This thesis explores the improvement of computational efficiency in simulating two-dimensional conveyor belt systems by applying model order reduction (MOR) techniques. Conveyor belts, crucial for material handling in various industries, are traditionally modeled using finite element methods (FEM), which can be computationally demanding, particularly for long-term simulations with a high number of grid elements. To address this, MOR techniques aim to reduce high-dimensional models into lowerdimensional models.
The study investigates three MOR approaches—modal decomposition, Proper Orthogonal Decomposition (POD), and Dynamic Mode Decomposition (DMD)—to apply on existing software built by VORtech that simulates two-dimensional conveyor belt systems. When considering MOR to reduce the two-dimensional system, challenges arise related to nonlinearities, differential algebraic equations (DAEs), and complex modeling steps. Among the methods tested, DMD proved to be the most effective, offering significant reductions in computational time while maintaining accuracy. POD also demonstrated accuracy but had less impact on speed due to the time-consuming complex modeling steps in the simulation software. These complex modeling steps are not investigated in detail in this thesis and therefore not reducible with the intrusive POD. Because of the non-intrusive nature of DMD, this method was able to incorporate these extra processes in the reduced order model.
The study concludes with recommendations for future research, emphasizing the need for optimization of the code segments that handle the complex modeling steps. In addition, conventional modeling approaches as alternatives to the complex interpolation step could be explored, to enhance the applicability of MOR. Furthermore, DMD with control or parametric DMD could be explored to obtain a reduced order model by interpreting misalignments of rollers as controls or parameters. Finally, a method is proposed to make modal decomposition useful for models, where the solutions depend highly on the external forces. Although this method is not applied to the model considered in this thesis, it would be interesting to explore this modified modal decomposition method on other models that are significantly influenced by the external forces.

To diagnose diseases using low-field MRI devices, automatic image recognition techniques are required to detect anomalies in a given image. A critical component of this process is image segmentation, which involves dividing the image into coherent regions with similar features to detect the anomalies. Low quality images make segmentation difficult by the existence of noise in the images. The Chan-Vese model is a segmentation technique to segment images into two regions. This report aims to segment the low quality images obtained by the low-field MRI devices with use of the Chan-Vese model. This model is found to be particularly well-suited for handling the challenges posed by low-quality images which the low-field MRI devices produce. Future research should investigate if it is possible to further segment the region with the objects, such that each object has a separate region.

Angiogenesis, i.e. the formation of blood vessels from existing ones, plays a vital role in bone or wound healing. The expansion of vascularization facilitates the healing process through the delivery of oxygen and nutrients to the injured site and through the removal of waste products. Clinical observations indicate that impaired angiogenesis can impede the healing process, or can result in non-healing outcomes.
The computational model developed in this thesis predicts tip/stalk cell patterning, marking the initial phase of sprouting angiogenesis. Growth factors signal endothelial cells to differentiate into tip and stalk cells. Tip cells branch from the existing vessel, leading the sprout, while stalk cells proliferate and follow behind, forming the newly emerged blood vessel. Understanding tip/stalk cell patterning is vital to ensure successful angiogenesis, as an excess or deficiency in tip cells leads to improper healing.
Despite several experimental studies and mathematical models exploring the signaling pathways behind tip cell selection, there is a noticeable gap regarding the effect of extracellular matrix (ECM) stiffness on this process. Given that alterations in stiffness occur in various physiological and pathological processes, comprehension of this effect is clinically relevant. This thesis aims to address the existing gap by investigating the specific influence of ECM stiffness on tip/stalk cell patterning.
A computational model is created that simulates a vessel sprout under stimulation of growth factors. This model is able to predict the cell patterning over various ECM stiffness levels, and highlights the relevance of incorporating ECM stiffness in the investigation of angiogenic treatments.
Enhancing the models’ accuracy and validating the ECM stiffness-dependent model predictions requires additional experimental data. However, further development of the model has great potential for deepening our understanding of angiogenesis dynamics and for facilitating the investigation of treatment strategies.

