Computational power is a challenge when it comes to the high-fidelity modeling of nuclear reactors. Detailed simulations of reactor physics involve complex calculations that require significant computing resources, which can be time-consuming and expensive. Reduced Order Modeling (ROM) allows for an approximation of a complex model by only capturing the essential features, thereby reducing the computational load. A reduced order model provides computationally efficient approximations of a system, but it requires still many evaluations of a high-fidelity model to capture all the dynamics. Using the adaptive sparse grid can reduce the number of evaluations needed, though the construction of the reduced order model is still computationally intensive. The aim is to minimize the computational workload involved in constructing a reduced-order model during the offline phase. This is achieved by decreasing the number of high-fidelity model evaluations necessary for building the reduced order model while maintaining accurate results. To this end, the existing adaptive proper orthogonal decomposition algorithm is enhanced by employing multi-fidelity techniques. Multi-fidelity methods aim to combine large amount of low-fidelity data with a limited amount of high-fidelity data to compute accurate, yet computationally inexpensive approximations. Two novel multi-fidelity reduced order model methods based on proper orthogonal decomposition are proposed; Filtered Bi-Fidelity Adaptive Proper Orthogonal Decomposition (FB-POD) algorithm and Adapted Bi-Fidelity Proper Orthogonal Decomposition (AB-POD). These models are evaluated on two different test cases, and the balance between the accuracy of each multi-fidelity ROM and the computational cost, measured by the number of high-fidelity evaluations, is investigated. In specific cases, the proposed methods significantly reduce the number of high-fidelity evaluations compared to the single high-fidelity ROM, while yielding comparable accuracy.

The aim of this thesis is to study the dosimetric feasibility of FLASH proton therapy for early-stage breast cancer patients. The biological effect of FLASH is seen under ultra-high dose-rates conditions and is beneficial in the damage to healthy tissue. The FLASH effect could enable the clinical feasibility of definitive radiotherapy for low-risk early-stage breast cancer patients. Some of these patients can not receive surgery for medical reasons and definitive conventional radiotherapy carries a high risk of toxicities. FLASH proton therapy could be a viable option for these patients. The biological FLASH effect enables to reduce the risk of toxicities by sparing healthy tissue and proton therapy enables to deliver dose to the cancer cells effectively. This thesis proposes a pencil beam scanning (PBS) ridge filter simulation set-up to achieve ultra-high dose-rates. The ridge filter generates spread-out Bragg peak (SOBP) beams that cover a broader depth of the tumor, avoiding the time-consuming energy switching of conventional PBS. An optimization is performed with intensity-modulated ridge filtered beams. SOBP beams with a width of 3 cm are implemented in the in-house treatment planning software Erasmus-iCycle to generate ridge filter treatment plans for two patients. In this study, the dosimetric feasibility is determined by comparing the ridge filter treatment plans to conventional intensity-modulated proton therapy (IMPT) treatment plans. The clinical acceptability of the treatment plans is assessed by dose uniformity and dose coverage. Furthermore, toxicity-model endpoints for fat necrosis and fibrosis are used to evaluate the treatment plans. The generated ridge filter treatment plans outperform the generated IMPT treatment plans for a FLASH enhancement ratio between 1.19-1.39 for one of the patients. The other patient required a FER in the order of magnitude of 2.0. The results of the study suggest that it is feasible to generate ridge filter treatment plans with the proposed set-up. However, the dosimetric feasibility comes with limitations in the tumor characteristics (size and position) and needs further investigation. More knowledge is needed of the optimal ridge filter, the optimization of SOBP beams, and the FLASH effect before clinically acceptable FLASH-compatible ridge filter treatment plans can be achieved.

