1. Delaunay tells the cancer cells their neighbours Philip K. Maini Centre for Mathematical Biology, Oxford, UK Michael Meyer-Hermann Frankfurt Institute for Advanced Studies (FIAS), Germany Gernot Schaller FIAS [now Dresden University of Technology, Germany] Outline: Modelling of biological cells Delaunay for neighbour detection Elastic spheres versus Voronoi Tumour growth in vitro Comparing Delaunay to PDE

2. Classical approach: PDE Continuous spatial distribution of cellular densities. Time dynamics defined for these densities Discreteness of cells is ignored Individuality of cells is ignored PDE are mean field models Fine for large numbers of identical cells Cancer develops from a single cell Every cell in a tumour further changes

3. Individual-based methods CA enlarged CA enl. Potts hyphasma Delaunay -Voronoi Delaunay -spheres isotropyno yes var neighbor-#no yes var cell volumenoyes contact surfaceno yes proliferation/volnoyes real cellsno yes noyes quantitative parsno yes in vitroyes noyes subcellular levelno yes no Overview of presently available theoretical methods in our group at FIAS

4. Delaunay & Voronoi concept Has been applied to 2d tissue [Weliky et al., Develop 113 (1991) 1231; Meineke et al., Cell Prolif. 34 (2001) 253] Simplices Empty circumsphere criterion Delaunay triangulation Dual: Voronoi tesselation [Schaller & M.-H., Comput. Phys. Commun. 2004]

5. Shape of 3-dimensional Voronoi Cells Convex polyhedra bounded by k polygons corresponding to k next neighbors Corners of the polytopes are centres of circumspheres of the Delaunay tetrahedra Volume, contact surfaces can easily be calculated! Suitable tool for pressure, cell growth, and adhesion!

6. Spheres or Voronoi ? or: Loose or dense tissue, a critical decision!  Take the minimum of sphere and Voronoi contact surface !  Voronoi-problems with in vitro and with borders  Sphere-problems with dense tissue overestimating surface

7. Weighted Voronoi tesselation Represent cells of different size: Variable radius of sphere Use radius as Delaunay weight Voronoi edge/plane coincides with edge/plane of overlapping spheres! In dense tissue Voronoi surface is a better approximation of cell surface Orthospheres instead of circumspheres!

8. Maintenance of Delaunay triangulations Start from established triangulation vertices may move, be inserted or deleted! and violate the Empty Circumsphere Criterion!  need for a dynamic triangulation update Possible approaches: Local (reduced) re-triangulation Restoration using flip-methods [Edelsbrunner / Shah, Algorithmica 15 (1996) 223]

9. Maintain Delaunay with 2-3 flips Maintain Delaunay with 2-3 flips 5 vertices can induce 2 or 3 simplices  different # of neighbours This can locally restore the empty circumsphere criterion. DE distant DE near Check for convexity with orientation test with all edges: Does a hyperplane through A,B exist such that CDE on same side of it

10. Insertion & deletion of vertices The fifth point E lies within the simplex ABCD! Insertion or deletion flips the number of simplices between 1 and 4

11. Location of simplices Find the simplex that contains an new vertex in its convex hull: Start at simplex 0: check for all vertices if the plane containing the other 3 separates the vertex from the new vertex If yes, switch to the simplex connected by that plane Repeat until no plane is found (at 15)

12. Insertion and local adjustment Red vertex inserted Is inside 3 circumspheres Use lifting and orientation for in-sphere check 3 simplices invalid (see dashed lines) Retriangulate this inner region  5 simplices All other simplices remain valid! This algorithm can be used for construction of the initial triangulation!

13. Vertex deletion by stepwise movement Open circle to be deleted  move in little steps towards NN Maintain Delaunay by 2  3 and 3  2 flips Stop when inner simplex (bold) can be savely removed  Vertex deletion is reduced to vertex movement! Step size control: Respect the orientation of neighboring simplices.

14. Performance: Simplex hopping pathfinder Scales with the distance of an arbitrary vertex from inserted vertex: n^1/3

15. Performance of insertion algorithm Includes pathfinder (N^4/3) and local re-triangulation (N) Starting from perturbed qubic lattice and using the center as starting point for the pathfinder or with the last inserted vertex as guess. Note: Naive search for neighbours scales with N^2!