HER2+ breast cancer patients, as observed by oncologist Agnes Jager from Erasmus Medical Centre (EMC), often achieve radiologic complete response (rCR) earlier than expected under standard treatments. To address this, Jager has partnered with Delft University of Technology to develop a computational model aimed at personalizing treatment schedules, potentially reducing chemotherapy cycles and minimizing side effects.

Building on previous MSc theses by Nathalie Oudhof, Eva Slingerland, and Rutger Engelberts, this thesis aims to improve the predictive capability of the Drug-Induced Mechanically Coupled Reaction-Diffusion (DI-MRCD) model and test its performance on a larger dataset consisting of 13 patients. The DI-MRCD model combines dynamic contrast-enhanced (DCE) and diffusion-weighted imaging (DWI) magnetic resonance imaging (MRI) data with patient-specific parameters to simulate the reaction of breast cancer tumours on chemotherapy. Key improvements include optimizing and generalizing the pre-processing pipeline for a larger patient cohort and enhancing input reliability by independently computing apparent diffusion coefficients (ADC).

Further refinements to the DI-MRCD model include updates to chemotherapy and shear modulus parameters, switching to a Trust Region Reflective (TRF) optimization method, and introducing a tissue-specific proliferation rate and natural cell death term. Despite these enhancements providing more insight, control, and making the model biologically more realistic, the model struggled to converge, highlighting the challenge of fitting patient-specific parameters with limited data points.

Future improvements could include resolving the convergence issues, incorporating additional calibration parameters, allowing the proliferation rate to travel with tumour cells as they diffuse, incorporating chemotherapy doses into the chemotherapy term, using Bayesian optimization for better parameter estimation, and making the results more explanatory by integrating other patient data.

Every year, 180000 new cases of hydrocephalus are diagnosed among infants in Sub-Saharan Africa. Unfortunately, more than two-thirds of the population in this region lacks access to essential medical imaging technologies, such as magnetic resonance imaging (MRI). To address this issue, a collaborative effort between the TU Delft, Leiden University Medical Center, Penn State, and Mbarara University of Science and Technology has led to the development of a low-cost, portable, low-field MRI system. However, images obtained from this scanner are often noisy and distorted and might contain artefacts, therefore, need preprocessing before they can be utilized in diagnostics. The enhancement of their quality can be achieved through both hardware calibration and optimization, as well as the application of filtering, enhancement, and segmentation techniques. In this master's project, we propose a two-step PDE-based segmentation approach. Additionally, we compare it with the modified approach where presegmentation in the initial phase of the standard algorithm is introduced. Both approaches yield segmentation results comparable to the ground truth or manually performed segmentation. Nonetheless, there remains room for further improvement in both denoising and segmentation techniques.

With breast cancer being the leading cause of death in the Netherlands, while also being expected to have double the amount of cases in the next ten years, it is vital that treatment is optimised. Over the last decade, research has been done to incorporate mathematical modelling in this process, especially in the case of HER2+ breast cancer patients. This aggressive form has poor chances of survival, but responds well to chemotherapy and is expected to be quite predictable. In previous works of N. Oudhof and E. Slingerland, a mechanically coupled reaction-diffusion model with an extension of chemotherapy was implemented in 2D, using three MRI scans to predict tumour response. The first two scans, taken right before and during treatment, are used to find patient-specific parameters corresponding to proliferation, tumour movement and chemotherapy efficacy. This calibrated model is then used to predict the third scan, taken at the end of treatment. Calibrating this model to individual patients takes up to a day, however. \\

