## Abstract

In Part 1 of this work, we formulated and analyzed a mathematical model for our fibroblast-populated collagen microsphere (FPCM) assay of cell traction forces (Moon and Tranquillo, 1993). In this assay, the FPCM diameter decreases with time as the cells compact the gel by exerting traction on collagen fibrils. In Part I we demonstrated that the diameter reduction profdes for varied initial cell concentration and varied initial FPCM diameter are qualitatively consistent with the model predictions. We show here in Part 2 how predictions of a model similar to that of Part 1, along with the determination of the growth parameters of the cells and the viscoelastic parameters of the gel, allow us to estimate the magnitude of a cell traction parameter, the desired objective index of cell traction forces. The model is based on a monophasic continuum-mechanical theory of cell-extracellular matrix (ECM) mechanical interactions, with a species conservation equation for cells (1), a mass conservation equation for ECM (2), and a mechanical force balance for the cell/ECM composite (3). Using a constant-stress rheometer and a fluids spectrometer in creep and oscillatory shear modes, respectively, we establish and characterize the linear viscoelastic regime for the reconstituted type 1 collagen gel used in our FPCM traction assay and in other assays of cell-collagen mechanical interactions. Creep tests are performed on collagen gel specimens in a state resembling that in our FPCM traction assay (initially uncompacted, and therefore nearly isotropic and at a relatively low collagen concentration of 2.1 mg/ml), yielding measurements of the zero shear viscosity, μ_{0} (7.4 × 10^{6} Poise), and the steady-state creep compliance, J^{0}_{c}. The shear modulus, G (155 dynes/cm^{2}), is then determined from the inverse of J^{0}_{c} in the linear viscoelastic regime. Oscillatory shear tests are performed in strain sweep mode, indicating linear viscoelastic behavior up to shear strains of approximately 10 percent. We discuss the estimation of Poisson’s ratio, V, which along with G and μ_{0} specifies the a.ssumed isotropic, linear viscoelastic stress tensor for the cell/collagen gel composite which appears in (3). The proliferation rate of fibroblasts in free floating collagen gel (appearing in (1)) is characterized by direct cell counting, yielding an estimate of the first-order growth rate constant, k (5.3 × 10^{−6} s^{−2}). These independently measured and estimated parameter values allow us to estimate that the cell traction parameter, τ_{0}, defined in the active stress tensor which also appears in (3), is in the range of 0.00007-0.0002 dyne · cm^{4}/mg collagen · cell. This value is in agreement with a reported measure of traction obtained directly via isometric force measurement across a slab of fibroblast-containing collagen gel.

Original language | English (US) |
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Pages (from-to) | 161-170 |

Number of pages | 10 |

Journal | Journal of biomechanical engineering |

Volume | 117 |

Issue number | 2 |

DOIs | |

State | Published - May 1995 |