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A Visco-Hyperelastic Constitutive Model for Human Spine Ligaments

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Abstract

Human spine ligaments show a highly non-linear, strain rate dependent biomechanical behavior under tensile tests. A visco-hyperelastic fiber-reinforced constitutive model was accordingly developed for human ligaments, in which the energy density function is decomposed into two parts. The first part represents the elastic strain energy stored in the soft tissue, and the second part denotes the energy dissipated due to its inherent viscous characteristics. The model is applied to various human spinal ligaments including the anterior and posterior longitudinal ligaments, ligamentum flavum, capsular ligament, and interspinous ligament. Material parameters for each type of ligament were obtained by curve-fitting with corresponding experimental data available in the literature. The results indicate that the model presented here can properly characterize the visco-hyperelastic biomechanical behavior of human spine ligaments.

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References

  1. Oza, A., Vanderby, R., & Lakes, R. (2006). Mechanics of Biological Tissue (pp. 379–397)., Creep and relaxation in ligament: Theory, methods and experiment Boca Raton: Taylor and Francis.

    Book  Google Scholar 

  2. Dvorak, J., Panjabi, M., Gerber, M., & Wichmann, W. (1987). CT-functional diagnostics of the rotatory instability of upper cervical spine: 1. An experimental study on cadavers. Spine, 12, 197–205.

    Article  CAS  PubMed  Google Scholar 

  3. Levine, A. M., & Edwards, C. C. (1989). Traumatic lesions of the occipitoatlantoaxial complex. Clinical Orthopaedics and Related Research, 239, 53.

    PubMed  Google Scholar 

  4. Alexander, R. M. N. (1984). Elastic energy stores in running vertebrates. American Zoologist, 24, 85–94.

    Google Scholar 

  5. Silver, F. H. (1987). Biological materials: structure, mechanical properties and modeling of soft tissues. New York: New York University Press.

    Google Scholar 

  6. White, A. A., & Panjabi, M. M. (1990). Clinical biomechanics of the spine (2nd ed.). Philadelphia: Lippincott.

    Google Scholar 

  7. Myklebust, J. B., Pintar, F., Yoganandan, N., Cusick, J. F., Maiman, D., Myers, T. J., et al. (1988). Tensile strength of spinal ligaments. Spine, 13, 526–531.

    Article  CAS  PubMed  Google Scholar 

  8. Fung, Y. C. (1973). Biorheology of soft tissues. Biorheology., 10, 139–155.

    CAS  PubMed  Google Scholar 

  9. Pioletti, D. P., & Rakotomanana, L. R. (2000). Non-linear viscoelastic laws for soft biological tissues. European Journal of Mechanics a-Solids., 19, 749–759.

    Article  Google Scholar 

  10. Gray, H. (1918). Anatomy of the human body. Philadelphia: Lea Febiger.

    Google Scholar 

  11. Tsuang, Y. H., Chiang, Y. F., Hung, C. Y., Wei, H. W., Huang, C. H., & Cheng, C. K. (2009). Comparison of cage application modality in posterior lumbar interbody fusion with posterior instrumentation—A finite element study. Medical Engineering & Physics, 31, 565–570.

    Article  Google Scholar 

  12. Pitzen, T., Geisler, F., Matthis, D., Müller-Storz, H., Barbier, D., Steudel, W. I., et al. (2002). A finite element model for predicting the biomechanical behaviour of the human lumbar spine. Control Engineering Practice, 10, 83–90.

    Article  Google Scholar 

  13. Toosizadeh, N., & Haghpanahi, M. (2011). Generating a finite element model of the cervical spine: Estimating muscle forces and internal loads. Scientia Iranica, 18, 1237–1245.

    Article  Google Scholar 

  14. Zhang, Q. H., Teo, E. C., Ng, H. W., & Lee, V. S. (2006). Finite element analysis of moment-rotation relationships for human cervical spine. Journal of Biomechanics, 39, 189–193.

    Article  PubMed  Google Scholar 

  15. Spencer, A. J. M. (1984). Continuum theory of the mechanics of fibre-reinforced composites. New York: Springer.

    Book  Google Scholar 

  16. Peng, X., Guo, Z., & Moran, B. (2006). An anisotropic hyperelastic constitutive model with fiber-matrix shear interaction for the human annulus fibrosus. Journal of Applied Mechanics, 73, 815–824.

    Article  Google Scholar 

  17. Guo, Z., Peng, X., & Moran, B. (2006). A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. Journal of the Mechanics and Physics of Solids, 54, 1952–1971.

    Article  CAS  Google Scholar 

  18. Sun, D. D. N., & Leong, K. W. (2004). A nonlinear hyperelastic mixture theory model for anisotropy, transport, and swelling of annulus fibrosus. Annals of Biomedical Engineering, 32, 92–102.

    Article  PubMed  Google Scholar 

  19. Natali AN, Carniel EL, Pavan PG, Dario P, Izzo I (2006) Hyperelastic models for the analysis of soft tissue mechanics: definition of constitutive parameters. Biomedical Robotics and Biomechatronics, BioRob 2006 The First IEEE/RAS-EMBS International Conference on: IEEE; p. 188–91.

  20. Tang, H., Buehler, M. J., & Moran, B. (2009). A constitutive model of soft tissue: from nanoscale collagen to tissue continuum. Annals of Biomedical Engineering, 37, 1117–1130.

    Article  PubMed  Google Scholar 

  21. Tang, Y., Ballarini, R., Buehler, M. J., & Eppell, S. J. (2010). Deformation micromechanisms of collagen fibrils under uniaxial tension. Journal of the Royal Society, Interface, 7, 839–850.

