This bachelor thesis is about blood vessel network formation and is based on a model from the paper ‘Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation’ by A. Gamba et al ...Show moreThis bachelor thesis is about blood vessel network formation and is based on a model from the paper ‘Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation’ by A. Gamba et al (March 21, 2003) [3]. In this paper, a model consisting of three partial differential equations (PDE’s) was said to describe the formation of a stable blood vessel network. There are several models to predict the formation of a blood vessel network, and it is important to keep on improving those and comparing them to each other. When these comparisons are done properly, biologists know which models are most accurate for what biological aspects and can predict the outcome of a certain experiment best. They can also compare the results from their experiments to the simulations given by the different models to indicate if their experiment gives proper results. According to the paper, the cells form stable structures on the long term. We are interested in the long term behaviour of this model because we would like to see whether or not these stable structures would indeed form. To simulate the given model of PDE’s, we used the program Matlab and the integration method Euler Forward (EF). Once this worked in a one-dimensional case, we extended the code and used the more complex method of integration Runge-Kutta 4 (RK4), first in one-dimension and later on in the final twodimensional case. After adding a density dependent pressure term to the model, just like was done in the paper [3], the results were not stable enough to draw any hard conclusions. The cells did seem to move towards each other to form a network, but blew up within the simulation of 500 seconds. This is probably due to our use of RK4, which is not a very accurate method of numerical integration. We can therefore conclude that the given model, extended with the pressure term, could give a credible prediction of the formation of a blood vessel network. We are however not sure because of instabilities most likely caused by our integration method. Therefore, more research needs to be done with simulations in more advanced programs using more accurate techniques than RK4Show less
Villi formation in the intestine is an example of tissue folding and a poorly understood mechanism. The Cellular Potts Model is a computational model to simulate cellular structures. A mathematical...Show moreVilli formation in the intestine is an example of tissue folding and a poorly understood mechanism. The Cellular Potts Model is a computational model to simulate cellular structures. A mathematical model for epithelial folding using the Cellular Potts Model was implemented to analyze the behaviour of a three layered tissue for different cell properties. The centers of mass and the forces on the cells were used to analyse the simulation results. The forces on the cells are higher during simulations than at the end, and higher growth ratios result in larger forces. Various relations are found between homotypic bond energies, resistance to compression or growth increments of the cells and the amplitude, frequency or size of the tissue. Resistance of the medium and the behaviour of the boundary cells is responsible for epithelial folding and only a small part of the parameter space yields tissues that have a wavelike shape and unbroken cell layers.Show less
