Skip to main content

Advertisement

SpringerLink
Log in

Modelling Young’s modulus for porous bones with microstructural variation and anisotropy

  • Published:
Journal of Materials Science: Materials in Medicine Aims and scope Submit manuscript

Abstract

A structural model with three compositional phases and two levels of hierarchical organization is proposed for predicting Young’s modulus of porous bones with microstructural variations and anisotropy based on their geometric similarity to metal foams. It has been shown that the proposed single model provides predictions of Young’s modulus with high accuracy up to ±30% for cortical and cancellous bones compared with measured data from the literature. In addition, the conversion of the solid bone shape from “Plate-like” to “Rod-like” at a porosity of 70% or higher (BV/TV 30% or lower)—verified by observations—can be predicted using the proposed model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Fratzl P, Weinkamer R. Nature’s hierarchical materials. Prog Mater Sci. 2007;52:1263–334.

    Article  CAS  Google Scholar 

  2. Torquato S. Random heterogeneous materials. 1st ed. New York: Springer; 2002. Chapt. 1, 1–16; Chapt. 21–22, 552–631.

  3. Maxwell JC. A treatise on electricity and magnetism. 3rd ed. New York: Dover Publications Inc.; reprinted 1954. Chapter 9.

  4. Landauer R. The electrical resistance of binary metallic mixtures. J Appl Phys. 1952;23:779–84.

    Article  CAS  ADS  Google Scholar 

  5. Wang JF, Carson JK, North MF, Cleland DJ. A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases. Int J Heat Mass Transf. 2008;51(9–10):2389–97.

    Article  CAS  Google Scholar 

  6. Wang JF, Carson JK, North MF, Cleland DJ. A new approach to modelling the effective thermal conductivity of heterogeneous materials. Int J Heat Mass Transf. 2006;49(17–18):3075–83.

    Article  Google Scholar 

  7. Wang JF, Carson JK, Willix J, North MF, Cleland DJ. A symmetric and interconnected skeleton structural (SISS) model for predicting thermal and electrical conductivity and Young’s modulus of porous foams. Acta Mater. 2008;56(18):5138–46.

    Article  CAS  Google Scholar 

  8. Currey JD. Bones-structure and mechanics. 1st ed. Princeton University Press; 2006.

  9. Gibson LJ, Ashby MF. Cellular solids—structure and properties. 2nd ed. London: Cambridge; 1997.

    Google Scholar 

  10. Helgason B, Perilli E, Schileo E, Taddei F, Brynjolfsson S, Viceconti M. Mathematical relationships between bone density and mechanical properties: A literature review. Clin Biomech. 2008;23:135–46.

    Article  Google Scholar 

  11. Sandino C, Planell JA, Lacroix D. A finite element study of mechanical stimuli in scaffolds for bone tissue engineering. J Biomech. 2008;41:1005–14.

    Article  CAS  PubMed  Google Scholar 

  12. Jones AC, Arns CH, Hutmacher DW, Milthorpe BK, Sheppard AP, Knackstedt MA. The correction of pore morphology, interconnectivity and physical properties of 3D ceramic scaffolds with bone ingrowth. Biomaterials. 2009;30:1440–51.

    Article  CAS  PubMed  Google Scholar 

  13. Hildebrand T, Laib A, Muller R, Dequeker J, Ruegsegger P. Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest and calcaneus. J Bone Miner Res. 1999;14(7):1167–74.

    Article  CAS  PubMed  Google Scholar 

  14. Lazenby RA, Cooper DML, Angus S, Hallgrimsson B. Articular constraint, handedness, and directional asymmetry in the human second metacarpal. J Hum Evol. 2008;54:875–85.

    Article  PubMed  Google Scholar 

  15. van Ruijven LJ, Giesen EBW, Mulder L, Farella M, van Eijden TMGJ. The effect of bone loss on rod-like and plate-like trabeculae in the cancellous bone of the mandibular condyle. Bone. 2005;36:1078–85.

    Article  PubMed  Google Scholar 

  16. Ding M, Odgaard A, Danielsen CC, Hvid I. Mutual associations among microstructural, physical and mechanical properties of human cancellous bone. J Bone Joint Surg. 2002;84B(6):900–7.

