Graduation Term

Spring 2026

Degree Name

Master of Science (MS)

Department

Department of Mathematics

Committee Chair

Olcay Akman

Committee Member

Mehdi Karimi

Committee Member

Thomas Hammond

Abstract

Multi-drug resistance is an evolutionary process in which treatment eliminates sensitive cells, allowing resistant clones to dominate. This thesis investigates this process using a framework integrating population dynamics, evolutionary game theory, and optimal control theory. We develop a two-population logistic growth model describing competition between drug-sensitive and drug-resistant cells under treatment, construct dose-dependent payoff matrices and replicator dynamics to characterize evolutionary competition, and derive a critical drug level Dcrit = (rS - rR)/(dS - dR) at which resistant cells gain a fitness advantage. An optimal control problem is formulated via Pontryagin's Maximum Principle to identify schedules balancing tumor suppression with resistance management. Numerical simulations comparing four strategies demonstrate that switching therapy reduces resistant enrichment by approximately 46-fold relative to continuous treatment, while using only half the cumulative drug exposure.

The framework is applied to longitudinal PSA data from three patients in the adaptive abiraterone trial of Zhang et al., yielding patient-specific parameter estimates, payoff matrices, critical drug thresholds Dcrit = 0.075 - 0.098, replicator dynamics trajectories, optimal control schedules, eigenvalue bifurcation analysis, and a discrete-time Markov chain model. The clinical dose (D = 1) exceeds each patient's Dcrit by roughly an order of magnitude, explaining quantitatively why maximum-dose therapy accelerates resistance. Notably, the standard Pontryagin's formulation prescribes aggressive continuous treatment - precisely the strategy evolutionary theory warns against - motivating evolutionary-aware cost functionals that explicitly penalize the resistant fraction. Overall, this work demonstrates that evolutionary game theory combined with optimal control and stochastic modeling provides a systematic framework for designing sustainable treatment protocols that delay resistance.

Access Type

Thesis-Open Access

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