Mathematical modelling of movement and glioma invasion

Modelling movement is an important topic in fields ranging from ecology to medicine. In particular, glioma, an often fatal brain tumour is characterised by its diffuse invasion into the surrounding normal brain tissue, enabling the tumour to escape therapy. In this thesis we focus on the mathematical modelling of glioma and movement in general. We begin by exploring the basic structure of the brain, describing glioma classification and the hallmarks of cancer as well as reviewing mathematical models of glioma. In Chapter 2 an Ordinary Differential Equation model is presented to describe the interaction between healthy and mutated cells in vivo and vitro scenarios. The model is extended to a Partial Differential Equation to cover the spatial dynamics of interaction and the possibility of travelling wave solutions. A leading hypothesis suggests that malignant glioma cells switch between proliferating and migrating phenotypes, a mechanism known as the “go or grow” hypothesis. Although the molecular mechanisms that control this switch are uncertain, it is generally assumed to depend on micro-environmental factors. In Chapter 3 we propose a simple mathematical model based on the go or grow hypothesis for brain tumours (gliomas). The model describes the competition between healthy glial cells and malignant cells, with the latter subdivided into invasive and proliferating subpopulations. Simulation and stability analysis is performed for spatial and non-spatial versions of the model. The model incorporates two types of switch between migration and proliferation glioma cells: a constant switch form and a density dependent form. In Chapter 4 we present a framework for modelling a different characteristic movement lengths based on a biased random walk in response to external control species. We use the model to understand different strategies by which a population may locate some resource in its environment. Further we consider a pilot application to glioma, showing how it can be used to model movement along different brain structures. Finally we conclude with a brief discussion that summarises the main results and highlights directions for future work