Spatial segregation across travelling fronts in individual-based and continuum models for the growth of heterogeneous cell populations

TL;DR

This paper develops a mathematical model for heterogeneous cell population growth, showing how differences in cell mobility lead to spatial segregation during invasion, supported by analytical and numerical results.

Contribution

It introduces a continuum PDE model derived from an individual-based model that captures phenotype-dependent movement and pressure effects in cell populations.

Findings

Cells with different mobility segregate spatially at the invasion front.

The continuum model accurately reproduces the behavior of the individual-based model.

Inter-cellular variability in mobility drives spatial segregation.

Abstract

We consider a partial differential equation model for the growth of heterogeneous cell populations subdivided into multiple distinct discrete phenotypes. In this model, cells preferentially move towards regions where they feel less compressed, and thus their movement occurs down the gradient of the cellular pressure, which is defined as a weighted sum of the densities (i.e. the volume fractions) of cells with different phenotypes. To translate into mathematical terms the idea that cells with distinct phenotypes have different morphological and mechanical properties, both the cell mobility and the weighted amount the cells contribute to the cellular pressure vary with their phenotype. We formally derive this model as the continuum limit of an on-lattice individual-based model, where cells are represented as single agents undergoing a branching biased random walk corresponding to phenotype-dependent and pressure-regulated cell division, death, and movement. Then, we study travelling wave solutions whereby cells with different phenotypes are spatially segregated across the invading front. Finally, we report on numerical simulations of the two models, demonstrating excellent agreement between them and the travelling wave analysis. The results presented here indicate that inter-cellular variability in mobility can provide the substrate for the emergence of spatial segregation across invading cell fronts.