A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage

[+] Author and Article Information
Michael A. Soltz, Gerard A. Ateshian

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

J Biomech Eng 122(6), 576-586 (Jul 10, 2000) (11 pages) doi:10.1115/1.1324669 History: Received November 30, 1999; Revised July 10, 2000
Copyright © 2000 by ASME
Your Session has timed out. Please sign back in to continue.


Kempson,  G. E., Freeman,  M. A., and Swanson,  S. A., 1968, “Tensile Properties of Articular Cartilage,” Nature (London), 220, pp. 1127–1128.
Woo,  S. L., Lubock,  P., Gomez,  M. A., Jemmott,  G. F., Kuei,  S. C., and Akeson,  W. H., 1979, “Large Deformation Nonhomogeneous and Directional Properties of Articular Cartilage in Uniaxial Tension,” J. Biomech., 12, pp. 437–446.
Roth,  V., and Mow,  V. C., 1980, “The Intrinsic Tensile Behavior of the Matrix of Bovine Articular Cartilage and Its Variation With Age,” J. Bone Jt. Surg., Am. Vol., 62A, pp. 1102–1117.
Akizuki,  S., Mow,  V. C., Muller,  F., Pita,  J. C., and Howell,  D. S., 1987, “Tensile Properties of Human Knee Joint Cartilage. II. Correlations Between Weight Bearing and Tissue Pathology and the Kinetics of Swelling,” J. Orthop. Res., 5, pp. 173–186.
Jurvelin,  J. S., Buschmann,  M. D., and Hunziker,  E. B., 1996, “Mechanical Anisotropy of Human Knee Articular Cartilage in Compression,” Trans. Annu. Meet.—Orthop. Res. Soc., 21, p. 7.
Schinagl,  R. M., Gurskis,  D., Chen,  A. C., and Sah,  R. L., 1997, “Depth-Dependent Confined Compression Modulus of Full-Thickness Bovine Articular Cartilage,” J. Orthop. Res., 15, pp. 499–506.
Kempson,  G. E., Freeman,  M. A., and Swanson,  S. A., 1971, “The Determination of a Creep Modulus for Articular Cartilage From Indentation Tests of the Human Femoral Head,” J. Biomech., 4, pp. 239–250.
Hayes,  W. C., and Bodine,  A. J., 1978, “Flow-Independent Viscoelastic Properties of Articular Cartilage Matrix,” J. Biomech., 11, pp. 407–419.
Grodzinsky,  A. J., Lipshitz,  H., and Glimcher,  M. J., 1978, “Electromechanical Properties of Articular Cartilage During Compression and Stress Relaxation,” Nature (London), 275, pp. 448–450.
Woo,  S. L., Simon,  B. R., Kuei,  S. C., and Akeson,  W. H., 1980, “Quasi-Linear Viscoelastic Properties of Normal Articular Cartilage,” ASME J. Biomech. Eng., 102, pp. 85–90.
Mow,  V. C., Kuei,  S. C., Lai,  W. M., and Armstrong,  C. G., 1980, “Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments,” ASME J. Biomech. Eng., 102, pp. 73–84.
Setton,  L. A., Zhu,  W., and Mow,  V. C., 1993, “The Biphasic Poroviscoelastic Behavior of Articular Cartilage: Role of the Surface Zone in Governing the Compressive Behavior,” J. Biomech., 26, pp. 581–592.
Mizrahi,  J., Maroudas,  A., Lanir,  Y., Ziv,  I., and Webber,  T. J., 1986, “The ‘Instantaneous’ Deformation of Cartilage: Effects of Collagen Fiber Orientation and Osmotic Stress,” Biorheology, 23, pp. 311–330.
Cohen,  B., Lai,  W. M., and Mow,  V. C., 1998, “A Transversely Isotropic Biphasic Model for Unconfined Compression of Growth Plate and Chondroepiphysis,” ASME J. Biomech. Eng., 120, pp. 491–496.
Bursac,  P. M., Obitz,  T. W., Eisenberg,  S. R., and Stamenovic,  D., 1999, “Confined and Unconfined Stress Relaxation of Cartilage: Appropriateness of a Transversely Isotropic Analysis,” J. Biomech., 32, pp. 1125–1130.
