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The development of resistance to anticancer agents Coldman, Andrew James 1986

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THE DEVELOPMENT OF RESISTANCE TO ANTICANCER AGENTS by ANDREW JAMES COLDMAN B.Sc, The University of Sussex, 1974 M.A., The University of Western Ontario, 1975 Dip. Ep., McGill University, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF STATISTICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1986 ©Andrew James Coldman, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. A.J. Coldman Department of S t a t i s t i c s The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e 1st A p r i l , 1987 DF-firVft-n i i ABSTRACT The mechanism of resistance of tumor c e l l s to chemotherapeutic agents i s explored using p r o b a b i l i s t i c methods where i t i s assumed that r e s i s t a n t c e l l s a r i s e spontaneously with a defined frequency. The resistance process i s embedded i n a discrete time Markov branching process which models the growth of the tumor and contains three seperate c e l l types: stem, t r a n s i t i o n a l and end c e l l s . Using the asymptotic properties of such models i t i s shown that the proportion of each type of c e l l converge to constants almost surely. It i s shown that the parameters r e l a t i n g to stem c e l l behaviour determine the asymptotic behaviour of the system. I t i s argued that for b i o l o g i c a l l y l i k e l y parameter values, cure of the tumor w i l l occur i f , and only i f , a l l stem c e l l s are eliminated. A model i s developed for the a c q u i s i t i o n of resistance by stem c e l l s to a single drug. P r o b a b i l i t y generating functions are derived which describe the behaviour of the process a f t e r an a r b i t r a r y sequence of drug treatments. The p r o b a b i l i t y of cure, defined as the p r o b a b i l i t y of ultimate e x t i n c t i o n of the stem c e l l compartment, i s characterised as the cen t r a l quantity r e f l e c t i n g the success of therapeutic intervention. Expressions for t h i s function are derived for a number of experimental s i t u a t i o n s . The e f f e c t s of v a r i a t i o n i n the parameter values are examined• The model i s extended to the case where two anticancer drugs are av a i l a b l e and formulae for the p r o b a b i l i t y of cure are developed. The problem of therapeutic scheduling i s examined and under s i t u a t i o n s where drugs are of "equal" effectiveness, but may not be given together, i t i i i i s shown that the mean number of tumor c e l l s i s minimised by sequential a l t e r n a t i o n of the drugs. The models are applied to data c o l l e c t e d on the L1210 leukemia treated by the drugs Cyclophosphamide and Arabinosylcytosine. In both cases the analysis of the data provide evidence that r e s i s t a n t c e l l s a r i s e spontaneously with a frequency of approximately I O - 7 per d i v i s i o n . When applied to human breast cancer, the model indicates that neo-adjuvant therapy i s u n l i k e l y to greatly influence the l i k e l i h o o d that the patient w i l l die from the growth of drug-resistant c e l l s . iv TABLE OF CONTENTS page Abstract i i List of Tables v i List of Figures ix Acknowledgement x Chapter 1. Introduction 1 1.1 Resistance in Other Biological Systems 7 Chapter 2. A Model for Tumor Growth 12 2.1 Properties of the Growth Model 16 Chapter 3. The Development of Resistance to a 31 Single Chemotherapeutic Agent 3.1 Calculating the Probability Generating 33 Function 3.2 Effects of Drug Treatment 39 3.3 Effects on the Normal Tissue 43 3.4 Modelling Treatment Effects on the Tumor 44 Cells 3.5 Summarizing Treatment Effects 47 3.6 Conditioning on N(t) - Approximation 1 53 3.7 Conditioning on N(t) - Approximation 2 59 3.8 Conditioning on N(t) - Approximation 3 71 3.9 Comparing the Three Approximations 80 3.10 Variation in the Resistance Parameters a, |3 83 and y V page Chapter 4. Resistance to Two or More Chemotherapeutic Agents 89 4.1 P r o b a b i l i t y Generating Function for Double 91 Resistance 4.2 Modelling Treatment E f f e c t s 101 4.3 Optimal Scheduling 110 4.4 Optimum Scheduling for Two Equivalent Agents 113 4.5 Discussion 130 4.6 V a r i a t i o n i n the Mutation Rates 132 4.7 Extensions 144 Chapter 5. Applications of the Theory 147 5.1 The E f f e c t of Treatment Strategies on C u r a b i l i t y 14 8 5.2 F i t t i n g the Model to Experimental Data 178 5.3 Neo-adjuvant Chemotherapy 194 Chapter 6. Conclusion 210 Bibliography 219 Index of Notation 222 v i LIST OF TABLES Table I Transitions Occurring i n the Stem C e l l Compartment which have P r o b a b i l i t y of Order At for the Interval [t, t+At] for the I n i t i a l State ( i , j ) . Table II The P r o b a b i l i t y of Diagnosis D i s t r i b u t i o n g(J)=a 1q J 1+a 2q2=P{N=j} where E[N]=10 1 0. Table III Table IV Table V Table VI Table VII Table VIII Table IX Table X The P r o b a b i l i t y of Cure, Expected Number and Standard Deviation of the Number of Resistant C e l l s . Transitions Occurring i n the Stem C e l l Compartment which have P r o b a b i l i t y of Order At for the Interval (t,t+At) for the I n i t i a l State {R Q(t)=i, R 1 ( t ) = j , R 2(t)=k, R 1 2 ( t ) - A } . Parameter Values for Simulations Presented i n Tables VI-X. Pr o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment for Parameter Values given i n Table V for Strategy S ( l ) = ( l , l , l , l , l , l , l , l ) . P r o b a b i l i t y of Ex t i n c t i o n of Cells at Times of Treatment for Parameter Values given i n Table V for Strategy S(2)=(1,1,1,1,2,2,2,2). P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment f o r Parameter Values given i n Table V for Strategy S ( 3)-(l , 2,l , 2,l , 2,l , 2 ) . P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment for Parameter Values given i n Table V for Strategy S(4)=(3,3,3,3,3,3,3,3) and ,= 10 — \ %i i=TCi o = 10 -2, TI 2 i2 =l< 7C 3,0" l3,l _ l l3,2" P r o b a b i l i t y of Ex t i n c t i o n of C e l l s at Times of Treatment for Parameter Values given i n Table V for Strategy S(4)=(3,3,3,3,3,3,3,3) and u 3 , 0 = 1 0 ~ 2 ' *3 , r u3,2" 1 0 » u3,12 = 1-page 34 77 81 92 160 161 162 163 164 165 v i i Table XI Parameter Values for Simulations Presented i n Table XII. page 166 Table XII P r o b a b i l i t y of Ex t i n c t i o n of C e l l s at Times of 167 Treatment for Parameter Values given i n Table XI for the Strategies S ' ( l ) = ( l , 2 , l , 2 ) , S'(2)=(l,2,2,l) and S'(3)=(l,2,2,2). Table XIII Mass Points for the Approximation to the Beta- 173 D i s t r i b u t i o n with E[Aj[]=S.D. [A j L]=5xlO - 5. Table XIV P r o b a b i l i t y of Ex t i n c t i o n of C e l l s at Times of 174 Treatment for Parameter Values given i n Table V for the Strategy S(l)=(l,1,1,1,1,1,1,1) where the Mutation Rates are Equal with P r o b a b i l i t y 1 and have the D i s t r i b u t i o n given i n Table XIII. Table XV P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of 175 Treatment for Parameter Values given i n Table V for the Strategy S(2)=(l,l,l,l,2,2,2,2) where the Mutation Rates are Equal with P r o b a b i l i t y 1 and have the D i s t r i b u t i o n given i n Table XIII. Table XVI P r o b a b i l i t y of Ex t i n c t i o n of C e l l s at Times of 176 Treatment for Parameter Values given i n Table V for the Strategy S(3)=(l,2,l,2,l,2,l,2) where the Mutation Rates are Equal with P r o b a b i l i t y 1 and have the D i s t r i b u t i o n given i n Table XIII. Table XVII Response of Int r a p e r i t o n e a l l y (IP) and Intravenously 191 (IV) Innoculated L1210 Leukemia to Single Doses of Cyclophosphamide. Table XVIII Observed (Obs) and Predicted Values for the P r o b a b i l i t y 192 of Cure for IP and IV Innoculated L1210 Leukemia Treated with Cyclophosphamide using the Maximum L i k e l i -hood Parameter Estimates for Fixed Rates (Predl) and for Variable Rates (Pred2). Table XIX Observed and Predicted Rates of Cure for Intravenously Innoculated L1210 Leukemia Treated with Repetitive Courses of Ara-C. 193 Table XX D i s t r i b u t i o n of Post-Surgical Tumor Burden for 716 Cases of Breast Cancer as a Function of Nodal Status and Menopausal C l a s s i f i c a t i o n . Table XXI Table of Values of C±..k, the P r o b a b i l i t y of Be r n o u l l i Parameter 0 ,^ for the Six Prognostic Categories. 205 206 v i i i page Table XXII Predicted D i s t r i b u t i o n of Residual Tumor Burden 207 a f t e r Surgery using the Values of 0^, and C^.^ given i n Table XXI. Table XXIII Predicted C u r a b i l i t y of Breast Cancer for Pre- 208 menopausal Disease as a Function of a and the Increase i n C u r a b i l i t y Associated with an Extra (Neo-adjuvant) Cycle. Table XXIV Predicted C u r a b i l i t y of Breast Cancer for Post- 209 menopausal Disease as a Function of a and the Increase i n C u r a b i l i t y Associated with an Extra (Neo-adjuvant) Cycle. i x LIST OF FIGURES page Figure 1. P r o b a b i l i t y of Cure for Approximations 1 and 2. 63 Figure 2. Schematic Representation of the Two Phase Growth Process for Sensitive C e l l Growth Used i n Approximation 2. 66 Figure 3. P r o b a b i l i t y of Cure Rate i s Present. when Va r i a t i o n i n the Mutation 87 Figure 4. P r o b a b i l i t y of Cure when C e l l Loss i s Present. 107 Figure 5. P r o b a b i l i t y of Cure when Va r i a t i o n i s Present - 1. 138 Figure 6. P r o b a b i l i t y of Cure when V a r i a t i o n i s Present - 2. 139 Figure 7. P r o b a b i l i t y of Cure when Va r i a t i o n i s Present - 3. 141 Figure 8. P r o b a b i l i t y of Cure when Va r i a t i o n i s Present - 4. 142 Figure 9. Expected Numbers of Ce l l s for Treatment Strategy S ( l ) . 168 Figure 10. Expected Numbers of Ce l l s for Treatment Strategy S(2). 169 Figure 11. Expected Numbers of Cel l s for Treatment Strategy S(3). 170 Figure 12. Expected Numbers of C e l l s for S(4) - 1. 171 Figure 13. Expected Numbers of Ce l l s for S(4) - 2. 172 X ACKNOWLEDGEMENT I would l i k e to acknowledge the assistance and support of a l l the members of my research committee, Drs. P. Band, J . Goldie, A. Marshall and J. Petkau, i n the development of th i s i n t e r d i s c i p l i n a r y t h e s i s . In p a r t i c u l a r I wish to thank John Petkau for his c r i t i c i s m and suggestions on e a r l i e r drafts of t h i s manuscript and for his considerable Input which help shaped i t s f i n a l form. I acknowledge my debt to James Goldie, without whom th i s research would not have been possible. I wish to thank both Howard Skipper and Roger Day for th e i r i n t e r e s t i n th i s research which was both stimulating and g r a t i f y i n g . I wish to thank Shirl e y Morton for her excellent work i n typing t h i s manuscript and i t s e a r l i e r d r a f t s . I also wish to acknowledge the assistance of the Cancer Control Agency of B r i t i s h Columbia, who have a c t i v e l y supported my work in this area. -1-1. INTRODUCTION Resistance i s a general term i n cancer therapy meaning i n s e n s i t i v i t y to treatment [1]. This term can be applied to any of the three arms of cancer therapy, surgery, radiotherapy and chemotherapy, but i t i s usually reserved for the l a t t e r two. Here we w i l l be concerned pr i m a r i l y with resistance to chemotherapy which has recently assumed greater importance with the increased use of this modality i n c l i n i c a l cancer therapy. Resistance may be either absolute (no e f f e c t of the drug) or p a r t i a l (reduced e f f e c t of the drug). In the discussion which follows we w i l l consider the development of resistance, whether p a r t i a l or absolute, to chemotherapeutic agents. Resistance to cancer chemotherapy i s known to be m u l t i f a c t o r i a l and there i s no reason to believe that a l l forms have yet been i d e n t i f i e d . Probably the simplest way i n which resistance can a r i s e i s that of pharmacologic sanctuary. In t h i s s i t u a t i o n tumor c e l l s a r i s e , or are transported to a s i t e which i s not accessible to the drug by the usual route of administration. For example, a number of drugs administered intravenously w i l l not gain access to the brain. A second mechanism i s the metabolic conversion of the drug to a non-active form. For example, the half l i f e of 5 - f l u o r o u r a c i l (5-FU), a drug commonly used i n the treatment of g a s t r o - i n s t e s t i n a l malignancies, has a measured systemic h a l f - l i f e of six to twenty minutes [2]. Therefore, the tumor exposure time to 5-FU administered by i n j e c t i o n i s l i k e l y to be short and many c e l l s may be expected to escape unaffected. C e l l s located d i s t a n t l y from the c a p i l l a r y bed are known to experience lower drug l e v e l s than those which are c l o s e r . Therefore, tumors may display resistance to -2-chemotherapy because many c e l l s are not exposed to therapeutic doses of the drug. Another mechanism of tumor resistance can re s u l t from the phase s p e c i f i c or p r e f e r e n t i a l a c t i v i t y of a drug. Tumor c e l l s , l i k e other d i v i d i n g c e l l s , move through the various phases of the c e l l c ycle, G ^ , S (synthesis), and M (mitosis) where G ^ and G ^ are intervening periods between the states of chromosomal synthesis and c e l l d i v i s i o n . In some cases drugs act p r e f e r e n t i a l l y or exclus i v e l y on the c e l l s i n p a r t i c u l a r phases of the c e l l cycle and thus c e l l s i n other phases w i l l appear r e s i s t a n t . Related to this i s the r e l a t i v e i n s e n s i t i v i t y of the state Go which i s used to designate viable c e l l s not a c t i v e l y i n the c e l l c y c l e . C e l l s i n th i s state are non- p r o l i f e r a t i n g and considerably less s e n s i t i v e to chemotherapeutic agents than a c t i v e l y p r o l i f e r a t i n g c e l l s . C e l l s i n GQ may l a t e r re-enter the c e l l cycle and continue to p r o l i f e r a t e . Therefore tumors with substantial numbers of c e l l s i n protected phases w i l l not respond to chemotherapy. This i s the main mechanism by which the normal hemopoietic system (which has many c e l l s i n GQ) survives the e f f e c t s of chemotherapy aimed at a tumor. A further, and important type of resistance i s the existence of a subpopulation of c e l l s within the tumor population on which administration of an agent has no or reduced e f f e c t when compared to the rest of the tumor c e l l s . This resistance i s i n t r i n s i c to the c e l l s themselves and p e r s i s t s when such c e l l s are transferred to another host. In - v i t r o studies of r e s i s t a n t c e l l s have associated the development of resistance with genetic and biochemical differences within these c e l l s when compared to the parent s e n s i t i v e c e l l s . One other form of resistance should be mentioned. Certain drugs -3-show virtually no effect in some types of tumor, whilst they are extremely active in others. Similar variation in response is also seen in different classes of non-tumor cells and i t is worth emphasizing the obvious that cells, whether normal or malignant, have varying biochemical properties, and this can be expected to influence their sensitivity to a drug. It is our objective here to develop a mathematical model for the growth of c e l l populations where individual cells show the intrinsic differential sensitivity to chemotherapy. It is recognised that we w i l l have to use data from passaged animal tumors even though what we desire to model is the therapy of human malignancy. This phenomenon is of interest since the existence of resistant cells w i l l obviously influence the short term and long term behaviour of tumors treated with chemotherapy. Before proceeding further i t is wise to ask whether the mechanism we intend to model is present in human malignancy to any significant extent. Consider the following common c l i n i c a l observation. A tumor is treated with an agent (or several agents simultaneously) and appears to shrink. It may even be no longer c l i n i c a l l y detectable. Therapy is continued, but i t later becomes obvious that the tumor is growing again. Experience indicates that continued therapy with the same agents is fruitless as the tumor is now c l i n i c a l l y resistant to these agents. Can any of the previous mechanisms explain this observation? If there are increases in proportion of cells in Go, or in the average intermitotic time (the time to go through the c e l l cycle), then this would imply that the tumor has become resistant since cells w i l l -4-spend a longer time in resistant phases of the c e l l cycle. However, i t would also imply that the growth rate of the tumor would slow considerably, which does not appear to be the case [3]. Also, i f there are fewer cycling cells or the cells have longer cycle times then proportionately fewer cells are in a sensitive state but also fewer cells need be k i l l e d to control growth. Although i t is not necessarily true that these two effects w i l l move in tandem precisely compensating for one another, they must tend to, to some degree. From this reasoning, and the lack of observation of significantly slower growth rates, i t seems reasonable to conclude that this mechanism is not a major cause of tumor regrowth during treatment. Changes in the host, so that the drug is more rapidly metabolised, also seems an unlikely explanation for tumor regrowth during treatment. Such changes would also imply that the toxic effect frequently seen in normal tissue should decline as the treatment continues, but this does not seem to be the case. Neither the mechanism of pharmacologic sanctuary or total resistance would seem to apply as the tumor responded in the f i r s t instance and is regrowing at the original site. Both distance from the capillary bed or the existence of resistant cells provide a plausible self-consistent explanation for the observation of relapse during ( i n i t i a l l y successful) therapy. Both predict the existence of a subpopulation of resistant cells which upon the application of therapy w i l l be "revealed" and repopulate the tumor. The regrowing tumor can then be expected to be resistant to the drug. Studies of both experimental and human malignancy have shown that resistant tumors contain cells which exhibit structural differences from -5-the original sensitive c e l l s . Therefore intrinsic cellular resistance provides a logical explanation of this commonly observed phenomenon which is consistent with observation in passaged animal tumors. Resistance to chemotherapy is thus an important concept whose understanding may better explain the response of tumors to chemotherapy. The variability in response (either survival time or proportion cured) to a fixed treatment protocol of an inbred strain of animals implanted with the same tumor line suggests that the development of resistance involves some random process. In what follows we w i l l thus use stochastic models for tumor growth and the development of resistance. Earlier work by Goldie and Coldman [4], in which drug resistant mutants were assumed to arise spontaneousely, provided a basic model of this phenomenon. This model provided "quantitative" predictions about the behaviour of tumors which are in broad agreement with experience from experimental and c l i n i c a l chemotherapy [5]. However, this basic model could not be f i t to much experimental data because i t assumed: (i) that there was no tumor c e l l differentiation or loss, ( i i ) that the drug was only applied once, ( i i i ) that a l l sensitive cells were ki l l e d by the drug and (iv) that resistant cells were absolutely resistant. In Chapter 3 a more general model w i l l be presented in which these assumptions are relaxed. This model w i l l then be fitted to experimental data and the results presented in Chapter 5. For human cancer the age of the tumor is seldom known. In order to use this model of resistance (which is para-meterized by time) in human data this parameter must be removed and three methods of accomplishing this, involving differing assumptions, are discussed in Chapter 3. -6-Experience with both experimental and c l i n i c a l tumors has shown that for almost a l l cases there exists a combined chemotherapy (the use of several drugs) which is superior to a single drug in curing disease or increasing survival time. This observation is not surprising since the addition of further anti-cancer agents seems likely, a-priori, to increase the efficacy of any single drug protocol. However, the reason for such an improvement in response is not well understood. These observations may be "explained" by assuming the various drugs in the combination to have differing phase-specific activity so that the combined therapy is more effective than any of the individual agents. However protocols which have attempted to combine agents with differing phase-specific activity generally have not been successful (in improving response), suggesting that other factors may be responsible for the benefits associated with combination chemotherapy. The superiority of combination chemotherapy is "naturally" explained i f we assume that the tumor contains subpopulations of cells resistant to particular drugs. The use of combination chemotherapy w i l l thus lead to the preferential selection of those cells which are resistant to a l l drugs in the protocol, which w i l l usually represent a smaller proportion of the total tumor than that which are resistant to only one of the drugs. In circumstances where the proportion of cells resistant to the combination is smaller than the proportion resistant to any one of the drugs, use of the combination w i l l yield superior results. In order to further model the response of tumors to several drugs i t is necessary to consider the joint distribution of multiple types of resistant c e l l s . In Chapter 4 the model developed in Chapter 3 is generalized to two drugs and measures -7-of the effectiveness of protocols involving two drugs are developed. This leads directly to considerations of maximizing the therapeutic effect of protocols, and results are given indicating the increase in the likelihood of cure obtained in two-drug protocols as compared to single-drug protocols. Examples are developed in Chapter 5 where i t is shown that the effects of different protocols depend on the choice of the outcome measure (survival time or proportion cured). Before continuing, i t is worthwhile to emphasize two points. F i r s t l y , in any complex biological system where many, possibly competing, processes are at work, and where any one may produce the same crude end point, i t is unrealistic to believe that consideration of one process, no matter how complete, w i l l lead to a comprehensive description of the observed phenomena. However, the consideration of a single process can give important indications of expected behaviour and may provide a framework for the incorporation of other mechanisms. Secondly, mathematical models of processes are seldom, i f ever, unique to that process. In particular, the model we w i l l develop can also be used for some of the other resistance mechanisms discussed earlier in this chapter. 1.1 Resistance in Other Biological Systems Analogous processes were f i r s t observed in the study of bacteria exposed to v i r a l infection. In a series of experiments investigating the infection of bacteria by viruses i t was found that after chronic exposure to a virus, a subpopulation of the i n i t i a l l y sensitive bacterial population, was no longer sensitive to infection by the same virus [6]. In most cases infection by the virus resulted in c e l l death. Furthermore although morphologic differences i n the c e l l s could sometimes be detected, this was frequently not so, and these r e s i s t a n t bacteria seldom displayed any resistance to i n f e c t i o n by other viruses. This observation led to two experimentally i n d i s t i n g u i s h a b l e hypotheses regarding the o r i g i n of r e s i s t a n t subtypes. It was not u n t i l 1943 that the pioneering work of Luria and Delbruck [6] permitted the two main competing theories to be compared and experimentally separated. These investigators summarised these two hypotheses as follows: "1) F i r s t hypothesis (mutation): There i s a f i n i t e p r o b a b i l i t y f o r any bacterium to mutate during i t s l i f e t i m e from ' s e n s i t i v e ' to 're s i s t a n t ' . Every o f f s p r i n g of such a mutant w i l l be r e s i s t a n t , unless reverse mutation occurs. The term ' r e s i s t a n t ' means here that the bacterium w i l l not be k i l l e d (absolute resistance) i f exposed to v i r u s , and the p o s s i b i l i t y of i t s i n t e r a c t i o n with virus i s l e f t open. 2) Second hypothesis (acquired hereditary immunity): There i s a small f i n i t e p r o b a b i l i t y for any bacterium to survive an attack by the v i r u s . Survival of an i n f e c t i o n confers immunity not only to the i n d i v i d u a l , but also to i t s o f f s p r i n g . The p r o b a b i l i t y of s u r v i v a l i n the f i r s t instance does not run i n clones. If we f i n d that a bacterium survives an attack, we cannot from t h i s information i n f e r that close r e l a t i v e s to i t , other than descendants, are l i k e l y to survive the attack." Using simple mathematical analysis, Luria and Delbruck showed that for both hypotheses the mean number of r e s i s t a n t c e l l s was proportional to the t o t a l number of c e l l s , N, but that the variance of the number of -9-r e s i s t a n t c e l l s was proportional to for hypothesis 1 and to N for hypothesis 2. By constructing a suitable experimental method, known as the f l u c t u a t i o n t e s t , they were able to show that th e i r data was incompatible with hypothesis 2 and supportive of hypothesis 1. Assuming hypothesis 1 to be true, they also discussed ways to estimate the mutation rate, which they defined to be the p r o b a b i l i t y that a c e l l would become r e s i s t a n t . The work of Luria and Delbruck spurred a great deal of research i n both experimental and mathematical analysis of this problem. Lea and Coulson [7], using the p r o b a b i l i t y gerating function and expanding i n powers, were the f i r s t to derive expressions for the d i s t r i b u t i o n function of the number of r e s i s t a n t c e l l s . This derivation assumed that the growth rates of s e n s i t i v e and r e s i s t a n t c e l l s were equal and constant, that mutations only occurred from s e n s i t i v i t y to resistance, and that the mutation rate was constant. An error i n t h e i r d e r i v a t i o n was pointed out by B a r t l e t t [8] and a correct solution was given by Armitage [9], who permitted d i f f e r e n t i a l growth rates between s e n s i t i v e and r e s i s t a n t c e l l s , and back mutations from resistance to s e n s i t i v i t y . A theme also explored at t h i s time was the possible e f f e c t of a phenomena known as phenotypic delay. This e f f e c t related to a possible delay a f t e r mutation u n t i l the resistance was expressed by the c e l l , which was modelled by assuming this time to be either f i x e d , or to depend upon the size of the r e s i s t a n t clone (population of c e l l s from a single parent). These processes were also examined by Kendall [10] who was interested i n t h e i r a p p l i c a t i o n to carcinogenesis. Crump and Hoel [11] u t i l i s e d the theory of f i l t e r e d Poisson -10-processes, and found a n a l y t i c r e s u l t s s i m i l a r to those previously obtained. They also c r i t i c a l l y examined the properties of estimators for the mutation rate which had been proposed elsewhere i n the l i t e r a t u r e . This approach was more recently extended by Tan [12] to e x p l i c i t l y model mutants at the hypoxanthine-guanine phospheribosil transferase locus i n Chinese hamster ovary c e l l s . Considerable research has been conducted recently i n the general theory of branching processes, of which mutational processes are but one s p e c i a l a p p l i c a t i o n . Much progress has been made i n the asymptotic theory of branching processes and l i m i t i n g d i s t r i b u t i o n s have been derived for cases of fixed t r a n s i t i o n a l rates for both single and multi-type branching processes. A comprehensive survey of r e s u l t s i n t h i s area i s contained i n Athreya and Ney [13]. These re s u l t s have found wide a p p l i c a t i o n i n physical problems where large numbers of p a r t i c l e s are present (e.g. chemical and nuclear r e a c t i o n s ) . In this thesis we w i l l be concerned with the d i s t r i b u t i o n of small numbers of r e s i s t a n t c e l l s where asymptotic analysis i s not appropriate. In the following chapters we present and explore the implications of mutation to resistance on the treatment of patients with cancer. Chapter 2 describes a model for tumor growth i n order to e s t a b l i s h a framework for the development of resistance. Chapter 3 contains a treatment of resistance to a single drug. Chapter 4 establishes a framework for the consideration of more general cases and presents a deta i l e d analysis of the s i t u a t i o n when two drugs are a v a i l a b l e . Chapter 5 presents cal c u l a t i o n s based on the previously developed theory and discusses some applications of this model both to experimental and human - 1 1 -cancer. The f i n a l chapter summarizes the main re s u l t s and discusses areas for future research. - 1 2 -2. A MODEL FOR TUMOR GROWTH In this chapter we w i l l discuss a model for tumor growth i n discrete time. Results w i l l be presented for the computation of the p r o b a b i l i t y generating function of the tumor growth model and i t s asymptotic d i s t r i b u t i o n w i l l be derived. We w i l l also discuss how the model parameters can be estimated from experimental observations and indic a t e how p a r t i c u l a r aspects of the model can be modelled i n continuous time, an idea that i s used i n subsequent chapters. Despite (or perhaps because of) the extensive research on models f o r tumor growth, there does not exist a single commonly accepted model. This i s due i n part to the fact that two broad, and d i f f e r i n g , approaches we w i l l r e fer to here as "empirical" and " b i o l o g i c a l " have been taken. In the empirical approach, use i s made of s e r i a l measurements of tumor size and various mathematical functions are used to f i t a model. In the b i o l o g i c a l approach, assumed processes of c e l l u l a r d i v i s i o n and i n t e r a c t i o n with the host are synthesized to give a model for the o v e r a l l tumor growth. Empirical growth functions have great value i n determining useful treatment parameters which cannot be d i r e c t l y observed. For example, knowledge of the growth curve permits the estimation of re s i d u a l disease a f t e r a therapeutic intervention by observing the time at which the disease recurs. However, for human malignancy the requirement of a large number of s e r i a l observations has severely l i m i t e d t h e i r usefulness. Further, these mathematical functions may contain parameters which have no obvious b i o l o g i c a l i n t e r p r e t a t i o n . A l t e r n a t i v e l y , the b i o l o g i c a l approach uses processes observed i n -13-severely l i m i t e d t h e i r usefulness. Further, these mathematical functions may contain parameters which have no obvious b i o l o g i c a l i n t e r p r e t a t i o n . A l t e r n a t i v e l y , the b i o l o g i c a l approach uses processes observed i n d i v i d i n g populations of c e l l s and re s u l t s i n models where the e f f e c t s of single mechanisms can be examined and evaluated independently. However, these models are frequently c r i t i c i s e d for f a i l i n g to take account of a l l processes, giving r e s u l t s which do not adequately f i t data, or y i e l d i n g models with so many parameters that they could be made to f i t almost any data. The l a t t e r c r i t i c i s m stems mainly from the fact that many processes, while well-understood i n general terms, are not uniquely s p e c i f i e d so that any attempt to use them requires the a - p o s t e r i o r i s p e c i f i c a t i o n of parameter values. In t h i s discussion we favour the b i o l o g i c a l approach since we are interested i n properties acting at the c e l l u l a r l e v e l . Our aim i s to develop a model which w i l l incorporate several known c h a r a c t e r i s t i c s of human malignant growth. In p a r t i c u l a r , we require a model which recog-nises that not a l l tumors are a homogenous c o l l e c t i o n of c e l l s with the same p r o l i f e r a t i v e c a p a b i l i t i e s . Examination of many s o l i d tumors, both experimental and c l i n i c a l , has shown them to contain c e l l s which are fu n c t i o n a l l y dead, i . e . c e l l s which are incapable of d i v i s i o n . Since tumors are believed to grow from microscopic f o c i , these dead malignant c e l l s represent the descendents of d i v i d i n g malignant c e l l s . In many populations of d i v i d i n g c e l l s i t i s recognised that not a l l c e l l s are capable of unlimited p r o l i f e r a t i o n . C e l l s capable of unlimited p r o l i f e r a t i o n are referred to as stem c e l l s and represent a variable f r a c t i o n (depending upon the tumor type) of the d i v i d i n g c e l l s i n the -14-tumor. The model we w i l l use here i s a s l i g h t l y modified version of one described by Mackillop et a l [14], which i s a stem c e l l model analogous to that used to describe the growth of normal tissue systems such as the hemopoietic system. This model assumes that c e l l s can be c l a s s i f i e d into one of three mutually exclusive classes based on t h e i r p r o l i f e r a t i v e p o t e n t i a l . In common with other work i n this area, we w i l l cast this model i n a d i s c r e t e framework i n which c e l l s are assumed to divide with a f i x e d i n t e r m i t o t i c i n t e r v a l with d i v i s i o n taking place at the beginning of each i n t e r v a l . This b i o l o g i c a l l y u n r e a l i s t i c assumption must be viewed as a f i r s t approximation to a complex process i n which the i n t e r m i t o t i c time can be expected to vary as a function of a large number of f a c t o r s . Part of this model w i l l be recast i n a continuous framework i n subsequent chapters, when the behaviour of stem c e l l s alone are considered. The three compartments consist of stem c e l l s , t r a n s i t i o n a l c e l l s and end  c e l l s defined as follows: 1. Stem c e l l s denoted (C^); c e l l s capable of unlimited p r o l i f e r a t i o n . At each d i v i s i o n a stem c e l l w i l l give r i s e to two stem c e l l s with p r o b a b i l i t y p, two t r a n s i t i o n a l c e l l s with p r o b a b i l i t y q and one of each with p r o b a b i l i t y 1-p-q. 2. T r a n s i t i o n a l c e l l s (C2,••,C n+i); c e l l s capable of l i m i t e d p r o l i f e r a t i o n . This class i s comprised of d i s j o i n t subclasses C2, Cn+1 where n i s referred to as the c l o n a l expansion number. T r a n s i t i o n a l c e l l s which are the immediate re s u l t of a stem c e l l d i v i s i o n are entered i n subclass C2« Upon d i v i s i o n a single C2 c e l l gives r i s e to two -15-C 3 c e l l s . These processes are repeated for C 3 , . . . , C n + i . 3. End c e l l s (Cn+2)> These are f u n c t i o n a l l y dead c e l l s incapable of further p r o l i f e r a t i o n . Two end c e l l s are formed by the divison of a single C n +^ t r a n s i t i o n a l c e l l . D ividing c e l l s (C^,...,Cn+^) are assumed to divide with a fixed and common i n t e r d i v i s i o n i n t e r v a l . A l l c e l l s are assumed to behave independently. For the purpose of this analysis the paramters p, q and n w i l l be considered to be fixed throughout the growth of the tumor, although i t i s a r e l a t i v e l y simple matter to calculate the quantities of i n t e r e s t i f these parameters are varied i n a systematic way. The occurrence of metastasis and measurement of experimental tumor systems indicate that substantial numbers of tumor c e l l s are l o s t from the primary tumor. C e l l loss from the primary tumor w i l l be modelled by assuming each c e l l i n compartment C^ to have a fixed p r o b a b i l i t y (i=l,...,n+2) that i t w i l l be l o s t per i n t e r m i o t i c i n t e r v a l , where for the purposes of c a l c u l a t i o n loss w i l l be assumed to occur at the end of the i n t e r v a l . Losses of c e l l s w i l l be assumed to occur independently and at a fixed rate per i n t e r m i t o t i c i n t e r v a l even for the non-dividing c e l l s i . e . Cn+2- l n t h i s s i t u a t i o n loss may be viewed to include l y s i s of dead c e l l s or migration outside the primary tumor. This model d i f f e r s from that of Mackillop et a l [14] who assumed that p+q = 1. This difference w i l l be shown to have important implications when we l a t e r consider stem c e l l resistance. An example where p+q<l had previously been considered by Moolgavkar and Venzon [15] i n t h e i r model of carcinogenesis. -16-Some constraints are placed on the choice of p, q and n by the nature of malignant growth. F i r s t l y , from the observation that few, i f any, c l i n i c a l l y detectable malignancies ever spontaneously become ex t i n c t , i t seems reasonable to l i m i t n to be less than 30. This i s chosen because 2 3 0 (=10 9) c e l l s represents the lower l i m i t of detection of primary tumors and since spontaneous complete regression i s almost never seen, the l i k e l i h o o d of tumors of t h i s size being composed of t o t a l l y t r a n s i t i o n a l c e l l s i s remote. S i m i l a r l y , observation of experimental tumors indicates that single c e l l s either have unlimited p r o l i f e r a t i v e p o t e n t i a l (stem c e l l s ) or can grow to produce clones of no more than 10 6 c e l l s . However, there i s i n theory no upper l i m i t on n since for any value i t i s always possible to choose 1^ (i=2,...,n+l) to give a model that i s consistent with the previous observations. 2.1 Properties of the Growth Model For the tumor to continue to grow (on the average), the stem c e l l compartment must grow. Thus the mean number of stem c e l l s produced by a d i v i s i o n of a single stem c e l l must exceed one. From t h i s we have the requirement (1-JLp (2p+l-p-q)>l or P-q> jya-^). ••• ( 2 , 1 ) The growth model, although very simple to define, has a complex structure. It i s nevertheless a straightforward exercise to write recursive r e l a t i o n s h i p s which w i l l give the j o i n t p r o b a b i l i t y generating function of the process. Let C (t) i=l,...,n+2, be random variables representing the number of c e l l s i n compartment at time t where t i s measured i n units of i n t e r d i v i s i o n times. Let $(s;t) be the j o i n t p r o b a b i l i t y generating function of the random vector C(t)=(C.(t),...,C , 0 ( t ) ) : C,(t) C n + 2 ( t ) * ( s ; t ) = E [ S i x ...x s n + 2 ], where s = (s,, s ,„). Let ~ 1 n+2 C (1) C (1) ^ ( s ) = E [ S ] L x ...x s n + ^ Z |C(0)=e.] where e^= (0,0,..,1,..,0) (the vector with 1 i n the i - t h p o s i t i o n and 0 elsewhere); <l^(s) i s the probabality generating function a f t e r one d i v i s i o n of a single c e l l i n state at time 0. Then i t can be shown that 2 2 (^(s) = A 1 + ( l - A 1 ) [ p s 1 + ( l - p - q ) s 1 s 2 + q s 2 ] , ^ ( s ) = V ( 1 " " V S i + l for i=2, n+1, From t h i s we obtain $(s;t+l) = $ ( ( K s);t), ... (2.2) where c|;(s )= f (J>, ( s ) , c|> l 0 ( s ) l . Equation (2.2) follows from a well ~ ~ i ~ n+z ~ known res u l t [16] for the p r o b a b i l i t y generating function for the sum of a random number of random v a r i a b l e s . Let X. .=(X... X T. .) ~ i j l i j ' J i j ' (i = l ,...,<», j = l , . . . ,J) , Y=(Y1,...,Y ) and Z=(Z1,...,Z ) be non-negative J Y. integer valued random vectors with Z = \ p X... Assuming X.. are j-1 i - l ~ 1 J ^ 2 independent for a l l i , j , X „ a r e i d e n t i c a l l y d i s t r i b u t e d for a l l i ( f o r each j ) , X^ _. and Y^ are independent for a l l i , j , then cP z(s)= 4,Y(<J;x(s)), ... (2.3) -18-where, Z l Z J <\>z(s) = E[ SX x . . . x S j ] , Y l Y J c|>Y(s) = E[ s L x . . . x S j ] , and X X (Px (s) = E[ s ^ ^ x . . . x S j J l j ] . j Equation (2.2) follows using (2.3) with Y=C(t) and X, .=C(t+l) condi t i o n a l on C(t)=ej (then unconditionally Z=C(t+l)). After s p e c i f i c a t i o n of $(s;0) i t i s possible to d i r e c t l y c a l c u l a t e $(s;t) by recursive use of (2.2). However, t h i s s o l u t i o n i s not very tractable and i s of l i m i t e d use since t i s seldom, i f ever, known for human malignancy. Three quantities of Interest which are measurable for human cancer, are the growth rate (GR) of the tumor, the proportion of stem c e l l s (Pg) and the p r o p o r t i o n of d i v i d i n g c e l l s (Prj)- Consider the following d e f i n i t i o n s : GR(t) = C(t) / C ( t - 1 ) , P s ( t ) = c l ( t ) / c ( t ) , _ ( 2 > 4 ) P D ( t ) = i - c n + 2 ( t ) / c ( t ) n+2 and C(t) = I C ( t ) . i = l As defined these quantities are random variables which are functions of a possibly unknown parameter t. Consider the l i m i t i n g q u a n t i t i e s : -19-GR = l i m GR(t), t-*» P G = lim P ( t ) , t-*» P D = lim P ( t ) . We w i l l now show that the l i m i t s GR, Pg and PQ e x i s t . In order to do t h i s we w i l l use asymptotic theory developed for multitype branching processes (of which the growth model considered here i s one example). Consider the matrix M, where M. . = E[C.(l)|C(0)=e ] for Ki,j<n+2. In t h i s case M i s given by (1-A )(l+p-q) (1-A )(l-p+q) 0 0 . . 0 0 0 0 2(1-Jl 2) 0 . . 0 0 0 0 0 2(1-I 3) . . 0 0 M = 0 0 0 0 . . 2(1-A ) 0 n 0 0 0 0 . . 0 2(1-1 ) n+1 0 0 0 0 . . 0 (1-A n+2 ...(2.6) (k) k Let ^ denote the ( i , j ) element of M . Two compartments C^ and C^ (Ki,j<n+2) are said to communicate i f and only i f there e x i s t integers k,m (>0) such that M,(k). >0 and M ( . M J >0. By convention M =1 (the i d e n t i t y matrix) and thus every compartment communicates with i t s e l f . Examination of (2.6) shows that the growth model considered here consists of n+2 communicating classes each -20-consi s t i n g of a single c e l l type. The eigenvalues, X, of M s a t i s f y the c h a r a c t e r i s t i c polynomial det|XI-M|=0, which i n th i s case i s ( X - A ) \ n ( \ - ( l - A n + 2 ) ) = 0 , where A = M^  ^=(l-£^)(1+p-q). The maximal eigenvalue i s X=A>1 (from (2.1)) which i s of m u l t i p l i c i t y one. Let v be the l e f t eigenvector of M associated with A, that i s , v M=Av, ...(2.7) n+2 where Y v.=l. Examination of (2.6) reveals that Mn _>0, M l 0 ,„>0, l 1,1 n+Z,n+Z (1=1,...,n+l) and by Theorem 4.1, page 66 i n Mode [17], for C(0)=e 1 we have — — •* wv almost surely, ...(2.8) A where w i s a non-negative scalar random v a r i a b l e . It i s e a s i l y seen that E[C^.(1) log C..