Purpose: We report on the development of the open-source cross-platform radiation treatment planning toolkit matRad and its comparison against validated treatment planning systems. The toolkit enables three-dimensional intensity-modulated radiation therapy treatment planning for photons, scanned protons and scanned carbon ions. Methods: matRad is entirely written in Matlab and is freely available online. It re-implements wellestablished algorithms employing a modular and sequential software design to model the entire treatment planning workflow. It comprises core functionalities to import DICOM data, to calculate and optimize dose as well as a graphical user interface for visualization. matRad dose calculation algorithms (for carbon ions this also includes the computation of the relative biological effect) are compared against dose calculation results originating from clinically approved treatment planning systems. Results: We observe three-dimensional c-analysis pass rates ≥ 99.67% for all three radiation modalities utilizing a distance to agreement of 2 mm and a dose difference criterion of 2%. The computational efficiency of matRad is evaluated in a treatment planning study considering three different treatment scenarios for every radiation modality. For photons, we measure total run times of 145 s-1260 s for dose calculation and fluence optimization combined considering 4-72 beam orientations and 2608-13597 beamlets. For charged particles, we measure total run times of 63 s-993 s for dose calculation and fluence optimization combined considering 9963-45574 pencil beams. Using a CT and dose grid resolution of 0.3 cm 3 requires a memory consumption of 1.59 GB-9.07 GB and 0.29 GB-17.94 GB for photons and charged particles, respectively. Conclusion: The dosimetric accuracy, computational performance and open-source character of matRad encourages a future application of matRad for both educational and research purposes.

The sensitivity of intensity-modulated proton therapy (IMPT) treatment plans to uncertainties can be quantified and mitigated with robust/min-max and stochastic/probabilistic treatment analysis and optimization techniques. Those methods usually rely on sparse random, importance, or worst-case sampling. Inevitably, this imposes a trade-off between computational speed and accuracy of the uncertainty propagation. Here, we investigate analytical probabilistic modeling (APM) as an alternative for uncertainty propagation and minimization in IMPT that does not rely on scenario sampling. APM propagates probability distributions over range and setup uncertainties via a Gaussian pencil-beam approximation into moments of the probability distributions over the resulting dose in closed form. It supports arbitrary correlation models and allows for efficient incorporation of fractionation effects regarding random and systematic errors. We evaluate the trade-off between run-time and accuracy of APM uncertainty computations on three patient datasets. Results are compared against reference computations facilitating importance and random sampling. Two approximation techniques to accelerate uncertainty propagation and minimization based on probabilistic treatment plan optimization are presented. Runtimes are measured on CPU and GPU platforms, dosimetric accuracy is quantified in comparison to a sampling-based benchmark (5000 random samples). APM accurately propagates range and setup uncertainties into dose uncertainties at competitive run-times (GPU [Formula: see text] min). The resulting standard deviation (expectation value) of dose show average global [Formula: see text] pass rates between 94.2% and 99.9% (98.4% and 100.0%). All investigated importance sampling strategies provided less accuracy at higher run-times considering only a single fraction. Considering fractionation, APM uncertainty propagation and treatment plan optimization was proven to be possible at constant time complexity, while run-times of sampling-based computations are linear in the number of fractions. Using sum sampling within APM, uncertainty propagation can only be accelerated at the cost of reduced accuracy in variance calculations. For probabilistic plan optimization, we were able to approximate the necessary pre-computations within seconds, yielding treatment plans of similar quality as gained from exact uncertainty propagation. APM is suited to enhance the trade-off between speed and accuracy in uncertainty propagation and probabilistic treatment plan optimization, especially in the context of fractionation. This brings fully-fledged APM computations within reach of clinical application.

12Purpose: Robust optimization (RO) methods are applied to intensity-modulated proton therapy (IMPT) 13 treatment plans to ensure their robustness in the face of treatment delivery uncertainties, such as 14 proton range and patient setup errors. However, the impact of those uncertainties on the biological 15 effect of protons has not been specifically considered. In this study, we added biological effect-based 16 objectives into a conventional RO cost function for IMPT optimization to minimize the variation in 17 biological effect. 18 Methods: One brain tumor case, one prostate tumor case and one head & neck tumor case were 19 selected for this study. Three plans were generated for each case using three different optimization 20 approaches: planning target volume (PTV)-based optimization, conventional RO, and RO incorporating 21 biological effect (BioRO). In BioRO, the variation in biological effect caused by IMPT delivery 22 uncertainties was minimized for voxels in both target volumes and critical structures, in addition to a 23 conventional voxel-based worst-case RO objective function. The biological effect was approximated by 24 the product of dose-averaged linear energy transfer (LET) and physical dose. All plans were normalized 25 to give the same target dose coverage, assuming a constant relative biological effectiveness (RBE) of 1.1. 26 Dose, biological effect, and their uncertainties were evaluated and compared among the three 27 optimization approaches for each patient case. 28Results: Compared with PTV-based plans, RO plans achieved more robust target dose coverage and 29 reduced biological effect hot spots in critical structures near the target. Moreover, with their sustained 30 robust dose distributions, BioRO plans not only reduced variations in biological effect in target and 31 normal tissues but also further reduced biological effect hot spots in critical structures compared with 32 RO plans. 33 Conclusion:Our findings indicate that IMPT could benefit from the use of conventional RO, which would 34 reduce the biological effect in normal tissues and produce more robust dose distributions than those of 35 PTV-based optimization. More importantly, this study provides a proof of concept that incorporating 36 biological effect uncertainty gap into conventional RO would not only control the IMPT plan robustness 37 in terms of physical dose and biological effect but also achieve further reduction of biological effect in 38 normal tissues. 39