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A model for sprouting angiogenesis. using a semi-stochastic cell-based formalism Literature review for the graduation thesis by Frans Bookholt. Overview. Sprouting angiogenesis Process Driving forces In vitro experiments at VUMC Models for angiogenesis Cellular Potts model (CPM)
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A model for sprouting angiogenesis using a semi-stochastic cell-based formalism Literature review for the graduation thesis by Frans Bookholt
Overview • Sprouting angiogenesis • Process • Driving forces • In vitro experiments at VUMC • Models for angiogenesis • Cellular Potts model (CPM) • Semi-discrete cell-based model • Continuum models • Toy CPM implementation • PDE’s for semi-discrete cell-based FEM
Angiogenesis is the formation of new vasculature from existing vessels Tip cell selection MMP uPA Lowered oxidative stress VEGF
Driving forces in cell movement • Chemotaxis • The movement towards a gradient of a chemical • Haptotaxis • The movement along stress lines caused by strain of cells themselves • Contact forces • Elastic forces upon collision • Cell-Cell adhesion • Cell-Matrix adhesion
In vitro sprouting assay at the VUMC dermatology department • Fibrinogen + Thrombin = Fibrin gel • Dissolved Ecs (~20.000), stimulated with different concentrations Vascular Endothelial Growth Factor (VEGF) • Microscopic images, both from above and fixated coupes
Models for angiogenesis describe chemicals and cell behavior • All models use PDE’s to model the chemicals • The cell behavior is modeled differently • CPM: Lattice updates • Semi-discrete cell-based: Mechanical approach • Continuum models: Cells as densities
Cellular Potts model (Glazer, Graner and Merks) • Lattice based update algorithm like cellular automaton models • Lattice sites get assigned a cell ID. A biological cell usually contains up to 50 lattice sites with the same cell ID. • Cell ID changes occur according to an energy function
Semi-discrete Cell-based model(Vermolen and Gefen) • Spherical cells that • Act straining forces upon the matrix to induce haptotaxis • Have random movement • Have stochastic viability • Act contact mechanical forces upon each other • Chemotaxis not yet implemented
Continuum models(Anderson, Chaplain, Alarcon, Schugart, Xue, Vermolen, Javierre) • Describe cells as distributions • Unable to track sprout development • Works on very large scales • Sometimes analytical solutions exist
Simple cellular Potts implementation • Same PDE’s Merks used • Tip vs. Stalk for sourcing and reaction • Defined Hamiltonian • Chemotaxis to gradient VEGF • Adhesion to matrix • Cell-Cell adhesion • Compact shape constraint • Gravity • Stochastic degeneration of the matrix Movie of the CPM simulation
Almost finished my FEM for improved chemicals • Continuous degeneration of the matrix • Improved diffusive terms • Nonlinear terms and cross terms Movie of the FEM simulation