In this talk we present both continuous and discrete mathematical models which describe two aspects of cancer growth: angiogenesis and tumour invasion. The continuous models consist of systems of coupled nonlinear partial differential equations describing the migratory response of cells (endothelial cells or tumour cells) to external chemical stimuli and with their substratum (extracellular matrix). We then use a discretized form of the partial differential equations to develop biased random walk models which enable us to track the motion of individual cells. This technique also enables us to incorporate into the model many processes that occur only at the cellular level. This technique of generating a biased random walk model from a PDE model (i.e. deriving a discrete model from a continuous model) has a much wider potential application to other biological and ecological systems e.g. animal migration.
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