Cancer Drug Delivery in Three Dimensions For a Distributed Parameter Control Model Using Finite Elements

Optimal distributed parameter control and a Galerkin finite element method are used to develop procedures for an ideal model of optimal drug delivery to brain tumors. The mathematical model comprises of a system of three coupled reaction diffusion models, involving the density of tumor cells, normal tissue and also the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameter method based on a simple application of variational calculus is used on a pseudo-Hamiltonian, which is then used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure. Finally, an ideal three dimensional test case is considered and partitioned into a set of brick finite elements in spherical coordinates, with tri-linear basis functions. Non-uniqueness of nodes in spherical coordinates is removed by combining like nodes, such as at the origin, at the poles and at the polar angle discontinuity. The Galerkin ODEs are solved by a combination of Crank-Nicolson and predictor-corrector methods