Börgers C, Larsen E W
Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA.
Med Phys. 1996 Oct;23(10):1749-59. doi: 10.1118/1.597832.
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.
电子束剂量计算通常基于诸如费米 - 艾格斯公式之类的笔形束公式。费米 - 艾格斯公式给出了费米方程的精确解。费米方程可以从一个更基本的数学模型——线性玻尔兹曼方程分两步推导得出。首先，线性玻尔兹曼方程由福克 - 普朗克方程近似。其次，福克 - 普朗克方程由费米方程近似。在本文中，我们研究这些近似。我们使用一个简化的模型问题，但选择的参数值与电子束治疗中相关的值非常相似。我们的主要结论是：（1）福克 - 普朗克近似的不准确性主要是由于忽略了大角度散射。（2）通过传输过程的蒙特卡罗模拟计算福克 - 普朗克方程的近似解时，应使极散射角为确定性的。（3）在浅深度处，线性玻尔兹曼方程与福克 - 普朗克方程之间的差异远比福克 - 普朗克方程与费米方程之间的差异重要。这些结论中的第一个肯定不是新的，但我们比以前的工作更严格地陈述并证明了它。