This thesis aims to extend this model to a higher resolution 3D setting, which should also hand doctors predictions within a working day. To increase speed, the linear-elastic sub-problem of finding shear stress due to tissue types and tumour growth was first optimised. With a novel Laplacian preconditioner in the Conjugate Gradient method, using FFT's and a tridiagonal solver, the time needed was vastly improved. Second, the maximum order of error needed for accurate temporal integration was confirmed to be quite high. Hence a state-of-the-art Parareal implementation was made, using Runge-Kutta 4 and Crank Nicholson as the respective fine and course solver, which succeeded in being both faster and more accurate than simple first-order methods. Then it was found that the underdeterminedness of the problem is best tackled using Total Variation Regularisation on the proliferation parameter and Tikhonov Regularisation on the other two parameters. This ensures that unique solutions can be found in reasonable time, that properly reflect expectations of these parameters in practical settings. The best set of parameters were found fastest with Powell's Dog-leg method, for which a novel way of finding the Jacobian analytically was used. \\

On simulated data, the error in the amount of tumour cells in the third image went down to single-digit percentage rate, with a maximal shape correlation coefficient. This prediction can be made within a few hours, which means that the feasibility of solving this problem in practical settings has been established successfully. On the real data, the calibration succeeded in calibrating the model on the first two scans, but the predictions still have room of improvement. The most important suggestion of this work is that more research must be done in the verification of the suitability of this model to the available real data, using the techniques presented here. Further improvements can be made by exploring more possibilities of using parallelisation, and by obtaining more data. Before obtaining more data, one should investigate the impact of the timing of the scans on the calibration results.

This thesis aims to develop an advanced numerical solver capable of efficiently computing the resonant states of quantum mechanical two-body and three-body problems, thereby expanding our understanding of these complex systems. The quantum three-body problems feature at least two dimensions, which necessitates substantial computational efforts. Therefore, in order to tackle these challenging computations, we need to seek assistance from supercomputers. By harnessing the capabilities of high-performance computing, we can significantly reduce the amount of time spent waiting for programs to run for hours.

In this thesis, we first introduce some basic knowledge about quantum few-body problems and resonant states, showing how the physical problem gives rise to a mathematical problem, the quadratic eigenvalue problem (QEP). Building upon the physical background, our journey in developing the methodology begins with two fundamental components: discretization and eigensolver. The pseudo-spectral methods are introduced to represent the quadratic eigenvalue problem as a matrix problem, by which we can solve the problem numerically through some eigensolvers. We describe a classical approach called linearization for solving QEPs, which transforms the quadratic problem into a generalized eigenvalue problem. Following the linear transformation, we apply the Jacobi-Davidson QZ (JDQZ) method, an iterative eigensolver, to solve the linearized problem. Alternatively, we could also use the Jacobi-Davidson (JD) method to approximate the quadratic eigenvalue problem's eigenpairs directly. In this thesis, we provide an outline of the Jacobi-Davidson process for solving both linear and quadratic problems. Two routes for solving QEPs are utilized and compared: linearization combined with the Jacobi-Davidson QZ method, and the quadratic Jacobi-Davidson method. Through our research and analysis, we demonstrate that the Jacobi-Davidson algorithm exhibits superior computational efficiency when adapted to solve QEPs directly.

Another significant objective of this thesis is to parallelize the eigensolver on supercomputers. We implement a hybrid distributed/shared memory parallelization of the Jacobi-Davidson algorithm to solve quadratic eigenvalue problems that arise from one-dimensional three-body problems. We leverage the tensor structure inherent in the three-body problem to optimize computational efficiency. Specifically, we implement an efficient tensor product scheme for the application of the stiffness and damping operators, which are realized as dense matrix-matrix products. By incorporating a preconditioner that also preserves the tensor structure, we enhance the performance of our Jacobi-Davidson algorithm in computing three-body resonance poles within an acceptable speed.

In our research, we have made a significant advancement in predicting the clinical outcome of high-risk non-muscle invasive bladder cancer (HR-NMIBC) by combining clinicopathological data with image-related features. This integrated approach has shown remarkable improvements in the accuracy of artificial intelligence techniques for outcome prediction.

We developed a novel methodology that effectively combines information from cell nuclei per patient, resulting in enhanced classification accuracy. By integrating clinicopathological data with image-related features extracted from medical imaging, we demonstrated the power of AI in more accurately predicting clinical outcomes for HR-NMIBC.

