Travelling wave analysis of cellular invasion into surrounding tissues

TL;DR

This paper investigates a coupled reaction-diffusion model of malignant invasion, revealing novel properties of travelling wave solutions and linking multi-species invasion models to moving boundary problems through analysis and simulations.

Contribution

It introduces a reaction-diffusion model explicitly describing tissue degradation and invasion, extending classical single-species equations with new properties and insights.

Findings

Travelling waves resemble features of Fisher-KPP and Porous-Fisher equations.

Wave solutions are well approximated by Fisher-KPP phase plane trajectories.

Links established between multi-species reaction-diffusion models and moving boundary problems.

Abstract

Single-species reaction-diffusion equations, such as the Fisher-KPP and Porous-Fisher equations, support travelling wave solutions that are often interpreted as simple mathematical models of biological invasion. Such travelling wave solutions are thought to play a role in various applications including development, wound healing and malignant invasion. One criticism of these single-species equations is that they do not explicitly describe interactions between the invading population and the surrounding environment. In this work we study a reaction-diffusion equation that describes malignant invasion which has been used to interpret experimental measurements describing the invasion of malignant melanoma cells into surrounding human skin tissues. This model explicitly describes how the population of cancer cells degrade the surrounding tissues, thereby creating free space into which the cancer cells migrate and proliferate to form an invasion wave of malignant tissue that is coupled to a retreating wave of skin tissue. We analyse travelling wave solutions of this model using a combination of numerical simulation, phase plane analysis and perturbation techniques. Our analysis shows that the travelling wave solutions involve a range of very interesting properties that resemble certain well-established features of both the Fisher-KPP and Porous-Fisher equations, as well as a range of novel properties that can be thought of as extensions of these well-studied single-species equations. Of particular interest is that travelling wave solutions of the invasion model are very well approximated by trajectories in the Fisher-KPP phase plane that are normally disregarded. This observation establishes a previously unnoticed link between coupled multi-species reaction diffusion models of invasion and a different class of models of invasion that involve moving boundary problems.