Graph burning is a process that models the spread of information through a network. This process is divided into rounds and is visualized as a spreading fire. Each round, a source of fire may be chosen from where the fire spreads and burns surrounding points. From every burned point, the fire propagates through adjacent points, eventually covering the entire network. The burning number of a graph, b(G), was introduced to measure the time required to burn the entire network.

It was proven that path graphs on n vertices have burning number ⌈√n⌉. It was then conjectured that all connected graphs have at most this burning number. The burning number has been estimated or identified for several classes of graphs. We provide a lower and upper bound for the burning number of a new class of graphs with a path-like structure, namely necklace graphs. Necklace graphs are constructed by concatenating smaller graphs, called pearls. To obtain bounds, we define lower-bound paths and upper-bound paths, which are paths connecting two designated vertices. We show that a lower-bound path always exists in the form of a shortest path, while finding an upper-bound path for an arbitrary necklace is NP-complete. For necklace graphs where a shortest path is an upper-bound path, we show that the burning number can take only two values.

Finally, we consider cycle-forests, which are disconnected graphs whose components are cycle graphs. We prove that burning cycle-forests is NP-complete.

Vascular calcification, the deposition of calcium in the vessel wall, is associated with several vascular diseases, including atherosclerosis, diabetes mellitus, and hypertension.
Fluid-structure interaction (FSI) is recommended to simulate blood flow incorporating vascular calcification. However, FSI applied to a three-dimensional (3D) model takes several days to simulate.
To reduce the computational complexity, 1D reduced order models (ROMs) are often used instead.

Reduced order modeling decreases the computational complexity of a model by removing dimensions of the coordinate system within a model. The cylindrical coordinate system is used in hemodynamics, especially in ROMs. The 1D ROM for hemodynamics is obtained by removing the azimuthal dimension (accomplished by assuming axial symmetry for all properties within arteries) and the radial dimension (accomplished by applying a predefined velocity profile to blood flow) from the 3D model. However, incorporating vascular calcification can make the geometry of arteries and flow within arteries asymmetric. A 2D ROM can increase the accuracy of the 1D ROM by including one of the two removed dimensions. Research regarding 2D blood flow mainly focuses on including the radial dimension, which cannot implement asymmetric calcification since axisymmetry is assumed.

This study obtains a 2D ROM for blood flow by removing the dimension corresponding to the radial distance from the three-dimensional model and by assuming that axial velocity is continuous in the neighborhood near the artery's origin. The 2D ROM obtains axisymmetric velocity by only allowing a single velocity profile. However, enabling a family of velocity profiles can make flow within arteries asymmetric. Hence, this study contributes to hemodynamics by studying blood flow that allows a family of velocity profiles.

A non-physiological steady-state solution has been obtained analytically, in which the volumetric flow rate vanishes, and numerical methods are developed to simulate the 2D ROM, which incorporates dimensional (Godunov) splitting, linear approximate solvers, and high-resolution methods. Jump-discontinuities within the mechanical properties of the vascular walls are smoothened for the 2D simulations. Numerical methods for the 2D ROM yield significant errors within the smoothening region for simulations with coarse grids.
The numerical method obtains the non-physiological steady-state solutions for arteries without calcification and has a relative error of O(Δx^1.500) for arteries with axisymmetric calcification. The 2D ROM cannot numerically obtain the non-physiological steady-state solution for arteries with asymmetric calcification due to the numerical errors within the smoothening range.

 3D and 2D numerical simulations with pulsatile blood flow are compared. The 3D simulation without calcification has a significantly higher diastolic pressure, larger inner wall radii, and larger volumetric flow rates than the 2D simulation. The differences in blood flow observed between pulsatile blood flow without calcification and with calcification match decently between the 3D simulations and the 2D simulations, except for locations within the smoothening region.

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is crucial for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study investigates the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations for predicting post-burn wound evolution. We trained DeepONets on various wound shapes, enhancing the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The most sophisticated model achieved an R² score of 0.9960, indicating strong predictive accuracy. Additionally, the model generalised well to convex combinations of basic shapes, with an R² score of 0.9944, and provided reliable predictions over an extended period of up to one year. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Achtergrond: Migraine is een veel voorkomende hoofdpijnaandoening waar neuropeptiden een belangrijke rol lijken te spelen. Dit onderzoek richt zich op het begrijpen van de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming bij vrouwen met en zonder migraine. Hiervoor worden LSCI en LDPI metingen getransformeerd met een Fourier transformatie. Doel: Het doel van dit onderzoek is om verschillen te identificeren in de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming tussen vrouwen met en zonder migraine. Methode: LSCI metingen van de VASCULAR studie worden gebruikt. Fourier analyse wordt toegepast om de invloed van zenuwen en gladde spiercellen te vinden in verschillende fasen: baseline, piek en plateau. Vergelijkingen worden gemaakt tussen vrouwen met migraine (met en zonder aura) en controlepersonen zonder migraine. Resultaten: In de baseline fase zijn minimale verschillen in de invloed van de zenuwen en de gladde spiercellen gevonden tussen vrouwen met en zonder migraine. Tijdens de piek fase lijkt er een verhoogde invloed van de zenuwen en de gladde spiercellen op de bloeddoorstroming bij vrouwen zonder migraine. Conclusie: In ons onderzoek zijn geen evidente verschillen gevonden. Wel lijken er enkele trends in de data te zitten. Verdere studies zijn nodig om deze bevindingen te bevestigen en uit te breiden.

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 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.