Technical Briefs

A Triphasic Orthotropic Laminate Model for Cartilage Curling Behavior: Fixed Charge Density Versus Mechanical Properties Inhomogeneity

[+] Author and Article Information
Leo Q. Wan, X. Edward Guo

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

Van C. Mow1

Department of Biomedical Engineering, Columbia University, New York, NY 10027vcm1@columbia.edu


Corresponding author.

J Biomech Eng 132(2), 024504 (Jan 29, 2010) (5 pages) doi:10.1115/1.4000942 History: Received May 09, 2009; Revised December 21, 2009; Posted January 05, 2010; Published January 29, 2010; Online January 29, 2010

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.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

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

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Figure 2

Geometry of laminate deformation. Based on the classic lamination theory, the line (e.g., AD) initially perpendicular to the midplane remains a straight line and remains perpendicular to midplane after deformation. Here, u, v, and w are displacements in x, y, and z directions, respectively, and u0, v0, and w0 are displacements for the midplane. Nx and Ny are resultant forces and Mx and My are resultant being moments.

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Figure 3

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

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Figure 4

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

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Figure 5

Variation in average curvature κ¯(=(κx+κy)/2) with external ion concentration for the case with uniform FCD distribution and the case with the absence of SL (also shown the results for the base case)




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