Our study provides a comprehensive view of the disease, taking into account both macroscopic characteristics and microscopic details observed at the cellular level. By aggregating information from thousands of cell nuclei for each patient, we transformed raw data into a format suitable for machine learning algorithms, improving the performance of AI techniques in clinical outcome prediction.

However, it is essential to address the potential biases and imbalanced variables present in the dataset. We noticed gender imbalance, differences in tumor size, and uneven grading levels, which may affect the generalizability of our conclusions.

To enhance our analysis, we retrained a convolutional neural network (CNN) using our image dataset, achieving high accuracy in segmenting hematoxylin and eosin stained images and accurately identifying cell nuclei boundaries. Additionally, we implemented an innovative clustering technique called FlowSOM, enabling us to group and classify millions of cell nuclei based on their characteristics, providing valuable insights into cellular heterogeneity.

Our AI models exhibited high performance metrics, particularly the random forest algorithm, which proved most suitable for the task. We also conducted a variable importance analysis, revealing specific cell clusters with significant impact on predicting clinical outcomes, emphasizing the relevance of cellular size and shape in disease progression and treatment response.

Proton irradiation therapy is a powerful form of cancer treatment, promising better dose conformity as compared to conventional radiotherapy. Due to the complex scattering properties of protons, the optimization process that is needed to accurately target the cancer cells whilst causing minimal damage to
the surrounding healthy tissue and minimizing detrimental effects, is very time-consuming. This means that the CT scan it is based on has lost part of its accuracy in describing the to be irradiated tissue. To account for this, the surrounding tissue is irradiated more to ensure effective treatment, reducing dose conformity and thus increasing the probability of detrimental side effects.
To solve this problem, a new proton therapy method concept, called Online Adaptive Proton Therapy, or OAPT, calls for a CT scan to be taken about 30 seconds before the treatment, allowing the planned treatment to be adapted to any anatomical changes that have occurred since the previous scan where the original treatment planning was based on. The computations required for this adaptation, and the required quality assurance, currently still take significantly longer than 30 seconds on current computational hardware, making the concept not yet viable in its current form. However, using GPU-acceleration, parts of the algorithm that is used for this can be significantly sped up to reduce overall computational time, potentially making the concept viable for real world application. In this thesis, the research question ”How can GPU-offloading decrease computation time for proton therapy dose calculations?” is answered by accelerating two model algorithms representative of two time-consuming steps in the proton therapy optimization process and analyzing the performance, on
an NVIDIA V100S GPU, of the accelerated code. Furthermore, a model is postulated and validated to characterize and predict the performance of a GPU accelerated algorithm. Using OpenACC, both algorithms achieved speedups between 30x and 440x excluding data transfer time, and between 0.88x and 40x including data transfer time, with both values depending on the problem size, with larger problems yielding larger speedups. Further research is needed on the validity of the derived model on different hardware and for different algorithms. Furthermore, additional research on the effect of implementing these accelerated algorithms on the total computation time of the real world algorithm is advised.

Intensity modulated proton therapy is an advanced radiotherapy technique that is used to treat cancer patients. In order to successfully treat a patient, sufficient dose to the tumor is required. However, during the fractionated treatment, multiple errors can cause a difference between the planned and actual dose delivery. To ensure adequate dose delivery in potential error scenarios, robust treatment plans are acquired as these are less sensitive to uncertainties inherent to proton therapy. However, robust optimization is challenging.

First, as robust optimization accounts for multiple error scenarios, the time needed to generate optimal treatment plans increases significantly. Therefore, it is investigated in this thesis if the optimization time can be reduced while preserving treatment plan quality. This is investigated through two different methods. In the first method, the number of error scenarios accounted for during optimization is reduced. We found that this can significantly reduce optimization time, while improved target coverage and lower risk on side effects are obtained. However, the near-maximum dose to the tumor was found to be less favourable. The second method investigated is variance optimization. This method significantly reduces optimization time. However, for similar target coverage, the risk of side effects increases.

