This example is the second in a series describing the "cell growth" material in FEBio. To better understand the material presented here, please refer to the example on Single Cell Growth.

The cell growth material is a continuum model that may be equally used to describe the growth of a single cell, or an aggregate of cells. Cell aggregates may form tissues and organs, and the cell growth model may be used for such aggregates as long as the extracellular matrix in those tissues represents a small fraction of the tissue volume. The assumption implicit in this approach is that all the cells of an aggregate are exposed to the same environmental conditions (same external osmolarity). As cells grow and divide, there is no need to explicitly account for boundaries between the cells of a cell aggregate in this continuum representation. Therefore, the model only needs to explicitly describe the growth process, not cell division.

In the example of Single Cell Growth, there was no solid matrix associated with the cell, and thus no explicit description of the stiffness of the cell cytoskeleton. As a result of that assumption, the spherical cell maintained its spherical shape during growth. In this example, we examine how the elasticity of the cell cytoskeleton (and/or the extracellular matrix, in the case of cell aggregates forming tissues or organs) imposes internal constraints that might influence the shape of the growing structure.

For example, consider a highly cellularized tissue that has a tubular form, such as the media layer (tunica media) of an artery. The growth of a tube may be modeled in FEBio by creating this tubular geometry, and assigning to it a "solid mixture" material consisting of a "cell growth" material, and an "orthotropic elastic" material. The "orthotropic elastic" material represents the stiffness of the solid matrix (the cell cytoskeleton and/or the extracellular matrix). Three cases are considered below, each representing a different choice of properties for the solid matrix. In all these cases, the "cell growth" model is set up to produce a fivefold increase in the intracellular membrane-impermeant solute content, cr, and the intracellular solid content, phir.

Isotropic Solid Matrix

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TubeGrowth.feb

In this case, consider that the solid matrix is isotropic. As expected, the resulting growth process is isotropic, and the tube grows in size equally in all directions. The volume of the tissue does not necessarily increase fivefold, depending on the magnitude of the solid matrix stiffness relative to the osmotic driving forces produced by the growth process. In this example, the modulus of the solid matrix is on the order of 1 kPa, and the relative volume at the end of the growth is 4.995.Orthotropic Solid Matrix with Greater Axial Stiffness

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In this second case, the modulus along the length of the tube is taken to be 100 times greater than in the circumferential and radial directions (10 kPa versus 0.1 kPa). Most of the growth thus manifests itself as a radial and circumferential expansion, with negligible change in length along the axial direction.Orthotropic Solid Matrix with Greater Radial and Circumferential Stiffness

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This third case is the reverse of the previous one: The axial modulus is 100 times smaller than in the circumferential and radial directions (0.1 kPa versus 10 kPa). Most of the growth thus manifests itself as an axial expansion, with negligible change along the radial and circumferential directions.

These examples illustrate the fact that the anisotropy of the solid structures in the cells or their surrounding extracellular matrix may have a significant influence on the shape of the growing cells, tissues, or organs. The absolute value of the solid matrix stiffness only restricts the size of the growth, and controls the magnitude of residual stresses that might develop in the solid matrix. However, the shape is controlled by the relative magnitudes of the stiffness along various directions.

The next example in this series illustrates the influence of External Constraints to Growth.

Gerard