Mathematical modeling of drug resistance and cancer stem cells dynamics

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In this dissertation we consider the dynamics of drug resistance in cancer and the related issue of the dynamics of cancer stem cells. Our focus is only on resistance which is caused by random genetic point mutations. A very simple system of ordinary differential equations allows us to obtain results that are comparable to those found in the literature with one important difference. We show that the amount of resistance that is generated before the beginning of the treatment, and which is present at some given time afterward, always depends on the turnover rate, no matter how many drugs are used. Previous work in the literature indicated no dependence on the turnover rate in the single drug case while a strong dependence in the multi-drug case.

We develop a new methodology in order to derive an estimate of the probability of developing resistance to drugs by the time a tumor is diagnosed and the expected number of drug-resistant cells found at detection if resistance is present at detection. Our modeling methodology may be seen as more general than previous approaches,

in the sense that at least for the wild-type population we make assumptions only on their averaged behavior (no Markov property for example). Importantly, the heterogeneity of the cancer population is taken into account. Moreover, in the case of chronic myeloid leukemia (CML), which is a cancer of the white blood cells, we are able to infer the preferred mode of division of the hematopoietic cancer stem cells, predicting a large shift from asymmetric division to symmetric renewal. We extend our results by relaxing the assumption on the average growth of the tumor, thus going beyond the standard exponential case, and showing that our results may be a good approximation also for much more general forms of tumor growth models.

Finally, after reviewing the basic modeling assumptions and main results found in the mathematical modeling literature on chronic myeloid leukemia (CML), we formulate a new hypothesis on the effects that the drug Imatinib has on leukemic stem cells. Based on this hypothesis, we obtain new insights on the dynamics of the development of drug resistance in CML.