Soulhat,  J., Buschmann,  M. D., and Shirazi-Adl,  A., 1999, “A Fibril-Network Reinforced Model of Cartilage in Unconfined Compression,” ASME J. Biomech. Eng., 121, pp. 340–347.
Huang,  C.-Y., Stankiewicz,  A., Ateshian,  G. A., Flatow,  E. L., Bigliani,  L. U., and Mow,  V. C., 1999, “Anisotropy, Inhomogeneity, and Tension-Compression Nonlinearity of Human Glenohumeral Cartilage in Finite Deformation,” Trans. Annu. Meet.—Orthop. Res. Soc., 24, p. 95.
Soltz,  M. A., Palma,  C., Barsoumian,  S., Wang,  C. C.-B., Hung,  C. T., and Ateshian,  G. A., 2000, “Multi-Axial Loading of Bovine Articular Cartilage in Unconfined Compression,” Trans. Annu. Meet.—Orthop. Res. Soc., 25, p. 888.
Kwan,  M. K., Lai,  W. M., and Mow,  V. C., 1990, “A Finite Deformation Theory for Cartilage and Other Soft Hydrated Connective Tissues—I. Equilibrium Results,” J. Biomech., 23, pp. 145–155.
Holmes,  M. H., and Mow,  V. C., 1990, “The Nonlinear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration,” J. Biomech., 23, pp. 1145–1156.
Ateshian,  G. A., Warden,  W. H., Kim,  J. J., Grelsamer,  R. P., and Mow,  V. C., 1997, “Finite Deformation Biphasic Material Properties of Bovine Articular Cartilage From Confined Compression Experiments,” J. Biomech., 30, pp. 1157–1164.
Frank,  E. H., and Grodzinsky,  A. J., 1987, “Cartilage Electromechanics—II. A Continuum Model of Cartilage Electrokinetics and Correlation With Experiments,” J. Biomech., 20, pp. 629–639.
Armstrong,  C. G., and Mow,  V. C., 1982, “Variations in the Intrinsic Mechanical Properties of Human Articular Cartilage With Age, Degeneration, and Water Content,” J. Bone Jt. Surg., Am. Vol., 64A, pp. 88–94.
Soltz,  M. A., and Ateshian,  G. A., 1998, “Experimental Verification and Theoretical Prediction of Cartilage Interstitial Fluid Pressurization at an Impermeable Contact Interface in Confined Compression,” J. Biomech., 31, pp. 927–934.
Soltz,  M. A., and Ateshian,  G. A., 2000, “Interstitial Fluid Pressurization During Confined Compression Cyclical Loading of Articular Cartilage,” Ann. Biomed. Eng., 28, pp. 150–159.
Mow,  V. C., Gibbs,  M. C., Lai,  W. M., Zhu,  W. B., and Athanasiou,  K. A., 1989, “Biphasic Indentation of Articular Cartilage—II. A Numerical Algorithm and an Experimental Study,” J. Biomech., 22, pp. 853–861.
Athanasiou,  K. A., Rosenwasser,  M. P., Buckwalter,  J. A., Malinin,  T. I., and Mow,  V. C., 1991, “Interspecies Comparison of in Situ Intrinsic Mechanical Properties of Distal Femoral Cartilage,” J. Orthop. Res., 9, pp. 330–340.
Armstrong,  C. G., Lai,  W. M., and Mow,  V. C., 1984, “An Analysis of the Unconfined Compression of Articular Cartilage,” ASME J. Biomech. Eng., 106, pp. 165–173.
Brown,  T. D., and Singerman,  R. J., 1986, “Experimental Determination of the Linear Biphasic Constitutive Coefficients of Human Fetal Proximal Femoral Chondroepiphysis,” J. Biomech., 19, pp. 597–605.
Spilker,  R. L., Suh,  J. K., and Mow,  V. C., 1990, “Effects of Friction on the Unconfined Compressive Response of Articular Cartilage: a Finite Element Analysis,” ASME J. Biomech. Eng., 112, pp. 138–146.
