The Relationship between Cell and Tissue Strain in Three-Dimensional Bio-Artificial Tissues

Abstract

Continuum constitutive laws are needed to ensure that bio-artificial tissue constructs replicate the mechanical response of the tissues they replace, and to understand how the constituents of these constructs contribute to their overall mechanical response. One model designed to achieve both of these aims is the Zahalak model, which was modified by Marquez and co-workers to incorporate inhomogeneous strain fields within very thin tissues. When applied to reinterpret previous measurements, the modified Zahalak model predicted higher values of the continuum stiffness of fibroblasts than earlier estimates. In this work, we further modify the Zahalak model to account for inhomogeneous strain fields in constructs whose cell orientations have a significant out-of-plane component. When applied to reinterpret results from the literature, the new model shows that estimates of continuum cell stiffness might need to be revised upward. As in this article's companion, we updated the average cell strain by defining a correction factor ("strain factor"), based upon the elastic response. Three different cell orientation distributions were studied. We derived an approximate scaling model for the strain factor, and validated it against exact and self-consistent (mean-field) solutions from the literature for dilute cell concentrations, and Monte Carlo simulations involving three-dimensional finite element analyses for high cell concentrations.

Whole cell mechanics of contractile fibroblasts: relations between effective cellular and extracellular matrix moduli

“…C can be interpreted as a measure of the ratio of cell length to cell spacing, b: C = (l c /b) 3 (Marquez et al 2005b). …”

“…The observation of a reduction in the cell modulus with increasing cell concentration can be viewed with confidence, especially for the highest cell concentrations: estimates of cell moduli were the best for the high cell concentrations (N > 55 million cells ml −1 ) because the scatter in the Monte Carlo simulations was very small for cases in which the cells were more compliant than the ECM, and any spatial variations in mechanical properties can be expected to diminish rapidly beyond the 'percolation threshold' of between 20 and 30 million cells ml −1 , at which cells form a contiguous network (Marquez et al 2005b). Since the Monte Carlo estimates for the ratio between the actual elastic modulus of the ECM and the measured tangent modulus of the deoxycholate-treated construct show very little scatter, the estimates of ECM tangent moduli and hence cell moduli contain little error from this source.…”

A Multiscale Approach to Modeling the Passive Mechanical Contribution of Cells in Tissues

“…[1,3]) considered both the cell and the matrix as continuous materials. Evolution of such models has generally involved incorporation of microstructural detail and/or cellular phenomenon to develop improved constitutive laws, e.g., models that incorporate fiber orientation distributions to capture anisotropic behavior of the ECM surrounding the cells [4][5][6], incorporation of "anisotropy tensors" to account for cell orientation within a tissue [7][8][9][10], or a mathematical model motivated by localized variations in the pericellular region [11,12]. In addition, multilevel finite element approaches have been used to account for tissue inhomogeneity by discretizing cells from the surrounding ECM to consider cells as separate entities [13].…”

The fibrous cellular microenvironment, and how cells make sense of a tangled web

“…Seminal models in the field of biomechanics, such as the Y. C. Fung "quasilinear viscoelastic model" (8), draw upon continuum representations of tissues that account, under certain cases (9,10), for the ways that fibers and their solvent interact, but hide the fibrous nature of the tissue. Flory-type treatment of tissues such as polymers with prescribed orientation distributions, as established for musculoskeletal tissues by Lanir (11), Sacks and colleagues (12), and Zahalak and colleagues (13), are a mainstay of modeling of fibrous collagenous tissues and their attachments to bone (14,15). Furthermore, biphasic mixture models underlie nearly all modern successes in cartilage tissue engineering (16,17).…”

“…Continuum models are well established for estimating the stresses and strains that cells feel (13); however, these models have failed to predict experimental results (7). Specifically, when cells were cultured on fibrous mats of nanofibers, their focal adhesions under certain circumstances did the opposite of what the models predicted they would do.…”