The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage

Abstract

Mixture models have been successfully used to describe the response of articular cartilage to various loading conditions. Mow et al. (J. Biomech. Eng. 102 (1980) 73) formulated a biphasic mixture model of articular cartilage where the collagen-proteoglycan matrix is modeled as an intrinsically incompressible porous-permeable solid matrix, and the interstitial fluid is modeled as an incompressible fluid. Lai et al. (J. Biomech. Eng. 113 (1991) 245) proposed a triphasic model of articular cartilage as an extension of their biphasic theory, where negatively charged proteoglycans are modeled to be fixed to the solid matrix, and monovalent ions in the interstitial fluid are modeled as additional fluid phases. Since both models co-exist in the cartilage literature, it is useful to show how the measured properties of articular cartilage (the confined and unconfined compressive and tensile moduli, the compressive and tensile Poisson's ratios, and the shear modulus) relate to both theories. In this study, closed-form expressions are presented that relate biphasic and triphasic material properties in tension, compression and shear. These expressions are then compared to experimental findings in the literature to provide greater insight into the measured properties of articular cartilage as a function of bathing solutions salt concentrations and proteoglycan fixed-charge density.

Figure 1

Rate of change of osmotic pressure with dilatation, Π, as a function of (a) c^* and (b) $c_r^F$ [Eq.(23)]. $H_{\pm A}^{\mathit{eff}}=H_{\pm A}+\mathrm{\Pi }$ and ${\lambda }_2^{\mathit{eff}}={\lambda }_2+\mathrm{\Pi }$ can be obtained by offsetting the response of Π with the appropriate constant.

Figure 2

Free-swelling strain, $E_{zz}^0$, in laterally confined configuration, as a function of (a) c^* and (b) $c_r^F$ [Eq.(24)].

Figure 3

Axial strain, $E_{zz}^0$, in unconfined free swelling configuration, as a function of (a) c^* and (b) $c_r^F$ [Eq.(34)].

Figure 4

Effective Young’s modulus in compression, $E_{-Y}^{\mathit{eff}}$, as a function of (a) c^* and (b) $c_r^F$ [Eq.(36), with $H_A^{\mathit{eff}}\{E_{zz}\}=H_{-A}+\mathrm{\Pi }$ and $H_A^{\mathit{eff}}\{E_{rr}\}=H_{+A}+\mathrm{\Pi }$)].

Figure 5

Effective Poisson’s ratio in compression, ${\nu }_{-}^{\mathit{eff}}$, as a function of (a) c^* and (b) $c_r^F$ [Eq.(37), with $H_A^{\mathit{eff}}\{E_{rr}\}=H_{+A}+\mathrm{\Pi }$)].

Figure 6

Effective Poisson’s ratio in tension, ${\nu }_{+}^{\mathit{eff}}$, as a function of (a) c^* and (b) $c_r^F$ [Eq.(37), with $H_A^{\mathit{eff}}\{E_{rr}\}=H_{-A}+\mathrm{\Pi }$)].