Lindsay, Danika2018-09-212018-09-212018-07https://hdl.handle.net/11299/200305University of Minnesota Ph.D. dissertation.July 2018. Major: Mathematics. Advisor: Jasmine Foo. 1 computer file (PDF); x, 122 pages.The evolution of resistance to therapy remains a significant challenge to the clinical treatment of cancer. As a tumor evolves, new genetic variants possessing a fitness advantage over normal cancer cells may be produced, thus leading to the development of resistance. Because the underlying biological processes driving this evolution are inherently random, they may be modeled using stochastic processes. This thesis consists of two main projects in which branching process models are employed to study the evolution of resistance to cancer therapy. In the first project, we develop a stochastic model of a non-small cell lung tumor undergoing treatment with a combination of two drugs. One drug is the current standard therapy used in the clinic to treat this disease, which has proven to be ineffective in the long term due to the inevitable development of resistance. The other is a novel drug specifically aimed at targeting oxygen-deprived regions in tumors. Using this model, we show that a therapeutic regimen in which both drugs are given may greatly improve patient outcomes over the current standard monotherapy. We also use our model to predict the optimal combination treatment protocol. The goal of the second project is to understand the impact of different resistance mechanisms on tumor recurrence. We define two separate branching process models to compare the case in which resistance arises via a single gene mutation with the case in which resistance arises via a gradual gene amplification process. We prove law of large numbers results regarding the convergence of the stochastic tumor recurrence times in both models and use these results to compare the two resistance mechanisms.enThesis or Dissertation