Objective. Uncertainties in treatment planning are typically managed using either margin-based or robust optimization. Margin-based methods expand the clinical target volume (CTV) towards a planning target volume, which is generally unsuited for proton therapy. Robust optimization considers worst-case scenarios, but its quality depends on the chosen uncertainty (scenario) set: excluding extremes reduces robustness, while including too many make plans overly conservative. Probabilistic optimization overcomes these limitations by modeling a continuous scenario distribution, enabling the use of statistical measures. Approach. We propose a novel approach to probabilistic optimization that steers plans towards individualized probability levels, to control CTV and organs-at-risks (OARs) under- and overdosage. Voxel-wise dose percentiles (d) are estimated by expected value (E) and standard deviation (SD) as E[d] ± δ ⋅ SD[d], where δ is iteratively tuned to match the target percentile of the underlying probability distribution (given setup and range uncertainties). The approach involves an inner optimization of E[d] ± δ ⋅ SD for fixed δ, and an outer optimization loop that updates δ. Polynomial chaos expansion provides accurate and efficient dose estimates during optimization. We validated the method on a spherical CTV (prescribed 60Gy) abutted by an OAR in different directions and a horseshoe-shaped CTV surrounding a cylindrical spine, under Gaussian-distributed setup (3mm) and range (3%) uncertainties. Main results. For spherical cases with similar CTV coverage, P (D _2% > 30 Gy) dropped by 10%–15%; for matched OAR dose, P (D _98% > 57 Gy) increased by 67.5%–71%. In spinal plans, P (D _98% > 57 Gy) increased by 10%–15% while P (D _2% > 30 Gy) dropped by 24%–28% in the same plan. Probabilistic and robust optimization times were comparable for spherical (hours) but longer for spinal cases (7.5–11.5h vs 9–20min). Significance. Compared to discrete scenario-based optimization, the probabilistic approach offered better OAR sparing or target coverage, depending on individualized priorities.

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently increasingly Deep Learning (DL) techniques. In this paper, we propose an autoregressive and data-driven method using the analogy with classical numerical solvers for time-dependent, parametric and (typically) nonlinear PDEs. We present how Dimensionality Reduction (DR) can be coupled with Neural Ordinary Differential Equations (NODEs) in order to learn the solution operator of arbitrary PDEs accounting both for (continuous) time and parameter dependency. The idea of our work is that it is possible to map the high-fidelity (i.e., high-dimensional) PDE solution space into a reduced (low-dimensional) space, which subsequently exhibits dynamics governed by a (latent) Ordinary Differential Equation (ODE). Solving this (easier) ODE in the reduced space allows avoiding solving the PDE in the high-dimensional solution space, thus decreasing the computational burden for repeated calculations for e.g., uncertainty quantification or design optimization purposes. The main outcome of this work is the importance of exploiting DR as opposed to the recent trend of building large and complex architectures: we show that by leveraging DR we can deliver not only more accurate predictions, but also a considerably lighter and faster DL model compared to existing methodologies.

Objective: Scenario-based evaluation in proton therapy often relies on a small number of error scenarios, leading to limited insight into the DVH values under uncertainty and suboptimal trade-offs. In this study, we investigated if re-optimization based on probabilistic evaluation improves the trade-off between OAR sparing and target coverage in neuro-oncological patients. Materials and methods: 22 neuro-oncological patients were included. 18 met their original target goals (group A), while in 4, target coverage was compromised to spare OARs (group B). The probabilistic goal for the CTV was calibrated to be consistent with PTV-based photon plans, resulting in D_99.8%,CTV = 0.95D_pres with a 90 % confidence level. The probabilistic OAR constraints were set to meet the clinical constraints with a 95 % confidence level. For both groups, the clinical plans were re-optimized, keeping the clinical objectives and constraints, but reducing robustness for the CTV objective (group A) to meet the probabilistic goal, or for the dose-limiting OAR objectives (group B) without exceeding the constraints. For the original and re-optimized plans, polynomial chaos expansion was applied to simulate 10,000 fractionated treatments, deriving probability distributions for relevant DVH parameters. Results: For group A, re-optimization resulted in a population median decrease of 8.2 (range: 0.4–20.8) Gy RBE in the total OAR-related clinical goal values. For group B, re-optimization resulted in a population median increase of 2.7 (range: 1.3–6.8) Gy RBE in the D_99.8%,CTV. The population median V_95%,CTV improved from 97.4 % to 99.1 %. Conclusion: We demonstrated that probabilistic evaluation guided IMPT planning enables either OAR sparing or target coverage enhancement.

