Constitutive Model

In embryonic tissues, cells readily rearrange themselves, and, as a result, the stress-strain characteristics of embryonic tissues are quite different from those of mature tissues. Analytical studies and computer simulations show that the constitutive relationship is greatly affected by cell-level activities such as reshaping, rearrangement, and mitosis [Brodland et al., 2000; Chen and Brodland, 2000; Brodland et al., 2005], a finding that is supported by experimental results [Brodland and Wiebe, 2004].

The model on which the analysis and simulations are based involves two primary assumptions:

1. Interfacial tension, y, is assumed to be a primary driving force of cell-cell interactions. This tension is generated by circumferential microfilament bundles (CMBs), membrane-associated proteins, and cell membranes, and cell-cell adhesion generated by cell adhesion molecules (CAMs) and other mechanisms reduce the net contraction [Chen and Brodland, 2000].

2. Cell cytoplasm is assumed to be incompressible and characterized by an effective viscosity, /Л. To model cells with these characteristics using the finite element method, each n-sided cell is

divided into n triangular elements (Fig. 2b) and rod-like elements are employed to model the interfacial tension. To account for the volume change that would occur in individual triangular elements due to motion of cytoplasm from one triangular element to another Poisson’s ratio for the cytoplasm is set to zero. To keep the total volume in each cell constant, a volume constraint is applied to each cell.

Standard finite element approaches are used to determine element stiffness matrices and equivalent nodal loads and to assemble these results into a system of simultaneous equations [Chen and Brodland 2000],

CU = f, (1)

where C is the damping matrix of the system and is derived from the triangular elements, f is the vector of driving forces produced by the rod-like elements carrying y, and U is the vector of nodal velocities. Using a forward difference scheme, this nonlinear equation can be rewritten as

CU » C = – LCAu = f, (2)

At At

and

^ C(U,+ 1 – U, ) = f, – (3)

Solution of these equations gives the time course of the resulting cell-cell interactions.

Insights from finite element models of this kind have lead to the development of constitutive equations for “patches” of such cells (Fig. 3a). The cells in a patch are characterized by a composite cell (Fig. 3b) that captures average aspect ratio, к, long-axis mean angle, a, and cell density, P [Brodland and Wiebe, 2004].

(b)

Figure 3. A cell aggregate and its composite cell. [Brodland, 2004]

Brodland and Wiebe (2004) have demonstrated that the principal stresses in the sheet are generated by interfacial tensions according to Equations 4 and 5:

Ultimately, the stress-strain characteristics of the tissue depend on the five parameters, у, /и, к, a and fd [Brodland et al., 2005]. These equations can be incorporated into a tissue-level finite element model used to model an entire embryo (Fig. 4). Values for all parameters can be obtained from suitable images of real embryos and from mechanical property tests on tissue specimens. Tests have shown that these parameters vary with tissue type, location, orientation, and developmental stage [Brodland and Wiebe, 2004; Wiebe and Brodland, 2005].

Figure 4. Finite element implementation of the constitutive equations.