Mathematical Analysis of a Kinetic Model for Cell Movement in Network Tissues

TL;DR

This paper analyzes a kinetic model for mesenchymal cell movement in fiber networks, proving global existence of solutions and identifying steady states, with explicit solutions in specific cases and convergence to a parabolic limit.

Contribution

It extends the analysis of a mesenchymal motion model to higher dimensions, providing mathematical proofs of solution existence, steady states, and explicit solutions for constant fiber distributions.

Findings

Existence of global classical solutions

Existence of network-type steady states

Explicit solutions and convergence in specific cases

Abstract

Mesenchymal motion describes the movement of cells in biological tissues formed by fiber networks. An important example is the migration of tumor cells through collagen networks during the process of metastasis formation. We investigate the mesenchymal motion model proposed by T. Hillen (J. Math. Biol. 53:585-616, 2006) in higher dimensions. We formulate the problem as an evolution equation in a Banach space of measure-valued functions and use methods from semigroup theory to show the global existence of classical solutions. We investigate steady states of the model and show that patterns of network type exist as steady states. For the case of constant fiber distribution, we find an explicit solution and we prove the convergence to the parabolic limit.