Optimal control of drug delivery to brain tumors for a distributed parameters model

The growth and treatment of brain tumors is mathematically examined using a distributed parameters model. The model is a system of three coupled reaction diffusion equations involving the tumor cells, normal tissue and the drug concentration. An optimal control problem is designed, with the drug delivery rate as the control and solved to obtain the state and co-state equations as well as the regular control. This gives rise to a coupled system of equations with a forward state equation and a backward co-state equation, which are solved using a modified double shot, forward-backward method. A numerical procedure based upon the Crank-Nicolson method is used to solve the coupled nonlinear system of six one-dimension partial differential equations, along with a quasilinear approximation of the nonlinearities using extrapolator-predictor-corrector iteration techniques.