In this report, the conversion from spin-echo signals, obtained with a low-field hand-held MRI scanner that was designed and built at the Leiden University, to images of the proton density within the sample is considered. This scanner does not make use of switchable gradient coils, but instead relies solely on the natural inhomogeneity of the field and on translations of the sample over this field for its spatial encoding. Specifically, an attempt is made to answer the question of how we can reconstruct a phantom using this kind of scanner. This is done by deriving a signal model, discretising it and writing it as a linear least squares problem. Then, we can make use of the techniques of Cojugate Gradient for Least Squares (CGLS) wih `2-regularization and Generalized Conjugate Gradient Minimal Error (GCGME) with `1-regularization for the difference between neighbouring pixels in order to solve this inverse problem. Firstly, we theoretically consider combinations for magnetic field geometry and measurement strategy for their usability for image reconstruction. After this, the obtained strategies are tested in three experiments, with two magnets and two samples. We start by doing this numerically, using a simulated phantom in combination with a measured magnetic field and the translation strategy. By doing this, we can determine if reconstruction is possible using that combination of field and strategy. Finally, the strategy is tested on real samples. Using numerical phantoms in combination with the magnetic field and translation strategy used in the measurements, we were able to correctly reconstruct the phantoms. However, the reconstruction broke down when data from real samples was considered. A variety of possible improvements is discussed. The improvement that would have the most impact would be to design a magnet that has a less uniform gradient in the z-direction, and instead has some locations in the xy-plane where the field falls slowly as function of z, but quickly in other locations.

Geothermal energy can be promising to help the energy transition. Warm water is pumped up from a geothermal well and after it is cooled down, the cold water is reinjected into the hot aquifer underground. The temperature change could cause chemical balances to be disturbed and result in precipitation on the porous medium of the aquifer. This process of clogging of an aquifer is not desirable since it re- duces the efficiency of the production of warm water. This research is a stepping stone for modelling these reactions and clogging of the aquifer. The reactions rates depend on the temperature and the concentration of the reactants in the aquifer. For this reason the temperature distribution of the aquifer was numerically derived with finite difference methods. The velocity field is important for modelling both the temperature and the concentrations. Therefore the pressure and velocity field were modelled with use of mass conservation and the Darcy model. The models were solved numerically with finite difference methods and some results were compared to the analytical solution of the model. It was concluded that the numerical model for the velocity field is accurate when compared to analytical solu- tions of the model equations. In the model used, an increase in porosity resulted in a delay of cooling the aquifer. This is because there was no relation between porosity and permeability implemented in the model. A lower flow rate in the aquifer delays the cooling of the aquifer as well. For further research it is recommended to implement the chemical reactions into the model and use this model to look into how to prevent the clogging in the aquifer.

In medical images intratumour heterogeneity well translates specific cancer cells properties in a fast and non-invasive way. Medical community and researchers have developed proper tools (texture features, shape features…) permitting the quantification and the analysis of this tumour heterogeneity in particular in PET images. Yet so far, both doctors and researchers have mainly focused on the use of those tools before assessing their relevance and robustness. In this study, a new heterogeneous PET phantom allowing an extensive understanding of the key concept of heterogeneity quantification in PET images is designed and partially developed. This phantom aims at representing a complex enough environment mimicking clinical conditions to properly challenge PET heterogeneity data extraction and quantification methods further than commercial phantom already do. This study focuses, first, on defining sound specifications, design methodology and manufacturing method for this new object. Second, the study focuses on designing and developing the defined solution. Third, a proof-of-concept analysis is conducted within the study to test and validate the developed PET phantom prototype. It can be concluded that the produced PET phantom was successfully designed and developed and can be used by other operators; yet, there is still room for improvement. Finally, besides developing a novel PET phantom, this study yields recommendations to improve the work done toward a more handy, flexible and realistic tool.

