Optimal control for the bioheat problem in multilayer tissues using infinite-dimensional systems theory

This paper presents a semi-analytical approach to solving the optimal control problem associated with thermal propagation in multilayer biological tissues during hyperthermia cancer treatment. The heat source at the tumor site is modeled as a distributed control function, and the problem is analyzed using a five-layer composite structure comprising skin, fat, muscle, tumor, and muscle layers. The approximate controllability of the multilayer bioheat problem is proved through infinite-dimensional systems theory, and the optimal control problem is formulated as a constrained Bolza problem in a one-dimensional framework. A hybrid solution method is proposed, integrating infinite-dimensional systems theory to model the multilayer bioheat problem as a system with inputs and outputs, along with strongly continuous semigroups theory to solve the corresponding abstract differential equation. Novel convolution operators are introduced for each layer, facilitating the computation of hybrid solutions for state and co-state problems. Additionally, a numerical Laplace inversion technique is applied to determine interface functions between layers. The effectiveness of the proposed approach is demonstrated through numerical simulations, providing valuable insights into the controllability of thermal propagation in multilayer tissues. These findings contribute to the optimization of hyperthermia treatment strategies for tumor therapy, improving precision and efficacy in clinical applications.