Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis

TL;DR

This paper mathematically proves the existence of slow and fast traveling wave solutions in a tumor invasion model, explaining the origin of the experimentally observed interstitial gap through geometric and asymptotic analysis.

Contribution

It introduces a rigorous mathematical proof for traveling wave solutions in the Gatenby--Gawlinski tumor model, including the novel explanation of the interstitial gap's origin.

Findings

Existence of slow and fast traveling wave solutions.

Mathematical derivation of the interstitial gap.

Geometric interpretation linked to bifurcation analysis.

Abstract

In this manuscript, we prove the existence of slow and fast traveling wave solutions in the original Gatenby--Gawlinski model. We prove the existence of a slow traveling wave solution with an interstitial gap. This interstitial gap has previously been observed experimentally, and here we derive its origin from a mathematical perspective. We give a geometric interpretation of the formal asymptotic analysis of the interstitial gap and show that it is determined by the distance between a layer transition of the tumor and a dynamical transcritical bifurcation of two components of the critical manifold. This distance depends, in a nonlinear fashion, on the destructive influence of the acid and the rate at which the acid is being pumped.