| One of the biggest challenges in treatment of cancer is the emergence of drug-resistant mutants during therapy. Combining several drugs may increase chances of treatment success, by reducing the probability of production of fully-resistant cells. Sometimes, one mutation can confer resistance to more than one drug. For example, there are currently three drugs available for treating Chronic Myeloid Leukemia (CML); however, the mutant T315I confers resistance to all three drugs; this is known as cross-resistance. We develop a stochastic model to study various treatments regimes, such as cyclical and combination treatment in presence and absence of cross-resistance. The microevolution of tumor is described by means of a linear birth-death process with mutations. The first-order linear partial differential equation for the probability generating function can be solved using the method of characteristics. The coefficients in the resulting equations are time-dependent quantities which reflect different treatment strategies. Also, ordinary differential equations for the average numbers of mutants of different types are formulated and analyzed. Our studies are divided into three chapters. (i) In the first chapter, we develop the stochastic model for drug resistance in cancer with the inclusion of cross-resistance and show that there is no advantage in combining more than two cross-resistant drugs. (ii) In the second chapter, we extend Roger Day's work on two-drug cyclic treatment, and in particular, we revisit the famous "worst drug rule", by including cross-resistance and using a more systematic methodology. We find that in most circumstances, it is advantageous to start treatment with the better drug, but use the worse drug for longer durations. (iii) In chapter 3, we apply the theory to existing, in vitro, data on mutant outgrowth in CML cancer cells. We find that combinations of different numbers of drugs with specific concentrations can give similar treatment outcomes. From this, we produce a counting and sorting technique that may be performed on any future inhibitors to find the best treatment options, maximizing the success rate of the treatment while minimizing the number and concentrations of the drugs. |
