Reducing phenotype-structured PDE models of cancer evolution to systems of ODEs: a generalised moment dynamics approach

TL;DR

This paper introduces a flexible mathematical framework that reduces complex phenotype-structured PDE models of cancer evolution to simpler ODE systems using moment generating functions, enabling easier analysis of phenotypic heterogeneity.

Contribution

It extends existing reduction methods by removing distribution shape assumptions, allowing for arbitrary order moment equations in cancer phenotypic evolution models.

Findings

The method generalizes previous approaches with no distribution shape constraints.

It effectively reduces PDE models to ODE systems for phenotypic moments.

Applications demonstrate the framework's flexibility and potential in cancer modeling.

Abstract

Intratumour phenotypic heterogeneity is nowadays understood to play a critical role in disease progression and treatment failure. Accordingly, there has been increasing interest in the development of mathematical models capable of capturing its role in cancer cell adaptation. This can be systematically achieved by means of models comprising phenotype-structured nonlocal partial differential equations, tracking the evolution of the phenotypic density distribution of the cell population, which may be compared to gene and protein expression distributions obtained experimentally. Nevertheless, given the high analytical and computational cost of solving these models, much is to be gained from reducing them to systems of ordinary differential equations for the moments of the distribution. We propose a generalised method of model-reduction, relying on the use of a moment generating function, Taylor series expansion and truncation closure, to reduce a nonlocal reaction-advection-diffusion equation, with general phenotypic drift and proliferation rate functions, to a system of moment equations up to arbitrary order. Our method extends previous results in the literature, which we address via two examples, by removing any \textit{a priori} assumption on the shape of the distribution, and provides a flexible framework for mathematical modellers to account for the role of phenotypic heterogeneity in cancer adaptive dynamics, in a simpler mathematical framework.