A mathematical analysis of a model for tumour angiogenesis

Abstract

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.

Modelling in vitro lung branching morphogenesis during development

“…Chaplain et al, 1995;Byrne, 1999; for a review see : Friedmann, 2004), it is assumed that the diffusing chemicals (here FGF) are in a quasi-steady state (i.e. the time derivative qcðx; tÞ=qt for the diffusing chemical vanishes) as diffusion processes are usually faster than growth.…”

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

“…Since the early work of (Balding and McElwain, 1985), many models of the process of capillary formation in response to a tumour-derived growth factor have been constructed. Examples include (Chaplain and Stuart, 1993;Chaplain et al, 1995;Byrne and Chaplain, 1995;Holmes and Sleeman;. These models are macroscopic in that they deal with continuous quantities, such as endothelial cell density, via a system of partial differential equations.…”

Stability Analysis of the Steady-State Solution of a Mathematical Model in Tumor Angiogenesis

“…(5) can be written as ∂ ∂ y (η y − ηG) = 0, which results in η y − ηG = 0, by the boundary conditions given by Eq. (2). By solving the last equation one obtains…”