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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Maroudas, A., Bayliss, M. T., and Venn, M. F., 1980, “Further Studies on the Composition of Human Femoral Head Cartilage,” Ann. Rheum. Dis., 39(5), pp. 514–523. [CrossRef] [PubMed]
Brocklehurst, R., Bayliss, M. T., Maroudas, A., Coysh, H. L., Freeman, M. A. R., Revell, P. A., and Ali, S. Y., 1984, “The Composition of Normal and Osteoarthritic Articular Cartilage From Human Knee Joints,” J. Bone Jt. Surg. Am., 66(1), pp. 95–106. http://jbjs.org/content/66/1/95.abstract
Asanbaeva, A., Tam, J., Schumacher, B. L., Klisch, S. M., Masuda, K., and Sah, R. L., 2008, “Articular Cartilage Tensile Integrity: Modulation by Matrix Depletion is Maturation-Dependent,” Arch. Biochem. Biophys., 474(1), pp. 175–182. [CrossRef] [PubMed]
Kiviranta, P., Rieppo, J., Korhonen, R. K., Julkunen, P., Töyräs, J., and Jurvelin, J. S., 2006, “Collagen Network Primarily Controls Poisson's Ratio of Bovine Articular Cartilage in Compression,” J. Orthop. Res., 24(4), pp. 690–699. [CrossRef] [PubMed]
Ficklin, T., Thomas, G., Barthel, J. C., Asanbaeva, A., Thonar, E. J., Masuda, K., Chen, A. C., Sah, R. L., Davol, A., and Klisch, S. M., 2007, “Articular Cartilage Mechanical and Biochemical Property Relations Before and After In Vitro Growth,” J. Biomech., 40(16), pp. 3607–3614. [CrossRef] [PubMed]
Williams, G. M., Dills, K. J., Flores, C. R., Stender, M. E., Stewart, K. M., Nelson, L. M., Albert, C. C., Masuda, K., Hazelwood, S. J., Klisch, S. M., and Sah, R. L., 2010, “Differential Regulation of Immature Articular Cartilage Compressive Moduli and Poisson's Ratios by In Vitro Stimulation With IGF-1 and TGF-β1,” J. Biomech., 43(13), pp. 2501–2507. [CrossRef] [PubMed]
Maroudas, A., and Bannon, C., 1981, “Measurement of Swelling Pressure in Cartilage and Comparison With the Osmotic Pressure of Constituent Proteoglycans,” Biorheology, 18(3–6), pp. 619–632. http://europepmc.org/abstract/med/6799013 [PubMed]
Asanbaeva, A., Masuda, K., Thonar, E. J.-M. A., Klisch, S. M., and Sah, R. L., 2007, “Mechanisms of Cartilage Growth: Modulation of Balance Between Proteoglycan and Collagen In Vitro Using Chondroitinase ABC,” Arthritis Rheum., 56(1), pp. 188–198. [CrossRef] [PubMed]
Maroudas, A., 1976, “Balance Between Swelling Pressure and Collagen Tension in Normal and Degenerate Cartilage,” Nature, 260(5554), pp. 808–809. [CrossRef] [PubMed]
Thomas, G. C., Asanbaeva, A., Vena, P., Sah, R. L., and Klisch, S. M., 2009, “A Nonlinear Constituent Based Viscoelastic Model for Articular Cartilage and Analysis of Tissue Remodeling Due to Altered Glycosaminoglycan-Collagen Interactions,” ASME J. Biomech. Eng., 131(10), p. 101002. [CrossRef]
Kuettner, K. E., 1992, “Biochemistry of Articular Cartilage in Health and Disease,” Clin. Biochem., 25(3), pp. 155–163. [CrossRef] [PubMed]
Williamson, A. K., Chen, A. C., and Sah, R. L., 2001, “Compressive Properties and Function—Composition Relationships of Developing Bovine Articular Cartilage,” J. Orthop. Res., 19(6), pp. 1113–1121. [CrossRef] [PubMed]
Buckwalter, J. A., and Mankin, H. J., 1998, “Articular Cartilage Repair and Transplantation,” Arthritis Rheum., 41(8), pp. 1331–1342. [CrossRef] [PubMed]
Noyes, F. R., and Stabler, C. L., 1989, “A System for Grading Articular Cartilage Lesions at Arthroscopy,” Am. J. Sports Med., 17(4), pp. 505–513. [CrossRef] [PubMed]
