Cell-Based Models of Tumour Angiogenesis

Michael John Plank

PhD Thesis

The University of Leeds, Department of Applied Mathematics

July 2003


It is now accepted that a tumour must induce the growth of new blood vessels, a process called angiogenesis, in order to grow beyond a diameter of approximately 2 mm. Knowledge of angiogenesis has increased massively over the last 30 years, enabling mathematical models of the process to be constructed and analysed. Initially, modelling was done at the continuum (i.e. cell density) level but, more recently, the appreciation that angiogenesis is an inherently discrete process has led to increasing research using individual cell-based models.

In this thesis, tumour angiogenesis is modelled at the individual cell level, using techniques based on the theory of reinforced random walks, and cell-based simulations are carried out. The purpose of this work is to develop the modelling techniques, to improve understanding of angiogenesis and to highlight potential strategies for therapeutic intervention.

Chapter 1 and Chapter 2 contain an introduction to the relevant biological and mathematical literature respectively. In Chapter 3, the theory of reinforced random walks is introduced. A result regarding the large time behaviour of one of the key governing equations is proved and a generalisation of the basic random walk model is presented, allowing a wider range of biological scenarios to be modelled.

In Chapter 4, the basic model for cell movement is developed in close conjunction with experimental data. Chemokinesis, chemotaxis, haptotaxis are included and their relative contributions to cell movement discussed. In Chapter 5, a model of tumour angiogenesis is formulated and used to simulate the growth of capillary networks. The model is also used to assess the potential of anti-angiogenic strategies. Chapter 6 develops a new model that includes the role of the angiopoietins, a recently discovered family of growth factors which have emerged as critical regulators of angiogenesis. In Chapter 7, a model that frees the cells from geometric constraints (a non-lattice model) is developed, and critically compared to existing lattice-based models.

Front matter

Chapter 1. Biological introduction

Chapter 2. Mathematical introduction

Chapter 3. Reinforced random walks

Chapter 4. A model of an experiment on endothelial cell migration

Chapter 5. A model of tumour angiogenesis and anti-angiogenic strategies

Chapter 6. A model for the role of the angiopoietins in angiogenesis

Chapter 7. Lattice and non-lattice models

Appendices and references