Discontinuous finite element space-angle treatment of the first order linear Boltzmann transport equation with magnetic fields: Application to MRI-guided radiotherapy

Aubin, J St.; Keyvanloo, Amir; Fallone, B. G.

doi:10.1118/1.4937933

Cited by 14 publications

(14 citation statements)

References 16 publications

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“…In the 2015 publication by St-Aubin et al (2015), very accurate results were shown comparing deterministic dose calculations in magnetic fields to Monte Carlo calculations, but it was stated that the iterative solution method used, coupled with the discretization method, produced an unstable iterative scheme in low density media with magnetic fields. In 2016, St-Aubin published a space-angle discontinuous finite element discretization with magnetic fields that was shown to alleviate the iterative instability of the 2015 work (St-Aubin et al 2016). However, a rigorous iterative stability analysis for these novel methods including magnetic fields has not been presented, and is the focus of this work.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a deterministic dose calculation for MRI-guided radiotherapy

2017

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Modern effort in radiotherapy to address the challenges of tumor localization and motion has led to the development of MRI guided radiotherapy technologies. Accurate dose calculations must properly account for the effects of the MRI magnetic fields. Previous work has investigated the accuracy of a deterministic linear Boltzmann transport equation (LBTE) solver that includes magnetic field, but not the stability of the iterative solution method. In this work, we perform a stability analysis of this deterministic algorithm including an investigation of the convergence rate dependencies on the magnetic field, material density, energy, and anisotropy expansion. The iterative convergence rate of the continuous and discretized LBTE including magnetic fields is determined by analyzing the spectral radius using Fourier analysis for the stationary source iteration (SI) scheme. The spectral radius is calculated when the magnetic field is included (1) as a part of the iteration source, and (2) inside the streaming-collision operator. The non-stationary Krylov subspace solver GMRES is also investigated as a potential method to accelerate the iterative convergence, and an angular parallel computing methodology is investigated as a method to enhance the efficiency of the calculation. SI is found to be unstable when the magnetic field is part of the iteration source, but unconditionally stable when the magnetic field is included in the streaming-collision operator. The discretized LBTE with magnetic fields using a space-angle upwind stabilized discontinuous finite element method (DFEM) was also found to be unconditionally stable, but the spectral radius rapidly reaches unity for very low-density media and increasing magnetic field strengths indicating arbitrarily slow convergence rates. However, GMRES is shown to significantly accelerate the DFEM convergence rate showing only a weak dependence on the magnetic field. In addition, the use of an angular parallel computing strategy is shown to potentially increase the efficiency of the dose calculation.

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Section: Introductionmentioning

confidence: 99%

Stability analysis of a deterministic dose calculation for MRI-guided radiotherapy

2017

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“…the sign of Ω α •n k will not change for each face over the entire sweep octant). In the previous work (St-Aubin et al 2016) utilizing unstructured tetrahedral spatial meshes, spatial element faces are directed randomly and small angular elements were required to minimize the impact when the sign of Ω α •n k changed over the angular element α. Thus a large number of angular elements were required to span the full unit-sphere leading to an overly complex calculation.…”

Section: Interplay Between Space-angle Discretization Scheme and Spatmentioning

confidence: 99%

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

et al. 2018

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Angular discretization impacts nearly every aspect of a deterministic solution to the linear Boltzmann transport equation, especially in the presence of magnetic fields, as modeled by a streaming operator in angle. In this work a novel stabilization treatment of the magnetic field term is developed for an angular finite element discretization on the unit sphere, specifically involving piecewise partitioning of path integrals along curved element edges into uninterrupted segments of incoming and outgoing flux, with outgoing components updated iteratively. Correct order-of-accuracy for this angular framework is verified using the method of manufactured solutions for linear, quadratic, and cubic basis functions in angle. Higher order basis functions were found to reduce the error especially in strong magnetic fields and low density media. We combine an angular finite element mesh respecting octant boundaries on the unit sphere to spatial Cartesian voxel elements to guarantee an unambiguous transport sweep ordering in space. Accuracy for a dosimetrically challenging scenario involving bone and air in the presence of a 1.5 T parallel magnetic field is validated against the Monte Carlo package GEANT4. Accuracy and relative computational efficiency were investigated for various angular discretization parameters. 32 angular elements with quadratic basis functions yielded a reasonable compromise, with gamma passing rates of 99.96% (96.22%) for a 2%/2 mm (1%/1 mm) criterion. A rotational transformation of the spatial calculation geometry is performed to orient an arbitrary magnetic field vector to be along the z-axis, a requirement for a constant azimuthal angular sweep ordering. Working on the unit sphere, we apply the same rotational transformation to the angular domain to align its octants with the rotated Cartesian mesh. Simulating an oblique 1.5 T magnetic field against GEANT4 yielded gamma passing rates of 99.42% (95.45%) for a 2%/2 mm (1%/1 mm) criterion.

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“…As such, a key step in the treatment flow is effective planning, where absorbed dose is prescribed to the tumour and upper dose limits are imposed on associated organs at risk (OARs). Dose calculation methods have evolved from correctionbased techniques (Lu 2013, Johns and Cunningham 1983, Papanikolaou et al 2004, Batho 1964, Young and Gaylord 1970 to model-based methods (Karlsson et al 2010, Khan and Gibbons 2014, Mackie et al 1984, Sievinen et al 2005, to state of the art modern methods such as Monte Carlo (Ma et al 2002, Fippel 2013, Gifford et al 2006, St-Aubin et al 2016 and deterministic solvers (St-Aubin et al 2015, Vassiliev et al 2010, Gifford et al 2006. Many modern dose calculation algorithms attempt to solve or approximate the solution of the linear Boltzmann transport equation (LBTE), an integropartial differential equation modelling radiation transport.…”

Section: Introductionmentioning

confidence: 99%

“…the original unmodified weighting function used in the DGFEM, κ is the multigroup magnetic field parameter, given by(Yang et al 2018, St-Aubin et al 2016 …”

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confidence: 99%

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

Swan¹,

Yang²,

Zelyak³

et al. 2020

Biomed. Phys. Eng. Express

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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.

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Discontinuous finite element space-angle treatment of the first order linear Boltzmann transport equation with magnetic fields: Application to MRI-guided radiotherapy

Cited by 14 publications

References 16 publications

Stability analysis of a deterministic dose calculation for MRI-guided radiotherapy

Stability analysis of a deterministic dose calculation for MRI-guided radiotherapy

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

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

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