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.

In this thesis the nonlinear spring system is considered. This system contains a semi-infinite string that is modelled by the wave equation with a pair of inititial conditions and a nonlinear boundary condition. The goal of this thesis is to find a good approximation of this system. Furthermore, the behaviour of the string is studied by plotting the reflected waves. Two methods are considered for estimating the solution. These are the Multiple Scales Perturbations method and the Fourth Order Runge Kutta method. The approximations of the two methods are compared to each other. From this, conclusions have been drawn on the accuracy of these approximations.