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In this chapter we will begin to discuss resistance in quantitative terms. Chapter 3 contained a discussion of the molecular and cellular processes by which resistance arises; this will not be addressed here. The purpose here is to describe the development of resistance using formulae so that predictions regarding the distribution and onset of resistance can be made. The beautiful thing about formulae is that they tell you everything and nothing about the nature of a system. Everything, in that a correct complete formulae tells you exactly how the process will evolve and what affects it. Nothing, in that identical formulae may apply to quite different mechanisms of effect so that it is not possible, in general, to discern the ‘how’ from the structure of an equation. Quite different mechanisms of resistance may (or may not) have formulae describing their development that are functionally the same. Formulae describing a system may be derived, in general, in two ways.
One method is to build a model for the system of interest using known characteristics of the system. As an example consider building a model to describe the length of a steel spring. Hook's law indicates that the extension of a steel spring is proportional to the force applied to it. Also, steel expands upon heating so that we would expect the length to depend upon the ambient temperature. Other physical mechanisms with known modes of action may be postulated and piece by piece a model for the system constructed.
The phenomenon of resistance to environmental toxins has probably been present ever since life first evolved on earth. Any early living organism that happened to produce chemicals that were toxic to its competitors would have had a significant survival advantage in the struggle for existence. Competing species that failed to evolve a satisfactory protective mechanism against these toxins would have become extinct, but those that were able to circumvent successfully the toxins produced by other organisms would have been able to survive. Over the billions of years that life has evolved, organisms have developed an immense variety of chemical weapons against competitors and predators, who have in turn evolved mechanisms to permit their own survival.
The development of antibiotics and other chemical compounds for the treatment of infectious disease has been one of the triumphs of 20th century medicine. However, it is not clear at this point whether the gains made against many pathogenic organisms can be maintained. Strains of disease-producing bacteria that are resistant to most or even all of the available therapeutic agents are being increasingly encountered. The lay press is filled with stories about ‘super bugs’ that have ‘learned’ to overcome antibiotics. These popular accounts somehow manage to convey the picture of bacteria sitting down around a conference table and consciously planning their battle strategy against human beings. As if dealing with pathogenic microorganisms was not enough, the human race also has to contend with the evolution of its own aberrant cells, in the form of cancer, becoming resistant to the agents that are available for systemic treatment.
The underlying theme of this book is that there is a common thread to all of these problems.
Drug resistance – the phenomenon whereby malignant tumours lose their responsiveness to therapeutic agents – is recognized as being the major obstacle to be overcome during the systemic therapy of cancer. In the 1980s and early 1990s an enormous amount of information was developed concerning the molecular mechanisms in the cell that can lead to resistance. In addition, these studies have provided insights into why resistance development is such a common property of cancer cells compared with normal cells.
We have been particularly interested in the processes that underlie the evolution of drug resistance within malignant cell populations and in the mathematical and biological models that have been developed to describe these processes. These models provide a greater intuitive understanding of drug resistance as well as providing insights into the more effective use of our available therapeutic agents.
Mathematical relationships in models may tell us little about specific mechanisms involved in various processes but they are often highly generalizable in terms of their inferences and usually lead to testable hypotheses.
Since we are concerned in this book with quantitative and mathematical models, any review of our own and related studies has to include some of the mathematics involved. The authors are aware of the reaction that is likely to engender in many readers (clinicians and biologists in particular) and the advice that was given to Professor Hawking (‘Each equation in a book decreases its sales by half’) as well as the assessment of the schoolboy diarist and commentator, Nigel Molesworth (‘All maths is friteful and mean 0, unless you are a grate brane’).
Much of this book has supported the notion that cancer cells resistant to anti-cancer drugs arise via random undirected mechanisms. The basis of this conclusion is not an exhaustive study of in vivo and in vitro tumour systems treated with all available drugs but a matter of generalization from a large body of evidence including the results of experiments of the fluctuation test type. As our review in Chapter 4 showed, a positive result in a fluctuation test implies that there exists a source of variation that cannot be attributed to the sampling variation expected under the directed mutation model. In terms of two possible processes that may result in resistance, directed mutation and random mutation, the fluctuation test can point to the existence of random mutation but not to the absence of directed mutation. Furthermore, since the essence of the fluctuation test is quantitative rather than qualitative, results from such experiments apply to the most common mechanism for the development of resistance under the experimental conditions. It is well known in cancer that many distinct genetic alterations can lead to resistance; as a result, the fluctuation test will provide information about the one that occurs most frequently (unless two or more mechanisms give rise to resistant cells with a similar frequency, in which case results will relate to the combination of these mechanisms).
The question of possible directed mutation in E. coli
Many investigators have considered the origin of resistant subtypes in different systems.
