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A Mathematical Model for the Diffusion of Tumour Angiogenesis Factor into the Surrounding Host Tissue

Chaplain, M. A. J. and Stuart, A. M. (1991) A Mathematical Model for the Diffusion of Tumour Angiogenesis Factor into the Surrounding Host Tissue. Mathematical Medicine and Biology, 8 (3). pp. 191-220. ISSN 1477-8599. https://resolver.caltech.edu/CaltechAUTHORS:20170612-104202241

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Abstract

Unless they are furnished with an adequate blood supply and a means of disposing of their waste products by a mechanism other than diffusion, solid tumours cannot grow beyond a few millimetres in diameter. It is now a well-established fact that, in order to accomplish this neovascularization, solid tumours secrete a diffusable chemical compound known as turnour angiogenesis factor (TAF) into the surrounding tissue. This stimulates nearby blood vessels to migrate towards and finally penetrate the tumour. Once provided with the new supply of nutrient, rapid growth takes place. In this paper, a mathematical model is presented for the diffusion of TAF into the surrounding tissue. The complete process of angiogenesis is made up of a sequence of several distinct events and the model is an attempt to take into account as many of these as possible. In the diffusion equation for the TAF, a decay term is included which models the loss of the chemical in the surrounding tissue itself. A threshold distance for the TAF is incorporated in an attempt to reflect the results from experiments of corneal implants in test animals. By formulating the problems in terms of a free boundary problem, the extent of the diffusion of TAF into the surrounding tissue can be monitored. Finally, by introducing a sink term representing the action of proliferating endothelial cells, the boundary of the TAF is seen to recede, and hence the position and movement of the capillaries can be indirectly followed. The changing concentration gradient observed as the boundary recedes may offer a possible explanation for the initiation of anastomosis. Several functions are considered as possible sink terms and numerical results are presented. The situation where the turnour. (i.e. the source of TAF) is removed is also considered.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1093/imammb/8.3.191DOIArticle
https://academic.oup.com/imammb/article-abstract/8/3/191/689707/A-Mathematical-Model-for-the-Diffusion-of-Tumour?redirectedFrom=fulltextPublisherArticle
Additional Information:© 1991 Oxford University Press. Received 18 May 1991 and in revised form 6 September 1991.
Subject Keywords:tumour angiogenesis factor; endothelial cells; free boundary; anastomosis
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ17
Issue or Number:3
Record Number:CaltechAUTHORS:20170612-104202241
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170612-104202241
Official Citation:M. A. J. CHAPLAIN, A. M. STUART; A Mathematical Model for the Diffusion of Tumour Angiogenesis Factor into the Surrounding Host Tissue. Math Med Biol 1991; 8 (3): 191-220. doi: 10.1093/imammb/8.3.191
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78102
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:12 Jun 2017 21:03
Last Modified:03 Oct 2019 18:05

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