Delaunay tells the cancer cells Delaunay tells the cancer cells their neighbourstheir neighbours

Philip K. MainiCentre for Mathematical Biology, Oxford, UK

Michael Meyer-Hermann Frankfurt Institute for Advanced Studies (FIAS), Germany

Gernot SchallerFIAS [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

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

Individual-based methodsIndividual-based methods

CA

enlargedCA

enl.Potts hyphasma

Delaunay-Voronoi

Delaunay -spheres

isotropy no no no no yes yes

var neighbor-# no no yes yes yes yes

var cell volume no yes yes yes yes yes

contact surface no no yes yes yes yes

proliferation/vol no yes yes yes yes yes

real cells no no yes yes no yes

quantitative pars no no no yes yes yes

in vitro yes yes yes yes no yes

subcellular level no no yes yes no no

Overview of presently available theoretical methods in our group at FIAS

Delaunay & Voronoi conceptDelaunay & Voronoi concept

Has been applied to 2d tissue[Weliky et al., Develop 113 (1991) 1231;

Meineke et al., Cell Prolif. 34 (2001) 253]

SimplicesEmpty circumsphere criterionDelaunay triangulationDual: Voronoi tesselation

[Schaller & M.-H., Comput. Phys. Commun. 2004]

Shape of 3-dimensional Voronoi CellsShape 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 forpressure, cell growth,and adhesion!

Spheres or Voronoi ?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 bordersSphere-problems with dense tissue overestimating surface

Weighted Voronoi tesselationWeighted Voronoi tesselationRepresent 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!

Maintenance of Delaunay triangulationsMaintenance 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]

Maintain Delaunay with 2-3 flipsMaintain Delaunay with 2-3 flips5 vertices can induce 2 or 3 simplices different # of neighboursThis 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

Insertion & deletion of verticesInsertion & deletion of vertices

The fifth point E lies within the simplex ABCD!

Insertion or deletion flips the number of simplices between 1 and 4

Location of simplicesLocation of simplicesFind 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)

Insertion and local adjustmentInsertion 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!

Vertex deletion by stepwise movementVertex deletion by stepwise movement

• Open circle to be deleted move in little steps towards NN• Maintain Delaunay by 23 and 32 flips• Stop when inner simplex (bold) can be savely removedVertex deletion is reduced to vertex movement!Step size control: Respect the orientation of neighboring simplices.

Performance: Simplex hopping pathfinderPerformance: Simplex hopping pathfinder

Scales with the distance of an arbitrary vertex from inserted vertex: n^1/3

Performance of insertion algorithmPerformance 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!

Performance of deletion by movementPerformance of deletion by movement

Strictly linear behaviour

Performance of moving verticesPerformance 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

The code as it isThe 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]

Equation of motionEquation of motion

Over-damped approximation

Neglect intercellular drag forces(depending on cell-velocities)

Adhesive friction:

Forces between cellsForces between cells

(Hertz-model)

Nutrient uptakeNutrient uptakeClean reaction: C6H12O6 + 6O2 6H20 + 6CO2+ 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!

Cell cycleCell 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

In vitro 3D tumour growth: glucoseIn vitro 3D tumour growth: glucose

[Freyer and Sutherland Cancer Res. 46 (1986) 3504; Schaller & M.-H. Phys Rev E 71 (2005) 051910.]

In vitro 3D tumour growth: oxygenIn vitro 3D tumour growth: oxygen

[Freyer and Sutherland Cancer Res. 46 (1986) 3504;Schaller & M.-H. Phys Rev E 71 (2005) 051910.]

Proliferating rimProliferating rim

glucose

oxygen0.07-0.28 mM

0.8-16.5 mM

NecroticQuiescent

Proliferating

Limited nutrients induce a necrotic core

Cut through 3D

Cell tensionCell tension

glucose

oxygen0.07-0.28 mM

0.8-16.5 mM

In abundance of nutrients cell tension is limiting tumour growth

Cut through 3D

PDE model for tumour growthPDE model for tumour growthConcentration of oxygen and glucose, viable and necrotic cells:

Nutrient diffusion depends on cell density:

Cell diffusion mimicks cell repulsion and adhesion for m>0:

Cell proliferation and necrosisCell 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)]

Travelling wave solutionTravelling wave solution

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

Tumour growth data are describedTumour growth data are described

Dashed line is with m=0Full line with m as fit parameter

The little difference to agent-basedThe 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)]

Diffusion mimicks explicit forcesDiffusion 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

ConclusionsConclusions

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

Thanks to ...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.... and you.

FIASFinancial support:

EU Marie Curie IntraeuropeanFellowship within the Sixth EU Framework Program

ALTANA AGALTANA AG