ABC efflux transporters are a key factor leading to multidrug resistance in cancer. Overexpression of these transporters significantly decreases the efficacy of anti-cancer drugs. Along with selection and induction, drug resistance may be trans- ferred between cells, which is the focus of this dissertaion. Specifically, we consider the intercellular transfer of P-glycoprotein (P-gp), a well-known ABC transporter that was shown to confer resistance to many common chemotherapeutic drugs.

In a recent paper, Dura ́n et al. studied the dynamics of mixed cultures of resistant and sensitive NCI-H460 (human non-small cell lung cancer) cell lines [1]. As expected, the experimental data showed a gradual increase in the percentage of resistance cells and a decrease in the percentage of sensitive cells. The experimental work was accompanied with a mathematical model that assumed P-gp transfer from resistant cells to sensitive cells, rendering them temporarily resistant. The mathematical model provided a reasonable fit to the experimental data.

In this dissertation we develop three new mathematical model for the transfer of drug resistance between cancer cells. Our first model is based on incorporating a resistance phenotype into a model of cancer growth [2]. The resulting model for P-gp transfer, written as a system of integro-differential equations, follows the dynamics of proliferating, quiescent, and apoptotic cells, with a varying resistance phenotype. We show that this model provides a good match to the dynamics of the experimental data of [1]. The mathematical model further suggests that resistant cancer cells have a slower division rate than the sensitive cells.

Our second model is a reaction-diffusion model with sensitive, resistant, and temporarily resistant cancer cells occupying a 2-dimensional space. We use this model as another extension of [1]. We show that this model, with competition and diffusion in space, provides an even better fit to the experimental data [1]. We incorporate a cytotoxic drug and study the effects of varying treatment protocols on the size and makeup of the tumor. We show that constant infusion leads to a small but highly resistant tumor, while small doses do not do enough to control the overall growth of the tumor.

Our final model extends [3], an integro-differential equation with resistance modeled as a continuous variable and a Boltzmann type integral describing the transfer of P-gp expression. We again extend the model into a 2-dimensional spatial domain and incorporate competition inhibited growth. The resulting model, written as a single partial differential equation, shows that over time the resistance transfer leads to a uniform distribution of resistance levels, which is consisten with the results of [3]. We include a cytotoxic agent and determine that, as with our second model, it alone cannot successfully eradicate the tumor. We briefly present a second extension wherein we include two distinct transfer rules. We show that there is no qualitative difference between the single transfer rule and the two-transfer rule model.