Exact traveling wave solutions of one-dimensional models of cancer invasion

TL;DR

This paper derives exact analytical traveling wave solutions for one-dimensional mathematical models of cancer invasion, providing insights into tumor growth dynamics and validating results with numerical simulations.

Contribution

It introduces exact solutions for a system of coupled nonlinear PDEs modeling cancer invasion, extending previous numerical findings with explicit analytical expressions.

Findings

Exact traveling wave solutions derived for the model

Solutions match numerical profiles for short time intervals

Provides analytical insight into tumor invasion dynamics

Abstract

In this paper we consider continuous mathematical models of tumour growth and invasion based on the model introduced by Chaplain and Lolas \cite{Chaplain&Lolas2006}, for the case of one space dimension. The models consist of a system of three coupled nonlinear reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the tissue. For these models under certain conditions on the model parameters we obtain exact analytical solutions in terms of traveling wave variables. These solutions are smooth positive definite functions for some of which whose profiles agree with those obtained from numerical computations \cite{Chaplain&Lolas2006} for not very large time intervals.