(1) |£(0)=^ ]<» for a l l i , j and we thus have from Theorem 4.1, page 66, i n Mode [17] that E[w|C(0)=e^]>0. Thus for r e a l i z a t i o n s of the process C of in t e r e s t ( i . e . those for which C(t)-*>° as t-*°) i t follows from (2.7) that c(t) = -> wv. 1 = w a. s., t t ~ ~ A A where 1^  i s the vector where each element i s 1. Thus, C(t)/C(t) + v a.s. From t h i s we see d i r e c t l y that P $ and PQ exist and are degenerate. To see that GR exists consider C(t) - C ( t - l ) A t A ^ 1 ' Then we have a.s. -21-o = l i m (Sit! _ ccizii) t - H » A A -iii.a|=}i(A--aa-). t-*» A C ( t - l ) Thus for r e a l i z a t i o n s of in t e r e s t we have GR = A a.s. Notice that the asymptotic growth rate of the tumor i s e n t i r e l y determined by parameters which control the growth of the stem c e l l compartment, that i s A. Furthermore, the random variable w relates to the growth of the stem c e l l compartment. To see t h i s , we note that i f 0 ( 0 ) = ^ (i=2,... ,n+2), then: C(t)<2 n for a l l t , where n i s the cl o n a l expansion number. Thus for A>1, A* Therefore for any r e a l i z a t i o n C(t) with C(t')=C', lim t - t " = W * ~ a • s • » t->ro A where w* depends only on C]_(t'). Because of the independent behaviour of the stem c e l l s , w* i s the convolution of w given i n (2.8). In attempting to f i t t h i s model to human disease we are faced with s i t u a t i o n s where only comparatively crude data are a v a i l a b l e . The f r a c t i o n of d i v i d i n g c e l l s can be currently estimated with l i m i t e d p r e c i s i o n [18]. Estimates of stem c e l l f r a c t i o n are i n the range of 0.001 and above [19]. Therefore when there are at least 10 9 c e l l s , the number of stem c e l l s exceeds 10 6. At the lower l i m i t of the number of stem c e l l s , one or both of the other compartments w i l l be large. The number of c e l l s growing from a single c e l l (of any type) has a f i n i t e mean and variance for a f i n i t e time period. Since c e l l s behave inde--22-pendently, f l u c t u a t i o n s i n the proportion of c e l l s i n each of the compartments w i l l be small with high p r o b a b i l i t y . In cases where the proportion of stem c e l l s i s very high, the proportion of the non-stem c e l l s w i l l be small i n comparison to the pre c i s i o n with which i t can be measured. From these considerations, we expect the l i m i t i n g values GR, P and P w i l l apply to a mature c l i n i c a l or experimental tumor where we w i l l assume n<20. We can then use expressions for GR, Pg and Prj to estimate the parameter values of the tumor. These expressions can be d i r e c t l y calculated by solving (2.7), where M i s given by (2.6), and lead to: V2 d+p-q) V ...(2.9.1) 2(1-1 ) Vi+1 = — A V f o r 1 = 2 n ' •••(2.9.2) and Vn+2 = A - ( l - l , 9) V i ' -..(2.9.3) n+2 n+2 The constraint £  vi~^ a n ( * equations (2.9.1-3) y i e l d : i-1 n i 2(1-A.) . n+1 .2(1-1.) v 1 = (i+P-q) [2+(i-P+q){I ( n + i i = g l ) A n 1 1=2 j-2 n+2 j=2 A -1 .. .(2.10) Using (2.9.1-3) and (2.10) we w i l l now calculate P s , P D and GR for several s p e c i a l cases of the n+2 element vector 1 of loss rates. We w i l l not indicate the sp e c i a l cases which ar i s e when both !=...=! =1 and 2 n 2(1-!)=A. -23-(I) If A = (A A, ^ n + 2 ) then GR = ( 1 - 1 ^ (1+p-q) = A, p ( V A ) + ( 1 - A i ) M ( l ( i ^ ) n n+2 (1-A) (1+p-q) and P g = - P . (j ^ - j l ) + rllp. ) ( 2 ( 1 ^ . ^ (II) If A = (A, 1, .... A, A n + 2 ) then GR = (1-A) (1+p-q), P A" 1 + An+2 D 2(1-1) - ( l - A n + 2 ) n and P s - ( i ± F ) F D. (III) If A = (A, .... A) then GR = (1-A) (1+p-q), and P S - ( i ± p ) n P D . (IV) If A = (0,0, 0, A) then GR = (1+p-q), D 1+A > (V) If A = (A, A, •.•, A,0) then G = (1-A) (1+p-q), p = (l-A)(p-g) -A D 1-2A and P, n 1 2 > D -24-Example (I) represents the s i t u a t i o n where the three types of c e l l s (stem, t r a n s i t i o n a l and end) are l o s t at three d i f f e r e n t rates. Example (II) represents the case where a l l d i v i d i n g c e l l s are l o s t at the same rate, and non-dividing c e l l s are l o s t at a d i f f e r e n t rate. ( I l l ) , (IV) and (V) represent s p e c i a l cases where a l l c e l l s are l o s t at the same rate, d i v i d i n g c e l l s are not l o s t and end c e l l s are not l o s t r e s p e c t i v e l y . In the absence of s p e c i f i c information for a p a r t i c u l a r tumor system, example II seems to be a reasonable compromise between complexity of the general case and the l i k e l y processes which cause c e l l loss i n a tumor. For example, a l l d i v i d i n g c e l l s can be expected to be shed at s i m i l a r rates into the blood and lymphatic systems because of growth pressure. End c e l l s w i l l be l o s t at a d i f f e r e n t rate because-they w i l l die at a higher rate (than d i v i d i n g c e l l s ) by t h e i r very nature. However, even t h i s model contains f i v e unknown parameters, (p, q, n, JL and 1 ,„) which cannot be uniquely i d e n t i f i e d from GR, P^ and P„. To n+2 D S estimate the parameters of t h i s model requires either the a - p r i o r i s p e c i f i c a t i o n of some of the parameters or the c o l l e c t i o n of data on other tumor c h a r a c t e r i s t i c s s p e c i f i e d by the parameters of t h i s model. Examination of (2 .9.1-3) shows that the parameters p and q only appear as the d i f f e r e n c e ±(p-q). Therefore even i f X and n were known, p and q cannot be i n f e r r e d (except i n the t r i v i a l case p-q=l) from experiments measuring GR, Pg and P Q. This problem i s not e a s i l y resolved since the i d e n t i f i c a t i o n of stem c e l l s , both t h e o r e t i c a l l y and experimentally, i s based upon t h e i r p r o l i f e r a t i v e p o t e n t i a l , and at present i t i s not easy to separate stem c e l l s from other d i v i d i n g c e l l s and carry out experiments on them. However, a closer analysis of experiments c a r r i e d out to measure Pg shows that further information can be gained. In order to measure Pg i n a human.tumor, a biopsy specimen of the tumor i s f i r s t homogenized and then a sample, N, of c e l l s , are plated out i n d i v i d u a l l y onto a medium supplying nutrients and a suitable matrix for growth. After an incubation period the number of c e l l s , r say, which have gone on to form colonies of c e l l s greater than some fixed s i z e , say, are counted. Then the proportion r/N i s reported as the f r a c t i o n of stem c e l l s . If i s chosen too small (as determined by n, the c l o n a l expansion number), some of the colonies generated may be the product of t r a n s i t i o n a l c e l l s . If i s chosen large enough, the counted colonies w i l l consist e n t i r e l y of colonies generated by stem c e l l s . However, i f i s chosen too large many stem c e l l s present w i l l not form colonies of size because the stem c e l l may i n i t i a l l y (or subsequently) divide to form only t r a n s i t i o n a l c e l l s . Thus to design experiments to measure Pg we must know (or have a good idea of) n. However, i f i s chosen very large then i t i s possible to obtain an approximate expression for E[r/N] as follows. Using the same notation for colonies as previously used for tumors, l e t C*(t) be the state vector for the i - t h colony at time t and set n+2 . i v i C (t) = I C . ( t ) . In t h i s experimental s i t u a t i o n we w i l l assume that j - i 3 A=0, that i s there i s no loss from the colonies. If i s chosen very large then almost a l l sample paths for which C 1(t')>S M (for some t') w i l l grow a r b i t r a r i l y large, that i s , these paths w i l l s a t i s f y l i m C > 0. a.s. t-*» A -26-Since the proportion of stem c e l l s approaches v^, we have for such paths lim P{C*(t)=k}=0 for f i n i t e non-zero k. Thus the p r o b a b i l i t y K=P{C 1(t')>S M}, i s approximately given by K=1-9 where the e x t i n c t i o n p r o b a b i l i t y 6 = lim P{C^(t)=0}. The p r o b a b i l i t y generating function f o r t -XX) stem c e l l growth <)>(s), i s given by (s) i n (2.2) with SL =0 and s = l I ~ (s,0,...,0), that i s <t>(s) = q+(l-p-q)s+ps 2. ...(2.11) It i s a well known res u l t (p. 397 i n K a r l i n and Taylor [20]) that 9 i s given by the minimum solution of 9=<j>(9). Solving t h i s equation y i e l d s 9=q/p and thus K=l-q/p. The proportion of stem c e l l s w i l l be approximately given by Pg(since the c e l l s are sampled from a mature tumor) and i f S^ i s very large: E[r/N]-P s(l-q/p). ...(2.12) The righ t hand side of (2.12) depends on the value of p and not just on the difference p-q. Thus by carrying out a series of experiments at values of S M i t i s possible to obtain information on the values of p and q-A further property of this model which i s important i n the subsequent development i s that the stem c e l l compartment functions autonomously; that i s , the size of the stem c e l l compartment i s determined by the h i s t o r y of stem c e l l d i v i s i o n s and not by any of the other compartments. Assuming that disease i s diagnosed at a r e l a t i v e l y early stage, then, except i n extremely rare cases, elimination of the stem c e l l compartment i s a necessary and s u f f i c i e n t condition for cure of the tumor. This statement i s based on the following assumptions: -27-( i ) Diagnosis i s made at approximately 1 0 1 0 c e l l s and death w i l l occur at no less than 1 0 1 2 c e l l s ; ( i i ) The proportionate k i l l of chemotherapy i s the same for a l l d i v i d i n g c e l l s ( i . e . stem and t r a n s i t i o n a l c e l l s ) ; ( i i i ) P s(t)(=P s)>10 - l + i . e . at least one i n I0k c e l l s are stem c e l l s ; ( i v ) A c l o n a l expansion number (n) i n excess of 15 i s u n l i k e l y ; (v) (q/p)<0.95, that i s the r a t i o of stem c e l l d i v i s i o n s forming only t r a n s i t i o n a l c e l l s compared to those forming only stem c e l l s , i s not too large. By ( i i i ) and ( i ) there are at least 1 0 6 ( 1 0 - l t x l 0 1 0 ) stem c e l l s i n the tumor. By assumption the stem c e l l s are eliminated and th i s implies that only s u f f i c i e n t numbers can survive the e f f e c t s of treatment so that they go spontaneously extinct ( y i e l d progeny which are t r a n s i t i o n a l and end c e l l s only). The p r o b a b i l i t y a single stem c e l l w i l l go spontaneously extinct i s (q/p) and because c e l l s behave independently the p r o b a b i l i t y that k would go extinct i s (q/p) • By (v) (q/p)<0.95 and thus i f n>100 then (q/p) 1 0 0<0.01. This implies that the p r o b a b i l i t y a stem c e l l w i l l survive therapy i s <10 - i t( 10 6/10 2). Thus by ( i i ) the expected number of surviving t r a n s i t i o n a l c e l l s i s < 1 0 6 ( 1 0 - l t x l 0 1 0 ) . By ( i v ) each t r a n s i t i o n a l c e l l can give r i s e to no more than 2 1 5=3.3xl0 1* c e l l s . Thus the maximum size the r e s i d u a l tumor can achieve ( i f a l l stem c e l l s are eliminated) i s 10 6x3.3xl0 1 +=3.3xl0 1 0 which i s less than the minimum s i z e which can cause patient death by ( i ) . As indicated i n the previous discussion, the long-term behaviour of the tumor (that i s whether i t i s curable or not) can be assessed by considering whether the stem c e l l compartment can be eliminated or not. -28-However, the short-term response of tumors to therapy w i l l n a t u r a l l y be a function of the response of a l l tumor c e l l s . In attempting to describe tumor behaviour i n terms of this model, we w i l l r e s t r i c t our analysis to considerations of long-term response, based on the behaviour of the stem c e l l compartment. By the nature of the growth model presented here, not every sample path passes through the point k (not every path s a t i s f i e s C(t)=k for some t ) . In p a r t i c u l a r i f C^(t')>k for a p a r t i c u l a r path we cannot conclude that there exists a t<t' such that C^(t)=k for the path. In l a t e r chapters we wish to consider t as a continuous parameter, to be able to condition on C^(t) and require that every path for which C^(t')>k s a t i s f y C^(t)=k for some t<t". In order to do th i s we require a model for growth which only changes by increments of +1 or -1. A convenient process which has this property i s the l i n e a r b i r t h and death process. In examining long-term response we w i l l u t i l i z e a b i r t h and death model for the stem c e l l compartment. In th i s model a l l losses from the stem c e l l compartment (to t r a n s i t i o n a l c e l l s , c e l l deaths, e t c ) w i l l be termed deaths. Additions of new stem c e l l s by d i v i s i o n w i l l be referred to as b i r t h s . We w i l l assume that for a single c e l l i n a time i n t e r v a l [t,t+At) d i v i s i o n s r e s u l t i n g i n two stem c e l l s occur with p r o b a b i l i t y bAt+o(At), d i v i s i o n s r e s u l t i n g i n one stem c e l l and one t r a n s i t i o n a l c e l l with p r o b a b i l i t y cAt+o(At) and deaths occur with p r o b a b i l i t y rate dAt+o(At). We make the correspondence between the discre t e and continuous models by requiring that b, c and d s a t i s f y the following con s t r a i n t s : -29-b ...(2.13.1) b+c+d c (1-p-q) ( 1 - ^ ) , ...(2.13.2) b+c+d and b-d i n [ ( 1 - 1 ^ (1+p-q)]. .. .(2.13.3) Conditions (2.13.1) and (2.13.2) re s u l t from requiring that the events associated with b, c and d occur i n the appropriate l i m i t i n g frequency with respect to each other. Equation (2.13.3) guarantees that the net mean growth rate w i l l be the same i n both formulations. A continuous Markov model i s a better, although imperfect, model of c e l l u l a r d i v i s i o n than one i n which the i n t e r - m i t o t i c times are constant. A more r e a l i s t i c model of i n t e r - d i v i s i o n times would have support on [x,=°] x>0, thus implying a non-zero mode. However the growth process i s of secondary i n t e r e s t i n t h i s analysis and the mathematically tractable exponential d i s t r i b u t i o n for i n t e r d i v i s i o n times w i l l be used. The relevance of the growth model considered here i s that i t i s b i o l o g i c a l l y plausible and contains parameters which allow the number of stem c e l l s to be varied f or a fixed t o t a l number of tumor c e l l s . This i s important since the porportion of stem c e l l s i s suspected to d i f f e r greatly between tumor systems. In the next chapter we w i l l consider the spontaneous evolution of variant stem c e l l s which display resistance to one or more chemotherapeutic agents. Relationships w i l l be developed which r e l a t e the c u r a b i l i t y by chemotherapy of the tumor to the k i n e t i c parameters of the tumor and other parameters r e f l e c t i n g the development of resistance. The growth model developed here w i l l not be e x p l i c i t l y considered i n l a t e r chapters but i s assumed to apply. In what follows we w i l l -30-concentrate on the stem c e l l development which, as has been shown, determines the growth and c u r a b i l i t y of tumor. -31-3. THE DEVELOPMENT OF RESISTANCE TO A SINGLE CHEMOTHERAPEUTIC AGENT In Chapter 1 we discussed various mechanisms which lead to resistance to chemotherapy used i n the treatment of cancer. In that chapter we discussed how drug r e s i s t a n t c e l l s are known to a r i s e i n experimental tumors where they are one of the p r i n c i p a l causes of treatment f a i l u r e . Resistant c e l l s are also thought to be a primary cause of treatment f a i l u r e i n human malignancy although the evidence i s not as strong as i n the experimental case. We w i l l now consider the development of permanently r e s i s t a n t stem c e l l s within the context of the growth model developed i n Chapter 2. In t h i s chapter we w i l l develop expressions which r e f l e c t the development of resistance and the long-term response of tumors treated with a single drug. We w i l l only consider the primary tumor and not the status of any c e l l s contained i n distant metastatic deposits. Because of the nature of the tumor growth model presented i n Chapter 2 we need only consider the behaviour of stem c e l l s since they alone influence the long-term c u r a b i l i t y of the tumor. Stem c e l l s w i l l be considered to be i n one of two states with respect to a drug: s e n s i t i v e or r e s i s t a n t . Resistant c e l l s w i l l not be assumed to necess a r i l y be t o t a l l y r e s i s t a n t , that i s , r e s i s t a n t c e l l s may show some response to the drug but t h i s response w i l l be q u a n t i t a t i v e l y less than that exhibited by s e n s i t i v e c e l l s . The two states are therefore defined with respect to one another and are generally not defined i n absolute terms. This d e f i n i t i o n i m p l i c i t l y involves a notion of the environment of the experiment, which includes the c e l l l i n e , the drug and the dosage under consideration. A more general d e s c r i p t i o n would include a number of states which show -32-varying s e n s i t i v i t y to the drug. For reasons, which w i l l l a t e r become apparent, such a multitype model i s d i f f i c u l t to analyze and we w i l l only consider a two state model. We w i l l assume that i n a time i n t e r v a l of length At the p r o b a b i l i t y that a single stem c e l l divides to form two stem c e l l s , i s bAt + o(At), that i t divides to form a stem and t r a n s i t i o n a l c e l l i s cAt + o(At) and that i t migrates, dies or forms two t r a n s i t i o n a l c e l l s i s dAt + o(At) (see Chapter 2). These events w i l l be referred to as b i r t h s , renewals and deaths r e s p e c t i v e l y . The p r o b a b i l i t y of two or more events occuring i n a time i n t e r v a l of length At w i l l be assumed to be o(At). In what follows b, c and d w i l l be assumed to be constants for a p a r t i c u l a r tumor. In common with the t h e o r e t i c a l model of Luria and Delbruck [6], we assume that there i s a fixed p r o b a b i l i t y a that a b i r t h event i n a s e n s i t i v e c e l l w i l l r e s u l t i n the addition of a single r e s i s t a n t c e l l and p r o b a b i l i t y 1-a that a s e n s i t i v e c e l l i s added. S i m i l a r l y , we assume that there i s a p r o b a b i l i t y B that a renewal event to a s e n s i t i v e c e l l w i l l r e s u l t i n the replacement of a s e n s i t i v e stem c e l l by a r e s i s t a n t stem c e l l and a p r o b a b i l i t y 1-8 that there i s no change i n the number of s e n s i t i v e stem c e l l s . We also assume that a s e n s i t i v e stem c e l l may spontaneously mutate from s e n s i t i v i t y to resistance with p r o b a b i l i t y yht + o(At) i n an i n t e r v a l of length At. Resistant stem c e l l s are assumed to have the same parameters b, c and d but a l l progeny of r e s i s t a n t c e l l s are assumed to remain r e s i s t a n t , that i s t r a n s i t i o n s from the r e s i s t a n t to the s e n s i t i v e state are assumed not to occur. In the next section we derive the p r o b a b i l i t y generating function of the process and use i t to deduce some quantities which describe the -33-behaviour of the system. We then give a basic d e s c r i p t i o n of the e f f e c t of drugs on both normal and malignant c e l l s . These are then integrated into the model for the development of resistance and equations developed for the p r o b a b i l i t y generating function of the d i s t r i b u t i o n of stem c e l l s a f t e r an a r b i t r a r y sequence of treatments by a single drug. Subsequently we discuss three approaches to developing the p r o b a b i l i t y generating function for the numbers of s e n s i t i v e and r e s i s t a n t stem c e l l s when the time paramter t i s unknown. F i n a l l y , we examine the e f f e c t of random v a r i a t i o n i n the resistance parameters on the d i s t r i b u t i o n of r e s i s t a n t and s e n s i t i v e c e l l s . 3.1 Calculating the Probability Generating Function Let Rg(t) = number of s e n s i t i v e stem c e l l s at time t, R^(t) = number of r e s i s t a n t stem c e l l s at time t, N(t) = R Q ( t ) + R 1 ( t ) and p i > : j ( t ) = P{ V t ) = i > R l ( t ) = J 1' f o r t > 0 ' Table I indicates t r a n s i t i o n s between states and t h e i r associated p r o b a b i l i t i e s . Referring to Table I we may now use the Kolmogorov forward equations [21] to obtain the following family of d i f f e r e n t i a l equations for P. . ( t ) : 1»2 d P i , j ( t ) = - [(b+d)j + (b+d+c+y)!] P. .(t) + b(l-o) ( i - l ) 2 ( t ) dt ' J ' J + c(l-B) i P , ,(t) + d(i+l) P... ,(t) + a b i P (t) + (Bc+y)(i+l) P i + i ^ C t ) + b ( j - l ) P ^ ^ C t ) + d(j+l) P i > j + 1 ( t ) ...(3.0) -34-TABLE I Transitions Occurring i n the Stem Cell Compartment i n the interval [t,t+At) which have Probability of Order At. In i t i a l State Final State Probability ( i , j ) (i+l,j) ib(l-a)At+o(At) ic(l-B)At +jcAt+o(At) ( i , j ) (i-l,j) idAt+o(At) (i,j+l) iabAt+jbAt +o(At) (i,j) U-l,j+l) i(6c+ Y)At+o(At) ( i , j ) (i,j~l) jdAt+o(At) for i,j>0 where P .(t) = 0 for i<0 or j<0. Let 4(s , s ;t) be the i > 3 u i p r o b a b i l i t y generating function of {R^(t),R^(t)}, that i s x 00 00 < D ( s Q , s ;t) = I I P (t) s 1 s J . u 1 i=0 j=0 1 , J u 1 In what follows we w i l l specify the i n i t i a l d i s t r i b u t i o n of c e l l s by the p r o b a b i l i t y generating function at time 0, that i s 4 ( s 0 , s 1 ; 0 ) = ( K s 0 , s 1 ) . Then using (3.0) we can show (by multiplying by SQS^ and summing over i and j and interchanging the order of d i f f e r e n t i a t i o n and summation) that the p r o b a b i l i t y generating function s a t i s f i e s , d K s Q . s ^ t ) d A C s ^ s ^ t ) 5 6 ( s 0 , s 1 ; t ) E = [ b S o - d ] [ s 0 - l ] - ^ - + [ a b s 0 + v ] [ S l - s 0 ] — ^ o 6 ( s n , s 1 ; t ) + [ b S l - d ] [ S ; L - l ] ^ — i ...(3.1) where v = Be + y. Using the method of c h a r a c t e r i s t i c s (see for example John, p. 9 [22]), s o l u t i o n of (3.1) can be reduced to solving the following set of ordinary d i f f e r e n t i a l equations: ^ - = 1 , ...(3.2.1) dX n(u) [ X 0 ( u ) - l ] [ d - b X ( ) ( u ) ] - [abx 0(u)+v ] [ x 1(u ) - x 0(u)], ...(3.2.2) dx L(u) = [ X 1 ( u ) - l ] [ d - b X l ( u ) ] , ...(3.2.3) du I A ^ V - / where X Q ( U ) and X ^ ( U ) a r e dummy va r i a b l e s . From (3.2.1) we have, t=u, ...(3.3.1) where, without loss of generality we have set the constant of i n t e g r a t i o n to zero. Solving (3.2.3) we obtain cp(6u) .. .(3.3.2) d l l - X ^ O ) ] + [b X l(0)-d] exX 1 ( U ) = b [ l - X l ( 0 ) ] + [b X l ( 0 ) - d ] exp(6u) ' where 6=b-d and we assume that b>d so that the process i s s u p e r c r i t i c a l , that i s , i t represents a growing tumor. To solve the d i f f e r e n t i a l equation (3.2.2), f i r s t notice that X Q ( U ) = X ^ ( U ) i s a p a r t i c u l a r s o l u t i o n . Substituting X Q ( U ) = X ^ ( u ) + 1 / V ( U ) i n (3.2.2) y i e l d s the following d i f f e r e n t i a l equation for y(u): y(u)[b+d+v-b(2-a ) x 1(u)]= b ( l - a ) . The so l u t i o n for y(u) i s given by { X 0 ( 0 ) - X 1 ( 0 ) } _ 1 + b(l-o ) / J F(x)dx Y ( U ) = FOT) ' where F(x)=exp/Q g(v)dv and g(v)=b+d+v-b( 2-oc) x^(v). Writing y(u) = [ X Q ( u ) - X ^ ( U ) ] ^ y i e l d s the following expression for X Q ( U ) , X Q ( u ) - X L ( u ) + F ( u ) [ { x 0 ( 0 ) - x 1 ( 0 ) } " 1 + b(l-a ) / J F(x)dx]~} ..(3.3.3) where _,, . E2-a (&+ad+v)xr, ,, 6x -2+a ,„ . n. F(x)=6 e v [b[l - x 1(0 ) ] + [ b x 1(0)-d]e ] . ...(3.4.1) It follows from the method of c h a r a c t e r i s t i c s that i f the substitutions X 0 ( u ) = s Q , x 1 (u)=s 1 and u=t ...(3.4.2) are made i n (3.3.2) and (3.3.3) then the solu t i o n of (3.1) i s given by < K s 0, S l;t) = c K x 0 ( 0 ) , X 1 ( 0 ) ) . ...(3.5) Carrying out these substitutions leads to e x p l i c i t expressions f o r X Q(0) and x^(0) as functions of s^, s^ and t. To emphasize the dependence of -37-X Q(0) and X-^ O) on t we w i l l write w Q(t)=x 0(0) and w 1(t)=x 1(0). Using the substitutions (3.4.2) i n (3.3.2) we obtain d(l-s 1)+(bs 1-d)exp(-6t) w i ( t ) = x n(0) = m N , T T ; — x * ...(3.6) l v ' * l v ' b(l-s 1)+(bs 1-d)exp(-6t) v ' S i m i l a r l y using the substitutions (3.4.2) i n (3.3.3) y i e l d s , a f t e r some algebra, w Q(t) = x Q(0) = W l ( t ) + , ...(3.7) 1 [ 6 2 a ( s 0 - S l ) ] 1-b(l-a)/Jf(v)dv ' where f(v)=exp{-(6+ad+v)v} [b(\-* )+(bs -d)e 6 v ] " 2 + a . Notice that (3.6) i s the p r o b a b i l i t y generating function for the b i r t h and death process with fixed parameters b and d. As expected, the su b s t i t u t i o n of S Q = S ^ = S i n (3.7) yi e l d s w^(t)= w^(t). Thus the development of the stem c e l l compartment as a whole i s a b i r t h and death process with parameters b and d. S i m i l a r l y , s u b s t i t u i t i o n of s^=l i n (3.7) shows that the s e n s i t i v e stem c e l l compartment grows as a b i r t h and death process with parameters b(l-oc) and (d+v). For future use we w i l l now calculate some elementary properties of the process {R ( t ) , R ^ ( t ) } . By d i f f e r e n t i a t i n g (3.1) with respect to s and s^, s e t t i n g S Q = S^=1 and interchanging the order of d i f f e r e n t i a t i o n we obtain the following ordinary d i f f e r e n t i a l equations for mQ(t)=E[RQ(t)] and m^(t)=[R^(t)] r e s p e c t i v e l y : d m 0 ( t ) = (6-ab-v)m f.(t), dt d m l ( t ) = 6m 1(t)+(ab+v)m 0(t), dt which y i e l d s -38-m 0(t) = m0 exp{(6-ab-v)t}, ...(3.8) L ( t ) = [m 1+m 0(l-exp{-(ab+v)t})]e 6 t, m where m^^n^O), m^=m^(0) are obtained d i r e c t l y from the p r o b a b i l i t y generating function at t=0, 4>(SQ,S^). From (3.8) we see E[N(t)]=( m i+m 0)e 6 t. In a si m i l a r fashion we can derive ordinary d i f f e r e n t i a l equations which the variances and covariance must s a t i s f y . Let V^(t) and V^(t) be variances of R g ( t ) and R^(t) respectively and l e t V g i ^ O he t h e i r covariance. Then d V 0 ( t ) = (b+d-ab+v)m Q(t) + 2 ( 6 - a b - v ) V Q ( t ) , dt d V 0 1 ( t ) = -v m Q(t) + ( 2 6 - a b - v ) V Q 1 ( t ) + (ab+v)V 0(t), dt d V l ( t ) = (ab+v)m Q(t) + (b+d)m L(t) + 2(a b+v)V Q 1(t) + 26 V x ( t ) . dt These equations have the following s o l u t i o n s : V Q ( t ) - [V + A 1(l-exp{-(6-ab-v)t}] exp{2(6 -ab-v)t}, V Q 1 ( t ) = [V Q 1+ [V Q + A 1](l-exp { - ( a b+v)t}) -A 2(l-exp(-6t))]exp{(26 - a b-v)t}, and V 1 ( t ) = [V1 + 2 [ V Q 1 + V Q + A x - A 2](l-exp{-(ab+v)t}) - [V Q + A 1](l-exp{-2 ( a b+v)t}) + (b+d)[m Q + m 1](l-exp(-6t))/6 mn A [(b+d -ab-v)-2(ab+v)—] (1-exp{-( 6+ab+v)t}) ]exp {26t}, (6+ab+v) m^ (b+d-ab+v) . [(ab+v)(l-a) + (ad+v)] . where A = -Vs—Z \ ™A » A o = 5 T 5 — Z — \ b m n » a n d 1 (6-ab-v) 0 2 o(o-ab-v) 0 V 0=V Q(0), V -V (0) and v 0 1 = v 0 1 ( ° ) a r e calculated from ( K s^s^. -39-F i n a l l y we note that the p r o b a b i l i t y that a single stem c e l l , present at some time t=t", w i l l not have any surviving progeny at t=°°, i s given by e=d/b (see K a r l i n and Taylor p.147 [20]). This event w i l l be referred to as spontaneous e x t i n c t i o n . S i m i l a r l y , since the stem c e l l compartment grows as a b i r t h and death process with parameters b(l-a) and (d+v), the p r o b a b i l i t y that a single s e n s i t i v e stem c e l l w i l l not any have any surviving s e n s i t i v e progeny at time t=<*> i s (d+v)/b(l-a) • In order to consider the behaviour of a tumor subject to therapy we must f i r s t examine the e f f e c t s of therapy on the tumor c e l l s and on the normal t i s s u e . 3.2 E f f e c t s of Drug Treatment As mentioned previously the development of resistance to a drug can aris e as a mutational process. Evidence for some drugs from experimental tumors shows that resistance can be e f f e c t i v e l y absolute. An example of thi s i s resistance to Arabinosylcytosine i n the L1210 mouse leukemia system [23]. That i s , treatment with any dosage of the drug on a c e l l r e s i s t a n t to i t w i l l have no e f f e c t . In other cases t h i s i s not true, and c e l l s may be i d e n t i f i e d that show reduced s e n s i t i v i t y when compared to the parent s e n s i t i v e l i n e . To model the resistance phenomenon we f i r s t consider the response of a single c e l l to chemotherapy. A large body of experimention, notably by Skipper and h i s associates [23], has indicated a l i n e a r r e l a t i o n s h i p between a single delivered dose and the logarithm of the f r a c t i o n of c e l l s over a large range of dosages. Repeated courses of chemotherapy to the same population of c e l l s s a t i s f y the same r e l a t i o n s h i p with the same constant of p r o p o r t i o n a l i t y as long as re s i s t a n t c e l l s do not emerge. This -40-r e l a t i o n s h i p has been found to hold for a number of d i f f e r e n t (non-phase s p e c i f i c ) drugs, i n several types of tumors and for a range of tumor sizes [24]. From these observations, Skipper and his co-investigators have postulated that tumor c e l l s subject to chemotherapy at dose D have an i n d i v i d u a l fixed p r o b a b i l i t y , TC(D) say, of surviving chemotherapy, which may be expressed as 7i(D)=exp{-kD} where k i s a constant of p r o p o r t i o n a l i t y and that the response of each c e l l i s independent of that of the others. For drugs with phase s p e c i f i c e f f e c t t h i s r e l a t i o n s h i p also applies providing c e l l s are i n the s e n s i t i v e phase of the c e l l c y cle. We w i l l use t h i s model of chemotherapeutic action i n the development that follows. Consider the binary random variable X, which indicates whether the c e l l survives (X=l) administration of a single course of the drug or not (X=0). If £(s) i s the p r o b a b i l i t y generating function of X, then S(s)=l-n;(D)+T!(D)s ...(3.9.1) for a non-phase s p e c i f i c agent, and C(s)=l-pn(D)+p7t(D)s ...(3.9.2) for a phase s p e c i f i c agent where p i s the p r o b a b i l i t y that the c e l l i s i n the s e n s i t i v e phase of the c e l l c ycle. This model for drug action was constructed for agents administered over a short period where the drug i s r a p i d l y degraded or excreted so that the e f f e c t of the drug may be considered as an instantaneous one. In general the dose at some time t, D(t), i s defined as D(t) = C(u)du ...(3.10) where the drug i s introduced at time t=0 and C(t) i s the concentration of -41-the drug (at the tumor c e l l under consideration). We assume, without loss of generality, that C(t)=0 for t<0. If therapy i s phase-specific and i s given over an extended period then the l i k e l i h o o d that a c e l l i s i n the s e n s i t i v e phase of the c e l l cycle, at some time during the therapeutic period, w i l l increase as the duration of therapy i s lengthened. Let I ( t ) = 1 i f the c e l l i s i n the s e n s i t i v e phase at time t, = 0 otherwise. Let C'(t) be the e f f e c t i v e concentration for the c e l l at time t, then C'(t) = C ( t ) I ( t ) and the e f f e c t i v e dose experienced by the c e l l , D ' ( t ) , i s D'(t)= /QC'(u)du . C l e a r l y the use of the indica t o r function I ( t ) represents an i d e a l i z a t i o n as the t r a n s i t i o n between phases of the c e l l cycle w i l l not be instantaneous. However, since the time spent i n t r a n s i t i o n between phases i s small compared to the time spent within each phase t h i s approximation seems reasonable. The form of C(t) i s dependent on the method of administration of the drug and w i l l be strongly peaked for a single i n j e c t i o n but w i l l be f l a t t e r for i n f u s i o n therapy. A further p r a c t i c a l problem to the c a l c u l a t i o n of e f f e c t i v e dose i s that some agents tend to block c e l l s from proceeding through the c e l l cycle however this phenomena w i l l not be modelled here. In the c a l c u l a t i o n of drug dose i t also may be that i f C(t)<k* (say) then the drug has no e f f e c t . This may be simply taken into account by considering C*(t) i n the c a l c u l a t i o n of dose where -42-C*(t) = C(t) i f C(t) >k* = 0 i f C(t) <k* It w i l l be noticed that none of the these considerations a l t e r the form of £(s) given i n (3.9.1-2). They a f f e c t the value of the binomial parameter and induce a possible complex time dependency. Assuming that the e f f e c t on a c e l l at time t^ only depends on the dose p r i o r to time t^ , and that the re l a t i o n s h i p s known for instantaneous doses apply, we may calculate the e f f e c t of drugs when C(t) varies slowly. To do t h i s we define the instantaneous doses at time t^ as follows: t. D. = J\ C(u)du where 0=t n<t 1...<t =t. l Jt. , 0 1 J l - l Then J D(t) = I D . i = l Let £(s;t) be the p r o b a b i l i t y generating function for the i n d i c a t i n g random variable of c e l l s u r v i v a l at time t and l e t £^(s) be the pr o b a b i l i t y generating functions for the ind i c a t o r random variables of c e l l s u r v i v a l for the instantaneous doses . Using (3.9.1) we have £(s;t) = l - T t ( D ( t ) ) + T i(D(t))s and F,.(s) = l - T t(D . ) + 7 c(D i)s. Then 5(s;tj) = S 1 ( S 2 - - ' C j ( s ) . . ) for j = l , . . . , J , i f j < D ( t ) ) = n TC(D ) . 3 i = l This condition holds i f the logarithm of the p r o b a b i l i t y of c e l l s u r v i v a l i s i n proportion to dose as has been found for chemotherapeutic agents -43-In cases where C(t) varies slowly in time, i t s effect may be computed using a series of instantaneous effects. In what follows we w i l l assume a single instantaneous effect with the understanding that i f this assumption were not appropriate we would consider a series of instantaneous doses as discussed above. This approach w i l l be useful i n cases additional different treatments are applied at times t^ (i=l,...,J). In human malignancy the concentration of drug, C(t), is frequently measured by noting the amount of drug in the serum and not at the tumor. As noted before (Chapter 1) the exposure of a c e l l to the drug is a function of i t s distance from the capillary bed and thus may vary between cel l s . The model of tumor growth we use here does not account for such an effect and incorporation of this feature must be deferred for further research. 3.3 E f f e c t s on the Normal Tissue The effects of treatment regimens are not necessarily specific to the tumor system but can also include the host's normal tissue. To account for these define a random variable: T=T{(D ,t ), ieN} where, T = 1 i f host suffers unacceptable toxicity for any t, = 0 otherwise, which reflects the toxicity of the regimen {(D^,t^), ieN} where is the dose given at time t^. Unacceptable toxicity may reflect death when considering animal experimentation and w i l l reflect a (complex) combination of objective and subjective measurements for human disease. A common objective (althougn not necessarily theoretically optimal) -44-i n experimental and human disease i s to select {(D^,t^), ieN} so that P{T=l}< P^ for the whole population where P^ i s some constant which depends on the experimenter or c l i n i c i a n . Frequently experimenters use PT=0.1, the so-called LD 1 Q' An assumption commonly made i n experimental research i s that the l i k e l i h o o d of t o x i c i t y depends upon the cumulative dose D=J^D^. We w i l l refer to t h i s as a "cumulative dose t o x i c i t y model". The "model" of t o x i c i t y used for chemotherapy i n c l i n i c a l medicine i s less e x p l i c i t . In general regimens are constructed so that the D^(i=l,...J) and t ^ + i - ^ ( i = l , • , J - l ) a r e fixed for a pre-determined series of J cycles of therapy. Here P^ for the complete regimen may be chosen to be quite high since the and t^ may be modified dynamically i f t o x i c i t y occurs. This w i l l be referred to as the " c l i n i c a l t o x i c i t y model" and w i l l be assumed when considering c l i n i c a l disease. This approach has i t s l i m i t a t i o n s since regimens are constructed using the frequency of acute t o x i c i t y with escalating dose and the influence of the timing on the t o x i c i t y response surface i s not usually examined. Having examined the e f f e c t of chemotherapy on tumor c e l l s and how doses are modified because of toxic side e f f e c t s we w i l l now discuss how the tumoricidal e f f e c t s of chemotherapy may be incorporated into the process (R^Ct),R^(t)}. We w i l l assume that the dosage schedule has been constructed so that t o x i c i t y i s at an "acceptable" l e v e l . 3.4 Modelling Treatment E f f e c t s on the Tumor C e l l s In modelling the e f f e c t s of treatment i t i s necessary to separate the primary from the secondary malignancies. By primary we refer to a c l i n i c a l l y detectable l e s i o n which i s subject to treatment. Secondary -45-disease w i l l r e f e r to any disease present o r i g i n a t i n g from the same i n i t i a l malignancy as the primary, but which i s not c l i n i c a l l y detect-able. Primary and secondary disease may be located at multiple s i t e s . Radiation therapy i s usually aimed at primary disease but i n c e r t a i n s i t u a t i o n s i t may be used upon secondary disease. The mathematical de s c r i p t i o n of the mechanism of action of ra d i a t i o n i s s i m i l a r to that of chemotherapy. That i s , c e l l s behave independently and the s u r v i v a l of each c e l l can be modelled as a Be r n o u l l i t r i a l . Tumors have been i d e n t i f i e d which are termed r a d i o - r e s i s t a n t and show a reduced s e n s i t i v i t y to the a p p l i c a t i o n of therapy. I n s e n s i t i v i t y to r a d i a t i o n i s believed to a r i s e as a r e s u l t of i n s u f f i c i e n t oxygen because oxygen i s known to enhance the c e l l k i l l i n g e f f e c t of r a d i a t i o n . Tumors with poor vascular supply, or tumor c e l l s within a region of poor v a s c u l a r i s a t i o n , w i l l tend to be re s i s t a n t because of the lower oxygen tension i n such regions. As we are mainly concerned with modelling chemotherapy we w i l l not be greatly concerned with the modelling of ra d i o - r e s i s t a n t c e l l s . We w i l l consider radiotherapy to be a non-selective treatment ( i . e . act equally on chemosensitive and chemoresistant c e l l s ) and model i t s e f f e c t by considering i t to act to increase d, the death rate of c e l l s , over the period of r a d i a t i o n treatment. Surgery i s almost exclu s i v e l y concerned with the therapy of primary disease. The response of i n d i v i d u a l c e l l s to surgery may not be as simple as for other modalities. For example, data on the surgery of breast cancer indicates that the variance of the res i d u a l number of tumor c e l l s i s much greater than would be expected using a binomial model [25]. This "extra-binomial" v a r i a t i o n may be modelled by assuming that the number of surviving c e l l s i s a binomial variable where the parameter i s a random v a r i a b l e . In t h i s case the binomial parameter w i l l be a function of the histology, l o c a t i o n and extension of the tumor. This p a r t i c u l a r model retains independence but offers great v e r s a t i l i t y . When modelling the e f f e c t of surgery we w i l l assume that the binomial parameter has been observed, so that the model of t h i s treatment regimen w i l l be s i m i l a r to the others. We w i l l use t h i s model when we consider data from breast cancer i n Chapter 5. When a single drug i s given alone v i a i n j e c t i o n , we w i l l assume that i t s e f f e c t i s instantaneous and independent of other treatments (see Section 3.2). If t j i s the time of the j-th treatment ( j = l , • • • , J ) , then by (2.3) we have <t>(s0,s1;tj) = * ( C 0 ( s 0 ) , C 1 ( s 1 ) ; t j ) , ...(3.11.1) where £ Q ( S Q ) , | ^ ( S ^ ) are the p r o b a b i l i t y generating functions for the i n d i c a t o r random variables of c e l l s u r v i v a l for the s e n s i t i v e and r e s i s t a n t c e l l s r e s p e c t i v e l y and t ^ represents the time immediately before treatment. At any time t * where tj<t*<t^ +^, the p r o b a b i l i t y generating function for the number of c e l l s i s given by <K8 ( J,8 1;t*) = * ( w ( ) ( t * - t j ) , w 1 ( t * - t j ) ; t j ) , ...(3.11.2) where w^(t*-t^) and w^(t*-t^) are given r e s p e c t i v e l y by (3.7) and (3.6). In p a r t i c u l a r , the continuity of the functions 6 and w^(t*-t^) (i=0,l) i n t* imply that 6 ( s Q , s ^ ; t I s given by the righ t hand side of (3.11.2) with t=t ^. Notice that these equations also apply to phase s p e c i f i c agents since the ^ ( s ^ ) a r e °f the same form. In addition <j>(sQ,s^;t^) i s -47-given by (3.5) with t=t^. These relationships may be used recursively to calculate the probability generating function for {Rrj(t) ,R^(t)} after several courses of the same agent. The expected number of resistant and sensitive cells may be recursively calculated using m Q(t ) = TI (D )m 0(t") and m^t ) = TC (DJm^t"). From (3.8) we also have m O ( t j + l ) = m 0 ( t j ) e x P { ( ° - « b - v ) ( t j + 1 - t )}, ...(3.12.1) m l ( t j + l ) = [m 1(t j)+m 0(t : j)(l-exp{-(ab+v)(t j + 1-t j)})]exp{6(t j + 1-t j)}. ...(3.12.2) If chemotherapy is not injected but is given continuously over some fi n i t e period, then i t s effect may be computed as discussed in Section 3.2 . The probability generating functions and expected values may be calculated using (3.11.1-2) and (3.12.1-2) where now the t^ are the times of the approximating instantaneous doses as discussed in Section 3.2. The effects of surgery or radiation on the joint probability generating function can be assessed using the same techniques i f i t is assumed that the survival of the individual cells are independent Bernoulli t r i a l s . The complex form of (3.7) and (3.5) and the recursive nature of the operation needed to determine * ( S Q , S ^ ; t ) , when treatments have been applied, indicate the need for some simple measure which summarizes the effects of treatment. The expected values ^ ( t ) and m^(t) provide one such summary, however we w i l l now develop a more useful summary measure. 3.5 Summarising Treatment Effects Using the previously described recursive relationships i t i s possible to calculate the probability generating function -48-A(s^,S2;t) for a r b i t a r y t. However, the relationships are d i f f i c u l t to i n v e r t and i n order to obtain the d i s t r i b u t i o n of c e l l counts at time t. We therefore consider some quantities which w i l l provide a useful summary of the behaviour of the system at time t. The expected values n ^ t ) , m^(t) are two useful measures. Another quantity of some i n t e r e s t i s p{N(t)=R Q(t)+R 1(t )=0} since this i s the p r o b a b i l i t y that there are no stem c e l l s at time t. Since the elimination of the stem c e l l s implies that the tumor w i l l eventually become extinct (or not grow s u f f i c i e n t l y to k i l l the patient or animal) t h i s may be thought of as the p r o b a b i l i t y that the tumor can no longer cause the death of the patient. The p r o b a b i l i t y that there are no stem c e l l s at time t i s given by 6 ( 0 , 0;t) and may be e a s i l y calculated from (3.11.1-2). However, 6 ( 0 , 0 ,;t) does not represent the p r o b a b i l i t y that the tumor has been cured by the treatment regimen, for i f t j i s the time of the l a s t treatment and t'>t'>t then, for the model under consideration x. Z. \j < K 0 , 0;tp=P{R 0(tp= 0 , R 1(tp= 0}>P{R ( )(tp= 0 , R 1(tp= 0 } = 6 ( 0 , 0 ; t p with equality i f d=0. This motivates consideration of P = E[P{N(~)=0|N(t )} ] . tJ J We w i l l r e f e r to P as the p r o b a b i l i t y of cure, which w i l l of course t J depend on the regimen being used. Since each c e l l has a p r o b a b i l i t y e=d/b of spontaneous e x t i n c t i o n (see the discussion i n Section 3.1) and c e l l s behave independently, the p r o b a b i l i t y n c e l l s w i l l go spontaneously extinct i s e • It follows that N(t ) R (t )+R (t ) P = E[e ] = E[e U ] = 6 ( e , e ; t T ) ...