    Article  PubMed Central  PubMed  Google Scholar 

  22. Pioletti, D. P., Rakotomanana, L. R., & Leyvraz, P. F. (1999). Strain rate effect on the mechanical behavior of the anterior cruciate ligament-bone complex. Medical Engineering & Physics, 21, 95–100.

    Article  CAS  Google Scholar 

  23. Ivancic, P. C., Coe, M. P., Ndu, A. B., Tominaga, Y., Carlson, E. J., & Rubin, W. (2007). Dynamic mechanical properties of intact human cervical spine ligaments. Spine Journal, 7, 659.

    Article  PubMed Central  PubMed  Google Scholar 

  24. Mattucci, S. F. E., Moulton, J. A., Chandrashekar, N., & Cronin, D. S. (2012). Strain rate dependent properties of younger human cervical spine ligaments. Journal of the mechanical behavior of biomedical materials, 10, 216–226.

    Article  PubMed  Google Scholar 

  25. Pioletti, D. P., Rakotomanana, L., Benvenuti, J. F., & Leyvraz, P. F. (1998). Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons. Journal of Biomechanics, 31, 753–757.

    Article  CAS  PubMed  Google Scholar 

  26. García, J. J., & Cortés, D. H. (2006). A nonlinear biphasic viscohyperelastic model for articular cartilage. Journal of Biomechanics, 39, 2991–2998.

    Article  PubMed  Google Scholar 

  27. García, J. J., & Cortés, D. H. (2007). A biphasic viscohyperelastic fibril-reinforced model for articular cartilage: Formulation and comparison with experimental data. Journal of Biomechanics, 40, 1737–1744.

    Article  PubMed  Google Scholar 

  28. Schroeder, Y., Elliott, D., Wilson, W., Baaijens, F., & Huyghe, J. (2008). Experimental and model determination of human intervertebral disc osmoviscoelasticity. Journal of Orthopaedic Research, 26, 1141–1146.

    Article  CAS  PubMed  Google Scholar 

  29. Limbert, G., & Middleton, J. (2004). A transversely isotropic viscohyperelastic material: Application to the modeling of biological soft connective tissues. International Journal of Solids and Structures, 41, 4237–4260.

    Article  Google Scholar 

  30. Limbert, G., & Middleton, J. (2006). A constitutive model of the posterior cruciate ligament. Medical Engineering & Physics, 28, 99–113.

    Article  Google Scholar 

  31. Zhurov, A. I., Limbert, G., Aeschlimann, D. P., & Middleton, J. (2007). A constitutive model for the periodontal ligament as a compressible transversely isotropic visco-hyperelastic tissue. Computer Methods in Biomechanics and Biomedical Engineering, 10, 223–235.

    Article  PubMed  Google Scholar 

  32. Ogden, R. (1984). Non-linear elastic deformations. New York: Halsted Press.

    Google Scholar 

  33. Yang, L., Shim, V., & Lim, C. (2000). A visco-hyperelastic approach to modelling the constitutive behaviour of rubber. International Journal of Impact Engineering, 24, 545–560.

    Article  Google Scholar 

  34. Humphrey, J., Strumpf, R., & Yin, F. (1990). Determination of a constitutive relation for passive myocardium: I. A new functional form. Journal of Biomechanical Engineering, 112, 333.

    Article  CAS  PubMed  Google Scholar 

  35. Holzapfel, G. A., Eberlein, R., Wriggers, P., & Weizsäcker, H. W. (1996). Large strain analysis of soft biological membranes: Formulation and finite element analysis. Computer Methods in Applied Mechanics and Engineering, 132, 45–61.

    Article  Google Scholar 

  36. Hirokawa, S., & Tsuruno, R. (2000). Three-dimensional deformation and stress distribution in an analytical/computational model of the anterior cruciate ligament. Journal of Biomechanics, 33, 1069–1077.

    Article  CAS  PubMed  Google Scholar 

  37. Rao, S., Daniel, I. M., & Gdoutos, E. E. (2004). Mechanical properties and failure behavior of cord/rubber composites. Applied Composite Materials, 11, 353–375.

    Article  CAS  Google Scholar 

  38. Beda, T. (2007). Modeling hyperelastic behavior of rubber: A novel invariant-based and a review of constitutive models. Journal of Polymer Science Part B: Polymer Physics, 45, 1713–1732.

    Article  CAS  Google Scholar 

  39. Panjabi, M. M., Crisco, J. J., Lydon, C., & Dvorak, J. (1998). The mechanical properties of human alar and transverse ligaments at slow and fast extension rates. Clinical Biomechanics, 13, 112–120.

    Article  PubMed  Google Scholar 

  40. Natali, A., Pavan, P., Carniel, E., & Dorow, C. (2004). Viscoelastic response of the periodontal ligament: an experimental-numerical analysis. Connective Tissue Research, 45, 222–230.

    Article  CAS  PubMed  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the support of the National Natural Science Foundation of China (50975236, 11172171) and Key Scientific and Technological Project Foundation of Chongqing, China (CSTC2012gg-yyjs10011).

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Authors and Affiliations

  1. School of Materials Science and Engineering, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai, 200030, China

    Yugang Jiang & Xiongqi Peng

  2. Pediatric Department, Southwest Hospital of the Third Military Medical University, Chongqing, 400038, China

    Yu Wang

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  1. Yugang Jiang
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  2. Yu Wang
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Correspondence to Xiongqi Peng.

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Jiang, Y., Wang, Y. & Peng, X. A Visco-Hyperelastic Constitutive Model for Human Spine Ligaments. Cell Biochem Biophys 71, 1147–1156 (2015). https://doi.org/10.1007/s12013-014-0322-9

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  • DOI: https://doi.org/10.1007/s12013-014-0322-9

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