Angiogenesis: A Model of Cell Differentiation

Abstract

Angiogenesis is the formation of blood vessels, and is of great importance in the growth of tumours. Attempts have been made to desgin experiments in petri-dishes that mimic the 'Conditions of tumour growth. The first of the experiments is the 'matrigel' assay. Matrigel provides a matrix for the endothelial cells to grow on, and contains all the nutrients that the cells need. It is found that in the matrigel assay blood vessels didn't form, although some transient strucutres formed at early times in the experiment. The second experiment is the 'biocure' assay. In this experiment the petri dish is filled with both endothelial and fibroblast cells. The fibroblasts form a strucutal supporting network for the endothelial cells. Tubules resembling blood-vessels formed after about ten days in the biocure asssay.

The process of cell differentiation is thought to be important in the growth of blood vessels. Cells can sense that they are part of a blood vessel, and change their shape to form tubules. Also it is likely that they change their chemical messaging properties, and their abilities to bind to other endothelial cells.

A model is developed that describes cell differentiation, and separates cells into different classes. For simplicity the spatial distribution of cells in different classes is ignored. Using simple population dynamics, a set of coupled non-linear ODEs is developed to describe the dynamics of the system. The system is found to have two different long-time states, one corresponding to the formation of blood vessels and one where vessels did not form. The ratio of the cell proliferation rate to the cell maturity rate (the time it takes to realise that it is part of a blood vessel) is critical in determining which is the final state of the system.