Another challenge related to robust optimization is the increase in delivered dose to healthy tissues surrounding the tumor, which increases the risk of side effects. Therefore, it is investigated in this thesis if the risk of side effects can be lowered by allowing higher maximum dose to the tumor. It is found that this method indeed reduces the risk of side effects. However, the increased maximum dose to the tumor may not be clinically desired as the increase may lead to higher risks of other side effects: edema and fibrosis. The clinically desired trade-off between near-maximum dose and normal tissue sparing should be established.

A common problem in wireless communication is the existence of multipath propagation. This means that a transmitted signal is received multiple times because of reflections caused by the environment. We present two ways of modeling multipath propagation of an acoustic underwater signal. We discretise these models to solve them numerically. During the solving process, we are presented with inconsistent, overdetermined systems of linear equations. We investigate two methods to go about these systems: the ordinary least squares method and the total least squares method. We reconstruct a signal using both of these methods and compare their results. The method of least squares reconstructs the signal moderately well. For the total least squares method this is not the case. It turns out that it is not straightforward to formulate a total least squares problem in the corresponding model. We suspect that, in part, this is why the signal reconstruction does not work well for the total least squares method.

Fog plays a major role in chain collisions. Proper fog detection is essential for the Dutch road authority to anticipate foggy weather conditions. Dozens of stations in the Netherlands can measure fog. However, fog can be a very local phenomenon. Therefore, more local measurements are needed. There are about 5,000 traffic cameras in the Netherlands. Several studies on detecting fog on traffic cameras have been done. The most successful studies used machine learning classification models to detect fog. The biggest challenge they face is the extreme imbalance, limited diversity, and limited accuracy of the dataset. Obtaining adequate precision is one of the primary challenges since the extreme imbalance of the dataset significantly impacts precision. The main objective of this research is to improve the dataset and investigate many machine learning configurations. Another objective is to examine the possibilities of generating synthetic data.

This thesis uses a clever (re)labeling method, significantly improving the dataset's quality. However, it turned out that the dataset still has its limitations. A large portion of false positives are caused by labeling errors. After comparing several machine learning models, it follows that a 9-layer ResNET model is optimal. Adding more layers will not result in better performance. Unexpectedly, initializing ResNET with pre-trained weights actually decreases performance. In addition, the effect of oversampling and/or using a weighted binary cross-entropy loss is investigated. Just oversampling leads to overfitting, but using a weighted binary cross-entropy loss isn't ideal either. The best performance is achieved by combining weighted binary cross-entropy loss with oversampling. Decision threshold optimization substantially improved the results. The experiments allowed for selecting the ideal configuration, which substantially increased performance. The best-performing configuration achieved a strong correlation in the Matthews correlation coefficient.

Finally, the possibilities of generating synthetic data are investigated. ADASYN and SMOTe seem attractive at first sight, but from a recent study, it follows that they don't work better than random oversampling. One of the most promising ideas for generating synthetic data is to add fog to clear images. In this thesis, a conceptual algorithm is designed to add artificial fog to clear images. Most generated images look convincing, but there is much room for improvement.

The aim of this thesis was to test the efficiency in practice of an analytical propagator with collision detection for N-body Keplerian systems. This can be used to simulate the evolution of a protoplanetary disk, which gives insight into how planetary systems form. The analytic propagator calculates collisions one by one, while a numerical propagator would compute each time step. The idea of using the analytic propagator is that collisions are rare in astronomical scales, such that jumping from collision to collision and calculating it, is more efficient than calculating all the time steps that are between collisions. Simplifying the orbits of the planetesimals into perfect Keplerian orbits, analytical solutions exist which are used by the analytic propagator.

