To accurately model dose in a magnetic field, the Lorentz force must be included in the traditional linear Boltzmann transport equation (LBTE). Both angular and spatial stabilization are required to deterministically solve this equation. In this work, a streamline upwind Petrov-Galerkin (SUPG) method is applied to achieve angular stabilization of the LBTE with magnetic fields. The spectral radius of the angular SUPG method is evaluated using a Fourier analysis method to characterize the convergence properties. Simulations are then performed on homogeneous phantoms and two heterogeneous slab geometry phantoms containing water, bone, lung/air and water for 0.5 T parallel and 1.5 T perpendicular magnetic field configurations. Fourier analysis determined that the spectral radius of the SUPG scheme is unaffected by magnetic field strength and the SUPG free parameter, indicating that the Gauss-Seidel source iteration method is unconditionally stable and the convergence rate is not degraded with increasing magnetic field strength. 100% of simulation points passed a 3D gamma analysis at a 2%/2 mm (3%/3 mm) gamma criterion for both magnetic field configurations in the homogeneous phantom study, with the exception of the 1.5 T perpendicular magnetic field in the pure lung phantom where a 77.4% (87.0%) pass rate was achieved. Simulations in the lung slab geometry phantom resulted in 100% of points passing a 2%/2 mm gamma analysis in a 0.5 T parallel magnetic field, and 97.7% (98.8%) of points passing a 2%/2 mm (3%/3 mm) gamma criterion in a 1.5 T perpendicular magnetic field. For the air slab geometry phantom, 72.1% (79.2%) of points passed a 2%/2 mm gamma criterion in a 0.5 T parallel magnetic field and 90.3% (92.8%) passed the same gamma criterion in a 1.5 T perpendicular magnetic field. While the novel SUPG angular stabilization method shows feasibility in some cases, it was found that the accuracy of this method was degraded for very low density media such as air.

为了准确模拟磁场中的剂量，必须将洛伦兹力包含在传统的线性玻尔兹曼传输方程 (LBTE) 中。确定性地求解这个方程需要角度和空间稳定性。在这项工作中，应用流线型逆风 Petrov-Galerkin (SUPG) 方法来实现具有磁场的 LBTE 的角稳定。角 SUPG 方法的谱半径使用傅立叶分析方法进行评估，以表征收敛特性。然后对同质体模和两个含有水、骨、肺/空气和水的异质板几何体模进行模拟，用于 0.5 T 平行和 1.5 T 垂直磁场配置。傅里叶分析确定SUPG方案的谱半径不受磁场强度和SUPG自由参数的影响，表明Gauss-Seidel源迭代方法无条件稳定，收敛速度不会随着磁场强度的增加而降低。100% 的模拟点通过了 2%/2 mm (3%/3 mm) 伽马标准的 3D 伽马分析，适用于均匀体模研究中的两种磁场配置，纯中的 1.5 T 垂直磁场除外肺体模通过率达到 77.4% (87.0%)。肺板几何模型中的模拟导致 100% 的点在 0.5 T 平行磁场中通过了 2%/2 mm 伽马分析，97.7% (98.8%) 的点通过了 2%/2 mm (3%/ 3 mm) 1.5 T 垂直磁场中的伽马判据。对于空气板几何体模，72.1% (79.2%) 的点在 0.5 T 平行磁场中通过了 2%/2 mm 伽马准则，90.3% (92.8%) 在 1.5 T 垂直磁场中通过了相同的伽马准则. 虽然新的 SUPG 角稳定方法在某些情况下显示出可行性，但发现该方法的准确性对于非常低密度的介质（例如空气）会降低。