The delineation of the regions of interest (ROIs) plays an important role in the radiotherapy workflow. The ROIs include the gross tumor volume (GTV), the clinical target volume (CTV), the planning target volume (PTV) and organs at risk (OARs). There are however uncertainties related to the delineation of the ROIs due to inter­observer variability between radiation oncologists caused by e.g., a lack of consensus on anatomic definition or a lack of contrast in medical images. The largest inter­observer variability has been found in esophageal cancers, head and neck cancers, and lung cancers. To pre­ vent inter­observer variability, auto­contouring software can be used to delineate the ROIs. There are however also uncertainties in the delineations made by auto­contouring software. The purpose of this study was to perform an accurate evaluation of the dosimetric effect of delineation uncertainties. To do so, first a method to characterize the delineation uncertainties had to be found. The delineations uncertainties were characterized with principal component analysis (PCA). By performing PCA on a set of delineations, the eigenmodes which represent the variations between the delineations can be found and new random delineations based on the eigenmodes can be formed. To determine the dosimetric impact of the delineation uncertainties, two cases were investigated: (1) the dosimetric impact of delineation uncertainties for a fixed dose distribution; and (2) the dosimetric impact of delin­ eation uncertainties for a dose distribution reoptimized for every possible realization of a delineation. For the fixed dose distribution, Polynomial Chaos Expansion (PCE) was used as a meta­model for the dose volume histogram (DVH) of the target, and for the reoptimized dose distribution PCE served as a meta­model for the total dose distribution and the DVHs of the target and other ROIs. The delineation uncertainties of two data sets were analyzed: (1) 12 manual delineations of the GTV of a hepatocellular carcinoma patient; and (2) 90 auto­contours of the CTV and brainstem of a head and neck patient. The variation of the manual delineations of the GTV could be accurately described by 5 eigenmodes, while 45 eigenmodes were needed to describe the variations of the brainstem auto­ contours. The delineation uncertainties in the auto­contours of the CTV could not be characterized by PCA due to the shape of the CTV. The dosimetric effect of the delineation uncertainties of the manual delineations for a fixed dose distri­ bution was investigated for both an intensity modulated proton therapy (IMPT) plan and a volumetric­ modulated arc therapy (VMAT) plan for 10,000 random delineations of the CTV. For this patient the CTV was equal to the GTV. In the VMAT plan, the CTV received sufficient dose for all delineations, but the PTV was underdosed in 17.1% of the delineations. In the IMPT plan, the CTV was underdosed in 69.0% of the delineations. This percentage of underdosed CTV delineations in the IMPT plan was reduced when the plan was made robustly. The results for the fixed IMPT dose distribution should however be verified with a more accurate PCE model. A proof of principle for the PCE as a meta­model for the reoptimized dose distribution and DVHs of the target and other ROIs was shown. However, a more accurate PCE model with a higher polynomial order would be needed to analyze the effects of the reoptimization on the dose delivered to the ROIs. It has been shown that the dosimetric impact of delineation uncertainties can be modelled using PCE. This is a first step towards systematically and quantitatively taking into account delineation uncertainties in radiotherapy treatment planning. In future research the analysis of the dosimetric impact of the un­ certainties can e.g., be used in adaptive radiotherapy in which auto­contouring is used. By knowing the dosimetric impact of the uncertainty, treatment plans could be optimized such that a structure receives its target dose with a certain probability or critical areas where the dosimetric impact of the delineation uncertainties is large could be flagged such that these areas are checked before treatment.

Radiotherapy (RT) is an important modality in treatment against cancer. Developments in this modality are necessary for improvement of patient complications and survival. Aside from fractionation and precise irradiation, FLASH radiotherapy could be a third way to significantly increase the therapeutic window of radiotherapy.The FLASH effect is a biological, healthy tissue sparing effect that is found in tissue that receive a high dose (>10 Gy) within a very-short irradiation time (<100 ms). Although the mechanics behind the FLASH effect remain unknown, it is found to benefit a large variety of tissues and organs. Nevertheless, FLASH-RT still lacks clinical translation and implementation of FLASH irradiation is limited by current technology. However, the extremely high dose rates required for FLASH can readily be achieved using cyclotron accelerated proton transmission beams. Cancer patients with small lung tumours are a preferred starting point for clinical translation of FLASH because (i) transmission beams are of special benefit for lung tumours since they mitigate for range uncertainties and contain a sharp lateral penumbra, (ii) high doses are required for the FLASH effect to occur and hypofractionation is not uncommon for lung patients, and (iii) the very short irradiation times and high doses in combination with gating allow smaller treatment margins for moving tumours.In this study, we took a first step towards intensity modulated proton therapy (IMPT) with FLASH treatment plan optimisation by (i) investigating the most delivery-time effective way to irradiate a field, (ii) proposing an optimisation method to maximise FLASH within a treatment plan, and (iii)analysing and evaluating FLASH treatment plans. Because increased pencil-beam (spot) separation showed to benefit the pencil-beam scanning(PBS) delivery time of a field, spot overlap minimisation was proposed as optimisation method for FLASH. Treatment plans using 3, 5 and 7 transmission beams were generated with and without FLASH optimisation for 6 small lung tumour patients. Although spot overlap minimisation doesoccasionally improve delivery time for a limited number of beam angles, the overall treatment plan does not benefit from the optimisation. Therefore, spot overlap minimisation cannot be used for FLASH optimisation in clinical IMPT treatment planning. Recommendations for future research include optimisation on PBS delivery time and PBS pattern optimisation.

