ORCID Identifier(s)

0000-0001-5280-093X

Graduation Semester and Year

2023

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Gaik Ambartsoumian

Second Advisor

Barbara Shipman

Third Advisor

Souvik Roy

Abstract

In this work, we present an optimal control approach for the assessment of treatments in prostate cancer. For this purpose, we use two different approaches, based on differential equations, to model the dynamics of prostate cancer. For the first approach, we use a system of ordinary differential equations (ODE) that model androgen-dependent and independent prostate cancer cell mechanisms. Given some synthetic patient data, we then performed a parameter estimation process by formulating an optimization problem to obtain the coefficients in this model. A second optimal control problem was formulated to obtain optimal androgen suppression therapies. A theoretical analysis of both optimization problems was performed to prove the existence of the minimizers. The numerical implementation of the optimization problems was done using a non-linear conjugate gradient method. Several numerical experiments demonstrate the accuracy and robustness of our proposed ODE framework. The second approach involved extending a reduced version of the aforementioned ODE model to a Liouville partial differential equation model that captures more variabilities and randomness involved in clinical trials and formulating the corresponding parameter estimation and optimal control problems. The numerical implementation was done using a second-order spatially accurate finite volume scheme. First, the comparison of the ODE and the Liouville framework results of parameter estimation demonstrated that the Liouville modeling framework is more accurate in capturing the cancer cell dynamics. Results of the Liouville optimal control framework demonstrated the effectiveness in obtaining optimal therapies to combat prostate cancer.

Keywords

Optimal control, Parameter estimation, Liouville equation, Mathematical ODE model

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Available for download on Thursday, August 01, 2024

Included in

Mathematics Commons

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