Journal of Theoretical Medicine
Volume 4 (2002), Issue 4, Pages 251-270

A Mathematical Model of an In Vitro Experiment to Investigate Endothelial Cell Migration

1School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
2Molecular Medicine Unit St. James' University Hospital, Leeds LS9 7TF, UK

Received 10 October 2002; Accepted 10 June 2003

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Angiogenesis, the growth of new blood vessels from existing ones, is an important, yet not fully understood, process and is involved in diseases such as rheumatoid arthritis, diabetic retinopathy and solid tumour growth. Central to the process of angiogenesis are endothelial cells (EC), which line all blood vessels, and are capable of forming new capillaries by migration, proliferation and lumen formation. We construct a cell-based mathematical model of an experiment (Vernon, R.B. and Sage, E.H. (1999) “A novel, quantitative model for study of endothelial cell migration and sprout formation within three-dimensional collagen matrices”, Microvasc. Res. 57, 118–133) carried out to assess the response of EC to various diffusible angiogenic factors, which is a crucial part of angiogenesis. The model for cell movement is based on the theory of reinforced random walks and includes both chemotaxis and chemokinesis. Three-dimensional simulations are run and the results correlate well with the experimental data. The experiment cannot easily distinguish between chemotactic and chemokinetic effects of the angiogenic factors. We, therefore, also run two-dimensional simulations of a hypothetical experiment, with a point source of angiogenic factor. This enables directed (gradient-driven) EC migration to be investigated independently of undirected (diffusion-driven) migration.