A triphasic orthotropic laminate model for cartilage curling behavior: fixed charge density versus mechanical properties inhomogeneity

Abstract

Osmotic pressure and associated residual stresses play important roles in cartilage development and biomechanical function. The curling behavior of articular cartilage was believed to be the combination of results from the osmotic pressure derived from fixed negative charges on proteoglycans and the structural and compositional and material property inhomogeneities within the tissue. In the present study, the in vitro swelling and curling behaviors of thin strips of cartilage were analyzed with a new structural model using the triphasic mixture theory with a collagen-proteoglycan solid matrix composed of a three-layered laminate with each layer possessing a distinct set of orthotropic properties. A conewise linear elastic matrix was also incorporated to account for the well-known tension-compression nonlinearity of the tissue. This model can account, for the first time, for the swelling-induced curvatures found in published experimental results on excised cartilage samples. The results suggest that for a charged-hydrated soft tissue, such as articular cartilage, the balance of proteoglycan swelling and the collagen restraining within the solid matrix is the origin of the in situ residual stress, and that the layered collagen ultrastructure, e.g., relatively dense and with high stiffness at the articular surface, play the dominate role in determining curling behaviors of such tissues.

Figure 1

The layer structure of articular cartilage strip with its dimensions and the predominant orientation of collagen fibrils

Figure 2

Geometry of laminate deformation. Based on the classic lamination theory, the line (e.g., AD) initially perpendicular to the mid-plane remains a straight line and remains perpendicular to mid-plane after deformation. Here u, v, w are displacements in x, y and z direction, respectively, and u₀, v₀, w₀ are displacements for the mid-plane. N_x and N_y are resultant forces, and M_x and M_y are resultant being moments.

Figure 3

Variation of stretch (Λ = dx/dX) of cartilage strip in different external ion concentrations (0.015M, 0.05M, 0.15M, 0.5 M, and 2M). The stretch is defined as the ratio of the length after deformation (dx) and the original length (dX).

Figure 4

Variation of curvature of cartilage strip in different ion concentrations for the base case parameters listed in Table 1.

Figure 5

Variation of average curvature κ̄ (=(κ_x + κ_y)/2) with external ion concentration for the case with uniform fixed charge density (FCD) distribution and the case with the absence of superficial layer (SL) (also shown the results for the base case).