(3.13) fcJ J ' -49-At this point we should note that P w i l l not correspond exactly to J the c l i n i c a l l i k e l i h o o d of cure since i t includes the contribution of sample paths destined for e x t i n c t i o n , which may nevertheless grow s u f f i c i e n t l y to cause patient death. Such paths occur with i n s i g n i f i c a n t l y small p r o b a b i l i t y i n most p r a c t i c a l s i t u a t i o n s and P J w i l l be considered to be equal to the c l i n i c a l p r o b a b i l i t y of cure. In some cases, as i n the treatment of L1210 leukemia by the drug Ara-C [26], resistance can be e f f e c t i v e l y absolute for any drug concentration which does not r e s u l t i n animal death. In th i s case there exists the p o s s i b l i t y that a tumor cannot be cured by the drug no matter what dose i s used. If we also assume that at the therapeutic dosage H;Q(D)=0, then i t i s only necessary to apply a single course of the drug (since subsequent courses w i l l have no e f f e c t ) , and we have the p r o b a b i l i t y of cure, P , i s given by 1 P = 4(e,e;t )= <|>(l,e;t7). ...(3.14) 1 This expression may be viewed as an approximation to the p r o b a b i l i t y of cure for cases i n which ( D ) = l , ^ Q ( D ) =0 and the treatments are applied frequently. Using equations (3.5), (3.6) and (3.7) we have P. -<KG(t,),£), ...(3.15) 1 where G(t 1)=e + -(6+<xd+v)t (l-e)(6+ad+v)e _ , t, N K w i -(6+ad+v)t 1 1 (6+ad+v)-o ( l - a ) [ l - e 1] and (Ks 0 , s 1)=4 ( s 0 , s 1;0). If (6+ad+v)t 1»l, then P = <Ke,e). ...(3.16) 1 Thus for s u f f i c i e n t l y large t 1 , P i s approximately equal to <\>(e,e), the 1 -50-p r o b a b i l i t y , that the tumor w i l l go spontaneously e x t i n c t . Equation (3.15) may be used to assess the c u r a b i l i t y of an experimental tumor where the number of c e l l s implanted has p r o b a b i l i t y generating function I K S Q , S ^ ) , the drug parameters are HQ ( D)=0, TC^(D)=1 and the tumor i s treated at time t ^ where t ^ i s large. However i t also i l l u s t r a t e s that the theory developed to this point i s of lim i t e d use i n describing the treatment of large tumors (either c l i n i c a l or experimental) since i t includes spontaneous extinctions (which w i l l l a r g e l y have occurred i n the early h i s t o r y of the neoplasm). This deficiency i s e s p e c i a l l y marked f or human disease where the tumor originates with a single s e n s i t i v e stem c e l l i . e . 4 > ( S Q , S ^ ) = S Q and thus the p r o b a b i l i t y of spontaneous e x t i n c t i o n can be large ( i f e i s l a r g e ) . Before discussing modifications to exclude spontaneously extinct tumors we w i l l f i r s t consider an example which i l l u s t r a t e s an a p p l i c a t i o n of the theory developed to th i s point. Example: Consider the sp e c i a l case u^(D)=l for a l l D where the drug considered i s not phase s p e c i f i c . Let TiQ(D)=exp {-kD} as i n Section 3.2. Consider a tumor system where v=d=0 which follows the cumulative dose Consider the s p e c i a l case u^(D)=l for a l l D where the drug considered i s not phase s p e c i f i c . Let ^ (D^exp{-kD} as i n Section 3.2. Consider a tumor system where v=d=0 which follows the cumulative dose t o x i c i t y model. We wish to determine whether i t i s better to give a single dose of magnitude D at time t ^ or two doses and at times t ^ and t ^ where D^+D2=D and ^ ^ j / A regimen i s better i f i t has a higher p r o b a b i l i t y of cure. - 5 1 -Since 7i^(D)=l for a l l D we need not consider r e s i s t a n t c e l l s present at time t^ as these w i l l be unaffected by either regimen. Thus we w i l l only consider 6*(sQ) = (|>(sQ,0;tp and assume without loss of generality that there are no re s i s t a n t c e l l s present at time t ^ . If a l l the drug i s given i n a single dose at time t ^ , then the p r o b a b i l i t y of cure i s P t ^ 6 * ( l - 7 t 0 ( D ) ) . . . . ( 3 . 1 7 ) For the second regimen where two doses are used we must consider the intertreatment development of resistance. Using equations ( 3 . 6 ) and ( 3 . 7 ) we have -bu S *3 w x(u) = - - r - j - , . . . ( 3 . 1 8 . 1 ) l-s^+ s^e , , -2+a -bu r. , -bu, 6 I i. -s TS e I w Q(u) = w l (u) + * 1 _ B L K _ 1 + A ' . . - ( 3 . 1 8 . 2 ) r , -1 ~ l r / i . -bu.-l+a LSQ-SJ^] -S 1 [ ( 1 - S ^ S ^ ) - 1 ] where u=t 2-t^. The p r o b a b i l i t y of cure at time i s P = <D(0,0;t 2) = 6(l-7C 0(D 2),0;t2) since spontaneous death does not occur (d=0). Now by ( 3 . 1 1 . 2 ) we have * ( s ( ) , s 1 ; t 2 ) = 6(w 0(u),w 1(u);t 1), where w Q ( U ) and w^(u) are given by ( 3 . 1 8 . 1 - 2 ) . Taking the l i m i t as sL->0 i n ( 3 . 1 8 . 2 ) we have [ l - 7 C 0 ( D 2 ) ] e " b u p t - •( — ^T'0'^ Z2 l - [ l - 7 c 0 ( D 2 ) ] ( l - a ) [ l - e D U ] 7L ( D 1 ) ( l - u N ( D ))e b U = ^ ( I - T E Q C D ^ + ^ — - — - ) . . ( 3 . 1 9 ) l - ( l - 7 c 0 ( D 2 ) ) ( l - a ) ( l - e " b u ) a f t e r taking account of c e l l k i l l TIQ(D^) at time t ^ . Since <t>*(s) i s a p r o b a b i l i t y generating function i t i s monotonic -52-non-decreasing on [ 0 , 1 ] and we need only compare the arguments of 6 * i n equations ( 3 . 1 7 ) and ( 3 . 1 9 ) . But by assumption 7 ! Q ( D ) = T C Q ( D ^ ) T I Q ( D 2 ), and thus for u> 0 , 0 ( D l ) ( l - , 0 ( D 2 ) ) e - b u 1 - V V + — l - ( l - 7 i 0 ( D 2 ) ) ( l - a ) ( l - e b U ) -bu = l - n 0 ( D 1 ) T i ( ) ( D 2 ) - n 0 ( D 1 ) ( l - T i 0 ( D 2 ) ) [ l - 6 l - ( l - 7 t 0 ( D 2 ) ) ( 1-a) ( 1-e b u ) < 1 - U 0 ( D 1 ) U 0 ( D 2 ) = 1 - T X 0 ( D ) . . . . ( 3 . 2 0 ) Thus giving the t o t a l dose at t^ r e s u l t s i n a higher p r o b a b i l i t y of cure than s p l i t t i n g the dose into two parts given at t ^ and t ^ t ^ . ^ we set D ^ = 0 , D 2 = D we also see that giving the t o t a l dose l a t e r i s associated with a lower p r o b a b i l i t y of cure. More generally i t i s preferable to give a drug i n the highest possible dose at the e a r l i e s t time rather than spread the same dose over a series of smaller doses. This provides a p a r t i a l j u s t i f i c a t i o n for the strategy commonly employed i n c l i n i c a l medicine of using the highest possible doses that are t o l e r a b l e . These observations may also be generalized to cases where v>0,d>0 and 7 t ^ ( D)<l, since the underlying nature of the process Is unchanged although the computations become more complex. This completes consideration of t h i s example. When observing a c l i n i c a l or experimental tumor the number of r e s i s t a n t stem c e l l s at any point i n time i s usually unknown. The t o t a l number of stem c e l l s can be estimated either by d i r e c t experimentation or by applying the appropriate formula for Pg (the proportion of stem c e l l s ) developed i n Chapter 2 to the observed o v e r a l l tumor s i z e . In both these s i t u a t i o n s we w i l l refer to the number of stem c e l l s as being "observed" even though they may only have been i n f e r r e d from the observed tumor s i z e . As previously mentioned the theory developed i n Section 3.1 describes the growth of the sen s i t i v e and r e s i s t a n t stem c e l l s and includes cases where these c e l l s go spontaneously e x t i n c t . By the time a tumor has reached a si z e where i t i s c l i n i c a l l y detectable the l i k e l i h o o d of spontaneous e x t i n c t i o n i s small. This d i r e c t l y leads to the consideration of P{R^(t)|N(t)}. Unfortunately t h i s d i s t r i b u t i o n i s not e a s i l y obtained because the i n t e g r a l i n (3.7) cannot be expressed i n terms of standard functions. A further problem i n the consideration of human tumors i s ignorance of the age, t, of the tumor and i t i s therefore desirable to construct expressions independent of t h i s parameter. Since these problems are of ce n t r a l importance i n the construction of an appropriate d i s t r i b u t i o n for the number of r e s i s t a n t c e l l s we w i l l o u tline three seperate approaches which provide approximate solutions to this problem and w i l l be of use i n various experimental and c l i n i c a l s i t u a t i o n s . 3.6 Conditioning on N(t) - Approximation 1 As a f i r s t approximation to the problem of conditioning upon N(t) we w i l l examine the process where sample paths that correspond to tumors which go spontaneously extinct ( i n the absence of treatment) are excluded. The basic idea i n t h i s approximation w i l l be to consider the d i s t r i b u t i o n P{R (t),R (t)|N(t)>0} and to approximate i t by P{R Q(t),R 1(t)|N(»)>0} and substitute a pl a u s i b l e value for t derived from consideration of the observed d i s t r i b u t i o n of stem c e l l s . This approach has previously been used elsewhere [27]. In the absence of treatment, we have for any t - ^ t ^ w n e r e t 2>t^that N(t2)>0 + N(t1)>0. Thus we may exclude r e a l i z a t i o n s corresponding to tumors which go spontaneously extinct at any time by conditioning on N(=°)>0. This i s (approximately) equivalent to including only those r e a l i z a t i o n s corresponding to tumors which, i f l e f t untreated, could go on to r e s u l t i n patient or animal death ( i f we exclude r e a l i z a t i o n s which grow to a s u f f i c i e n t size to cause death but are nevertheless destined for e x t i n c t i o n ) . We w i l l now calculate the p r o b a b i l i t y generating function 6 " ( S Q , S ^ ;t) of the process {R^t) ,R^(t)} which consists of a l l sample paths {R Q(t) ,R 1(t) } for which N(°°)>0. Let CO oo <D'(s ,s ;t) = I l?{ R (t)=i,R (t)-j|N(«)>0}sJ . i=0 j=0 1 L {i,jM0,0} To evaluate t h i s p r o b a b i l i t y generating function we f i r s t note P{R Q(t)=i, R L ( t ) = j | N(»)>0} = (1-P{N(»)=0}) ~ 1 [ p { R 0 ( t ) = i , R L(t)=j} - p{R Q(t)=i, R t ( t ) = j , N(»)=0}]. Since the c e l l s behave independently, the p r o b a b i l i t y that a single c e l l (either s e n s i t i v e or r e s i s t a n t ) w i l l go spontaneously extinct i s equal to E=d/b and i t follows that p{R0(t)=i,R1(t)=j,N(o>)=0} = P { R 0 ( t ) = i , R 1 ( t ) = j } e i + j . After a l i t t l e algebra we obtain * ' ( s ( ) , s 1 ; t ) = CO CD (l-p{N(co)=0})" 1((D(s n,s 1;t)-P{N(t)=0}- I I P{R „(t)=i,R (t )=j } e ^ s j . s j) . i=0 j=0 UJMo.o} Since P{N(<»)=0}=<Ke, e)=<K0,0;<=°) i s the p r o b a b i l i t y that the stem c e l l compartment w i l l go e x t i n c t , the desired p r o b a b i l i t y generating function -55-may be expressed as 4 ' ( s 0 , s 1 ; t ) = [l - ( K e , e ) ] ~ 1 { ( t ) ( s 0 , s 1;t ) - ( | ) ( e s 0 , e s 1;t)}. ...(3.21) We may calculate the f i r s t moments m^Ct) and m^(t) of the process by d i f f e r e n t i a t i n g (3.21) with respect to s^ and resp e c t i v e l y and evaluating at S Q = S I = 1 . After some algebra we obtain m^(t) = E[RjJ(t)] = l - ^ e . e ) [m 0exp(6-ab-v)t-{^J^- | s = £ } e exp {-(6+ad+v)t} ], ...(3.22.1) and m'(t) = m£(t) + m£ (t) 1 r, . •> 6t ;5((i(s,e) i , C 4 ( E , S ) . i -6tn .„ = 1-4, ( £ > e ) l ( W e " H t ^ ls= e + - i s — ~ lS= £} e e J' -.-(3.22.2) where m^(t) i s given by the difference of these two expressions. Equation (3.21) shows that the p r o b a b i l i t y generating function for the condi t i o n a l (on N(°°)>0) process may be expressed i n terms of that of the unconditional process, and thus may be calculated using formulae (3.5) to (3.7). When modelling the e f f e c t s of treatment we would use (3.21) for the i n i t i a l growth period and (3.11.2) for growth i n the intertreatment i n t e r v a l s . We do not use (3.21) for intertreatment growth, as t h i s would have the e f f e c t of assuming that the tumor could not be cured. If we again consider ( a s i n the previous section) the s p e c i a l case 7 IQ(D)=0, I;^(D)=1 then we may cal c u l a t e P F C , the p r o b a b i l i t y of cure, for this process with p r o b a b i l i t y generating function given by (3.21). In analogy to (3.14) we have P =6'(l , e;t 1) and thus using (3.21) we obtain 1 P =[l - (Ke,e)] - 1 [<KG(t ),e)-<D(e,e2;t")] 1 where G(t^) i s a s s p e c i f i e d i n (3.15). For the case where I | ; ( S Q , S ^ ) = S Q , as i s l i k e l y i n human disease, t h i s s i m p l i f i e s to y i e l d h " L l + e ( l V 6 t l ) a b + v + fid-^Cti) g(t )h(t ) ], ...(3.23) e 1 ( l - e ) 1 - b ( l - a ) /Jlg(v)h(v)dv where g(t)=exp{-(6+ad+v)t} and h ( t ) = [ l + e ( l - e ~ 6 t ) ] " 2 + < x . Examination of (3.23) shows that as expected lim P =0, t 1^° 1 that i s , cure w i l l occur with vanishingly small p r o b a b i l i t y i f treatment i s delayed too long. This may be contrasted with (3.16) where, for d>0, the p r o b a b i l i t y of cure was always greater than zero since t h i s expression included the l i k e l i h o o d that the tumor did not e x i s t . As indicated previously, the age of a tumor i s only known i n c e r t a i n experimental s i t u a t i o n s and i s of course measured i n a r b i t r a r y u n i t s . For human disease we usually do not know the age of the tumor and thus we do not know the time of the f i r s t (or any subsequent) treatment measured on the scale where the tumor originated at time t=0. Once one treatment time i s s p e c i f i e d on t h i s scale then a l l other treatment times are known. It seems most natural to specify treatment times i n terms of t]., the unknown time of f i r s t treatment. A reasonable approach i n modelling treatment e f f e c t s on a s p e c i f i c tumor class i s to choose t]^ so that the d i s t r i b u t i o n of stem c e l l s ( i m p l i c i t i n (3.21)) at the time of f i r s t treatment approximates that observed i n the tumor type. If we l e t N'(t) be the random variable with the d i s t r i b u t i o n of N(t) condi t i o n a l on N(°°)>0, then we wish to choose t(=t^) so that the d i s t r i b u t i o n of N'(t) i s s i m i l a r to the observed d i s t r i b u t i o n at diagnosis for the tumor c l a s s . -57-For the s p e c i a l case < J ; ( S Q , S 1 ) = S Q the p r o b a b i l i t y generating function f or N'(t) i s given by (3.21) with s Q=s 1=s; that i s <t>'(s,s;t) = [l-£] - 1[ 6(s,s;t)-*(es, es;t)] .. .(3.24) Using A ( S Q , s ^ ;t) as given by (3.5) with S Q = S ^ = S , the right hand side of (3.24) may be expanded i n powers of s to y i e l d the d i s t r i b u t i o n for N'(t): , , i .1..,. - 6 t N i - l -6t P{N'(t)-l} = ( b " d ) 6 ( 1 " ^ > 6 for i-1,2, (3.25) (b-de o t ) 1 " r i For large i , such that e 1 « 1 we have P{N-(t)-i}- — ^ 2 [ 1 - ( l - ^ t ] i - 1 ( l - e ) e - 6 t , (1-ee 0Zy (1-ee 0 C ) and i f t i s also large, so that 6 t » l , then to leading order i n e p{N'(t)-i} = ( l - ( l - e ) e " 6 t ) i - 1 ( l - e ) e ~ 6 t . ...(3.26) Examination of (3.26) shows that the d i s t r i b u t i o n of N'(t) i s approximately geometric and only depends upon b and d through m'(t). This has three implications for the modelling of "large" tumors. F i r s t l y , the approximation to the d i s t r i b u t i o n of N'(t) has only one parameter, i t s mean value, and thus i n attempting to determine an appropriate value of t^ ( i n terms of given b and d) one need only employ one summary measure of the d i s t r i b u t i o n . Secondly, whatever summary measure i s used (mean, median e t c ) t h i s w i l l always r e s u l t i n choosing t ^ to s a t i s f y some r e l a t i o n s h i p i n terms of the mean m'(t^). Thir d l y , we may wish to compare the d i s t r i b u t i o n of r e s i s t a n t c e l l s f o r d i f f e r e n t tumor models with d i f f e r i n g b, c and d but the same a, 8 and y. If the d i f f e r e n t tumor models are required to have the same mean numbers of stem c e l l s , then they w i l l have approximately the same d i s t r i b u t i o n of stem c e l l s . Thus differences i n P between such models w i l l not be due 1 -58-to differences i n the d i s t r i b u t i o n of the number of stem c e l l s . The c o n d i t i o n a l process (with p r o b a b i l i t y generating function given by (3.21)) thus provides a convenient framework for comparing the e f f e c t s of various parameters (including treatment) on the c u r a b i l i t y of the tumor for a f i x e d d i s t r i b u t i o n of stem c e l l s at t ^ . However th i s approach w i l l not be suitable for the modelling of s i t u a t i o n s where the observed d i s t r i b u t i o n of stem c e l l s at diagnosis i s not well approximated by a geometric d i s t r i b u t i o n . We w i l l now examine some elementary properties of this process. Consider the expected f r a c t i o n of r e s i s t a n t c e l l s , which i s approximately given by m^(t)/m'(t). If we assume that 4 > ( S Q , S ^ ) = S Q and, using the mean as the summary measure, we choose t ^ so that N*=[(l-e) e l ] , (N* i s the mean size of the tumor at diagnosis), then from (3.22.1) and (3.22.2), m . ( t ^ = l - [ ( l - e ) N * ] ° . ...(3.27) From t h i s i t may be seen, as expected, that the f r a c t i o n of r e s i s t a n t c e l l s increases as any of a, 8, or y increase. Increases i n c (since v=Sc+y) or d also increase the f r a c t i o n of r e s i s t a n t stem c e l l s although these parameters are also related to P , the f r a c t i o n of stem c e l l s i n the tumor (Chapter 2). We can also examine the e f f e c t of the parameters a, 8, y» b, c, and d upon Pfc given by (3.23), where 7i;0(D)=0, n^{D)-l and as before t ^ i s chosen to be given at a fixed mean stem c e l l compartment size i . e . N*=m'(t^). As expected, increases i n a, 8 or y decrease the value of t h i s function for fixed N*. Increases i n c are also found to decrease -59-P but the value of t h i s function i s not influenced by changes i n d. 1 This function i s plotted for various values of the parameters i n Figure 1. Although increasing d increases the mean number of r e s i s t a n t c e l l s i t does not change the p r o b a b i l i t y that a fixed size stem c e l l compartment w i l l be curable because of the compensating e f f e c t of increases i n the spontaneous death rate of r e s i s t a n t c e l l s . The considerations presented here for the case 7IQ(D)=0, TC^(D)=1 carry over generally to the case TJ^(D)=1, IXQ(D)=TIQ>0 except, of course, that the magnitude of P w i l l depend upon the effectiveness and timing 1 of subsequent treatments. We w i l l now turn to consideration of a second approximation for conditioning on N(t). 3.7 Conditioning on N(t) - Approximation 2 In most cases of p r a c t i c a l i n t e r e s t a « l and v « b ( i . e . t r a n s i t i o n s to resistance proceed slower than growth) so that, for the majority of sample paths R (t)«R Q(t) and thus R Q(t)=N(t). This suggests that i t may be reasonable to approximate the d i s t r i b u t i o n P|R^(t)|N(t)} with the d i s t r i b u t i o n P|R^(t)|R^(t)}. This c a l c u l a t i o n i s complex for general ( | » ( S Q , S ^ ) and we w i l l only consider the s p e c i a l case (|J (SQ,S^ )=SQ. Thus 6(sQ,s^;t ) = W Q(t) as given i n equation (3.7). Since A(sQ,l;t) i s the p r o b a b i l i t y generating function of the number of s e n s i t i v e c e l l s at time t, the c o e f f i c i e n t of S Q i n the expansion of w^(t) (evaluated at s^=l) i n powers of s^ gives the p r o b a b i l i t y that there w i l l be i s e n s i t i v e c e l l s at time t . Performing t h i s expansion y i e l d s -60-P{R 0(t)=i} - ^ e ' X t [ b ( l - a ) ( l - e - X t ) ] f o r 1 - 1 > 2 f . . . , ... ( 3.28) [b(l-a)-(d+v)e A Z ] X ^ L where X=6-ab-v. S i m i l a r l y the c o e f f i c i e n t of s^ i n the expansion of W g ( t ) (for general s^) y i e l d s 00 -2+cc i-1 1 p { ( R 0 ( t ) - l , V O - j l B J - ' ^ ( t ) ^ ( ^ 1 + 1 i-1,2,..., ...(3.29) j=0 [6 + s 1 I ( t ) ] where f ( t ) i s given i n (3.7) and I ( t ) = b ( l - o c ) f ( v ) d v . Taking the r a t i o of (3.29) to (3.28) and setting s^=s then y i e l d s the p r o b a b i l i t y generating function C ±(s;t) of the d i s t r i b u t i o n P J R ^ t ) |R Q(t)=i} as .-2+a \tr ..,±-1 C.(s;t) - 6 2 £ ( t ) e . j f ( t " ' ...(3.30) \ A [ h ( t ) ] 1 + i where g(t) = I ( t ) / [ b ( l - a ) ( l - e _ X t ) ] and h(t) = [ 6~2+0t+ s I ( t ) ] / [ b ( l - a ) - ( d + v ) e " X t ] . We may use (3.30) to evaluate E [ R 1 ( t ) | R Q ( t ) = i ] , by d i f f e r e n t i a t i n g with respect to s and set t i n g s=l. However, the r e s u l t i n g expressions are rather complex involving the difference of a number of exponential functions. If 6»<xb+v (which implies a « l ) then we obtain E [ R 1 ( t , | R 0 ( t , . i 1 - l ^ l g ' f c ^ . ^ ' - l ] • H+H „ t [ e(«W-v)t . j j + ^ ^ ...(3.31) where L d±v j ((2-«) ( 6 - a b - v ) 2 } -( 6-2ab-2 v)t + - ( S - a b - v ) ^ 1 v b(l-a) ; v ( l - a ) o ( 2 6 - a b - v ) ; ; and _ b(2-a) (ab+v) 6t (ab+v)t L 2 ~ 6(26-ab-v) e + U ( e ) -For large t ( 6 t » l ) , i s dominated by the f i r s t term i n (3.31). If i> E(R_ ( t ) l then i e ^ < x b + V ^ t > e 6 t and L„ i s dominated by the f i r s t term of (3.31). If i«E[R^(t)] then may be comparable or larger than the f i r s t term i n (3.31) and the approximation Rg(t)=N(t) may not be a good one. However, th i s occurs with small p r o b a b i l i t y when a b + v « 5 . From (3.8), when m^(0)=0, as here, we have E [ E { R l ( t ) | R 0 ( t ) } ] = E [ R 0 ( t ) ] ( e ( a b + V ) t - l ) which shows that the terms and i n (3.31) have expectation 0 (approximately) with respect to R^. The f i r s t term of (3.31) w i l l i n most cases (where i=«E[RQ(t)]) be a reasonable approximation to E[R^(t)|Rg(t)] for large t except i n situa t i o n s where t i s such that E[R Q(t)]»i. For the s p e c i a l case it^(D)=l,7r,Q(D)=0 we may cal c u l a t e Pfc ( i ) , the p r o b a b i l i t y that a tumor with i s e n s i t i v e stem c e l l s w i l l be cured by a single course of therapy at time t ^ . Using the same argument as previously used i n deriving equation (3.14) we have P t 1 ( i ) = C i ^ V Using (3.30) we f i n d -(ab+ad+2v)t r(o+«d+v) [ b ( l - a ) - ( d + v ) e ~ ( 6 ~ g b " " v ) t l - . 2 1 1 6 /* v v r-i , v - (6+ad+v)t 1 . J(6-ab-v)[b +v -d(l-a)e v 1] z-t - (6+ad+v)t 1 w . . . . . . - ( 6-ab-v)t r ( l - e 1) (b( l-a)-(d+v)e 1) -ii-1 * L - ( 6-ab-v)t 1 w . . . . . v -(6+ad+v)t.. J (1-e ' l)(b+v-d(l-a)e 1) If ( 6-ab-v)t»l we obtain the approximation P . C D - C . U . O , e - ( ^ + o C d + 2 v ) t [ ( 6 + a d + v ) b ( l - o c ) ] 2 [ M ^ O ] - ! . _ T\ / ^>X\ Y' L (6-ab-v)(b+v) J L b+v J If i n addition the i n d i v i d u a l mutation rates are small so that (ab+d+2v)t«l and 6»<xb+v, then -(ctb+ocd+2v)t . e -1 (o+ocd+v)b(l-a) b(l-tx) . ' (6-ab-v)(b+v) ~l> b+v _ i a V / D and thus C,(e;t )=P ( i ) » [1-a-v/b] 1" 1. ...(3.33) 1 -62-This function i s plotted for p a r t i c u l a r a and v/b i n Figure 1. The form of this r e l a t i o n s h i p i s very simple and makes i n t u i t i v e sense as follows. In Section 3.6 we found that P did not vary with d 1 (for the co n d i t i o n a l process considered there) for f i x e d mean size N*. We also found that the d i s t r i b u t i o n of N'(t) was approximately geometric and was independendent of d once the mean was f i x e d . Since P i n Section 1 3.6 i s the average p r o b a b i l i t y of cure across the d i s t r i b u t i o n (approximately given by (3.26)) of the number of stem c e l l s and both are approximately independent of d for given mean s i z e , i t seems l i k e l y that the i n d i v i d u a l terms representing the c u r a b i l i t y at a given size are also independent of d. That i s exactly what i s indicated by equation (3.33) for N(t)=R n(t). This suggests an approximation to P as given by (3.23) 1 can be obtained by taking the product of the r i g h t hand sides of equations (3.26) and (3.33) and summing. Letting m'=[(l-e)e ^ 1 ] ^ we have CO CO P = I p{cure|N'(t )-i}p{N'(t ) - i } * I P (i)p{N'(t )-i} 1 1-1 1-1 1 CD = I ( l - a - v / b ) 1 " 1 m'"1 (1-m'" 1) 1" 1 i = l = [l-a-v/b+(a+v/b)m'] - i. ...(3.34) The d e r i v a t i o n of (3.34) uses (3.26) where N'(t) was assumed to be large. If m' i s large the p r o b a b i l i t y that N'(t) i s small w i l l be small and thus (3.34) can be expected to be a reasonable approximation. Numerical evaluation of (3.23) and (3.34) for a=10" 3,10 _ 1 +,... ,10 - 8 and v/b=10a, a, 1 0 - 1 a shows that the absolute difference between (3.34) and (3.23) i s less than 0.01 for 10<N', m'<109. Thus (3.34) provides a reasonable -63-Flgure 1 Probability of Cure for Approximations 1 and 2. 10° 101 102 103 104 105 106 107 N u m b e r of S t e m C e l l s The function P plotted as a function of N* where t 1 i s selected to t l s a t i s f y N*=[(l-E)e ^ 1 ] \ for various values of the mutation ra t e s . The s o l i d curves are for equation (3.23) and dashed curves are for equation (3.33). The two curves to the righ t have a=5xl0~ 6 and v/b=5xl0 - 6, and those to the l e f t have a=5xl0 - l t, v/b=5xl0 - 1*. These curves do not depend on b (which behaves as a constant for s c a l i n g time) and are e s s e n t i a l l y coincident for a l l e=d/b. approximation to (3.23) and gives a deeper understanding of the nature of V Conditioning on R g ( t ) appears to be reasonable i f t i s known, however we are also interested i n situ a t i o n s where i t i s not. The expression for E[R^(t)|Rg(t)] which i s approximated by (3.31), depends upon t and thus i t s d i s t r i b u t i o n depends upon the choice of t. Here we w i l l propose another more complex method for removing t than that which was presented i n Section 3.6 although t h i s w i l l again be approximate. The basis of t h i s approach i s to observe that when there are many stem c e l l s present t h e i r growth i s quite regular. The major contribution to the d i s t r i b u t i o n of the number of stem c e l l s at time t (when grown up from a single c e l l ) r e s u l t s from the v a r i a b i l i t y of growth when small numbers of c e l l s are present. This suggests that i t should be reasonable to approximate the growth process by a two phase model i n which the growth of s e n s i t i v e stem c e l l s i s f i r s t stochastic and l a t e r d e t e r m i n i s t i c . A schema i l l u s t r a t i n g this approach i s given i n Figure 2. Resistant c e l l s w i l l be assumed to grow s t o c h a s t i c a l l y i n both phases. In the stochastic phase, which i s r e s t r i c t e d to the i n t e r v a l [ 0 , f ] , we use (3.30) and t " i s chosen so that the p r o b a b i l i t i e s p{R 0(t')>U|R 0(f)*0} and p{R 0 (O<L|R 0(t')*0} are both small. L represents a lower l i m i t for which growth i s s u f f i c i e n t l y regular and U<N where N i s the observed siz e of the s e n s i t i v e stem c e l l compartment that we are interested i n conditioning upon. For example, i t i s easy to show (using (3.28)) that i f the mean of the geometric d i s t r i b u t i o n of R g ( t ) i s mQ(t)»l then we can choose U and L where U/L=103 so that P{U>R (t')>L|R n(t')*0}>0.99. Thus i n situ a t i o n s where the number of stem c e l l s , N ( = number of s e n s i t i v e stem c e l l s ) , at diagnosis s a t i s f i e s N>106, we may put U=106 and L=10 3 and choose t" so that P{U>R 0(t')>L|R 0(t')^0}>0 .99. Thus even at the lower l i m i t L, there w i l l be 1,000 s e n s i t i v e stem c e l l s and growth af t e r t " can be expected to be approximately regular. After time t ' , s e n s i t i v e stem c e l l s w i l l be assumed to grow exponentially with parameter 6-ab-v. We w i l l now calculate the p r o b a b i l i t y generating function, 6(s;t), for the number of r e s i s t a n t c e l l s i n the deterministic phase and examine some basic properties of t h i s process. Consider a model of this process where the s e n s i t i v e stem c e l l s grow exponentially. In p a r t i c u l a r we w i l l assume R Q C O = A r j e ^ V ^ which i s chosen to be the same as t h e i r expected growth under a stochastic model; see (3.8). Using a r e s u l t for f i l t e r e d Poisson processes [21], the p r o b a b i l i t y generating function <t>(s;t), of the number of r e s i s t a n t c e l l s i s given by « 6(s;t)=exp{/gk(u)[ri(s;t-u)-l] du}, ...(3.35) where 4(s;0)=l, k(u)= AQ(ab+v)e^ a b V ^ U i s the rate at which new mutations to resistance occur, and T|(s;t) i s the p r o b a b i l i t y generating function of the b i r t h and death process with parameters b and d. n(s;t) i s given by w^(t) (equation (3.6)) with s^=s. Equation (3.35) cannot be written i n terms of standard functions, however the mean may be obtained by d i f f e r e n t i a t i n g with respect to s and se t t i n g s=l; t h i s y i e l d s r r „ . (6-ob-v)t, (ab+v)t 1 N _, / 4 . w (ab+v)t 1 N / 0 E [ R x ( t ) J = A^e ( e v -1) = R Q ( t ) ( e v -1). ...(3.36) Further, evaluating (3.35) for s=e y i e l d s the p r o b a b i l i t y of cure at time t as -66-Flgure 2 Schematic Representation of the Two Phase Growth Process for Sensitive C e l l Growth Used i n Approximation 2. Sensit ive Ce l l Growth: Sensi t ive Ce l l Growth: Stochast ic Determinist ic t' t'+Tj (Time) - 6 7 -4>(e;t) = exp{- i| b±^iili>(R 0(t) - A Q)} = ( l - a - v / b ) R ° ( t ) ^ . . . ( 3 . 3 7 ) for 6»ab+v. We see that this deterministic model y i e l d s s i m i l a r expressions for E[R^(t)] and P t(when T I Q ( D ) = 0 , ^ ( D ) = l ) as for the process conditioned on Rg(t) developed previously i n this section; see ( 3 . 3 1 ) and ( 3 . 3 3 ) . We w i l l now construct a p r o b a b i l i t y generating function for the d i s t r i b u t i o n of r e s i s t a n t c e l l s for the two phase process which can then be used as an approximation to the p r o b a b i l i t y generating function for a tumor (at diagnosis) of known size but unknown age. Let N be the number of ( s e n s i t i v e ) stem c e l l s present when the tumor i s observed. Let be the time required for i s e n s i t i v e c e l l s present at time t " to grow d e t e r m i n i s t i c a l l y to size N (see Figure 2). That i s x = (6-ab-v)" 1 i n (N/i) for i = l N. . . . ( 3 . 3 8 ) Then the p r o b a b i l i t y generating function for the number of new re s i s t a n t c e l l s ( i . e . mutations from s e n s i t i v i t y and t h e i r progeny) i n the period [ t ' j t ' + T j J , <t>^ (s; T ^ ) , i s given by ( 3 . 3 5 ) with Ag=i. The number of re s i s t a n t c e l l s already present at time t' when there are i n s e n s i t i v e c e l l s has p r o b a b i l i t y generating function C^(s;t') given by ( 3 . 3 0 ) . During the deterministic phase of (s e n s i t i v e ) c e l l growth these w i l l grow so that the d i s t r i b u t i o n of the number of re s i s t a n t c e l l s present at t ' + T w whose progenitor mutation occurred p r i o r to t ' w i l l have p r o b a b i l i t y generating function C^(n(s;^)»t") where n(s;T^)=W ^ ( T ^ ) i s given by ( 3 . 6 ) . Thus the p r o b a b i l i t y generating function of the number of r e s i s t a n t c e l l s at time t"+x^ i n a tumor which has i s e n s i t i v e c e l l s at time t' i s -68-C i(T)(s;-c j L);t')(|) i(s;T i). Since there are i s e n s i t i v e c e l l s at time t ' with p r o b a b i l i t y P{Rg(t')=i} we form the o v e r a l l p r o b a b i l i t y generating function for the number of r e s i s t a n t c e l l s at size N, $(s ; N ) , as U *(s ; N ) = K I C.(TI(S;X );tO<t>,(s;T.)P{R n(t')=i}, ...(3.39) i=L where K i s chosen so that 3?(1;N)=1. In what follows we s h a l l set U=N and L=l to simplify the evaluation of (3.39). Immediately we have K= [l-p{R 0(t-)=0}-P{R 0 (O>N}] - 1. A number of improvements can be suggested to increase the accuracy of the approximation (for example include sample paths for which R^(t ' ) > N or adjust p{RQ(t')=i} for spontaneous extinctions i n the i n t e r v a l [ t ' j t ' + T j J ) however these w i l l not be discussed here as they complicate an already d i f f i c u l t computation. We w i l l now calculate approximate expressions for the mean number of c e l l s E[R^(N)] and the p r o b a b i l i t y of cure P^ for the random variable R^(N) which has p r o b a b i l i t y generating function given by (3.39). The mean i s then given by E [ R i ( N ) ] . M g i f f i s=l N = K I P{R Q(t')=i} L Z i oC,(s;t') i - l os 0 1 1 ( 8 ; ^ ) s=l 5 s + 5 ^ ( 8 ; ^ ) s=i as 8 = 1 We w i l l now evaluate the above function and i n order to do t h i s we w i l l assume that 6»ab+v and that t** i s large. Then P { R 0 ( 0 - 0 } - P{R 0(»)=0} = r ^ p j j = e+ae+v/b since the s e n s i t i v e c e l l s grow as a b i r t h and death process with paramters b(l-ct) and (d+v) i n the f i r s t phase. If N»E[R 0(t')] then K>(l-P{R 0(t')=0}) - 1 - (1-e-ae-v/b)" 1. -69-Also we have, dC.(s;t') 5s S T I C S ; ^ ) - i ( e ( a b + v ) t - l ) from ( 3 . 3 1 ) , 9s s=l s=l - 1 = ( N / i ) d - ( a b + v ) / 6 ) a N / i f r o m ( 3 > 6 ) > and j i * N 6t. (ab+v)x . 5 ( " i S ; T l ) = i e X [ e * - l ] from (3.36). 9s s=l Thus we have E[R.(N)] - K I P { R n ( t ' ) = i } [ N ( e ( 0 t b + v ) t ' - l ) + N ( e ( a b + v ) x i - 1 ) ] . 1=1 In most cases R (t ) « R Q ( t ) and thus ( e ( a b + v ) t - l ) « l and we w i l l approximate e ^ a b + V ^ t - l = (ab+v)t for t=t'and t=T^. Then N , E[R (N)]= K I P{R n(t')=i} N(ab+v) (t'+(6 -ab-v) to(N/i)}, 1-1 N = ^ N + K I [(6-«b-v)t'-Jta(l)]P{Ro(t')=!}]• In t h i s process we require that R ^ ( t ' ) > 0 and we have m*(t')=E[R 0(t')|R 0(tO>0]=(l- e-ae-v/b)" 1e ( 6" a b _ v ) t', and thus (6 - a b-v)t'= ln[(l-e-ae-v/b)m*(t')] • Using the above expression we obtain E [ R 1 ( N ) ] a (l-e"-«-v/b) [to{N(l-e-ae-v/b)} + D ] , where D= A n ( E [ R Q ( t ' ) | R Q ( t ' ) > 0 ] ) - E [ { * n R 0 ( t ' ) | R Q ( t ' ) > 0 } ] . By analogy with the discussion presented i n Section 3 . 6 , the d i s t r i b u t i o n of Rgtt') condi t i o n a l on Rg ( t ' ) > 0 i s geometric, and thus D does not depend on the parameters b, d, a and v a f t e r the mean i s f i x e d . If E[R^(t^)] = 1 0 6 , then d i r e c t c a l c u l a t i o n y i e l d s D = 1 . 1 5 . Thus for N=10 7 and - 7 0 -£+ae+v/b< 0 . 9 9 , then An [ N ( 1-E-ae-v/b)]> 1 2 > 1 0 D and we may approximate E t R ^ N ) ] by E [ R i ( n ) ] = i | ^ g ) N A n ( N ( l - e ) ) . . . . ( 3 . 4 0 ) This r e l a t i o n s h i p w i l l hold (approximately) for large N ( > 1 0 7 ) where N » E [ R Q ( t ' ) | R Q ( t ' ) > 0 ] as required by the o r i g i n a l assumptions of the development presented here. We w i l l now calculate P^='l>(e;N), the pr o b a b i l i t y the tumor i s curable at size N for the s p e c i a l case H Q ( D ) = 0 , u 1 ( D ) = l . Using ( 3 . 3 7 ) and ( 3 . 3 3 ) we have N P =®(e ; N)=(l-e-ae-v/b)" 1 I [ l - a - v / b j ^ ^ l - a - v / b ] ^ 1 P(R (t')=i} 1=1 = [ l - a - v / b ] N - 1 . . . . ( 3 . 4 1 ) The use of t h i s approach i s l i m i t e d for the modelling of experimental and human cancer because of the complex nature of the re s u l t i n g p r o b a b i l i t y generating function: equation ( 3 . 3 9 ) . However, i n contrast with the previous approximation (Section 3 . 7 ) i t does permit c a l c u l a t i o n of the p r o b a b i l i t y generating function c o n d i t i o n a l on a single value of R Q ( = N ) rather than for a fixed d i s t r i b u t i o n of N . We notice that ( 3 . 4 1 ) i s of the same form as ( 3 . 3 3 ) . This i s to be expected since the right hand side of ( 3 . 3 3 ) i s independent of t and thus the c u r a b i l i t y when t i s unknown w i l l be the same. It i s i n t e r e s t i n g to note that i f we use the deterministic model of se n s i t i v e c e l l growth presented here for the whole period [ 0 , t ] , we can choose A Q and t (see ( 3 . 3 5 ) ) so that the mean number of res i s t a n t c e l l s and p r o b a b i l i t y of cure Is approximately the same as that for the process with p r o b a b i l i t y generating function <T?(s;N) given by ( 3 . 4 0 ) and ( 3 . 4 1 ) r e s p e c t i v e l y . S p e c i f i c a l l y t h i s i s achieved by s e t t i n g A =(l-e) * and t=6 An ( N ( l - e ) ) . We w i l l use t h i s approximation of (deterministic) s e n s i t i v e stem c e l l growth when we consider drug resistance further i n Chapter k . We w i l l now consider our t h i r d approximation to the d i s t r i b u t i o n of r e s i s t a n t c e l l s at diagnosis. This method i s s i m i l a r to the f i r s t method, In that N(t) has a d i s t r i b u t i o n at diagnosis, but permits some f l e x i b i l i t y i n s e l e c t i n g t h i s d i s t r i b u t i o n . 3.8 Conditioning on N(t) - Approximation 3 The f i n a l approximation to be discussed here w i l l consider not only the growth of tumors but also the rate at which they are i n i t i a t e d . The basic approach w i l l be to 'integrate out' the time parameter present i n the previous discussions and develop formulae by summing across a d i s t r i b u t i o n for N(t). Again we w i l l only consider the s p e c i a l case <1>(SQ ) = S Q • Consider the following i d e a l i z a t i o n of the detection of a tumor. An i n d i v i d u a l i s selected at random and i s found to be of age t. The i n d i v i d u a l i s examined and a tumor i s diagnosed with a p r o b a b i l i t y which depends on the number, n, of stem c e l l s present. Notice that t now represents the age of the i n d i v i d u a l and not the age of the tumor (as p r e v i o u s l y ) . We wish to calculate P r | n ( t ) > t n e p r o b a b i l i t y that there are r r e s i s t a n t c e l l s i n a tumor containing n stem c e l l s detected i n an Individual of age t and w i l l show that for values of t of i n t e r e s t i t may be well approximated by P r| n(°°)= p r | n -We w i l l assume that a tumor i s created by the transformation of a single normal c e l l and that the number of transformations ( i n an i n d i v i d u a l ) i s a Poisson random v a r i a b l e , I ( t ) , with mean p.(t) (u{t)=0, t<0). At time t, co n d i t i o n a l on I ( t ) = i , define i-dimensional random vectors U(t), R(t), and N(t) with elements as follows: -72-U\(t) = time of i n i t i a t i o n of the j - t h tumor, K j < i , Rj ( t ) = number of re s i s t a n t c e l l s i n j-th tumor, K j < i , and Nj(t) = number of c e l l s i n j-th tumor, l<j<i. Notice that t - ^ ( t ) I s t n e (random) age of the j-th tumor. For each t the tumors are l a b e l l e d randomly (the u\.(t) are not ordered). Conditional upon I ( t ) = i we have, t t P{N(t)=n|l(t)=i} = J... J P{N(t)=n |u(t)=u,I(t)=i} d F u ( t ) | I ( t ) ( u ) . Assuming that each c e l l grows independently, i p{N(t)=n|U(t)=u,I(t)=i} - n P{N(t-u )=n }, where P[N(t-Uj)=n^} i s as defined i n Section 3.1. Conditional on I ( t ) = i we have d F u ( t ) | T ( 0 ( u ; - x [ 0 j t ] ( u . ) d u ( U . ) , where X [ 0 , t ] ( u ) = 1 i f u e [ 0 , t ] ' = 0 otherwise. Combining the above equations we obtain, i t 1 p{N(t)=n|l(t)=i} = II / P{N(t-u )=n } - — - d u ( u ). j=l 0 ^ J 3 S i m i l a r l y we have p{R(t)=r,N(t)=n|l(t)=i} i t = n / P (t-u.) _ d n ( u . ) , j=l 0 n j r j ' r j J u(t) J where P (t) i s as defined i n Section 3.1. If we assume that detection of the tumor depends only upon the size (number of stem c e l l s ) of the largest tumor i n the i n d i v i d u a l , N ( t ) , then we w i l l c a l c u l a t e the m c o n d i t i o n a l d i s t r i b u t i o n of the number of r e s i s t a n t c e l l s present, R ( t ) . m Then conditional on I ( t ) = i , P{R (t)=r,N (t)=n|l(t)=i} PtR m(t)=r|N m(t)=n,I(t)=i} = " p ^ , ^ , I ( t ) = i }  — '- for i>0, P{N(t-u)=n} du(u) = X [ 0 ) ( ) ] ( r ) for i - 0 . The above re s u l t follows from a simple consideration of the order s t a t i s t i c s for independent i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s . In most cases a r i s i n g i n human disease, tumors ar i s e quite infrequently so that the l i k e l i h o o d of an i n d i v i d u a l having more than one cancer i s small• The form of u(t) w i l l n a t u r a l l y depend upon the animal and tumor under consideration. We w i l l assume here that u.(t)=\t. This form i s assumed since i t leads to tractable r e s u l t s and i s not an unreasonable approximation for tumors which do not posess a strong age dependent I n i t i a t i o n rate. This form i s also of i n t e r e s t since i t provides a contrast with the two previous approaches where the i n i t i a t i o n time was i m p l i c i t l y assumed to be f i x e d . Substituting for p.(t) and l e t t i n g v=t-u we have, j ' P (v)dv P I (t) = PlR (t)=r|N (t)=n,I(t)>Ol = " n ~ r , r . ...(3.42) r n v 1 mv 1 mv ' v ' rt .- x v ' 1 J 0 P{N(v)=n}dv We now wish to remove the time parameter t i n order to obtain expressions for the d i s t r i b u t i o n of the number of r e s i s t a n t c e l l s c o n d i t i o n a l on the observed number of stem c e l l s , P . . For any f i n i t e r n non-zero n we have, lim J ™ P{N(v)=n}dv = 0 t-x» and thus, i f the age of the subject i s great ( t » 0 ) , we may put P{N(v)=n}dv « P{N(v)=n}dv. A s i m i l a r argument y i e l d s P (v)dv = C P (v)dv. •'O n-r,r •'O n-r,r Thus i f the age of the animal i s much greater than the l i k e l y time a tumor has taken to grow to size n a reasonable approximation to P r| n * s provided by P | = _^_JL_Li£ f o r r < n > ...(3.43) r ' n /QP{N(v)=n}dv and P i = 0 for r>n. r | n Unfortunately (3.43) may not be simply evaluated because P (v) n-r ,r i s complicated. However, as we w i l l now show, i f the number of stem c e l l s follows a p a r t i c u l a r d i s t r i b u t i o n at the time of diagnosis for a tumor c l a s s , i t i s possible to obtain the p r o b a b i l i t y generating function for the number of r e s i s t a n t c e l l s (at diagnosis). We w i l l now specify the form of the d i s t r i b u t i o n for the number of stem c e l l s at diagnosis and derive expressions for the r e s u l t i n g p r o b a b i l i t y generating function of the number of r e s i s t a n t c e l l s . When modelling c l i n i c a l disease there i s no unique s i z e , n, at which a tumor i s detected but rather a d i s t r i b u t i o n of such s i z e s . If we l e t g(n) be the p r o b a b i l i t y that a tumor w i l l be detected at si z e n (assuming no dependence on t, the age of the p a t i e n t ) ; g(j) = P{N=j|Tumor i s diagnosed}. The probability that a tumor w i l l have r resistant cells at detection, ?r» (where the dependence on g is suppressed) is given by oo P = y g(n) P i n=l 1 where g(0) = 0 implies that a tumor w i l l not be diagnosed i f i t has no stem ce l l s . We may pass g(n) through the integral sign in (3.43) to obtain P _ (t) V SI I 8 0 0 I " r > r , } dt. ...(3.44) n-l J0P{N(v)=n}dv But P{N(v)=n} is given by equation (3.28) with <x=v=0 and t=v, since this is then the probability distribution for a birth and death process with parameters b and d. Integration gives /™P{N(v)=n}dv = (bn)" 1 for n>0, and thus oo . 0 0 n dt. p = J n I bng(n)P (t) r J0 L . 6 V ' n-r,r v ' n=l For general g(n), P is d i f f i c u l t to evaluate because P (t) is o a\ / r n-r,r v ' not easily evaluated. However, consider the special case (which is of the same form as one previously considered by Day [34]), J g(n) = I a q n, ...(3.45) J n J J -1 °° where I a q n >0 for a l l n, q.<l, I a =0 and I a q (1-q ) = I g(n)=l. j=l J j-1 3 j=l J J J n-l If N is a random variable on the non-negative integers, where P{N=n}=g(n) (of the form (3.45)), then i t is easily shown that J ? E[N] = I'a q (1-q ) j=l J and -76-J E[N 2] = J a,q,(l-q.) 3 . In p a r t i c u l a r i f J=2 then j = l J j J E[N] = and var(N) = ( l - q i ) ( l - q 2 ) q l . q2 ( l - q i ) 2 ( l - q 2 ) 2 If E [ N]»0 and J=2 then i t i s straightforward to show that _ l / 2 C.V.