In proton therapy, the calculation of the dose distribution is of great importance in order to find the best treatment plan. For a treatment plan with high quality, several error scenarios are investigated, in order to come up with a general plan that suits best in these scenarios. All of the scenarios have a different simulated error, e.g., in patient displacement and proton beam range. In order to cope with a large number of errors scenarios, a quick calculation method is needed. In this work, a method that used Reduced Order Modelling (ROM) and subsequently polynomial regression to overcome the computational problem is presented, as well as its results on a real head and neck cancer patient. Before error scenarios are simulated, using inverse optimisation a treatment plan is calculated for a perfectly known position and organ structure of the patient, i.e. the nominal scenario. Then, several error scenarios are simulated and for each of these scenarios, the real dose on tissue is calculated, using the treatment parameters that were previously set for the nominal scenario. The dose distributions of all these scenarios are stored in a matrix, on which a Singular Value Decomposition (SVD) is executed. A reduced order of singular values and vectors is used, together with regression models on the right singular vectors, to reconstruct the dose distribution matrix. This is done in order to create a computationally cheap method for dose distribution calculations. The right singular vectors are now a function of the used parameters: positioning errors dx, dy, dz and proton beam range error dρ. The dose distribution matrix is reconstructed from the left singular vectors, first few singular values and the function for the right singular vectors. Range errors with a uniform distribution with a maximum of 3% and positioning errors with a normal distribution with a standard deviation of 3 mm in each direction were simulated. For 100 error scenarios, the dose distribution matrix could be reconstructed with an average accepted error of 1% on a voxel relative to the maximum dose for 96:5% of the voxels, using a voxel-by-voxel comparison. These results were obtained for a 17th order of the SVD and a 7th order polynomial in the regression. On unseen test data, the acceptance was 81:0% of the voxels with an allowed error of 1%. For an allowed error of 3%, 92:5% of the voxels of this test set were accepted. These results are promising and encouraging for future research. First of all, it is recommended to examine the regression of the right singular vectors more closely, especially the regression of the higher order right singular vectors. Further subjects of interest would be to perform the same procedure that is done on the dose distribution matrix on the dose influence matrix. Also, harder-to-reach error scenarios, like organ movement and tumour deformation could be taken into account. This research was conducted at the Medical Physics and Technology section, a part of the Department of Radiation Science and Technology at Delft University of Technology. The project was part of the Bachelor program of Applied Physics at the Technical University of Delft.

