Modeling of Bone Failure by Cohesive Zone Models

  • Thomas Siegmund
  • Matthew R. Allen
  • David B. Burr

Abstract

Cohesive zone models are a powerful tool for investigations of non-linear deformation and failure processes. For the nanoscale, the use of cohesive zone models is particularly attractive as the ratio of interface to volume is high, and because locally acting bonds between material components can become relevant. The present paper demonstrates the relevance of cohesive zone modelling approaches to the development of a nano-mechanical composite model of the mineralized collagen fibril, a fundamental building block of bone. As difficulties exist in determining the independent biomechanical effects of collagen cross-linking using in vitro and in vivo experiments, computational modeling can provide insight into the nanoscale processes. Stress-strain curves for mineralized collagen fibrils were obtained under tensile loading for various collagen cross-linking conditions. Our model predicts that the elastic deformation mode, the yield response and the final failure of the mineralized collagen fibril may depend significantly on the state of collagen cross-linking.

Keywords

Cohesive Zone Cohesive Zone Model Cohesive Interface Collagen Domain Human Cortical Bone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abdul-Baqi, A., van der Giessen, E.: Indentation-induced interface delamination of a strong film on a ductile substrate. Thin Solid Films 381, 143–154 (2001)CrossRefGoogle Scholar
  2. 2.
    Allen, M.R., Gineyts, E., Leeming, D.J., Burr, D.B., Delmas, P.D.: Bisphosphonates alter trabecular bone collagen cross-linking and isomerization in beagle dog vertebra. Osteoporosis Int. 19, 329–337 (2008)CrossRefGoogle Scholar
  3. 3.
    Allen, M.R., Burr, D.B.: Mineralization, microdamage, and matrix: How bisphosphonates influence material properties of bone. BoneKEy 4, 49–60 (2007)CrossRefGoogle Scholar
  4. 4.
    Arnoux, P.J., Bonnoit, J., Chabrand, P., Jean, M., Pithioux, M.: Numerical damage models using a structural approach: Application in bones and ligaments. Eur. Phys. J. – Appl. Phys. 17, 65–73 (2002)CrossRefGoogle Scholar
  5. 5.
    Bailey, A.J., Wotton, S.F., Sims, T.J., Thompson, P.W.: Post-translational modifications in the collagen of human osteoporotic femoral head. Biochem. Biophys. Res. Comm. 185, 801–805 (1992)CrossRefGoogle Scholar
  6. 6.
    Bailey, A.J., Paul, R.G., Knott, L.: Mechanisms of maturation and ageing of collagen. Mech. Ageing Dev. 106, 1–56 (1998)CrossRefGoogle Scholar
  7. 7.
    Banse, X., Sims, T.J., Bailey, A.J.: Mechanical properties of adult vertebral cancellous bone: correlation with collagen intermolecular cross-links. J. Bone Min. Res. 17, 1621–1628 (2002)CrossRefGoogle Scholar
  8. 8.
    Beyer, M.K.: The mechanical strength of a covalent bond calculated by density function theory. J. Chem. Phys. 112, 7307–7312 (2000)CrossRefGoogle Scholar
  9. 9.
    Boxberger, J., Vashishth, D.: Nonenzymatic glycation affects bone fracture by modifying creep and inelastic properties of collagen. Trans. Orthop Res. Soc. 29, 0491 (2004)Google Scholar
  10. 10.
    Bühler, M.J.: Atomistic and continuum modeling of mechanical properties of collagen: Elasticity, fracture, and self-assembly. J. Mat. Res. 21, 1947–1961 (2006)CrossRefGoogle Scholar
  11. 11.
    Burr, D.B.: The contribution of the organic matrix to bone’s material properties. Bone 31, 8–11 (2002)CrossRefGoogle Scholar
  12. 12.
    Catanese, J., Bank, R., Tekoppele, J., Keaveny, T.: Increased cross-linking by non-enzymatic glycation reduces the ductility of bone and bone collagen. Proc. Am. Soc. Mech. Eng. Bioeng. Conf. 42, 267–268 (1999)Google Scholar
