Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L 2 -stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L 2 -stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

Objective. To present an efficient uncertainty quantification method for range and set-up errors in Monte Carlo (MC) dose calculations. Further, we show that uncertainty induced by interplay and other dynamic influences may be approximated using suitable error correlation models. Approach. We introduce an importance (re-)weighting method in MC history scoring to concurrently construct estimates for error scenarios, the expected dose and its variance from a single set of MC simulated particle histories. The approach relies on a multivariate Gaussian input and uncertainty model, which assigns probabilities to the initial phase space sample, enabling the use of different correlation models. Through modification of the phase space parameterization, accuracy can be traded between that of the uncertainty or the nominal dose estimate. Main results. The method was implemented using the MC code TOPAS and validated for proton intensity-modulated particle therapy (IMPT) with reference scenario estimates. We achieve accurate results for set-up uncertainties (γ 2 mm/2% ≥ 99.01% (E[ d ]), γ 2 mm/2% ≥ 98.04% (σ( d ))) and expectedly lower but still sufficient agreement for range uncertainties, which are approximated with uncertainty over the energy distribution. Here pass rates of 99.39% (E[ d ])/ 93.70% (σ( d )) (range errors) and 99.86% (E[ d ])/ 96.64% (σ( d )) (range and set-up errors) can be achieved. Initial evaluations on a water phantom, a prostate and a liver case from the public CORT dataset show that the CPU time decreases by more than an order of magnitude. Significance. The high precision and conformity of IMPT comes at the cost of susceptibility to treatment uncertainties in particle range and patient set-up. Yet, dose uncertainty quantification and mitigation, which is usually based on sampled error scenarios, becomes challenging when computing the dose with computationally expensive but accurate MC simulations. As the results indicate, the proposed method could reduce computational effort while also facilitating the use of high-dimensional uncertainty models.