Numerical models are paramount in describing the complex physical world around us. They are often based on non-linear functions, which have to be solved in order to run a simulation. The first goal of this thesis is to compare the Jacobian-free Newton-Krylov method against the regular Newton-Raphson method for solving non-linear equations. When doing this, preconditioning was only used for the Newton-Raphson method. As a second goal, the optimal parameters for solving a coupled system with the Schur complement method are determined. For computational purposes, these analyses are performed using PETSc. The comparison between JFNK and Newton-Raphson were performed by solving a heat equation. The physical system which was analyzed is a one-dimensional radiating rod, with a heterogeneous thermal conductivity and Dirichlet boundary conditions. Firstly this rod was modelled as one single system. For this case, it was found that Newton-Raphson outperformed the JFNK method by a factor of 5-1700, depending on the type of preconditioning used. Secondly, the rod was modelled as a coupled system by solving two parts of the rod separately. In this case, it was found that the JFNK method outperformed the preconditioned Newton-Raphson method by a factor of 9.1± 0.3. It was concluded that the JFNK is only favorable over regular Newton-Raphson for coupled systems. To achieve the second goal, an incompressible Navier-Stokes coupled system was solved using the Schur complement method. The coupled system originated from incompressible flow in a back-step pipe, using a finite element method. For the Schur complement parameters, it was shown that using an approximation of the Schur complement offered a significant increase in efficiency. The momentum and pressure subsystems were analyzed separately. The momentum subsystem performed best with a relative tolerance of τr =10-4.5, while the pressure subsystem performed best with a tolerance of 10-2. With these parameters, the Schur complement method had the same computational time as the pressure-correction method, which was used as a benchmark. For future research, there are three main recommendations. Firstly, to use preconditioning for the JFNK method. This was not done in this thesis because of technical constraints, but it is theoretically possible. Secondly, if the JFNK method is preconditioned, it could be used for the incompressible Navier-Stokes simulation. Finally, there are combinations of parameters which were not tested for the Schur complement method. Trying out more combinations might result in finding an even more optimized method.

The freeze plug is a key safety component of the molten salt fast reactor (MSFR), one of six next-generation nuclear reactor technologies being developed under the Generation IV International Forum (GIF). It should be designed to melt if an accident occurs, allowing the MSFR to drain before it incurs structural damage. Two freeze plug concepts have been considered in recent years, in which the plug is melted either through the decay heat produced in the core, or through heat generated by special heating rings and stored in steel blocks adjacent to the freeze plug. Variations consisting of both a single freeze plug, and multiple smaller plugs contained in a metal plate, have been proposed. This work seeks to evaluate the feasibility of these designs and study how parameters such as the sub-cooling of the plug affect melting times. Additionally, an alternative, wedge-shaped freeze plug design is proposed for increased reliability.  Simulations performed in COMSOL showed that the decay heat plug melts within 600 s only if placed within 0.01 m of the mixed core flow. Because such a placement makes the plug vulnerable to temperature and velocity fluctuations in the core during regular operation of the reactor, this design is considered unfeasible and is not recommended for further study. On the other hand, melting times under 600 s were possible with the heating ring design for a range of sub-cooling amounts and plug configurations, suggesting that this design is promising. A thin frozen layer was shown to form on top of the metal grate in the multi-plug configurations, preventing heat transfer through the top of the plate. Although the melting behavior of this layer warrants further investigation, its insulating effect was found to generally cause the single-plug designs to melt faster than the multi-plug designs. A simplified, isothermal model of the wedge-shaped plug was simulated using the enthalpy-porosity approach to account for convection. To model the sinking of the wedge, an extended Darcy term approach was developed based on an analytical solution which was validated experimentally, with good agreement. This model shows that melting of the wedge is unsteady, and that melting times depend linearly on the wedge angle and sub-cooling. Unfortunately, melting times of the wedge plug could not be estimated with realistic, non-isothermal, time-dependent boundary conditions. For future study, a customizable numerical solver such as OpenFOAM is recommended, which would allow the sinking of the solid phase to be modeled more robustly through an immersed boundary method.