[N]>2 , where C V. i s • the c o e f f i c i e n t of v a r i a t i o n . Some examples of the d i s t r i b u t i o n g(n) (of the form (3.45) for J=2) are given i n Table I I . Unfortunately d i s t r i b u t i o n s g(n) of the form (3.45) do not constitute a s u f f i c i e n t l y r i c h set to accurately model an ab i t r a r y d i s t r i b u t i o n at diagnosis. The major l i m i t a t i o n a r i s e s because these d i s t r i b u t i o n s cannot give enough weight (99% or more p r o b a b i l i t y ) to a range of tumor stem c e l l sizes (Nmin.Nmax) where <102. This corresponds to a r e l a t i v e difference of 5 f o l d i n the l i n e a r dimensions of a spher i c a l tumor. However, for c l i n i c a l neoplasms, data i s f a i r l y coarse and we may use g(n) of th i s form to approximate the diagnostic d i s t r i b u t i o n . Let 0(s) be the p r o b a b i l i t y generating function for the d i s t r i b u t i o n P and l e t be the number of res i s t a n t c e l l s , i . e . a r 1 ' random variable where p{R^=r}=P r« It may be shown from the d e f i n i t i o n of the p r o b a b i l i t y generating function that oo J 9(s) = I P s r= I a q b J " r=0 j=l J J 5<t>(s0,s0s;t) d S 0 "0 - j dt, ...(3.46) s n=q. where A(sQ,SQs;t) is given by equation (3.5). We will only consider the -77-TABLE II The Probablity of Diagnosis Distribution g(n) - a ^ " + P{N=n}, where E[N]=10 1 0. 1-q. Range of N t N 4 »N ] L min maxJ N -1 max I 8(n) n=N min N -1 max I g(n) S.D.(N) n=0 [ 1 ,5xl0 8 ) 0.012 0.012 [5*10 8 l x l O 9 ) 0.028 0.039 [1x10 9 5 x l 0 9 ) 0.316 0.355 9.1xl0 9 1.111 1.000 j > 1 0 9 , l x l 0 1 0 ) 0.274 0.630 x l O " 1 0 x IO" 9 [ l x l O 1 0 ,5xl0 1 0) 0.366 0.996 [5x10 1 0 , l x l O n ) 0.004 1.000 [ l x l O 1 1 00 ) 0.000 1.000 [ 1 ,5xl0 8 ) 0.005 0.005 [ 5 x l 0 8 , l x l 0 9 ) 0.013 0.018 [ l x l O 9 ,5xl0 9 ) 0.251 0.269 1.667 2.500 [5xlO 9 , l x l 0 1 0 ) 0.328 0.598 7.2xl0 9 x i O " 1 0 x l O " 1 0 [ l x l O 1 0 ,5xl0 1 0) 0.402 0.999 [ 5 x l 0 1 0 . l x l O 1 1 ) 0.001 1.000 [ l x l O 1 1 oo ) 0.000 1.000 [ 1 ,5x10s ) 0.005 0.005 [5 x l 0 8 , l x l 0 9 ) 0.013 0.018 [ l x l O 9 ( 5 x l 0 9 ) 0.247 0.264 1.961 2.041 [5 x l 0 9 , l x l 0 1 0 ) 0.330 0.594 7.1xl0 9 x l O " 1 0 x l O " 1 0 [ l x l O 1 0 ,5xl0 1 0) 0.405 0.999 [ 5 x l 0 1 0 , l x l O n ) 0.001 1.000 [ l x l O 1 1 00 ) 0.000 1.000 [ 1 ,5xl0 8 ) 0.005 0.005 [ 5 x l 0 8 , l x l 0 9 ) 0.013 0.018 [ l x l O 9 ,5xl0 9 ) 0.247 0.264 1.996 2.004 [5xlO 9 , l x l 0 1 0 ) 0.330 0.594 7.1xl0 9 x l O " 1 0 x l O " 1 0 [ l x l O 1 0 , 5 x l 0 1 0 ) 0.406 1.000 [ 5 x l 0 1 0 , l x l O U ) 0.000 1.000 [ l x l O 1 1 oo ) 0.000 1.000 case ( ) ; ( S Q , S ^ ) = S Q and we have by integra t i n g (3.7) with S ^ = S Q S , J o 4 ( s 0 , s 0 s ; t ) 0 o s Q dt = G.(s)+H.(s), ...(3.47) J~U (s,t)V ( s , t ) dt -TT 5 »^ H - ( s ) — ~ *l~hS) 2 l-b(l-a)£ V.(s,t) c where G.(s) = J b ( * j 1 _T , / 1 _„N I T7 / jjj-[ l - e e ~ 6 t + q s ( l - a ) ( l - e " 6 t ) ] J Ti - o t - 6 t , i q..|l-ee - q..s(l-e ) J and „ , , N W 1 .2-a f l -6t - 6 t . r 2 + a -(6+ad+v)t V j ( s , t ) = q..(l-s)(l-e) [1-ee -q..s(l-e ) J e The term G^(s) i n (3.47) i s obtained by d i r e c t i n t e g r a t i o n of the deriv a t i v e (with respect to s) of the f i r s t term i n (3.7). The second term, R\(s), i s most simply obtained by interchanging the order of d i f f e r e n t i a t i o n and inte g r a t i o n of the second term i n (3.7). We may cal c u l a t e the p r o b a b i l i t y of cure, P (where g indicates dependence on the d i s t r i b u t i o n g) for the s p e c i a l case H Q (D)=0, i t ^ ( D ) = l by evaluating (3.46) for s=e. This function must be evaluated by 2 numerical methods. E[R^] and E[R^] may be calculated by d i f f e r e n t i a t i n g (3.46) with respect to s and evaluating at s=l. Carrying out this operation and interchanging the order of d i f f e r e n t i a t i o n and i n t e g r a t i o n y i e l d s l as s l j = 1 j j b ( l - q j ) / i J E [ R 2] = d W + M 5 l | 1 1 J ,2 's=l ds 's=l ds J 1+q. - I a q b[ L + i + i + 2 b ( l - a ) I I ], j-1 2 2 b(l-q^) 2 2 2 2 where -79-" ( l . t ) !Vll!ll s = 1dt, i J o J as s i . f { 2 a u . ( s > t ) | ^ a v . ( s > t ) | ^ + „ d . t j ^ ^ t ) , } d t i J 0 as 3s J . 2 as o aT For general d and v these i n t e g r a l s must be evaluated numerically. I - J jy-*-/ Q t < 3 J J n — L 8 = 1 When Q j - l * (which i s the usual case), close attention must be paid to the accuracy with which these i n t e g r a l s are evaluated. This i s necessary since most of the in t e g r a l s have large absolute value, however they do not have the same sign, and the differences ( i n numeric value) are comparatively small. Therefore i t i s of some p r a c t i c a l i n t e r e s t to determine whether s p e c i a l cases exist which lead to simple forms for (3.46). Inspection of (3.47) shows that the sp e c i a l case d=0 (no stem c e l l death) and v=0 (mutations occur only at d i v i s i o n ) permits considerable s i m p l i f i c a t i o n y i e l d i n g : 9(a) = 1 a q [ T I = ^ y + ^ ]. j = 1 J J U qjS) ( 1 _ q . s ) ^ a - ( i - s ) ( i - q . s ) Then we have the p r o b a b i l i t y , 9(0), that the tumor i s curable when TC Q (D)=0, T I^(D) = 1, i s J - i 0 0 - i 9(0) - I a q ( l - ( l - a ) q ) = £ g(n) ( l - a ) n \ ...(3.48) j=l J n-1 The mean and second moment are given by J a .q . : [ R ] = e'(i) = I 2 3 [ i - ( i - q ) a ] j - i d - q J 3 [R?] - I a j q j , { ( l + q , - 2 ( 1 " < l ^ a ^ 1 " ^ 1 " < l l > a > + 2aq (1-q )"}. j - i ( i -q , ) 3 3 and E[ L' i - i a-c.. j Table III gives computed values of the p r o b a b i l i t y of cure P and the -80-mean and variance of the number of resistant cells for several choices of a, v and e where J=2, q^=0.99, q^-0.9. Table III also contains the analogous quantities which would be obtained using the deterministic model previously presented (see discussion following (3.41) in Section 3.7) when the constant AQ=(1-E) ^ as suggested there. The probability of cure (written at P^  to emphasize its dependence on the distribution at diagnosis), mean and variance are calculated using the deterministic model for each tumor size n and are then averaged over g(n) so that these quantities may be compared using the same underlying distribution of tumor size. We see from Table III that the deterministic model has greater variance than the comparable " f u l l " model; this result probably arises from the condition R^ < N which is not satisfied by the deterministic model and the different distribution of i n i t i a t i o n times implicit in each model. On the other hand examination of this table shows that at least for the examples considered, the coefficient of variation is quite similar for both models. We w i l l now discuss the relative merits of the three approximations presented in this chapter. 3.9 Comparing the Three Approximations The main strength of the f i r s t approximation is that the resulting probability generating function of the number of resistant cells is a simple function of the probability generating function of the underlying process. It provides a reasonable framework for the comparison of treatment effects because of the approximate stability of the underlying geometric distribution of the number of stem c e l l s . However, because the -81-TABLE I I I The P r o b a b i l i t y of Cure P , Expected Number and Standard Deviation s of the Number of Resistant C e l l s . J=2 q1=0.99 q2=0.9 e a v/b P E(R ) S.D.(R.) 0.0 0.01 0.0 0.46 0.46 4.9 5.4 14.9 20.9 0.5 0.01 0.0 0.46 0.46 9.3 9.1 25.1 29.6 0.9 0.01 0.0 0.47 0.46 29.9 25.9 55.6 66.2 0.0 0.02 0.0 0.28 0.28 9.6 10.4 21.6 29.4 0.5 0.02 0.0 0.29 0.28 17.7 17.4 35.6 41.5 0.9 0.02 0.0 0.31 0.28 50.8 45.2 72.7 90.6 0.0 0.01 0.01 0.28 0.28 10.6 10.4 25.7 29.4 0.5 0.01 0.01 0.29 0.28 18.6 17.4 37.9 41.5 0.9 0.01 0.01 0.31 0.28 51.2 45.2 73.2 90.6 The l e f t hand column r e p r e s e n t s c a l c u l a t i o n s based on p r o b a b i l i t y generating function given by (3.46) and right hand column i s that based on the deterministic model given by (3.35) averaged over g(n); see (3.45). Pg i s the p r o b a b i l i t y of cure for the d i s t r i b u t i o n at diagnosis g(n). -82-underlying d i s t r i b u t i o n i s approximately fixed i t does not provide a s u i t a b l e framework for estimating the d i s t r i b u t i o n of r e s i s t a n t c e l l s when the true d i s t r i b u t i o n of stem c e l l s i s not geometric. This method i s the simplest of the three and for t h i s reason i t i s probably the most useful for estimating the e f f e c t s of d i f f e r e n t treatment regimens ( d i f f e r i n g timing and dosages) when the d i s t r i b u t i o n of the number of stem c e l l s at diagnosis i s unspecified. The second approximation provides the p r o b a b i l i t y generating function when the number of stem c e l l s i s fixed and addresses the problem of conditional d i s t r i b u t i o n of r e s i s t a n t c e l l s most d i r e c t l y . However, i t i s approximate and i t s c a l c u l a t i o n i s quite complex. This hybrid s t o c h a s t i c - d e t e r m i n i s t i c model of s e n s i t i v e c e l l growth may be approximated (to give the same mean number of r e s i s t a n t c e l l s and p r o b a b i l i t y of cure) by a purely deterministic model. In t h i s case the deterministic growth curve for the number of s e n s i t i v e stem c e l l s i s approximately the same as the mean value function of the number of stem c e l l s found for the f i r s t approximation. This suggests one can reasonably approximate the d i s t r i b u t i o n of r e s i s t a n t c e l l s using a deterministic model of s e n s i t i v e stem c e l l growth, and that the deterministic growth function should be the expected growth function under a stochastic model where e x t i n c t i o n has been eliminated. We w i l l use the purely deterministic model of s e n s i t i v e stem c e l l growth i n Chapter 4. The t h i r d approximation presents the most r e a l i s t i c model for the d i s t r i b u t i o n of r e s i s t a n t c e l l s i n spontaneously occurring human or animal tumors since i t i m p l i c i t l y incorporates the spontaneous incidence -83-rate of the tumor. However the c a l c u l a t i o n of the p r o b a b i l i t y generating function represents a considerable problem for cases other than the one considered, where u(t)=Xt ; even i n t h i s case the p r o b a b i l i t y generating function of the process i s complex when c e l l loss i s present. Table III shows that the t h i r d approximation and (a modified form of) the second approximation do not y i e l d the same d i s t r i b u t i o n of r e s i s t a n t c e l l s for the same d i s t r i b u t i o n of stem c e l l burden. The main contributor to t h i s difference i s , of course, the assumption ( i n the t h i r d approximation) that new tumors are being i n i t i a t e d uniformly i n time. In most experiments c e l l s are implanted and thus the t h i r d approximation w i l l not be s u i t a b l e . Human tumors appear to be i n i t i a t e d throughout l i f e and thus to accurately model resistance i n such tumors i t i s necessary to consider the appropriate d i s t r i b u t i o n of i n i t i a t i o n times. In conclusion, each approximation has i t s strengths and weaknesses and the choice of one of these w i l l depend upon the experimental or c l i n i c a l s i t u a t i o n to be modelled and on the ultimate object of the modelling. In Chapter 4 we w i l l use a deterministic model of s e n s i t i v e c e l l growth i n order to f a c i l i t a t e further development of this theory. Before completing our d e s c r i p t i o n of single drug resistance we w i l l consider the possible e f f e c t of v a r i a t i o n i n the paramters a, B and y. 3.10 V a r i a t i o n i n the Resistance Parameters a, 8 and y Up to this point we have assumed that a, B and y are f i x e d . In passaged animal tumor systems this assumption appears reasonable and has been assumed In a l l analyses of these systems. These tumor systems also possess l i t t l e v a r i a t i o n i n a number of other physical properties. This i s not unexpected since the process by which these tumors are chosen f o r -84-study tends to select those which maintain their characteristics after ser ia l passaging. Spontaneous tumors, whether animal or human, do not undergo such a selection process and exhibit a greater var iab i l i ty in a number of physical characteristics than do passaged tumors. For example, experimental tumors display quite regular growth rates especially when many cel ls are present. In contrast, human tumors of almost every type display considerable variation in growth rates. Possible variation in a,. 8 and y can be thought of as occurring in two distinct ways. F i r s t l y , these parameters may be considered to "evolve" (either deterministically or stochastically) as a tumor grows. One special case of this would be the possible effects of treatment on these parameters. Radiation and many drugs used in cancer therapy are known to be mutagenic and the values of a, 6 and y be expected to increase subsequent to treatment. Secondly, the parameters a, B and y may vary between tumors within the same class with each class having some distinct distribution of a, B and Y-Modelling the effect of mutagenicity of treatment is relat ively straight-forward i f we assume that the effect of treatment brings about a deterministic change in the value of the mutation rates for a l l the tumor ce l l s . Since the probability generating function for the appearance of new mutations to resistance is independent for disjoint time intervals , we may use recursive relationships such as (3.11.1-2) to determine the probability generating function after treatment. If the effect of treatment is to induce a random change in the mutation rates of a l l the cel ls in a tumor (for a f in i te or an inf in i te time period) then this is extremely complex to model. Here we w i l l only examine the effects of -85-v a r i a t i o n s i n the mutation rates between tumors of a given class which are constant i n time. As we know l i t t l e regarding the r e l a t i v e magnitude of a, 6 and y (most experiments measure the quantity a+v/b) i t i s not necessary to consider t h e i r j o i n t d i s t r i b u t i o n . If we l e t 6(s;t,a) be the p r o b a b i l i t y generating function of the d i s t r i b u t i o n of the number of r e s i s t a n t c e l l s (computed using the second of the three approximations previously presented) now viewed as con d i t i o n a l on a=a+v/b, we have that the unconditional p r o b a b i l i t y generating function, ¥(s;t), i s given by where F(a) i s the cumulative d i s t r i b u t i o n function for a. L i t t l e i s known about the d i s t r i b u t i o n F(a), since almost a l l experiments have assumed a to be f i x e d . We w i l l therefore choose a convenient d i s t r i b u t i o n which has support on a subset of [0,1]. An obvious choice for the d i s t r i b u t i o n of a i s to use the conjugate of p{R(t)|N(t),a}; however, t h i s p r o b a b i l i t y d i s t r i b u t i o n function has not been determined. We propose to use the beta d i s t r i b u t i o n which has support [0,1] and i s conjugate to the B e r n o u l l i d i s t r i b u t i o n . We have already shown that the p r o b a b i l i t y of cure at size N for fixed a where ¥(s;t) = /<D(s;t,a)dF(a), ...(3.49) u Q(D)=0, ^ ( D ) - ! , i s P N(a) = ( l - a ) N-l ; see equation (3.41). Then the cure p r o b a b i l i t y , P , for the class of tumors i s given by PN= J o ( l ~ a ) N 18(a;u,v)da, where {u,v} are the parameters of the beta d i s t r i b u t i o n , and we assume that a and N are independent. It follows that r(u+v)r(v+N-l) r(v)r(u+v+N-i) N-2 v+x - n (; u+v+x .. .(3.50) x=0 where T i s the gamma function. -86-It i s a simple matter to evaluate ( 3 . 5 0 ) . In order to estimate the s i g n i f i c a n c e of v a r i a t i o n s i n a on t h i s P^, i t i s necessary to f i x a frame of reference. We choose here to assume that for some s p e c i f i e d reference size there i s a constant cure rate. Then we explore the e f f e c t of d i f f e r e n t choices of u and v at sizes other than the reference point. Examples are presented i n Figure 3 , where i t may be seen that the values of u and v can e f f e c t the shape of the curve considerably. Figure 3 shows that v a r i a t i o n i n a w i l l a f f e c t the p r o b a b i l i t y of no r e s i s t a n t c e l l s and thus a f f e c t the l i k e l i h o o d that the tumor w i l l be curable as a function of s i z e . This observation seems important since not only does t h i s formula r e l a t e to the p r o b a b i l i t y of cure i n c l i n i c a l disease but i t also relates to current methods used to estimate (assumed fixed) mutation rates i n animal tumors. Experimental estimation of mutation rates i s frequently based on destructive t e s t i n g where i t i s assumed that H Q ( D ) = 0 , U ^ ( D ) = 1 . The percentage of surviving animals i s measured for various tumor burdens, and the mutation rate i s estimated using an equation l i k e ( 3 . 4 1 ) . Thus the f i t t i n g of t h i s type of data to equation ( 3 . 4 9 ) allows one to estimate the v a r i a b i l i t y present i n the mutation rates. However, other factors which a f f e c t c u r a b i l i t y may also cause s i m i l a r departures from the form ( 3 . 4 1 ) and thus i t i s not possible to uniquely i d e n t i f y v a r i a b i l i t y i n mutation rates as the only cause. The curves i n Figure 3 are, of course, strongly dependent upon the assumption of the b e t a - d i s t r i b u t i o n . If the d i s t r i b u t i o n of a i s not adequately approximated by a beta d i s t r i b u t i o n the curves of P N may be quite d i f f e r e n t . The e f f e c t of v a r i a t i o n of a i n the mean number of r e s i s t a n t c e l l s i s e a s i l y c a l c u l a t e d . From ( 3 . 4 0 ) the corresponding -87-Flgure 3 Probability of Cure when Variation i n the Mutation. Rate i s Present. Number of Stem Cells Plots of the probability of cure, P J J , as a function of the number of stem cel ls where the mutation rates are assumed to follow a distr ibut ion (3.50). The parameters were chosen so that the mean and standard deviation of a were as given. Each curve has been constructed to pass through the point N=10\ P N =0.25. -88-number of r e s i s t a n t c e l l s i s given by ( 1 JL ) A r i ( N ( l - e ) ) /J aB(a;u,v)da = Jul ( N ( l - e ) ) . This concludes our treatment of single drug resistance. In the next chapter we w i l l consider the problem of resistance to two drugs. -89-4. RESISTANCE TO TWO OR MORE CHEMOTHERAPEUTIC AGENTS The previous chapter considered the development of resistance to a single drug by tumor stem c e l l s . In the chemotherapy of many human malignancies several active drugs are a v a i l a b l e . Where possible these drugs may be combined to form regimens which are more e f f e c t i v e than eit h e r of t h e i r i n d i v i d u a l constituents. Here we w i l l consider the development of resistance to two drugs. The possible combinations ( i n d i v i d u a l drugs and t h e i r dosages) are li m i t e d because of t h e i r e f f e c t s on the host normal tissue systems. The construction of combined regimens depends on a var i e t y of considerations, which include consideration of the a c t i v i t y of p o t e n t i a l drugs on each component of the normal system of the host, pharmacokinetics of the drugs and other factors which r e l a t e to the " a c c e p t a b i l i t y " of the r e s u l t i n g regimen. The f i n a l regimen may also include r a d i a t i o n or surgery. The construction of regimens ( e s p e c i a l l y i n the l i g h t of the r e s t r i c t e d and imperfect information a v a i l a b l e ) requires consideration of factors which we do not propose to model here. Therefore, we w i l l consider that the drugs, t h e i r dosages and the timings of administration are f i x e d . We w i l l consider a general framework for the development of resistance i n stem c e l l s and w i l l provide a detailed examination of the case of two drugs. Consider the case where there are n d i f f e r e n t antitumor agents ava i l a b l e , T^, T^. An i n d i v i d u a l tumor c e l l may then be character-ized as being i n one of 2 n mutually exclusive states with respect to these agents, according to which therapies i t i s r e s i s t a n t to and which not. As before a c e l l w i l l be defined as re s i s t a n t i f the p r o b a b i l i t y of -90-c e l l death a f t e r administration of chemotherapy i s lower than i n the parent ( s e n s i t i v e ) l i n e . Let R . . ( t ) be the number of stem c e l l s at time t which are ij...m r e s i s t a n t to the set of drugs {T^, T\, T^} and not r e s i s t a n t to any i n the set {T. T } {T., T T } and refer to such c e l l s as 1 1 n J 1 i 2 m being i n the state R . . . Those stem c e l l s s e n s i t i v e to a l l drugs w i l l i j . . .m be i d e n t i f i e d as members of R ^ , (6 i s the empty s e t ) , which w i l l be written R Q . The possible states for the i n d i v i d u a l tumor c e l l s w i l l be written as R ^ , where Q ^ , i=0, 1,..., 2 n - l (QQ=6) are the 2 n d i s t i n c t subsets of {l,2,...,n}. We w i l l assume that when a stem c e l l i n R ^ divides to form two new stem c e l l s , one of them w i l l be i n R and the other w i l l be i n R with i j 2 n - l p r o b a b i l i t y a where E a =1. As i n the single drug case these p r o b a b i l i t i e s w i l l depend on the tumor type, the drug concentration, and the length of time the drug i s administered. S i m i l a r l y , we w i l l define 8 as the p r o b a b i l i t y that a stem c e l l i» j t r a n s i t s from R to R when the c e l l divides forming a stem c e l l and a Q i Q j t r a n s i t i o n a l c e l l . Also l e t y At + o(At) be the p r o b a b i l i t y that a stem c e l l mutates from R to R i n the i n t e r v a l [t, t+At). Transitions i j from the s e n s i t i v e state RQ to the r e s i s t a n t state R _ w i l l have as j parameters a. _ , 8, _ and v. n for the three d i f f e r e n t types of t r a n s i t i o n . We w i l l write these rates a 8 and y r e s p e c t i v e l y . 3 3 3 To si m p l i f y notation we w i l l omit braces i n the rate parameters and -91-use 0 to represent the empty set. For example a ^ - j . {12} be written a l 12' ^{1} a s ^1' Y{1} 4 a S Y l 0' e t c * ^ e vm n o w concentrate attention on the special case n=2, that i s , two drugs. This case is both tractable and informative. As before we w i l l assume that the probability of two transitions between states occurring in a time interval of length At is of the order o(At). As in Chapter 3 we w i l l assume that the acquisition of resistance is permanent. This implies 0^  Q=P-J_ 0 = Y1 0=^' a2,0 = P2,0 = Y2,0 = O' a12,0 = P12,0 = Y12,0 = 0 , al,2 = Pl,2 = Y1,2 = a2,1 = P2,1 = Y2,1 = 0 and a 1 2 1 = P 1 2 1=Y 1 2 l = a i 2 2=^12 2 = Y12 2 =°' ^ i n C n a P t e r 3 w e w i l 1 o n l y "keep track" of stem cells and the development of transitional and end cells (irrespective of their resistance status) w i l l not be considered ex p l i c i t l y . Similarly we w i l l assume that the growth parameters of a l l cells are the same. This assumption appears reasonable for some drugs and tumor types but others display differential growth rates for the sensitive and resistant c e l l s . We w i l l now discuss the calculation of the probability generating function for the process. 4.1 Probability Generating Function for Double Resistance Define P^ j ( k > = P{R 0(t)-i, R^t) = j , R £(t) = k, R 1 2 ( t ) = Jl} and N(t) = R ( )(t)+R 1(t)+R 2(t)+R 1 2(t). Table IV indicates the permitted transitions with their associated probabilities. We continue by writing down the Kolmogorov forward equations [21] for the process which yields the following family of differential equations: a F i . . i . k . » ( t ) d t -92-TABLE IV Transitions Occurring i n the Stem C e l l Compartment i n the i n t e r v a l [t,t+At) which have P r o b a b i l i t y of Order At. I n i t i a l State F i n a l State P r o b a b i l i t y ( i . j . k . A ) (i+l.j.k.A) i b ( l - a 2 - a 2 - a 1 2 ) A t + o ( A t ) ( i , j , k , A ) (i.J.k.A) i c ( l - p 1 - p 2 - p i 2 ) A t + j c ( l - p i 1 2)At+AcAt +kc(l-p 2' 1 2)At+o(At) (i , j , k , A ) (i-1,j,k,A) idAt+o(At) (i , j , k , A ) ( i , j + l , k , i ) i b a At + j b ( l - a l j l 2 ) A t + o ( A t ) (i . j . k . A ) ( i - l , j + l , k , A ) i ( P l C + Y l ) A t + o ( A t ) (i, j , k , A ) ( i , j , k + l , I ) i b a 2 A t +kb(l-a 2 1 2)At+o(At) (l. j . k . A ) ( i - l , j , k + l , J l ) i ( P 2c+Y 2)At+o(At) (l . j . k . A ) ( i , j , k , i + l ) i b a 1 2 A t + j b a 1 At +kba 2 1 2At+ibAt+o(At) (i, j , k , A ) ( l - l , j , k , J t f l ) i ( P 1 2c+Y 1 2)At+o(At) (i. j . k . A ) ( i , j - l , k , J l ) jdAt+o(At) ( i . j . k . A ) ( i , j - l , k , ! + l ) j ( P l , 1 2 C + Y l , 1 2 ) A t + ° ( A t ) (I,j,k,A) ( i , j , k - l , A ) kdAt+o(At) (1, j,k,A) ( i . j . k - l . A + l ) k ( P 2 , 1 2 C + Y 2 , 1 2 ) A t + 0 ( A t ) ( i , j.k.A) ( i , j , k , A - l ) MAt+o( At) = -[( b +d+c)(i+j+k+A) + y 1 > 1 2 j + Y 2 > 1 2 k + ( Y ^ n ) 1 ^ , j , k , A ( t ) + b ( l - V a 2 - a 1 2 ) ( i - l ) P i _ 1 > j > k > A ( t ) + ^ b i P ^ . _ 1 > k > A ( t ) + « 2 b i P l f J . k - l . A ^ + ^ ^ i + l . j . k . j ^ + c ( l - 6 1 - B 2 - B 1 2 ) i P l i j > k f A ( t ) + (8,0+ Y ^ C i + D P ^ ^ . ! ^ , ACt) + (P 2 c+Y 2 )( i+DP 1 + l f J f k _ l t J l ( t ) + ^ " i . J . k . X - l ^ + ( 8 1 2 c + y 1 2 ) ( i + l ) P . + 1 ^ ^ ( t ) + b ( l - « l f l 2 ) ( j - l ) P l f j . l f k f A ( t ) + b a l , 1 2 3 P i , j , k ^ - l ( t > + d ( J + 1 ) P i , j + l , k , J l ( t ) + b ( l - a 2 f l 2 ) ( k - l ) P l f J f k _ l j J l ( t ) + b a ^ k P . ^ . ^ ^ ( t ) + ^ ^ i . j . k + l , * ^ + «l-h,U>**l,SM™ + ( c P 2 , 1 2 ^ 2 , 1 2>( k + 1 ) Pi,j,k,A-l ( t) + b ^ - 1 ) P i , 3 , k , A - l ( t ) + ^ ^ i . j . k . A + l ^ + ' " i . j . k . A ^ • • • ( 4 ' 1 ) f o r a l l of i,j,k,£ >0 and where P . , 0 ( t ) = 0 for any of i , j , k or KO. We w i l l assume that P . , (0) i s known. Let <(>(s;t) be the p r o b a b i l i t y generating function for the process, that i s <t>(s;t) = 6 ( s 0 , s 1 , s 2 , s 3 ; t ) CO 00 CO 00 , i 1 k X I I I I P ± j k A ( t ) s j s s* s j . L= 0 j-0 k=0 A=0 1»J» l t» J t u 1 z i i k A Then multiplying (4.1) by SQ s, S 2 S-J and summing i,j,k,£ over 0 to yi e l d s a*(s;t) 3 o<t>(s;t) = ±l0 (bSi_d) <Si_1) 2 o*(s;t) + J j C ^ b s ^ v ^ ) ( s 3 - s . ) — g - r — S4(s;t) a* ( s ; t ) + ( ^ b s ^ ) ( S ; L - s 0 ) - ^ - — } + ( a 1 2 b s 0 + v 1 2 ) ( s 3 - s 0 ) 5 S q (4.2) where v ^ c P ^ , \il2=ch,U+y±,l2 f o r i = 1 > 2 a n d V 1 2 = C p 1 2 + Y 1 2 ' We may use the method of c h a r a c t e r i s t i c s [22] to reduce the so l u t i o n of (4.2) to the so l u t i o n of the following set of f i v e ordinary d i f f e r e n t i a l equations: dt(u) _ du L t d X l ( U ) - d - X i ( u ) ) ( b x i ( u ) - d ) + ( X i ( u ) - X 3 ( u ) ) ( b a i ) 1 2 x i ( u ) + v i ) 1 2 ) , du ' ' • for i-1,2, d X 3 ( u ) = ( l - x 3 ( u ) ) ( b x 3 ( u ) - d ) , du d X 0 ( u ) = (l-X 0(u))(bx 0(u)-d) du 2 + I ( a i b x 0 ( u ) + v j L ) ( x 0 ( u ) - x i ( u ) ) + ( a 1 2 b x Q ( u ) + v 1 2 ) ( x Q ( u ) - x 3 ( u ) ) , where u, x ^ ( u ) > X 2 ( u ) i ^3^ u) a n d X ^ ( u ) a r e dummy v a r i a b l e s . Unfortunately, although the f i r s t four equations are straightforward to solve, the f i n a l equation (involving X Q ( u ) ) i s complicated and a closed form s o l u t i o n i s not apparent. However, we have already shown ( i n the case of single resistance) that i f t and Rrj(t) are known, then the d i s t r i b u t i o n of the number of re s i s t a n t c e l l s can be reasonably well approximated (Section 3.7) by using a continuous deterministic function for the growth of the sen s i t i v e c e l l s . From t h i s point on i n t h i s chapter we w i l l assume that s e n s i t i v e stem c e l l s grow d e t e r m i n i s t i c a l l y and to emphasize t h i s we w i l l set Rg(t)=B(t); the compartments R^, R 2 and R^2 w i l l grow as before. A less general form of th i s model has previously been considered by Coldman et a l [27]. Let P i , j , k ( t ) = P{R 1(t)=i,R 2(t)=j,R 1 2(t)=k|R 1(0)=0,R 2(0)=0,R 1 2(0)=0} and -95-a oo , i J > 00 00 00«(s;t) = «(s ,s ,s ,s ;t) = £ £ I P * . ( t ) a s u 1 ^ J i-0 j=0 k=0 ' J ' be the j o i n t p r o b a b i l i t y generating function for the number of r e s i s t a n t ( R ^ , R ^ and R ^ 2 ) c e l l s derived from s e n s i t i v e stem c e l l s ( R Q ) a f t e r time t=0, excluding c e l l s i n R ^ , R 2 O R R^2 P r e s e n t a t t i m e t = 0 , that i s PJ^QQ(0)=1 and thus $(s;0)=l. This generating function i s dependant on the function B ( t ) , but this dependence w i l l not be e x p l i c i t l y indicated-The assumption of deterministic s e n s i t i v e c e l l growth a l t e r s the form of the t r a n s i t i o n p r o b a b i l i t i e s given i n Table IV and thus i n (4.1). The e f f e c t i s to delete the state R Q and set to zero a l l p r o b a b i l i t i e s which applied to changes i n the numbers of c e l l s i n R Q alone ( i . e . those without changes i n the numbers of c e l l s i n ei t h e r R ^ , R ^ or R ^ as w e l l ) . The p r o b a b i l i t i e s for t r a n s i t i o n involving the number of c e l l s i n R Q and the numbers i n either R ^ , R ^ or R ^ are unchanged except that i i s replaced by B ( t ) . Transitions between other states are as before. We may then derive the following p a r t i a l d i f f e r e n t i a l equation for $(s;t) i n the same way as (4.2) was obtained: 3<£(s;t) 3 9$(s;t) - o T — = JL [ b s i " d l I V 1 ! - O q -2 o$(s;t) + J 1 ^ a i , 1 2 b s i + V i , 1 2 ) ( s 3 - S i > - ^ ~ + ( " l ^ l X V 1 ) W O C ^ D } + ( a 1 2 b + v 1 2 ) ( s 3 - l ) B ( t ) * ( s ; t ) . ...(4.3) si S4(s;t) The r e u l t can also be obtained by s e t t i n g S Q ^ ' <t)(g,;t)=®(s;t) and as 0 = B(t) $ ( s ; t ) i n (4.2), Using the method of c h a r a c t e r i s t i c s [22] the sol u t i o n of (4.3) i s obtained by solving the following s e r i e s of d i f f e r e n t i a l equations: -96-...(4.4.1) d X i ( U ) - d-X,(u)) (b X.(u)-d) du + ( X i ( u ) - x 3 ( u ) ) ( b a i ) 1 2 ^ i ( u ) + v i , i 2 ) ' i = 1' 2> •••(4.4.2) dX 3(u) , ( 1 _ x ( u ) ) ( b ( u ) _ d ) > ...(4.4.3) du 2 ^ ( ^ ( u ) ; u ) _ 1 d ^ ( ^ ( u ) ; u ) = {E (a ib+v i) ( ^00-1) du 1=1 + ( a 1 2 b + v 1 2 ) ( x 3(u)-l)}B(u). ...(4.4.4) Now we note that the equation (4.4.3) for X 3 ( u ) i s simply solved as before (see equation (3.2.3)): X 3(u)= d H - X 3 ( 0 ) ] + [ b X 3 ( 0 ) - d ] e 5 u m _ ( 4 > 5 ) b [ l " X 3 ( 0 ) ] + [ b x 3(0)-d]e 6 u Noting that x ^ ( u ) = X 3 ( u ) i s a p a r t i c u l a r s o l u t i o n for (4.4.2) we have ( i n analogy to the solu t i o n of (3.2.2)) X ±(u) =X 3(u) + , for i-1,2 ...(4.6) .-1 [X i(0)-X 3(0)] + b ( l - a i > 1 2 ) J F ±(x)dx where 2-a. 1 0 x,12 F 1 ( x ) - 6 X > exp{(6+a 1 > 1 2d+v. ) 1 2)x} . -2+a *[b [ l - X 3 ( 0 ) ] + [ b x 3(0)-d]e O X] 1 , i Z . Equation (4.4.4) may then be solved d i r e c t l y by sub s t i t u t i n g (4.6) for X i(u)(i=l,2) and (4.5) for X 3 ( u ) a n d int e g r a t i n g the l e f t and ri g h t hand sides d i r e c t l y . The required s o l u t i o n 5>(s;t) i s then obtained by se t t i n g u=t and x i(u)=s i (for 1=1,2,3) and in v e r t i n g (4.5) and (4.6) so that X^(0) (i=l,2,3) are expressed i n terms of s^  and t. These values are then substituted into the expression obtained by integ r a t i o n of the ri g h t hand side of (4.4.4). Carrying out this s u b s t i t u t i o n we obtain, a f t e r some s i m p l i f i c a t i o n , the following expression for $ ( s j t ) : 2 In S(s;t) = I 0 B Q + B Q £ ( o i b+v i)I 1 ( s 1 ) , ...(4.7) i = l where I 0 = { « 1 2 b + v 1 2 + I^b+v.)} 5 ( s 3 - l ) /' ^ ( t ~ V ) d V _ 6 v . l L i-1 1 J 0 b ( l - s 3 ) + (bs 3-d)e o v t B'(t-u) g i(u) du I i ( 8 ) = =L u ' 0 [6 i ' 1 2 ( s - s 3 ) ] - b ( l - a ± ) 1 2 ) / g.(v) dv with -(6+a 1 0d+v. )v . -2+a, 1 0 / \ 1>12 i,12 r, ,, N . -ov. i,12 g ±(v) = e [ b ( l - s 3 ) + (bs 3-d)e ] B'(u) = B ( U ) / B Q and B ^ = B ( 0 ) . Equation (4.7) generalizes a previous r e s u l t found by Coldman et a l [27]. The function $ ( s ; t ) , given by (4.7), i s the p r o b a b i l i t y generating function for the number of s i n g l y and doubly r e s i s t a n t c e l l s derived from the growth of the s e n s i t i v e c e l l s over the i n t e r v a l [0,t] conditional on R (0)=R o(0)=R (0)=0. Our objective i s to derive ¥(s;t) the unconditional p r o b a b i l i t y generating function for an a r b i t r a r y d i s t r i b u t i o n of s e n s i t i v e and r e s i s t a n t c e l l s at t=0. We w i l l now examine the development of r e s i s t a n t c e l l s from s i n g l y r e s i s t a n t c e l l s present at t=0. This i s quite straightforward since the development of double resistance i n c e l l s already r e s i s t a n t to one agent i s analogous to the development of single resistance i n s e n s i t i v e c e l l s considered i n Chapter 3. Let <l>^(s;t), i=l,2, be the p r o b a b i l i t y generating functions of the number of r e s i s t a n t c e l l s derived from (progeny of) a single c e l l i n R^, at time t=0, i . e . c o n d i t i o n a l on R Q(t)=0, R i(0)=l, R 3_ i(0)=0 and R 1 2(0)=0. Then -98-^ ( s ; t ) = w Q ( t ) , 1=1,2, ...(4.8) where w Q(t) i s given by (3.7) with s Q=s i, s^-s^, 0=0^ ^ and v-v^ 1 2 « S i m i l a r l y , l e t <}>3(s;t) be the p r o b a b i l i t y generating function of the number of r e s i s t a n t c e l l s derived from (progeny of) a single c e l l i n R , 2 at time t=0, i . e . c o n d i t i o n a l on R Q(t)=0, R 1(0)=0, R 2 ( ° ) a n d R 1 2 ^ = 1 " Then <J»3(s;t) = w L ( t ) , ...(4.9) where w,(t) i s given by (3.6) with s^-s^. For future use i t i s convenient to include a term i n the unconditional generating function r e f l e c t i n g the number of s e n s i t i v e c e l l s at time t. To do t h i s we w i l l multiply the generating function by s 0 ^ ^ t ^ ' w n ^ c n "^y be viewed as the approximate generating function for the number of s e n s i t i v e c e l l s . Using the general r e s u l t (2.3), Y ( s ; t ) , the unconditional p r o b a b i l i t y generating function of { B ( t ) , R 1 ( t ) , R 2 ( t ) , R 1 2 ( t ) } , i s given by ¥(s;t) =(),(l,(|) 1(s;t),<)) 2(s;t ) , ( t. 3(s;t))$(s;t)s^ B ( t ) ] ...(4.10) where i K l , s , , s 2 , s 3 ) = ¥(l,s,,s 2 >s 3;0) i s the p r o b a b i l i t y generating function for the d i s t r i b u t i o n of { R ^ O ) , R 2 ( 0 ) , R 1 2 ( 0 ) } . For future reference we w i l l now c a lculate m,(t)=E [ R,(t)], m 2 ( t ) = E [ R 2 ( t ) ] and m 1 2 ( t ) = E [ R 1 2 ( t ) ] . D i f f e r e n t i a t i n g (4.10) with respect to s , (i=l,2,3) and s e t t i n g s = (1,1,1,1) y i e l d s the following i ~ r e l a t i o n s h i p s : (6-a „)t t -(6-a ? ) u m (t) = e ' (m.(0) + a J e ' B(u)du), 1=1,2, ...(4.11.1) 0 -99-where a =a.b+v., a. ,~=ba. 1 0+v. 1 0 and i i i i,12 i,12 i,12 2 - a t m ( t ) = e 6 t (m (0) + \ m,(0) [1-e 1 , 1 2 ] i = l 2 t . -a. 1 0 t t _ ( 6 - a . 1 0 ) u + I a [ J B(u)e du - e 1 , 1 2 / B(u)e 1 , 1 2 du] i-1 0 0 t + a J B(u)e du}, ...(4.11.2) 0 where a 1 2 = < X 1 2 b + V 1 2 * The s p e c i a l case B(u)=Bgexp(ku), (k^S-a^ ^ 2) i s of p a r t i c u l a r i n t e r e s t since i t i s the mean growth function for a b i r t h and death process with fixed rates. In t h i s case the expected values are (k-6+a )t a B [e 1 , i Z -1] m^t) = e x p { ( 6 - a i j l 2 ) t } (m^O) + [ k - 5 + a . j ), -..(4.12.1) i , 12 and m (t) = e 6 t {m (0) + £ m (0) [ l - e ^ ' " ' j Li. i = 1 1 -a, + B Y a i a i , 1 2 r e ^ ' - l (1-e i > 1 2 ) ]  B 0 J 1 [ k - 6 ^ l f l 2 ] L (k-6) " a 1 § 1 2 J + T&4r [ e ( k " 6 ) t - 1 ] } ' ...(4.12.2) The choice k=6-a^-a 2-a^ 2 y i e l d s the same expected numbers of sin g l y r e s i s t a n t c e l l s as i n the f u l l y stochastic case, i . e . that with j o i n t p r o b a b i l i t y generating function s a t i s f y i n g (4.2). This may be shown by d i f f e r e n t i a t i n g (4.2) with respect to s^ (1=0,1,2,3) s e t t i n g s=l and obtaining d i f f e r e n t i a l equations for m ^ t ) , m^(t), m 2(t) and m^ 2 ( t ) . In p a r t i c u l a r the ordinary d i f f e r e n t i a l equation for ^ ( t ) = E[Rg(t)] obtained from (4.2) i s d m 0 ^ = ( 6 - a 1 - a 2 - a 1 2 ) m ( ) ( t ) , dt -100-which has s o l u t i o n ^ ( t ) = mgexp {( 6-a,-a2_a,2) }t • Repeating the procedure for m,(t), ^ ( t ) and m^Ct) shows that (4.12.1-2) are solutions to the appropriate d i f f e r e n t i a l equations when k=6-a,-a2_a,2* I n the 0 e ku following analysis of the two drug case we w i l l assume that B(u) =Bn  where k=6-a,-a2_a,2* As discussed i n Chapter 3 we w i l l be interested i n s i t u a t i o n s of growth from a single s e n s i t i v e stem c e l l where the tumor siz e (stem c e l l s ) N i s observed, but t i s unknown. We w i l l then use the approximation suggested i n Section 3.7, equations (3.38) and (3.40), and assume that the o v e r a l l growth of the stem c e l l compartment i s given by BQe^ where Bg=(l-e) Thus we w i l l set t-S ^ J l n ( N ( l - e ) ) , ...(4.13) where the term (1-e) arises from excluding tumor growth paths i n which the stem c e l l compartment goes spontaneously e x t i n c t . This factor i s retained since, although the stem c e l l compartment cannot go extinct (because the s e n s i t i v e c e l l s are growing d e t e r m i n i s t i c a l l y ) , i t y i e l d s a better approximation to the f u l l y stochastic model. Now i f we observe R* s e n s i t i v e c e l l s at some time t (which may not be known on the scale where t=0 i s the o r i g i n time of the progenitor stem c e l l ) then we would use t given by (4.13) i n (4.10) and set the l a s t factor on the righ t hand side of (4.10) to be, [B(t)] R* ,, s0 s0 ...(4.14) In most cases of p r a c t i c a l i n t e r e s t N i s observed and R* i s unknown. In such cases we w i l l set R* -[N-m^ O-n^ O-m^t)] (=N) where t i s given by (4.14). We w i l l now consider the modelling of treatment e f f e c t s i n the two drug case. -101-4.2 Modelling Treatment E f f e c t s Radiotherapy and surgery w i l l be modelled i n the same way (with c e l l s u r v i v a l as B e r n o u l l i random variables) as presented i n Section 3.4 and the e f f e c t of each w i l l be the same for a l l r e s i s t a n t subtypes. To model the e f f e c t s of chemotherapy upon stem c e l l s we w i l l assume that the drugs obey the same laws of k i l l as outlined i n Section 3.2 [26] and define the following quantities for Qe{{0},{1},{2},{12}}: TZ. n(D) = Pja c e l l i n R w i l l survive administration of a single 1 > X X course of the drug T^ at dose D} for i=l,2. We w i l l generally omit the dependence of %. n(D) on D where i t i s 1 » X understood to rel a t e to some fixed but possibly unspecified dose. We define the variable X as follows: I » x X. = 1 i f a c e l l i n R survives administration of T , I > X X I = 0 otherwise. Then £ n ( s ) , the p r o b a b i l i t y generating function for X , i s given by ^i,Q ( s> = ^ i . Q * *i.Q S ' For s i m p l i c i t y , as before, we w i l l write TZ ,.. , as 7t e t c Now i f treatment T^ i s given at time t ^ then Y ( s ; t 1 ) = n^CsJjt"), ...(4.15) where h& = «i , 0 < 8 0 > » 5 l , l < 8 l > ' ^i,2 ( s2>' h t 1 2 < s 3 » ' .-.(4.16) This r e s u l t follows from (2.3) and i s s i m i l a r to re s u l t (3.11.1) for the single drug case. Equation (4.15) deserves some comment since Y ( s j t p contains one part i n which the number of se n s i t i v e c e l l s i s deterministic and another i n which i t i s random. This arose because we assumed Rg(t)=B(t) i n order -102-to derive the p r o b a b i l i t y generating function for the number of resistance c e l l s derived from s e n s i t i v e c e l l s . We have also written a p r o b a b i l i t y generating function for the number of s e n s i t i v e c e l l s at time t]_, (4.14), and used i t to derive the p r o b a b i l i t y generating function of the number of s e n s i t i v e c e l l s a f t e r treatment, (4.15). We have done t h i s to obtain a better approximation to the behaviour of the f u l l y stochastic model. In intertreatment i n t e r v a l s we may consider stem c e l l growth to be stochastic, but to calculate the d i s t r i b u t i o n of r e s i s t a n t c e l l s which a r i s e from s e n s i t i v e c e l l s ( i n that i n t e r v a l ) we use the deterministic growth model for R ( t ) . We know from Section 3.1 that i n the case of single resistance the stem c e l l compartment grows ( s t o c h a s t i c a l l y ) as a b i r t h and death process with parameters b(l-a) and v+d. Since the ultimate destination of c e l l s leaving the s e n s i t i v e compartment i s i r r e l e v a n t to the growth of this compartment we deduce that, i n the f u l l y stochastic model for resistance to two agents, the s e n s i t i v e c e l l compartment w i l l grow as a b i r t h and death process with parameters b(l-ai-a2 _a,2) and (d+v^+v^+v,^)• If we l e t ^ ( s j t ) be the p r o b a b i l i t y generating function of the number of s e n s i t i v e stem c e l l s i n t h i s f u l l y stochastic model * 0 ( s ; t ) = w L ( t ) , ...(4.17) where w,(t) i s given by (3.6) with S , = S Q , b replaced by b ( l - a ^ - c ^ - a ^ ) and d by (d+v^+v^+v,^)• We may use the stochastic model for the growth of the s e n s i t i v e c e l l compartment to "update" the p r o b a b i l i t y generating function for newly r e s i s t a n t stem c e l l s as follows. In deriving the p r o b a b i l i t y generating function (4.7) we assumed that B was a constant. If instead we consider - 1 0 3 -B Q to be a random variable with d i s t r i b u t i o n not dependent on t, then $(s;t) can be viewed as being condi t i o n a l on B Q. If we emphasize t h i s by wr i t i n g ( s ; t ) , then we see from ( 4 . 7 ) that B 0 ~ B 0 4 (sjO-I^Cs;!:)] U. Furthermore i f BQ has a d i s t r i b u t i o n with support on the non-negative integers with p r o b a b i l i t y generating function 0(s) say, then the unconditional p r o b a b i l i t y generating function of the number of c e l l s i s given by 0(<&,(s;t)). In p a r t i c u l a r t h i s w i l l be useful here since a f t e r treatment the number of stem c e l l s i s random. Using ( 2 . 3 ) , we may write an expression for the p r o b a b i l i t y generating function i n an intertreatment i n t e r v a l as n s;t j 4^)=¥($ 1(s;v)* 0(s;v),A 1(s;v), * 2 ( s ; v ) , < t ) 3 ( s ; v ) ; t j ) , . . . ( 4 . 1 8 ) where t^<t^+v<t^ +,, t ^ ( j = l , . . . , J ) are treatment times, <£,(s;v) i s given by ( 4 . 7 ) with B = 1 , A.(s;v) ( 1 = 1 , 2 ) i s given by ( 4 . 8 ) , A,(s;v) i s given by ( 4 . 9 ) , and 6 n ( s ; v ) i s given by ( 4 . 1 7 ) . u ^ We may therefore use equations ( 4 . 1 5 ) and ( 4 . 1 8 ) to calculate r e c u r s i v e l y the r e s u l t i n g p r o b a b i l i t y generating function for the growth process corresponding to various treatment sequences by s e t t i n g v=t for the i n t e r v a l [t..,t^ +,) w n e r e the i n i t i a l p r o b a b i l i t y generating function at time t , i s given by ( 4 . 1 0 ) . Notice that we may use ( 4 . 