Multiscale modelling of cancer invasion and metastasis

: integrating stochastic and continuum approaches

Abstract

By integrating key biological mechanisms at multiple scales, this thesis focuses on the development and analysis of mathematical models for cancer growth, invasion, and metastasis. We introduce a hybrid modelling framework that combines individual-based modelling techniques to capture the behaviour of migrating cancer cells with continuum approaches that describe the collective dynamics of tumour progression. This hybrid approach allows for a more comprehensive representation of tumour heterogeneity, phenotypic plasticity, and tumour-environment interactions.

One of the primary goals of this thesis is to investigate the mathematical connections between microscopic and macroscopic scales in cancer modelling. The invasion-metastasis cascade, a multiscale process involving epithelial-like cancer cells (ECCs) that drive the growth of the tumour and mesenchymal-like cancer cells (MCCs) that more migrate, is analysed through hybrid modelling techniques. This thesis examines how stochastic differential equation (SDE) models—at the individual cell level—correspond to macroscopic partial differential equation (PDE) models of tumour invasion, providing, hence, a mathematical foundation for linking cellular-scale interactions to large-scale tumour dynamics.

Analytical and numerical investigations indicate how heterogeneity at the single-cell level influences macroscopic tumour behaviour. Numerical simulations indicate the impact that phenotypic plasticity has on the invasion speed, the morphology of the tumour, and the response to environmental stimuli. Comparisons with experimental data validate the models' predictive capabilities and demonstrate its potential for exploring targeted therapeutic strategies.

The main contribution of this thesis to the field of Mathematical Oncology is through the development of a generic modelling framework that captures key aspects of tumour progression. By linking individual-cell behaviour with macroscopic tumour evolution, the models developed in this thesis shed mathematical light into cancer invasion mechanisms and propose new directions for future modelling efforts. These findings underscore the importance of multiscale mathematical modelling in the effort to understand tumour dynamics and develop improved strategies for cancer treatment.

Date of Award	2 Dec 2025
Original language	English
Awarding Institution	University of St Andrews
Supervisor	Nikolaos Sfakianakis (Supervisor) & Mark Chaplain (Supervisor)