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.

Dose calculations in proton therapy require the fast and accurate solution of a high-dimensional transport equation for a large number of (pencil) beams with different energies and directions. Deterministically solving this transport problem at a sufficient resolution can however be prohibitively expensive, especially due to highly forward peaked scattering of the protons. We propose using a model order reduction approach, the dynamical low-rank approximation (DLRA), which evolves the solution on the manifold of low-rank matrices in (pseudo-)time. For this, we compare a collided-uncollided split of the linear Boltzmann equation and its Fokker-Planck approximation. We treat the uncollided part using a ray-tracer and combine high-order phase space discretizations and a mixture model for materials with DLRA for the collided equation. Our method reproduces the results of a full-rank reference code at significantly lower rank, and thus computational cost and memory, and further makes computations feasible at much higher resolutions. At higher resolutions, we also achieve good accuracy with respect to TOPAS MC in homogeneous as well as heterogeneous materials. Finally, we demonstrate that several beam sources with different angles can be computed with little cost increase compared to individual beams.

Ultra-High Dose-Rate (UHDR) Proton Therapy is an area of active research due to its potential to target cancer cells while sparing healthy tissues. To deepen the knowledge of underlying biological mechanisms of FLASH effect, pre-clinical experiments are required, necessitating large uniform field achieving dose rates above 40 Gy/s. To achieve this, in the Research & Development (R&D) fixed horizontal proton beam line of HollandPTC, a fully 3D printed contoured passive scattering system has been developed. This system is designed to shape a 250 MeV proton beam into a suitable configuration for pre-clinical radiobiological experiments, achieving the necessary dose rates and uniform field distribution. The beamline configuration was initially modelled using the Monte Carlo-based Tool for Particle Therapy Simulation (TOPAS). Subsequently, the contoured passive scattering system was optimized through simulations to generate a sufficiently uniform field for future radiobiological experiments. To validate simulations, the system was fabricated using advanced 3D printing technology. A tungsten-heavy filament blend, consisting of approximately 75% tungsten by mass, was employed in a fused deposition modelling (FDM) process. Simulations were validated against experimental data. The measured dose distributions demonstrated a lateral field uniformity exceeding 95%, corresponding to a dose homogeneity within ± 3% over a field diameter of 2.8 cm. Dose rates above 40 Gy/s were achieved under experimental conditions, with measured values reaching up to approximately 60 Gy/s at the beam entrance, and 100 Gy/s within the Spread-Out Bragg Peak (SOBP). These results confirm the system’s design and performance, thus opening multiple possibilities for UHDR radiobiological experiments.

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.

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.

Objective. A dynamic model is developed to predict the impact of radiotherapy on circulating lymphocyte counts in women with locally advanced cervical cancer (LACC). This study aims to compare the effects of photon and proton therapy, as well as the influence of bone marrow sparing (BMS) techniques, on relative lymphocyte preservation over time. Approach. A dynamic lymphocyte flow model was developed to simulate the migration of lymphocytes based on seven compartments. Biological cell death and lymphocyte production were integrated across compartments. The lymphocyte flow model was applied to 19 LACC patients. Volumetric modulated arc therapy (VMAT) and intensity modulated proton therapy (IMPT) treatment plans were created for each patient without BMS and with BMS. The model calculated radiation dose to lymphocytes to estimate radiation-induced cell death over time. The output of the model was the relative lymphocyte count relative to baseline (RLC) over time and the RLC nadir in the blood and total body. Main results. According to the model, IMPT resulted in lower doses to lymphocyte and higher RLC nadirs compared to VMAT for all 19 patients. The total RLC nadir (mean ± SD) was 48.4% ± 4.0% for VMAT and 62.5% ± 5.1% for IMPT. In the blood compartment, the RLC nadir was 32.7% ± 3.5% for VMAT and 47.7% ± 5.9% for IMPT. The RLC nadir in the blood compartment improved with 3Gy BMS from 32.7% ± 3.5% to 33.0% ± 3.5% , while it decreased for IMPT from 47.7% ± 5.9% to 46.6% ± 6.0%. Total RLC nadir decreased with BMS for VMAT from 48.4% ± 4.0% to 48.2% ± 3.9% and for IMPT from 62.5% ± 5.1% to 60.9% ± 5.3%. Significance. By incorporating a dynamic flow model, we predicted the RLC over time. The model predicted a substantial sparing effect IMPT has on the lymphocytes compared to VMAT. This sparing was both present in the blood and the total body. Sparing the bone marrow showed only a minimal effect on the RLC.

HollandPTC is an independent outpatient center for proton therapy, scientific research, and education. Patients with different types of cancer are treated with Intensity Modulated Proton Therapy (IMPT). Additionally, the HollandPTC R&D consortium conducts scientific research into the added value and improvements of proton therapy. To this end, HollandPTC created clinical and pre-clinical research facilities including a versatile R&D proton beamline for various types of biologically and technologically oriented research. In this work, we present the characterization of the R&D proton beamline of HollandPTC. Its pencil beam mimics the one used for clinical IMPT, with energy ranging from 70 up to 240 MeV, which has been characterized in terms of shape, size, and intensity. For experiments that need a uniform field in depth and lateral directions, a dual ring passive scattering system has been designed, built, and characterized. With this system, field sizes between 2 × 2 cm² and 20 × 20 cm² with 98 % uniformity are produced with dose rates ranging from 0.5 Gy/min up to 9 Gy/min. The unique and customized support of the dual ring system allows switching between a pencil beam and a large field in a very simple and fast way, making the HollandPTC R&D proton beam able to support a wide range of different applications.

