Parameter identification problems and analysis of the impact of porous media in biofluid heat transfer in biological tissues during thermal therapy

Abstract

This paper considers time-dependent identification problems governed by some generalized transient bioheat transfer type models in biological systems with porous structures and directional blood flow. The analysis and the real applications of the porous media models and biofluid heat transfer in living tissues are relatively recent, however the blood perfusion rate and the porosity parameter affect considerably the effects of thermal physical properties on the transient temperature of biological tissues. The control considered in this work estimates simultaneously these two parameters. 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 the treatment of cancer. First, the mathematical models are introduced and the existence, the uniqueness and the regularity of the solution of the state equation are proved as well as stability and maximum principle under extra assumptions. Afterwards the identification problems with Tichonov regularization are formulated, in different situations, in order to control the online temperature given by radiometric measurement. An optimal solution is proven to exist and finally necessary optimality conditions are given. Some strategies for numerical realization based on the adjoint variables are provided.