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Research Papers

# Modeling the Matrix of Articular Cartilage Using a Continuous Fiber Angular Distribution Predicts Many Observed Phenomena

[+] Author and Article Information
Gerard A. Ateshian1

Department of Mechanical Engineering, and Department of Biomedical Engineering, Columbia University, New York, NY 10027ateshian@columbia.edu

Vikram Rajan

Department of Mechanical Engineering, Columbia University, New York, NY 10027

Center for Micro and Nano Technology, Lawrence Livermore National Laboratory, Livermore, CA 94550

Clare E. Canal, Clark T. Hung

Department of Biomedical Engineering, Columbia University, New York, NY 10027

A complete enzymatic digestion of the proteoglycans is an option, albeit imperfect.

The tensor dyadic products $⊗$, $⊗̱$, and $⊗¯̱$ are described by Curnier et al. (35).

For comparison purposes, the elasticity tensor of a linear isotropic Hookean elastic solid is $λI⊗I+2μI⊗¯̱I$, where $λ$ and $μ$ are the Lamé constants. Thus, in the range of small strains, $λ≡−(π+J∂π/∂J)$ and $μ≡π$.

When modeling anisotropic fiber bundle properties, it is arguably simpler to assume equal fiber fractions in all directions and allow the $m$ material coefficients $ξj$ and $αj$ to vary with orientation.

The traction-free initial configuration, denoted with 0, generally differs from the traction-free reference configuration, denoted with $r$, because of the osmotic swelling pressure $(p0≠0,pr=0)$.

For example, $ν12≡−∂E22/∂E11$.

1

Corresponding author.

J Biomech Eng 131(6), 061003 (Apr 21, 2009) (10 pages) doi:10.1115/1.3118773 History: Received May 13, 2008; Revised January 09, 2009; Published April 21, 2009

## Abstract

Cartilage is a hydrated soft tissue whose solid matrix consists of negatively charged proteoglycans enmeshed within a fibrillar collagen network. Though many aspects of cartilage mechanics are well understood today, most notably in the context of porous media mechanics, there remain a number of responses observed experimentally whose prediction from theory has been challenging. In this study the solid matrix of cartilage is modeled with a continuous fiber angular distribution, where fibers can only sustain tension, swelled by the osmotic pressure of a proteoglycan ground matrix. It is shown that this representation of cartilage can predict a number of observed phenomena in relation to the tissue’s equilibrium response to mechanical and osmotic loading, when flow-dependent and flow-independent viscoelastic effects have subsided. In particular, this model can predict the transition of Poisson’s ratio from very low values in compression $(∼0.02)$ to very high values in tension $(∼2.0)$. Most of these phenomena cannot be explained when using only three orthogonal fiber bundles to describe the tissue matrix, a common modeling assumption used to date. The main picture emerging from this analysis is that the anisotropy of the fibrillar matrix of articular cartilage is intimately dependent on the mechanism of tensed fiber recruitment, in the manner suggested by our recent theoretical study (Ateshian, 2007, ASME J. Biomech. Eng., 129(2), pp. 240–249).

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## Figures

Figure 1

(a) Representative stress-strain responses of human glenohumeral joint cartilage in tension and compression, from the superficial zone (SZ) and middle zone (MZ). Tensile responses are reported for cartilage strips harvested parallel and perpendicular to the local split-line direction; compressive responses are for cylindrical specimens loaded axially along the direction normal to the articular surface. From Huang (26), with permission. (b) Theoretical predictions for a continuous fiber angular distribution.

Figure 2

Theoretical prediction of Poisson’s ratio for uniaxial tension and compression using (a) a continuous fiber distribution, or (b) three orthogonal fiber bundles. SZ: superficial zone; MZ: middle zone.

Figure 3

Representative experimental tensile response data from Fig. 2 of Elliott (25), converted from infinitesimal strain to Lagrangian strain using the formula given by these authors, and replotted as lateral versus axial strain. The solid curve is the least-squares fit of a quadratic polynomial to the data. The slope of this curve is the incremental Poisson’s ratio.

Figure 4

Average experimental response from the study of Chahine (24) on immature bovine cartilage at various NaCl bathing solution concentrations, for compressive loading along the split-line direction, and lateral contraction perpendicular to it. (a) Stress versus strain. (b) Poisson’s ratio versus strain.

Figure 5

Theoretical predictions for testing conditions similar to the study of Chahine (24), using a continuous fiber angular distribution

Figure 6

(a) Stress-strain response of normal and proteoglycan-depleted bovine articular cartilage in uniaxial tension, adapted from Fig. 6 of Schmidt (45), with permission. (b) Theoretical simulation of similar testing conditions using a continuous fiber angular distribution.

Figure 7

(a) Equilibrium uniaxial tensile stress response of bovine articular cartilage in bathing solutions of various NaCl concentrations, under 20% isometric strain; adapted from Fig. 8 of Grodzinsky (46), with permission. (b) Theoretical predictions for similar testing conditions, at isometric tensile strains varying from 0.7% to 16%. NaCl concentration is in units of molarity (M). Stresses are normalized to the corresponding value achieved in the most hypotonic bath (10−4M NaCl). Black curves represent the strains for which the isometric stress decreases as the bathing solution is switched from distilled water to isotonic saline (0.15M), whereas gray curves represent those conditions where the stress correspondingly increases; this complex response is consistent with the findings of Akizuki (27).

Figure 8

Parametric variation of the material parameter ξi from 2 MPa to 10 MPa, by increments of 2 MPa. (a) Sensitivity of the stress-strain response. (b) Sensitivity of the Poisson’s ratio-strain response.

Figure 9

Parametric variation of the material parameter αi from 2.0 to 3.0, by increments of 0.25. (a) Sensitivity of the stress-strain response. (b) Sensitivity of the Poisson’s ratio-strain response.

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