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Different numerical approaches and algorithms arising in the context of modelling of cellular tissue evolution are discussed in this thesis. Being suited in particular to off-lattice agent-based models, the numerical tool of three-dimensional weighted kinetic and dynamic Delaunay triangulations is introduced and discussed for its applicability to adjacency detection. As there exists no implementation of a code that incorporates all necessary features for tissue modelling, algorithms for incremental insertion or deletion of points in Delaunay triangulations and the restoration of the Delaunay property for triangulations of moving point sets are introduced. In addition, the numerical solution of reaction-diffusion equations and their connection to agent-based cell tissue simulations is discussed. In order to demonstrate the applicability of the numerical algorithms, biological problems are studied for different model systems: For multicellular tumour spheroids, the weighted Delaunay triangulation provides a great advantage for adjacency detection, but due to the large cell numbers the model used for the cell-cell interaction has to be simplified to allow for a numerical solution. The agent-based model reproduces macroscopic experimental signatures, but some parameters cannot be fixed with the data available. A much simpler, but in key properties analogous, continuum model based on reaction-diffusion equations is likewise capable of reproducing the experimental data. Both modelling approaches make differing predictions on non-quantified experimental signatures. In the case of the epidermis, a smaller system is considered which enables a more complete treatment of the equations of motion. In particular, a control mechanism of cell proliferation is analysed. Simple assumptions suffice to explain the flow equilibrium observed in the epidermis. In addition, the effect of adhesion on the survival chances of cancerous cells is studied. For some regions in parameter space, stochastic effects may completely alter the outcome. The findings stress the need of establishing a defined experimental model to fix the unknown model parameters and to rule out further models.
The goal of this project is to develop a framework for a cell that takes in consideration its internal structure, using an agent-based approach. In this framework, a cell was simulated as many sub-particles interacting to each other. This sub-particles can, in principle, represent any internal structure from the cell (organelles, etc). In the model discussed here, two types of sub-particles were used: membrane sub-particles and cytosolic elements. A kinetic and dynamic Delaunay triangulation was used in order to define the neighborhood relations between the sub-particles. However, it was soon noted that the relations defined by the Delaunay triangulation were not suitable to define the interactions between membrane sub-particles. The cell membrane is a lipid bilayer, and does not present any long range interactions between their sub-particles. This means that the membrane particles should not be able to interact in a long range. Instead, their interactions should be confined to the two-dimensional surface supposedly formed by the membrane. A method to select, from the original three-dimensional triangulations, connections restricted to the two-dimensional surface formed by the cell membrane was then developed. The algorithm uses as starting point the three-dimensional Delaunay triangulation involving both internal and membrane sub-particles. From this triangulation, only the subset of connections between membrane sub-particles was considered. Since the cell is full of internal particles, the collection of the membrane particles' connections will resemble the surface to be obtained, even though it will still have many connections that do not belong to the restricted triangulation on the surface. This "thick surface" was called a quasi-surface. The following step was to refine the quasi-surface, cutting out some of the connections so that the ones left made a proper surface triangulation with the membrane points. For that, the quasi-surface was separated in clusters. Clusters are defined as areas on the quasi-surface that are not yet properly triangulated on a two-dimensional surface. Each of the clusters was then re-triangulated independently, using re-triangulation methods also developed during this work. The interactions between cytosolic elements was given by a Lennard-Jones potential, as well as the interactions between cytosolic elements and membrane particles. Between only membrane particles, the interactions were given by an elastic interaction. For each particle, the equation of motion was written. The algorithm chosen to solve the equations of motion was the Verlet algorithm. Since the cytosol can be approximated as a gel, it is reasonable to suppose that the sub-cellular particles are moving in an overdamped environment. Therefore, an overdamped approximation was used for all interactions. Additionally, an adaptive algorithm was used in order to define the size of the time step used in each interaction. After the method to re-triangulate the membrane points was implemented, the time needed to re-triangulate a single cluster was studied, followed by an analysis on how the time needed to re-triangulate each point in a cluster varied with the cluster size. The frequency of appearance for each cluster size was also compared, as this information is necessary to guarantee that the total time needed by to re-triangulate a cell is convergent. At last, the total time spent re-triangulating a surface was plotted, as well as a scaling for the total re-triangulation time with the variation. Even though there is still a lot to be done, the work presented here is an important step on the way to the main goal of this project: to create an agent-based framework that not only allows the simulation of any sub-cellular structure of interest but also provides meaningful interaction relations to particles belonging to the cell membrane.
Spatio-temporal dynamics of primary lymphoid follicles during organogenesis and lymphneogenesis
(2007)
Primary lymphoid follicles are structures which are important for adaptive immune responses in mammals. Within the follicles follicular dendritic cells (FDC) are maintained by constant stimuli provided by B cells. It is thought that the FDC are important for immune response. It is of interest to know how lymphoid follicles are regulated in order to understand their role in various autoimmune diseases in which these follicles are created ectopically. With the help of a tissue simulation relying on an agent-based cell model on top of a regular triangulation various scenarios suggested by the available experimental data have been investigated. In order to cope with the complexity in the simulation of immune tissue the regular triangulation has been implemented for the use on parallel computers. The algorithms for kinetic and dynamic regular triangulation have been created newly. Also the cell model underlying the simulation has been designed newly in many aspects. The simulations allowed to identify common factors that regulate the formation of lymphoid follicles normally during organogenesis in development and lymphneogenesis in the course of diseases. The generation of FDC from local stromal populations under the influence of B cell aggregates is shown to be possible with the given experimental parameters. The sequence of the organogenesis and lymphneogenesis can be described with regard to the morphology of the B and T zone. Tests for the stability of the primary lymphoid follicle system constraints the regulation of the B cell efflux. The required lymphatic vessels around the lymphoid follicle are shown to be negatively correlated with the FDC network. Moreover it is shown that the adjacent T zone consisting of its own stromal population and T cells has similar regulation principles. This easily explains the intermediate ring of B cells found around the T zone during development and certain signaling molecule deficiencies. A major result of this thesis is that the generation of FDC needs negative regulation while a number of other possible mechanisms is incompatible with the available experimental data. Moreover the observed microanatomy was brought into a functional relationship with data on the cellular level finally culminating in the proposal of new experiments that shed light on the dynamics of the primary lymphoid follicle. One conclusion is that the FDC directly or indirectly influence the angiogenesis and lymphangiogenesis processes in secondary lymphoid tissues. The work presented here may help to guide experiments with the help of computers in order to reduce the amount of experiments and design them in a way to maximize the amount of information about biological systems.