Mathematical modelling of malignant growth and invasion
The work presented in this thesis is concerned with the growth and development of malignancy. Such development can be thought of in terms of cell proliferation and associated morphological developments as well as in terms of active migration of malignant cells. Consequently this thesis can broadly be divided into two parts, one concerning growth dynamics in tumours, and the other active invasion of tissue by the malignancy. The first part of this thesis is concerned with the development of growth induced stresses within a multi-cell tumour spheroid (MCS), and associated structural changes. In particular, the growth and development of necrotic regions within a MCS is studied. Traditionally necrotic regions are considered to arise from the accumulation of necrotic cell debris, and as such form under chemically adverse conditions e.g. in hypoxic or nutrient deficient regions. However, it has been observed that the connection between such conditions and necrosis formation is not so simple. In particular, necrosis formation can precede or follow hypoxia. Therefore, in this thesis we examine a novel mechanism for necrosis formation, by allowing necrotic regions to arise under conditions of adverse mechanical stress. We consequently develop a model for spheroid growth in which necrosis forms in areas of mechanical tension but does not assume this formation a priori, and show that under the right conditions such a spheroid will support necrosis formation pre-hypoxia. Models in which the MCS is composed of a viscous, an elastic, and a viscoelastic material are all considered, and it is concluded that both biologically and mathematically a tumour spheroid is best modelled as a viscoelastic medium. The second part of this thesis is concerned with active migration of cells across a substrate via haptotaxis, and the application of this motility mechanism to glioma invasion of the central nervous system. A novel model for receptor mediated haptotaxis is developed which allows adhesion, proteolysis of extra-cellular matrix (ECM) components and subsequent migration of a cell to be modelled in a biochemically accurate manner. This single cell framework is then used to derive an average cell continuum velocity and flux, and these in turn are used to examine cell population migration via receptor mediated haptotaxis. Under appropriate limits the model presented is shown to reduce to a well known class of models, and as such provides a sound biochemical basis for these previous modelling attempts. Invasion of glioma cells into the central nervous system is studied with particular attention being paid to the effects of glioma-host interactions in modulation of migration velocity and interface shape. It is concluded that, under certain circumstances, an up-regulation of pro-migratory ECM components by the brain can inhibit glioma migration by slowing cell migration speed, and by sharpening the glioma-host interface. The phenomena of interface sharpening is seen as important, since gliomas often show diffuse boarders which present problems for their surgical resection within reasonable limits. The model outlined therefore suggests potential avenues for pre-surgical treatment which may prove very fruitful.