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Bulletin of Mathematical Biology

, Volume 72, Issue 7, pp 1696–1731 | Cite as

Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics

  • Martin Bock
  • Amit Kumar Tyagi
  • Jan-Ulrich Kreft
  • Wolfgang Alt
Original Article

Abstract

Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein–Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.

Keywords

Cell tissue model Circular Voronoi power diagram 

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Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Martin Bock
    • 1
  • Amit Kumar Tyagi
    • 1
  • Jan-Ulrich Kreft
    • 1
    • 2
  • Wolfgang Alt
    • 1
  1. 1.IZMB, Theoretische BiologieUniversität BonnBonnGermany
  2. 2.Centre for Systems Biology, School of BiosciencesUniversity of BirminghamBirminghamUK

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