Analysis of linearized elasticity models with point sources in weighted Sobolev spaces: applications in tissue contraction

¹ Laboratory for Modeling and Scientific Computing MOX, Politecnico di Milano, p.za Leonardo da Vinci 32, Milano 20133, Italy
² Computational Mathematics Group, Department of Mathematics and Statistics, University of Hasselt, Agoralaan Building D, 3590 Diepenbeek, Belgium

^* Corresponding author: wietsemarijn.boon@polimi.it

Received: 10 February 2022
Accepted: 14 June 2023

Abstract

In order to model the contractive forces exerted by fibroblast cells in dermal tissue, we propose and analyze two modeling approaches under the assumption of linearized elasticity. The first approach introduces a collection of point forces on the boundary of the fibroblast whereas the second approach employs an isotropic stress point source in its center. We analyze the resulting partial differential equations in terms of weighted Sobolev spaces and identify the singular behavior of the respective solutions. Two finite element method approaches are proposed, one based on a direct application and another in which the singularity is subtracted and a correction field is computed. Finally, we confirm the validity of the modeling approach, demonstrate convergence of the numerical methods, and verify the analysis through the use of numerical experiments.