C Börgers | PubMed Health

PubMed1998

Complexity of Monte Carlo and deterministic dose-calculation methods.

Grid-based deterministic dose-calculation methods for radiotherapy planning require the use of six-dimensional phase space grids. Because of the large number of phase space dimensions, a growing number of medical physicists appear to believe that grid-based deterministic dose-calculation methods are not competitive with Monte Carlo methods. We argue that this conclusion may be premature. Our results do suggest, however, that finite difference or finite element schemes with orders of accuracy greater than one will probably be needed if such methods are to compete well with Monte Carlo methods for dose calculations.

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PubMed1996

On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport.

Electron beam dose calculations are often based on pencil beam formulas such as the Fermi-Eyges formula. The Fermi-Eyges formula gives an exact solution of the Fermi equation. The Fermi equation can be derived from a more fundamental mathematical model, the linear Boltzmann equation, in two steps. First, the linear Boltzmann equation is approximated by the Fokker-Planck equation. Second, the Fokker-Planck equation is approximated by the Fermi equation. In this paper, we study these approximations. We use a simplified model problem, but choose parameter values closely resembling those relevant in electron beam therapy. Our main conclusions are: (1) The inaccuracy of the Fokker-Planck approximation is primarily due to neglect of large-angle scattering. (2) When computing an approximate solution to the Fokker-Planck equation by Monte Carlo simulation of a transport process, one should let the polar scattering angle be deterministic. (3) At shallow depths, the discrepancy between the linear Boltzmann and Fokker-Planck equations is far more important than that between the Fokker-Planck and Fermi equations. The first of these conclusions is certainly not new, but we state and justify it more rigorously than in previous work.

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