This paper presents an experimental study for the transient growth of an ice layer in a square channel under laminar flow conditions and a mixed convection heat transfer regime. The ice layer was grown from a cold plate located at the bottom of the channel, capable of reaching temperatures between 0 and −20 °C. The onset of ice formation was marked by a sudden sharp increase of the cold plate temperature followed by a rapid spreading of the ice over the cold plate surface. This was attributed to subcooling effects within the thermal boundary layer of the flow. The flow field was measured using particle image velocimetry (PIV) and the ice profiles were measured at several instances of time after the onset of freezing by a visual tracing of the solid–liquid interface. In addition, a parametric study was performed regarding the effect of the cold plate temperature and the flow rate on the ice growth rate. Suitable approximations to the experimental boundary conditions were found after a detailed analysis of the cold plate's transient temperature response, which could be readily implemented in numerical software. An important novelty of the present work is the measurement of the transient ice development of the ice-layer near the inlet of the channel, in addition to the centre of the channel where the flow is more developed. As such, a comprehensive and well-described experimental data set was generated for transient freezing in laminar internal flow. With this approach, a very good agreement was obtained between the experimental results and numerical simulations which were included to indicate the suitability of the current experimental campaign for numerical benchmarking purposes.

Objective.The integration of proton beamlines with x-ray imaging/irradiation platforms has opened up possibilities for image-guided Bragg peak irradiations in small animals. Such irradiations allow selective targeting of normal tissue substructures and tumours. However, their small size and location pose challenges in designing experiments. This work presents a simulation framework useful for optimizing beamlines, imaging protocols, and design of animal experiments. The usage of the framework is demonstrated, mainly focusing on the imaging part.Approach.The fastCAT toolkit was modified with Monte Carlo (MC)-calculated primary and scatter data of a small animal imager for the simulation of micro-CT scans. The simulated CT of a mini-calibration phantom from fastCAT was validated against a full MC TOPAS CT simulation. A realistic beam model of a preclinical proton facility was obtained from beam transport simulations to create irradiation plans in matRad. Simulated CT images of a digital mouse phantom were generated using single-energy CT (SECT) and dual-energy CT (DECT) protocols and their accuracy in proton stopping power ratio (SPR) estimation and their impact on calculated proton dose distributions in a mouse were evaluated.Main results.The CT numbers from fastCAT agree within 11 HU with TOPAS except for materials at the centre of the phantom. Discrepancies for central inserts are caused by beam hardening issues. The root mean square deviation in the SPR for the best SECT (90 kV/Cu) and DECT (50 kV/Al-90 kV/Al) protocols are 3.7% and 1.0%, respectively. Dose distributions calculated for SECT and DECT datasets revealed range shifts <0.1 mm, gamma pass rates (3%/0.1 mm) greater than 99%, and no substantial dosimetric differences for all structures. The outcomes suggest that SECT is sufficient for proton treatment planning in animals.Significance.The framework is a useful tool for the development of an optimized experimental configuration without using animals and beam time.

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.

We present a finite volume adaptive mesh refinement method for solid-liquid phase change problems with convection. The refinement criterion consisted of three different error estimators for the solid-liquid interface, the flow field, and the temperature field respectively. For the solid-liquid interface, the cells undergoing phase change were refined based on the maximum difference in the liquid fraction over the cell faces. For the flow field and the temperature field, an error indicator was used based on the cell residual method. To maintain a high parallelization efficiency, a dynamic load balancing procedure was used. The adaptive mesh refinement strategy was verified through three different test cases, these are the gallium melting in both 2D and 3D cavities, and the molten salt reactor freeze valve. For all three cases, very good agreement was obtained between the adaptive mesh results and the reference solutions. In addition, more accurate results were obtained with the adaptive meshes compared to static meshes with a similar amount of mesh cells. This illustrated the potential of the current approach for developing computationally efficient numerical methods for solid-liquid phase change problems.

This work presents two color LIF temperature measurements for the transient freezing in a square channel under laminar flow conditions. This is the first time non-intrusive temperature measurements were performed within the thermal boundary layer during the transient growth of an ice layer in internal flow. A combination of a local outlier factor algorithm and a smoothing operation was used to remove the top to bottom striations and reduce the other measurement noise. The temperature uncertainty in our measurements was between σ=0.3∘C and σ=0.5∘C. For the largest temperature difference between the bulk and the melting point of 14.6 °C, good results were obtained. As such, the current campaign demonstrates the potential of LIF as a non-intrusive temperature measurement technique for solid–liquid phase change experiments. However, some artefacts were present within the vicinity of the ice-layer due to the scattering of the laser light, especially near the inlet of the channel where the ice-layer is curved instead of flat. LIF measurements taken within a short time span prior to the onset of ice freezing showed approximately 2 °C of subcooling, consistent with previous findings. In addition, an anomalous behavior within the thermal boundary layer was observed, with a much smaller temperature gradient within the first few mm above the cold plate and a point of inflection in the temperature profile. The anomalous temperature behavior is possibly attributed to enhanced natural convection as a result of the subcooling at the cold plate surface.