Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation

TL;DR

This paper analyzes a PDE model of cancer cell growth structured by age and telomere length, revealing how telomere restoration influences tumor proliferation and stability of cell populations.

Contribution

It introduces a novel PDE model with non-local boundary conditions for clonal cancer evolution, analyzing solution existence, growth dynamics, and effects of telomere restoration and crowding.

Findings

Without telomere restoration, cell population extinguishes.

Telomere restoration leads to exponential growth.

Crowding effects influence stability and steady states.

Abstract

We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the clonal evolution model of cancer cell growth. This model has a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. We study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population extinguishes. With telomere restoration, exponential growth occurs in the linear model. We further characterise the specific growth behaviour of the cell population for special cases. We also study the effects of crowding induced mortality on the qualitative behaviour, and the existence and stability of steady states of a nonlinear model incorporating crowding effect. We present examples and extensive numerical simulations, which illustrate the rich dynamic behaviour of the linear and nonlinear models.