The main challenge of radiotherapy, used for treating cancer, is to deposit the prescribed dose in the tumor volume while sparing the surrounding tissue.The depth-dose distribution of protons makes proton therapy an alternative to conventional radiotherapy for some tumor sites. A better knowledge of the proton radiobiological mechanisms can improve the effectiveness of radiotherapy treatments. Holland Proton Therapy Center (HPTC) is one of the Dutch proton therapy centers. One of the purposes of its experimental beamline is to perform radiobiological experiments. To conduct different types of pre-clinical experiments, the beamline must be equipped to provide large field irradiation with precise dose characterization. Moreover, having a reliable Monte Carlo(MC) model of the system allows to perform in silico verification of the beamline design and contribute to its optimization. In this context, the goals of this project were: implementing a dual-ring scattering system in the experimental room of HPTC to produce homogeneous fields of different sizes; creating a MC model of the HPTC passive beamline able to reproduce the experimental setup.A dual-ring double scattering system was implemented in the HPTC horizontal beamline, starting from a single 150MeV pencil beam. The resulting passive irradiation fields were characterized by measuring and analyzing the lateral beam profiles, the depth-dose distributions and the relative dose at target position. Moreover, the beam characteristics and setup were implemented in theTOPAS MC code. The beam source parameters, input of the MC model, were found by comparing the experimental and simulated beam envelope and depth-dose distributions of the pencil beam. Then, the passive system model was benchmarked with experimental data by evaluating the lateral profiles, Braggcurves and dose distributions.The results show that the implemented passive system can achieve dose uniformity between 96% and 99% for field sizes between 4x4cm2 and 20x20cm2for a 150 MeV proton beam. Moreover, using a collimator with a 5x5cm2aperture, uniformity of at least 97% in the different Bragg peak regions is achieved. Good uniformity is also obtained for beam energies in the range 115MeV-150 MeV, showing robustness of the setup. Furthermore, the range of the proton beams traversing the beam-shaping elements, as well as the energy arriving to target, were studied. Moreover, with a ridge modulator, Spread-OutBragg peaks were obtained with a width up to 3,4cm and of uniformity 98,5%.The MC model produced in TOPAS was first benchmarked against a 150MeVproton beam in air. Secondly, the experimental data of the large fields were compared with the simulated ones. The simulation of depth-dose distributions and lateral beam profiles agreed with the experimental data. Furthermore, the model was used to estimate the number of initial protons required to achieve an experimental dose and to estimate the location of the Bragg peaks.All in all, the dual-ring passive scattering setup has been successfully implemented and is ready to be used to perform radiobiological experiments. Also, the TOPAS MC model can be used to assist in the preparation of these experiments and to further optimize the beamline design.

Radiotherapy is one of the main treatment modalities available to treat cancer. Radiotherapy treatment plans are created based on CT scans of the patient. In such scans the macroscopic tumor is visible, but microscopic disease present in the surrounding tissue cannot be observed. To achieve an optimal clinical outcome, both the macroscopic and the microscopic disease must be treated. Currently, the macroscopic tumor is extended by a margin into the Clinical Target Volume (CTV) to include the microscopic disease in the treated volume. The same margin is used for all patients, although the extent of microscopic disease is patient-specific and can vary largely among patients. In this study, probabilistic treatment planning was investigated as a method to replace the margin concept. Probabilistic models were created by explicitly modeling uncertainties in the microscopic disease into an objective function used in the treatment plan optimization. By optimizing either the expected Tumor Control Probability (ETCP) or the expected Logarithmic Tumor Control Probability (ELTCP), optimal dose distributions could be obtained. Two different one-dimensional models for probabilistic treatment planning were investigated. In the first model, the uncertainty in the extent of the microscopic disease was modeled into an objective function. This was done using a function that describes the probability of finding microscopic disease at a certain distance from the macroscopic disease. In the second model, the uncertainty in the tumor cell density in the microscopic disease area was modeled into an objective function. The uncertainty was modeled by defining the tumor cell density field as a random field and generating different realizations of the tumor cell density field using a Karhunen-Loève (KL) expansion. For the first model, both the ETCP and the ELTCP were used as objective functions and in the second model, only the ETCP was used as an objective function. Furthermore, a penalized ETCP objective function was investigated for both models. In this penalized objective function a penalty on the dose was used to allow for controlling the balance between tumor control and sparing of normal tissue. Using the first model, two different types of dose distributions were found. When the ETCP was optimized, the maximum dose was given to as large a volume as possible and no dose was given in the rest of the investigated volume. When the ELTCP was optimized, dose was given throughout the volume, so that the whole volume received as much dose as possible. Optimization of both objectives resulted in good tumor control. When the penalized ETCP was optimized, dose was given to a much smaller part of the volume than with the unpenalized objective, while the tumor control was still good. Using the second model, it was shown that the KL-expansion is a promising method to model the uncertainty in tumor cell density. Different shapes of the input mean tumor cell density field were investigated. Optimizing the ETCP resulted in realistic dose distributions. Good tumor control was obtained for the different shapes of the input mean tumor cell density field. Furthermore, using the penalized ETCP, good tumor control was retained, while the dose deposited in the volume was decreased. In conclusion, probabilistic treatment planning promises to be a good alternative to the current margin concept. It was shown that good tumor control could be achieved in the microscopic disease area using probabilistic objective functions. Both models showed promising results and the penalized objectives showed that it is possible to balance between tumor control in the microscopic disease area and sparing of normal tissue. Additional research is necessary to extend the one-dimensional KL-model into a more detailed three-dimensional model. Furthermore, the objectives need to be implemented in treatment planning systems to create real patient plans. Such studies should be performed in cooperation with clinicians and radiologists.

