Borgers 9626696 The investigator studies grid-based computational methods for the simulation of steady particle beams penetrating passively scattering backgrounds. Such beams are of interest in several fields; the application motivating the parameter choices in this project is radiation therapy using electron beams. Mathematically, the problem is modeled by a linear Boltzmann equation with inflow boundary conditions. The linear Boltzmann equation resembles a convection-diffusion equation in six-dimensional position/velocity space. Particles are convected, while their direction vector undergoes a process resembling diffusion on the unit sphere, and their energy undergoes a process resembling convection-diffusion. The boundary data in a beam problem have singularities corresponding to sharp beam edges and monodirectionality of incident beams. The first half of the project concerns accurate ways of dealing with these singularities. Difference schemes designed for convection problems with discontinuous solutions are used to resolve beam edges. Explicit-implicit schemes with implicit treatment of the directional diffusion are used to deal with the monodirectionality of incident beams. The second half of the project concerns the treatment plan optimization problem in radiation therapy. Specifically, the investigator studies the application of methods using second derivatives and of multilevel methods. At its present stage, the project focuses on model problems in two space dimensions with one additional independent variable denoting particle direction. Computational techniques are extensively used for radiation therapy treatment planning in current clinical practice. The numerical methods underlying current treatment planning systems have been developed by the medical physics community; the radiotherapy planning problem has received little attention from mathematicians and numerical analysts. However, numerical techniques developed for particle b eam problems by the nuclear engineering community, and techniques developed in different contexts by numerical analysts, have a strong potential of being helpful in the development of more accurate treatment planning algorithms. In the judgement of medical physicists, advances in the accuracy of treatment planning algorithms may translate into greater success in the radiation treatment of cancer. The investigator, whose background is in numerical analysis, studies the application of approaches developed by numerical analysts to the radiotherapy planning problem.
