Effective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes

TL;DR

This paper develops effective interface conditions for continuum models describing the invasion of multiple cell populations through thin membranes, simplifying complex tissue interactions into a manageable limiting transmission problem.

Contribution

It introduces a novel asymptotic method to derive biophysically consistent transmission conditions replacing thin membranes with effective interfaces in continuum models.

Findings

Derived effective interface conditions for multiple cell populations.

Reduced complex membrane interactions to a simplified transmission problem.

Applicable to biological and medical research areas like cancer and immunology.

Abstract

We consider a continuum mechanical model for the migration of multiple cell populations through parts of tissue separated by thin membranes. In this model, cells belonging to different populations may be characterised by different proliferative abilities and mobility, which may vary from part to part of the tissue, as well as by different invasion potentials within the membranes. The original transmission problem, consisting of a set of mass balance equations for the volume fraction of cells of every population complemented with continuity of stresses and mass flux across the surfaces of the membranes, is then reduced to a limiting transmission problem whereby each thin membrane is replaced by an effective interface. In order to close the limiting problem, a set of biophysically-consistent transmission conditions is derived through a formal asymptotic method. Models based on such a limiting transmission problem may find fruitful application in a variety of research areas in the biological and medical sciences, including developmental biology, immunology and cancer growth and invasion.