Effective interface conditions for a porous medium type problem

TL;DR

This paper derives effective interface conditions for a porous-medium model of tumor invasion through thin membranes, focusing on the zero-thickness limit to simplify complex biological tissue interactions.

Contribution

It provides a rigorous mathematical derivation of transmission conditions for the zero-thickness membrane limit, extending previous formal results with a detailed analysis.

Findings

Derived effective interface conditions for zero-thickness membranes

Established a priori estimates and compactness for the limit process

Constructed an extension operator to handle degeneracy in mobility

Abstract

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.