# American Institute of Mathematical Sciences

2006, 1(3): 399-439. doi: 10.3934/nhm.2006.1.399

## Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity

 1 The SIMBIOS Centre, Division of Mathematics, University of Dundee, Dundee DD1 4HN, United Kingdom, United Kingdom

Received  February 2006 Revised  June 2006 Published  July 2006

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.
Citation: M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399
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