5. Optimal control of free boundary tumor growth model
In this project, we study the optimal control of treatment strategies in a mathematical model of tumor growth governed by a free boundary PDE system. The goal is to design a control strategy, such as a drug dosage schedule, that effectively inhibits tumor expansion while minimizing harmful side effects to healthy tissue.
To achieve this, we derive the optimality system associated with the control problem and establish necessary conditions for optimality using techniques from optimal control theory.
A key challenge lies in the moving boundary, which causes the domain of the PDE system to change when small perturbations are applied to the control. To address this, we employ a change of variables to reformulate the free boundary problem on a fixed reference domain. This transformation simplifies the handling of the domain variations but comes at the cost of making the resulting differential equations more complex.
Numerical simulations are performed to illustrate the effectiveness of the proposed control strategy and to analyze its impact on tumor dynamics under various scenarios.
This research provides insights into how mathematical modeling can inform treatment planning and improve our understanding of tumor progression and control.
Journal papers
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X. E. Zhao, Y. Wu, R. Leander, W. Ding and S. Lenhart, Optimal control of treatment in a free boundary problem modeling multilayered tumor growth, 2024. [ arXiv ]
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Y. Wu, X. E. Zhao, R. Leander and W. Ding, Optimal control for a free boundary tumor growth model, 2024. [ SSRN ]
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X. E. Zhao, Analysis and optimization of tumor inhibitor treatments in a free boundary tumor growth model, Nonlinear Analysis: Real World Applications, 2025. [ DOI ]
Presentations