Curl, W. W., Krome, J., Gordon, E. S., Rushing, J., Smith, B. P., and Poehling, G. G., 1997, “Cartilage Injuries: A Review of 31,516 Knee Arthroscopies,” Arthroscopy: J. Arthroscopy Relat. Surg., 13(4), pp. 456–460. [CrossRef]
Botter, S. M., van Osch, G. J. V. M., Waarsing, J. H., van der Linden, J. C., Verhaar, J. A. N., Pols, H. A. P., van Leeuwen, J. P., and Weinans, H., 2008, “Cartilage Damage Pattern in Relation to Subchondral Plate Thickness in a Collagenase-Induced Model of Osteoarthritis,” Osteoarthritis Cartilage, 16(4), pp. 506–514. [CrossRef] [PubMed]
Temple-Wong, M. M., Bae, W. C., Chen, M. Q., Bugbee, W. D., Amiel, D., Coutts, R. D., Lotz, M., and Sah, R. L., 2009, “Biomechanical, Structural, and Biochemical Indices of Degenerative and Osteoarthritic Deterioration of Adult Human Articular Cartilage of the Femoral Condyle,” Osteoarthritis Cartilage, 17(11), pp. 1469–1476. [CrossRef] [PubMed]
Wong, B. L., Kim, S. H. C., Antonacci, J. M., McIlwraith, C. W., and Sah, R. L., 2010, “Cartilage Shear Dynamics During Tibio-Femoral Articulation: Effect of Acute Joint Injury and Tribosupplementation on Synovial Fluid Lubrication,” Osteoarthritis Cartilage, 18(3), pp. 464–471. [CrossRef] [PubMed]
Novakofski, K. D., Williams, R. M., Fortier, L. A., Mohammed, H. O., Zipfel, W. R., and Bonassar, L. J., 2014, “Identification of Cartilage Injury Using Quantitative Multiphoton Microscopy,” Osteoarthritis Cartilage, 22(2), pp. 355–362. [CrossRef] [PubMed]
Maroudas, A., and Venn, M., 1977, “Chemical Composition and Swelling of Normal and Osteoarthrotic Femoral Head Cartilage. II. Swelling,” Ann. Rheum. Dis., 36(5), pp. 399–406. [CrossRef] [PubMed]
Saarakkala, S., Julkunen, P., Kiviranta, P., Mäkitalo, J., Jurvelin, J. S., and Korhonen, R. K., 2010, “Depth-Wise Progression of Osteoarthritis in Human Articular Cartilage: Investigation of Composition, Structure and Biomechanics,” Osteoarthritis Cartilage, 18(1), pp. 73–81. [CrossRef] [PubMed]
Cotofana, S., Buck, R., Wirth, W., Roemer, F., Duryea, J., Nevitt, M., Eckstein, F., and Osteoarthritis Initiative Investigators Group, 2012, “Cartilage Thickening in Early Radiographic Knee Osteoarthritis: A Within-Person, Between-Knee Comparison,” Arthritis Care Res., 64(11), pp. 1681–1690. [CrossRef]
Rodríguez, J. F., Cacho, F., Bea, J. A., and Doblaré, M., 2006, “A Stochastic-Structurally Based Three Dimensional Finite-Strain Damage Model for Fibrous Soft Tissue,” J. Mech. Phys. Solids, 54(4), pp. 864–886. [CrossRef]
Calvo, B., Peña, E., Martinez, M. A., and Doblaré, M., 2007, “An Uncoupled Directional Damage Model for Fibred Biological Soft Tissues. Formulation and Computational Aspects,” Int. J. Numer. Methods Eng., 69(10), pp. 2036–2057. [CrossRef]
Rodriguez, J. F., Alastrue, V., and Doblare, M., 2008, “Finite Element Implementation of a Stochastic Three Dimensional Finite-Strain Damage Model for Fibrous Soft Tissue,” Comput. Methods Appl. Mech. Eng., 197(9), pp. 946–958. [CrossRef]
Gajewski, T., Weisbecker, H., Holzapfel, G. A., and Lodygowski, T., 2013, “Implementation of a Hyperelastic Model for Arterial Layers Considering Damage and Distributed Collagen Fiber Orientations.” http://www.ikb.poznan.pl/tomasz.gajewski/cad.put.poznan.pl/TG/cmm2013_gajewski_weisbecker_holzapfel_logydowski.pdf
