USING MATHEMATICS TO STUDY SOLID TUMOUR GROWTH

1. Introduction. Hardly a day goes without the appearance of a press release claiming that a new cure for cancer has been discovered. Such media of interest is unsurprising given that cancer is now poised to overtake heart disease as the major cause of premature death in the Western World. Whilst many of these deaths are indirectly the result of improvements in healthcare (as life expectation rises the chances of succumbing to cancer increase), it is also true that treatment for many forms of cancer are still alarmingly ineffective. In the face of such news, biologists, clinicians, and pharmaceutical companies are now investing considerable effort in trying to improve the prognosis of patients diagnosed with cancer. In order to develop effective treatments, it is important to identify the mechanisms controlling cancer growth, how they interact, and how they can most easily be manipulated to eradicate (or manage) the disease. In order to gain such insight, it is usually necessary to perform large numbers of time-consuming and intricate experiments—but not always. Through the development and solution of mathematical models that describe different aspects of solid tumour growth, applied mathematics has the potential to prevent excessive experimentation and also to provide biologists with complementary and valuable insight into the mechanisms that may control the development of solid tumours. Whilst the application of mathematics to problems in industry has been an active area of research for many years, its application to medicine and biology is still a relatively new development. For example, most of the models of solid tumour growth were written in the last twenty years. In this paper, I review some of the major developments in modelling of solid tumour growth that have taken place over the past twenty years and indicate what I believe are the main directions for future mathematical research in this field. By reading the article I hope that you will, at least, learn some biology and, at best, be stimulated to learn more about the ways in which mathematics can help in the battle against cancer. The outline of the remainder of the paper is as follows. Section 2 contains a brief description of the key stages of cancer growth and introduces the biological terminology. Sections 3, 4, and 5 contain reviews of different, but interrelated , models of avascular tumour growth. The paper concludes in Section 6