  13. 13.
    Currey, J.D.: Effects of differences in mineralization on the mechanical properties of bone. Phil. Trans. Royal Soc. London Series B, Bio. Sci. 304, 509–518 (1984)CrossRefGoogle Scholar
  14. 14.
    Eyre, D.R., Dickson, I.R., Van Ness, K.: Collagen cross-linking in human bone and articular cartilage. Age-related changes in the content of mature hydroxypyridinium residues. Biochem. J. 252, 495–500 (1988)Google Scholar
  15. 15.
    Eyre, D.R., Wu, J.J.: Collagen cross-links. Top Curr. Chem. 247, 207–229 (2005)Google Scholar
  16. 16.
    Fritsch, A., Hellmich, C.: ‘Universal’ microstructural patterns in cortical and trabecular, extracellular bone materials, Micromechanics-based prediction of anisotropic elasticity. J. Theor. Bio. 244, 597–620 (2007)CrossRefGoogle Scholar
  17. 17.
    Garnier, L., Gauthier-Manuel, B., van der Vegte, E.W., Snijders, J., Hadziioannou, G.: Covalent bond force profile and cleavage in a single polymer chain. J. Chem. Phys. 113, 2497–2503 (2000)CrossRefGoogle Scholar
  18. 18.
    Garnero, P., Borel, O., Gineyts, E., Duboeuf, F., Solberg, H., Bouxsein, M.L., Christiansen, C., Delmas, P.D.: Extracellular post-translational modifications of collagen are major determinants of biomechanical properties of fetal bovine cortical bone. Bone 38, 300–309 (2006)CrossRefGoogle Scholar
  19. 19.
    Gourrier, A., Wagermaier, W., Burghammer, M., Lammie, D., Gupta, H.S., Fratzl, P., Riekel, C., Wess, T.J., Paris, O.: Scanning X-ray imaging with small-angle scattering contrast. J. Appl. Crystal 40, S78-S82 (2007)Google Scholar
  20. 20.
    Gupta, H.S., Seto, J., Wagermaier, W., Zaslansky, P., Boesecke, P., Fratzl, P.: Cooperative deformation of mineral and collagen in bone at the nanoscale. Proc. Nat. Acad. Sci. USA 104, 17741–17746 (2006)CrossRefGoogle Scholar
  21. 21.
    Hellmich, C., Barthélémy, J.F., Dormieux, L.: Mineral–collagen interactions in elasticity of bone ultrastructure – a continuum micromechanics approach. Eur. J. Mech. A 23, 783–810 (2004)MATHCrossRefGoogle Scholar
  22. 22.
    Hernandez, C.J., Tang, S.Y., Baumbach, B.M., Hwu, P.B., Sakkee, A.N., van der Ham, F., DeGroot, J., Bank, R.A., Keaveny, T.M.: Trabecular microfracture and the influence of pyridinium and non-enzymatic glycation-mediated collagen cross-links. Bone 37, 825–832 (2005)CrossRefGoogle Scholar
  23. 23.
    Hutchinson, J.W., Evans, A.G.: Mechanics of materials: Top-down approaches to fracture. Acta. Mat. 48, 125–135 (2000)CrossRefGoogle Scholar
  24. 24.
    Jäger, I., Fratzl, P.: Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particle. Biophys. J. 79, 1737–1746 (2000)CrossRefGoogle Scholar
  25. 25.
    Jäger, I.: A model for the stability and creep of organic materials. J. Biomech. 38, 1459–1467 (2005)CrossRefGoogle Scholar
  26. 26.
    Ji, B., Gao, H.: Mechanical properties of nanostructure of biological materials. J. Mech. Phys. Solids 52, 1963–1990 (2004)MATHCrossRefGoogle Scholar
  27. 27.
    Katz, J., Ukraincik, K.: On the anisotropic elastic properties of hyroxyapatite. J. Biomech. 4, 221–227 (1971)CrossRefGoogle Scholar
  28. 28.
    Keaveny, T.M., Morris, G.E., Wong, E.K., Yu, M., Sakkee, A.N., Verzijl, N., Bank, R.A.: Collagen status and brittleness of human cortical bone in the elderly. J. Bone Mineral Res. 18(supp. l2), S307 (2003)Google Scholar
  29. 29.
    Knott, L., Bailey, A.J.: Collagen cross-links in mineralizing tissues: a review of their chemistry, function, and clinical relevance. Bone 22, 181–187 (1998)CrossRefGoogle Scholar