The Molten Salt Fast Reactor (MSFR) is a next generation nuclear reactor. The MSFR may play an important role in combatting global warming whilst ensuring energy safety considering fossil fuels’ finiteness. A key safety mechanism for the MSFR is the freeze plug, a valve underneath the reactor core designed to melt during reactor malfunction, letting the fuel salt drain into an emergency draining system. In this thesis, the effect of the inclination angle of the freeze plug on its melting behaviour was investigated. This was done via the Linearised Enthalpy Method for phase change in OpenFOAM 8, a Finite Volume Method software package. First, two benchmark cases - one of inclined gallium melting and one of a heater copper fin in air - were performed. These showed reasonable agreement with established results. A cylindrical freeze plug consisting of LiF-ThF4 salt was defined, with copper and hastelloy annuli around it to simulate the cooling and the draining pipe, respectively. Steady-state simulations were performed for inclination angles of 𝜃 = 0°, 15°, 30°, 45°, 60°, and 90°. Via simulations on meshes with 40,500 and 1.1 million cells, it was shown that the freeze plug did not form at 𝜃 = 60° and 90°. For the other four inclination angles, the resulting freeze plug was found to be mesh convergent via a third mesh of 3.6 million cells. Then, a transient simulation mimicking reactor malfunction was performed for the inclination angles 0° ≤ 𝜃 ≤ 45°. It was found that the start and end opening times - the times where the melt front first and last reached the freeze plug’s bottom - were reduced by 74% and 52%, respectively for 𝜃 = 45° with respect to 𝜃 = 0°. The other two 𝜃’s also showed lower opening times compared to 𝜃 = 0°. The inclination angle thus was found to have a significant effect on the freeze plug’s melting behaviour, shortening the opening times. Physical improvements for the model include the incorporation of molten salt leakage after start opening time, and settling. Computational improvements are also possible, such as the refinement of the mesh near the salt boundary or adaptive mesh refinement near the solid-liquid boundary. Symmetry can be exploited to speed up results.