Weisbecker, H., Pierce, D. M., Regitnig, P., and Holzapfel, G. A., 2012, “Layer-Specific Damage Experiments and Modeling of Human Thoracic and Abdominal Aortas With Non-Atherosclerotic Intimal Thickening,” J. Mech. Behav. Biomed. Mater., 12, pp. 93–106. [CrossRef] [PubMed]
Famaey, N., Vander Sloten, J., and Kuhl, E., 2013, “A Three-Constituent Damage Model for Arterial Clamping in Computer-Assisted Surgery,” Biomech. Model. Mechanobiol., 12(1), pp. 123–136. [CrossRef] [PubMed]
Hosseini, S. M., Wilson, W., Ito, K., and van Donkelaar, C. C., 2014, “A Numerical Model to Study Mechanically Induced Initiation and Progression of Damage in Articular Cartilage,” Osteoarthritis Cartilage, 22(1), pp. 95–103. [CrossRef] [PubMed]
Hollander, A. P., Pidoux, I., Reiner, A., Rorabeck, C., Bourne, R., and Poole, A. R., 1995, “Damage to Type II Collagen in Aging and Osteoarthritis Starts at the Articular Surface, Originates Around Chondrocytes, and Extends Into the Cartilage With Progressive Degeneration,” J. Clin. Invest., 96(6), pp. 2859–2869. [CrossRef] [PubMed]
Rolauffs, B., Muehleman, C., Li, J., Kurz, B., Kuettner, K. E., Frank, E., and Grodzinsky, A. J., 2010, “Vulnerability of the Superficial Zone of Immature Articular Cartilage to Compressive Injury,” Arthritis Rheum., 62(10), pp. 3016–3027. [CrossRef] [PubMed]
Stender, M. E., Raub, C. B., Yamauchi, K. A., Shirazi, R., Vena, P., Sah, R. L., Hazelwood, S. J., and Klisch, S. M., 2012, “Integrating qPLM and Biomechanical Test Data With an Anisotropic Fiber Distribution Model and Predictions of TGF-β1 and IGF-1 Regulation of Articular Cartilage Fiber Modulus,” Biomech. Model. Mechanobiol., 12(6), pp. 1073–1088. [CrossRef] [PubMed]
Schröder, J., and Neff, P., 2003, “Invariant Formulation of Hyperelastic Transverse Isotropy Based on Polyconvex Free Energy Functions,” Int. J. Solids Struct., 40(2), pp. 401–445. [CrossRef]
Davison, L., Stevens, A. L., and Kipp, M. E., 1977, “Theory of Spall Damage Accumulation in Ductile Metals,” J. Mech. Phys. Solids, 25(1), pp. 11–28. [CrossRef]
Lanir, Y., 1983, “Constitutive Equations for Fibrous Connective Tissues,” J. Biomech., 16(1), pp. 1–12. [CrossRef] [PubMed]
Lei, F., and Szeri, A. Z., 2006, “The Influence of Fibril Organization on the Mechanical Behaviour of Articular Cartilage,” Proc. R. Soc. A, 462(2075), pp. 3301–3322. [CrossRef]
Ateshian, G. A., 2007, “Anisotropy of Fibrous Tissues in Relation to the Distribution of Tensed and Buckled Fibers,” ASME J. Biomech. Eng., 129(2), pp. 240–249. [CrossRef]
Ateshian, G. A., Rajan, V., Chahine, N. O., Canal, C. E., and Hung, C. T., 2009, “Modeling the Many Observed Phenomena,” ASME J. Biomech. Eng., 131(6), p. 061003. [CrossRef]
Shirazi, R., Vena, P., Sah, R. L., and Klisch, S. M., 2011, “Modeling the Collagen Fibril Network of Biological Tissues as a Nonlinearly Elastic Material Using a Continuous Volume Fraction Distribution Function,” Math. Mech. Solids, 16(7), pp. 706–715. [CrossRef] [PubMed]
Williamson, A. K., Chen, A. C., Masuda, K., Thonar, E. J.-M. A., and Sah, R. L., 2003, “Tensile Mechanical Properties of Bovine Articular Cartilage: Variations With Growth and Relationships to Collagen Network Components,” J. Orthop. Res., 21(5), pp. 872–880. [CrossRef] [PubMed]
Buschmann, M., and Grodzinsky, A., 1995, “A Molecular Model of Proteoglycan-Associated Forces in Cartilage Mechanics,” ASME J. Biomech. Eng., 117(2), pp. 179–192. [CrossRef]