Background and purpose: In online-adaptive proton therapy planning based on cone beam computed tomography (CBCT), CT number errors can pose challenges. We propose an approach for coping with CT number uncertainties by increasing range robustness settings (RRS) in online-adaptive planning. This was compared to our trigger-based offline (TB-Offline) adaptive approach, and to daily replanning using in-room CT-on-rails (CTOR). Material and methods: For 23 head-and-neck cancer patients, a CTOR and CBCT were acquired in a single fraction. CTOR contours were copied rigidly onto the CBCT. CBCT-based plans were generated with 3, 6, 8, 10, and 12 % RRS, each with 1 mm setup-RS, followed by a forward dose calculation on the reference CTOR. This was compared to dose distributions from our TB-Offline approach (3 mm/3% SRS/RRS), also recomputed on the CTOR. Coverage (voxelwise-minimum) of the primary clinical target volume (CTV₇₀₀₀) and elective lymph nodes (CTV₅₄₂₅) and grade ≥ II normal tissue complication probabilities were compared between strategies. Results: When going from RRS = 3 % to RRS = 10 %, the population 90th percentiles of CTV₅₄₂₅ V_94% improved from 89.6 % to 96.4 %, and CTV₇₀₀₀ V_94% from 92.8 % to 96.4 %. Substantial coverage loss (V_94%<95 %) with CBCT-based online adaptive and RRS = 10 % was observed in 1/23 evaluated patients for CTV₇₀₀₀ and 2/23 for CTV₅₄₂₅. This was an improvement compared to 3/23 and 4/23 with TB-Offline. Moreover, for RRS = 10 % the average risk of xerostomia improved by 2.4 percentage point compared to TB-Offline. Conclusions: Robust optimization with increased range robustness settings effectively mitigated dose degradation from CT number errors in CBCT-based online-adaptive proton therapy.

Objective. To assess the viability of a physics-based, deterministic and adjoint-capable algorithm for performing treatment planning system independent dose calculations and for computing dosimetric differences caused by anatomical changes. Approach. A semi-numerical approach is employed to solve two partial differential equations for the proton phase-space density which determines the deposited dose. Lateral hetereogeneities are accounted for by an optimized (Gaussian) beam splitting scheme. Adjoint theory is applied to approximate the change in the deposited dose caused by a new underlying patient anatomy. Main results. The dose engine’s accuracy was benchmarked through three-dimensional gamma index comparisons against Monte Carlo simulations done in TOPAS. For a lung test case, the worst passing rate with (1 mm, 1%, 10% dose cut-off) criteria is 94.55%. The effect of delivering treatment plans on repeat CTs was also tested. For non-robustly optimized plans the adjoint component was accurate to 5.7% while for a robustly optimized plan it was accurate to 4.8%. Significance. Yet anOther Dose Algorithm is capable of accurate dose computations in both single and multi spot irradiations when compared to TOPAS. Moreover, it is able to compute dosimetric differences due to anatomical changes with small to moderate errors thereby facilitating its use for patient-specific quality assurance in online adaptive proton therapy.

