Mathematical modeling of drug resistance and cancer stem cells dynamics

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

Why tyrosine kinase inhibitor resistance is common in advanced gastrointestinal stromal tumors

Background: Most patients with advanced gastrointestinal stromal tumors (GIST) develop drug resistance to tyrosine kinase inhibitors (TKIs) within two years of starting therapy, whereas most chronic myeloid leukemia (CML) patients in chronic phase still exhibit disease control after a decade on therapy. This article aims to explain this divergence in long term outcomes. Methods and results: By combining clinical and experimental observations with mathematical formulas we estimate that, in advanced GIST, the genetic changes responsible for resistance are generally already present at disease detection. Conclusion: This result has relevant clinical implications by providing support for the exploration of combination therapies

The (not so) immortal strand hypothesis

Background: Non-random segregation of DNA strands during stem cell replication has been proposed as a mechanism to minimize accumulated genetic errors in stem cells of rapidly dividing tissues. According to this hypothesis, an “immortal” DNA strand is passed to the stem cell daughter and not the more differentiated cell, keeping the stem cell lineage replication error-free. After it was introduced, experimental evidence both in favor and against the hypothesis has been presented. Principal findings: Using a novel methodology that utilizes cancer sequencing data we are able to estimate the rate of accumulation of mutations in healthy stem cells of the colon, blood and head and neck tissues. We find that in these tissues mutations in stem cells accumulate at rates strikingly similar to those expected without the protection from the immortal strand mechanism. Significance: Utilizing an approach that is fundamentally different from previous efforts to confirm or refute the immortal strand hypothesis, we provide evidence against non-random segregation of DNA during stem cell replication. Our results strongly suggest that parental DNA is passed randomly to stem cell daughters and provides new insight into the mechanism of DNA replication in stem cells

Evaluating cancer etiology and risk with a mathematical model of tumor evolution

International audienceRecent evidence arising from DNA sequencing of healthy human tissues has clearly indicated that our organs accumulate a relevant number of somatic mutations due to normal endogenous mutational processes, in addition to those caused by environmental factors. A deeper understanding of the evolution of this endogenous mutational load is critical for understanding what causes cancer. Here we present a mathematical model of tumor evolution that is able to predict the expected number of endogenous somatic mutations present in various tissue types of a patient at a given age. These predictions are then compared to those observed in patients. We also obtain an improved fitting of the variation in cancer incidence across cancer types, showing that the endogenous mutational processes can explain 4/5 of the variation in cancer risk. Overall, these results offer key insights into cancer etiology, by providing further evidence for the major role these endogenous processes play in cancer

Dependence of the slope on the degree of heterogeneity in the number of required driver genes across tissues.

<p>(A) Regression line (red), for predicted cancer risk if it is assumed that all cancer types require exactly the same number of driver gene mutations, n, as assumed by Little et al. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0175535#pone.0175535.ref001" target="_blank">1</a>]. The slope of the regression line is 0.5. (B) Regression line (red), for predicted cancer risk if it is assumed that different cancer types require different numbers n of driver gene mutations (two in bone, ovarian, thyroid, gallbladder, brain, and lung cancer, and three drivers in all other cancer types.) To make the data points homogeneous in n, in order to regress on them, their risk was modified according to an Armitage and Doll model, for simplicity, with a mutation rate u = 5x10^-7 per gene per cell division. All other variables were identical to those in A. The slope of the regression line is 1.8.</p