    Article  Google Scholar 

  17. Nicholson PHF, Muller R, Cheng XG, Puegsegger P, van der Perre G, Dequeker J, et al. Quantitative ultrasound and trabecular architecture in the human calcaneus. J Bone Miner Res. 2001;16(10):1886–92.

    Article  CAS  PubMed  Google Scholar 

  18. Day JS, Sumner DR, Waarsing JH, Weinans H. Inter-individual and gender differences in bone adaptation strategy. In: Proceedings of 50th annual meeting of the Orthopaedic Research Society, Paper No. 0400; 2004.

  19. Hildebrand T, Ruegsegger P. Quantification of bone microarchitecture with the structure model index. Comput Methods Biomech Biomed Eng. 1997;1(1):15–23.

    Article  Google Scholar 

  20. van der Linden JC, Day JS, Verhaar JAN, Weinans H. Altered tissue properties induce changes in cancellous bone architecture in aging and diseases. J Biomech. 2004;37:367–74.

    Article  PubMed  Google Scholar 

  21. Seto J, Gupta HS, Zaslansky P, Wagner HD, Fratzl P. Tough lessons from bone: extreme mechanical anisotropy at the mesoscale. Adv Funct Mater. 2008;18(13):1905–11.

    Article  CAS  Google Scholar 

  22. Sevostianov I, Kovacik J, Simancik F. Elastic and electric properties of closed-cell aluminium foams: cross-property connection. Mater Sci Eng A. 2006;420(1–2):87–99.

    Google Scholar 

  23. Goodall R, Weber L, Mortensen A. The electrical conductivity of microcellular metals. J Appl Phys. 2006;100:AN044912.

    Google Scholar 

  24. Hatta H, Taya M. Effective thermal conductivity of a misoriented short fibre composite. J Appl Phys. 1985;58(7):2478–86.

    Article  CAS  ADS  Google Scholar 

  25. Jager I, Fratzl P. Mineralized collagen fibril: a mechanical model with a staggered arrangement of mineral particles. Biophys J. 2000;79:1739–46.

    Article  Google Scholar 

  26. White ET, Tan WH, Ang JM, Tait S, Litster JD. The density of a protein crystal. Power Technol. 2007;179:55–8.

    Article  CAS  Google Scholar 

  27. Hedges RW, Jacob AE. Phonons and the elastic moduli of collagen and muscle. Nature. 1977;267:285–7.

    Article  ADS  Google Scholar 

  28. Snyders R, Music D, Sigumonrong D, Schelnberger B, Jensen J, Schneider JM. Experimental and ab initio study of the mechanical properties of hydroxyapatite. Appl Phys Lett. 2007;90:193902-1,3.

    Google Scholar 

  29. Snyders R, Bousser E, Music D, Jensen J, Hocquet S, Schneider JM. Influence of the chemical composition on the phase constitution and the elastic properties of RF-sputtered hydroxyapatite coatings. Plasma Process Polym. 2008;5:168–74.

    Article  CAS  Google Scholar 

  30. Viswanath B, Raghavan R, Ramamurty U, Ravishankar N. Mechanical properties and anisotropy in hydroxyapatite single crystals. Scripta Mater. 2007;57:361–4.

    Article  CAS  Google Scholar 

  31. Sha MC, Li Z. Single-crystal elastic constants of fluorapatite, Ca5F(PO4)3. J Appl Phys. 1994;75(12):7784–7.

    Article  CAS  ADS  Google Scholar 

  32. Katz JL, Ukraincik K. On the anisotropic elastic properties of hydroxyapatite. J Biomech. 1971;4:221–7.

    Article  CAS  PubMed  Google Scholar 

  33. Katz JL. Hard tissue as a composite material I. bounds on the elastic behaviour. J Biomech. 1971;4:455–73.

    Article  CAS  PubMed  Google Scholar 

  34. Grant CA, Brockwell DJ, Radford SE, Thomson NH. Effects of hydration on the mechanical response of individual collagen fibrils. Appl Phys Lett. 2008;92:233902-1,3.