Very high energy electron (VHEE) radiotherapy is being investigated as a potential replacement of photon therapy. VHEE pencil beams have a small penumbra and strong depth dependence for radiotherapy when compared to photons. This allows a lower dose to healthy tissue. Generating these high electron energy beams could be achieved by using laser accelerators. These accelerators allow the equipment to be smaller than currently possible and make it possible to fit them in a standard radiotherapy treatment bunker. This makes VHEE a potential middle ground between photon and proton therapy in relation to equipment costs and treatment quality. The purpose of this project is to compare photon based Intensity Modulated Ra­diation Therapy (IMRT) and VHEE treatment plans for treatment of prostate cancer.  For 10 prostate cancer patients IMRT treatment plans were generated with 23 beams. These plans were optimized using Erasmus MC in-house developed Erasmus-iCycle automated treatment plan optimization tool. The dependence of VHEE plan quality on beam energy (200, 300 and 400 MeV) and number of equi-angular beams (9, 18 and 36) was investigated. The treatment plans were optimized using the multi-criterial optimizer Erasmus-iCycle. For each patient VHEE pencil beam dose distributions were pre-calculated using TOPAS MC, a Monte Carlo simulation program based on Geant4.  Results: VHEE treatment plans show either a reduced or similar mean OAR dose when compared to IMRT treatment plans except for the 9 beam 200 MeV plan which was worse compared to IMRT. All treatment plans were normalized to a PTV coverage of 99 % V57Gy.  It is found that the mean rectum dose reduces from 13.5 Gy for IMRT to between 9.2 and 11.9 (p=0.002-0.004) for the VHEE plans. For the anus dose a reduction in mean dose was found for all VHEE treatment plans except for the 9 beam 200 MeV and the 18 beam 200 MeV VHEE treatment plans. The mean anus dose reduced from 12.4 Gy for IMRT to 7.0 - 10.6 Gy (p=0.002-0.014) for the VHEE treatment plans. The bladder dose reduced from 20.1 Gy to between 15.2 and 18.2 Gy (p=0.002).  Increasing the number of VHEE beams in a treatment plan reduces the OAR dose. Comparing 9, 18 and 36 beam treatment plans with 300 MeV. The mean rectum dose reduces from 14.0 Gy for the 9 beam plan to 11.1 Gy (p=0.002) and 9.9 Gy (p=0.002) for the 18 and 36 beam plan, respectively. The same pattern is found for the anus and bladder.  Treatment plans with a higher beam energy reduces the dose to OAR. The mean rectum dose reduces from 13.4 Gy for the 200 MeV plan to 11.1 (p=0.002) and 10.0 (p=0.002) Gy for the 300 and 400 MeV plans. The same pattern is found in the mean bladder dose with 18.2 Gy, 16.8 Gy (p=0.002) and 15.9 Gy (p=0.002). The right femoral head maximum dose increases from 28.0 Gy for 200 MeV to 29.4 Gy(p=0.002) and 29.6 Gy (p=0.049) for the 300 and 400 MeV treatment plans.  VHEE is a potential replacement for IMRT due to the reduced dose to healthy tissue while maintaining similar target coverage compared to IMRT. By increasing the number of beams and/or the electron beam energy we can further reduce doses to the healthy tissue.  

The goal of this project is to see if we can improve the treatment plan for proton therapy by using reduced order models and adjoint theory for proton therapy. We shall use a singular value decomposition on a dose distribution matrix to obtain the modes from which we can reconstruct every dose distribution. Using adjoint methodologies for proton therapy, we will define a response from which we can find the sensitivities, or gradient, in order to fit a Hermite interpolation polynomial on multidimensional simplices. The results show that the Hermite interpolation polynomial is a useful tool to find responses for low dimensional problems. For errors in one or two dimensions, the Hermite polynomial was able to reconstruct all dose distributions with R²=1. However, for an error in three dimensions the Hermite polynomial sometimes fails to reconstruct the dose distribution to within acceptable margins. We conclude that the combination of a ROM and adjoint method to find Hermite interpolation polynomial is a promising tool in order to further improve the proton therapy treatment plan. Further research should be done in order to determine whether Hermite interpolation can be used in every scenario, or if it fails if the grid becomes irregular. Finally, the extrapolating qualities of the Hermite polynomial should be tested.

In proton therapy the development of dose calculation methods is of great importance to optimize the dose. In this work a method is presented to solve the Fokker-Planck transport equation using ray-tracing techniques and nite element methods. The idea of the method is dividing the Fokker-Planck equation in two seperate equa- tions, one without scattering and the other with scattering. The result of the unscat- tered equation is then used to function as source for the second one. The equations are discretized in all free variables: space, angle and energy. The spatial domain is discretized such that each spatial element has its own angular and energy mesh. The main solution technique is the discontinuous Galerkin (dG) method in combination with ray-tracing techniques. The results from the ray-tracing method for calculating the unscattered dose correspond to known physical phenomena. The method for Gaussian sampling however shows a great sensitivity to the number of sampling beams used and yields consistent results only for a numer of sampling beams in the order of 103. In future work the sampling needs to be done by using the quadrature method, which results in a smaller error and uses only 4 sampling beams, resulting in a more favorable computation time.

In Intensity Modified Proton Therapy, the number of energy layers and the number of beamlets determine the radiation time and the plan calculation time. The purpose of this research project is to test sparsity inducing terms to reduce both the energy layers and the beamlets.