A Cell-centered, Agent-based Method that utilizes a Delaunay and Voronoi Environment in 2- and 3-Dimensions Ryan C. Kennedy 1, Glen E.P. Ropella 2, C. Anthony Hunt 1 1 Department of Bioengineering and Therapeutic Sciences, University of California, San Francisco, CA 2 Tempus Dictum, Inc., Portland, OR
Background Postdoctoral Researcher, UCSF Department of Bioengineering and Therapeutic Sciences Modeling and Simulation (M&S) Student, University of Notre Dame Department of Computer Science and Engineering PhD, MS, BS Bioinformatics M&S (GIS-aware ABM)
Background Motivation Develop a method to model individual agent behavior from an agent s perspective in an off-lattice environment Use case: biological cells Goal Facilitate study of more advanced agent (cell) behavior through a dynamic off-lattice environment Deeper explanatory mechanistic insight into how mammalian cells restructure and use themselves during morphogenesis and tissue repair Ability for cells to dynamically adjust during a simulation
Cell-centered Models Numerous techniques Cellular Potts Model (CPM) Widely used to study morphogenesis and tumors Grid-based perspective Cells exist in a lattice connected by bonds largely governed by equations
Off-lattice vs Lattice Environments Lattice Discrete Less computationally expensive Examples: CPM Off-lattice Continuous More computationally expensive Calculate neighbors, update spatial representation Dynamic Examples: Vertex, force-based, subcellular element method
Why Off-lattice? Agents not limited to predefined spatial boundaries Agents can take a variety of shapes and sizes Agents can have a varying number of neighbors Can be more realistic Can come close to approximating the shape and pattern of biological cells
Comparison of Grid Types Grid vs Hexagonal vs D/V
Delaunay/Voronoi Representation Limited work has been performed in ABM cell-centered realm Meyer-Hermann group utilizes dynamic force-based models utilizing Delaunay/Voronoi (D/V) grids to study tumor spheroids [1] D/V grids to represent butterfly wing epidermis [2] [1] Schaller, G., Meyer-Hermann, M., Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model, Physical Review E, 71(5), 2005. [2] Honda, H., Tanemura, M., Yoshida, A., Differentiation of Wing Epidermal Scale Cells in a Butterfly Under the Lateral Inhibition Model Appearance of Large Cells in a Polygonal Pattern, Acto Biotheoretica, 48:121-136, 2000.
Voronoi-like Examples http://www.lochnesswatergardens.com/pondblog/wp-content/uploads/2012/03/dragonfly-wing-closeup.jpg http://i282.photobucket.com/albums/kk273/brucedstaylor/voronoi/voronigiraffe.jpg http://i282.photobucket.com/albums/kk273/brucedstaylor/voronoi/voronoimapleleaf2.jpg
Method Framework MASON Open-source, Java Multiagent Simulation Toolkit Distinct separation of simulation from display No built-in D/V support Visualization Toolkit (VTK) Open-source, C++ (with wrappers for other languages) Advanced system for image processing and 3D graphics Triangulation support, but no built-in support for calculating Voronoi tessellations Must build from the triangulation
Agent-Centered Model to Study the Behavior of Agents Represent the behavior of many individual agents to study the collective behavior of a system of agents Space as Delaunay and Voronoi Grids Each point exists in an area; its corresponding V-Cell encompasses all points closer to said point than any other point Abstraction from regular grids
Visualization/Terminology Voronoi-cells (V- Cells) distinguished by color V-Cell centers are the dots within each V-Cell Delaunay triangulation shown with white lines White circumcircles intersect each V-Cell center (triangles)
Interactive Example http://www.cs.cornell.edu/home/chew/delaunay.html
V-Cell Representation Method supports and arbitrary number of V- Cell types with varying behaviors V-Cell decisions influenced by neighborhood Neighbors are considered as V-Cells 1-hop away from a given V-Cell (V-Cell center to V- Cell center) Calculated from triangulation Triangulation recalculated as V-Cells move Changes V-Cell shapes
2D Process Given a set of V-Cell center points: Triangulate with VTK Calculate extent of V-Cell using VTK s circumcircle method on each triangle System adapted from Bowyer: Bowyer, A., Computing Dirichlet tessellations, The Computer Journal, 24(2):162-166, 1981.
3D Process Given a set of V-Cell center points: Generate each "face" (membrane) of the V-Cell through the triangulation (which in 3D returns tetrahedra) For a given V-Cell, find tetrahedra that share the center point and share faces with the given tetrahedron (adjacent tetrahedra- there will be 2) For an adjacent tetrahedron, calculate the circumsphere Using the circumspheres from the 2 adjacent tetrahedra and the given tetrahedron, form a triangle (what you see on the surface) Together, these triangles form the surface membrane of the cell
2D Voronoi Cells http://blog.kleinproject.org/wp-content/uploads/2012/05/voronoi.png
3D Voronoi Cells http://www-math.mit.edu/dryfluids/gallery/voronoi_huge.png
Use Case: Biological Cells Initial focus on modeling cysts and 3 main cell types: Lumen, Luminal epithelial (LEC), Myoepithelial (MEC) Several behaviors captured Behavior governed by axioms developed by domain experts Model movement, growth, division, and death in 2D and 3D Cell rotation implemented in 2D
2D Voronoi Cells Key: Lumen LEC MEC Triangulation
3D Voronoi Cells
3D Voronoi Cells with Triangulation
Demonstrations 2D Cell Movement, Division, Death 2D Cell Growth 2D Cell Rotation 3D Cell Movement, Division, Death 3D Cell Growth
Applications Cell-centered Modeling Explore the impact of cell layouts and types on cell behavior Predict the impact of a cyst on surrounding cells, modeling growing or rotating cysts Networks Nodes naturally map to V-Cell centers Model routes between agents Simulate ad-hoc cell phone networks
Summary New framework within the realm of agent-centered modeling Demonstrated basic cell behavior D/V environment a viable alternative to traditional lattice-based environments
Questions or Comments?