| Summary: | The mechanism of resistance of tumor cells to chemotherapeutic agents is explored using probabilistic methods where it is assumed that resistant cells arise spontaneously with a defined frequency. The resistance process is embedded in a discrete time Markov branching process which models the growth of the tumor and contains three seperate cell types: stem, transitional and end cells. Using the asymptotic properties of such models it is shown that the proportion of each type of cell converge to constants almost surely. It is shown that the parameters relating to stem cell behaviour determine the asymptotic behaviour of the system. It is argued that for biologically likely parameter values, cure of the tumor will occur if, and only if, all stem cells are eliminated.
A model is developed for the acquisition of resistance by stem cells to a single drug. Probability generating functions are derived which describe the behaviour of the process after an arbitrary sequence of drug treatments. The probability of cure, defined as the probability of ultimate extinction of the stem cell compartment, is characterised as the central quantity reflecting the success of therapeutic intervention. Expressions for this function are derived for a number of experimental situations. The effects of variation in the parameter values are examined.
The model is extended to the case where two anticancer drugs are available and formulae for the probability of cure are developed. The problem of therapeutic scheduling is examined and under situations where drugs are of "equal" effectiveness, but may not be given together, it is shown that the mean number of tumor cells is minimised by sequential alternation of the drugs.
The models are applied to data collected on the L1210 leukemia treated by the drugs Cyclophosphamide and Arabinosylcytosine. In both cases the analysis of the data provide evidence that resistant cells arise spontaneously with a frequency of approximately 10⁻⁷ per division. When applied to human breast cancer, the model indicates that neoadjuvant
therapy is unlikely to greatly influence the likelihood that the patient will die from the growth of drug-resistant cells. === Science, Faculty of === Statistics, Department of === Graduate
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