Multigrid FEMs in Clinical Cancer Therapy Planning

Peter Deuflhard
Konrad-Zuse-Zentrum (ZIB) and Freie Universität Berlin


The author describes the role of various multigrid methods in the context of a recent planning system for the cancer therapy hyperthermia. Within the rather sophisticated integrated software environment HyperPlan, three types of PDEs have to be solved to high efficiency and medical reliability: (I) high frequency Maxwell's equations for the total system {individual cancer patient, water bolus, radiofrequency applicator, air}, (II) the linear Bioheat-Transfer Equation (BHT) that models the distribution of heat in the human body, and (III) nonlinear extensions of the BHT equation, which include systemic physiological effects of the patient body. For the indefinite functional in case (I), a multiplicative MG method has been designed (in cooperation of the Berlin and the Augsburg multigrid groups), which involves a hybrid smoother to take special care of the unwanted nullspace of the curl-operator; even though numerical experiments confirm the typical multigrid complexity, the number of MG cycles needed is regarded as still too high. As for the purely elliptic case (II), additive multigrid methods (KASKADE with BPX preconditioner) are applied; recently, a possible coupling of domain decomposition methods with subdomain CCG methods has been studied by Lipnikov and the author, which are presently under further investigation. As for the nonlinear models (III), two approaches are followed: (a) a recently improved adaptive Rothe method (due to Lang) is used to solve the time dependent problem up to the stationarity, (b) the recently suggested global adaptive multigrid solver for nonlinear elliptic problems (NEWTON-KASKADE by the author and Weiser) is directly applied to the stationary problem. In the latter case, a slight difficulty arises, since the underlying functional is not globally convex, but only decent in a neighborhood of the solution. Finally, optimal temperature distributions for different virtual patients (corresponding to anonymized real cancer patient data) are visualized.