Derivation and Application of Effective Interface Conditions for Continuum Mechanical Models of Cell Invasion through Thin Membranes

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.

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

“…Building on [13], we consider N populations of cells moving through a region of space that is filled with a porous embedding medium, e.g. the extracellular matrix.…”

“…In [13], we developed a formal asymptotic method that enables the derivation of a set of biophysically consistent transmission conditions to close such an equivalent transmission problem for a continuum mechanical model of cell invasion through tissues separated by thin porous membranes. This method applies to biological scenarios in which there is only one single population of invading cells.…”

“…Hence, modelling such processes requires to describe the evolution in time and space of multiple cell populations, characterised by different proliferation, migration and invasion abilities. For this reason, here we generalise the derivation method presented in [13] to the case of multiple cell populations, thus making the application domain of the transmission conditions resulting therefrom significantly wider.…”

Effect of a Membrane on Diffusion-Driven Turing Instability

“…In the last twenty years, biological applications of membrane problems have increased. Furthermore, they can describe phenomena on several dierent scales: from the nucleus membrane [4,8,26] to thin interfaces traversed by cancer cells [5,11] and to exchanges in bloody vessels, numerically studied in [23]. Also semi-discretization of mass diusion problems requires numerical treatment in adjoint domains coupled at the interface (see [3]).…”

Existence of a global weak solution for a reaction–diffusion problem with membrane conditions

“…This kind of problem is described by the so called Kedem-Katchalsky conditions [15] and has been used in mathematical biology recently. They can describe transport of molecules through the cell/nucleus membrane [25], the flux of cancer cells through thin interfaces [10] or solutes absorption processes through the arterial wall [23].…”