Streamline Upwind Petrov-Galerkin Angular Stabilization of the Linear Boltzmann Transport Equation with Magnetic Fields

• Author / Creator

  Swan, Amanda

• Dose modelling is an important component of radiotherapy treatment planning, as clinicians prescribe a dose to a tumour while requiring certain adjacent organs to receive at or below a given tolerance dose. Mathematical models such as the linear Boltzmann transport equation, or LBTE, provide a method for calculating the dose received at each location within a patient for a given beam configuration. Solution of this model can be complicated, and deriving accurate and efficient numerical solution methods continues to be an active area of research. Stochastic solutions remain the "gold standard" in terms of accuracy, though many deterministic numerical solvers have been shown to achieve the same level of accuracy as Monte Carlo in a computationally efficient manner.

  The development of the combination linear accelerator/magnetic resonance imaging system has created the need for a modified LBTE model that incorporates the Lorentz force and its influence on secondary electrons. Such a model was previously derived mathematically where a new angular advection term was introduced. Finding computationally efficient methods to stabilize the angular advection term so that numerical solutions are both accurate and efficient is a difficult problem. Previously, a numerical method was developed capable of solving the modified LBTE, however the upwinding scheme used creates a magnetic field dependency in the spectral radius, thus reducing the convergence rate for increasing magnetic field strengths.

  In this work, a linear streamline upwind Petrov-Galerkin (SUPG) method is applied in angle as a potential stabilization scheme. A spectral radius analysis shows that application of this method eliminates any magnetic field strength dependency in the convergence rate and the resulting system is unconditionally stable. Simulation results of the discretized system confirm these findings. Phantom simulations also confirm that the SUPG method stabilizes the angular advection term. However, while the method proved accurate in the 0.5 T parallel magnetic field case, advection dominated too strongly in the 1.5 T perpendicular magnetic field and the results were over-diffusive in low density media, negatively impacting the accuracy in these regions. A non-linear SUPG method was then derived and proposed as a possible remedy to the accuracy problems in low density media.

• Subjects / Keywords

  • Streamline Upwind Petrov-Galerkin
  • Linear Boltzmann Transport Equation

• Graduation date

  Spring 2020

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/r3-de69-ev81

• License

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