In this thesis, the runtimes of simulations were measured as well as other properties directly related to the runtime. The overall efficiency of the algorithm with respect to N seemed to be O(N³), which is one power less than previously predicted. The prediction was that the runtime of the full simulation is O(N²ε + N⁴s³/Ia³). Here ε is the maximum eccentricity, s/a is the ratio of a planetesimal's radius to the semi-major axis of its orbit, and I is the maximum inclination. This was calculated by estimating the total number of collisions to be O(N²s²/Ia²) and the runtime for each collision to be O(N²s/a). But the number of collisions turns from quadratic to linear in N, implying that above a certain N almost all planetesimals collide, which reduces the power of N by one. For comparison, the octree code has an algorithmic efficiency of O(N logN) per time step, and the number of steps for a fixed integration time grows as O(N^4/3 log N).

HER2+ breast cancer patients suffer from aggressive tumours that often respond well to treatment compared to HER2- patients. Currently, patients are all treated with the same number of anti-tumour treatments, although the required number to eradicate all tumour cells varies between patients. The goal is thus to have an individualised model based on an MRI before the start of therapy and an MRI after several rounds of therapy that predicts how many treatments are needed to eradicate all tumour cells. In the previous MSc thesis by N. Oudhof, a two-dimensional spatiotemporal mechanically coupled reaction-diffusion model was implemented. This was used to model the number of tumour cells in each position of the breast, where the mechanics are included to simulate the behaviour of the surrounding tissue.

The goal of this thesis is to improve that model by improving the calibration in which the patient-specific parameters are optimised, by taking into account the treatment schedule of the patients and by extending the model to three dimensions. For calibration, the first and second MRI scans are used and a third MRI is used for validation. To simulate the chemotherapy, both the Kety-Tofts model and a Normalised Blood Volume Map are applied, of which the latter appeared the best option. In 2D, four models are implemented: the reaction-diffusion models with and without mechanics and with and without chemotherapy term. These were compared by analysing the predicted and measured tumour densities for a cohort of three HER2+ patients. For 3D, only the basic reaction-diffusion and chemotherapy-incorporated reaction-diffusion models are implemented. The models are compared in terms of mean squared error, global relative error and concordance correlation coefficient. For both 2D and 3D, the models without chemotherapy gave slightly better results in terms of these measures, although the differences between the models were rather small. To determine the required number of therapy rounds, it is however better to use the chemotherapy-incorporated models because they offer more possibilities to simulate different treatment strategies. The computation time of the 3D implementation should be reduced before the mechanics can be included, which will allow conclusions to be drawn on which model is most accurate. Other directions for future improvements include using the second MRI to calculate the chemotherapy concentrations after calibration, adding an immunotherapy term and increasing the number of patients.

Topology optimization is a branch of mechanical engineering in which the topology of a structure is created and optimized to certain conditions and restrictions. In the last few decades, the demand for highly accurate and complex models of these structures has increased and it has a big effect on the computational power needed. To ease the computational load for the dynamical systems one can use model order reduction methods to reduce the size of the models.
Classic Arnoldi is a widely used method for model order reduction (MOR) with topology optimization. In this thesis, we discuss two-sided Arnoldi and IRKA to help find a suitable moment-matching MOR method for topology optimization. These two reduction methods are implemented and improved to create a high fidelity reduced model. For improvements in the accuracy, the use of orthogonalization methods is analysed and discussed as well as including rigid body modes for IRKA and a preconditioner for two-sided Arnoldi. Lastly, a participation factor is discussed and improved to help reduce the model created with two-sided Arnoldi.
In the end, we find that two-sided Arnoldi in combination with the participation factor performs better than IRKA by creating a smaller and more accurate reduced model.

Parkinson’s Disease (PD) is the fastest growing neurodegenerative disease, currently affecting two to three percent of the population over 65. Studying functional connectivity (FC) in PD patients may provide new insights into how the disease alters brain organization in different subjects. We explored persistent homology (PH) as a method for studying FC based on the functional magnetic resonance imaging (fMRI) recordings of 63 subjects, of which 56 were diagnosed with PD.

