A novel derivation of rigorous macroscopic limits from a micro-meso description of signal-triggered cell migration in fibrous environments

TL;DR

This paper develops a unified mathematical approach to derive macroscopic models from microscopic descriptions of cell migration in fibrous environments, accounting for signal influences without relying on orthogonality.

Contribution

It introduces a novel scaling method that unifies parabolic and hyperbolic limits for kinetic transport equations in cell migration modeling.

Findings

Rigorous derivation of macroscopic limits from kinetic models

Unified treatment of different scaling regimes

Conditions for measure-valued fiber distributions to ensure validity

Abstract

In this work we upscale a prototypical kinetic transport equation which models a cell population moving in a fibrous environment with a chemo- or haptotactic signal influencing both the direction and the magnitude of the cell velocity. The presented approach to scaling does not rely on orthogonality and treats parabolic and hyperbolic scalings in a unified manner. It is shown that the steps of the formal limit procedures are mirrored by rigorous operations with finite measures provided that the measure-valued position-direction fiber distribution enjoys some spacial continuity.