Asymptotic Analysis of a Two-Phase Model of Solid Tumour Growth

TL;DR

This paper analyzes the stability of a two-dimensional tumor growth model based on a one-dimensional continuum framework, using asymptotic methods to derive conditions for instability and supporting findings with numerical simulations.

Contribution

It extends a one-dimensional tumor growth model to two dimensions and derives analytical stability conditions using asymptotic analysis.

Findings

Derived an asymptotic limit for perturbations in tumor growth

Established analytical conditions for tumor growth instability

Validated stability analysis with numerical simulations

Abstract

We investigate avascular tumour growth as a two-phase process consisting of cells and liquid. Based on the one-dimensional continuum moving-boundary model formulated by (Byrne, King, McElwain, Preziosi, Applied Mathematics Letters, 2003, 16, 567-573), we defined boundary conditions for the analogous model of tumour growth in two dimensions. We investigate linear stability of one dimensional time-dependent solution profiles in the moving-boundary formulation of a limit case (with negligible nutrient consumption and cell drag). For this, we obtain an asymptotic limit of the two-dimensional perturbations for large time (in the case where the tumour is growing) by using the method of matched asymptotic approximations. Having characterised an asymptotic limit of the perturbations, we compare it to the time-dependent solution profile in order to analytically obtain a condition for instability. Numerical simulations are mentioned.