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.
Börgers
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Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA.
Larsen
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D000465:Algorithms; D006801:Humans; D008962:Models, Theoretical; D009010:Monte Carlo Method; D011878:Radiotherapy / Q000295:instrumentation / Q000379:methods*; D011879:Radiotherapy Dosage; D011880:Radiotherapy Planning, Computer-Assisted; D015203:Reproducibility of Results; D012542:Scattering, Radiation