Identifying a response parameter in a model of brain tumour evolution under therapy

A nonlinear conjugate gradient method is derived for the inverse problem of identifying a treatment parameter in a nonlinear model of reaction-diffusion type corresponding to the evolution of brain tumours under therapy. The treatment parameter is reconstructed from additional information about the tumour taken at a fixed instance of time. Well-posedness of the direct problems used in the iterative method is outlined as well as uniqueness of a solution to the inverse problem. Moreover, the parameter identification is recasted as the minimization of a Tikhonov type functional and the existence of a minimizer to this functional is shown. Finite-difference discretization of the space and time derivatives are employed for the numerical implementation. Numerical simulations on full 3D brain data are included showing that information about a spacewise-dependent treatment parameter can be recovered in a stable way.