1 8 ) r e c u r s i v e l y at times where treatment i s not given i n order to improve the approximation to the f u l l y s tochastic model. In general we would not do t h i s p r i o r to t , as t h i s would then induce (a non-degenerate) d i s t r i b u t i o n for Rg(t^) with a l l the attendant problems t h i s produces (as extensively discussed i n Chapter 3 ) . -104-In the s i t u a t i o n to be considered l a t e r (Chapter 5) we w i l l only use (4.18) at times of treatment, that i s v = t ^ + ^ - t j . We w i l l use (4.10) for the i n t e r v a l [ 0 , t ^ ) , chose t ^ as given i n (4.13) and use (4.14) with R * = [ N - m 1 ( t ~ ) - m 2 ( t ~ ) - m 1 2 ( t ~ ) ] (the integer part) where m^t") (1-1,2), m ^ 2 ( t j ) are calculated from (4.11.1-2) and N i s the "observed" stem c e l l compartment s i z e . The incorporation of a stochastic element to the growth of the s e n s i t i v e stem c e l l s i s somewhat ' a r t i f i c i a l ' however i t does improve the approximation of the model to the f u l l y stochastic one. It also allows a reasonable determination of P {N(t)=0} which would otherwise be i d e n t i c a l l y zero i f R()(t) were l e f t purely dete r m i n i s t i c . The model can be expected to be a reasonable r e f l e c t i o n of r e a l i t y since when there are large numbers of s e n s i t i v e stem c e l l s , growth can be expected to be quite regular and thus well approximated by the deterministic assumption. When the number of s e n s i t i v e c e l l s i s small, the l i k e l i h o o d that new r e s i s t a n t c e l l s w i l l a r i s e (from R Q ) i s small and thus the assumption of deterministic growth should not cause a great d i s t o r t i o n to the d i s t r i b u t i o n of the number of r e s i s t a n t c e l l s . As i n Chapter 3 we w i l l now consider some s p e c i a l cases which i l l u s t r a t e the behaviour of the model• In many cases two drugs may not be given together because of t h e i r overlapping t o x i c i t y on normal t i s s u e . Consider the s p e c i a l case where the drugs act independently and can be given together, with N(0)=RQ(0)=1 (the tumor originates from a single s e n s i t i v e stem c e l l ) , ic^ i 2 ~ n 2 12 =^' and %^ Q=ll2 0 = K1 2~T"2 1 =^ ^ 1 i a n ( * T"2 2 a r e a r b i t r a r y ) i«e. when the two drugs are given together a l l stem c e l l s are k i l l e d except those i n R 1 0 . -105-When N(0)= RQ(0)=1 we w i l l set B Q=(1-E ) ^ as described i n the discussion leading to (4.13). If both drugs are given at t , then the p r o b a b i l i t y the tumor w i l l be cured, P , i s given by ¥(1,1,1,e;t 1), (see (3.14)), 1 where ¥(s;t) i s given by (4.10). This reduces to P = $ ( l , l , l , e ; t ~ ) , 1 since (Kl,s,,s 2,s.j) = !• To simplify notation we w i l l write P F C for P F C . Examining the terms i n (4.7) we have for s^ = e that u -2+<x. ,„ -(6+a. 1 0d+v. ,„)u / g.(v)dv = 6 [1-e x » 1 2 x ' 1 2 ]• 0 ( o + a l i l 2 d + v l f l 2 ) For 1=1,2 and s^=e we also have ^ ( 1 ) = 6(c+a. 1 0d+v. 1 0 ) t -a*/* \ A i,12 i,12 r B (t-u) du . b Jr> / \ (6+a. 1 0 d+v. 1 0 ) u , E / 1 N 0 ( a i ) 1 2 b + v i , i 2 ^ e 1 , 1 2 1 , 1 2 + 6 ^ 1 - a i , 1 2 ^ After some s i m p l i f i c a t i o n we obtain 2 a.a 6 t . (6+a* 1 0 ) u l n P = - I ' j (e i ' 2 ~1) B(t~u) du C 1=1 k ^ (6+a* n ) u , ./., N a i , 1 2 e 1 , 1 2 + 6 ( 1 - < X i , 1 2 ) , t a12 / B ( t - u ) du, ...(4.19) b 0 where a^, a^ ^(i=l,2), a,^ are as given i n (4.11.1-2) and a l , 1 2 = a i , 1 2 _ a i , 1 2 6 -Using B ( t ) = B Q exp {( 6-a,-a 2-a, 2)t}, the formula for P F C may be numerically evaluated. As we are primarily interested i n treatment applied at some fixed s i z e but unknown time, we w i l l r e s t r i c t attention to the c a l c u l a t i o n of P where t = 6 '''Jin [N(l-e)] and i n th i s case we w i l l designate the p r o b a b i l i t y of cure as P N » P ^ i s plotted as a function of N for various mutation rates i n Figure 4. In most cases of -106-i n t e r e s t we w i l l have 6»a^+a2+a^2 a n d 6 » a ^ ^ f ° r 1=1»2. Figure 4 shows that for some sample values of a^, a^, ai2> a ± 12 ^ = ^ ' ^ t n e s n a P e of the r e s u l t i n g curves of against N are s i m i l a r to those obtained f o r the analogous case i n single resistance (Figure 1). This suggests that i n analogy to (3.37) and (3.41) i t may be possible to approximate ?^ by a N—1 - l function of the form (1-a*) , or exp{-a*(N-l)}, (which are numerically s i m i l a r for a * « l ) where a* i s a function of a^, a^, a ^ . a ^ ^ a r u * a„ 1 9 . We w i l l thus attempt to approximate (4.19) for fixed N; to do 2,1/ t h i s we w i l l f i r s t bound P^. To simplify further presentation we note that the scale of measurement of t i s unimportant i n the c a l c u l a t i o n of Pjj. Thus we w i l l choose a scale for which b=l and assume that the other rates are a l l r e l a t i v e to this time scale. This w i l l be emphasized by writing e(=d/b) rather than d. Thus using B(t) = (1-e) *exp {(l-e-a^-a2~a :| L2) t} and a^a^ra^ra^ i n (4.19) we obtain - I a±a± 1 2 N [ N ( l - £ ) ] - a ( 1 - £ ) J Q - ± ^ 1 — du 1-1 * .-1 (1-e+a* 1 0 ) u . . (1-e) a i ) 1 2 e ^ 1,12' + - _ a l _ 2 _ _ { [ N ( l - e ) ] 1 " a ( 1 " e ) -1}, ...(4.20) (1-e-a) where t = ( l - e ) - 1 A n [ N ( l - e ) ] . We w i l l now bound AnP N« For i=l,2 l e t h^(u) be the integrand on the righ t hand side of (4.20). Then i f U-e- 2 ( 1- £- a ) u}  U ±(u) = , -1 (l-e-a)u , ,, v - ( l - e - a ) u (1-e) a ± i l 2 + ( l - a ± f l 2 ) e -107-Flgure 4 Probability of Cure when Loss i s Present. tor 0.8 > =! 0.6 oo g0 .4h CC Q_ 0.21-0.0 10 0 101 102 103 104 105 106 Number of Stem Cells-N 107 P r o b a b i l i t y of cure Pfj as a function of stem c e l l burden at diagnosis where and are given simultaneously at that time, %^ Q=0> ^ ^ a r b i t r a r y , \ 1 2 = 1 f ° r 1 = 1 , 2 \ 2 ="2 1 = 0 , a i + v i / b = a i i 2 + v i 1 2 / b = 1 0 ~ 3 f o r i = 1 » 2 a n d b = 1 * For x i = = a £ + v i / b < 1 0 - 2 and x.^  i 2 = a ± 1 2 + V i i 2 ^ ^ ® ~ 2 > P N D E P E N D S ( U P T O T B E fourth decimal place) on x. and x. , „ only and not on the i n d i v i d u a l <x.,v, r i i , 12 i i etc. that sum to x^. P N as given i n (4.19) with t=6 1 l n [ N ( l - e ) ] i s pl o t t e d for three values of e: ( i ) e=0 ( i i ) E==0.5 ( i i i ) e=0.9 L ±(u) = we have -108-{ 1 _ e - ( l - e - a ) u } ( l e; a . j l 2 e +i a±^2 U^u) - h^u)>0, i f u>0 and l-e-2a>a* 1 2 > L ±(u) - h i(u)<0, i f u>0, and thus U i(u)>h 1(u)>L i(u) for u>0 and l-e-2a>a* 1 2 « ...(4.21) By integr a t i n g the bounding functions over [0,t] we obtain k 1 / 2 k 1 / 2 k 1 / 2 - t a n " 1 ^ ) e " X t ) ] + i [ e " X t - l ] } K0 K l and ^ L i ( u ) d u = i { ( - ^ ) *n [ ° ^ + i [ e " X t - l ] } , u 1 K k Q+ kje 1 where \=l-e-a, ^ Q=a^ \ and k,= l - ^ ,2> Now t=(l-e) 1An(N(l-e)) and thus r x T / i \ i l ~ a ( l - e ) 1 e =[N(1-e)] v ' For large N, e X t=0. If also (l-e)»a, ^ 2 + a 2 ^ 2 then k^ i s small and k ^ - l . I f i n addition k ^ ^ e X t i s small then, J 0 u i v u , , a u = ^J-~t-a> and ^U 1( )du - (1-e-a) l{ | [ a i ) 1 2 d - e ) "l} jJjl^ OOdu » (1-e-a) 1 { A n [ ( l - e ) / a ± j u]-l }. To this order of approximation, we have ( 1 - e - a f M | [ a 1 ) 1 2 ( l - £ ) " l ] " l / 2 - l ) > /Qh i(u)du > ( l - e - a ) _ 1 { A n [ ( l - e ) / a i ) 1 2 ] - l } . ...(4.22) In Chapter 3 we found that P J J ( N f i x e d , single resistance) did not -109-depend upon e. However J Q I K C U ^ U does depend upon e since i f e=0, a « l , we have j j u l ( » > d . . i [a l j l 2r1 / 2 - 1 . 1/2 whereas i f l-e<[a^ ^ ] , we have 'oVU> = t a i , l 2 r 1 / 2 [ - I ^ ^±,12^' Since a^ , ^2«1» the lower bound can exceed the upper bound (for d i f f e r e n t e). For example consider the i n e q u a l i t i e s for the two cases e=l-10 - 3 and e=0 where a± 1 2 = 1 0 - 6 , a = 1 0 ~ 5 * Equation (4.22) implies / ^ h ^ u ^ u , and therefore P^, varies with e ( i n contrast to what was found for s i n g l e r e s i s t a n c e ) . Numeric evaluation of (4.20) for 10~ 9<a<10 - 1, 10<N<109 reveals that the lower bound JgL^(u)du i s close to / ^ ^ ( u j d u , and can be used i n approximating P N for most cases a r i s i n g i n p r a c t i c e . Using the l-a( l-e)"''" r i g h t hand side of (4.22) and approximating N v ' by N, - a ( l - e ) " 1 (1-e) ^ ' by 1, and 1-e-a by 1-e and using (4.20) i n the r e s u l t i n g expression for AnP Nin (4.20) we obtain AnP N « -(1-e) N { a 1 2 ( l - e ) + ^ ^ a i a j L > 1 2 [ A n ( ( l - e ) / a i j l 2 ) - l ] } . ...(4.23) This approximation i s of the form expected i . e . P^= exp{-a*N}, where -1 ? a* = a 1 2+ (1-e) X [ A n ( ( l - e ) / a 1 > 1 2 ) - l ] . ...(4.24) A s i m i l a r approximation has been derived previously i n a less general se t t i n g [27]. Note that a* depends upon e ( i . e . d). If e i s large, say 0.9, the e f f e c t on P„ can be considerable. A s i m i l a r e f f e c t on P„ would N N be seen for cases i n which it^, n;2, rc^ 2'^2 1 a r e n 0 t n e c e s s a r i l v 0 although the c u r a b i l i t y of the tumor w i l l then depend upon the t o t a l treatment -110-protocol. The structure exhibited i n equation (4.23) has implications for the general analysis of these processes. Resistance to some drugs appears to a r i s e from a single d i s c r e t e change i n the genetic material. In such cases resistance may be almost absolute. In other cases resistance may a r i s e incrementally, such as i n processes involving gene a m p l i f i c a t i o n [28]. In these circumstances the a c q u i s i t i o n of each gene copy may be viewed as a separate stage. Therefore the d i s t r i b u t i o n of the numbers of c e l l s possessing a s p e c i f i e d l e v e l of resistance ( i . e . some minimum number of gene copies) w i l l be that of a multistage process and not that of a single stage process. This c l e a r l y represents a d i f f f i c u l t problem when attempting to analyze experiments designed to estimate mutation rates to drug resistance. Indeed i n a multistage process there i s no single parameter to estimate but rather a variable number depending on the number of stages involved. The number of stages would also be needed to be estimated ( i f not known) from such experiments and given the extremely variable nature of the basic process, i t seems that estimation of parameters w i l l be quite d i f f i c u l t . Furthermore, even when the number of stages i s known, i t i s not possible ( i n general) to write down expressions for the d i s t r i b u t i o n functions for the multistage process. This problem i s i n need of much more d e t a i l e d exploration. We w i l l now consider the problem of planning treatment and how this model may be used i n th i s context. 4.3 Optimal Scheduling In attempting to f i n d an optimal treatment plan i t i s necessary to consider two fa c t o r s : a c r i t e r i a which q u a n t i t a t i v e l y measures the value -111-of a treatment plan, and the set of treatment regimens which are to be considered. Ideally the c r i t e r i o n would include measurement of both the therapeutic and toxic e f f e c t s of a treatment plan on the subject. Unfortunately, the side e f f e c t s of various treatment regimens are often d i f f i c u l t to describe i n a quantitative form. We s h a l l assume that each regimen to be considered has acceptable side e f f e c t s and that the "value" of the therapy may be measured by i t s (tumor s p e c i f i c ) therapeutic e f f e c t s . A natural c r i t e r i o n for the value of any regimen i s the p r o b a b i l i t y of cure, since cure i s the usual object of therapy. In t h i s case P J J , the p r o b a b i l i t y of cure for a tumor f i r s t treated at size N, w i l l be defined as the l i m i t t-*» i f P{N(t)=0|N(t 1)=N}. When a l l the tumor and drug parameters are known i t i s possible to examine the e f f e c t of various dosages and schedules of administration on the p r o b a b i l i t y of cure for the tumor using equations (4.10), (4.15) and (4.18). In cases where cure i s u n l i k e l y another "natural c r i t e r i o n " i s the expected number of c e l l s at some time a f t e r the completion of therapy E[N(t)], t>tj where t j i s the time of the l a s t treatment i n the regimen. This quantity may be simply evaluated using (4.12.1-2) i n conjunction with equations generalizing (3.12.1-2). For a given set of therapeutic regimens i t seems desirable that the optimal regimen be the same for e i t h e r c r i t e r i o n (Pjj or E [ N ( t ) ] ) . Unfortunately this i s not always the case, although i n many cases of i n t e r e s t the optimal strategies are the same (as w i l l be discussed l a t e r ) . One way to r e s t r i c t the set of possible protocols i s to consider those of some fixed length, that i s , those where there are a f i x e d number of times at which treatments are applied (protocols of fixed length). -112-Notice that i t i s always possible to "improve" a protocol, that i s , increase P J J or decrease E[N(t)], by adding further treatment applications to the end of the regimen. By th i s reasoning any protocol of length J - l (number of cycles of therapy) w i l l be no better than at least two protocols of length J ( i . e . those which add a single cycle of eith e r or T 2 to the protocol of length J - l ) . The length of the regimen w i l l therefore depend on a decision about the value of any further increase i n the p r o b a b i l i t y of cure versus the "costs" (both human and f i n a n c i a l ) associated with extra cycles of treatment. Protocols of fixed length are of some i n t e r e s t since they correspond to the structure of many c l i n i c a l protocols. Another way to r e s t r i c t the set of possible treatment regimens i s to consider only those which s a t i s f y some constraint placed on the measure of the therapeutic e f f e c t . That i s , we can r e s t r i c t attention to protocols for which Pfl>A (0<A<1) or E[N(t)]<k (k>0, where care must be taken i n the s e l e c t i o n of t used i n t h i s case). Sets of protocols (when not empty) s a t i s f y i n g such a condition are of some i n t e r e s t when i t i s desired to reduce the duration and quantity of therapy without unduly in f l u e n c i n g therapeutic r e s u l t s . The optimal regimen w i l l then be one where the number of treatments J i s minimal, among regimens s a t i s f y i n g the condition imposed. Notice, that once J i s determined then the optimal protocol of length J (determined from the set of protocols of length J) w i l l be an optimal protocol by th i s c r i t e r i o n . Thus the optimal protocol of length J i s of quite general i n t e r e s t . Examination of the e f f i c a c y of the optimal protocol of length J, for a range of values of J, i s thus useful for determining both the length and "content" -113-of the protocol of c l i n i c a l i n t e r e s t . In p r i n c i p l e the s p e c i f i c a t i o n of the c r i t e r i o n for e f f i c a c y and the set of permissable protocols permit i d e n t i f i c a t i o n of the optimum regimen for a given set of tumor parameters, although this w i l l usually be rather a lengthy exercise. Many tumor parameters are not under con t r o l either i n the laboratory or i n the c l i n i c and thus i t i s not necessary to analyze the ef f e c t of changing these parameters on the optimal regimen (for a p a r t i c u l a r type of tumor). One parameter which i s under control i n the experimental s e t t i n g i s the size of the tumor at f i r s t treatment. In the c l i n i c i n d i v i d u a l patients, with tumors of the same type, present with d i f f e r i n g tumor burdens. It i s thus of some i n t e r e s t to know whether regimens which are optimal for one size (at f i r s t treatment) are optimal at other s i z e s . In general the optimal regimen depends upon the size of the tumor when the f i r s t treatment i s applied. Thus when i d e n t i f y i n g the optimal treatment plan for a p a r t i c u l a r s i t u a t i o n , care must be taken to v e r i f y that the plan i s optimal at a l l sizes l i k e l y to be encountered. A p r a c t i c a l problem a r i s e s i n the therapy of c l i n i c a l disease when few of the relevant tumor parameters are known with any accuracy. C l e a r l y ignorance of the parameters makes i t d i f f i c u l t to evaluate optimal s t r a t e g i e s . However, i t i s possible to derive optimal rules i n the p a r t i c u l a r case, where two drugs are of equal ef f e c t i v e n e s s . Since this case i s of some p r a c t i c a l Interest, we w i l l now examine i t i n some d e t a i l . 4.4 Optimum Scheduling f o r Two Equivalent Agents One s p e c i a l case which i s of some p r a c t i c a l i n t e r e s t i s the -114-s i t u a t i o n where two drugs (or combinations) are a v a i l a b l e which are of approximately equal e f f i c a c y . This appears to a r i s e i n the treatment of Hodgkin's Disease where two combinations, MOPP (Nitrogen Mustard, Oncovin, Procarbazine and Prednisone) and ABVD (Adriamycin, Bleomycin, Vinblastine and Dacarbazine) produce s i m i l a r cure rates and remission rates when delivered over the same time i n t e r v a l [29]. These observations suggest that the development of resistance to each combination proceeds at the same rate and that c e l l k i l l s of each combination are s i m i l a r . The a v a i l a b l e evidence also suggests that each combination i s equally successful i n producing remissions and cures i n tumors which have previously f a i l e d with the other therapy. This implies that each combination's e f f e c t i s approximately the same i n c e l l s r e s i s t a n t to the other. As a f i r s t approximation we may consider the two drug combinations as having equal values for the model parameters. In th i s s i t u a t i o n we w i l l r e f e r to the two combination as being equivalent, and by that we w i l l mean that each drug has i d e n t i c a l values for a l l parameters. In what follows we w i l l model two agents as two i n d i v i d u a l drugs. When an agent consists of a combination of drugs this model must be considered a f i r s t approximation since resistance to multiple agents i s more complex than that to a single agent (see discussion i n Section 4.2). E x p l i c i t l y two agents w i l l be said to be equivalent i f Q~%2 0' \,l=%2,2> %l, 12=^2,12' r l , 2 = 7 t 2 , l ' W W V1,12 = V2,12 a n d t h e intertreatment times t^+^-ty j = l , . . . , J - l are constant. In t h i s case, i f ¥(s ( ),s 1,s 2,s 3;0) = ¥ ( s Q , s 2 , s 1 , s 3 ; 0 ) , ...(4.25) that i s -115-p{R 1(0)=i,R 2(0)=j|R ( )(0),R 1 2(0)}=P{R 1(0)=j >R 2(0)=l|R 0(0),R 1 2(0)} then for t<t^ (the time of f i r s t treatment), we have ¥(s Q,s 1,s 2,s 3;t) = ¥(s ( ),s 2,s 1,s 3;t). ...(4.26) Here we w i l l assume (4.25) holds, which i s reasonable since otherwise we would expect the response of the tumor to therapy by (alone) to be d i f f e r e n t from the response to T2 (alone) and thus the agents would not appear equivalent. This d e f i n i t i o n of equivalent agents has been used previously i n the consideration of the e f f e c t s of cancer therapy [30,31]. Intertreatment times are usually selected to be the minimum times necessary for the recovery of normal tissues between cycles of treatment. By assuming that intertreatment times are the same for each treatment we indicate that the minimum recovery time for each treatment i s the same. The term "equivalent" i s motivated by the observation that i f either of the drugs i s used alone then the d i s t r i b u t i o n of the t o t a l number of c e l l s w i l l be the same for each drug. Note that from the general d e f i n i t i o n of the re s i s t a n t states, we have it. „<7i. 1 0 and %. ~ .<n. . for i=l,2. The i,0 i , i i,12 i»3-i i , i tumor parameters b, c and d are fixed and w i l l not be e x p l i c i t l y s p e c i f i e d . As noted before (Section 3.3), chemotherapy i s given i n repeating cycles for c l i n i c a l disease i n which the doses and drugs used are fi x e d i n advance [32]. The intervening time between repeat applications i s determined by the recovery time of the patients' normal t i s s u e s . This recovery time i s selected to be the minimum time for the necessary recovery. Protocols which administer the cycles at greater than the minimum i n t e r v a l w i l l be less e f f e c t i v e than those giving the same drugs -116-at the same dose i n the same sequence as frequently as permissable, since longer intertreatment times allow more time for regrowth. We w i l l now consider the construction of optimal rules f o r sequencing the administration of two equivalent agents. In th i s section we w i l l consider the construction of the optimal treatment regimen within the set of protocols of fixed length J (number of times a treatment i s applied). We w i l l r e f e r to the treatment plan as a strategy, which represents the sequence i n which treatments are administered (the times of administration being already s p e c i f i e d ) . F i r s t we w i l l f i x J, the number of times of administration of treatments i n the regimen. A therapeutic strategy, S, w i l l be represented by a vector which consists of a sequence of J l ' s or 2's with each number r e f e r r i n g to the subscript of the treatment given (either or T2), and the sequence i n d i c a t i n g the order i n which they are given. There w i l l be 2 3 such strategies and we w i l l write S(v) when we wish to refer to a p a r t i c u l a r strategy i n the set. A sol u t i o n to the fixed length problem, which of course w i l l depend on J , w i l l be referred to as an optimal strategy for each c r i t e r i a of treatment e f f i c a c y . At least one optimal strategy e x i s t s because the number of strategies of fix e d length J Is f i n i t e . When a tumor i s treated with strategy S(v) we w i l l write the p r o b a b i l i t y of cure as P ^ ( S ( v ) ) and the expected number of c e l l s as E [ N s ( v ) ( t ) ] . Having defined the set of strategies to be considered i t remains to specify the c r i t e r i o n for the e f f i c a c y of the therapy. As before, two natural candidates are PJJ and E [ N ( t ) ] . From ( 4 . 2 5 ) and general considerations of the behaviour of the process, at least two d i s t i n c t -117-s t r a t e g i e s have the same value of because the drugs are equivalent and each strategy has a "mirror image" ( i . e . l ' s and 2's interchanged). Similar considerations apply to E[N(t)] and to any symmetric functional (with respect to R ^ ( t ) and R ^ ( t ) ) of the d i s t r i b u t i o n of { R 0 ( t ) , R 1 ( t ) , R 2 ( t ) , R 1 2 ( t ) } . We wish to show that there e x i s t optimal strategies which are independent of the drug and tumor parameters for any pair of equivalent drugs. Such optimal strategies do e x i s t for the c r i t e r i o n E[N(t)] as we w i l l show subsequently. Unfortunately these s t r a t e g i e s are not necessarily optimal for P J J , as w i l l be shown by producing a counterexample (see Chapter 5). We may formally l i n k minimizing E[N(t)] and maximizing under p a r t i c u l a r circumstances as follows. For two strategies S ( i ) and S ( j ) , i f P{N s ( i )(t)>k}>P{N s ( j )(t)>k} for a l l k, ...(4.27) then i t follows immediately that E[N ( t ) ] > E[N ( t ) ] and P N<S(i)) < P N ( S ( j ) ) . Thus the i n t u i t i v e idea of minimising E[N(t)] w i l l also be formally equivalent to maximising Pfl(S) i n s i t u a t i o n s where the rather strong condition (4.27) of stochastic ordering a p p l i e s . However, i t i s doubtful that t h i s condition could ever be v e r i f i e d i n p r a c t i c e . The p a r t i c u l a r s i t u a t i o n of equivalent drugs permits the consideration of c r i t e r i a of e f f i c a c y other than P^ and E [ N ( t ) ] . A second quantity which can be minimized may be motivated by the consideration of the r e s i s t a n t subcompartments of the tumor. I n i t i a l l y we observe that c e l l s i n RQ "see only" one drug, since each drug has the same e f f e c t on RQ c e l l s . Thus the only component of the -118-strategy which a f f e c t s the d i s t r i b u t i o n of the number of c e l l s i n R Q at time t > t : j > i s the length J of the strategy. S i m i l a r l y the e f f e c t of the strategy on c e l l s already i n R , 2 at t , depends only on the length J and not on the order i n which the drugs are given. Therefore the "value of" s t r a t e g i e s r e s u l t from t h e i r d i f f e r e n t i a l e f f e c t on the c e l l s i n R ^ and R 2 » For any strategy to be of some value i t must be capable of causing a net o v e r a l l decline i n the mean number of singly r e s i s t a n t c e l l s . This, of course, does not follow from any formal constraints placed on t h i s model but from a consideration of what this would imply about the regrowth of the r e s i s t a n t c e l l s . It i s possible for treatments to eliminate s i n g l y r e s i s t a n t compartments (when treatment i s given) with a non-negligible p r o b a b i l i t y , even though the net mean growth of these c e l l s may be p o s i t i v e because of a very large regrowth between treatments. There i s no evidence to suggest that this occurs i n c l i n i c a l disease although there are probably many cases where small c e l l k i l l s are 'balanced' by regrowth between treatments. An i n t e r e s t i n g case of t h i s has been i d e n t i f i e d by Skipper i n his analysis of the response of a mouse mammary tumor to treatment by the CAF (Cyclophosphamide, Adriamycin and 5-Fluorouracil) regimen [33]. The objective of the therapeutic strategy i s to cause a net decline (to e x t i n c t i o n ) of the s i n g l y r e s i s t a n t c e l l s i n such a way as to minimize the number of t r a n s i t i o n s to double resistance. C l e a r l y , the cases of greatest importance are those where the growth of the c e l l s already i n R ^ 2 cannot be made s u b c r i t i c a l and i t i s necessary to plan the strategy so that the l i k e l i h o o d of t r a n s i t i o n s from the s i n g l y -119-r e s i s t a n t to the doubly r e s i s t a n t state i s minimized. Even i n cases where the chemotherapy can make the growth of doubly r e s i s t a n t c e l l s s u b c r i t i c a l (over the treatment plan) one would wish to minimize the number of new doubly r e s i s t a n t c e l l s since they are by d e f i n i t i o n the most d i f f i c u l t to tr e a t . We w i l l now develop an expression for the mean number of tr a n s i t i o n s from single to double resistance during the treatment period and subsquently derive the form of strategies which minimize this quantity. Consider the number of c e l l s i n at any time tz(ty which are derived from c e l l s i n R, at time t . . Conditional on R,(t.) the 1 J 1 3 expected number of such c e l l s at time t, g^(t^, t ) , i s given by the f i r s t term on the right hand side of (4.12.1) with m^O^R^t..): g 1 ( t j , t ) = R ^ t j ) exp { ( 6 - a 1 ) 1 2 ) ( t - t j ) } . Conditional on R^ ( t ^ ) , the expected number of t r a n s i t i o n s to double resistance i n the i n t e r v a l ( t ^ , t ) by these c e l l s , ( t ^ . t ) , i s given by u ^ ( t .,t) = a 1 ) 1 2 / ^ g l ( t .,u)du. The simplest way to obtain the above r e l a t i o n s h i p i s to consider c e l l s i n R^ as being s e n s i t i v e (to T 2) and i n R^2 as being r e s i s t a n t (to T 2) and use the d i f f e r e n t i a l equations leading to (3.8) as follows. Set mQ(t)=g^(tj , t ) , <xb+v=a^  ^ 2 and m^(t)=u^ ( t j , t ) . Then solve the d i f f e r e n t i a l equation f o r ( t ^ , t ) noting that the term 6u^ (t_.,t)sO because here we are counting the number of tr a n s i t i o n s (which have no i n t r i n s i c growth). L e t t i n g |i-(t . ,t)=E[ ( t . , t ) ] , i t follows that t (6-a ) (u-t ) u x ( t , t ) - ax J m (t )e L * L * J du. . . . ( 4 . 2 8 ) -120-S i m i l a r l y we may define \i^{tyt) for t r a n s i t i o n s from to R,2' The expected number of t r a n s i t i o n s to double resistance i n [ t . , t . ,) from J J + l s i n g l y r e s i s t a n t c e l l s at t . i s thus u . ( t . , t . , - )+u„(t.,t.., ). Thus the J 1 j J+l 2 V j ' j + l ' mean number of those events which occur i n some [t.., f ° r j =0»...»J (t^-0) i s given by J 2 M'" I I U.(t ,t ), ...(4.29) j-0 i - l J J where for s i m p l i c i t y we set t j + , = t y K t j ~ t j_,) • We seek strategies which minimize (4.29). In seeking to minimize (4.29) we may minimize any function of the form KM'+C where K (>0) and C are not dependent upon the strategy. In p a r t i c u l a r we may replace each term u ^ ( t j , t ,) of the form (4.28) by (k-6+a )(u-t ) t a l M o ( t )[e ' 3 -1] (6-a )(u-t ) a i , l 2 J t . K(tj> + [k-6+a, > e d u j I»1^ = a i , 1 2 ^ t J + l m i ( u ) d u 3 from (4.12.1). The added terms do not depend upon the strategy and thus minimizing M' is'equivalent to minimizing M*, where J t ~ M* ml / J [m 1(u)+m 2(u)]du. ...(4.29) j-1 J Minimization of this quantity has previously been considered for the s p e c i a l case c=d=0, it, ±~1Z± 1 2 = 1 [^1' ^ e W H 1 n o w proceed to characterize the strategies which minimize (4.29) and then show that these strategies also minimize E[N(t)], t>t . J In order to do t h i s we now define some new q u a n t i t i e s . Let E, .(t) be the expected number of c e l l s r e s i s t a n t to T^ alone at time t, which derive from (have grown from, including the e f f e c t s of treatment) "new -121-mutations" ( t r a n s i t i o n s from Rn to ) i n the i n t e r v a l [ t . ,, t.) f o r 0 V j - l ' y j=l,...,J+l. Define E .(t)=0 i f t<t . Singly r e s i s t a n t c e l l s present 1>J J — i at t=0 w i l l be included i n the i n t e r v a l [ t Q , t 1 ) = [ 0 , t 1 ) . Then for i=l,2 and u < t J + 1 J+l m (u)= I E (u), i k=l 1 , K and we may then write (4.29) as J t ~ j+l M*= I J 2 I [\ k(u)+E (u)]du, j=l t , k=l l ' K Z ' K which gives M*= I I / t j + 1 [E, k(u)+E (u)]du j=l k=l t . L'* l , K J J t ~ + I S 3 t E l , j + l < u ) + E 2 , j + l ( u ) ] d u ' ...(4.30) The second term on the right hand side of (4.30) represents "new mutations" a r i s i n g from c e l l s i n RQ between cycles of treatment which have not been exposed to either drug. Conditional on the treatment times and the number of treatment cycles, the d i s t r i b u t i o n of c e l l s i n R Q i s the same for a l l treatment strategies (because the two drugs are equivalent) for ab i t r a r y t, and thus the second term i s the same for a l l such s t r a t e g i e s . Thus minimizing M* i s equivalent to minimizing the f i r s t term on the right hand side of (4.30). The growth of E ,(u) over [ t . , t . .] for a l l j , k where j>k i s i>k j j+l exponential with parameter (6-a^ ^ ) (see f i r s t term i n (4.12.1)). Thus E i , k < u > d u " Ei,k (V< 6- ai,12> - 1 i - P [ ( 6 - a i ) 1 2 ) ( t j + 1 - t j ) ] - l } . -122-Using (4.30) we have M* = ( 6 - a 1 ) 1 2 ) - 1 e x p { ( 6 - a 1 ) 1 2 ) ( t 2 - t 1 ) - l } M + K (since a, , 0=a 0 1 0 and t . . , - t . i s constant for j=l,...,J) where K i s a 1,12 2,12 j+l j constant given by the second term of (4.30) and J j 2 M = I I I E (t ). ...(4.31). j=l k=l i = l 1 , K J Thus minimizing M* i s equivalent to minimizing M. We w i l l now proceed to develop our notation i n order to e x p l i c i t l y minimize M and thus minimize M". Define 2 J : (S(v)) = I I E (t ), * i = l j=k 1 , 1 C 3 where the E,^(t) are calculated for the treatment strategy S(v); then J M = I C (S(v)) . k=l fc Define 6j(v) = 1 i f the j - t h treatment i n S(v) i s T,, = 0 otherwise, ...(4.32) and l e t I X k(S(v),A) = | I 6 (v) - [1-6 (v)] |, for 1 <k<KJ, j=k J J the modulus of the number of times T^ i s given minus the number of times T^ i s given between the k-th and A-th times of treatment. For k = l ) • • • y J y l e t K {S(v)}= {i:JL ( S ( v ) , i ) = max 3C (S( v), I) }, ...(4.33) K K k<KJ K the indices of treatment times where the modulus of the difference i n the number of T]_'s and T2*s i s maximized commencing at k. Let B ={S(v): max X, (S(v),A)=l} for k=l,...,J, ...(4.34) fc k<KJ K the set of strategies where, commencing at the k-th time of treatment, -123-the maximum modulus of the difference i n the number of times and T^are given equals 1. Let g = expiCS^ 1 2 ) t^)} for j = l , ..., J . As before, .^r ^ = P{a c e l l i n R . w i l l survive one cycle of T. } for i = 1,2 i ,Q 1 Q l ' To s i m p l i f y notation, l e t TCQ = i ( = 7 t 2 2^ a n d %l = \ 2^=%2 1^ w n e r e U^<UQ from the general d e f i n i t i o n of resistance. By equivalence E, .(tT) = E_ .(tT) and we w i l l l e t E .=E, .(tT). Define i , j y 2 , j v y 2 i,3 J k n.(k) = I 6 ( v ) , J i - j (see 4.32), the number of times T^ i s given between the j-th and k-th cycles of therapy where reference to the strategy, indexed by v, i s supressed for s i m p l i c i t y . Using this notation i t i s then straightforward to show J n.(k) k-j+l-n.(k) k-j+l-n.(k) n.(k) C.(S(v))= E I gk~2 [%Q2 * 2 + nQ 2 r^2 ]. ...(4.35) J Jk=j We see from (4.35) that, as expected, mirror image strategies ( i . e . 1 and 2's interchanged) have the same value of C^(S(v)) since E^ does not depend on S(v). Having developed the required notation we w i l l now show that M given by (4.31) and thus M' given by (4.29) i s minimized by the a l t e r n a t i n g strategies {l,2,1,2,...}, {2,1,2,1,... } amongst those of fixed length J . THEOREM 1 J Among a l l strategies of fixed length J, M = £ C.(S(v)) i s minimized only j - l 2 by the two strategies which alternate therapy at each cy c l e . Proof: -124-The proof w i l l be achieved by characterizing the strategies S(v) which minimize Cj(S(v)) for ab i t r a r y j . We w i l l then show that the al t e r n a t i n g strategies minimize Cj(S(v)) for a l l j . The proof w i l l consist of three parts: ( i ) Choose a r b i t r a r y j . For any S(v) not i n (see (4.34)), there ex i s t s S(v*) e B^ such that Cj(S(v*)) < C^(S(v)). ( i i ) If S(v), S(v*) £ B^ then C ( S ( v ) ) = C..(S(v*)). ( i i i ) If S(l) = (1,2,1,2,...) and S(2) = (2,1,2,1,...) then S ( l ) and J S(2) minimize £ C.(S(v)) among a l l strategies of length J . j-1 2 ( i ) Choose a r b i t r a r y j . If S(v) i s not i n B^ choose one k e K j ( S ( v ) ) , as defined i n (4.33). Consider f i r s t the case k<J and 6 k ( v ) = l , that i s the k-th cycle i s T,. Let be the operator which interchanges the k-th and k+l-st elements of a strategy. Now k e K^(S(v)) and k<J implies that S^ + 1(v)=2. Consider the strategy o kS(v). Using (4.35) we have C ( S ( v ) ) - C (cr kS(v)) k-j n ( k ) - l k-j+l-n (k) = Ejg [nQ2 nl ( l l o~ 1 tl ) k-j+l-n (k) n ( k ) - l + nQ (\-*0)] where n j ( k ) i s calculated for S(v). Thus we may write C j ( S ( v ) ) - C j ( a k S ( v ) ) n ( k ) - l k-j+l-n (k) k-j+l-n (k) n ( k ) - l - E.g 2[*Q 2 nl - 7 t0 \ 3 1 k - i where E_.g J does not depend on S(v). Now since S(v) i s not i n B^ and 6 k ( v ) = l , we have n.(k) > (k-j+2)/2, n ( k ) - l k-j+l-n (k) k-j+l-n (k) n ( k ) - l Since ^Q>^ we have TC, > UQ 1 -125-and thus CjCSCv)) - C ( o ^ S C v ) ) ^ . For k<J and 6^(v)=0 then we can also show the above r e s u l t using s i m i l a r considerations. For {J}=K_.(S( v)) consider the strategy S(v') where the J-th treatment i s replaced by the other treatment and obtain a s i m i l a r i n e q u a l i t y for C,.(S( v) )-C..(S( v') ) . We may now apply the same considerations to the new strategy which we have created (either a^S(v) or S(v')) and obtain a sequence of d i s t i n c t s t r a t e g i e s , {S(v)} say, which have s t r i c t l y decreasing C^.(S(v)). Now the number of possible strategies i s f i n i t e (for f i n i t e J ) , and t h i s process of producing new strategies must terminate since each strategy i s d i s t i n c t . Since there i s at least one v such that S(v) e B . (and the J process of improving strategies i s v a l i d for a l l v such that S(v) i s not i n B J ) we conclude that the sequence of strategies terminates with the l a s t member being contained i n B ^ . This proves the desired r e s u l t . ( i i ) For a l l j , K j < J , B . . contains 22+^J~2^2^ elements and therefore consider the n o n - t r i v i a l case S(v)*S(v*). Using (4.35) we have Cj(S(v)) - C j(S(v*)) = J k-j n.(k) k-j+l-n.(k) k-j+l-n.(k) n.(k) E I g |> 2 K 2 + 7!Q 2 T t 1 2 J k=j n*(k) k-j+l-n*(k) k-j+l-n*(k) n*(k) ~ ^0 ^ %l ^ ~ ^0 ^ \ 2 ~\' ...(4.36) where n*(k) i s calculated for strategy S(v*). Since S(v), S(v*) e B ^ we have and n^(k) = k " ^ + 1 = n*(k) for k-j+1 even, n^.(k) = n*(k) or n^(k) = k-j+l-n*(k) for k-j+1 odd. -126-Thus each term i n the sum (4.36) i s zero, and therefore C^(S(v)) = C\(S(v*)), proving the required r e s u l t . ( i i i ) Now S ( l ) , S(2) e B.. for a l l K j < J . Furthermore only S ( l ) and S(2) have t h i s property. But S(l) and S(2) minimize C^(S(v)) for a l l j and thus only S ( l ) and S(2) minimize J M = I C ( S ( v ) ) . j - l 2 The proof i s complete. The proof of a s p e c i a l case of t h i s theorem (c=d=0, it. .=iz. , n = l ) 1 , 1 i,12 has been presented previously [31]. We w i l l now show that the a l t e r n a t i n g strategies minimize E[N(t)] for t > t j . Theorem 2 Among the strategies of fixed length J, S ( l ) and S(2) minimize E[N(t)] for a r b i t r a r y t > t j . Proof: We w i l l evaluate E[N(t)] at time t J + 1 = t j + ( t j ' t j ^ ) ( a s before) without loss of generality. Consider the development of doubly r e s i s t a n t c e l l s i n the i n t e r v a l [tj> '•j+i^ ^ o r j=0»«'«>J from c e l l s which were not doubly r e s i s t a n t at time t ^ . Each such c e l l must have grown from one of three types of progenitor at time t ^ , i . e . either a R Q , a R ^ or a R ^ c e l l . The treatment sequence does not a f f e c t the d i s t r i b u t i o n of Rg(t) (only the length does because of equivalent treatments), so the number of doubly r e s i s t a n t c e l l s at t . , . derived from R _ c e l l s at t . does not J+l 0 j depend on the treatment sequence. Thus the d i f f e r e n t i a l e f f e c t of various strategies on the number of doubly r e s i s t a n t c e l l s at time t, t j < t < t j + ^ , r e s u l t s from i t s d i f f e r e n t i a l e f f e c t on singly r e s i s t a n t c e l l s -127-present at the treatment times t , , . . . t ^ . We w i l l now cal c u l a t e the expected number of doubly r e s i s t a n t c e l l s which have ari s e n from si n g l y r e s i s t a n t c e l l s present at treatment times. Let R 1 2 ( t , t ' , t " ) . ( t " > t ' , t>t') be the number of doubly r e s i s t a n t c e l l s present at time t whose progenitor ( f i r s t doubly r e s i s t a n t c e l l ) o r i g i n a t e d as a mutation from a singly r e s i s t a n t c e l l (either or R 2) i n [ t ' , t " ) . Using (4.12.2) we can write K i R i 2 < t k + i » V W l W * W i = [ \ ( \ ) + R 2 ( t k ) ] h ' f ° r k = 1 ' - - * . J . ...(4.37) „ „ ^ W V M - a l,12 ( t k + l H : k \ j , u where h=e (1-e ). Let T ^ - T C , 12 ^~r"2 12^ e n E [ R 1 2 ( t . ) ] = n2 E [ R 1 2 ( t j ) ] for J-1....J. 5 ( tk+r tk ) If we l e t g*=e then we have by (4.12.2) E t R 1 2 ( t J + l » V W ' W ' R 2 ( t k > l = [ R 1 ( t k ) + R 2 ( t k ) ] h ( T i 2 g * ) J _ k , for k - l , . . . J . ...(4.38) From the same considerations used i n deducing (4.35) we have - I V V + R 2 ( t k ) ] k k-j n.(k) k-j+l-n (k) k-j+l-n.(k) n.(k) = ^ Ejg {*0 3 \ 3 +*0 J rcx J }, -..(4.39) where E^ and n j ( k ) are the same as i n the proof of Theorem 1. Thus using (4.37), (4.38) and (4.39) we have E I R12 ( t J+l» t l » t J+l ) l J k k-j n.(k) k-j+l-n.(k) = I h (Tt g * ) J ~ * I E g {n 2 it 2 k-1 j-1 2 L k-j+l-n (k) n (k) + T t Q TI, } • ...(4.40) Now E [ R 1 2 ( t J + 1 ) ] - E [ R 1 2 ( t J + 1 ) t l , t J + 1 ) ] + E [ R 1 2 ( t J + l j t 0 , t l ) ] , where E [ R , 2 ( t J + , , t Q , t , ) ] does not depend on the strategy S(v). Also from (4.39) -128-we have E [ R l ( t j + 1 ) + R 2 ( t J + 1 ) ] J+l n (J) J-j+l-n (J) J-j+l-n (J) n (J) = I E g ITCQ \ 2 + 2 * 2 }, ...(4.41) j - l where n^(k)=0 i f j>k. Using (4.40) and (4.41) we obtain E [ N ( t J + 1 ) ] = E [ R 0 ( t J + 1 ) ] + E [ R l ( t J + 1 ) ] + E [ R 2 ( t J + 1 ) ] + E [ R 1 2 ( t J + 1 ) ] J+l J-j+1 n (J) J-j+l-n.(J) J-j+l-n.(J) n.(J) = K+ I E g {it 2 * 2 + n 2 * J } j - l J J-k k k-j n (k) k-j+l-n.(k) k-j+l-n.(k) n.(k) + I h (*g*) I E g {u 2 u 2 + u J 71 J } k=l j=l J ...(4.42) where K does not depend on the strategy S(v). The terms within the summations i n (4.42) have both been seen to be minimized by strategies belonging to Bj; i t follows that the summations are uniquely minimized by S(l) and S(2). Thus S( l ) and S(2) minimize E[N(t)] for t > t j . This completes the proof of the theorem. We have found that there i s one "pattern" of strategies which i s optimal ( i n terms of minimizing E[N(t)]) for any treatment parameters providing the two drugs are equivalent. This property i s extremely convenient since i n any s i t u a t i o n where treatment must be stopped early ( i . e . patient t o x i c i t y or r e f u s a l ) , the truncated regimen i s then optimal for the number of treatments given. S i m i l a r l y i f i t i s decided to increase the treatment regimen we may s t i l l construct the optimal plan of the required length by adding cycles of the drugs to the pre-existing regimen. As previously indicated, however, the p r o b a b i l i t y of cure P^, i s not -129-ne c e s s a r i l y maximized by those strategies which minimize E[N(t)] for t>tj (or minimize M') when treatments are equivalent. An example of th i s i s given i n Chapter 5 and the accompanying discussion suggests that t h i s phenomena w i l l only occur i n the p a r t i c u l a r set of circumstances when regrowth between treatments i s large and the composite process of treatment and regrowth (for si n g l y r e s i s t a n t c e l l s ) i s not strongly s u b c r i t i c a l . This s i t u a t i o n i s u n l i k e l y to be encountered i n human disease since growth over periods of one month (which i s greater than most intertreatment Intervals) i s modest for the majority of human tumors. However, such conditions may be encountered i n several experimental cancers where doubling times i n the order of twelve hours are not uncommon. The two theorems, with the preceding discussion, indicate that i n cases of human cancer where two equivalent agents are a v a i l a b l e , which may not be used concurrently, the best way to use these two w i l l be i n an a l t e r n a t i n g strategy. This r e s u l t i s of i n t e r e s t both because of i t s generality ( i t does not depend on the p a r t i c u l a r parameter values) and because i t i s not current c l i n i c a l p r a c t i c e . In c l i n i c a l medicine protocols are developed whereby active agents are combined, as much as possible, into regimens which are then repeated a fixed number of cycles. Where two such regimens are a v a i l a b l e the common pract i c e i s to use one continuously u n t i l there i s evidence of relapse when the other regimen i s employed. Conversely, although al t e r n a t i n g strategies represent a departure from c l i n i c a l p r a c t i c e , they are compatible with the c l i n i c a l concept of combination chemotherapy. Combination chemotherapy uses drugs given at constant times during a -130-cycle and this cycle i s repeated a fixed number of times. In each regimen the drugs are frequently not given simultaneously but on d i f f e r e n t days. An a l t e r n a t i n g regimen can be viewed as combination chemotherapy with repeated cycles of the regimen T^T^ (or T^Tj) over a longer intertreatment i n t e r v a l . 4.5 Discussion The i d e n t i f i c a t i o n of optimal strategies ( i . e . those which maximize P^) represents a considerable problem i n computation when the parameters 12 are known. For example, when J=12 there are 2 possible s t r a t e g i e s . Thus i t i s desirable to seek h e u r i s t i c s to reduce the set of strategies which must be considered. For a strategy to be e f f e c t i v e the treatments must be able to make the net growth of Rg(t), R^(t) and s u b c r i t i c a l (over the treatment period); otherwise no cure i s possible. In p a r t i c u l a r the c e l l s present at time t^ i n R^, R^  and R^ must be eliminated with a "large" p r o b a b i l i t y . Following this reasoning we i n f e r that the expected number of these c e l l s should be small at completion of the treatment regimen. That i s , "reasonable" strategies would be expected to s a t i s f y , n (J) J-n (J) E [ R i ( t 1 ) ] [Ttlt± 7 C 2 j l ] < k, ...