An XFEM‐based numerical strategy to model mechanical interactions between biological cells and a deformable substrate

Abstract

SUMMARYContractile cells are known to constantly probe and respond to their mechanical environment through mechanosensing. Although the very mechanisms responsible for this behavior are still obscure, it is now clear that cells make full use of cross‐talks between mechanics, chemistry, and transport to organize their structure and generate forces. To investigate these processes, it is important to derive mathematical and numerical models that can accurately capture the interactions between cells and an underlying deformable substrate. The present paper therefore introduces a computational framework, based on the extended FEM (XFEM) and the level set method, to model the evolution of two‐dimensional (plane stress) cells lying on an elastic substrate whose properties can be varied. Cells are modeled with a continuum mixture approach previously developed by the authors to describe key phenomena of cell sensing, such as stress fiber formation, mechanosensitive contraction, and molecular transport whereas cell–substrate adhesion is formulated with a linear elastic cohesive model. From a numerical viewpoint, cell and substrate are discretized on a single, regular finite element mesh, whereas the potentially complex cell geometry is defined in terms of a level set function that is independent of discretization. Field discontinuities across the cell membrane are then naturally enforced using enriched shape functions traditionally used in the XFEM formulation. The resulting method provides a flexible platform that can handle complex cell geometries, can avoid expensive meshing techniques, and can potentially be extended to study cell growth and migration on an elastic substrate. In addition, the XFEM formalism facilitates the consideration of the cell's cortical elasticity, a feature that is known to be important during cell deformation. The proposed method is illustrated with a few biologically relevant examples of cell–substrate interactions. Generally, the method is able to capture some key phenomena observed in biological systems and displays numerical versatility and accuracy at a moderate computational cost.Copyright © 2012 John Wiley & Sons, Ltd.