The random mutation model of drug resistance explored in Chapters 5 and 6 can be used to derive a number of unambiguous predictions about the behaviour of clinical tumours in response to chemotherapy. These are basically the same as the ones that were developed from the consideration of the experimental tumours described in Chapter 5 but with the necessary qualifications. The clinical malignancies are much more heterogeneous and complex than the experimental systems. One important distinction is that it is not possible to stage clinical tumours with anything like the accuracy that can be achieved in the laboratory. The number of actual clonogenic cells in a transplanted tumour can be measured with precision whereas only a very rough approximation can be made for clinical malignancies. The strongest predictions made by the model include:
there will be an unpredictable variation in response to treatment in what appears to be identical cases of malignancy
there will be an inverse relationship between tumour mass and likelihood of cure
combination chemotherapy will be superior to single-agent treatment with respect to the production of cures
the sequence of drug administration influences outcome.
There are a number of other inferences that can be made from the model. The above predictions, however, can be considered to be strong predictions which easily lead from the model and which can be subjected to experimental and clinical tests.
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.
Analysis of the process of drug resistance indicates that it appears to be only too easy for malignancies to generate many varieties of drug-resistance mechanisms. Once a tumour has reached a certain critical size a combination of large numbers of candidate cells together with genetic instability will ensure that the collective molecular heterogeneity of the neoplasm will be immense. In retrospect, the surprising thing is not that cancers are difficult to treat with drugs but that some are highly sensitive to drug therapy and indeed are potentially curable. Although the addition of new cytotoxic agents provides a reasonably steady incremental improvement in cancer therapy outcome, these increments are small and are for the most part confined to the classes of tumour that are known already to be drug responsive. Malignancies such as pancreatic carcinoma, renal cell cancer, melanoma and nonsmall cell lung cancer are only minimally responsive to cytotoxic agents and it seems unlikely that random drug searches will yield single agents that are going to be much more effective than the drugs we currently possess. A major problem with these types of cancer seems to be the large number of different multidrug resistant mechanisms that they express. Modulation of one or two of these mechanisms is unlikely to be sufficient to render the advanced forms of the tumour curable. The greater the number of discreet mechanisms that need to be disabled then, of course, the more logistically cumbersome the treatment protocol becomes.
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.
Much of what has been learned concerning the properties of cancer cells has been developed from studies of a variety of cell lines that have been adapted for growth in tissue culture or in appropriate experimental animals. A variety of mammalian (chiefly rodent) and human cancer cell lines have been developed for this purpose. It has to be kept in mind that the properties of these highly selected experimental systems may differ from those that one might expect to find occurring in primary tumours in patients. Despite this caveat, it is apparent that many principles that have been learned from the experimental systems have been valuable in the understanding of human malignancy.
Any line of tumour cells that has adapted to growing in tissue culture or through serial transplantation in animals will have been subject to an extremely rigorous selection process. Normal (i.e. nonmalignant) fibroblasts in tissue culture will die out after they have undergone a number of sequential cell divisions (approximately 50). This appears to be the case no matter how carefully the culture conditions are established. This also appears to be true for many cancer cells in that there appears to be an upper limit to the number of sequential divisions they will undergo before becoming senescent and dying. Some tumour cells, however, become ‘immortalized’ and they will replicate indefinitely in the right type of environment. Recently, it has been suggested that an important step in the process whereby cells become immortal is related to the expression of the enzyme telomerase.
There are approximately 60 different chemical compounds generally available for the treatment of cancer (not including hormones or biological response modifiers). They are of diverse structure and from a variety of sources. They do not readily fit into a single classification system and are usually described partly on the basis of their chemical structure, partly on their primary source (fungi, plant, etc.) and partly on what is thought to be their general mechanism of action (antimetabolite, alkylating agent, etc.) (Table 3.1). We can generalize by stating that all of the drugs appear to exert their therapeutic effect by interfering with the processes involved in cell division. This interference results in the cell being physically disrupted or rendered permanently sterile. We can further say that cancer cells have the potential to become resistant to any of the drugs in our inventory and the cell can, moreover, express resistance to a great many agents simultaneously. Although we are not aware of the experiment actually having been done, it seems more than probable that an individual cancer cell could display resistance to all 60 available drugs concurrently. As we will see, this capacity to express resistance to many agents is, in part, related to the fact that there are a large number of mechanisms that once expressed by the cell will generate broad degrees of cellular resistance.
It would be well beyond the scope of this book to discuss in any detail the molecular changes that have been described for all of the various cytotoxic agents. However, we will mention the general processes involved in drug action and how these may be modified in the drug-resistant state.
Drug resistance in cancer, whereby a proportion of cancer cells evades chemotherapy, poses a profound and continuing challenge for the effective treatment of cancer. The principles underlying the biological mechanisms behind this phenomenon are clearly understood and explained in this volume. However, a deeper understanding of drug resistance requires a quantitative appreciation of the dynamic forces that shape tumour growth, including spontaneous mutation and selection processes. The authors seek to explain and to simplify these complex mechanisms, and to place them in a clinical context. Clearly explained mathematical models are used to illustrate the biological principles and provide an insight into tumour development and the effectiveness and limitations of drug treatment. It is suitable for those with a non-mathematical background and aims to enhance the effectiveness of cancer therapy.