  30. 30.
    Kotha, S.P., Guzelsu, N.: Effect of bone mineral content on the tensile properties of cortical bone: experiments and theory. J. Biomech. Eng. 125, 785–793 (2003)CrossRefGoogle Scholar
  31. 31.
    Landis, W.J.: The strength of a calcified tissue depends in part on the molecular structure and organization of its constituent mineral crystal in their organic matrix. Bone 16, 533–544 (1995)CrossRefGoogle Scholar
  32. 32.
    Lees, S.: Considerations regarding the structure of the mammalian mineralized osteoid from the viewpoint of the generalized packing model. Connect Tissue Res. 16, 281–303 (1987)CrossRefGoogle Scholar
  33. 33.
    Lees, S., Eyre, D.R., Barnard, S.M.: BAPN dose dependence of mature crosslinking in bone matrix collagen of rabbit compact bone: Corresponding variation of sonic velocity and equatorial diffraction spacing. Connect Tissue Res. 24, 95–105 (1990)CrossRefGoogle Scholar
  34. 34.
    Monnier, V.M.: Toward a Maillard reaction theory of aging. Prog. Clin. Bio. Res. 304, 1–22 (1989)Google Scholar
  35. 35.
    Needleman, A.: A continuum model for void nucleation by inclusion debonding. J. Appl. Mech. 54, 525–531 (1987)MATHCrossRefGoogle Scholar
  36. 36.
    Nyman, J.S., Roy, A., Tyler, J.H., Acuna, R.L., Gayle, H.J., Wang, X.: Age-related factors affecting the postyield energy dissipation of human cortical bone. J. Orthop. Res. 25, 646–655 (2007)CrossRefGoogle Scholar
  37. 37.
    Odetti, P., Rossi, S., Monacelli, F., Poggi, A., Cirnigliaro, M., Federici, M., Federici, A.: Advanced glycation end products and bone loss during aging. Annals New York Acad. Sci. 1043, 710–717 (2005)CrossRefGoogle Scholar
  38. 38.
    Oxlund, H., Barckmann, M., Ortoft, G., Ancreassen, T.T.: Reduced concentrations of collagen cross-links are associated with reduced strength of bone. Bone 17, 365S-371S (1995)Google Scholar
  39. 39.
    Oxlund, H., Mosekilde, L., Ortoft, G.: Reduced concentration of collagen reducible cross links in human trabecular bone with respect to age and osteoporosis. Bone 19, 479–484 (1996)CrossRefGoogle Scholar
  40. 40.
    Petruska, J.A., Hodge, A.J.: A subunit model for the tropocollagen macromolecule. Proc. Nat. Acad. Sci. USA 51, 871–876 (1964)CrossRefGoogle Scholar
  41. 41.
    Saito, M., Marumo, K., Fujii, K., Ishioka, N.: Single-column high performance liquid chromatographic fluorescence detection of immature, mature and senescent crosslinks of collagen. Annals. Biochem. 253, 26–32 (1997)CrossRefGoogle Scholar
  42. 42.
    Silver, F.H., Christiansen, D.L., Snowhill, P.B., Chen, Y.: Transition from viscous to elastic-based dependency of mechanical properties of self-assembled type I collagen fibers. J. Appl. Polym. Sci. 79, 134–142 (2001)CrossRefGoogle Scholar
  43. 43.
    Silver, F.H., Freeman, J.W., Seehra, G.P.: Collagen self-assembly and the development of tendon mechanical properties. J. Biomech. 36, 1529–1553 (2003)CrossRefGoogle Scholar
  44. 44.
    Steiner, T.: The hydrogen bond in the solid state. Angewandte Chemie – Int. Ed 41, 48–76 (2002)CrossRefGoogle Scholar
  45. 45.
    Tan, H., Jiang, L.Y., Huang, Y., Liu, B., Hwang, K.C.: The effect of van der Waals-based interface cohesive law on carbon nanotube-reinforced composite materials. Comp. Sci. Techn. 67, 2941–2946 (2007)CrossRefGoogle Scholar
  46. 46.
    Tang, S., Bank, R., Tekoppele, J., Keaveny, T.: Nonenzymatic glycation causes loss of toughening mechanisms in human cancellous bone. Trans. Orthop. Res. Soc. 30 (2005)Google Scholar
  47. 47.