We used PH to translate each set of fMRI recordings into a stable rank. Stable ranks are homological invariants that are amenable for statistical analysis. The pipeline has multiple parameters, and we explored the effect of these parameters on the shape of the stable ranks. Moreover, we fitted functions to reduce the stable ranks to points in two or three dimensions. We clustered the stable ranks based on the fitted parameter values and based on the integral distance between them.

For some of the parameter combinations, not all clusters were located in the space covered by controls. These clusters correspond to patients with a topologically distinct connectivity structure, which may be clinically relevant. However, we found no relation between the clusters and the medication status or cognitive ability of the patients.

It should be noted that this study was an exploration of applying persistent homology to PD data, and that statistical testing was not performed. Consequently, the presented results should be considered with care. Furthermore, we did not explore the full parameter space, as time was limited and the data set was small. In a follow-up study, a measurable desired outcome of the pipeline should be defined and the data set should be expanded to allow for optimizing over the full parameter space.

This Bachelor End Project Thesis is about creating parallel variants of time-dependent methods. To achieve this, the time-dependent problem is formulated as a large system of linear equations that is solved iteratively. This linear system is determined from the time-dependent linear heat equation in 1D that all the analysis is about. The experiments are tested for different boundary condition combinations. Namely, for a mixture of Dirichlet, Neumann and Robin conditions. Also, three different time integration methods are used, namely Forward Euler, Backward Euler and the θ-Rule for θ = 0.2, 0.4, 0.6 and θ = 0.8. Since time is continuous and moves in the positive direction, time integration methods need the previous solution in order to find the current solution. Executing a calculation in parallel time is therefore a complex and tricky phenomenon. This research considers two different iterative methods, the block Jacobi method and the block GaussSeidel method. Both have a lot in common but the main difference lies in the use of different parts of the matrix. Where the Jacobi method uses just the diagonal in order to approximate the inverse, the Gauss-Seidel method also uses the upper or lower triangular part of the matrix. Therefore, creating a parallel variant for the block Jacobi method is easily possible, but for the block Gauss-Seidel method it is not. In order to get to a valid conclusion, the stability of each time integration method is considered. The block Jacobi method converges slowly to the solution since it uses the diagonal for every step instead of the whole matrix. Therefore, in this research, combinations of the block Gauss-Seidel method have been created and tested. Three different variants are discussed. The first one is the step-wise combination method which uses a block Jacobi iteration for every step size chosen, and uses the sequential Gauss-Seidel time steps for all the other iterations. The second variant is called the Red-Black Ordering and is known for its odd and even iterations. Here every odd iteration is done in parallel and similar for its even iterations. Since the equation only needs its previous solution, this method would work very well in parallel. The last variant uses the number of processors to manipulate the maximum number of iterations. Here the method converges earlier or at the number of processors chosen. What can be concluded from the results is that the way a method converges all depends on how many Jacobi iterations take place during the computation. The Gauss-Seidel calculation only takes one iteration, so the time until the solution converges all depends on how many Jacobi iterations the method computes. Having a computer with more processors than number of Jacobi iterations performed, will also speed-up the computation. Our predictions show in particular that with only implicit time integration methods the speed-up can really be achieved. For the block Jacobi method, this equals that the maximum number of iterations is also the number of time steps nt. The Gauss-Seidel variants are all more efficient, since less Jacobi iterations are used. For variant 1, the step-wise combination method, the maximum number of iterations equals nt/step . For the Red-Black Ordering method, every other iteration, a Jacobi step is used so the maximum number of iterations equals nt/2. The last variant, the processor-wise combination method, is a special variant of the first variant. This method has the characteristic that the maximum number of iterations equals the chosen number of processors, np and therefore has, in terms of the first variant, a ’step’-size of nt/np. This project has been conducted with Matlab which did not fully cooperate when using the parallel time function. In order to really test, and therefore not predict, the obvious way to perform a parallel computation is to implement this program in another programming language and later on test it on the DelftBlue supercomputer. The predictions should still hold but the results will be more clearly visualised.

