A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields

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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“…The stationary CSD linear Boltzmann transport equation in magnetic fields is derived in the following form (St-Aubin et al 2015,…”

Section: Continuous Slowing Down (Csd) Linear Boltzmann Transport Equmentioning

confidence: 99%

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

2017

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“…Discrete ordinates (DO) represents a classical angular technique where the LBTE is solved along discrete transport directions which are chosen from numerical quadrature sets (Lewis and Miller 1993). It has been successfully applied to radiotherapy (Vassiliev et al 2010, however in the presence of magnetic fields, the solution stability was found to be conditional on many parameters including magnetic field strength, and the material interaction cross section (St-Aubin et al 2015, Zelyak et al 2018.…”

Section: Introductionmentioning

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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“…Although machine specifications, magnetic field orientations and strengths differ in these studies, all have found that the dose is distorted by the magnetic field, especially at air-tissue-interfaces in lung and skin. However, dedicated dose calculation algorithms, which are based either on Monte Carlo methods or on solving of the linear Boltzmann transport equation, are able to account for the presence of magnetic fields [107][108][109]. Several studies have shown that using such dose calculation algorithms to account for the ERE during treatment plan optimization allows for the design of clinically acceptable lung SBRT treatments [104][105][106].…”

Section: Delineation Dose Calculation and Treatment Planningmentioning

confidence: 99%

MRI-guided lung SBRT: Present and future developments

2017

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Stereotactic body radiotherapy (SBRT) is rapidly becoming an alternative to surgery for the treatment of early-stage non-small cell lung cancer patients. Lung SBRT is administered in a hypo-fractionated, conformal manner, delivering high doses to the target. To avoid normal-tissue toxicity, it is crucial to limit the exposure of nearby healthy organs-at-risk (OAR). Current image-guided radiotherapy strategies for lung SBRT are mostly based on X-ray imaging modalities. Although still in its infancy, magnetic resonance imaging (MRI) guidance for lung SBRT is not exposure-limited and MRI promises to improve crucial soft-tissue contrast. Looking beyond anatomical imaging, functional MRI is expected to inform treatment decisions and adaptations in the future. This review summarises and discusses how MRI could be advantageous to the different links of the radiotherapy treatment chain for lung SBRT: diagnosis and staging, tumour and OAR delineation, treatment planning, and inter-or intrafractional motion management. Special emphasis is placed on a new generation of hybrid MRI treatment devices and their potential for real-time adaptive radiotherapy.

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A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields

Cited by 24 publications

References 52 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

MRI-guided lung SBRT: Present and future developments

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