Uncertainty quantification and numerical methods in charged particle radiation therapy

Radiation therapy is applied in approximately 50% of all cancer treatments. To eliminate the tumor without damaging organs in the vicinity, optimized treatment plans are determined. This requires the calculation of three-dimensional dose distributions in a heterogeneous volume with a spatial resolution of 2-3mm. Current planning techniques use multiple beams with optimized directions and energies to achieve the best possible dose distribution. Each dose calculation however requires the discretization of the six-dimensional phase space of the linear Boltzmann transport equation describing complex particle dynamics. Despite the complexity of the problem, dose calculation errors of less than 2% are clinically recommended and computation times cannot exceed a few minutes. Additionally, the treatment reality often differs from the computed plan due to various uncertainties, for example in patient positioning, the acquired CT image or the delineation of tumor and organs at risk. Therefore, it is essential to include uncertainties in the planning process to determine a robust treatment plan. This entails a realistic mathematical model of uncertainties, quantification of their effect on the dose distribution using appropriate propagation methods as well as a robust or probabilistic optimization of treatment parameters to account for these effects. Fast and accurate calculations of the dose distribution including predictions of uncertainties in the computed dose are thus crucial for the determination of robust treatment plans in radiation therapy. Monte Carlo methods are often used to solve transport problems, especially for applications that require high accuracy. In these cases, common non-intrusive uncertainty propagation strategies that involve repeated simulations of the problem at different points in the parameter space quickly become infeasible due to their long run-times. Quicker deterministic dose calculation methods allow for better incorporation of uncertainties, but often use strong simplifications or admit non-physical solutions and therefore cannot provide the required accuracy. This work is concerned with finding efficient mathematical solutions for three aspects of (robust) radiation therapy planning: 1. Efficient particle transport and dose calculations, 2. uncertainty modeling and propagation for radiation therapy, and 3. robust optimization of the treatment set-up.