McCormack, T., and Mansour, J. M., 1997, “Reduction in Tensile Strength of Cartilage Precedes Surface Damage Under Repeated Compressive Loading in vitro,” J. Biomech., 31(1), pp. 55–61. [CrossRef]
Bae, W. C., Temple, M. M., Amiel, D., Coutts, R. D., Niederauer, G. G., and Sah, R. L., 2003, “Indentation Testing of Human Cartilage: Sensitivity to Articular Surface Degeneration,” Arthritis Rheum., 48(12), pp. 3382–3394. [CrossRef] [PubMed]
Temple, M. M., Bae, W. C., Chen, M. Q., Lotz, M., Amiel, D., Coutts, R. D., and Sah, R. L., 2007, “Age- and Site-Associated Biomechanical Weakening of Human Articular Cartilage of the Femoral Condyle,” Osteoarthritis Cartilage, 15(9), pp. 1042–1052. [CrossRef] [PubMed]
Wilson, W., van Donkelaar, C. C., van Rietbergen, B., and Huiskes, R., 2005, “A Fibril-Reinforced Poroviscoelastic Swelling Model for Articular Cartilage,” J. Biomech., 38(6), pp. 1195–1204. [CrossRef] [PubMed]
Wilson, W., Huyghe, J. M., and Donkelaar, C. C., 2006, “Depth-Dependent Compressive Equilibrium Properties of Articular Cartilage Explained by Its Composition,” Biomech. Model. Mechanobiol., 6(1–2), pp. 43–53. [CrossRef] [PubMed]
Davol, A., Bingham, M. S., Sah, R. L., and Klisch, S. M., 2007, “A Nonlinear Finite Element Model of Cartilage Growth,” Biomech. Model. Mechanobiol., 7(4), pp. 295–307. [CrossRef] [PubMed]
Wilson, W., van Donkelaar, C. C., van Rietbergen, R., and Huiskes, R., 2005, “The Role of Computational Models in the Search for the Mechanical Behavior and Damage Mechanisms of Articular Cartilage,” Med. Eng. Phys., 27(10), pp. 810–826. [CrossRef] [PubMed]
Buck, R. J., Wyman, B. T., Le Graverand, M. P., Hudelmaier, M., Wirth, W., Eckstein, F., and A9001140 Investigators, 2009, “Does the Use of Ordered Values of Subregional Change in Cartilage Thickness Improve the Detection of Disease Progression in Longitudinal Studies of Osteoarthritis?,” Arthritis Rheum., 61(7), pp. 917–924. [CrossRef] [PubMed]
Lai, W. M., Hou, J. S., and Mow, V. C., 1991, “A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage,” ASME J. Biomech. Eng., 113(3), pp. 245–258. [CrossRef]
Loret, B., and Simões, F. M., 2004, “Articular Cartilage With Intra-and Extrafibrillar Waters: A Chemo-Mechanical Model,” Mech. Mater., 36(5), pp. 515–541. [CrossRef]
Oungoulian, S., 2007, “Articular Cartilage Constitutive Modeling: A Polyconvex Strain Energy Function for Proteoglycan and Validation of a Growth Mixture Model With Collagen Remodeling: A Thesis,” Master's dissertation, California Polytechnic State University, San Luis Obispo, CA.
Sasazaki, Y., Shore, R., and Seedhom, B. B., 2006, “Deformation and Failure of Cartilage in the Tensile Mode,” J. Anat., 208(6), pp. 681–694. [CrossRef] [PubMed]
Asanbaeva, A., Masuda, K., Thonar, E. J.-M. A., Klisch, S. M., and Sah, R. L., 2007, “Regulation of Immature Cartilage Growth by IGF-I, TGF-β1, BMP-7, and PDGF-AB: Role of Metabolic Balance Between Fixed Charge and Collagen Network,” Biomech. Model. Mechanobiol., 7(4), pp. 263–276. [CrossRef] [PubMed]
Weightman, B., 1976, “Tensile Fatigue of Human Articular Cartilage,” J. Biomech., 9(4), pp. 193–200. [CrossRef] [PubMed]
Weightman, B., Chappell, D. J., and Jenkins, E. A., 1978, “A Second Study of Tensile Fatigue Properties of Human Articular Cartilage,” Annals of the Rheumatic Diseases, 37(1), pp. 58–63.
