Introduction

In this chapter, we further develop the description of resistance in quantitative terms. Chapter 4 detailed how the quantitative analysis of the directed and random mutation models for the development of drug resistance provided a method, the fluctuation test, to distinguish between them. Evidence was cited that favoured the random mutation model, implying that the onset of resistance would be a variable process and would occur prior to application of the drug. However, it is worth noting that many of the deductions that we will make will be equally true under a directed mutation model.

In the development that follows, we will assume that resistance arises via a random mutation process. The purpose of this chapter is to develop equations which describe the evolution of the resistant cell compartment as the tumour grows when resistance is caused by random mutations.

In order to discuss resistance in tumour systems one must first have some basic model for the functioning of the tumour system. A basic model for functioning will mean widely different things to different people, but here we will be concerned about the way in which the tumour system grows. We will be concentrating on a quantitative description of the growth of resistant and sensitive cell populations without detailed description of what regulates or stimulates such growth. We believe this approach is justified in that it permits concentration on the dynamics of the resistance process; however, as with any other model, such simplification is justified only if it does not lead to erroneous conclusions. We use the term model to imply some construct intended to describe reality.