(4.43) for 1=0,1,2 where k i s chosen as a function of d ( i . e . i t w i l l be larger i f the death rate i s larger; a possible choice i s k=0.5(l-e) ^"). In c e r t a i n cases the set of i n e q u a l i t i e s (4.43) may provide useful lower and upper bounds on n^(J) ( i . e . not 0 and J ) , thus eliminating some strategies from consideration. These i n e q u a l i t i e s may also indicate that J i s too small so that the search for an optimal rule of length J may not be of great use. -131-The search for optimal st r a t e g i e s , using P N as the c r i t e r i o n , has been examined i n considerable d e t a i l by Day [34], who considered 16 strategies (chosen to "span" the set of possible s t r a t e g i e s ) for the case J=12, and calculated t h e i r e f f e c t on the p r o b a b i l i t y of cure for 256 d i f f e r e n t combinations of drug and tumor parameters. He showed that i t i s possible to i d e n t i f y c e r t a i n patterns i n the best (of the 16) treatment strategies as the degree of asymmetry i n the parameters of the two drugs increases. In a p a r t i c u l a r c l i n i c a l problem strategies "close" to the best of the 16 determined by Day could be examined. The d e t a i l s of such a search remain to be worked out and we w i l l return to t h i s problem i n Chapter 5. It should be remarked that the assumption of a fixed number of treatments may not be a reasonable model for the c l i n i c a l s i t u a t i o n when the two drugs have d i f f e r e n t recovery times before further therapy i s possible. In such cases i t may be more reasonable to f i x the t o t a l treatment i n t e r v a l [t.. ,t'] where J w i l l be chosen so that t <t". If the tumor parameters are known then i t i s straightforward, although computationally demanding, to calculate the optimum strategy. In order to treat the problem of optimizing strategies comprehensively, we need a precise statement of the r e l a t i o n s h i p between dose and t o x i c i t y for each of the drugs. If t h i s were s p e c i f i e d then i t would be possible to construct optimum dosages as well as optimum schedules. However l i t t l e t h e o r e t i c a l work has been undertaken i n t h i s area and at present i t i s not possible to include considerations of t o x i c i t y i n modelling the e f f e c t s of treatment. This concludes the consideration of optimizing treatment s t r a t e g i e s . We w i l l now consider v a r i a t i o n i n mutation rates -132-on the development of double resistance. 4.6 Variation i n the Mutation Rates In the previous chapter dealing with resistance to a single agent we examined the e f f e c t of v a r i a t i o n i n the rate a=ort-v/b (Section 3.10). Here ( i n analogy to the case of single resistance) we w i l l consider v a r i a t i o n s i n mutation rates where the rates for an i n d i v i d u a l tumor are fixed but follow a d i s t r i b u t i o n f or tumors of that type. In p a r t i c u l a r we w i l l consider v a r i a t i o n s i n the vector of parameters A*, where A* = ( A ^ , , A ^ , A^,A^) = ( a 1b+v 1, a 2b+v 2, a x ^ 1 2b+v 2 ^  u , o ^ b + v ^ ). We w i l l assume a ^ 2 b + v i 2 = ^ s i n c e w e a r e primarily interested i n examining the e f f e c t of v a r i a t i o n i n rates on the two step development of double resistance; the one step process having been e s s e n t i a l l y covered i n Section 3.10. Thus we w i l l consider A, the f i r s t four elements of A*, at t h i s point although we w i l l consider A* l a t e r i n a d i f f e r e n t context. Also, because the d i s t r i b u t i o n function of {R Q(t), R^(t), R 2 ( t ) , R 1 2 ( t ) } cannot be obtained i n e x p l i c i t form, we w i l l (as i n Section 3.10) consider the e f f e c t of v a r i a t i o n s i n A on the p r o b a b i l i t y of cure. The scale of measurement of t i s , of course, a r b i t r a r y . In order to s i m p l i f y presentation we w i l l assume, without any loss of generality, that t i s measured on a scale for which b=l. The p r o b a b i l i t y of cure depends on the treatment strategy for a r b i t r a r y u . In analogy with the case for single resistance we w i l l I»x only consider the s p e c i a l case TC^ Q = U 2 Q=%\ 2~%2 %1 12~%2 12=^ ^ \ 1 and TC2 2 are a r b i t r a r y ) and assume that both drugs are given together. Thus a l l c e l l s , except the doubly r e s i s t a n t ones, are eliminated by the -133-f i r s t a p p l i c a t i o n of the combination of the two drugs. In th i s case the p r o b a b i l i t y of cure depends only on the f i r s t time of administration of the combination since subsequent a p p l i c a t i o n has no e f f e c t on the remaining doubly r e s i s t a n t stem c e l l s . Even i n th i s case the p r o b a b i l i t y of cure i s a complicated function (involving i n t e g r a l s ) and thus we w i l l use the approximation given by (4.23). In what follows we w i l l assume that A i s random and w i l l indicate the dependence of ?^ on A by wr i t i n g writing P„(a). We wish to select a d i s t r i b u t i o n for A which leads to an expression for E [ P ^ ( A ) ] which i s reasonably simple to c a l c u l a t e . We assume that there e x i s t s a density function for the random variable A, f( a ) say. Unfortunately, l i t t l e information i s ava i l a b l e as to the form of f(a) since no experiments have been undertaken to attempt to i d e n t i f y i t . Given our ignorance on the form of f(a) i t seems reasonable to require that f(a) have structure which accords with our physical understanding about the nature of the processes involved. We have, generally, f(a) = g ( a 3 , a 4 \ a l t a 2 ) h ' C a , ^ ) , where g (a^ ,a^ |a, ,a 2) i s the density of (A^A^) condit i o n a l on (A,=a, yk 2=& 2) and h(a, a2) i s the marginal density of (A,»A 2). We postulate here that (A^.A^) are c o n d i t i o n a l l y (on (A,,A^)) independent: g ( a 3 , a 4 | a 1 , a 2 ) = g ^ a ^ a , ^ ) g 2 ( a 4 l a , ,a 2 ), where g,(a 3|a,,a 2) and g 2 ( a 4 | a , , a 2 ) are the marginal densities of A 3 and A 4 r e s p e c t i v e l y , c o n d i t i o n a l upon (A,=a,,A 2=a 2). Also we postulate that g 1 ( a 3 | a 1 , a 2 ) = g 1 ( a 3 | a 2 ) , g 2 ( a 4 | a , , a 2 ) = g 2 ( a 4 | a , ) , that i s , the development of resistance to Tj i n c e l l s r e s i s t a n t to T^ -134-i s dependent only on the r e a l i z e d parameter for the a c q u i s i t i o n of resistance to i n se n s i t i v e c e l l s . Combining the above postulates we have, f ( a ) = g 1 ( a 3 | a 2 ) g ^ a j a ^ h ( a 1 , a 2 ) . ...(4.44) This implies that i f Ai and A 2 are marginally independent then (A^jA^) and (A^,A^) are independent. The structure for f(a) expressed i n (4.44) seems a reasonable s i m p l i f i c a t i o n to impose since i t implies that the pairs (A^,A^) and (A^,A^) are independent i f , and only i f , A^ and A 2 are independent. Also the d i s t r i b u t i o n of rates to double resistance depends only on the analogous rates to single resistance. In common with the single resistance case (section 3.10) we w i l l use a beta d i s t r i b u t i o n to model v a r i a t i o n i n the mutation rates as detai l e d below. Reference to (4.44) shows that there are three seperate densities whose form must be s p e c i f i e d . We would l i k e to model h(a^,a 2) by a bi v a r i a t e beta d i s t r i b t u i o n . The "natural" b i v a r i a t e beta d i s t r i b u t i o n (which i s obtained by conditioning on sums of gamma random variables) has a negative c o r r e l a t i o n for a l l parameter values. Since i n s t a b i l i t y i n the stem c e l l genome i s l i k e l y to lead to higher mutation rates of a l l kinds, mutation rates to drug resistance are more l i k e l y to be p o s i t i v e l y , than negatively correlated. Rather than attempt to construct a p o s i t i v e l y correlated b i v a r i a t e d i s t r i b u t i o n with beta marginals, we w i l l consider two p a r t i c u l a r forms for h(a^,a 2) as follows: ( i ) independence: h(a^,a 2) = h ^ ( a ^ ) h 2 ( a 2 ) where h ^ ( a ^ ) , h 2 ( a 2 ) are both univariate b e t a - d i s t r i b u t i o n s . ...(4.45) ( i i ) dependence: ^2=A^ with p r o b a b i l i t y 1 where A^ has a beta d i s t r i b u t i o n . ...(4.46) -135-To motivate the choice of the densities g^a^\a^) and g 2 ( a 4 | a , ) , i t i s h e l p f u l to consider some underlying structure for t h e i r expected values. A convenient form i s the l i n e a r model, that i s E[Ai+2|A..=a..] = n ^ + k ^ a j - i i j ) 1-1,2, j-3 - i , ...(4.47) where \is= E [Aj] and thus E 1=^n-2* S * n c e w e m u s t have 0 < E [ A J , _ I A.-a . ]<1, we require that i+2 1 J J m a x C-iziT » ~TT-> < k j < m i n ("Rir » —>• 3 3 3 3 We w i l l consider two d i f f e r e n t forms for the d i s t r i b u t i o n s g,(.) and g^(') which exhibit t h i s l i n e a r structure as follows. ( i ) F i r s t Form Ai+2 = l l i + 2 + k j ( A j " t i j ) w i t h p r o b a b i l i t y 1-This may be viewed as the l i m i t of a beta d i s t r i b u t i o n (for the con d i t i o n a l d i s t r i b u t i o n given Aj=a^) with parameters (u,v) where u-*», V -KO i n such a way that - Z T - = ^ A O * . ( - i = l » 2 , j=3-i. u+v ^i+2 j j j L e t t i n g PN = E I P N ( A - ) ] = / f(fL>*fL > then from (4.24) we have I i . 1 2 P = J 1 f exp{-(l-e) N I a (u +k (a -u )) ™ 0 0 1=1 J J J in [ ( l - e ) / e ( ^ 1 + 2 + k j ( a j - j i j ) ) ] } h f a ^ a ^ d a ^ , ...(4.48) where j=3-i. As previously mentioned, two forms for h(a,,a 2) w i l l be used: (4.45) and (4.46). In c a l c u l a t i n g (4.48) we w i l l be concerned mainly with cases where the standard deviations of A, and A 2 are small, since i t i s clear (by analogy with the case of single resistance, Section 3.10) that when the standard deviation i s large, P J J w i l l vary slowly -136-with N. By examining the cases where S.D.(A^) i s small, we w i l l be able to examine the e f f e c t s upon c u r a b i l i t y of v a r i a b i l i t y i n the mutation rates which l i e close to the l e v e l of d e t e c t a b i l i t y even i n experimental systems. Figures 5 and 6 plot equation (4.48) as a function of N for d=0, p 3=u 4=u 1=n 2=10~ 3, S.D. ( A ^ S . D . (A 2)=10~ 3 and where k 1 = k 2 = 0 and k^=k 2=l re s p e c t i v e l y . As may be seen the most marked e f f e c t of v a r i a b i l i t y i n the rates i s to produce a pronounced t a i l i n P J J (for increasing N) which i s not evident when the rates are f i x e d , ( i i ) Second Form Here we w i l l assume that A ^ + 2 (i=l,2) have beta d i s t r i b u t i o n s (for the cond i t i o n a l d i s t r i b u t i o n s given A^= a^) where E[Ai+2\k.=a.] = u . ^ + k ^ a j - U j ) for 1-1,2, j-3-1. This does not uniquely specify the beta d i s t r i b u t i o n (which has two parameters) and thus we w i l l also require that the c o e f f i c i e n t s of v a r i a t i o n are the same, that i s , C.V.[A |A.] = C V . [A.], for i=l,2, j=3-i. ...(4.49) We assume that the conditi o n a l c o e f f i c i e n t of v a r i a t i o n i s constant since v a r i a t i o n i n mutation rates are l i k e l y to be proportional to th e i r absolute magnitude. The integr a l s to be calculated to evaluate P ^ for the second form are more complex than the f i r s t form and involve the numerical c a l c u l a t i o n of one more nested i n t e g r a l . Examples are presented i n Figures 7 and 8. Examination of these figures shows a sim i l a r t a i l for P N to that seen previously where A ^ + 2 ( i = l , 2 ) was a degenerate function of A^ (j=3-i). For the most pronounced case, where k^= k 2 = l , a considerable change i s produced from the case where the rates are fixed (Figure 8). -137-In summary we can conclude that even modest v a r i a t i o n s (S.D. (A)=|J.(A)) i n the mutation rates can lead to su b s t a n t i a l changes i n the function P„ for the s p e c i a l case TC. „=0, TC. .=0 ( i * i ) and TE. n«=l N i,0 i , j i,12 ( i = l , 2 ) . C l e a r l y these e f f e c t s w i l l apply to other s i t u a t i o n s where the re's are a r b i t r a r y , however the e f f e c t s are then more d i f f i c u l t to ca l c u l a t e because they depend on the f u l l treatment protocol. An example (with further discussion) of a case where TC, ^ ^0 and TC. ( i , j = l , 2 ) i s v i,0 i , j ' J given i n Chapter 5. However i f we assume that we may use the example presented as a model for the (more complex) si t u a t i o n s encountered i n r e a l tumor systems, we may make some tentative observations. If a p a r t i c u l a r tumor type has a small, but s i g n i f i c a n t , cure rate when treated at an advanced stage (large bulk of tumor), then the predicted c u r a b i l i t y at lesser tumor burdens (of the same type) w i l l be a function of the amount of v a r i a b l i t y i n the mutation rates. For example a f i v e -f o l d reduction i n siz e would imply a large increase i n c u r a b l i t y and the size of t h i s increment w i l l decrease as the degree of v a r i a b l i t y i n the rates increases. This observation has implications for the therapy of human disease where the c u r a b l i t y of a regimen i s observed and l i t t l e i s known of the mutation rates. The e f f e c t of v a r i a t i o n i n A on the mean number of doubly r e s i s t a n t c e l l s i s more e a s i l y evaluated. Using (4.12.2) and approximating a l l exponentials (except e^fc) by the f i r s t three terms i n t h e i r expansion y i e l d s 2 2 » 1 2(t)-e 6 t{m 1 2 ( 0 ) + I - i W t l a - ! • « £ ] +' B Q t 2 / 2 I a.a }. i = l i = l Taking the expected value of this expression (with respect to A) and -138-Plgure 5 Probability of Cure when Variation i s Present - 1. Number of Stem Cells-N P r o b a b i l i t y of cure P^ plotted as a function of stem c e l l burden at diagnosis where and T 2 are given simultaneously at time of diagnosis, \ 0 = 0 , ^ i i a r b i t r a r v » \ 1 2 = 1 ' ° f o r i = 1 » 2 , \ 2 = 7 I2 1 = 0 > b = 1 a n d d = 0 * T n e function i s plotted for three separate cases where k,=k2=0 i n (4.48): ( i ) A,=A2=A.j=A4=10-3, mutation rates f i x e d . ( i i ) A 3=A 4=10~ 3, A, and A 2 independent with B - d i s t r i b u t i o n with E[A ±]= S.D. [ A ^ - I O - 3 for i=l , 2 . ( i i i ) A 3=A 4=10~ 3, A,=A2 with p r o b a b i l i t y 1, where A, has a B - d i s t r i b u t i o n with E ^ J - S . D . [ A ^ - 1 0 - 3 . -139-Figure 6 Probability of Cure when Variation i s Present - 2. 10 o 101 102 103 104 105 106 107 Number of Stem Cells-N P r o b a b i l i t y of cure P^ plotted as a function of stem c e l l burden at diagnosis where and T 2 are given simultaneously at time of diagnosis, • ^ i 0=0, Tt^ i a r b i t r a r y , 1 2=1.0 for 1=1,2, T C ^ 2=%2 1 = 0 , b = 1 a n d d = 0 * 7 1 1 6 function i s plotted for three separate cases where k^=k 2=l i n (4.48): ( i ) A,=A =A =A,=10"3, mutation rates f i x e d . 1 2 3 4 ( i i ) A i + 2 = A 3 - I W i t h P r o b a D i l i t y 1 f o r 1 = 1 >2> A ! a n d A 2 independent and follow a 8 - d i s t r i b u t i o n with E[A ]«S.D.[A ]-10 - 3 for i-1,2. ( i i i ) A i + 2 = A 3 - i a n d ^ ^ 2 w i t h P r o b a D i l i t y 1 f o r i = 1 » 2 » where A^ ^ has a B d i s t r i b u t i o n with E[A 1]=S.D. [A 1]=10~ 3 -140-assuming the structure expressed by (4.44), (4.47) and (4.49) we have t 2 t 2 2 V a r ( V E t a i , 1 2 - 2 a i , 1 2 ^ = ^i+2" 2 ^ i + 2 + k j v a r ( A j ) ] [ 2 + 1 ] and E [JX a i a i , 1 2 ] - J 1 I » i l » 1 i+2 + k J < c o v ( A 1 ' A 2 ) ] -From this we have E [ R 1 2 ( t ) ] = e 6 t{m 1 2(0) 2 t 7 7 var(A.) + t I m.(0) [u - | + k var (A 2 2 + B t 12 I cov (A ,A )]} where j=3-i. i = l J For c l i n i c a l disease, where we assume m,(0) = m 2(0) = m, 2(0) = 0> the net e f f e c t of v a r i a t i o n i n mutation rates on E [ R , 2 ( t ) ] w i l l depend on k,, k 2 and cov(A, A 2 ) . Thus even when A, and A 2 have a small c o r r e l a t i o n , i f th e i r variance i s large the mean number of doubly r e s i s t a n t c e l l s may be quite d i f f e r e n t from when the rates are f i x e d . It i s natural to consider whether i t i s possible to generalize the notion of equivalent agents (where each component of A* i s fi x e d , as i n Section 4.3) to include the s i t u a t i o n where A* (A^O) has a nondegenerate d i s t r i b u t i o n . Assuming A* to have a density function f . . . A ( x ) say, then a natural d e f i n i t i o n of equivalent agents i s A, ,A2 ,A.j , fA A A A A ( ~ } = fA A A A A ( X ) ' ...(4.50) 1 2 3 4 5 2 1 4 3 5 where as before * 1 > ( ) - * 2 > 0 , * l f l - * 2 , 2 ' * l , 2 = 7 t 2 , l * \ , 12 = 7 I2,12 a n d ( t j + , - t j ) are fixed for j = l , . . . , J - l . We may extend Theorem 2 to t h i s s i t u a t i o n . However, Theorem 1 may not be simply extended ( i n general) since the r a t i o n a l e behind i t s construction (minimizing t r a n s i t i o n s from -141-Figure 7 Probablity of Cure when Variation i s Present -3 . 10° 101 102 103 104 105 106 10 Number of Stem Cells-N 7 P r o b a b i l i t y of cure P^ plotted as a function of stem c e l l burden at diagnosis where and T 2 are given simultaneously at time of diagnosis, TCi 0 = 0 , ^ i i a r D i t r a r v > \ 1 2 = 1 ' ° f o r i = 1 , 2 » %i 2 = 7 t2 1 = 0 ' b = 1 a n d d = 0 " ' I h e function i s plotted f o r three separate cases where k^=k2=0 i n (4.48): ( i ) ( i i ) A^=A2=A.j=A^=10-3, mutation rates f i x e d . - - - - A^ and A^ independent and follow a beta d i s t r i b u t i o n where E[A ± + 2]=S.D.[A i + 2]=10" 3 for i-1,2, A^ and A 2 independent with beta d i s t r i b u t i o n and E[A^]=S.D.[A^]=10 - 3, for i-1,2. (A^,A^) are independent of (A^,A 2). ( i i i ) A^ and A^ independent and follow a beta d i s t r i b u t i o n where E [ A 1 + 2 ] = S . D . [ A i + 2 ] = 1 0 - 3 for i-1,2. A ]=A 2 with p r o b a b i l i t y 1 where has a beta d i s t r i b u t i o n with E[A^]=S.D.[A^]=10 - 3. (A^,A^) are independent of (A^,A 2). -142-Figure 8 Probability of Cure when Variation i s Present - 4. ,8 Number of Stem Cells-N P r o b a b i l i t y of cure P^ plotted as a function of stem c e l l burden at diagnosis where and T 2 are given simultaneously at time of diagnosis, ^ i 0 = 0 > U i i a r D i t r a r y » 111 1 2 = 1 * ° f ° r i = 1 ' 2 , u i 2 = T t2 1 = 0 , b = 1 a n d d = 0 " T h e function i s plotted for two cases where k ^ - l ^ - l i n (4.48): ( i ) A =A =A = A .=10 - 3, mutation rates f i x e d . 1 2 3 4 ( i i ) - - - - - A . J and A ^ follow a 8 - d i s t r i b u t i o n where E I A i + 2 I A 3 - i = a 3 - i ] = a 3 - i • S ' ° ' I A i + 2 I A 3 - i = a 3 - i 1 ~a2-i ^ o r 1=1»2- A i = A 2 w * t n p r o b a b i l i t y 1 where A ^ has a 6 - d i s t r i b u t i o n with E [ A ] = S . D . [ A ]=10~ 3. -143-R, and R 2 to R, 2) i m p l i c i t l y assumed that = A^ with p r o b a b i l i t y 1. C o r o l l a r y 1 (to Theorem 2) For strategies of fixed length J, i f A l , A 2 , A 3 ' A 4 ' A 5 ~ ^2,A,,A^,A^,A^ ~ then the a l t e r n a t i n g strategies S(l) and S(2) minimize E[N(t)] for a r b i t r a r y t>t^. Proof: Without loss of generality assume that t i s measured on a scale where b=l. To proceed we w i l l f i r s t condition on A*=a*. F i r s t l y we note that E. .(t.) (1=1,2, j=l,«««,J) depends upon a* and thus i n general 1»J J ~ E. .=E. .(t.)*E .(t.)=E . for j = l , . . . , J . S i m i l a r l y the terms g and h 1 » j 1>3 3 ^ > 3 3 ^»3 (used i n Theorem 2) depend on a* and are not i n general the same for R, s define g i = e x p { ( 6 - a 1 + 2 ) ( t k + 1 - t k ) } and R^' Thus define and h^=e [1-e J for k=l,...,J, i=l,2. Examining (4.42) we see that the constant K depends on a* but i s not dependent on the treatment strategy. The two summations i n (4.42) only depend on a* through E. ., g. and h.. Carrying out the appropriate ~ l , j l I substitutions of E. ., g. and h. for E., g and h i n (4.42) we have i,3 i 1 3 E[N(t J + 1)|A*=a*] = K(a*) J+l J-j+l n.(J) J-j+l-n . (J) J-j+l J-j+l-n . (J ) n .(J) + .ME1»3 8 1 ^0 3 \ 3 + E 2 , j S 2 ^0 3 \ 3 * j=l J J-k k k-j n.(k) k-j+l-n.(k) k-j k-j+l-n(k) n.(k) +J$*28*> J ) E l , j h l g l V *1 3 + E 2 , j h 2 S 2 *0 V } fc—1 j—J-...(4.51) -144-where g* i s as used i n (4.38). Conditional on A*=a* we may use (4.12.1) to c a l c ulate E. .. In order to do t h i s we use k=6-a -a„-a r and notice 1 , 3 1 2 5 that m.(0)=0 (by the d e f i n i t i o n of E. . ) , which gives I 1 > J 6Atr -a„,. At -(a1+a0+a,.)At a. B ( t . ,) e e 2+i - e l a, + a, + a - a„., 2^5 } a Z ] where j = l , . . , J , A t = t 0 - t 1 and i=l,2 for a l l a* such that a+a.+a.-a. *0. Z J. ~ 1 Z j Z+I We w i l l assume that a.j+a2+a,--a2+.j,*0 with p r o b a b i l i t y 1 and thus from the d e f i n i t i o n of equivalence (4.50), E [ E 1 } j g l J " j + 1 ] = E [ E 2 ) . g 2 J _ j + 1 ] ...(4.52.1) and E t E ^ . h l g l k " j ] = E [ E 2 > . h 2 g 2 k _ j ] . ...(4.52.2) From (4.51) and (4.52.1-2) we have E [ N ( t J + 1 ) ] = E [ E [ N ( t J + 1 ) | A ] ] i s of the same form as (4.42) and thus the c o r o l l a r y i s proved. Notice that the proofs of Theorems 1 and 2 and Corollary 1 do not require the assumption of deterministic growth of se n s i t i v e stem c e l l s since (conditional on a*) the E.. have the same values under the f u l l y stochastic model. This concludes our consideration of v a r i a t i o n i n mutation rates. We w i l l now b r i e f l y consider extensions of the proposed model. 4.7 Extensions Generalizing t h i s model to n drugs i s possible i n p r i n c i p l e , however the complexity of the process increases r a p i d l y as a function of n. For n drugs there are 2 n r e s i s t a n t states and 3n2 n ^ parameters. Thus e x p l i c i t s o l u t i o n of the f u l l problem w i l l l i k e l y be of l i t t l e p r a c t i c a l -145-value i n human disease for n>2 because of the large number of paramaters which must be s p e c i f i e d . Even i n the case where n=2, the large number of parameters require that we s i m p l i f y the problem and carry out s e n s i t i v i t y analyses to assess the e f f e c t s of assuming d i f f e r e n t choices of these parameters [34]. Under the strong assumption of equivalent agents (using the natural extension of i t s d e f i n i t i o n to multiple agents) the s p e c i f i c a t i o n of 3n parameters would be required. A f r u i t f u l approach therefore seems to examine multidrug therapies and determine whether i t i s possible to consider them as two drugs. In multidrug regimens (n>2) for c l i n i c a l cancer i t i s frequently possible to i d e n t i f y one of the drugs as being much more e f f e c t i v e than the others. We may thus attempt to model the regimen by considering i t to be composed of two drugs (the most e f f e c t i v e and the others) and try to approximate the e f f e c t of the regimen using the case n=2. We would argue that t h i s approach i s reasonable, e s p e c i a l l y i n l i g h t of the p o s s i b i l i t y that resistance to any one of the drugs may a r i s e i n a series of stages anyway. This approach i s not of great use i n the construction of protocols where i t i s desired to choose drugs and the dosages that are to be used. However, the major obstacle to using these models i n the planning of protocols i s a comprehensive description of the nature of t o x i c i t y associated with drug combinations and how this depends on the i n d i v i d u a l dosages used. This i s an extremely important problem which has not been extensively explored. Moreover, since most drugs overlap i n t o x i c i t y on only two or three normal tissue systems ( i . e . hemopoietic, gastro-i n t e s t i n a l , e t c . ) , i t may be possible to summarize the toxic e f f e c t s of -146-drugs using a vector with as l i t t l e as two or three elements (one for each system). This completes the consideration of multitype drug resistance. In the next chapter we w i l l present some applications of the theory developed i n Chapter 3 and 4 to experimental and c l i n i c a l cancer. -147-5 . APPLICATIONS OF THE THEORY In the previous chapters we have presented theory for the development of resistance to one or two drugs as a re s u l t of spontaneous t r a n s i t i o n s from the se n s i t i v e state. As remarked i n Chapter 1, t h i s i s one of many mechanisms which can lead to c l i n i c a l resistance and thus the model presented here can only be considered to be tentative for the response of c l i n i c a l disease to chemotherapy. Nevertheless, i t i s possible to examine observations on c l i n i c a l and experimental cancer i n the context of th i s model and assess th e i r " f i t " . The model presented i s c l e a r l y not comprehensive, since i t ignores many processes, but i t i s intended to be of general a p p l i c a b i l i t y to a large v a r i e t y of experimental and c l i n i c a l tumor systems. However, even within the context of the process of resistance considered, further generalization may s t i l l be required i n order to accurately model the process i n c l i n i c a l and experimental cancer. For example, we have assumed that the rates a and v do not vary with time. If these rates vary continuously i n time, we may approximate the r e s u l t i n g process by p a r t i t i o n i n g the growth and treatment periods into a number of i n t e r v a l s and assuming that the rates are fixed within each i n t e r v a l . The r e s u l t i n g o v e r a l l p r o b a b i l i t y generating function may then be constructed using the recursive r e l a t i o n s h i p s presented i n Chapters 3 and 4. The i n t e r d i v i s i o n time of c e l l s has been assumed to be exponentially d i s t r i b u t e d with a common parameter. This i s not an accurate r e f l e c t i o n of r e a l i t y where very small d i v i s i o n s times ( i n r e l a t i o n to the mean) may not occur. Although we may vary the values of b, c and d throughout the growth period ( i n the same way as for a and v), we cannot relax the d i s t r i b u t i o n a l assumption -148-included i n this model. Furthermore we have assumed that the growth parameters for se n s i t i v e and r e s i s t a n t c e l l s are the same. Keeping these l i m i t a t i o n s i n mind, we now propose to examine the ap p l i c a t i o n of the model presented i n three d i f f e r e n t cases. F i r s t l y , using the theory presented i n Chapter 4, we w i l l present c a l c u l a t i o n s of the e f f e c t of various treatment strategies on c u r a b i l i t y . Secondly, we w i l l examine experimental data c o l l e c t e d on the treatment of a mouse leukemia with two chemotherapeutic agents. T h i r d l y , we w i l l examine the concept of neo-adjuvant chemotherapy i n the l i g h t of th i s model. 5.1 The E f f e c t of Treatment Strategies on C u r a b i l i t y A computer program was written which incorporates the r e l a t i o n s h i p s presented i n (4.11), (4.17) and (4.18). Numerical i n t e g r a t i o n i s performed using Simpson's r u l e . The in t e g r a l s are generally well-behaved and may be evaluated to 8 figure accuracy by p a r t i t i o n i n g the i n t e r v a l of integ r a t i o n into no more than 100 subintervals. Input consists of parameters which define the behaviour of the tumor and of the drugs and are described i n more d e t a i l below. The basic treatment parameters are Tt , the p r o b a b i l i t y of a c e l l I > X i n compartment R Q surviving administration of treatment i , and T ( i ) , the recovery time a f t e r treatment i , i . e . the minimum time before any further treatment may be "safely " administered. Five treatments are considered as follows: i=l,2 correspond to s p e c i f i c chemotherapeutic agents T^ and T , i=3 corresponds to the two agents (T„) being given together TC„ = u i n 7 1? n» ^"=4 represents a non-chemotherapeutic treatment (T.) which 1 > X ^ » X ^ a f f e c t s a l l stem c e l l s equally, that i s TE, =k (0<k<l) for a l l Q , and i=5 '4, x represents a n u l l treatment (T ) where TE,. n=l«0 for a l l Q . It i s assumed -149-that no treatment may be administered within the (minimum) recovery time for the preceding treatment. T,. i s included so that other treatments may be applied at a r b i t r a r y times after the minimum recovery time. In the examples which follow treatments w i l l be applied at the minimum recovery times. The following parameters are also input: N = number of stem c e l l s at diagnosis, DT = the doubling time of the tumor, e = d/b the r e l a t i v e rate of c e l l death, c* = c/b the r e l a t i v e rate of c e l l renewal, an » n = t r a n s i t i ° n parameters for resistance to the drug T. q i ' q j V q j 1 (1-1,2) where O^ .Q e {0, 1, 2, 12}, k n n = Y A / b = r e l a t i v e rates for spontaneous development of Q i ' q j V ^ j resistance Q ± e {0, 1, 2, 12}, J = number of times treatments are administered. There i s no i m p l i c i t time scale used but each parameter r e f l e c t i n g times (DT and T ( i ) ) must be entered using the same scale i . e . days, hours etc. In a l l cases the tumor i s assumed to have grown from a single stem c e l l . The output from the program includes E [ R Q ( t ) ] , E f R ^ t ) ] , E [ R 2 ( t ) ] , E [ R 1 2 ( t ) ] evaluated at t ^ , and t ^ , for j = l , . . . , J . The following p r o b a b i l i t i e s are also calculated: P^t.. )=<|>( e, 1,1,1; t_. ), P ^ t )-<|>(l,e,l,l;t ), P 2 ( t )-4>(l,l,e,l;t ) , P ^ C t j J - ^ l . l . l . e j t j ) and P(tj)=<|>( e, e, e, e;t..), (the p r o b a b i l i t y of cure a f t e r the j - t h treatment), fo r j = l , . . . , J . The f i r s t four of these quantities correspond to the -150-marginal p r o b a b i l i t i e s that c e l l s i n R Q , R ^ , R ^ and R ^ respectively at time t ^ w i l l go spontaneously extinct at some l a t e r time. ^Oj) ^ s t n e p r o b a b i l i t y of cure. Notice that P Q ^ , . ) i s the p r o b a b i l i t y that the s e n s i t i v e c e l l s at time t j w i l l go spontaneously extinct ( a l l c e l l s derived from these c e l l s go extinct) and not the p r o b a b i l i t y that there w i l l be no s e n s i t i v e stem c e l l s at time t=°°. This observation also applies (for the appropriate states) to P . ( t . ) , P»(t .) and P 1„(t.). •*• 3 ^3 i^ 3 We w i l l present an example with parameters chosen to be i n the range of those seen i n passaged experimental tumors. The parameter values are indicated i n Table V. The parameters e and c* were chosen to be zero, implying that a l l c e l l s are stem c e l l s which seems to be approximately true for a number of experimental tumors. The doubling times (DT) and intertreatment times ( T ( i ) ) were chosen to be 5 and 3 days re s p e c t i v e l y . This doubling time represents the upper l i m i t for most experimental tumors and the lower l i m i t of those measured for human disease. However, as noted previously, the unit of measurement i s i r r e l e v a n t to these computations and i t i s only the r a t i o (5/3) of the quantities which i s important. As noted i n Chapter 4, when a +v /b i s f i x e d , the 1, j 1. 3 various values of a and v have l i t t l e r e a l e f f e c t on the ^ i , q j ^i,3 p r o b a b i l i t y of cure. Thus for s i m p l i c i t y we have chosen v =0 and Y* „ = 6 „ =0. For s i m p l i c i t y we have assumed that a.. =v 1 9=0, that i s d i r e c t t r a n s i t i o n s from s e n s i t i v i t y to double resistance do not occur. The therapeutic parameters have been chosen so that r e s i s t a n t c e l l s are absolutely r e s i s t a n t to the p a r t i c u l a r drug. The number of times therapy i s administered, J, has been set to 8. Parameters were chosen so that -151-the drugs s a t i s f y the d e f i n i t i o n of equivalence given i n Section 4.4. In a l l simulations which follow the intertreatment i n t e r v a l has been assumed to be the minimum permitted by the recovery time of the previous treatment ( i n th i s case 3 days). Tables VI, VII and VIII show the ef f e c t of three treatment strategies on c u r a b i l i t y : S(l)=(l,1,1,1,1,1,1,1), S(2)=(l,l,l,l,2,2,2,2) and S ( 3 ) = ( l , 2 , l , 2 , l , 2 , l , 2 ) . That i s , S ( l ) , represents eight cycles of T given at 3 day i n t e r v a l s with the f i r s t cycle being given when the tumor consists of 10 7 stem c e l l s e t c Since the treatments are equivalent, each strategy has i t s mirror image which has the same p r o b a b i l i t y of cure. Figures 9, 10 and 11 plot the expected number of c e l l s for the treatment strategies S ( l ) , S(2) and S(3). Tables VI, VII, and VIII show, for this example, that among the three strategies of length J=8 which give a single drug per treatment, the p r o b a b i l i t y of cure i s maximized by the al t e r n a t i n g strategy S(3). As can be seen by r e f e r r i n g to Tables VI-VIII, a l l three strategies control (eliminate with high p r o b a b i l i t y ) the s e n s i t i v e c e l l s but the strategies have d i f f e r e n t i a l e f f e c t i n c o n t r o l l i n g the various r e s i s t a n t compartments. S(2) and S(3) successfully control both the sin g l y r e s i s t a n t compartments but have a d i f f e r e n t i a l e f f e c t on c e l l s i n R-j.2* Furthermore since neither T^ or T^ have any e f f e c t on c e l l s i n R.^ further treatment (a f t e r t 0 with either T or T ) cannot increase the o I L p r o b a b i l i t y of cure to a value which exceeds P „(t ). The question 12 O arises as to whether S(3) i s best, i n the sense that i t maximizes P(tg) over a l l strategies with J=8 which use either T± or T 2 at the minimum permissable treatment times? Since the treatments are equivalent there -152-are at most 27=128 strategies with d i s t i n c t p r o b a b i l i t i e s of cure (since each strategy has a mirror image). We w i l l now consider general arguments to reduce the set of strategies which must be considered, i n order to determine the optimal one. or We have assumed, i n this example, that TC^ \2=n2 \2~^~ a n d W e n a v e ^ any strategy that P(t)<P^ 2(t') where t>t'. Examining Table VIII we see that the a l t e r n a t i n g strategies of length J=8 have P,„(t o)=0.569. The 12 o strategies considered i n Table VI and Table VII are not optimal since a f t e r 3 consecutive applications of the same drug, P 1 2(t 3)=0.487<0.568. When TS^ 1 2 = T ! :2 12 t h e v a l u e o f P i 2 ^ t j ^ d o e s n o t depend on which drug i s given at time t ^ (either T^ or T^) but only depends on preceding applications of therapy. Thus a strategy whose f i r s t three cycles are (1,1,2) have the same value for P -^C^) a s that of a strategy commencing with (1,1,1). Thus P 1 2(t 3)=0.487 i f (or T 2) i s given as the f i r s t two cycles of the strategy. From t h i s we conclude that the optimum strategy must begin with the a l t e r n a t i o n of T^ and T 2. Examination of Tables VI and VII also show that strategies which include four cycles of T^ and four cycles of T 2 are s u f f i c i e n t to eliminate the si n g l y r e s i s t a n t stem c e l l s with p r o b a b i l i t y >0.999. If there are only three cycles of T 2 (T x) then the l i k e l i h o o d that the ^ (R 2) c e l l s w i l l be eliminated i s s i g n i f i c a n t l y reduced; for example compare P^(t^) and P 1 ( t g ) i n Tables VII or VIII. Thus we need only consider ( 3)=20 strategies to determine the best of length J=8, i . e . those that begin with (1,2) and have four cycles of T^. Simulations of these 20 strategies indicates that there i s l i t t l e to choose between st r a t e g i e s which commence with either (1,2,1,2) or (1,2,2,1) and have four cycles of -153-T, i n the t o t a l treatment strategy. Although these considerations only apply to the model with the p a r t i c u l a r values of the parameters s p e c i f i e d , arguments s i m i l a r to these may usually be applied to reduce the number of strategies which must be considered to determine the optimal one. Figures 9-11 present plots of the mean number of c e l l s (for each of the r e s i s t a n t subcompartments and o v e r a l l ) for the tumor model with parameters given i n Table V for the three strategies S ( l ) , S(2) and S(3) r e s p e c t i v e l y . Judged by E [ N ( t Q ) ] , S ( l ) i s c l e a r l y i n f e r i o r to S(2) and o S(3), however there i s l i t t l e difference between the l a t t e r two (E[N(t Q)]=131.6 and 122.5 r e s p e c t i v e l y ) . The r e l a t i v e l y small diff e r e n c e o i n E [ N ( t Q ) l for the two strategies can be contrasted with the large o d i f f e r e n c e i n predicted c u r a b i l i t y between S(2) and S(3) (Tables VII and VIII). This indicates that the e f f e c t s of strategies on c u r a b i l i t y may not be r e f l e c t e d by s i m i l a r proportionate changes i n E [ N ( t Q ) ] and o this has c l i n i c a l implications as follows. In the analysis of c l i n i c a l and experimental chemotherapy two measures of e f f i c a c y are i n common use: cure rate ( p r o b a b i l i t y of cure) and s u r v i v a l time or time to relapse. Time to relapse (or s u r v i v a l time) depends on the growth rate of the neoplasm and the post-treatment tumor burden. If the e f f e c t s of treatment are s i m i l a r on the two p r o l i f e r a t i v e compartments of the tumor ( i . e . stem c e l l s and t r a n s i t i o n a l c e l l s ) and produce a large net reduction i n the number of tumor c e l l s , then the tumor w i l l regrow at the rate determined i n Chapter 2 and the time taken for i t to reach some predetermined size w i l l depend on the post-treatment stem c e l l burden. Thus i n an experiment where recurrence times are -154-measured In g e n e t i c a l l y i d e n t i c a l animals, the times w i l l be a function of the post-treatment stem c e l l burden. It i s common to view these two measures of treatment e f f i c a c y (cure rate and relapse time) as measuring the same underlying e f f i c a c y of the treatment protocol. Indeed we have argued i n Chapter 4 that t h i s i s l i k e l y to be so, that i s , P(S(v)) i s maximized and E[N . At)] i s minimized by the same strategy. However, £>( v) even when these two c r i t e r i a do induce a s i m i l a r ordering on the set of strategies, this does not imply that differences between strategies w i l l be q u a n t i t a t i v e l y s i m i l a r using either measure of e f f i c a c y . In the previous example we saw that the P ( t Q ) for S(3) was 0.569 and for S(2) o was 0.275 whereas the corresponding values of E [ N ( t 0 ) ] were 122.5 and o 131.6 r e s p e c t i v e l y . In an experiment c a r r i e d out on a tumor where t h i s model was appropriate and the parameter values were as given i n Table V the large difference i n cure p r o b a b i l i t i e s would be r e a d i l y apparent. However, consider the same example except that <x^=5xl0-1+ (=ct2=a^ i2=a2 12^* ^ n t b :*" s c a s e t n e p r o b a b i l i t y of cure i s n e g l i g i b l e for a l l strategies which use T^ and/or T^ only, since for TC^ \2=T*2 12 =* we have for t>t^ P ( t ) < P 1 2 ( t 1 ) < 1 0 - 1 0 . If we apply S(2) and S(3) we f i n d that E [ N ( t g ) ] = 13,064 and 12,158 respectively. The e f f e c t i v e extension of the time to relapse i s hi [13,064/12,158] v _ ... „, n v . , 1 — ^ r 2 j — 1 L * Doubling time = 0.10 x 5 days =0.5 day When th i s i s compared against an estimated 86 days from time of f i r s t treatment to relapse (at 10 8 c e l l s ) we see that improvements of the order of 0.5 day w i l l be very d i f f i c u l t to detect. Thus even i n cases where Theorem 2 applies ( i . e . treatments are equivalent), increases i n -155-disease-free s u r v i v a l may be d i f f i c u l t to d i s t i n g u i s h experimentally. We may continue t h i s example and consider the s u r v i v a l time under various s t r a t e g i e s . I f , for s i m p l i c i t y , we assume that death occurs at 1 0 1 0 c e l l s , then we s t i l l have a mean difference of 0.5 day (between S(2) and S(3)) and a mean s u r v i v a l time of approximately 119 days. We may contrast t h i s with the protocol where T\ i s given u n t i l relapse (at 10 8 c e l l s ) when the treatment i s switched to T 2 which i s continued u n t i l death (at 1 0 1 0 c e l l s ) . In this case an animal has an approximate mean s u r v i v a l time (from f i r s t treatment) of 120 days. Thus the approximate difference i n mean s u r v i v a l time between the l a s t strategy and S(3) (the best strategy) i s 1 day. Given possible uncontrolled var i a t i o n s i n experimental conditions, var i a t i o n s i n the number of re s i s t a n t c e l l s and the i n t r i n s i c p r e c i s i o n of measurement, i t w i l l be extremely d i f f i c u l t to detect differences of this order i n r e a l systems. From consideration of th i s example we see that the value of strategies as r e f l e c t e d by t h e i r a b i l i t y to produce cures ( i n cases where this i s possible) may not be equally r e f l e c t e d i n mean disease-free i n t e r v a l s or s u r v i v a l times when cure i s u n l i k e l y . The strategies considered up to th i s point have a l l assumed that T^ and T 2 may not be given simultaneously. We w i l l now consider cases where they can be given together. In each of the following two cases the parameters values are as given i n Table V except as indicated. Table IX and Figure 12 contain d e t a i l s of the e f f e c t of the strategy S(4)=(3,3,3,3,3,3,3,3) where Tt =Tt u and T(3)=T(1)=T(2). Table X and Figure 13 contain the same information for the strategy S(4) where T t 3 Q= 10 - 2, T t 3 , = 1 ^ 2=10 - 1, T t 3 1 2=1 and T(3)=T(1)=T(2). The parameter -156-values chosen for treatment 3 i n the c a l c u l a t i o n s presented i n Table IX correspond to a case where T^ and T 2 have no overlapping t o x i c i t y and thus may be given i n f u l l dose together. The parameter values chosen for treatment 3 i n Table X correspond to a case where t o x i c i t y overlaps on one or more normal tissues and i n order to give them together the drug dose of each i s halved. As expected, when there i s no overlapping t o x i c i t y , S(4) has the highest p r o b a b i l i t y of cure (of a l l strategies considered); t h i s Indicates that where possible active drugs should be combined (Table IX). Comparison of Tables VIII and X indicates that when the i n d i v i d u a l drug dosages are reduced ( i n order to combine them) the r e s u l t i n g strategy can be better than c y c l i c administration of the two agents s i n g l y . Notice that i n t h i s case we have assumed that the n e t - k i l l per cycle of the combination i s the same to s e n s i t i v e c e l l s as that of e i t h e r of the drugs given alone i n f u l l dose. If t h i s were not true then such regimens might not be superior to one or more strategies involving c y c l i c administration of each drug at f u l l dose. As discussed i n Chapter 4 minimizing E[N(t)] i s not neces s a r i l y equivalent to maximizing the p r o b a b i l i t y of cure, P ( t ) . We w i l l now present an example where P(t) i s not maximised by a l t e r n a t i o n of two equivalent drugs (where of couse E[N(t)] i s minimized). Table XI contains the parameter values and Table XII the r e s u l t s of three strategies S'(l)-(1,2,1,2), S'(2)=(l,2,2,l) and S'(3)=(l,2,2,2) for t h i s example. It can be seen that the a l t e r n a t i n g strategy i s c l e a r l y i n f e r i o r and that S'(3) i s a superior strategy. C a l c u l a t i o n shows, as expected, that the a l t e r n a t i n g strategy S'(l) minimizes the expected -157-tumor size at time t ^ . Calcu l a t i o n also shows that S'(3) i s the best of the sixteen strategies of length J=4, i . e . that which maximizes P(t4). Examination of Table XII shows that extending the length of the strategies may improve the c u r a b i l i t y of the regimens since V^)~>V(t^) f o r each of the three s t r a t e g i e s . However, notice that P j ^ C t ^ ) ^ o r i s much less than P(t^) for either S'(2) or S'(3), and thus a l l a l t e r n a t i n g regimens (of length J>4) w i l l have a lower p r o b a b i l i t y of cure P(t) than at least two other strategies (those that begin with either S'(2) or S'(3)). Examination of Table XII shows that the reason the a l t e r n a t i n g strategy does not maximise P(t4) i s because the R 2 c e l l s are eliminated, with p r o b a b i l i t y 0.912, by the f i r s t course of T]_. Because of the fast regrowth of the c e l l s several courses of T 2 must be given to eliminate c e l l s i n R]_. This combination of circumstances seems u n l i k e l y to occur i n the treatment of human cancer, but could a r i s e i n the therapy of experimental neoplasms. In Chapter 4 we examined the e f f e c t s of v a r i a b i l i t y i n mutation rates on the p r o b a b i l i t y of cure, for the s p e c i a l case where both drugs were given together and eliminated a l l but the R,2 c e l l s . We w i l l now examine the e f f e c t s on more general treatment s t r a t e g i e s . In t h i s example we w i l l use the parameter values as given i n Table V except that the mutation rates follow a d i s t r i b u t i o n . We w i l l assume that A (A,_=0) s a t i s f i e s A,=A2=A3=A4 with p r o b a b i l i t y 1 (see Section 4.6) where A^ follows a beta d i s t r i b u t i o n with E(A,)=S.D.