A CLASS OF GAME-THEORETIC AND FOKKER-PLANCK OPTIMAL CONTROL FRAMEWORKS IN COLON AND ESOPHAGEAL CANCER

Abstract

In this dissertation, we first present a new stochastic framework for parameter estimation and uncertainty quantification in colon cancer-induced immune responses. A stochastic process that captures the system's inherent randomness determines the dynamics of colon cancer. The stochastic framework is based on the Fokker-Planck equation, which represents the evolution of the probability density function corresponding to the stochastic process. We formulate an optimization problem that takes individual patient data with randomness present and solves it to obtain the unknown parameters corresponding to the individual tumor characteristics. Furthermore, we perform a sensitivity analysis of the optimal parameter set to identify the parameters that require control, thereby revealing the types of drugs suitable for treatment. Afterward, we introduce a differential game framework for assessing the protocols used in the administration of these drugs in colon cancer. In this specific setting, we discuss a new framework for a non-cooperative evolutionary game involving colon cancer and the oncologist. A novel mathematical model considering the dynamical progression of colon cancer, including resistance mechanisms and the various therapies delivered, forms the basis of this framework. To determine the optimal course of action for the patient, this model is used to formulate and compute two equilibrium game theoretic strategies: Stackelberg and Nash's equilibria. Our study concludes with a model of esophageal cancer signaling pathways and monoclonal antibodies engaged in a game of evolutionary competition. This structure illustrates the extent to which evolutionary game theory may provide novel, efficient methods for cancer therapy assessment. Within this framework, we analyze and calculate two game-theoretic techniques, Stackelberg and Nash's equilibria, to find the best possible result for the patient. Numerous numerical experiments are presented to prove the efficacy of the theoretical investigations.