Cardiac perfusion magnetic resonance imagining (MRI) is often used for ischemic diagnostics. However, the currently most popular technique uses an intravenous contrast agent: gadoliniumchelate. As this injection can cause problems in patients with renal problems, which are more prevalent in cardiac patients, a non-contrast cardiac perfusion technique would be ideal. Arterial spin labelling (ASL) is such a non-contrast technique. However, ASL requires using separate labelling pulses, complicating timing and sequence design. Concurrent labelling and imaging would solve this, which is why in this project simultaneous multi-slice (SMS) ASL was investigated. SMS is an imaging method during which multiple slices are excited at once and after receiving their signals the images are separated. If SMS were to be applied to two slices that contain the same arteries, the upstream slice would be functioning as a labelling and imaging site at the same time, as the blood in the upstream slice inevitably gets saturated during imaging. This is a way around the aforementioned downsides of ASL. Since blood flow in the arteries is highly pulsatile, SMS-cine was used, a dynamic SMS method which is more robust to motion. To isolate the saturated blood from background tissue and to compensate for bias from magnetization transfer between boound pools, the subtraction between upstream blood saturation and downstream blood saturation was taken. The hypothesis of this report is that by continuously labelling and imaging at the same time with SMS-cine, a signal can be produced directly related to the dynamic cardiac perfusion of the arteries. The viability of the proposed technique is investigated and optimized using numerical models and phantom experiments. From the phantom experiments and model predictions, it follows that for SMSASL a large flip angle and large slice thickness are optimal. The relative flow enhancement seen in the phantom experiments, using tubes with flowing water, responded to the varying of these parameters as predicted by the simulations. Yet, the relative flow enhancement was much higher than predicted by the model. This is probably caused by the fact that the tubes and water have a different T1 than blood and tissue. The model predicts relative flow enhancements of more than 90 % caused by pulsatile flow, regardless of the dicretization grid in z used, which indicates that it would be fruitful to further explore the potential of SMS-ASL in vivo. It is recommended to use an SPGR sequence and both a large flip angle and large slice thickness for this. Unbalancing the flip angle and slice thickness could also be explored using an expanded version of the numerical model and later phantom and in vivo experiments. Finally, a large potential improvement could be made to the numerical model by applying a strict conservation of total magnetization when moving spins to simulate flow, which is currently not guaranteed by the interpolation embedded in flow simulation.

MRI has been used worldwide as an imaging modality for decades due to its ability to distinguish between soft tissues. However, MRIs cannot be used in all situations due to their extensive hard- and software. For this reason, research in the field of low-field MRI has been conducted. Low-field MRI systems do not use superconducting magnets but restive, or in this research, permanent magnets. This reduces the costs of these scanners, but these same changes result in a poorer signal-to-noise ratio (SNR) than high-field scanners. Based on this hardware, research in single-sided scanners has emerged, and various research groups have been able to reconstruct images with these scanners. They are based on conventional MRI hardware including a radiofrequency (RF) coil, permanent magnets and gradients coils for encoding. In collaboration with the Technical University (TU) Delft, a handheld scanner has been constructed in the Leiden University Medical Center (LUMC). This scanner consists of an RF coil and a permanent magnet and does not have a gradient coil. The inhomogeneity of the 50 can be used for spatially encoding the signal based on translations. The SNR in images is improved with model-based image reconstruction. Although, in theory, this set-up should work, the image results so far have been somewhat disappointing. Therefore, in this thesis, we have investigated a different approach, using a neural network. In this research, the correlation between the received signal of a measurement and a simulation of the same shape is found to be greater than the correlation between other simulations and the received signal. Due to this proven correlation, an image reconstruction deep learning algorithm is constructed based on the AUTOMAP model. Simulated signals are reconstructed into images with promising results; however, the algorithm cannot reconstruct mea- sured data. Therefore, suggestions for future improvement include improving the simulated data set and adding more real measurements that would allow for training on a measured data set.