A system of reaction diffusion equations arising in the theory of reinforced random walks
SIAM Journal on Applied Mathematics
Aggregation, blowup, and collapse: the abc's of taxis in reinforced random walks
SIAM Journal on Applied Mathematics
A stochastic model of tumor angiogenesis
Computers in Biology and Medicine
Multiscale cancer modeling: In the line of fast simulation and chemotherapy
Mathematical and Computer Modelling: An International Journal
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Tumour angiogenesis is the process whereby a capillary network is formed from a preexisting vasculature in response to tumour secreted growth factors (TAFs). The capillary network is largely composed of migrating endothelial cells (EC) which organise themselves into dendritic structures. In this paper, we model angiogenesis via the theory of reinforced random walks, whereby the chemotactic response of the endothelial cells to TAF and their haptotactic response to the matrix macromolecule fibronectin is accomplished through transition probability rate functions. These functions essentially assign directional probabilities for the movement of endothelial cells. Under a pseudosteady state hypothesis, whereby the TAF and fibronectin gradients are specified a priori, simulations in one, two, and three space dimensions are performed. Realistic elementary capillary networks are obtained and analysed. In addition, even without EC proliferation, the formation of microloops or anastomoses is observed. The model provides a ''bridge'' linking microcellular and macrocellular events. As such, it indicates a new step towards understanding tumour angiogenesis and possible mechanisms for its control.