Background and purpose: In the Netherlands, 2 protocols have been standardized for PT among the 3 proton centers: a robustness evaluation (RE) to ensure adequate CTV dose and a model-based selection (MBS) approach for IMPT patient-selection. This multi-institutional study investigates (i) inter-patient and inter-center variation of target dose from the RE protocol and (ii) the robustness of the MBS protocol against treatment errors for a cohort of head-and-neck cancer (HNC) patients treated in the 3 Dutch proton centers. Materials and methods: Clinical treatment plans of 100 HNC patients were evaluated. Polynomial Chaos Expansion (PCE) was used to perform a comprehensive robustness evaluation per plan, enabling the probabilistic evaluation of 100,000 complete fractionated treatments. PCE allowed to derive scenario distributions of clinically relevant dosimetric parameters to assess CTV dose (D_99.8%/D_0.2%, based on a prior photon plan calibration) and tumour control probabilities (TCP) as well as the evaluation of the dose to OARs and normal tissue complication probabilities (NTCP) per center. Results: For the CTV_70.00, doses from the RE protocol were consistent with the clinical plan evaluation metrics used in the 3 centers. For the CTV_54.25, D_99.8% were consistent with the clinical plan evaluation metrics at center 1 and 2 while, for center 3, a reduction of 1 GyRBE was found on average. This difference did not impact modelled TCP at center 3. Differences between expected and nominal NTCP were below 0.3 percentage point for most patients. Conclusion: The standardization of the RE and MBS protocol lead to comparable results in terms of TCP and the NTCPs. Still, significant inter-patient and inter-center variation in dosimetric parameters remained due to clinical practice differences at each institution. The MBS approach is a robust protocol to qualify patients for PT.

Objective. In head-and-neck cancer intensity modulated proton therapy, adaptive radiotherapy is currently restricted to offline re-planning, mitigating the effect of slow changes in patient anatomies. Daily online adaptations can potentially improve dosimetry. Here, a new, fully automated online re-optimization strategy is presented. In a retrospective study, this online re-optimization approach was compared to our trigger-based offline re-planning (offline _TB re-planning) schedule, including extensive robustness analyses. Approach. The online re-optimization method employs automated multi-criterial re-optimization, using robust optimization with 1 mm setup-robustness settings (in contrast to 3 mm for offline _TB re-planning). Hard planning constraints and spot addition are used to enforce adequate target coverage, avoid prohibitively large maximum doses and minimize organ-at-risk doses. For 67 repeat-CTs from 15 patients, fraction doses of the two strategies were compared for the CTVs and organs-at-risk. Per repeat-CT, 10.000 fractions with different setup and range robustness settings were simulated using polynomial chaos expansion for fast and accurate dose calculations. Main results. For 14/67 repeat-CTs, offline _TB re-planning resulted in <50% probability of D _98% ≥ 95% of the prescribed dose (D _pres) in one or both CTVs, which never happened with online re-optimization. With offline _TB re-planning, eight repeat-CTs had zero probability of obtaining D _98% ≥ 95%D _pres for CTV ₇₀₀₀, while the minimum probability with online re-optimization was 81%. Risks of xerostomia and dysphagia grade ≥ II were reduced by 3.5 ± 1.7 and 3.9 ± 2.8 percentage point [mean ± SD] (p < 10 ⁻⁵ for both). In online re-optimization, adjustment of spot configuration followed by spot-intensity re-optimization took 3.4 min on average. Significance. The fast online re-optimization strategy always prevented substantial losses of target coverage caused by day-to-day anatomical variations, as opposed to the clinical trigger-based offline re-planning schedule. On top of this, online re-optimization could be performed with smaller setup robustness settings, contributing to improved organs-at-risk sparing.

