Extraction of macroscopic and microscopic adjoint concepts using a lattice Boltzmann method and discrete adjoint approach

Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).