    Tang, S., Zeenath, U., Vashishth, D.: Effects of non-enzymatic glycation on cancellous bone fragility. Bone 40, 1144–1151 (2007)CrossRefGoogle Scholar
  48. 48.
    Tvergaard, V.: Effect of fiber debonding in a whisker-reinforced metal. Mat. Sci. Eng. A 125, 203–213 (1990)CrossRefGoogle Scholar
  49. 49.
    Vasan, S., Foiles, P., Founds, H.: Therapeutic potential of breakers of advanced glycation end product-protein crosslinks. Arch. Biochem. Biophys. 419, 89–96 (2003)CrossRefGoogle Scholar
  50. 50.
    Vashishth, D., Gibson, G.J., Khoury, J.I., Schaffler, M.B., Mimura, J., Fyhrie, D.P.: Influence of non-enzymatic glycation on biomechanical properties of cortical bone. Bone 28, 195–201 (2001)CrossRefGoogle Scholar
  51. 51.
    Vashishth, D., Wu, P., Gibson, G.: Age-related loss in bone toughness is explained by non-enzymatic glycation of collagen. Trans. Orthop. Res. Soc. 29 (2004)Google Scholar
  52. 52.
    Vashishth, D.: The role of collagen matrix in skeletal fragility. Curr. Osteoporos Rep. 5, 62–66 (2007)CrossRefGoogle Scholar
  53. 53.
    Viguet-Carrin, S., Garnero, P., Delmas, D.P.: The role of collagen in bone strength. Osteoporos Int. 17, 319–336 (2006)CrossRefGoogle Scholar
  54. 54.
    Viguet-Carrin, S., Roux, J.P., Arlot, M.E., Merabet, Z., Leeming, D.J., Byrjalsen, I., Delmas, P.D., Bouxsein, M.L.: Contribution of the advanced glycation end product pentosidine and of maturation of type I collagen to compressive biomechanical properties of human lumbar vertebrae. Bone 39, 1073–1079 (2006)CrossRefGoogle Scholar
  55. 55.
    Viswanath, B., Raghavan, R., Ramamurty, U., Ravishankar, N.: Mechanical properties and anisotropy in hydroxyapatite single crystals. Scripta. Mat. 57, 361–364 (2007)CrossRefGoogle Scholar
  56. 56.
    Wang, X., Shen, X., Li, X., Agarwal, C.M.: Age-related changes in the collagen network and toughness of bone. Bone 31, 1–7 (2002)CrossRefGoogle Scholar
  57. 57.
    Wang, X., Li, X., Shen, X., Agrawal, C.M.: Age-related changes of noncalcified collagen in human cortical bone. Ann. Biomed Eng. 31, 1365–1371 (2003)CrossRefGoogle Scholar
  58. 58.
    Wang, X., Qian, C.: Prediction of microdamage formation using a mineral-collagen composite model. J. Biomech. 39, 595–602 (2006)MATHCrossRefGoogle Scholar
  59. 59.
    Wilson, E.E., Awonusi, A., Morris, M.D., Kohn, D.H., Tecklenburg, M., Beck, L.W.: Highly ordered interstitial water observed in bone by nuclear magnetic resonance. J. Bone Min. Res. 20, 625–634 (2005)CrossRefGoogle Scholar
  60. 60.
    Wu, P., Koharski, C., Nonnenmann, H., Vashishth, D.: Loading on non-enzymatically glycated and damaged bone results in an instantaneous fracture. Trans. Orthop Res. Soc. 28, 404 (2003)Google Scholar
  61. 61.
    Xu, X.P., Needleman, A.: Numerical simulation of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1415 (1994)MATHCrossRefGoogle Scholar
  62. 62.
    Zioupos, P., Currey, J.D., Hamer, A.J.: The role of collagen in the declining mechanical properties of aging human cortical bone. J. Biomed Mat. Res. 45, 108–116 (1999)CrossRefGoogle Scholar

Copyright information

© © Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Thomas Siegmund
    • 1
  • Matthew R. Allen
    • 2
  • David B. Burr
    • 2
    • 3
    • 4
  1. 1.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.Department of Anatomy and Cell BiologyIndiana University School of MedicineIndianapolisUSA
  3. 3.Department of Orthopaedic SurgeryIndiana University School of MedicineIndianapolisUSA
  4. 4.Biomedical EngineeringIndiana University-Purdue University at IndianapolisIndianapolisUSA

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