A congestion model for cell migration

TL;DR

This paper introduces a macroscopic congestion model for two-species cell migration, analyzing both passive and chemotactic active movement, with a focus on theoretical formulation and numerical simulation.

Contribution

It develops a gradient flow framework for the model and provides a numerical strategy for simulating cell migration with congestion effects.

Findings

Model effectively captures cell migration dynamics under congestion.

Numerical simulations demonstrate model behavior for prescribed velocities.

Framework applies to Keller-Segel type chemotaxis models.

Abstract

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant. We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.