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Research Papers

An Equilibrium Constitutive Model of Anisotropic Cartilage Damage to Elucidate Mechanisms of Damage Initiation and Progression

[+] Author and Article Information
Michael E. Stender

Department of Mechanical Engineering,
University of Colorado,
Boulder, CO 80309

Richard A. Regueiro

Department of Civil, Environmental,
and Architectural Engineering,
University of Colorado,
Boulder, CO 80309

Stephen M. Klisch

Department of Mechanical Engineering,
California Polytechnic State University,
San Luis Obispo, CA 93407

Virginia L. Ferguson

Department of Mechanical Engineering,
University of Colorado,
427 UCB, Boulder, CO 80309
e-mail: virginia.ferguson@colorado.edu

1Corresponding author.

Manuscript received November 24, 2014; final manuscript received May 19, 2015; published online June 16, 2015. Assoc. Editor: James C Iatridis.

J Biomech Eng 137(8), 081010 (Aug 01, 2015) (13 pages) Paper No: BIO-14-1582; doi: 10.1115/1.4030744 History: Received November 24, 2014; Revised May 19, 2015; Online June 16, 2015

Traumatic injuries and gradual wear-and-tear of articular cartilage (AC) that can lead to osteoarthritis (OA) have been hypothesized to result from tissue damage to AC. In this study, a previous equilibrium constitutive model of AC was extended to a constitutive damage articular cartilage (CDAC) model. In particular, anisotropic collagen (COL) fibril damage and isotropic glycosaminoglycan (GAG) damage were considered in a 3D formulation. In the CDAC model, time-dependent effects, such as viscoelasticity and poroelasticity, were neglected, and thus all results represent the equilibrium response after all time-dependent effects have dissipated. The resulting CDAC model was implemented in two different finite-element models. The first simulated uniaxial tensile loading to failure, while the second simulated spherical indentation with a rigid indenter displaced into a bilayer AC sample. Uniaxial tension to failure simulations were performed for three COL fibril Lagrangian failure strain (i.e., the maximum elastic COL fibril strain) values of 15%, 30%, and 45%, while spherical indentation simulations were performed with a COL fibril Lagrangian failure strain of 15%. GAG damage parameters were held constant for all simulations. Our results indicated that the equilibrium postyield tensile response of AC and the macroscopic tissue failure strain are highly dependent on COL fibril Lagrangian failure strain. The uniaxial tensile response consisted of an initial nonlinear ramp region due to the recruitment of intact fibrils followed by a rapid decrease in tissue stress at initial COL fibril failure, as a result of COL fibril damage which continued until ultimate tissue failure. In the spherical indentation simulation, damage to both the COL fibril and GAG constituents was located only in the superficial zone (SZ) and near the articular surface with tissue thickening following unloading. Spherical indentation simulation results are in agreement with published experimental observations. Our results indicate that the proposed CDAC model is capable of simulating both initial small magnitude damage as well as complete failure of AC tissue. The results of this study may help to elucidate the mechanisms of AC tissue damage, which initiate and propagate OA.

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Figures

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Fig. 1

The COL fibril constituent stress strain response with damage for a single COL fibril. All COL fibrils are hypothesized to initially respond with a linear elastic response and then fail completely if the COL fibril Lagrangian damage strain in direction N, END, value is exceeded. Thus, the maximum COL fibril yield stress in direction N is the product of the COL fibril modulus, Ef and the COL fibril Lagrangian failure strain, END in that direction.

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Fig. 2

Initial highly anisotropic COL fibril area density in the SZ and MZ of newborn bovine AC from the patellofemoral groove in a plane perpendicular to the articular surface. Initial COL fibril area fraction distributions are determined via quantitative polarized light microscopy measurements from Stender et al. [32] and adjusted for the biochemical COL fibril volume fraction from Williams et al. [6]. Note that angle = 90 deg corresponds to the direction perpendicular to the articular surface.

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Fig. 3

GAG damage parameter, dGAG, plots showing (a) dGAG as a function of GAG damage ISV, β, for several values of the GAG damage scaling variable, η, with dmaxGAG=0.5. (b) GAG damage, dGAG, plotted for values of dmaxGAG=0.5 and η=0.25 as used in this study as a function of the minimum J=detF value in the material loading history.

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Fig. 4

Finite-element models showing (a) uniaxial stress in tension finite-element model with dashed lines denoting the initial configuration. (b) Spherical indentation model with superficial and MZs of AC. For spherical indentation simulations, the spherical indenter was modeled as an analytical rigid surface while the cartilage block was modeled using 10,625, eight-node linear hexahedral C3D8 elements with displacements at the bottom surface constrained in all directions. The adjacent SZ and MZ of regions were constrained at the shared surface using a tied contact formulation.

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Fig. 5

Predicted equilibrium values of AC ultimate tensile failure strain for COL fibril Lagrangian failure strain values END=0.15,0.30,0.45 in the SZ and MZ of bovine AC. Macroscopic AC tissue failure strain values were determined as when the tissue axial tensile stress reached 1% of the max stress.

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Fig. 6

Predicted equilibrium uniaxial tension axial stress strain response in the SZ and MZ for COL fibril Lagrangian failure strain values of END=0.15,0.30,0.45. Vertical axis range is uniform for ease of comparison.

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Fig. 7

Predicted equilibrium COL fibril damage, Dcol (%), in the SZ and MZ for COL fibril Lagrangian failure strain values of END=0.15,0.30,0.45. Vertical axis range is uniform for ease of comparison.

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Fig. 8

Comparisons of COL fibril area fraction distribution in the MZ and SZ in undamaged tissue, and at the point of ultimate macroscopic tensile failure for a Lagrangian failure strain of END=0.30 . Note that uniaxial tensile load was aligned parallel to the articular surface in the 180 deg direction. Vertical axis range is the same for ease of comparison.

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Fig. 9

Contour plots of finite-element analysis results for (a) GAG damage parameter, dGAG, (b) COL fibril damage parameter, Dcol, (c) maximum principal LE strain, and (d) maximum principal Cauchy stress (MPa) in a bilayered AC model with discrete SZ and MZ. A rigid spherical indenter was displaced into the articular surface while displacements in all directions at the bottom of the MZ were held fixed. Damage parameters of END=0.15, dmaxGAG=0.5, and η=0.25 were used. Mesh lines removed for clarity.

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Fig. 10

Contour plots of (top) GAG damage parameter, dGAG, and (bottom) COL fibril damage parameter, Dcol with progressive spherical indentation loading in a bilayered AC model with discrete SZ (SZ and MZ). A rigid spherical indenter was displaced into the articular surface while displacements in all directions at the bottom of the MZ were held fixed. Damage parameters of END=0.15, dmaxGAG=0.5, and η=0.25 were used. Mesh lines removed for clarity.

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Fig. 11

Finite-element results. (a) A contour plot of LE strain perpendicular to the articular surface in the SZ showing cartilage thickening postdamage. Displacements are shown 100× for clarity. (b) Force versus displacement plots for two cycle indentation loading showing hysteresis and a decrease in reaction force in the second cycle compared to the first cycle.

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