On the application of strain factors for approximation of the contribution of anisotropic cells to the mechanics of a tissue construct

Bio-artificial tissue constructs consisting of fibroblast cells embedded in a collagenous matrix are valuable in vitro systems in which to study cellular mechanics. Deriving cellular mechanics from the results of experimentation on tissue constructs requires a mathematical relationship that delineates amongst the contributions of the constituents of a tissue construct. A scaling between the average strain in a uniformly stretched tissue and the axial strain in isotropic cells was used in earlier work to study relations between cell mechanics and the overall mechanics of a tissue construct. That work showed that a scaling factor called a "strain factor" provided an accurate representation of the average axial strain in isotropic cells. The present study analyzes such relationships for anisotropic cells. We incorporate Eshelby's (1957; Proceedings of the Royal Society of London A 241, 376; 1959; Proceedings of the Royal Society of London A 252, 561) exact solution for the strain field in isolated ellipsoidal inclusions into the Zahalak (Biophysical journal 79, 2369) constitutive model for tissue constructs. Results showed that, for the case of prolate cells, the strain along the major cell axis is mostly influenced by the remote strain projected along that axis; off-axis cell mechanics plays only a small role in most tissues. The strain factor approximation is shown to be accurate for anisotropic cells to within a few percent for the vast majority of tissues. The results presented in this paper provide an explicit measure of the effects of cellular anisotropy, and a mechanism for calculating the contributions of these effects to overall tissue mechanics when these effects are important.