(A,). This corresponds to a p a r t i c u l a r example of the dependent case (4.46) for g i ( . ) and g 2(-) of the second form (4.49). This single s p e c i a l i z e d example i s considered because of the complexity of the ca l c u l a t i o n s involved. Even i n th i s -158-s i t u a t i o n where A i s e s s e n t i a l l y a scalar random va r i a b l e , i t i s necessary to approximate i t s true d i s t r i b u t i o n . In order to provide comparability with the previous example (Table V) we w i l l assume that v i=v i 1 2=0 for i=l,2, v 1 2=0 and E ^ ]=5xl0~ 5 for i=l,...,4. The e f f e c t of v a r i a t i o n i n A i s d i f f i c u l t to compute exactly because of the recursive nature of the relatio nships involved (see (4.11), (4.17) and (4.18)) where E[<|>(s;t,a)] i s not of closed form. We w i l l therefore approximate the beta d i s t r i b u t i o n by a set of 10 d i s c r e t e mass points of weight 0.1 placed at the 5,15,...,95 percentiles of the beta d i s t r i b u t i o n . The points are given i n Table XIII. Tables XIV-XVI give the r e s u l t s of applying the strategies S ( l ) , S(2) and S(3) to the tumor system. A s i m i l a r c a l c u l a t i o n using 20 mass points (at the 2.5,...,97.5 percentiles) yielded r e s u l t s which were the same (to four decimel places) as those presented and thus the d i s c r e t e approximation to the beta d i s t r i b u t i o n can be expected to be reasonable for the puposes of t h i s c a l c u l a t i o n . As i s to be expected, the p r o b a b i l i t y of no doubly r e s i s t a n t c e l l s at the commencement of therapy, ^^2^1^' ^ s d i f f e r e n t from the s i t u a t i o n when the mutation rates were f i x e d . However, the difference i s quite small. We f i n d that, as when mutation rates were f i x e d , the a l t e r n a t i n g regimen S(3) i s superior to either S ( l ) or S(2); i n fact i t maximizes P(tg) among the strategies which only give one treatment per treatment time. However, there are differences i n the e f f e c t s of the strategies on the two d i f f e r e n t tumor systems. Comparison of Tables VI-VIII and Tables XIV-XVI, shows that the p r o b a b i l i t y of e x t i n c t i o n of the s e n s i t i v e stem c e l l s i s v i r t u a l l y the same i n the two series of computations. -159-S i m i l a r l y , differences i n the p r o b a b i l i t y of e x t i n c t i o n of the s i n g l y r e s i s t a n t c e l l s are small and of the type expected (see Figure 3). That i s , when the mutation rates are va r i a b l e , P i ( t ) and P2(t) increase e a r l i e r i n the treatment regimen but require approximately the same number of treatments to approach unity. Comparison of Tables VII and XV shows that changes i n P-^O occur more slowly during the treatment period when v a r i a t i o n i s present. This behaviour i s to be expected as may be seen from the following observation. From (4.24) we may approximate the p r o b a b i l i t y of no doubly r e s i s t a n t c e l l s p r i o r to treatment, P ^ ^ i ^ ' * n t n e r o r m e x P ( -a*N). Thus the e f f e c t on P ^ ^ l ^ of any v a r i a t i o n i n a* w i l l have the same e f f e c t as an analogous v a r i a t i o n i n N when a* i s f i x e d . Thus we may consider P^Ct^) a s t n e weighted sum of points of the function P N given i n Figure 4. As time increases (and the tumor grows) each point w i l l experience a d i f f e r e n t rate of change of P ^ . In the example considered, the fixed mutation rate case experiences a high ( i n absolute value) rate of change of P N and P ^ w i l l decline comparatively quickly. For the variable mutation rate P-^Ct,) roav be considered to decline as a mixture of variable rates of change i n P J J (some large and some small i n absolute value), and thus P 1 2 ^ t l ^ w m decline more slowly than when the mutation rates are f i x e d . This argument also indicates that the p r o b a b i l i t y of cure w i l l not always change more slowly (during the treatment period) when v a r i a b i l i t y i n mutation rates i s present than when i t i s not. When the mutation rates are fixed and the rate of change i n P N i s small then P , ^ (t,) may decline at a faster rate when mutation rates have considerable v a r i a b i l i t y . This behaviour may be of some p r a c t i c a l i n t e r e s t . Consider a class -160-TABLE V Parameter Values for Simulations Presented i n Tables VI-X. Parameter Value N 10 7 DT 5 days £ 0 c* 0 a (=a =a =a ) 5 x IO" 5 1 2 1,12 2,12 v ( = v =v =v ) 0 1 2 1,12 2,12 a 0 12 v 0 12 Tt ( = TC =TC =TC ) 10~ 2 1.0 2,0 1,2 2,1 Tt ( = TC =Tt =TC ) 1 1.1 2,2 1,12 2,12 T(1)(=T(2)) 3 days J 8 -161-TABLE VI P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment f o r Parameter Values given i n Table V f o r Strategy S(l)=(l,1,1,1,1,1,1,1). Time t Treatment ?n(t) p x ( t ) P12^ t^ ? (- z^ 0 0 0 0.641 0 t x T 1 0 0 0 0.641 0 t 2 T1 0 0 0.500 0.573 0 t 3 T x 0 0 0.984 0.487 0 t 4 T x 0.707 0 1.000 0.386 0 t 5 T x 0.995 0 1.000 0.277 0 t 6 T x 1.000 0 1.000 0.172 0 t 7 T x 1.000 0 1.000 0.087 0 t 0 T n 1.000 0 1.000 0.033 0 -162-TABLE VII Probability of Extinction of Cells at Times of Treatment for Parameter Values given In Table V for Strategy S(2)«(l,l,1,1,2,2,2,2). Time t Treatment ^o^) P]_(t) p 2 ( t ) P12^ t^ P^ t) t ^ 0 0 0 0.641 0 tx Tj_ 0 0 0 0.641 0 t 2 Tj_ 0 0 0.500 0.573 0 t 3 T x 0 0 0.984 0.487 0 t 4 Tj_ 0.707 0 1.000 0.386 0 t 5 T 2 0.995 0 1.000 0.277 0 t 6 T 2 1.000 0.059 1.000 0.275 0.022 t-j T 2 1.000 0.934 1.000 0.275 0.263 t g T 2 1.000 0.999 1.000 0.275 0.275 -163-TABLE VIII P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment f o r Parameter Values given i n Table V for Strategy S ( 3 ) = ( l , 2 , l , 2 , l , 2 , l , 2 ) . Time t Treatment P Q ( t ) P L ( t ) P 2 ( t ) 0 0 0 0.641 0 t L TX 0 0 0 0.641 0 t 2 T 2 0 0 0 0.573 0 t 3 TL 0 0 0.369 0.571 0 t 4 T 2 0.707 0.254 0.368 0.569 0.044 t 5 T x 0.995 0.254 0.968 0.569 0.155 t 6 T 2 1.000 0.955 0.968 0.569 0.537 t 7 TL 1.000 0.955 0.999 0.569 0.550 t Q To 1.000 0.999 0.999 0.569 0.568 -164-TABLE IX P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment f o r Parameter Values given i n Table V f o r Strategy S(4)=(3,3,3,3,3,3,3,3) and TC3,0=10~ » w 3 , l = 1 l 3 , 2 = 1 0 ~ 2 » *3,12 - 1' Time t Treatment ^o^) P12^ t^ P^ f c^ t ^ 0 0 0 0.641 0 t , T 3 0 0 0 0.641 0 t 2 T 3 0.859 0.515 0.515 0.640 0.163 t 3 T 3 1.000 0.985 0.985 0.640 0.627 t 4 T 3 1.000 1.000 1.000 0.640 0.639 t 5 T 3 1.000 1.000 1.000 0.640 0.640 t 6 T 3 1.000 1.000 1.000 0.640 0.640 t 7 T 3 1.000 1.000 1.000 0.640 0.640 t Q To 1.000 1.000 1.000 0.640 0.640 -165-TABLE X Probability of Extinction of Cells at Times of Treatment for Parameter Values given in Table V for Strategy S(4)=(3,3,3,3,3,3,3,3) and Time t Treatment ^nit) P ^ t ) P 2 ( t ) p12^ t^ P ^ t ^ t ^ 0 0 0 0.641 0 t x T 3 0 0 0 0.641 0 t 2 T 3 0 0 0 0.627 0 t 3 T 3 0 0 0 0.624 0 t 4 T 3 0.707 0.266 0.266 0.624 0.036 t 5 T 3 0.995 0.781 0.781 0.624 0.406 t 6 T 3 1.000 0.956 0.956 0.624 0.584 t 7 T 3 1.000 0.992 0.992 0.624 0.618 t Q To 1.000 0.999 0.999 0.624 0.623 -166-TABLE XI Parameter Values f o r Simulations Presented i n Table XII. Parameter Value N 10 7 DT 0.3 days e 0 c* 0 a l ( = a 2 = a l , 1 2 = a 2 , 1 2 > 1 0 -5 v l ( = v 2 = = v l , 1 2 = v 2 , 1 2 ) a12 v 1 2 "1,0 <=1t2,0> *1,2 < = 1 t2,l> r t l , l ( = 1 l2,2 =' r vl,12 = = 7 l2,12 ) T(1)(=T(2)) 3 days 0 0 0 1 0 - 5 lO" 1* 1 -167-TABLE XII Pr o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment for Parameter Values given i n Table XI for the Strategies s ' ( i ) = -(1,2,1,2), S'(2) =(1,2,2,1) and S'(3) =(1,2,2,2). Strategy Time t Treatment P l ( t ) V ^ P 1 2 ( t ) P(t) fcl 0 0 0 0.979 0.0 S'(l) H T l 0 0 0.912 0.979 0.0 T2 0.363 2.7x10" 1 1 0.329 0.009 7.5x10" 13 fc3 T l 0.990 2.7x10" 1 1 0.712 0.005 4.3x10" 12 H T2 1.000 2.7x10" 1 1 0.712 1.9x10" 1 1 4.3x10" 12 S'(2) T l 0 0 0.912 0.979 0.0 H T2 0.363 2.7x10- 1 1 0.329 0.009 7.5x10" 13 C 3 T2 0.990 0.012 0.326 0.005 1.0x10" •k H T l 1.000 0.012 0.326 0.002 1.1x10" •4 S'(3) ti T l 0 0 0.912 0.979 0.0 H T2 0.363 2.7x10" 1 1 0.329 0.009 7.5x10" •13 fc3 T2 0.990 0.012 0.326 0.005 1.0x10" •4 H T2 1.000 0.512 0.326 0.002 0.001 -168-Flgure 9 Plot of expected number of stem c e l l s In each of the r e s i s t a n t compartments for the tumor with parameters given i n Table V treated with S(1)={1,1,1,1,1,1,1,1}. -169-Flgure 10 Expected Numbers of Cells for Treatment Strategy S(2). Plot of expected number of stem cel ls in each of the resistant compartments for the tumor with parameters given in Table V treated with S(2)={l,l,l,l,2,2,2,2}. -170-Figure 11 Expected Number of Cells for Treatment Strategy S(3). O jQ 107 10e 101 10' | 10; z "D £ 10' o -OJ * 10 >3-1 _ LLI io°L^  J ; !< N Ro R, R ? R,? 0 3 6 9 12 15 18 21 Ti T 2 T, T 2 Ti T 2 T, T 2 T i m e in D a y s 24 Plot of expected number of stem c e l l s i n each of the r e s i s t a n t compartments for the tumor with parameters given i n Table V treated with S<3)={1,2,1,2,1,2,1,2}. -171-Flgure 12 Expected Number of Cells for Treatment Strategy S(4) - 1. E Z5 10' 10F 1 105 104 10C "D £ 10' o <D CL X LU 101 io°Lv N R o R i , R 2 R , 2 T 3 3 T 3 6 T 3 9 12 15 T 3 T3 T 3 T i m e in D a y s 18 T 3 21 T 3 Plot of expected number of stem cells in each of the resistant compartments for the tumor with parameters given in Table V treated with S(4)={3,3,3,3,3,3,3,3} where * r J » x 1>H >Q -172-Flgure 13 Expected Number of Cel ls for Treatment Strategy S(4) - 2. 107 10( CD o 10 10 103 0) JO E z "O £ 10 o a „ £ 1° 10' I . — • I N Ro R , o r R ? R w 0 T 3 3 T3 6 T 3 9 12 15 T 3 T 3 T 3 Time in Days 18 21 24 T, T 3 Plot x of expected number of stem cells in each of the resistant compartments for the tumor with parameters given in Table V treated with S(4)={3,3,3,3,3,3,3,3} where u ^ - H r 2 , %3 ^ l C T 1 and T C 3 1 2=1 -173-TABLE XIII Mass Points for the Approximation to the Beta Distribution with E[k±]=5xl0-5, S.D. [ A J - S x l O - 5 . The parameters of the beta distribution B(a;u,v) are u=l-10 _ l t, v K l - l O - ^ X Z x l O - 4 -1). Point Percentile Mass 2 . 6 x l 0 - 6 0.05 0.10 8.1x l 0 - 6 0.15 0.10 1.4xl0 - 5 0.25 0.10 2.2xl0- 5 0.35 0.10 3.0xl0- 5 0.45 0.10 4.0x10-5 0 > 5 5 o a o 5.3x10-5 0 > 6 5 0 > 1 0 6.9xl0- 5 0.75 0.10 9.5x10-5 0.85 0.10 1.5X10-4 0.95 0.10 -174-TABLE XIV Probability of Extinction of Cells at Times of Treatment for Parameter Values given in Table V for the Strategy S ( l ) = ( l , l , l , l , l , l , l , l ) where the Mutation Rates are Equal with Probability 1 and have the Distribution Given i n Table XIII. Time t Treatment P Q ( t ) P-^t) P 2 ( t ) P ( t ) 0 0 0 0.676 0 t , T, 0 0 0.032 0.676 0 t 2 Tj_ 0 0 0.591 0.639 0 t 3 T, 0 0 0.985 0.597 0 t 4 T x 0.707 0 1.000 0.551 0 t 5 T, 0.995 0 1.0 0.501 0 t, I, 1.000 0 1.0 0.449 0 6 1 t y T, 1.000 0. 1.0 0.398 0 t Q T, 1.000 0 1.0 0.349 0 -175-TABLE XV Probability of Extinction of Cells at Times of Treatment for Parameter Values given in Table V for the Strategy S(2)=(l,l,l,l,2,2,2,2) where the Mutation Rates are Equal with Probability 1 and have the Distribution Given in Table XIII. Time t Treatment ^o^) ^ ( t ) P 2^ t^ P12^ t^ ? ^ 0 0 0 0.676 0 t , T, 0 0 0.032 0.676 0 t 2 T, 0 0 0.591 0.639 0 t 3 T, 0 0 0.985 0.597 0 t 4 T, 0.707 0 1.000 0.551 0 t 5 T 2 0.995 0.002 1.000 0.501 0.002 t 6 T 2 1.000 0.259 1.000 0.500 0.232 t ? T 2 1.000 0.938 1.000 0.500 0.491 t Q T 0 1.000 0.999 1.000 0.500 0.500 -176-TABLE XVI P r o b a b i l i t y of E x t i n c t i o n of C e l l s at Times of Treatment for Parameter Values given i n Table V for the Strategy S(3)=(l,2,l,2,l,2,l,2) where the Mutation Rates are Equal with P r o b a b i l i t y 1 and have the D i s t r i b u t i o n Given i n Table XIII. Time t Treatment V ^ P i ( t ) p 2 ( t ) P12^ t) P ^ t ) f[ 0 0 0 0.676 0 tx T L . 0 0 0.032 0.676 0 t 2 T 2 0 0.019 0.028 0.639 0 t 3 Tx 0 0.019 0.500 0.638 0 t 4 T 2 0.707 0.421 0.500 0.638 0.197 t 5 T L 0.995 0.421 0.969 0.638 0.366 t 6 T 2 1.000 0.957 0.969 0.638 0.619 t 7 T x 1.000 0.957 0.999 0.638 0.626 t Q T 0 1.000 0.999 0.999 0.638 0.637 -177-of tumors treated with two agents, T, and T^, where the doubly r e s i s t a n t c e l l s are absolutely r e s i s t a n t and mutation rates are f i x e d . If i s the p r o b a b i l i t y that an i n d i v i d u a l tumor w i l l contain no doubly r e s i s t a n t c e l l s at the time of f i r s t treatment, then the effectiveness (as judged by the p r o b a b i l i t y of cure) of various treatment strategies r e s u l t from t h e i r a b i l i t y (or lack of i t ) to decrease the l i k e l i h o o d that double resistance w i l l develop i n the remaining proportion (1-qR) of tumors. Consider the same s i t u a t i o n , where again a proportion qjj of tumors contain no doubly r e s i s t a n t c e l l s , where now the mutation rates are nondegenerate random v a r i a b l e s . In t h i s case the tumors with e x i s t i n g doubly r e s i s t a n t c e l l s at diagnosis tend to contain a greater proportion of tumors with higher mutation rates and vice-versa. As before the e f f e c t of the treatment s t r a t e g i e s , which use only and T2, i s on those tumors where doubly r e s i s t a n t c e l l s have not emerged p r i o r to the commencement of treatment. Since these tumours w i l l tend to have lower mutation rates the rate of development of double resistance w i l l be "slower". This w i l l r e s u l t i n the differences i n c u r a b i l i t y between various strategies being diminished (compared to the s i t u a t i o n where rates are f i x e d ) . For example, i n the most extreme case, the mutation rates among the tumors without e x i s t i n g r e s i s t a n t c e l l s at the time of f i r s t treatment may be a l l i d e n t i c a l l y zero. In t h i s extreme case a l l strategies of fixed length which give T^ the same number of times w i l l be of equal ef f e c t i v e n e s s . Thus as the v a r i a b i l i t y i n mutation rates increases the differences i n the p r o b a b i l i t y of cure for strategies of the same length w i l l decline, possibly to the point where they become experimentally i n d i s t i n g u i s a b l e . Similar arguements apply to the e f f e c t s -178-of variation in the size of the tumor, N, at f i r s t treatment, which w i l l cause a similar reduction in the relative benefits of various strategies when compared to the case where N is fixed. In summary, random variation in parameters may act to decrease the differences in effectiveness among strategies and thus i t is necessary to consider such variation in the modelling of real systems. This completes our examination of response when two drugs are available. We w i l l now examine some experimental data to determine the appropriateness of the model presented here. 5.2 Fitting the Model to Experimental Data We w i l l examine experimental data collected by H. Skipper, F. Schabel and co-workers on the treatment of L 1 2 1 0 (mouse) leukemia by two drugs: Cyclophosphamide (Cyc) and Arabinosylcytosine (Ara-C ) [ 2 6 ] « This tumor and these drugs were chosen because of the extensive data collected on them by a single group of investigators in the same laboratory using the same breed of mouse. These drugs are also representative of two of the major types of drugs used in cancer chemotherapy, the alkylating agents (Cyc) and the antimetabolites (Ara-C). The data to be used in the examination of response to Cyclophosphamide alone is given in Table XVII; a l l data is for single doses given up to the L D ^ Q which occurs at about 3 0 0 mg/kg. This information has been compiled from a number of c l i n i c a l t r i a l s carried out by the investigators for intraperitoneally (IP) and intravenously (IV) implanted L 1 2 1 0 leukemia. The data is collected from experiments in which a fixed number (usually in the range of 1 0 0 - 1 0 0 0 ) of cells are implanted in an animal. The growth of the tumor is known to be -179-regular (for innoculums i n th i s range) and the size at any l a t e r time can be accurately estimated given the size of the o r i g i n a l innoculum [3]. Autopsies of animals indicate that 45 day survivors ( a f t e r the completion of any treatment) are free of any measurable L1210 leukemia [3]. The data presented i n Table XVII gives the number of 45 day survivors. The L1210 leukemia has been extensively studied and many of i t s physical properties are well known. Observation of the tumor (using thymidine l a b e l l i n g ) suggests that the median i n t e r m i t o t i c time of the tumor i s close to the median doubling time [26]. This implies that most c e l l s are a c t i v e l y d i v i d i n g and consequently that the end c e l l compartment i s small, and that c e l l loss i s small. Limiting d i l u t i o n assays (where a l i q u i d suspension of c e l l s are successively d i l u t e d and then injected into animals) suggests that a single c e l l i s s u f f i c i e n t to cause animal death (from the leukemia) [26]. This implies that almost a l l the c e l l s are stem c e l l s . We w i l l assume that a l l c e l l s are stem c e l l s and thus we have a model i n which c=d=0 (and there are no t r a n s i t i o n a l c e l l s ) . Data on c e l l s from this tumor which have been selected for Cyclophosphamide resistance suggests that such resistance i s e f f e c t i v e l y absolute ( r e s i s t a n t c e l l s survive administration of the drug with p r o b a b i l i t y 1), that i s TC^(D)=1.0 for a l l achievable doses D (see Section 3.1). Data on the mode of therapeutic action of Cyclophosphamide shows that i t has general a c t i v i t y on a l l phases of the c e l l cycle (see Section 3.2). If the c e l l s behave independently we see that the pro b a b i l i t y a tumor of size S se n s i t i v e stem c e l l s w i l l be cured by administration of the drug at dose D i s [ I - T C Q (D ) ] , where TCQ(D) i s the pro b a b i l i t y that a single s e n s i t i v e c e l l w i l l survive administration of -180-the drug. The form of ^ ( D ) be estimated from observation on growth delay curves (time taken to reach some fixed size a f t e r treatments of varying dosages c a r r i e d out at a common i n i t i a l s i z e ) . These observations indicate (assuming c e l l s behave independently) that 7i0(D)=exp{-kD}, ...(5.1) for a range of doses up to the L D ^ Q (Section 3.2). There i s some i n d i c a t i o n that (5.1) may not be accurate for doses approaching the L D ; L Q » where the therapeutic e f f e c t may be less than predicted by (5.1) [26]. This observation may be explained i n at least two ways. F i r s t l y , i t may be that the form of (5.1) should be modified at high doses because some mechanism (possibly drug transport into the c e l l ) becomes saturated so that the ef f e c t of increasingly large doses i s l i m i t e d . Secondly, we note that estimates of 7to(n) are based on observations of the whole tumor and not just on se n s i t i v e c e l l s . Since large therapeutic e f f e c t s can only be measured i n large tumors, i t i s possible that r e s i s t a n t c e l l s have emerged i n these large tumors and contribute to the regrowth of the tumor. Thus a deviation from (5.1) would be expected i n large tumors where estimates of TCQ(D) are based on the response of the t o t a l tumor. Since i t i s known that r e s i s t a n t c e l l s are present i n large tumors, we w i l l assume that the second explanation i s the true one. Let t^ be the time of treatment (only one cycle i s given) and N the number of stem c e l l s ( a l l the tumor i n th i s case). Then since c=0, d=0 we have P N= P{cure|N(t~)=N} = P{R Q(t 1)-0,R 1(t 1)-0|N(t~)=N}. -181-Since TE^(D)=1.0 we have P N= p{R ( )(t 1)=0,R 1(t^)=0|N(t^)=N}, = P{R 0( t l)=0|R 0(t^)=N} p{R 1(t^)=0|N(t~)=N}. ...(5.2) Assuming that the e f f e c t of therapy i n each c e l l i s independent, the f i r s t term of (5.2) i s given by P{R 0(t 1)=0|R 0(t^)=N} = [ 1 - T C Q ( D ) ] N , since the p r o b a b i l i t y a single stem c e l l w i l l survive therapy i s ^ ( D ) . Using the approximation suggested i n Section 3.7, that i s replacing N(t^) by Rg(t^) i n the second term i n (5.2), we have P{R 1(t^)=0|N(t~)=N}=P{R 1(t^)=0|R ( )(t^)=N} = ( l - a - v / b ) N _ 1 from (3.33). Since we cannot d i s t i n g u i s h a and v from one another (without further information) we set a=oc+v/b and obtain N-l N P ^ I l - a ] 1 " [1 -TC Q(D)] . From (5.1) we have N-l N P N = [ l - a ] W i [ l - e x p ( - k D ) ] J N . Equation (3.33) was derived under the assumption that the tumor grew from a single s e n s i t i v e stem c e l l . In the s i t u a t i o n under consideration a number (100-1000) of c e l l s are implanted and this formula must be viewed as approximate. We w i l l set P N = [ l - a ] N [ l - e x p ( - k D ) ] N ...(5.3) since N=N(t^) i s large i n a l l cases. The approximate l o g - l i k e l i h o o d , L(a,k), for the data i s given by I J L(a,k) = V J n. .[f, .JlnP. .+(l-f. .)An(l-P. .)] ...(5.4) ± ^ j = x ! J i J i j i J i j where N^ = siz e of tumor at treatment i = l , . . . , I , D. = dosage of treatment applied j = l , . . . , J , -182-n.. = number of animals tested at si z e N. and dosage D., i j i J s = number of animals cured among the n ^ treated, f. . = s. ./n. ., the observed proportion of animals cured, i j i j i j ' and P. . = p r o b a b i l i t y of cure at size N, and dose D., where, N N P ± j = [1-a] [1-expf-kD^] \ We may then d i f f e r e n t i a t e (5.4) to obtain equations i n a and k, which the maximum l i k e l i h o o d estimators a* and k* must s a t i s f y : M = y y n r ^ j . i3_. M i 4 5a 1 £ 1 . i , i j ^ . d - P , . ) 1 9a I J f . . (1-f, .) N.P, . / . i j L P , . d - P . . ) J d-a) U ' i = l j=l J i j xy ' I J f s . (1-f, .) M = v v „ r _ l i _ i l l i M ak V V r J-3 I J T 0^. • . \ .\n±3[?r~ O^PTO1 w i J i = l j=l i j i j I J f. . (1-f..) N.D.expl-kD.}P. . = y y „ r _ J J . _ 13 i 1 J 3 1 J = o i j L P . . (1-P. . ) J [l-exp{-kD. }] i = l j=l J i j i j 1 J The data for IP innoculation i n Table XVII were f i r s t modelled using the previous equations with a^O which yielded a maximum l i k e l i h o o d estimate k*=0.0678 and a corresponding l o g - l i k e l i h o o d of L(0,k*) =-1262.94. The f u l l model was then f i t and the maximum l i k e l i h o o d estimates were a*=l.04xl0 - 7, k*=0.0780 with L(a*,k*)=-530.20. Using the asymptotic x 2 d i s t r i b u t i o n for twice the difference i n the l o g -2 l i k e l i h o o d s a test of HQ:a=0 versus H^:a*0 has x^ =1465.48 providing strong evidence that a*0. The predicted values of P ^ using the maximum l i k e l i h o o d estimates a* and k* are given i n Table XVIII. The f i t of the model to t h i s data i s not good, as judged by a l o g - l i k e l i h o o d goodness-o f - f i t s t a t i s t i c of 453.26. Nevertheless the data analysis provides evidence for the development of drug r e s i s t a n t mutants. Coupled with observational evidence that drug (Cyclophosphamide) r e s i s t a n t c e l l s may -183-be selected from this tumor, we conclude that this analysis i s compatible with the idea that these drug r e s i s t a n t c e l l s a r i s e v i a spontaneous mutations although the goodness-of-fit indicates that t h i s model i s not a complete d e s c r i p t i o n of the data. Calculations based on growth delay curves indicate that the therapeutic e f f e c t ( p r o b a b i l i t y of se n s i t i v e c e l l s u r v i v a l ) of cyclophosphamide i s greater for IV implanted tumors than for IP implanted tumors [26]. Repeating the preceeding analysis for the data on IV implanted tumors i n Table XVII, we f i n d that when a=0 that the maximum l i k e l i h o o d estimate i s k*=0.0648 with a l o g - l i k e l i h o o d of L(0,k*)=-478.01. F i t t i n g the f u l l model we f i n d a*=1.06xl0 - 7, k*=0.0802 2 and L(a*,k*)=-175.65. A test of H0:a=0 has associated x1=604.72 providing strong evidence that a#0. Again the f i t of the model i s not good as assessed by the l o g - l i k e l i h o o d goodness-of-fit of X^ 3 = 141.99. The predicted values of P.. for the f u l l model are presented i n Table XVIII. The analysis presented thus provides some evidence that the therapeutic e f f e c t on sens i t i v e c e l l s , k, i s increased i n the IV innoculated tumors but the estimated values of a are almost i d e n t i c a l . By combining the data sets we may test whether the parameters a and k vary with route of implantation. Let a^, k^ ( i = l for IP and i=2 for IV) be the parameters for the two groups. We w i l l f i r s t f i t the model a^,k^=k2=k. Proceeding as before we obtain the maximum l i k e l i h o o d estimates a*=l.06xl0 - 7, a*=l.OlxlO - 7, k*=0.0784 with associated l o g - l i k e l i h o o d L(a*,a*,k*,k*)=-706.82. Using the l o g - l i k e l i h o o d s calculated from the separate models presented previously we have an -184-2 asymptotic test for H ^ i k ^ l ^ versus H^k^^k^. This y i e l d s \^ = 1-94 and thus we may conclude that there i s no evidence (from t h i s a n alysis) that the parameter k i s affected by the route of implantation of the tumor. A - p r i o r i we would postulate that the mutation rate, a, should be the same for both IP and IV innoculated tumors since i t has been assumed to be a property of the tumor c e l l s . This hypothesis may be tested by f i t t i n g the model a^-a^-a and k^=k2=k. F i t t i n g t h i s model we obtain the maximum l i k e l i h o o d estimates a*=l.048xl0 - 7 and k*=0.0784 with L(a*,a*,k*,k*)= -706.89. Comparing t h i s with the previous model we have a test for the 2 hypothesis U^ia^a^ versus H ^ i a ^ * ^ with associated x-^O'l^. On the basis of th i s analysis (and data), we conclude that the mutation rate does not vary with the route of implantation. The analysis presented so far has assumed that the mutation rates are f i x e d . In Section 3.10 we presented theory which modelled the mutation rates as random variables with beta d i s t r i b u t i o n s . We may use that development to determine whether t h i s data provides evidence for v a r i a b i l i t y i n the mutation rates (of a type which may be approximated by the beta d i s t r i b u t i o n ) and estimate the parameters of the d i s t r i b u t i o n . A technical problem arises because the p r o b a b i l i t y of no r e s i s t a n t c e l l s i s given by (3.50), which requires computing the product of 8x l 0 7 terms (the largest size i n the data), that i s P{R 1(t-)=0|R 0(t-)=N}, V (^). x=0 where (u,v) are the parameters of the beta d i s t r i b u t i o n . In the preceeding analysis, when rates were fixed we found a=10 - 7. We would therefore expect that the mean of this beta d i s t r i b u t i o n , u/(u+v), would -185-be small and thus that u « v . If this i s indeed the case, then we may approximate the product as follows: N-2 _,_ N-2 M n = - I M (1 + -£-) L n ^u+v+x;j L n v+x x=0 x=0 N-2 _ « - u J (v+x) x=0 r dw „ rV+N-2n - - u J — = -u An . W L V J V Using t h i s approximation to (3.50),then from (5.3) we have V P l T ^ I U [l-exp(-kD)] N. ...(5.5) F i t t i n g t h i s model to the IP data using the l o g - l i k e l i h o o d function (5.4) with v+N -2 -u N P ± j - [ ~ ~ ] [l-exp(-kD.)] \ yielded the maximum l i k e l i h o o d estimates u*= 0.301, v*=0.578xl0 5 and k*=0.0857 with associated l o g - l i k e l i h o o d L(u*,v*,k*)=-347.42. The fixed mutation rate model i s a s p e c i a l case (u-* 0, v-*» such that u/(u+v)+a) of the variable mutation rate model and we may construct a test assessing whether the f i t of the model i s improved by permitting v a r i a b i l i t y . This 2 y i e l d s x^ =365.56 which provides evidence that the f i t of the model i s considerably improved by permitting v a r i a b i l i t y . Despite t h i s improvement there s t i l l remains considerable r e s i d u a l v a r i a t i o n as judged 2 by the l o g - l i k e l i h o o d goodness-of-fit s t a t i s t i c of X ^ = 87.70. Repeating this analysis for the data on IV implanted tumors y i e l d s u*=0.633, v*=4.912xl0 5, k*=0.0846 with a l o g - l i k e l i h o o d L(u*,v*,k*) = -117.41. As i n the IP case we f i n d that permitting the rates to vary (with a beta d i s t r i b u t i o n ) improves the f i t of the model with an 2 associated =116.48. However, as before we f i n d that this model does -186-not adequately f i t the data as judged by the l o g - l i k e l i h o o d goodness-of-2 f i t s t a t i s t i c of x 1 2 = 25.49. If we l e t i (=1 for IP and =2 for IV) index the route of implantation, we may analyse the combined data set and test whether k^=k2=k. F i t t i n g this model yi e l d s L(u*,v*,u*,v*,k*)=-464.95 and thus a 2 test of H Q:k^=k 2 has x-^=0'24 providing no evidence for a difference i n the therapeutic parameters. F i t t i n g the model u^=u2=u, v^=v2=v, k^=k2=k we obtain L(u*,v*,k*)=-470.24 and a test of H Q : U ^ = U 2 , V ^ = V 2 (assuming 2 k^=k 2) i s given by x 2 =10*58 thus providing some evidence that the d i s t r i b u t i o n s of a may not be the same for IV and IP implanted tumors. The estimated values of the cure rates for the IP and IV implanted tumors using the maximum l i k e l i h o o d estimators u*, v* and k* (i=l,2) are given i n Table XVIII. Interpreting these r e s u l t s i s not straightforward since i f v a r i a b i l i t y i n mutation rates exists we would not expect i t to vary with route of implantation. The evidence that v a r i a b i l i t y e x i s t s must remain hypothetical and we can only say that the analysis of the data presented here i s compatible with this idea. This subject i s worthy of future (experimental) study although t h i s w i l l not be easy. Data on s u r v i v a l of animals having L1210 tumors treated with Ara-C i s given i n Table XIX. Ara-C i s e s p e c i a l l y active against c e l l s i n the S-phase of the c e l l cycle and thus i t s e f f e c t i s l i m i t e d by the proportion of c e l l s i n this phase during treatment [26]. This drug i s best administered i n doses far below the L U ^ Q since large doses have no greater tumoricidal e f f e c t . After much experimentation with this drug, Skipper and his associates have found that doses of 15mg/kg may be -187-repeated every 3 hours up to 8 times without r e s u l t i n g i n undue t o x i c i t y [26]. Observations on the growth delay of tumors treated with between 1 and 8 cycles of Ara-C (every 3 hours) suggest that the log of the f r a c t i o n of surviving c e l l s i s l i n e a r l y proportional to the number of cycles given. This would imply that the e f f e c t of each cycle of Ara-C i s the same (assuming independence) and that the c e l l s s u f f i c i e n t l y r e d i s t r i b u t e themselves about the c e l l cycle so that, approximately, a constant proportion of c e l l s are i n the S phase at each a p p l i c a t i o n of the drug. Further cycles of therapy beyond 8 (every 3 hours) produce considerable t o x i c i t y , however, i f therapy i s not given for three days the animal's normal tissues recover s u f f i c i e n t l y for therapy to be applied again. A regimen of 8 cycles of Ara-C given every 3 hours w i l l be referred to as a course [26]. Up to four courses may be given, with intervening three day recovery periods, without undue t o x i c i t y . Data from experiments using between one and four courses, for various i n i t i a l tumor burdens, are given i n Table XIX. We propose to model this data using the model presented i n Section 3.7. In what follows j=l,...,4 w i l l index the number of courses of Ara-C given. As e a r l i e r the l o g - l i k e l i h o o d , L, i s given by (5.4). In t h i s case, however, P i s of a more complex form. Here we w i l l use the approximation developed i n Section 3.7 where we assumed that RQ(t^)= N(t^) where t ^ i s the time of the f i r s t cycle of the f i r s t course. Since the death rate for this tumor i s assumed to be zero, the tumor i s cured i f , and only i f , {R^t j)=0,R^(t j)=0} where tj i s the time of the l a s t cycle of therapy. For s i m p l i c i t y we w i l l assume that v=0 and estimate a only, that i s we w i l l assume that t r a n s i t i o n s to resistance -188-occur only at c e l l d i v i s i o n . Observation on Ara-C re s i s t a n t c e l l s suggest that t h i s resistance i s e f f e c t i v e l y absolute, i . e . TC^(D)=1 for the doses of Ara-C used. If (|>(sQ,s^;t) i s the p r o b a b i l i t y generating function for the d i s t r i b u t i o n of s e n s i t i v e and r e s i s t a n t c e l l s i n the tumor, then the p r o b a b i l i t y of cure i s <j>(0,0;tj); see (3.13). Since TC^(D)=1, from (3.11.1) we have <t>(s0,0;tj) = <KS 0(s 0),0;t-j), f o r J = 1 » ' " » J > where from (3.9.1), V s o ) = 1 - V D ) + V D ) V From (3.11.2) we have, <Ks 0,0;t~ + 1) = < K w 0(tj + 1-tj),0;tj) for j = l , . . . , J - l , where w^(t)=0 for s^=0 and soe -bt 0 [ l - s 0 ( l - a ) (1-e b t ) ] since s^=0 and v=0 (see (3.6) and (3.7)). Using (3.30) we may write R 0 <t)(s 0,s 1;t 1) = s Q C R ( s ^ t ^ , where R Q i s the "observed" number of se n s i t i v e stem c e l l s at time t ^ and we w i l l set R Q = n » the t o t a l tumor siz e at f i r s t treatment. From (3.33) we have n - N .N-l <t>(s0,0;t1)=s() (1-a) Using the above equations we may estimate the p r o b a b i l i t y of cure for various values of the parameters a and T C Q ( D). Notice that the drug i s only given at a single dose l e v e l (15 mg/kg) so i t i s not necessary to specify the form of 7£Q(D). The complex form of P makes i t i n f e a s i b l e -189-to set up equations for the maximum l i k e l i h o o d estimates of a and UQ(15)=^Q- Thus a d i r e c t approach was taken by s e l e c t i n g " l i k e l y " values of the parameters and i t e r a t i n g i n d i r e c t i o n s so that the l o g - l i k e l i h o o d increases. I n i t i a l l y t his method was used on a version of t h i s model i n which a=0. In t h i s case i t proved d i f f i c u l t to compute the l o g -l i k e l i h o o d since for a l l choices of u_ either P. .=1 or P. .=0 for some 0 i j i j i , j . When the l o g - l i k e l i h o o d was calculated at least one term i n the sum overflowed y i e l d i n g the following i n e q u a l i t y for the l o g - l i k e l i h o o d : L(0, it*)<-10 3 8. However, j o i n t estimation of a and did produce e a s i l y computed l i k e l i h o o d s for a range of these two parameters. Maximum l i k e l i h o o d estimates were obtained for a number of s t a r t i n g values (10 - 9<a<10 - 5, 0.1<rvQ<0.3) and i n a l l cases ( i n which the l o g - l i k e l i h o o d did not overflow) each sequence converged to the same estimates. The maximum l i k e l i h o o d estimates were given by a*=l•791xl0 - 7, n*=0.186 with L(a*,n*)=-209.41. The l o g - l i k e l i h o o d goodness-of-fit of x 2 Q= 22.38 indicates that v a r i a t i o n e x i s t s which i s not explained by the model. The predicted estimates of P using t h i s model are given i n Table XIX. There i s thus considerable evidence that spontaneously r e s i s t a n t c e l l s do a r i s e with a frequency of the order of 1 0 - 7 . F i t t i n g a model incorporating variable mutation rates poses a considerable technical problem since the recursive nature of the r e l a t i o n s h i p s involved do not permit an approximation of the type used i n (5.5). We w i l l approximate the e f f e c t of v a r i a t i o n i n a (following a beta d i s t r i b u t i o n ) using a discrete d i s t r i b u t i o n s i m i l a r to that used i n Section 5.1. The number of mass points used i n the approximation was was varied (5,10 and 20) and each lead to e s s e n t i a l l y the same r e s u l t (to -190-s i x s i g n i f i c a n t figures i n the l i k e l i h o o d ) . Using the same notation as i n the analysis of Cyclophosphamide, the maximum l i k e l i h o o d estimates were u*=3.298, v*=1.574xl0 8 (where u and v are the parameters of the beta d i s t r i b u t i o n which generate the percentiles used i n the discre t e approximation) and TE*=0.186 with L (U * , V * , T I * ) = - 2 0 9 . 15. The asymptotic x 2 d i s t r i b u t i o n of the difference i n l o g - l i k e l i h o o d s yielded a test for the presence of v a r i a b i l i t y i n a of x 2 =0'52 providing no evidence for v a r i a t i o n i n a (following an approximate beta d i s t r i b u t i o n ) . The analysis of the data on two quite d i f f e r e n t drugs (one phase s p e c i f i c and one not) appear to provide evidence compatible with the hypothesis that drug r e s i s t a n t c e l l s do a r i s e as a r e s u l t of random mutations. In one case (Cyc) there was evidence that the mutation rate may be random, whereas the analysis of the data for Ara-C provided no evidence for t h i s . We cannot conclude that the mutation rate has been demonstrated to be random for resistance to Cyc i n the L1210 leukemia because there s t i l l e x i s t s considerable unexplained v a r i a t i o n i n the data. The existence of random v a r i a t i o n In mutation rates for spontaneous tumors cannot be determined from the analysis of passaged animal tumors because each spontaneous tumor i s unique whereas each animal implanted with a passaged tumor (L1210) should be considered to have a sample of a single tumor. By testing for random v a r i a t i o n i n mutation rates i n a single type of experimental tumor we are t e s t i n g whether these rates spontaneously evolve during the s e r i a l passaging of the tumor. In summary the presence of v a r i a t i o n i n mutation rates can properly be determined only by analyzing data from a series of de-novo spontaneous tumors. Since v a r i a t i o n can influence the value of various -191-TABLE XVII Response of In t r a p e r i t o n e a l l y (IP) and Intravenously (IV) Innoculated L1210 Leukemia to Single Doses of Cyclophosphamide.* IP IV Dose Size at .# of Animals # of # of animals # of mg/kg treatment Treated Survivors Treated Survivors 300 8 x l 0 7 94 7 80 4 8 x l 0 6 148 60 30 10 8 x l 0 5 39 30 20 14 250 8 x l 0 7 - - 66 1 8 x l 0 6 - - 30 3 8 x l 0 5 - - 30 17 230 8x10 6 50 7 - -8x10 5 40 10 - -8 x l 0 4 50 41 - -200 8 x l 0 7 109 3 60 0 8x10 6 160 11 40 3 8x10 5 60 11 10 0 8X101* 10 8 - -8 x l 0 3 10 10 - -150 8x10 7 30 0 245 0 8 x l 0 6 19 0 60 0 8 x l 0 5 20 1 50 3 100 8 x l 0 7 10 0 130 0 8 x l 0 6 20 0 30 0 8x10 5 144 0 20 0 * Abstracted from reference [26]. -192-TABLE XVIII Observed (Obs) and Predicted values f o r the P r o b a b i l i t y of Cure f o r IP and IV Innoculated L1210 Leukemia Treated with Cyclophosphamide Using the Maximum Likelihood Parameter Estimates f o r Fixed Rates (Predl) and f o r Variable Rates (Pred2). Dose Size at t IP IV mg/kg treatment Obs Predl Pred2 Obs Predl Pred2 300 8 x l 0 7 0.074 0.000 0.109 0.050 0.000 0.040 8 x l 0 6 0.405 0.435 0.223 0.333 0.428 0.165 8 x l 0 5 0.769 0.920 0.446 0.700 0.919 0.544 250 8 x l 0 7 - - - 0.015 0.000 0.038 8 x l 0 6 - - - 0.100 0.422 0.164 8 x l 0 5 - - - 0.567 0.917 0.544 230 8 x l 0 6 0.140 0.393 0.218 - - -8 x l 0 5 0.250 0.911 0.445 - - -8X104 0.820 0.992 0.778 - - -200 8 x l 0 7 0.028 0.000 0.008 0.000 0.000 0.001 8 x l 0 6 0.069 0.145 0.173 0.075 0.180 0.115 8 x l 0 5 0.183 0.824 0.435 0.000 0.843 0.524 8x10 4 0.800 0.981 0.776 - - -8 x l 0 3 1.000 0.998 0.964 - - -150 8 x l 0 7 0.000 0.000 0.000 0.000 0.000 0.000 8 x l 0 6 0.000 0.000 0.000 0.000 0.000 0.000 8x10 5 0.050 0.003 0.066 0.060 0.008 0.045 100 8x l 0 7 0.000 0.000 0.000 0.000 0.000 0.000 8x10 6 0.000 0.000 0.000 0.000 0.000 0.000 8 x l 0 5 0.000 0.000 0.000 0.000 0.000 0.000 -193-TABLE XIX Observed and Predicted Rates of Cure f o r Intravenously Innoculated L1210 Leukemia Treated with Repetitive Courses of A r a - C * Dose Size at # of Animals # of Observed Predicted Schedule treatment Treated Survivors Cure Rate Cure Rati q 3hr (x8) 8x10 6 10 0 0.000 0.000 1 course 8x10 5 60 2 0.033 0.021 8x10^ 20 11 0.550 0.681 q 3hr (x8) 8x10 7 20 0 0.000 0.000 2 courses 8 x l 0 6 40 3 0.075 0.223 8x10 5 19 11 0.579 0.860 q 3 hr (x8) 8 x l 0 6 9 3 0.333 0.224 3 courses 8x10 5 30 25 0.833 0.861 q 3 hr (x8) 8xl 0 7 59 0 0.000 0.000 4 courses 8 x l 0 6 80 25 0.313 0.224 8 x l 0 5 215 187 0.870 0.861 8xlOk 30 30 1.000 0.985 *Data abstracted from reference [26]. - 1 9 4 -therapeutic interventions t h i s subject i s worthy of further study. 