Background and purpose: In the Netherlands, head-and-neck cancer (HNC) patients are referred for proton therapy (PT) through model-based selection (MBS). However, treatment errors may compromise adequate CTV dose. Our aims are: (i) to derive probabilistic plan evaluation metrics on the CTV consistent with clinical metrics; (ii) to evaluate plan consistency between photon (VMAT) and proton (IMPT) planning in terms of CTV dose iso-effectiveness and (iii) to assess the robustness of the OAR doses and of the risk toxicities involved in the MBS. Materials and methods: Sixty HNC plans (30 IMPT/30 VMAT) were included. A robustness evaluation with 100,000 treatment scenarios per plan was performed using Polynomial Chaos Expansion (PCE). PCE was applied to determine scenario distributions of clinically relevant dosimetric parameters, which were compared between the 2 modalities. Finally, PCE-based probabilistic dose parameters were derived and compared to clinical PTV-based photon and voxel-wise proton evaluation metrics. Results: Probabilistic dose to near-minimum volume v = 99.8% for the CTV correlated best with clinical PTV-D_98% and VWmin-D_98%,CTV doses for VMAT and IMPT respectively. IMPT showed slightly higher nominal CTV doses, with an average increase of 0.8 GyRBE in the median of the D_99.8%,CTV distribution. Most patients qualified for IMPT through the dysphagia grade II model, for which an average NTCP gain of 10.5 percentages points (%-point) was found. For all complications, uncertainties resulted in moderate NTCP spreads lower than 3 p.p. on average for both modalities. Conclusion: Despite the differences between photon and proton planning, the comparison between PTV-based VMAT and robust IMPT is consistent. Treatment errors had a moderate impact on NTCPs, showing that the nominal plans are a good estimator to qualify patients for PT.

In this paper we propose a solution to the need for a fast particle transport algorithm in Online Adaptive Proton Therapy capable of cheaply, but accurately computing the changes in patient dose metrics as a result of changes in the system parameters. We obtain the proton phase-space density through the product of the numerical solution to the one-dimensional Fokker-Planck equation and the analytical solution to the Fermi-Eyges equation. Moreover, a corresponding adjoint system was derived and solved for the adjoint flux. The proton phase-space density together with the adjoint flux and the metric (chosen as the energy deposited by the beam in a variable region of interest) allowed assessing the accuracy of our algorithm to different perturbation ranges in the system parameters and regions of interest. The algorithm achieved negligible errors ((Formula presented.)) for small Hounsfield unit (HU) perturbation ranges (–40 HU to 40 HU) and small to moderate errors (3% to 17%)–in line with the well-known limitation of adjoint approaches–for large perturbation ranges (–400 HU to 400 HU) in the case of most clinical interest where the region of interest surrounds the Bragg peak. Given these results coupled with the capability of further improving the timing performance it can be concluded that our algorithm presents a viable solution for the specific purpose of Online Adaptive Proton Therapy.

Background: Fast dose calculation is critical for online and real-time adaptive therapy workflows. While modern physics-based dose algorithms must compromise accuracy to achieve low computation times, deep learning models can potentially perform dose prediction tasks with both high fidelity and speed. Purpose: We present a deep learning algorithm that, exploiting synergies between transformer and convolutional layers, accurately predicts broad photon beam dose distributions in few milliseconds. Methods: The proposed improved Dose Transformer Algorithm (iDoTA) maps arbitrary patient geometries and beam information (in the form of a 3D projected shape resulting from a simple ray tracing calculation) to their corresponding 3D dose distribution. Treating the 3D CT input and dose output volumes as a sequence of 2D slices along the direction of the photon beam, iDoTA solves the dose prediction task as sequence modeling. The proposed model combines a Transformer backbone routing long-range information between all elements in the sequence, with a series of 3D convolutions extracting local features of the data. We train iDoTA on a dataset of 1700 beam dose distributions, using 11 clinical volumetric modulated arc therapy (VMAT) plans (from prostate, lung, and head and neck cancer patients with 194–354 beams per plan) to assess its accuracy and speed. Results: iDoTA predicts individual photon beams in ≈50 ms with a high gamma pass rate of (Formula presented.) (2 mm, 2%). Furthermore, estimating full VMAT dose distributions in 6–12 s, iDoTA achieves state-of-the-art performance with a (Formula presented.) (2 mm, 2%) pass rate and an average relative dose error of 0.75 ± 0.36%. Conclusions: Offering the millisecond speed prediction per beam angle needed in online and real-time adaptive treatments, iDoTA represents a new state of the art in data-driven photon dose calculation. The proposed model can massively speed-up current photon workflows, reducing calculation times from few minutes to just a few seconds.

