Analysis of the Impact of Nonlinear Heat Transfer Laws on Temperature Distribution in Irradiated Biological Tissues: Mathematical Models and Optimal Controls

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.