces and cell boundary approximations to improve accuracy and computational efficiency.

Contribution

It introduces and evaluates different approaches, including immersed boundary and hole methods, for accurately computing stresses and displacements in tissue models with multiple cells.

Findings

Polygonal boundary approximation yields acceptable accuracy with low computational cost.

The hole approach increases accuracy but is computationally expensive and complex.

The immersed boundary approach offers a good balance for multi-cell tissue modeling.

Abstract

We consider a mathematical model for wound 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.