A model for sprouting angiogenesis

1. A model for sprouting angiogenesis using a semi-stochastic cell-based formalism Literature review for the graduation thesis by Frans Bookholt

2. 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

3. Sprouting Angiogenesis

4. Angiogenesis is the formation of new vasculature from existing vessels Tip cell selection MMP uPA Lowered oxidative stress VEGF

5. 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

6. 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

7. Models for angiogenesis

8. 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

9. 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

10. 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

11. 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

12. Cellular Potts implementation

13. 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

14. semi-stochastic cell-based FEM

15. 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