|dc.description.abstract||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.||en_US