Mathematics Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2793
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Item Dynamics, Networks, and Information: Methods for Nonlinear Interactions in Biological Systems(2021) Milzman, Jesse; Levy, Doron; Lyzinski, Vince; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this dissertation, we investigate complex, non-linear interactions in biological systems.This work is presented as two independent projects. The mathematics and biology in each differ, yet there is a unity in that both frameworks are interested in biological responses that cannot be reduced to linear causal chains, nor can they be expressed as an accumulation of binary interactions. In the first part of this dissertation, we use mathematical modeling to study tumor-immune dynamics at the cellular scale.Recent work suggests that LSD1 inhibition reduces tumor growth, increases T cell tumor infiltration, and complements PD1/PDL1 checkpoint inhibitor therapy. In order to elucidate the immunogenic effects of LSD1 inhibition, we create a delay differential equation model of tumor growth under the influence of the adaptive immune response in order to investigate the anti-tumor cytotoxicity of LSD1-mediated T cell dynamics. We fit our model to the B16 mouse model data from Sheng et al. [DOI:10.1016/j.cell.2018.05.052] Our results suggest that the immunogenic effect of LSD1 inhibition accelerates anti-tumor cytoxicity. However, cytotoxicity does not seem to account for the slower growth observed in LSD1 inhibited tumors, despite evidence suggesting immune-mediation of this effect. In the second part, we consider the partial information decomposition (PID) of response information within networks of interacting nodes, inspired by biomolecular networks.We specifically study the potential of PID synergy as a tool for network inference and edge nomination. We conduct both numeric and analytic investigations of the $\Imin$ and $\Ipm$ PIDs, from [arXiv:1004.2515] and [DOI:10.3390/e20040297], respectively. We find that the $I_\text{PM}$ synergy suffers from issues of non-specificity, while $I_{\text{min}}$ synergy is specific but somewhat insensitive. In the course of our work, we extend the $I_\text{PM}$ and $I_{\text{min}}$ PIDs to continuous variables for a general class of noise-free trivariate systems. The $I_\text{PM}$ PID does not respect conditional independence, while$I_{\text{min}}$ does, as demonstrated through asymptotic analysis of linear and non-linear interaction kernels. The technical results of this chapter relate the analytic and information-theoretic properties of our interactions, by expressing the continuous PID of noise-free interactions in terms of the partial derivatives of the interaction kernel.Item Mathematical Models of Tumor Heterogeneity and Drug Resistance(2015) Greene, James; Levy, Doron; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this dissertation we develop mathematical models of tumor heterogeneity and drug resistance in cancer chemotherapy. Resistance to chemotherapy is one of the major causes of the failure of cancer treatment. Furthermore, recent experimental evidence suggests that drug resistance is a complex biological phenomena, with many influences that interact nonlinearly. Here we study the influence of such heterogeneity on treatment outcomes, both in general frameworks and under specific mechanisms. We begin by developing a mathematical framework for describing multi-drug resistance to cancer. Heterogeneity is reflected by a continuous parameter, which can either describe a single resistance mechanism (such as the expression of P-gp in the cellular membrane) or can account for the cumulative effect of several mechanisms and factors. The model is written as a system of integro-differential equations, structured by the continuous ``trait," and includes density effects as well as mutations. We study the limiting behavior of the model, both analytically and numerically, and apply it to study treatment protocols. We next study a specific mechanism of tumor heterogeneity and its influence on cell growth: the cell-cycle. We derive two novel mathematical models, a stochastic agent-based model and an integro-differential equation model, each of which describes the growth of cancer cells as a dynamic transition between proliferative and quiescent states. By examining the role all parameters play in the evolution of intrinsic tumor heterogeneity, and the sensitivity of the population growth to parameter values, we show that the cell-cycle length has the most significant effect on the growth dynamics. In addition, we demonstrate that the agent-based model can be approximated well by the more computationally efficient integro-differential equations, when the number of cells is large. The model is closely tied to experimental data of cell growth, and includes a novel implementation of transition rates as a function of global density. Finally, we extend the model of cell-cycle heterogeneity to include spatial variables. Cells are modeled as soft spheres and exhibit attraction/repulsion/random forces. A fundamental hypothesis is that cell-cycle length increases with local density, thus producing a distribution of observed division lengths. Apoptosis occurs primarily through an extended period of unsuccessful proliferation, and the explicit mechanism of the drug (Paclitaxel) is modeled as an increase in cell-cycle duration. We show that the distribution of cell-cycle lengths is highly time-dependent, with close time-averaged agreement with the distribution used in the previous work. Furthermore, survival curves are calculated and shown to qualitatively agree with experimental data in different densities and geometries, thus relating the cellular microenvironment to drug resistance.Item Mathematical modeling of drug resistance and cancer stem cells dynamics(2010) Tomasetti, Cristian; Levy, Doron; Dolgopyat, Dmitry; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)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.