Anisotropic Diffusivity Tensor in Articular Cartilage: Effective Medium Approach

Abstract

Due to the avascular nature of articular cartilage, molecular transport occurs via interstitial fluid flow as well as via diffusion. Diffusion in cartilage has been studied experimentally, but no mathematical models have been developed to interpret the experimental results and the observed isotropy or anisotropy in the different cartilage zones. Here, we propose a model for the determination of the diffusivity tensor of uncharged macromolecules in articular cartilage, accounting for the inhomogeneity and anisotropy arising from fiber arrangement, volumetric fraction, and radius. We study a representative element of volume (REV) comprising a fiber surrounded by fluid-saturated proteoglycan matrix. The REV permeability tensor is evaluated using a previously developed model, while the REV diffusivity tensor is obtained by incorporating the hydrodynamic effect and the steric effect of the fiber-reinforced matrix. Both effects are represented by anisotropic second-order tensors. The overall diffusivity tensor is obtained as the averaging integral of the REV diffusivity, weighted by the probability distribution of fiber orientation. The model's predictions of the trend of the magnitude of the diffusivity of spheroidal macromolecules as a function of molecular radius agree with published experimental results. For large linear macromolecules, the model underestimates the diffusivity magnitude (i.e., the equivalent isotropic diffusivity). The model correctly predicts the anisotropic behavior for linear macromolecules, although it underestimates the numerical value of the diffusivity anisotropy ratio of large linear macromolecules in the superficial zone, and overestimates it in the deep zone. In summary, this model constitutes a first step toward understanding the relation between diffusivity and permeability in articular cartilage.