Feasibility of streamline upwind Petrov-Galerkin angular stabilization of the linear Boltzmann transport equation with magnetic fields

Abstract

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.