16. Performance of deletion by movement Strictly linear behaviour

17. Performance of moving vertices Is also strictly linear but no figure! Restore Delaunay property by flips Delaunay restoration is a factor 10 faster than retriangulation if vertices move a fraction 0.1 of the minimum observed vertex distance

18. The code as it is 3D Incremental insertion algorithm (for initial configuration) Simplex location by guess grid and (scales with N^1/3) Dynamical insertion and deletion with local maintenance of triangulation (scales with N) Kinetic vertices following Newtonian or Monte Carlo force equations (scales with N) [Schaller & M.-H. Comp Phys Commun 162 (2004) 9] Automatic stepsize control to avoid non-flippable configurations [Schaller, PhD thesis 2005] Weighted (regular) Delaunay triangulation for objects of different size (combined sphere-Voronoi model) Hybrid regular lattice for reaction-diffusion of solubles (ADI) Parallel version of the code (kinetics!!!) [Beyer et al. Comp Phys Commun 172 (2005) 86]

19. Equation of motion Over-damped approximation Neglect intercellular drag forces (depending on cell-velocities) Adhesive friction:

20. Forces between cells (Hertz-model) 

21. Nutrient uptake Clean reaction: C 6 H 12 O 6 + 6O 2  6H 2 0 + 6CO 2 + energy  Oxygen uptake 6 times faster than glucose uptake?  No! Induction of necrosis: Two critical concentrations for oxygen and glucose One combined critical concentration (product) Concentration dependend uptake rates (?) Waste products inducing necrosis (?) Nutrient diffusion with classical equation but space dependent diffusion constant!

22. Cell cycle Durations t G1 until mitotic radius is reached G0 phase entered if critical tension T is exceeded Reenter of S/G2 if tension fine Deterministic process Normal distributed duration of S/G2

23. In vitro 3D tumour growth: glucose [Freyer and Sutherland Cancer Res. 46 (1986) 3504; Schaller & M.-H. Phys Rev E 71 (2005) 051910.]

24. In vitro 3D tumour growth: oxygen [Freyer and Sutherland Cancer Res. 46 (1986) 3504; Schaller & M.-H. Phys Rev E 71 (2005) 051910.]

25. Proliferating rim glucose oxygen 0.07-0.28 mM 0.8-16.5 mM Necrotic Quiescent Proliferating Limited nutrients induce a necrotic core Cut through 3D

26. Cell tension glucose oxygen 0.07-0.28 mM 0.8-16.5 mM In abundance of nutrients cell tension is limiting tumour growth Cut through 3D

27. PDE model for tumour growth Concentration of oxygen and glucose, viable and necrotic cells: Nutrient diffusion depends on cell density: Cell diffusion mimicks cell repulsion and adhesion for m>0:

28. Cell proliferation and necrosis Proliferation smoothly depends on cell compression: Necrosis is entered in dependence of the nutrient product: [Schaller & M.-H. Philos Trans Roy Soc A 2006 (in press)]

29. Travelling wave solution Analytically for m=0 one gets 19.6 microns/day wave velocity Slower and steeper wave for m=2 (numerical)

30. Tumour growth data are described Dashed line is with m=0 Full line with m as fit parameter

31. The little difference to agent-based Only for high nutrient concentrations the tumour size is the same! The tumour size differs between pde and agent- based models for low nutrient concentrations! Viable cells in agent-based models m=0 m=0.73 [Schaller & M.-H. Philos Trans Roy Soc A 2006 (in press)]

32. Diffusion mimicks explicit forces For m=0 PDE exhibits equal tumour radius for all nutrient concentrations This is reduced for m>0 but the more realistic size dependence of IBM is not reached Adhesion and repulsion is mimicked by larger m but in the IBM adhesion can reverse the force direction in PDE the force is only reduced Increased adhesion induces saturation in IBM not in PDE Relaxation of pressure follows a diffusion equation in the PDE and should follow a wave equation in reality. Stress is better described in the IBM

33. Conclusions Macroscopic tumour-cell number measurements are equally described by IBM and PDE! Differences occur in the tumour size predictions Saturation possible in IBM with increased adhesion – not in PDE. The advantage of IBM turns relevant when microscopic interactions are measured  Need for quantitative microscopic experiments

34. Thanks to... Collaborators: Philip Maini, Oxford University, UK My research group: Tilo Beyer (Frankfurt) Hasnaa Fatehi (Frankfurt) Jakub Pijewski (Frankfurt,Munich) Gernot Schaller (Frankfurt,Dresden)... and you. FIAS Financial support: EU Marie Curie Intraeuropean Fellowship within the Sixth EU Framework Program ALTANA AG