# Derivation and application of effective interface conditions for continuum mechanical models of cell invasion through thin membranes

## TL;DR

This paper develops a mathematical framework to simplify models of cell invasion through thin membranes by replacing them with effective interfaces, validated through simulations and applied to cancer invasion scenarios.

## Contribution

It introduces a formal asymptotic method to derive biophysically consistent interface conditions for continuum models of cell invasion through thin membranes.

## Key findings

- The derived interface conditions accurately approximate the original problem as membrane thickness approaches zero.
- Numerical simulations confirm the validity of the limiting transmission problem.
- Application examples demonstrate the model's relevance to cancer cell invasion and metastasis.

## Abstract

We consider a continuum mechanical model of cell invasion through thin membranes. The model consists of a transmission problem for cell volume fraction complemented with continuity of stresses and mass flux across the surfaces of the membranes. We reduce the original problem to a limiting transmission problem whereby each thin membrane is replaced by an effective interface, and we develop a formal asymptotic method that enables the derivation of a set of biophysically consistent transmission conditions to close the limiting problem. The formal results obtained are validated via numerical simulations showing that the relative error between the solutions to the original transmission problem and the solutions to the limiting problem vanishes when the thickness of the membranes tends to zero. In order to show potential applications of our effective interface conditions, we employ the limiting transmission problem to model cancer cell invasion through the basement membrane and the metastatic spread of ovarian carcinoma.