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Mathematical Medicine and Biology 2003 20(2):135-181; doi:10.1093/imammb/20.2.135
© 2003 by Institute of Mathematics and its Applications
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A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies

M. J. Plank1 and B. D. Sleeman1

1 School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

It is now well accepted that the growth of a tumour beyond approximately 2 mm in diameter is dependent on its ability to induce the growth of new blood vessels, a process called angiogenesis. This has raised hope that an anti-angiogenic treatment may be effective in the fight against cancer. Here we formulate, using the theory of reinforced random walks, an individual cell-based mathematical model of tumour angiogenesis in response to a diffusible angiogenic factor. The early stages of angiogenesis, in which endothelial cells (EC) escape the parent vessel and invade the extra-cellular matrix, are included in the model, as are the action of a proteolytic enzyme, EC proliferation and capillary branching and anastomosis. The anti-angiogenic potential of angiostatin, a known inhibitor of angiogenesis, is also examined. The capillary networks predicted by the model are in qualitative agreement with experimental observations. Proteolysis and proliferation are shown to be crucial for vascularization, whilst angiostatin is seen to be capable of limiting capillary growth.

Keywords: angiogenesis; angiostatin; chemotaxis; haptotaxis; reinforced random walk; tumour

Received 23 October 2002. Revised 10 April 2003.


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J R Soc Interface, August 6, 2008; 5(25): 813 - 834.
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