Objective. In radiotherapy, the internal movement of organs between treatment sessions causes errors in the final radiation dose delivery. To assess the need for adaptation, motion models can be used to simulate dominant motion patterns and assess anatomical robustness before delivery. Traditionally, such models are based on principal component analysis (PCA) and are either patient-specific (requiring several scans per patient) or population-based, applying the same set of deformations to all patients. We present a hybrid approach which, based on population data, allows to predict patient-specific inter-fraction variations for an individual patient. Approach. We propose a deep learning probabilistic framework that generates deformation vector fields warping a patient's planning computed tomography (CT) into possible patient-specific anatomies. This daily anatomy model (DAM) uses few random variables capturing groups of correlated movements. Given a new planning CT, DAM estimates the joint distribution over the variables, with each sample from the distribution corresponding to a different deformation. We train our model using dataset of 312 CT pairs with prostate, bladder, and rectum delineations from 38 prostate cancer patients. For 2 additional patients (22 CTs), we compute the contour overlap between real and generated images, and compare the sampled and ‘ground truth’ distributions of volume and center of mass changes. Results. With a DICE score of 0.86 ± 0.05 and a distance between prostate contours of 1.09 ± 0.93 mm, DAM matches and improves upon previously published PCA-based models, using as few as 8 latent variables. The overlap between distributions further indicates that DAM’s sampled movements match the range and frequency of clinically observed daily changes on repeat CTs. Significance. Conditioned only on planning CT values and organ contours of a new patient without any pre-processing, DAM can accurately deformations seen during following treatment sessions, enabling anatomically robust treatment planning and robustness evaluation against inter-fraction anatomical changes.

Objective. The Dutch proton robustness evaluation protocol prescribes the dose of the clinical target volume (CTV) to the voxel-wise minimum (VWmin) dose of 28 scenarios. This results in a consistent but conservative near-minimum CTV dose (D98%,CTV). In this study, we analyzed (i) the correlation between VWmin/voxel-wise maximum (VWmax) metrics and actually delivered dose to the CTV and organs at risk (OARs) under the impact of treatment errors, and (ii) the performance of the protocol before and after its calibration with adequate prescription-dose levels.Approach. Twenty-one neuro-oncological patients were included. Polynomial chaos expansion was applied to perform a probabilistic robustness evaluation using 100,000 complete fractionated treatments per patient. Patient-specific scenario distributions of clinically relevant dosimetric parameters for the CTV and OARs were determined and compared to clinical VWmin and VWmax dose metrics for different scenario subsets used in the robustness evaluation protocol.Main results. The inclusion of more geometrical scenarios leads to a significant increase of the conservativism of the protocol in terms of clinical VWmin and VWmax values for the CTV and OARs. The protocol could be calibrated using VWmin dose evaluation levels of 93.0%-92.3%, depending on the scenario subset selected. Despite this calibration of the protocol, robustness recipes for proton therapy showed remaining differences and an increased sensitivity to geometrical random errors compared to photon-based margin recipes.Significance. The Dutch proton robustness evaluation protocol, combined with the photon-based margin recipe, could be calibrated with a VWmin evaluation dose level of 92.5%. However, it shows limitations in predicting robustness in dose, especially for the near-maximum dose metrics to OARs. Consistent robustness recipes could improve proton treatment planning to calibrate residual differences from photon-based assumptions.

