Introduction

We are coming to what some readers may find to be the most difficult section of the book because we will attempt to synthesize a number of the mathematical developments we have described previously into a more complex model that is intended to conform more closely to the behaviour of clinical malignancies. The most important elements in this synthesis will be the basic random mutation model of resistance (Chapters 4 and 5) and the stem cell model of tumour growth (Chapter 2). We will describe in more detail the birth/death processes that were introduced in Chapter 2 and indicate how they impact on the issue of drug resistance and the more general question of tumour heterogeneity.

It should be kept in mind that birth/death events are more than just convenient mathematical abstractions for they can provide a mathematical description of the effects of molecular processes that regulate movement through the cell cycle or signal differentiation and apoptosis.

In Chapter 5 we introduced and discussed the random mutation model for resistance to an anticancer drug. This model predicted that tumours which start sensitive would, as they grow, convert to drug resistance by the spontaneous evolution of drug-resistant cells whose population expands at a rate that exceeds that of the tumour as a whole. This model was developed within a framework in which cells divide with unlimited potential (stem cells) forming new stem cells at each expansion. Comparison with data from in vivo tumour systems showed that this model accurately simulated and explained the pattern of animal survival seen in some experiments.