    Google Scholar 

  35. Heim AJ, Mattews WG. Determination of the elastic modulus of native collagen fibrils via radial indentation. Appl Phys Lett. 2006;89:181902-1,3.

    Google Scholar 

  36. Wenger MPE, Bozec L, Horton MA, Mesquida P. Mechanical properties of collagen fibrils. Biophys J. 2007;93:1255–63.

    Article  CAS  PubMed  Google Scholar 

  37. Burger C, Zhou HW, Wang H, Sics I, Hsiao BS, Chu B, et al. Lateral packing of mineral crystals in bone collagen fibrils. Biophys J. 2008;95:1985–92.

    Article  CAS  PubMed  Google Scholar 

  38. Hodgskinson R, Currey JD. Young’s modulus, density and material properties in cancellous bone over a large density range. J Mater Sci: Mater Med. 1992;3:377–81.

    Article  Google Scholar 

  39. Morgan EF, Bayraktar HH, Keaveny TM. Trabecular bone modulus–density relationships depend on anatomic site. J Biomech. 2003;36:897–904.

    Article  PubMed  Google Scholar 

  40. Kang Q, An YH, Friedman RF. Mechanical properties and bone densities of canine trabecular bone. J Mater Sci: Mater Med. 1998;9:263–7.

    Article  CAS  Google Scholar 

  41. Cavani F, Giavaresi G, Fini M, Bertoni L, Terlizzi FD, Barkmann R, et al. Influence of density, elasticity, and structure on ultrasound transmission through trabecular bone cylinders. IEEE Trans Ultrason Ferroelectr Freq Control. 2008;55(7):1465–71.

    Article  CAS  PubMed  Google Scholar 

  42. Fox J, Newman MK, Turner CH, Guldberg RE, Varela A, Smith SY. Effects of treatment with parathyroid hormone 1–84 on quantity and biomechanical properties of thoracic vertebral trabecular bone in ovariectomized rhesus monkeys. Calcif Tissue Int. 2008;82:212–20.

    Article  CAS  PubMed  Google Scholar 

  43. Hoffmeister BK, Johnson DP, Janeski JA, Keedy DA, Steinert BW, Viano AM, et al. Ultrasonic characterization of human cancellous bone in vitro using three different apparent backscatter parameters in the frequency range 0.6–150 MHz. IEEE Trans Ultrason Ferroelectr Freq Control. 2008;55(7):1442–52.

    Article  CAS  PubMed  Google Scholar 

  44. Keller TS, Mao Z, Spengler DM. Young’s modulus, bending strength, and tissue physical properties of human compact bone. J Orthop Res. 1990;8(4):592–603.

    Article  CAS  PubMed  Google Scholar 

  45. Spatz HC, O’Leary EJ, Vincent TFV. Young’s moduli and shear moduli in cortical bone. Proc R Soc Lond B: Biol Sci 1996;263(1368):287–94.

    Article  CAS  ADS  Google Scholar 

  46. Schaffler MB, Burr DB. Stiffness of compact bone: effects of porosity and density. J Biomech. 1988;21(1):13–6.

    Article  CAS  PubMed  Google Scholar 

  47. Currey JD. The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. J Biomech. 1988;21(2):131–9.

    Article  CAS  PubMed  Google Scholar 

  48. Bevill G, Easley SK, Keaveny TM. Side-artifact errors in yield strength and elastic modulus for human trabecular bone and their dependence on bone volume fraction and anatomic site. J Biomech. 2007;40:3381–8.

    Article  PubMed  Google Scholar 

Download references

Acknowledgements

Thanks to Professor John D. Currey for helpful suggestions and to Dr. Yun-Long Chang for constructive discussion about the microstructure of porous bones.

Author information

Authors and Affiliations

  1. School of Engineering and Advanced Technology, Massey University, Private Bag 11-222, Palmerston North, New Zealand

    Jianfeng F. Wang

Authors
  1. Jianfeng F. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianfeng F. Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J.F. Modelling Young’s modulus for porous bones with microstructural variation and anisotropy. J Mater Sci: Mater Med 21, 463–472 (2010). https://doi.org/10.1007/s10856-009-3919-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10856-009-3919-6

Keywords

Search

Navigation