Mathematical modelling of tumour-induced angiogenesis

In this post Lucy Hutchinson, a first-year D.Phil. student at the WCMB, describes her research in mathematical modelling of angiogenesis. Update: Lucy has won the Microsoft Research Project Prize for the best DTC short project in her cohort.


The Motivation
Most people have been affected by cancer in some way. The loss of friends and family members can be devastating, but can also fuel a passion for research into this mysterious killer. Approaches to cancer research are becoming more creative, and indeed interdisciplinary research is more and more popular. There are numerous processes involved in the growth and spread of cancer, which are researched across the globe. My research takes a mathematical approach to understanding the formation of a network of blood vessels which can provide a tumour with oxygen and nutrients to fuel its growth and spread. By manipulating this vascular network, could tumours be "starved" to death? Or could a more stable vessel network be useful in delivering specific chemotherapeutic drugs to the tumour?

The System
When a tumour reaches a size of around 2mm, some regions are starved of oxygen, or hypoxic. In response to hypoxia, tumours release growth factors which lead to the development of new blood vessels from those which already exist. This process is called angiogenesis. Angiogenesis is also observed in wound healing and fetal development.

The interplay between several growth factors, and sub-processes, such as vessel maturation, can lead to a whole host of different behaviours of the new vessels. In order to get an overview of how the growth factors and cell types interact, I drew a schematic diagram of the system in question.

Schematic diagram showing the processes involved in tumour induced angiogenesis

Vascular Endothelial Growth Factor (VEGF) is released from tumour cells in response to hypoxia; it binds to receptors on vascular endothelial cells, which stimulates these cells to produce another growth factor, Ang2, and also to proliferate. Immature endothelial cells produce Platelet Derived Growth Factor (PDGF), leading to the recruitment of smooth muscle cells, known as pericytes, which attach themselves to the new endothelial cells to plug holes in and increase the stability of new blood vessels. Pericytes also produce Ang1, and the ratio of the binding of Ang1 and Ang2, respectively, determines the maturity of the vessel network. This is not a simple system!

The Model
In fact, in order to model this system mathematically, I use a set of 13 ordinary differential equations to represent the evolution of growth factors, receptors and cell densities in time which are based on those in a recent model from Trachette Jackson and co-workers (1). My mathematical model does not include spatial terms, thus the growth of individual blood vessels is not modelled explicitly. Rather, we look at an average behaviour of the densities of the variables. A benefit of using a spatially averaged model is the computational efficiency compared to a more complex spatial model. A drawback is the lack of information of how well perfused the tumour actually is.

The model includes biochemical interactions taking place on the surface of the ECs that constitute the blood vessels, as well as pericyte recruitment, endothelial cell proliferation and maturity evolution of endothelial cells. These equations include binding and unbinding terms based on Michealis-Menten kinetics and the decay of ligands.

My industrial collaborators, Hoffman la Roche, will later provide me with data which could better inform my parameter choices, and this data is likely to be in a similar format of average values of ligands.

The Future
Using in silico investigations of the effects of different anti-angiogenic drug schedules and combinations could guide biological experiments, and lead to a more efficient treatment plan which targets tumour vasculature. In addition, the combination of anti-angiogenic drugs with chemotherapeutic drugs could be explored using computational techniques to make suggestions for experiments in the lab.

References

1. T. Jackson, G. Y. Koh, & X. Zheng, "A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, platelet-derived growth factor-B, and pericytes." Discret. Contin. Dyn. Syst. - Ser. B 18, 1109–1154 (2013).