| In this thesis we study the optimal drug delivery to brain tumors using deterministic control theory and finite element methods. The mathematical model is a system of three reaction-diffusion equations involving the tumor cells, the normal tissues and the drug concentration. The control problem which is deterministic in nature is formulated in a way so as to reduce the tumor density and also to reduce the effect of toxicity. Calculus of variations is used on a pseudo-Hamiltonian and solved to obtain a coupled system of state, co-state and control equations. These equations are then solved using finite difference methods for one dimension and finite element methods for two and three dimensions. Test results are presented for a circular disk and spherical geometry for the two and three dimensional cases respectively. Finally, more general brain geometries are reviewed using brain atlases or maps as well as physical processes beyond diffusion and growth of reaction diffusion models. The results obtained were very encouraging from the analytical and computational point of view, while more practical clinical testing has been left for future plans. |
