A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage

Abstract

A biphasic mixture model is developed which can account for the observed tension-compression nonlinearity of cartilage by employing the continuum-based Conewise Linear Elasticity (CLE) model of Curnier et al. (J Elasticity 37:1-38, 1995) to describe the solid phase of the mixture. In this first investigation, the orthotropic octantwise linear elasticity model was reduced to the more specialized case of cubic symmetry, to reduce the number of elastic constants from twelve to four. Confined and unconfined compression stress-relaxation, and torsional shear testing were performed on each of nine bovine humeral head articular cartilage cylindrical plugs from 6 month old calves. Using the CLE model with cubic symmetry, the aggregate modulus in compression and axial permeability were obtained from confined compression (H −A =0.64±0.22 MPa, k z = 3.62 ± .97 × 10 −16 m 4 /N.s, r 2 =0.95 ±0.03), the tensile modulus, compressive Poisson ratio and radial permeability were obtained from unconfined compression (E +Y = 12.75 ± 1.56 MPa, ν − =0.03±0.01, k r =6.06±2.10×10 −16 m 4 /N.s, r 2 =0.99±0.00), and the shear modulus was obtained from torsional shear (µ=0.17±0.06 MPa). The model was also employed to successfully predict the interstitial fluid pressure at the center of the cartilage plug in unconfined compression (r 2 =0.98±0.01). The results of this study demonstrate that the integration of the CLE model with the biphasic mixture theory can provide a model of cartilage which can successfully curvefit three distinct testing configurations while producing material parameters consistent with previous reports in the literature.

Heterogeneous three-dimensional strain fields during unconfined cyclic compression in bovine articular cartilage explants

“…While it is difficult to infer constitutive relations from the strain components, nevertheless this study has provided appropriate data including applied normal stresses at the tissue boundary and strain components throughout the volume of the material that may be used to validate previous constitutive relations. Such relations may incorporate transversely isotropic symmetry [5,8,9,18], as well as other formulations [13,32].…”

A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

“…However, bimodular models have been used. As mentioned before, Ateshian and colleagues (Soltz and Ateshian 2000;Wang et al 2003) used a bimodular model for infinitesi mal strains. Also, the orthotropic model proposed for arterial tissue (Holzapfel et al 2004) and the transversely isotro pic model proposed for the intervertebral disc (Baer et al 2004) allowed for different mechanical properties in tension and compression.…”

“…The proteoglycans are negatively charged molecules that mainly resist compressive loads (Basser et al 1998;Lai et al 1991) while the collagen net work primarily resists tensile and shear loads (Mow and Ratcliffe 1997;Venn and Maroudas 1977). Due to this molecular structure, articular cartilage typically exhibits a mechanical response with marked anisotropy and tension-compression asymmetry (Akizuki et al 1986;Laasanen et al 2003;Soltz and Ateshian 2000;Wang et al 2003;Woo et al 1976Woo et al , 1979, and likely experiences finite, multi-dimensional strains due to typical in vitro and in vivo loads. Although MRI mea surements of in situ and in vivo joints have predicted that cartilage is subject to average strains of less than 10% under physiologic loading conditions (Eckstein et al 2000;Herberhold et al 1999), local strains may be much higher due to nonhomogeneous mechanical properties that depend on both anatomic location and depth from the articular surface (Schinagl et al 1997;Wang et al 2001).…”

“…For infinitesimal strains, Ateshian and colleagues (Soltz and Ateshian 2000;Wang et al 2003) have employed elastic and biphasic models with a bimodular stress constitutive equation that allows for different mechanical proper ties in tension and compression. Their model can describe the mechanical response in unconfined compression in three orthogonal directions while providing reasonable predictions for the other protocols (Wang et al 2003).…”

A nonlinear finite element model of cartilage growth

“…Due to the observed tension-compression asymmetry in cartilage, we use a bimodular stress constitutive equation for the collagen constituent. For infinitesimal strains, Ateshian and colleagues (Soltz and Ateshian 2000;Wang et al 2003) have employed bimodular stress constitutive equations that allow for different mechanical properties in tension and compression. Those models were based on a bimodular theory for infinitesimal strains (Curnier et al 1995) in which the material constants may be discontinuous (or jump) across a surface of discontinuity in strain space, provided that stress continuity conditions are satisfied at the surface.…”