5.3 Neo-Adjuvant Chemotherapy Adjuvant i s a term applied to chemotherapy which i s used i n addition to other forms of therapy ( i . e . radiotherapy or surgery) [32]. Adjuvant chemotherapy i s commonly used i n a large number of s o l i d human tumors and has proven successful i n increasing the c u r a b i l i t y of several of these tumors ( i . e . breast cancer) [35]. The use of adjuvant chemotherapy has been p a r t i c u l a r l y successful (and i n i t i a l l y somewhat controversial) when used i n in d i v i d u a l s with no observable disease (perhaps a f t e r surgery), but who are believed to have microscopic disease present (based on the experience of i n d i v i d u a l s with si m i l a r disease). In a l l these cases chemotherapy i s given subsequent to "curative" therapy (usually surgery). For some types of tumor, i n d i v i d u a l s may present with advanced disease where surgery, although desirable, i s not possible. In p a r t i c u l a r tumors ( i n i t i a l l y head and neck cancer) a new concept has been proposed, that of neo-adjuvant (or pre-operative) chemotherapy [36]. In thi s approach chemotherapy i s given f i r s t i n order to shrink the primary tumor so that surgery i s possible. After surgery the patient then receives the appropriate therapy. Like any good idea i t has been applied to a v a r i e t y of cases where i t i s more or less appropriate. In p a r t i c u l a r neo-adjuvant chemotherapy has been advocated, and i s currently being tested, i n si t u a t i o n s where surgery i s already possible without the neo-adjuvant chemotherapy. In breast cancer, which i s one such case, neo-adjuvant chemotherapy amounts to s t a r t i n g the programme of adjuvant therapy at a time p r i o r to surgery rather than a f t e r [37]. -195-In t h i s section we w i l l consider the case of human breast cancer i n some d e t a i l - Many advantages are espoused for the neo-adjuvant a p p l i c a t i o n of therapy, however, we w i l l be concerned with only one of them here; that neo-adjuvant therapy reduces the l i k e l i h o o d of treatment f a i l u r e from drug resistance. At th i s point we w i l l provide an overview of the approach to be taken. We w i l l assume that p r i o r to diagnosis the d i s t r i b u t i o n of tumor c e l l s Is given by the approximation of Section 3.6. We w i l l set up an ad-hoc model for the e f f e c t of surgery on the d i s t r i b u t i o n of tumor c e l l s and f i t i t to observations from human breast cancer. We w i l l not assume that we have a p a r t i c u l a r drug (with given a, v, e t c . ) , but require that the drug used i s "reasonably" e f f e c t i v e (against the se n s i t i v e c e l l s ) and examine the c u r a b i l i t y for various values of the mutation rates. Using the models developed for the e f f e c t of surgery and chemotherapy (Chapter 3) we w i l l then examine the e f f e c t on c u r a b i l i t y of an extra neo-adjuvant cycle of therapy. We know, from the example considered i n Section 3.5, that for a given treatment strategy the e a r l i e r the treatment i s begun the greater w i l l be the p r o b a b i l i t y of cure. The c r i t i c a l question i s the magnitude of the increase i n the p r o b a b i l i t y of cure produced by an extra neo-adjuvant cycle of therapy. We w i l l use breast cancer as an example although this approach can, i n p r i n c i p l e , be extended to other tumor types. The adjuvant therapy w i l l be assumed to be a single drug which i s given i n a fi x e d number of cycles. This a p p l i c a t i o n of the theory d i f f e r s from those considered previously since we are now considering the eff e c t of two modalities of therapy (chemotherapy and surgery) rather than chemotherapy alone. -196-As mentioned before (Chapter 3), the ef f e c t of surgery ( i n terms of tumor reduction) i s complex and depends on many fa c t o r s . One of the p r i n c i p a l d i f f i c u l t i e s i s that metastasis of the tumor to other s i t e s may not be apparent at diagnosis. For example i n breast cancer, the findi n g of lymph node involvement i s strongly i n d i c a t i v e of tumor dissemination to other s i t e s . This i s the idea i m p l i c i t i n the c l i n i c a l and pathological staging systems for tumors although other prognostic factors not included i n these systems have been i d e n t i f i e d . In breast cancer two prognostic factors which have been i d e n t i f i e d are commonly used i n the planning of c l i n i c a l t r i a l s : menopausal status (pre-menopausal or post-menopausal) and lymph node status (0 p o s i t i v e nodes, 1-3 p o s i t i v e nodes, 4+ p o s i t i v e nodes) [25]. We w i l l consider separately the s i x groups defined by menopausal status and nodal status for women with breast cancer. In order to estimate the eff e c t s of surgery within each of these groups i t i s necessary to analyse data on recurrence times of i n d i v i d u a l s with breast cancer treated by surgery. Ideally such data would include i n d i v i d u a l measurement of recurrence times and growth rates for women treated s u r g i c a l l y (using a standard procedure) for breast cancer. Unfortunately such data does not appear to be av a i l a b l e since i n d i v i d u a l growth rates are seldom reported. Here we propose to use the re s u l t s of an analysis by Skipper [38] of data of Valagussa et a l consisting of women treated s u r g i c a l l y for breast cancer [25]. In that analysis the following assumptions were made: ( i ) A l l premenopausal disease grows at a fixed rate, ( i i ) A l l postmenopausal disease grows at a fixed rate, -197-( i i i ) Recurrence occurs when the tumor burden at a single s i t e exceeds 10 9 c e l l s , ( i v ) Individuals not recurring within 10 years a f t e r surgery are cured, (v) A l l c e l l s are stem c e l l s . The f i r s t four assumptions are c e r t a i n l y not p r e c i s e l y true but are not unreasonable approximations. The f i f t h assumption i s not e x p l i c i t l y stated by the author but i s i m p l i c i t i n the development of the estimates of r e s i d u a l tumor burden. In t h i s development we would have preferred not to make th i s assumption however the raw data was not a v a i l a b l e for a n a l y s i s . For consistency, we have thus assumed that c=d=0 i n what follows although this i s not required by the subsequent development. The estimates of r e s i d u a l tumor burden (stem c e l l burden) subsequent to surgery are given i n Table XX. We w i l l also require the following further assumptions to continue with t h i s a n a l y s i s : ( v i ) The removal of c e l l s by surgery i s a random process and does not d i s t i n g u i s h between c e l l types i . e . drug s e n s i t i v e and drug r e s i s t a n t , ( v i i ) The f a i l u r e of drug therapy i s s o l e l y due to the presence of drug r e s i s t a n t c e l l s a r i s i n g by the process described i n Chapter 3, ( v i i i ) The two modalities (chemotherapy and surgery) do not i n t e r a c t with one another i . e . the e f f e c t of each modality for an i n d i v i d u a l i s independent of the time at which i t i s given, (ix) The number of tumor c e l l s at the s i t e of the f i r s t recurrence ( i f i t occurs) i s much greater than the number of tumor c e l l s at -198-any other s i t e s i n the same i n d i v i d u a l , (x) The r e s i s t a n t c e l l s survive chemotherapy with p r o b a b i l i t y 1, (xi) The s e n s i t i v e c e l l k i l l of the chemotherapy i s s u f f i c i e n t l y large and the therapy i s applied s u f f i c i e n t l y frequently so that the net growth of the se n s i t i v e c e l l s during the treatment period i s strongly s u b - c r i t i c a l , ( x i i ) The d i s t r i b u t i o n of the number of tumor c e l l s a f t e r surgery i s not related to the pre-surgery tumor burden. It seems appropriate, at th i s point, to indicate the reasons f o r these further assumptions. Assumption ( v i ) seems reasonable and considerably s i m p l i f i e s the l a t e r development. Assumption ( v i i ) r e l a t e s to the intended objective of this section which i s to examine the e f f e c t of neo-adjuvant chemotherapy i n preventing the development of drug resistance. Assumption ( v i i i ) permits analysis of the e f f e c t of timing and i s a reasonable s i m p l i f i c a t i o n of the behaviour of these two very d i f f e r e n t modalities. Assumption (ix) implies that we may approximate the t o t a l tumor burden of the i n d i v i d u a l by the number of tumor c e l l s at the s i t e of recurrence. We may then approximate the post-surgical p r o b a b i l i t y generating function for the t o t a l number of c e l l s by the p r o b a b i l i t y generating function for the number of c e l l s at the s i t e of recurrence. Also from (I) and ( i i ) we may make the preceedlng approximation at a l l times a f t e r the time of surgery. Assumptions (x) and (xi) are are si m p l i f y i n g assumptions which imply that the p r o b a b i l i t y of cure for the chemotherapy i s approximately equal to the p r o b a b i l i t y of cure at the f i r s t cycle of therapy. Thus the p r o b a b i l i t y of cure i s only weakly dependent on the d e t a i l s of the way i n which the chemotherapy i s -199-applied ( a f t e r the f i r s t c y c l e ) . Assumption ( x i i ) i s c l e a r l y i n c o r r e c t , however we are forced to make this assumption because of a lack of d e t a i l e d information on the pre-surgical tumor burden. We w i l l approximate the post-surgical d i s t r i b u t i o n of tumor c e l l s seperately for each of the s i x prognostic groups (menopausal x nodal combinations). Approximation ( x i i ) may not be as bad as i t f i r s t seems since the r e l a t i v e difference i n i n i t i a l tumor burden before surgery i s l i k e l y to be much smaller than the r e l a t i v e difference i n tumor burden a f t e r surgery. This assertion i s based on the assumption that the majority of the tumor burden p r i o r to therapy i s located i n the breast l e s i o n which i s (almost t o t a l l y ) excised i n a l l cases, thus leaving the more variable metastatic burden i n place. We are now i n a p o s i t i o n to determine a n a l y t i c expressions which summarise the e f f e c t s of applying the chemotherapy early (neo-adjuvant). The estimates of tumor burden a f t e r surgery ( i n the absence of neo-adjuvant therapy) derived by H. Skipper are given i n Table XX for the six prognostic groups. Cl e a r l y the v a r i a t i o n i n r e s i d u a l tumor burden i s quite large. We can now proceed to f i t a d i s t r i b u t i o n to the data given i n Table XX, however, the possible mathematical form of the d i s t r i b u t i o n which can be used i n subsequent analysis i s l i m i t e d . The reason i s that we do not have the d i s t r i b u t i o n function for the number of s e n s i t i v e and re s i s t a n t c e l l s i n e x p l i c i t form. The natural model for the response (removal or not) of a single c e l l to surgery i s a B e r n o u l l i v a r i a b l e , where the parameter, 9, i s a function of the i n d i v i d u a l (tumor) and the s u r g i c a l technique. The parameter i s unknown and cannot be estimated r e l i a b l y since we only have one observation per i n d i v i d u a l . We w i l l -200-assume that the parameters {9} only take a f i n i t e number of values and then f i t t h i s model to the observed data. This model i s quite ad-hoc, but our aim here i s to estimate the post-surgical d i s t r i b u t i o n of se n s i t i v e and r e s i s t a n t c e l l s and thus we only need to calculate the e f f e c t of surgery and not construct a v a l i d model of the mechanism of act i o n . We w i l l set e i = 1 0 _ 1 1=1,...,11, which spans the l i k e l y range of {9}. Let j(=l,2) and k(=l,2,3) index the prognostic groups and define C. =P{9=9.| i n d i v i d u a l i s i n prognostic group j,k}. 1JK. 1 We w i l l assume that the d i s t r i b u t i o n of the number of stem c e l l s p r i o r to surgery has p r o b a b i l i t y generating function, C(s;t), given by (3.24). The p r o b a b i l i t y generating function of the number of c e l l s a f t e r surgery given at time t ^ i n prognostic group j,k , C j k ( s ; t ^ ) , i s given by 11 C j k ( s ; t ^ ) = I C . j k C ( C i(s);t j f e) for j=l,2, k=l,2,3, where E.(s)=l-9,+9.s for 1=1,...,11. i i l Let N ( t ^ ) be the post-surgical number of stem c e l l s for j V i n d i v i d u a l s i n prognostic group j,k. Then P{Nj k(t^)=n} i s given by the c o e f f i c i e n t of s 1 1 i n the above expression. Expanding and i d e n t i f y i n g the c o e f f i c i e n t of s 1 1, we have n-rl 11 9.e *[0 (1-e *)] H N j k ( t , ) = n } = I C i j k ^ for n>0, [9.+(l -9.)e *] x 1 and 11 (1-9.)e b t * P{N k(t*)=0} = I C 1 -i - i 1 J K [e.+a-e^e^*] p{N j k(t*)=0} = I C. -201-11 ( i - e i ) e " b t * j k v *' > ^ i j k p. . -bt i = i 1 J K [ e^ i -e^e""*] Assuming that the mean number of c e l l s p r i o r to surgery i s 10 10 (approximately a 2 cm- diameter sph e r i c a l tumor), we set e b t*=10 1 0. Then values of be chosen and the post-surgical tumor burden examined and compared with the "observed" values (Table XX). The values selected for C ., are given i n Table XXI and the predicted pos t - s u r g i c a l i Jk d i s t r i b u t i o n i s given i n Table XXII. These values are not unique and their " f i t " i s not perfect as may be seen by comparing Tables XXI and XXII. These values were selected by an informal procedure of t r i a l and error u n t i l the f i t t e d values were within ±1% of the observed values. Given that the observed d i s t r i b u t i o n of post-surgical tumor burden has considerable random error (since i t was estimated from data on 716 cases) the f i t t e d model seems adequate. Let <(>(sQ,s^;t) be the p r o b a b i l i t y generating function for the number of s e n s i t i v e and re s i s t a n t c e l l s i n the tumor at time t (see Section 3.1). For the neo-adjuvant approach, chemotherapy i s applied f i r s t , at time t^ say. Then from (3.9.1) we have: 4 ) ( s 0 , s 1 ; t 1 ) = 4 > ( C 0(s 0),s 1;t 1 ), where £Q(S)=1—TCQ+TCQSQ. If surgery i s applied at time we have <t>(sQ,s]L;t2) = < t > ( w 0 ( t 2 - t 1 ) , w 1 ( t 2 - t 1 ) ; t 1 ) , where w Q(t) i s given by (3.7), w ^ t ) by (3.6) with c=d=0. The e f f e c t of surgery on the p r o b a b i l i t y generating function of the number of c e l l s for a tumor i n the prognostic group j , k i s then given by 11 4>(s 0, S l;t 2) = I C • ( 5 1 ( s 0 ) , 5 1 ( s 1 ) ; t ), 1=1 J for j=l,2, k=l,2,3. -202-To analyse the e f f e c t of neo-adjuvant therapy we must consider the parameters which are related to the chemotherapy- We w i l l assume that v=0 and c a l c u l a t e the c u r a b i l i t y for a number of values of a- From assumptions (x) and ( x i ) , i f the protocol i s s u f f i c i e n t l y long ( i . e . J lar g e ) , the c u r a b i l i t y of the regimen w i l l not depend very strongly on the parameter TCQ (the p r o b a b i l i t y of s e n s i t i v e c e l l s u r v i v a l for a cycle of chemotherapy). In his analysis of t h i s data, Skipper found that the doubling time was 56 days for premenopausal disease and 69 days for postmenopausal disease. We w i l l model conventional adjuvant chemotherapy as consisting of s i x cycles of chemotherapy where the f i r s t cycle i s given 28 days a f t e r surgery and then given i n cycles with 21 day i n t e r v a l . Calculations based on this model show that the c u r a b i l i t y i s approximately the same for a l l cases where 7CQ<0.1, J>4 (the number of treatment times) and the i n t e r v a l between cycles of chemotherapy i s le s s then t h i r t y days. Neo-adjuvant therapy w i l l be modelled by assuming that an single extra cycle of therapy i s given two days before surgery and then followed by the same post-surgical adjuvant therapy as above. In both cases the date of surgery i s the same, that i s , the Inclusion of the neo-adjuvant cycle does not a f f e c t the timing of other therapy- Tables XXIII-XXIV give the estimated c u r a b i l i t y of the tumor as a function of the mutation rate to resistance, a, for the conventional adjuvant protocol and the increase i n the p r o b a b i l i t y of cure associated with a neo-adjuvant cycle of therapy added to the same protocol for each of the prognostic groups. The most obvious r e s u l t which may be seen from examination of Tables XXIII-XXIV i s that i n no case does the calculated increase i n c u r a b i l i t y -203-exceed 0.01. Thus the l i k e l i h o o d of any measurable a f f e c t of neo-adjuvant therapy of the type described here for the development of resistance to a single drug for breast cancer i s n e g l i g i b l e . The modelling procedure i s not i d e a l , as has already been described, however i t would seem that inaccuracies i n the modelling of surgery or the e f f e c t s of chemotherapy are u n l i k e l y to cause an order of magnitude change i n the advantage of neo-adjuvant therapy. Secondly, i t can be seen that the c u r a b i l i t y of the tumor ( i n any of the prognostic groups) varies quite slowly with the mutation rate. Large improvements i n the cure rates obtained with adjuvant chemotherapy w i l l thus require s i g n i f i c a n t reductions i n the o v e r a l l mutation rate. For example, an improvement i n c u r a b i l i t y of 0.10 i n premenopausal negative node group requires a chemotherapy with a mutation rate of 10-1*. A further improvement i n c u r a b l i t y of 0.10 would require a chemotherapy with a mutation rate of 1 0 - 7 . The p r i n c i p l e reason that neo-adjuvant therapy i s predicted to have l i t t l e e f f e c t (on the development of resistance) i n t h i s tumor i s the highly variable p o s t - s u r g i c a l tumor burden. If the po s t - s u r g i c a l tumor burden l i e s i n a narrow range then the r e l a t i o n s h i p between c u r a b i l i t y and mutation rate w i l l be quite d i f f e r e n t from that displayed i n Tables XXIII-XXIV. In t h i s s i t u a t i o n c u r a b i l i t y w i l l r a p i d l y change (as a function of the mutation rate) i n the region where the inverse of the mutation rate i s approximately equal to the mean resi d u a l tumor burden. In such sit u a t i o n s an extra neo-adjuvant cycle of may have considerable impact i n preventing the development of resistance. In conclusion, i f neo-adjuvant chemotherapy i s to have any -204-measurable e f f e c t i n this tumor, i t s primary e f f e c t must be on other mechanisms of treatment f a i l u r e and not on the development of spontaneous resistance. This completes the consideration of applications of t h i s model. -205-TABLE XX Distribution of Post-Surgical Tumor Burden for 716 Cases of Breast Cancer as a Function of Nodal Status and Menopausal Classification.* Premenopausal Po s tmenopausal Number of Pos i t i v e Nodes Number of P o s i t i v e Nodes Tumor Burden 0 1-3 4+ 0 1-3 4+ [0] 0.69 0.31 0.12 0.74 0.35 0.15 [ 1 0 ° , 101) 0.07 0.22 0.11 0.05 0.10 0.08 [10 1, 102) 0.00 0.07 0.03 0.01 0.03 0.04 [10 2, 103) 0.04 0.02 0.03 0.02 0.00 0.03 [10 3, 0.03 0.04 0.09 0.02 0.08 0.04 [10\ 105) 0.01 0.07 0.14 0.02 0.06 0.05 [10 5, 106) 0.04 0.07 0.13 0.03 0.08 0.08 [10 6, 107) 0.08 0.09 0.07 0.04 0.09 0.20 [10 7, 108) 0.03 0.11 0.18 0.03 0.11 0.20 [ i o 8 , ro) 0.00 0.01 0.11 0.04 0.08 . 0.14 TOTAL 1.00 1.00 1.00 1.00 1.00 1.00 * Abstracted from reference [38]. -206-TABLE XXI Table of Values of C±^* t b e P r o b a b i l i t y of B e r n o u l l i parameter 8^, fo r the Six Prognostic Categories. Premenopausal Postmenopausal Ber n o u l l i Parameter Number of P o s i t i v e Nodes Number of P o s i t i v e Nodes 9 i 0 1-3 4+ 0 1-3 4+ 1 0 - 1 1 0.64 0.00 0.00 0.75 0.34 0.04 l O " 1 0 0.12 0.44 0.16 0.03 0.00 0.17 l O " 9 0.00 0.07 0.09 0.00 0.13 0.02 lO" 8 0.00 0.08 0.02 0.01 0.01 0.05 10 - 7 0.05 0.00 0.01 0.02 0.00 0.01 10~6 0.04 0.00 0.09 0.02 0.11 0.05 l O - 5 0.00 0.04 0.15 0.02 0.04 0.05 lo-1* 0.03 0.07 0.15 0.03 0.09 0.05 l O - 3 0.11 0.07 0.02 0.04 0.08 0.22 l O " 2 0.01 0.08 0.23 0.03 0.12 0.22 i o - i 0.00 0.14 0.09 0.05 0.08 0.13 TOTAL 1.00 1.00 1.00 1.00 1.00 1.00 -207-TABLE XXII Predicted Distribution of Residual Tumor Burden after Surgery using the values of 6^  and C^jk i n Table XXI. Tumor Burden Premenopausal Postmenopausal Number of Po s i t i v e Nodes Number of P o s i t i v e Nodes 1-3 4+ 1-3 4+ [0] [10°, 10 1) [10 1, 10 2) [10 2, 10 3) [10 3, 101*) [10\ 10 5) [10 5, 10 6) [10 6, 10 7) [10 7, 10 8) [10 8, ») 0.688 0.310 0.121 0.072 0.208 0.114 0.009 0.068 0.029 0.041 0.019 0.028 0.032 0.041 0.088 0.012 0.068 0.140 0.039 0.072 0.130 0.083 0.088 0.070 0.022 0.107 0.174 0.001 0.019 0.106 0.736 0.344 0.154 0.048 0.106 0.079 0.012 0.030 0.041 0.017 0.020* 0.024 0.018 0.085 0.042 0.023 0.057 0.051 0.034 0.081 0.077 0.037 0.088 0.194 0.032 0.107 0.203 0.044 0.082 0.134 TOTAL 1.000 1.000 1.000 1.000 1.000 1.000 * Observed and predicted tumor burden d i s t r i b u t i o n d i f f e r by more than 0.01. -208-TABLE XXIII Predicted C u r a b i l i t y of Breast Cancer f o r Premenopausal Disease as a Function of a and the Increase i n C u r a b i l i t y Associated with an Extra (Neo-adjuvant) Cycle. Mutation Rate a P r o b a b i l i t y of cure f o r adjuvant Chemotherapy Number of Nodes Increase i n P r o b a b i l i t y of Cure with Neo-adjuvant Therapy Number of Nodes 0 1-3 4+ 0 1-3 4+ 10" 0.994 0.968 0.897 4 x l 0 _ 1 + t 22xl0 _ 1 + 70xl0" 4 10 ,-8 0.960 0.872 0.736 16xl0 - l t 43xl0 _ l t 47xl0 _ l t 10 ,-7 0.891 0.773 0.616 15xl0 _ l t 22xl0 _ l t 29xl0 - l t 10 ,-6 0.851 0.694 0.475 4x l 0 - l + 1 3 x l 0 - 4 2 6 x l 0 - 4 10 -5 0.828 0.631 0.347 4 x l 0 _ 1 + 8 x l 0 - l t 15xl0 _ t t 10" 0.791 0.594 0.279 5xl0~k 4 x l 0 _ , + 5xl0"4 10 ,-3 0.763 0.545 0.246 2x l 0 _ l t 7 x l 0 _ l t 4 x l 0 _ l t 10" 0.742 0.447 0.189 3xl0~4 14xl0 - l t 7xl0 _ i t 10 -1 0.693 0.319 0.125 3x10-4 6 x l 0 _ I t 3 x l 0 _ l t Observed when no chemotherapy 0.692 0.309 0.120 -209-TABLE XXIV Predicted C u r a b i l i t y of Breast Cancer f o r Postmenopausal Disease as a Function of a and the Increase i n C u r a b i l i t y Associated with an Extra (Neo-adjuvant) Cycle. Mutation Rate a P r o b a b i l i t y of cure f o r adjuvant Chemotherapy Number of Nodes Increase i n P r o b a b i l i t y of Cure with Neo-adjuvant Therapy Number of Nodes 0 1-3 4+ 0 1-3 4+ 10 -9 0.965 0.924 0.872 19xl0 - l t 40xl0 _ l t 68xl0 _ l t 10 ,-8 0.923 0.812 0.660 1 2 X 1 0 - 1 1 SSxlO" 4 65xl0- 4 10 -7 0.882 0.708 0.470 8 x l 0 _ , + 19X10"1* 30xl0 _ 1 + 10 ,-6 0.848 0.625 0.379 4 x l 0 _ 1 + l l x l O - 4 l O x l O - 4 10 -5 0.824 0.551 0.325 2xl0~h 9X10"4 6 x l 0 - 1 + 10" 0.805 0.490 0.287 2x10-^ telO'** 3 x l 0 _ t t 10" 0.790 0.462 0.252 l x l O - 4 3xl0~k 3xl0 _ 1* 10 -2 0.773 0.402 0.207 2x l 0 _ 1 + 6 x l 0 _ 4 4 x l 0 _ l t 10" 0.740 0.347 0.158 2xl0" 1 + 2xl0~h 2xl0" l t Observed when no chemotherapy 0.736 0.353 0.154 -210-6. CONCLUSION In the previous chapters we developed a model for the resistance of tumor c e l l s to chemotherapeutic agents. This model i s predicated on the assumption that tumor c e l l s spontaneously acquire resistance to drugs as these c e l l s grow. This model uses a growth model (developed i n Chapter 2) which assumes that tumors, i n analogy to normal ti s s u e s , are composed of three types of c e l l s : stem c e l l s , t r a n s i t i o n a l c e l l s and end c e l l s . The growth of these c e l l s i s described by a discrete-time Markov model with constant t r a n s i t i o n p r o b a b i l i t i e s for each c e l l . Using known resu l t s the asymptotic d i s t r i b u t i o n of the number of c e l l s at time t was derived. For unbounded r e a l i s a t i o n s of tumor growth, i t was shown that the asymptotic d i s t r i b u t i o n depends only on the number of stem c e l l s at time tQ« For a l l unbounded r e a l i s a t i o n s , having the same growth parameters, the proportion of each type of c e l l converges almost surely to a fixed l i m i t . It was argued that, for most parameter values which are l i k e l y to ar i s e i n p r a c t i c e , this asymptotic d i s t r i b u t i o n would approximate the true d i s t r i b u t i o n for tumors of c l i n i c a l dimensions. In th i s case, the number of c e l l s of each type can be estimated from a knowledge of the parameter values and the observed size of the tumor. In p a r t i c u l a r the number of stem c e l l s can be estimated and c u r a b i l i t y of the tumor reduces to consideration of the stem c e l l s alone. The preceding model of tumor growth must be regarded as approximate since i t takes no account of l o c a l and systemic conditions which influence growth. Furthermore, the assumption that c e l l s grow independently must be considered a f i r s t approximation since interactions between c e l l s have been demonstrated i n a number of systems. -211-Further work i s needed to develop models describing the growth of tumors which preserve the disc r e t e nature of the process and incorporate the random nature of i n d i v i d u a l c e l l u l a r events. It i s u n l i k e l y that such models w i l l strongly influence the d i s t r i b u t i o n of r e s i s t a n t c e l l s unless there i s some, presently unrecognised, r e l a t i o n s h i p between parameters governing growth and those governing the development of resistance. In Chapter 3 a model was constructed for the development of stem c e l l resistance to a single drug. It was assumed that stem c e l l s behave independently and grow as a b i r t h and death process with f i x e d parameters. The p r o b a b i l i t y generating function of the number of se n s i t i v e and r e s i s t a n t stem c e l l s was derived for a tumor of known parameters that began with a single c e l l . It was shown that the mean proportion of r e s i s t a n t c e l l s increases i n time. Recursive r e l a t i o n s h i p s were developed for the c a l c u l a t i o n of the p r o b a b i l i t y generating function of the process a f t e r an a r b i t r a r y sequence of treatments. The formulation assumes that a l l c e l l s behave independently and that t h e i r i n t e r d i v i s i o n times are exponentially d i s t r i b u t e d with the same parameters. To model si t u a t i o n s where the growth rate of r e s i s t a n t and s e n s i t i v e c e l l s are d i f f e r e n t , i t could be necessary to use a model i n which this i s permitted: such a model has been described by Day [34]. Models which permit c e l l s to have i n t e r d i v i s i o n times which are not exponentially d i s t r i b u t e d are of i n t e r e s t . However these models w i l l generally not have the simple Markov structure of the one used here and t h e i r development w i l l be more complicated. Using the model developed i n Chapter 3 i t was shown that the best strategy for maximizing the p r o b a b i l i t y of cure for a given t o t a l dosage, -212-D, of a drug over a period [t^,") i s to give the whole dose at time t]_. Therapies which best approximate t h i s strategy ( i n r e a l systems) have previously been recommended, as a re s u l t of empirical research, on the basis that they maximise P{RQ(°°)=0} . In p a r t i c u l a r the knowledge that such a dosage schedule also maximizes P{R^(«>)=0} mandates i t s use (or the c l i n i c a l l y f e a s i b l e regimen which best "approximates" i t ) . This may be of p a r t i c u l a r importance since a number of d i f f e r e n t regimens may have s i m i l a r values for P{RQ(°°)=0} but divergent values for P{R^(»)=0} whereas the reverse i s not true (since RQ(°°)>0 implies R^(°>)>0). A ce n t r a l problem a r i s i n g i n the analysis of spontaneous tumors i s the s p e c i f i c a t i o n of the age of the tumor when f i r s t seen. Coupled to this i s the fact that c e r t a i n r e a l i s a t i o n s of the growth model have zero stem c e l l s at t=» and should not be included i n the consideration of large tumors. Three possible approaches were developed to address these problems: 1) Delete sample paths where N(t)-K) and choose t" so that the d i s t r i b u t i o n {N(t")|N(»)>0} approximates that observed, 2) Approximate the d i s t r i b u t i o n {Ri(t)|N(t)} by the d i s t r i b u t i o n {R^(t)|Rg(t)} over the early period of growth of the tumor, assume that the subsequent growth of R Q c e l l s i s deterministic and derive the r e s u l t i n g d i s t r i b u t i o n of R^(t*) for some observed Rg(t*), 3) Assume that tumors are i n i t i a t e d uniformly i n time and then c a l c u l a t e the r e s u l t i n g d i s t r i b u t i o n of r e s i s t a n t c e l l s for a tumor d i s t r i b u t i o n at diagnosis of a p a r t i c u l a r prescribed form. Each of these approaches represent solutions to d i f f e r e n t problems -213-and as such are generally not d i r e c t l y comparable to one another. Each case i s of use for a p a r t i c u l a r s i t u a t i o n . In terms of the model developed, the f i r s t two solutions can be generalized by redefining the concept of the size at diagnosis. One approach i s to define the c r i t i c a l tumor burden to have a d i s t r i b u t i o n across i n d i v i d u a l s (with that tumor) and assume that diagnosis w i l l occur when the size of the tumor f i r s t exceeds the c r i t i c i a l size i n that i n d i v i d u a l . This would require the consideration of f i r s t passage times and would be quite complex. The t h i r d approach can be generalised i n several d i r e c t i o n s . The r e s u l t i n g d i s t r i b u t i o n of r e s i s t a n t c e l l s can be examined for a va r i e t y of mean incidence functions, u(t), which do not have the simple form ( i . e . constant) assumed i n Chapter 3. Possible forms of t h i s function are ava i l a b l e from the mathematical modelling of carcinogenesis [15]. In such cases the modelling of resistance i s u n l i k e l y to y i e l d simple expressions for the p r o b a b i l i t y generating function and numerical evaluation w i l l be necessary. The use of incidence functions of t h i s type w i l l permit the examination of the d i s t r i b u t i o n of r e s i s t a n t c e l l s as a function of the age of the subject. In advance i t does not seem l i k e l y that a strong r e l a t i o n s h i p w i l l e x i s t , however, i t i s worthy of exploration. The major conclusion from the analysis of the three approaches i s that the quantitative d e s c r i p t i o n of resistance depends upon the de s c r i p t i o n of the system under consideration and that attention must be paid to the p a r t i c u l a r experimental s i t u a t i o n . However, q u a l i t a t i v e l y the systems behave s i m i l a r l y and one or other of the approaches presented i s l i k e l y to be of use i n most s i t u a t i o n s . In the l a s t section of Chapter 3 we introduce the concept of -214-i n t r i n s i c v a r i a b i l i t y i n the mutation. There i s l i t t l e d i r e c t evidence for such v a r i a b i l i t y however given the experimental complexity involved i n t e s t i n g for such v a r i a t i o n we analysed i t s e f f e c t assuming that the appropriate parameters to follow a beta d i s t r i b u t i o n . I t was shown that v a r i a b i l i t y i n the mutation rates a f f e c t the form of the p r o b a b i l i t y of cure and thus i t may be possible to i d e n t i f y this phenomenon i n experimental systems. This phenomena was examined i n Chapter 5 for the experimental data on the L1210 leukemia treated with Cyclophosphamide and Arabinosylcytosine. I n i t i a l l y a model was f i t where a l l c e l l s were considered s e n s i t i v e and the logarithm of the p r o b a b i l i t y of c e l l s u r v i v a l , a f t e r treatment, was proportional to the dose used. This model did not f i t the data well for either drug. Generalising t h i s model to permit the existence of r e s i s t a n t c e l l s considerably improved the f i t to the data for each drug. Allowing the mutation rates to vary improved the f i t of the model for the data on treatment with Cyclophosphamide, but not for Arabinosylcytosine. In both cases there s t i l l remained unexplained v a r i a t i o n . These considerations apply only to a single well behaved tumor system treated with two drugs. It i s quite possible that spontaneous tumors may have more variable mutation rates. In p a r t i c u l a r , we have analysed data on a single tumor, the L1210 leukemia, and we cannot generalise r e s u l t s from a p a r t i c u l a r leukemia to a l l leukemias ( i n the same animal). To determine whether v a r i a t i o n i n mutation rates exists i s necessary to compare estimates of the mutation rates for a v a r i e t y of experimental tumors of the same type. In addition, the data used i n the preceding analysis did not include the cause of death (whether due to r e s i s t a n t or s e n s i t i v e c e l l s ) . The -215-analysis of s i m i l a r data with cause of death information would allow more accurate determination of mutation and pharmacokinetic parameters. Such an analysis may also be useful i n determining the source of the r e s i d u a l v a r i a t i o n unexplained by the present model. Further analysis of such data i s desirable since the concepts developed from such experiments are used i n the construction of protocols for the treatment of human cancer. In Chapter 4 a model was developed for resistance to two drugs. Expressions were developed which enabled the j o i n t p r o b a b i l i t y generating function of the number of stem c e l l s to be calculated for an a r b i t r a r y treatment regimen. Although not e x p l i c i t l y d e t a i l e d , the e f f e c t s on the p r o b a b i l i t y of cure of the timing and dosage of a single drug are seen to carry over to t h i s s i t u a t i o n . However, the optimum use of two drugs remains an unresolved problem. The major problem i s that there i s no common scale of measurement for the e f f e c t s of drugs on normal t i s s u e . There i s a need for models of t o x i c i t y since the construction of protocols c r i t i c a l l y depends on them (both i n theory and p r a c t i c e ) . However, given that such a dosage and timing schedule have been described then i t i s possible to examine how the ordering of treatments may e f f e c t the p r o b a b i l i t y of cure. In p a r t i c u l a r , i t was shown that i f the treatments are "equivalent" ( i . e . each has the same e f f e c t on s e n s i t i v e c e l l s and c e l l s r e s i s t a n t to i t and are given at the same times) then the expected number of stem c e l l s i s minimized by giving these drugs i n an al t e r n a t i n g fashion. It was also argued that, i n most cases of p r a c t i c a l i n t e r e s t , the p r o b a b i l i t y of cure w i l l also be maximized by such st r a t e g i e s . Although equivalence may not usually a r i s e i n p r a c t i c e , i t s examination leads to the conclusion that treatments must be interspersed -216-to maximize the p r o b a b i l i t y of cure. For non-equivalent drugs the pattern of interspersement w i l l depend on a number of parameters which r e f l e c t the effectiveness of the drugs i n each of the stem c e l l sub-compartments. This problem has been extensively studied by Day [34] who has examined the r e l a t i o n s h i p between the tumor and drug parameters and the pattern of a p p l i c a t i o n of each drug i n the "optimal" s t r a t e g i e s . In cases where the parameters are known the optimal strategy may be determined. In cases where some parameters are not accurately known i t seems reasonable to give these parameters a d i s t r i b u t i o n r e f l e c t i n g the p r e c i s i o n with which they are known. Optimal strategies may then be determined for t h i s system. Such a c a l c u l a t i o n was presented for equivalent agents (using the generalized d e f i n i t i o n ) i n Chapter 5 where the mutation rates follow a d i s t r i b u t i o n . It was shown ( i n Section 4.6) that the optimal strategy (for E[N(t)]) i s the same as for the f i x e d mutation rate case (that i s , the drugs should be alternated). However, In other cases the optimal strategy may depend on the amount of v a r i a b i l i t y (or lack of p r e c i s i o n ) . This problem i s worthy of further exploration. The generalization of this model to more than two drugs represents a considerable technical problem. This s i t u a t i o n i s probably best approached using a model s i m i l a r to that developed by Day [34]. As i n the example of two drugs, an unresolved question i s the way i n which drugs may be combined. This requires a knowledge of t h e i r j o i n t e f f e c t on t o x i c i t y . Chapter 5 presented applications of the theory developed i n preceding chapters to experimental and c l i n i c a l tumors. In addition to -217-those s i t u a t i o n s already discussed, the model was applied to the neo-adjuvant chemotherapy of breast cancer. Using an ad-hoc model for the e f f e c t of surgery on the d i s t r i b u t i o n of stem c e l l s , we assessed the influence of an extra neo-adjuvant cycle of chemotherapy on the p r o b a b i l i t y of cure. Chemotherapy was assumed to consist of a s i n g l e drug with unspecified pharmacokinetic and mutation parameters. Generally i t i s found that the a p p l i c a t i o n of the extra neo-adjuvant cycle had l i t t l e e f f e c t on the p r o b a b i l i t y that the tumor i s cured. This lack of improvement r e s u l t s mainly from the high v a r i a b i l i t y i n the p o s t - s u r g i c a l tumor burden as estimated by Skipper [38]. In situa t i o n s where the v a r i a t i o n i n burden i s much smaller, the e f f e c t of neo-adjuvant therapy can be expected to be greater. However, i t should be emphasized that t h i s conclusion only applies to the development of spontaneous drug r e s i s t a n t c e l l s and i f other mechanisms of tumor s e n s i t i v i t y are influenced by this early cycle of therapy, then the resultant e f f e c t may be considerably l a r g e r . Of more general i n t e r e s t , this analysis i l l u s t r a t e s the s e n s i t i v i t y of th i s model to v a r i a t i o n i n the o v e r a l l stem c e l l burden. This i s not s u r p r i s i n g , at least i n retrospect, but i t does i l l u s t r a t e that the quantitative e f f e c t of therapeutic strategies determined for animal models may not translate simply to human disease where the v a r i a t i o n i n tumor burden at treatment i s much greater. Further work i n modelling human disease i s desirable, since an understanding of the parameters which influence the c l i n i c a l therapy of cancer i s the ultimate objective of such research. The greatest obstacle to such research i s the r e l a t i v e paucity of quantitative information a v a i l a b l e for human disease. At present a most f r u i t f u l approach would -218-seem to be to model c l i n i c a l systems where the parameters have considerable v a r i a t i o n which may be taken to r e f l e c t the heterogeneity or imprecision i n th e i r s p e c i f i c a t i o n . -219-BIBLIOGRAPHY [I] Caiman K.C., Smyth J.F. and T a t t e r s a l l M.H.N. Basic P r i n c i p l e s of Cancer Chemotherapy. MacMillan Press, London (1980). [2] Chabner B. Pharmacologic P r i n c i p l e s of Cancer Treatment. pl91, W. B. Saunders, Philedelphia (1982). [3] Skipper H.S., Schabel F.M. and Wilcox W.S. Experimental Evaluation of anti-cancer agents. XII. On the C r i t e r i a and Kinetics Associated with C u r a b i l i t y of Experimental Leukemia. Cancer Chemotherapy Reports 35:1-111 (1964). [4] Goldie J.H. and Coldman A.J. A Mathematic Model for Relating the Drug S e n s i t i v i t y of Tumors to Their Spontaneous Mutation Rate. Cancer Treatment Reports 63:1727-1733 (1979). [5] Skipper H.E. Some Thoughts Regarding a Recent Publication by Goldie and Coldman E n t i t l e d ' A Mathematic Model for Relating the Drug S e n s i t i v i t y of Tumors to t h e i r Spontaneous Mutation Rate.' Booklet #9, Southern Research I n s t i t u t e , Birmingham (1980). [6] Luria S.E. and Delbruck M. Mutations of Bacteria from Virus S e n s i t i v i t y to Virus Resistance. Genetics 28:491-511 (1943). [7] Lea D.E. and Coulson C.A. The D i s t r i b u t i o n of the Number of Mutants i n B a c t e r i a l Populations. Journal of Genetics 49:264-285 (1948). [8] B a r t l e t t M.S. mentioned i n following reference. [9] Armitage P. The S t a t i s t i c a l Theory of B a c t e r i a l Populations Subject to Mutation. Journal of the Royal S t a t i s t i c a l Society Series B 14:l-33 (1952). [10] Kendall D.G. Birth-and-Death Processes, and the Theory of Carcinogenisis. Biometrika 47 :13-21 (1960). [II] Crump K.S. and Hoel D.G. Mathematical Models for Estimating Mutation Rates i n C e l l Populations. Biometrika 61:237-252 (1974). [12] Tan W.Y. On the D i s t r i b u t i o n of the Number of Mutants at the Hypoxanthine-Guanine-Phosphoribosal-Transferase Locus i n Chinese Hamster Ovary C e l l s . Mathematical Biosciences 67:175-192 (1983). [13] Athreya K.B. and Ney P.E. Branching Processes. Springer Verlag, New York (1972). [14] Mackillop J ., Ciampi A., T i l l J.E. and Buick R.N. A Stem C e l l Model of Human Tumor Growth: Implications for Tumor C e l l Clonogenic Assays. Journal of the National Cancer I n s t i t u t e 70:9-16 (1983). -220-[15] Moolgavkar S.H. and Venzon D.J. Two Event Models for Carcinogenesis: Incidence Curves for Childhood and Adult Tumors. Mathematical Biosciences 47:55-77 (1979). [16] F e l l e r W. An Introduction to P r o b a b i l i t y Theory and i t s Applications. Volume 1, 2nd E d i t i o n . Wiley, New York (1957). [17] Mode C.J. Multitype Branching Processes. American E l s e v i e r , New York (1971). [18] Steel G.G., Growth Kinetics of Tumours, p 86 Clarendon Press, Oxford (1977). [19] Buick R.N., The C e l l Renewal Heirarchy i n Ovarian Cancer. In: Human Tumor Cloning, pp 3-13. Eds: Salmon S.E. and Trent J.M. (1984). [20] K a r l i n S. and Taylor H.M. A F i r s t Course i n Stochastic Processes. Academic Press, New York (1975). [21] Parzen E. Stochastic Processes. Holden Day, San Francisco (1962). [22] John F. P a r t i a l D i f f e r e n t i a l Equations. 4th E d i t i o n . Springer Verlag, New York (1982). [23] Skipper H.S., Schabel F.M. and Wilcox W.S. Experimental Evaluation of P o t e n t i a l Anti-Cancer Agents. XIV. Further Study of Certain Basic Concepts Underlying Chemotherapy of Leukemia. Cancer Chemotherapy Reports 4 5:5-28 (1975). [24] Bruce W.R., Meeker B.E. and Val e r i o t e F.A., Comparison of th S e n s i t i v i t y of Normal Hematopoietic and Transplanted Lymphoma Colony-Forming C e l l s to Chemotherapeutic Agents Administered In-Vivo. Journal of the National Cancer I n s t i t u t e 37:233-24 5 (1966). [25] Valagussa P., Bonadonna G. and Veronesi U. Patterns of Relapse and Survival i n Operable Breast Carcinoma with Po s i t i v e and Negative A x i l l a r y Nodes. Tumori 64:241-258 (1978). [26] Skipper H.S., Schabel F.M. and Lloyd H.H. Dose Response and Tumor C e l l Repopulation Rate i n Chemotherapeutic T r i a l s . In: Advances i n Cancer Chemotherapy, Volume I. Ed: A. Rozownki, pp 205-253, Marcel Dekker, New York (1979). [27] Coldman A.J., Goldie J.H. and Ng V. The E f f e c t of C e l l u l a r D i f f e r e n t i a t i o n on the Development of Permanent Drug Resistance. Mathematical Biosciences 74:177-198 (1985). [28] Schimke R.T. Gene Amplification, Drug resistance and Cancer. Cancer Research 44:1735-1742 (1984 ). -221-[29] Santoro A., Bonadonna G., Bonfanti V. and Valagussa P. Alt e r n a t i n g Drug Combinations i n the Treatment of Advanced Hodgkins Disease. New England Journal of Medicine 306:770-775 (1982). [30] Goldie J.H., Coldman A.J. and Gudauskas G. A ratio n a l e for the Use of Alternating Non-Crossresistant Chemotherapy. Cancer Treatment Reports 66:439-449 (1979). [31] Coldman A.J. and Goldie J.H. A Model for the Resistance of Tumor Ce l l s to Cancer Chemotherapeutic Agents. Mathematical Biosciences 65:291-307 (1983). [32] DeVita V.T. P r i n c i p l e s of Chemotherapy. In: Cancer, P r i n c i p l e s and Practice of Oncology, pl32-155. Eds DeVita V.T., Hellman S. and Rosenberg S.A. J. B. Lippincott Co, Philedelphia (1982). [33] Skipper H.E. Mammary 16/C. Are Experimental Neoplasms which Respond Temporarily and then Resume Growth During Treatment with a Combination of Non-Cross-Resistant Drugs, Resistant to a l l of the Individual Drugs i n the Combination? Booklet #19, Southern Research I n s t i t u t e , Birmingham (1984). [34] Day R. A Tumor Growth Model with Applications to Treatment P o l i c y and Protocol Choice. Ph.D t h e s i s . School of Public Health, Harvard University, Boston (1984). [35] Fisher B. Adjuvant Therapy for Breast Cancer: A b r i e f Overview of the NSABP Experience and Some Thoughts on Neo-adjuvant Chemotherapy. In: Pre-operative (Neo-adjuvant) Chemotherapy p54-68. Eds: Ragaz J. , Band P.R. and Goldie J.H. Springer-Verlag, New York (1986). [36] F r e i E., M i l l e r D., Clark J.R., Fa l l o n B.G. and Ervin T.J. C l i n i c a l and S c i e n t i f i c Considerations i n Pre-operative (Neo-adjuvant) Chemotherapy. In: Pre-operative (Neo-adjuvant) Chemotherapy p l - 5 . Eds: Ragaz J., Band P.R. and Goldie J.H. Springer-Verlag, New York (1986). [37] Ragaz J . Pre-operative (Neo-adjuvant) Chemotherapy for Breast Cancer: Outline of the B r i t i s h Columbia T r i a l . In: Pre-operative (Neo-adjuvant) Chemotherapy p69-78. Eds: Ragaz J . , Band P.R. and Goldie J.H. Springer-Verlag, New York (1986). [38] Skipper H.S. Repopulation Rates of Breast Cancer C e l l s a f t e r Mastectomy. Booklet #12, Southern Research I n s t i t u t e , Birmingham (1979). -222-INDEX OF NOTATION Index of f i r s t appearance of notation. Greek symbols are l i s t e d seperately. page a 99 a i , 1 2 9 9 a 1 2 99 a* 109 A 20 A* 132 b 32 B(t) 94 B Q 97 c 32 c* 14 9 C. 14 l C(t) 18 C ± ( t ) 16 C., ., 200 i jk C k(S(v)) ' 122 32 E i > k ( 0 121 g 123 g ± W3 g(n) 74 GR 19 h 127 h ± 14 3 I ( t ) 71 -223-L ( . ) page 181 m ±(t) M 37 19 n N(t) s e t ) p P p* p*(t) N p P^(i) PD PS 14 33 200 71 14 81 75 14 9 33 70 48 61 19 19 14 R ±(t) R(t) S(v) h U(t) 33 90 71 116 46 71 Y . ( t ) 71 w.(t) 37 -224-page a 32 aQ. ,Q . 9 0 i J 6 32 PQ i 5Q. 9 0 B(a;u?v) 85 Y 32 YQ.,Q. 9 0 I T YQ. ,q. 1 4 9 1 J 36 39 C ± ( s ; t ) 60 C j k ( - ) 200 e. 200 0(s) 76 X 60 l ± 15 H(t) 71 v 35 v. 94 1 0 94 I, 12 v 94 12 Us) 40 S i > Q ( s ) 101 TI(D) 4 0 *i,Q<D> 1 0 1 -225-page 67 43 4>(s 0,s 1;t) 35 4^(8;t) 98 $(s;t) 95 $ ~ ( s ; t ) 103 B 0 ~ <l'(s 0,s 1) 35 T(s;t) 98 

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