Kim,  Y. J., Bonassar,  L. J., and Grodzinsky,  A. J., 1995, “The Role of Cartilage Streaming Potential, Fluid Flow and Pressure in the Stimulation of Chondrocyte Biosynthesis During Dynamic Compression,” J. Biomech., 28, pp. 1055–1066.
Cohen,  B., Gardner,  T. R., and Ateshian,  G. A., 1993, “The Influence of Transverse Isotropy on Cartilage Indentation Behavior—A Study of the Human Humeral Head,” Trans. Annu. Meet.—Orthop. Res. Soc., 18, p. 185.
Lanir,  Y., 1987, “Biorheology and Fluid Flux in Swelling Tissues. II. Analysis of Unconfined Compressive Response of Transversely Isotropic Cartilage Disc,” Biorheology, 24, pp. 189–205.
Suh,  J. K., and Bai,  S., 1998, “Finite Element Formulation of Biphasic Poroviscoelastic Model for Articular Cartilage,” ASME J. Biomech. Eng., 120, pp. 195–201.
Mak,  A. F., 1986, “The Apparent Viscoelastic Behavior of Articular Cartilage—the Contributions From the Intrinsic Matrix Viscoelasticity and Interstitial Fluid Flows,” ASME J. Biomech. Eng., 108, pp. 123–130.
Khalsa,  P. S., and Eisenberg,  S. R., 1997, “Compressive Behavior of Articular Cartilage Is Not Completely Explained by Proteoglycan Osmotic Pressure,” J. Biomech., 30, pp. 589–594.
Curnier,  A., He,  Q.-C., and Zysset,  P., 1995, “Conewise Linear Elastic Materials,” J. Elast., 37, pp. 1–38.
Ateshian,  G. A., Wang,  H., and Lai,  W. M., 1998, “The Role of Interstitial Fluid Pressurization and Surface Porosities on the Boundary Friction of Articular Cartilage,” ASME J. Tribol., 120, pp. 241–251.
Cowin,  S. C., and Mehrabadi,  M. M., 1987, “On the Identification of Material Symmetry for Anisotropic Elastic Material,” Q. J. Mech. Appl. Math., 40, pp. 451–475.
Eisenberg,  S. R., and Grodzinsky,  A. J., 1985, “Swelling of Articular Cartilage and Other Connecitve Tissues: Electromechanical Forces,” J. Orthop. Res., 3, pp. 148–159.
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, pp. 245–258.
Maroudas,  A., and Venn,  M., 1977, “Chemical Composition and Swelling of Normal and Osteoarthrotic Femoral Head Cartilage. II. Swelling,” Ann. Rheum. Dis., 36, pp. 399–406.
Setton,  L. A., Tohyama,  H., and Mow,  V. C., 1998, “Swelling and Curling Behavior of Articular Cartilage,” J. Biomed. Eng., 120, pp. 355–361.
Mak,  A. F., Lai,  W. M., and Mow,  V. C., 1987, “Biphasic Indentation of Articular Cartilage—I. Theoretical Analysis,” J. Biomech., 20, pp. 703–714.
Ateshian,  G. A., Lai,  W. M., Zhu,  W. B., and Mow,  V. C., 1994, “An Asymptotic Solution for the Contact of Two Biphasic Cartilage Layers,” J. Biomech., 27, pp. 1347–1360.
Kvalseth,  T. O., 1985, “Cautionary Note About R2,,” The American Statistician, 39, pp. 279–285.
Buschmann,  M. D., Soulhat,  J., Shirazi-Adl,  A., Jurvelin,  J. S., and Hunziker,  E. B., 1998, “Confined Compression of Articular Cartilage: Linearity in Ramp and Sinusoidal Tests and the Importance of Interdigitation and Incomplete Confinement,” J. Biomech., 31, pp. 171–178.
Lanir,  Y., 1983, “Constitutive Equations for Fibrous Connective Tissues,” J. Biomech., 16, pp. 1–12.
Farquhar,  T., Dawson,  P. R., and Torzilli,  P., 1990, “A Microstructural Model for the Anisotropic Drained Stiffness of Articular Cartilage,” ASME J. Biomech. Eng., 112, pp. 414–425.
Wren,  T. A. L., and Carter,  D. R., 1998, “A Microstructural Model for the Tensile Constitutive and Failure Behavior of Soft Skeletal Connective Tissues,” ASME J. Biomech. Eng., 120, pp. 55–61.
Ateshian,  G. A., and Wang,  H., 1995, “A Theoretical Solution for the Frictionless Rolling Contact of Cylindrical Biphasic Articular Cartilage Layers,” J. Biomech., 28, pp. 1341–1355.
Kelkar,  R., and Ateshian,  G. A., 1999, “Contact Creep of Biphasic Cartilage Layers: Identical Layers,” ASME J. Appl. Mech., 66, pp. 137–145.
Akizuki,  S., Mow,  V. C., Muller,  F., Pita,  J. C., Howell,  D. S., and Manicourt,  D. H., 1986, “Tensile Properties of Human Knee Joint Cartilage: I. Influence of Ionic Conditions, Weight Bearing, and Fibrillation on the Tensile Modulus,” J. Orthop. Res., 4, pp. 379–392.
Macirowski,  T., Tepic,  S., and Mann,  R. W., 1994, “Cartilage Stresses in the Human Hip Joint,” ASME J. Biomech. Eng., 116, pp. 11–18.
McCutchen,  C. W., 1962, “The Frictional Properties of Animal Joints,” Wear, 5, pp. 1–17.
Forster,  H., and Fisher,  J., 1996, “The Influence of Loading Time and Lubricant on the Friction of Articular Cartilage,” ImechE, J. Eng. Med., 210, pp. 109–119.
Jurvelin,  J., Buschmann,  M., and Hunziker,  E., 1997, “Optical and Mechanical Determination of Poisson’s Ratio of Adult Bovine Humeral Articular Cartilage,” J. Biomech., 30, pp. 235–241.
Wang,  C. C.-B., Soltz,  M. A., Mauck,  R. L., Valhmu,  W. B., Ateshian,  G. A., and Hung,  C. T., 2000, “Comparison of the Equilibrium Strain Distribution in Articular Cartilage Explants and Cell Seeded Alginate Disks Under Compression,” Trans. Annu. Meet.—Orthop. Res. Soc., 25, p. 131.
Chang,  D. G., Lottman,  L. M., Chen,  A. C., Schinagl,  R. M., Albrecht,  D. R., Pedowitz,  R. A., Brossmann,  J., Frank,  L. R., and Sah,  R. L., 1999, “The Depth-Dependent, Multi-Axial Properties of Aged Human Patellar Cartilage in Tension,” Trans. Annu. Meet.—Orthop. Res. Soc., 24, p. 644.
Elliott,  D. M., Kydd,  S. R., Perry,  C. H., and Setton,  L. A., 1999, “Direct Measurement of the Poisson’s Ratio of Human Articular Cartilage in Tension,” Trans. Annu. Meet.—Orthop. Res. Soc., 24, p. 649.
Zhu,  W., Chern,  K. Y., and Mow,  V. C., 1994, “Anisotropic Viscoelastic Shear Properties of Bovine Meniscus,” Clin. Orthop., 306, pp. 34–45.


Grahic Jump Location
Interstitial fluid load, Pu(t), total load Fu(t), and ratio of fluid to total load support, Pu/Fu, as a function of time for unconfined compression stress-relaxation, using the average material properties listed in Table 1
Grahic Jump Location
For the octantwise orthotropic CLE model of the solid phase of cartilage, the three preferred directions of material symmetry are taken to be: a1 parallel to the cartilage split line direction, a2 perpendicular to the split line direction, and a3 normal to the articular cartilage surface
Grahic Jump Location
(a) Apparatus for performing confined and unconfined compression tests. A different testing chamber was employed for (b) confined, and (c) unconfined compression.
Grahic Jump Location
Apparatus for performing torsional shear tests
Grahic Jump Location
Experimental confined compression stress-relaxation response [Fc(t)/πr 02] and corresponding theoretical curve-fit for a typical specimen
Grahic Jump Location
Experimental unconfined compression stress-relaxation response [Fu(t)/πr 02] and corresponding theoretical curve-fit for the same specimen as in Fig. 4. The experimental interstitial pressure at the specimen center [p(r=0,t)] and corresponding theoretical prediction are also presented.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In