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Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity

Solid tumours grow through two distinct phases: the avascular and the 
vascular phase. During the avascular growth phase, the size of the 
solid tumour is restricted largely by a diffusion-limited nutrient 
supply and the solid tumour remains localised and grows to a maximum 
of a few millimetres in diameter. However, during the vascular growth 
stage the process of cancer invasion of peritumoral tissue can and 
does take place. A crucial component of tissue invasion is the 
over-expression by the cancer cells of proteolytic enzyme 
activity, such as the urokinase-type plasminogen activator (uPA) and 
matrix metalloproteinases (MMPs). uPA itself initiates the activation 
of an enzymatic cascade that primarily involves the activation of 
plasminogen and subsequently its matrix degrading protein 
plasmin. Degradation of the matrix then enables the cancer cells to 
migrate through the tissue and subsequently to spread to secondary 
sites in the body. 
 
In this paper we consider a relatively simple mathematical model of 
cancer cell invasion of tissue (extracellular matrix) which focuses on 
the role of a generic matrix degrading enzyme such as uPA. The model 
consists of a system of reaction-diffusion-taxis partial differential 
equations describing the interactions between cancer cells, the matrix 
degrading enzyme and the host tissue. The results obtained from 
numerical computations carried out on the model equations produce 
dynamic, heterogeneous spatio-temporal solutions and demonstrate the 
ability of a rather simple model to produce complicated dynamics, all 
of which are associated with tumour heterogeneity and cancer cell 
progression and invasion.