Derivation of a Hele-Shaw type system from a cell model with active motion

TL;DR

This paper derives a Hele-Shaw type free boundary model for tumor growth incorporating active cell motion, revealing smoother cell density and altered boundary dynamics due to active motion effects.

Contribution

It introduces a novel active motion component into the Hele-Shaw tumor growth model, derived from cell-level descriptions via asymptotic analysis.

Findings

Cell density becomes Lipschitz continuous with active motion.

Free boundary velocity deviates from pressure gradient due to 'mushy region'.

Active motion influences tumor invasion dynamics.

Abstract

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of 'mushy region' prepares the tumor invasion.