Bellucci, G., and Seedhom, B. B., 2002, “Tensile Fatigue Behaviour of Articular Cartilage,” Biorheology, 39(1), pp. 193–199. http://content.iospress.com/articles/biorheology/bir147 [PubMed]
Buehler, M. J., 2007, “Molecular Nanomechanics of Nascent Bone: Fibrillar Toughening by Mineralization,” Nanotechnology, 18(29), p. 295102. [CrossRef]
Tang, Y., Ballarini, R., Buehler, M. J., and Eppell, S. J., 2010, “Deformation Micromechanisms of Collagen Fibrils Under Uniaxial Tension,” J. R. Soc. Interface, 7(46), pp. 839–850. [CrossRef] [PubMed]
Pins, G. D., and Silver, F. H., 1995, “A Self-Assembled Collagen Scaffold Suitable for Use in Soft and Hard Tissue Replacement,” Mater. Sci. Eng.: C, 3(2), pp. 101–107. [CrossRef]
Jeffrey, J. E., Thomson, L. A., and Aspden, R. M., 1997, “Matrix Loss and Synthesis Following a Single Impact Load on Articular Cartilage In Vitro,” Biochim. Biophys. Acta, 1334(2), pp. 223–232. [CrossRef] [PubMed]
Bush, P., Hodkinson, P., Hamilton, G., and Hall, A., 2005, “Viability and Volume of Bovine Articular Chondrocytes? Changes Following a Single Impact and Effects of Medium Osmolarity,” Osteoarthritis Cartilage, 13(1), pp. 54–65. [CrossRef] [PubMed]
Lin, P., 2004, “Increased Stromelysin-1 (MMP-3), Proteoglycan Degradation (3B3- and 7D4) and Collagen Damage in Cyclically Load-Injured Articular Cartilage,” Osteoarthritis Cartilage, 12(6), pp. 485–496. [CrossRef] [PubMed]
Thibault, M., Robin Poole, A., and Buschmann, M. D., 2002, “Cyclic Compression of Cartilage/Bone Explants In Vitro Leads to Physical Weakening, Mechanical Breakdown of Collagen and Release of Matrix Fragments,” J. Orthop. Res., 20(6), pp. 1265–1273. [CrossRef] [PubMed]
Korhonen, R. K., Laasanen, M. S., Töyräs, J., Lappalainen, R., Helminen, H. J., and Jurvelin, J. S., 2003, “Fibril Reinforced Poroelastic Model Predicts Specifically Mechanical Behavior of Normal, Proteoglycan Depleted and Collagen Degraded Articular Cartilage,” J. Biomech., 36(9), pp. 1373–1379. [CrossRef] [PubMed]
Rieppo, J., Töyräs, J., Nieminen, M. T., Kovanen, V., Hyttinen, M. M., Korhonen, R. K., Jurvelin, J. S., and Helminen, H. J., 2003, “Structure-Function Relationships in Enzymatically Modified Articular Cartilage,” Cells Tissues Organs, 175(3), pp. 121–132.
Appleyard, R., 2003, “Topographical Analysis of the Structural, Biochemical and Dynamic Biomechanical Properties of Cartilage in an Ovine Model of Osteoarthritis,” Osteoarthritis Cartilage, 11(1), pp. 65–77. [CrossRef] [PubMed]
Rizkalla, G., Reiner, A., Bogoch, E., and Poole, A. R., 1992, “Studies of the Articular Cartilage Proteoglycan Aggrecan in Health and Osteoarthritis. Evidence for Molecular Heterogeneity and Extensive Molecular Changes in Disease,” J. Clin. Invest., 90(6), pp. 2268–2277. [CrossRef] [PubMed]
Peerlings, R. H. J., Geers, M. G. D., De Borst, R., and Brekelmans, W. A. M., 2001, “A Critical Comparison of Nonlocal and Gradient-Enhanced Softening Continua,” Int. J. Solids Struct., 38(44), pp. 7723–7746. [CrossRef]
Buehler, M. J., and Ackbarow, T., 2008, “Nanomechanical Strength Mechanisms of Hierarchical Biological Materials and Tissues,” Comput. Methods Biomech. Biomed. Eng., 11(6), pp. 595–607. [CrossRef]


Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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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