Skip to main content
SpringerLink
Log in

Elasticity–density and viscoelasticity–density relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

Cortical bone tissue is an anisotropic material characterized by typically five independent elastic coefficients (for transverse isotropy) governing shear and longitudinal deformations in the different anatomical directions. It is well established that the Young’s modulus in the direction of the bone axis of long bones has a strong relationship with mass density. It is not clear, however, whether relationships of similar strength exist for the other elastic coefficients, for they have seldom been investigated, and the results available in the literature are contradictory. The objectives of the present work were to document the anisotropic elastic properties of cortical bone at the tibia mid-diaphysis and to elucidate their relationships with mass density. Resonant ultrasound spectroscopy (RUS) was used to measure the transverse isotropic stiffness tensor of 55 specimens from 19 donors. Except for Poisson’s ratios and the non-diagonal stiffness coefficient, strong linear correlations between the different elastic coefficients \((0.7 < {r^{2}} < 0.99)\) and between these coefficients and density \((0.79 < {r^{2}} < 0.89)\) were found. Comparison with previously published data from femur specimens suggested that the strong correlations evidenced in this study may not only be valid for the mid-tibia. RUS also measures the viscous part of the stiffness tensor. An anisotropy ratio close to two was found for damping coefficients. Damping increased as the mass density decreased. The data suggest that a relatively accurate estimation of all the mid-tibia elastic coefficients can be derived from mass density. This is of particular interest (1) to design organ-scale bone models in which elastic coefficients are mapped according to Hounsfield values from computed tomography scans as a surrogate for mass density and (2) to model ultrasound propagation at the mid-tibia, which is an important site for the in vivo assessment of bone status with axial transmission techniques.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Ashman RB, Cowin SC, van Buskirk WC, Rice JC (1984) A continuous wave technique for the measurement of the elastic properties of cortical bone. J Biomech 17(5):349–361

    Article  Google Scholar 

  • Austman RL, Milner JS, Holdsworth DW, Dunning CE (2008) The effect of the density-modulus relationship selected to apply material properties in a finite element model of long bone. J Biomech 41(15):3171–3176

    Article  Google Scholar 

  • Bernard S, Grimal Q, Laugier P (2013) Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy. J Mech Behav Biomed Mater 18:12–19

    Article  Google Scholar 

  • Bernard S, Grimal Q, Laugier P (2014) Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid materials. J Acoust Soc Am 135(5):2601–2613

    Article  Google Scholar 

  • Bernard S, Marrelec G, Laugier P, Grimal Q (2015) Bayesian normal modes identification and estimation of elastic coefficients in resonant ultrasound spectroscopy. Inverse Probl. doi:10.1088/0266-5611/31/6/065010

  • Carter D, Hayes W (1977) The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg 59(7):954–962

    Google Scholar 

  • Crolet JM, Aoubiza B, Meunier A (1993) Compact bone: numerical simulation of mechanical characteristics. J Biomech 26(6):677–687

    Article  Google Scholar 

  • Currey JD (1988) Strain rate and mineral content in fracture models of bone. J Orthop Res 6(1):32–38

    Article  MathSciNet  Google Scholar 

  • Deuerling JM, Yue WM, Orias AAE, Roeder RK (2009) Specimen-specific multi-scale model for the anisotropic elastic constants of human cortical bone. J Biomech 42(13):2061–2067

    Article  Google Scholar 

  • Dong NX, Guo EX (2004) The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. J Biomech 37(8):1281–1287

    Article  Google Scholar 

  • Dong XN, Guo XE (2006) Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method. J Biomech Eng Trans ASME 128(3):309–316

    Article  MathSciNet  Google Scholar 

  • Espinoza Orías AA, Deuerling J, Landrigan M, Renaud J, Roeder R (2009) Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur. J Mech Behav Biomed Mater 2(3):255–263

    Article  Google Scholar 

  • Foiret J, