Application of a Probabilistic Microstructural Model to Determine Reference Length and Toe-to-Linear Region Transition in Fibrous Connective Tissue

[+] Author and Article Information
Christof Hurschler, Paolo P. Provenzano, Ray Vanderby

Department of Orthopedics and Rehabilitation and Department of Biomedical Engineering, University of Wisconsin, Madison, WI 53792-7375

J Biomech Eng 125(3), 415-422 (Jun 10, 2003) (8 pages) doi:10.1115/1.1579046 History: Received December 04, 2001; Revised November 19, 2002; Online June 10, 2003
Copyright © 2003 by ASME
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Lam,  T. C., Shrive,  N. G., and Frank,  C. B., 1995, “Variations in Rupture Site and Surface Strains at Failure in the Maturing Rabbit Medial Collateral Ligament,” ASME J. Biomech. Eng., 117(4), pp. 455–461.
Chimich,  D., Frank,  C., Shrive,  N., Dougall,  H., and Bray,  R., 1991, “The Effects of Initial End Contact on Medial Collateral Ligament Healing: A Morphological and Biomechanical Study in a Rabbit Model,” J. Orthop. Res., 9(1), pp. 37–47.
Przybylski,  G. J., Carlin,  G. J., Patel,  P. R., and Woo,  S. L., 1996, “Human Anterior and Posterior Cervical Longitudinal Ligaments Possess Similar Tensile Properties,” J. Orthop. Res., 14(6), pp. 1005–1008.
Sabiston,  P., Frank,  C., Lam,  T., and Shrive,  N., 1990, “Transplantation of the Rabbit Medial Collateral Ligament. I. Biomechanical Evaluation of Fresh Autografts,” J. Orthop. Res., 8(1), pp. 35–45.
Woo,  S. L., Gomez,  M. A., Inoue,  M., and Akeson,  W. H., 1987, “New Experimental Procedures to Evaluate the Biomechanical Properties of Healing Canine Medial Collateral Ligaments,” J. Orthop. Res., 5(3), pp. 425–432.
Hull,  M. L., Berns,  G. S., Varma,  H., and Patterson,  H. A., 1996, “Strain in the Medial Collateral Ligament of the Human Knee Under Single and Combined Loads,” [erratum appears in J. Biomech, 29(8), pp. 1115]; Hull,  M. L., Berns,  G. S., Varma,  H., and Patterson,  H. A.J. Biomech., 29 (2), pp. 199–206.
Fleming,  B. C., Beynnon,  B. D., Tohyama,  H., Johnson,  R. J., Nichols,  C. E., Renstrom,  P., and Pope,  M. H., 1994, “Determination of a Zero Strain Reference for the Anteromedial Band of the Anterior Cruciate Ligament,” J. Orthop. Res., 12(6), pp. 789–795.
Belkoff,  S. M., and Haut,  R. C., 1991, “A Structural Model Used to Evaluate the Changing Microstructure of Maturing Rat Skin,” J. Biomech., 24(8), pp. 711–720.
Belkoff,  S. M., and Haut,  R. C., 1992, “Microstructurally Based Model Analysis of Gamma-Irradiated Tendon Allografts,” J. Orthop. Res., 10(3), pp. 461–464.
Kastelic,  J., Palley,  I., and Baer,  E., 1980, “A Structural Mechanical Model for Tendon Crimping,” J. Biomech., 13(10), pp. 887–893.
Hurschler,  C., Loitz-Ramage,  B., and Vanderby,  R., 1997, “A Structurally Based Stress-Stretch Relationship for Tendon and Ligament,” ASME J. Biomech. Eng., 119(4), pp. 392–399.
Lanir,  Y., 1979, “A Structural Theory for the Homogeneous Biaxial Stress-Strain Relationships in Flat Collagenous Tissues,” J. Biomech., 12(6), pp. 423–436.
Lanir,  Y., 1963, “Constitutive Equations for Fibrous Connective Tissues,” J. Biomech., 16(1), pp. 1–12.
Kwan,  M. K., and Woo,  S. L., 1989, “A Structural Model to Describe the Nonlinear Stress-Strain Behavior for Parallel-Fibered Collagenous Tissues,” ASME J. Biomech. Eng., 111(4), pp. 361–363.
Viidik,  A., 1972, “Simultaneous Mechanical and Light Microscopic Studies of Collagen Fibers,” Z. Anat. Entwicklungsgesch, 136(2), pp. 204–212.
Sacks, M. S., 2001, “A Structural Constitutive Model for Planar Collagenous Tissues That Integrates Sals-Derived Fiber Orientation Data,” Advances in Bioengineering, 51 , ASME, New York.
Hurschler,  C., Provenzano,  P. P., Vanderby,  R., 1998, “Scanning Electron Microscopic Investigation of Healing and Normal Rat Medial Collateral Ligaments Fixed Under Slack and Loaded Conditions,” Trans. Orthop. Res. Soc., 23, pp. 1032.
Panjabi,  M. M., Yoldas,  E., Oxland,  T. R., and Crisco,  J. J., 1996, “Subfailure Injury of the Rabbit Anterior Cruciate Ligament,” J. Orthop. Res., 14(2), pp. 216–227.
Weibull,  W., 1951, “A Statistical Distribution Function of Wide Applicability,” ASME J. Appl. Mech., 18(3), pp. 293–297.
Hines, W. W., and Montgomery, D. C., 1980, Probability and Statistics in Engineering and Management Science, John Wiley and Sons, New York.
Abrahams,  M., 1967, “Mechanical Behavior of Tendon in Vitro: A Preliminary Report,” Med. Biol. Eng., 5, pp. 433–443.
Provenzano,  P. P., Heisey,  D., Haysashi,  K., Lakes,  R. S., and Vanderby,  R., 2002, “Sub-Failure Damage in Ligament: A Structural and Cellular Evaluation,” J. Appl. Physiol., (1), pp. 362–371.
Hansen,  K. A., Weiss,  J. A., and Barton,  J. K., 2002, “Recruitment of Tendon Crimp With Applied Tensile Strain,” ASME J. Biomech. Eng., 124(1), pp. 72–77.
Kato,  Y. P., Christiansen,  D. L., Hahn,  R. A., Shieh,  S.-J., Goldstein,  J. D., and Silver,  F. H., 1989, “Mechanical Properties of Collagen Fibers: A Comparison of Reconstituted and Rat Tail Tendon Fibers,” Biomaterials, 10, pp. 38–42.
Sasaki,  N., and Odajima,  S., 1996, “Elongation Mechanism of Collagen Fibrils and Force-Strain Relations of Tendon at Each Level of Structural Hierarchy,” J. Biomech., 29(9), pp. 1131–1136.
Sasaki,  N., and Odajima,  S., 1996, “Stress-Strain Curve and Young’s Modulus of a Collagen Molecule as Determined by The X-Ray Diffraction Technique,” J. Biomech., 29(5), pp. 655–658.


Grahic Jump Location
Example tendon stress-stretch data taken from Abrahams 21, retaining the units of the original paper. The open symbols represent data points, the dashed line represents the fit of the model to the data.
Grahic Jump Location
Simulated initially slack tendon stress-stretch data for a reference length of lr=9.9 length units (open circles). Adjusted stress-stretch data and corresponding model fit (filled circles, solid line) agree with the fit of the original data (lr=10 length units, dashed line).
Grahic Jump Location
Simulated initially preloaded tendon stress-stretch data for a reference length of lr=10.1 representing an initially preloaded specimen (open circles). Adjusted stress-stretch data and corresponding model fit (filled circles, solid line) agree well with the fit of the adjusted original data set (lr=10 length units, dashed line). Note that since some information in the toe-in region is lost, agreement is not as good as with the initially slack specimen (Fig. 2).
Grahic Jump Location
Tangent modulus computed from stress-stretch data based on different choices of reference length lr (white bars), and tangent modulus computed from the corresponding adjusted data (gray bars). The unadjusted modulus changes linearly with lr, the adjusted modulus remains relatively constant.
Grahic Jump Location
Model fits for control and subfailure stretched rat MCL data from 22. The structural model proposed by Hurschler et al. 11 fit each curve with R2>0.99. Note, the Weibull PDF has been scaled to the corresponding stress-stretch curve, see Methods. The Control and Damaged markers below the Stretch Ratio axis correspond to the stretch ratio calculated for a specific value of F (values appear in Table 1). The tick marks represent each of the five values of F examined, starting with 0.75 and moving to the right by increments of 0.05 until the value 0.95 is reached.
Grahic Jump Location
Model fits to data from (a) Viidik 15 in which changes in collagen fiber crimp were examined with polarized light during tissue strain (measured grip to grip) and (b) Hanson et al. 23 in which changes in collagen fiber crimp were examined with optical coherence tomography during tissue strain (measured grip to grip). In both cases the original units of the studies were maintained with the exception of strain being converted to stretch ratio. In the case of the Viidik data the model agrees very well, in the case of the Hanson et al. data the model slightly underpredicts the stretch at which complete fiber straightening occurs.




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