Objective. Next generation online and real-time adaptive radiotherapy workflows require precise particle transport simulations in sub-second times, which is unfeasible with current analytical pencil beam algorithms (PBA) or Monte Carlo (MC) methods. We present a deep learning based millisecond speed dose calculation algorithm (DoTA) accurately predicting the dose deposited by mono-energetic proton pencil beams for arbitrary energies and patient geometries. Approach. Given the forward-scattering nature of protons, we frame 3D particle transport as modeling a sequence of 2D geometries in the beam's eye view. DoTA combines convolutional neural networks extracting spatial features (e.g. tissue and density contrasts) with a transformer self-attention backbone that routes information between the sequence of geometry slices and a vector representing the beam's energy, and is trained to predict low noise MC simulations of proton beamlets using 80 000 different head and neck, lung, and prostate geometries. Main results. Predicting beamlet doses in 5 ± 4.9 ms with a very high gamma pass rate of 99.37 ± 1.17% (1%, 3 mm) compared to the ground truth MC calculations, DoTA significantly improves upon analytical pencil beam algorithms both in precision and speed. Offering MC accuracy 100 times faster than PBAs for pencil beams, our model calculates full treatment plan doses in 10-15 s depending on the number of beamlets (800-2200 in our plans), achieving a 99.70 ± 0.14% (2%, 2 mm) gamma pass rate across 9 test patients. Significance. Outperforming all previous analytical pencil beam and deep learning based approaches, DoTA represents a new state of the art in data-driven dose calculation and can directly compete with the speed of even commercial GPU MC approaches. Providing the sub-second speed required for adaptive treatments, straightforward implementations could offer similar benefits to other steps of the radiotherapy workflow or other modalities such as helium or carbon treatments.

Background and purpose: In intensity modulated proton therapy (IMPT), the impact of setup errors and anatomical changes is commonly mitigated by robust optimization with population-based setup robustness (SR) settings and offline replanning. In this study we propose and evaluate an alternative approach based on daily plan selection from patient-specific pre-treatment established plan libraries (PLs). Clinical implementation of the PL strategy would be rather straightforward compared to daily online re-planning. Materials and methods: For 15 head-and-neck cancer patients, the planning CT was used to generate a PL with 5 plans, robustly optimized for increasing SR: 0, 1, 2, 3, 5 mm, and 3% range robustness. Repeat CTs (rCTs) and realistic setup and range uncertainty distributions were used for simulation of treatment courses for the PL approach, treatments with fixed SR (fSR₃) and a trigger-based offline adaptive schedule for 3 mm SR (fSR₃OfA). Daily plan selection in the PL approach was based only on recomputed dose to the CTV on the rCT. Results: Compared to using fSR₃ and fSR₃OfA, the risk of xerostomia grade ≥ II & III and dysphagia ≥ grade III were significantly reduced with the PL. For 6/15 patients the risk of xerostomia and/or dysphagia ≥ grade II could be reduced by > 2% by using PL. For the other patients, adherence to target coverage constraints was often improved. fSR₃OfA resulted in significantly improved coverage compared to PL for selected patients. Conclusion: The proposed PL approach resulted in overall reduced NTCPs compared to fSR₃ and fSR₃OfA at limited cost in target coverage.

In this work, we present the results of a preliminary uncertainty quantification and sensitivity analysis study of the Molten Salt Fast Reactor (MSFR) behavior at steady-state performed by applying a non-intrusive Polynomial Chaos Expansion (PCE) approach. An in-house high-fidelity multi-physics simulation tool is used as reactor reference model. Considering several thermal-hydraulics and neutronics parameters as stochastic inputs, with a limited number of samples we build a PCE meta-model able to reproduce he reactor response in terms of effective multiplication factor, maximum, minimum, and average salt temperatures, and complete salt temperature distribution. The probability density functions of the responses are constructed and analyzed, highlighting strengths and issues of the current MSFR design. The sensitivity study highlights the relative importance of each input parameter, thus providing useful indications for future research efforts. The analysis on the whole temperature field shows that the heat exchanger can be a critical component, so its design requires particular care.