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TECHNICAL PAPERS: Cell

# A Three-dimensional Finite Element Model for the Mechanics of Cell-Cell Interactions

[+] Author and Article Information
Denis Viens

Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario N2T 2N4, Canada

G. Wayne Brodland1

Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario N2T 2N4, Canadabrodland@uwaterloo.ca

1

Corresponding author.

J Biomech Eng 129(5), 651-657 (Feb 14, 2007) (7 pages) doi:10.1115/1.2768375 History: Received June 06, 2006; Revised February 14, 2007

## Abstract

Technical challenges, including significant ones associated with cell rearrangement, have hampered the development of three-dimensional finite element models for the mechanics of embryonic cells. These challenges have been overcome by a new formulation in which the contents of each cell, assumed to have a viscosity $μ$, are modeled using a system of orthogonal dashpots. This approach overcomes a stiffening artifact that affects more traditional models, in which space-filling viscous elements are used to model the cytoplasm. Cells are assumed to be polyhedral in geometry, and each $n$-sided polygonal face is subdivided into $n$ triangles with a common node at the face center so that it needs not remain flat. A constant tension $γ$ is assumed to act along each cell-cell interface, and cell rearrangements occur through one of two complementary topological transformations. The formulation predicts mechanical interactions between pairs of similar or dissimilar cells that are consistent with experiments, two-dimensional simulations, contact angle theory, and intracellular pressure calculations. Simulations of the partial engulfment of one tissue type by another show that the formulation is able to model aggregates of several hundred cells without difficulty. Simulations carried out using this formulation suggest new experimental approaches for measuring cell surface tensions and interfacial tensions. The formulation holds promise as a tool for gaining insight into the mechanics of isolated or aggregated embryonic cells and for the design and interpretation of experiments that involve them.

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## Figures

Figure 7

Interactions between masses of heterotypic cells. (a) Initial configuration. The cells have properties 2γLL=γLD=2γDD=γLM=γDM. (b) Final configuration. The heavy dashed curve indicates the interface between the light and dark cells at the center of the mass.

Figure 8

Intracellular pressure versus inverse of cell radius for cells in an aggregate. Separate best fit lines are shown for the light and dark cell data as are the corresponding lines predicted by Eq. 8 based on form factors having a value of unity.

Figure 2

Cell rearrangement. (a) An edge pq spans between cells A and B, along the triple junction between cells C, D, and E (E has been removed to reveal the edge pq). (b) At the brief moment when the edge pq has shortened to zero length, cells A and B just touch each other. (c) As cells A and B continue to move closer together, a triangular interface fgh forms between them (A has been displaced upward to reveal the new interface fgh).

Figure 3

Rounding of a single cell. (a) Initial configuration taken from a Voronoi tessellation. (b) Final, rounded state of the cell. (c) Configuration produced when that shown in (b) is stretched along the horizontal axis of the figure to produce a prolate spheroid having κ=2. (d) Configuration produced when that shown in (b) is compressed along the horizontal axis to form an oblate spheroid having κ=2.

Figure 4

Kappa versus time for a single, isolated cell. With time, both prolate (Fig. 3) and oblate cells (Fig. 3) reshape toward a spherical geometry (κ=1).

Figure 5

Interactions between pairs of homotypic cells of similar size. (a) Initial configuration. (b) When γLM=γDM=4γLD, the cells form a tight sphere. See text for the definition of the surface and interfacial tensions. (c) When γLM=γDM=γLD, the cells form a “peanut” shape. (d) When γLM=γDM=0.5γLD, the cells nearly separate. In all cases, the interface between the cells is essentially flat.

Figure 1

The FE model. (a) A schematic representation of a typical cell in a close-packed aggregate. (b) Node placements and vectors used to calculate interfacial tension forces. (c) Dashpots and “slider” representing the viscosity of the cytoplasm. Similar dashpot and slider systems are placed in the y- and z-directions, as well.

Figure 6

Interactions between cells of dissimilar size. The cell on the right has approximately four times the volume of the cell on the left. In (a)–(c), the cell on the left is the dark (D) cell. (a) Homotypic cells having γLM=γDM=4γLD, as in Fig. 5, again produce a tight sphere. (b) Homotypic cells having γLM=γDM=γLD form an asymmetric peanut (cf. Fig. 5). ((c) and (d)) Heterotypic cells having γLM=1.7γDM=1.3γLD. The types of the large and small cells have been exchanged in the two parts of the figure. The heavy dashed curves indicate the interface between the cells as it would appear in a longitudinal cross section.

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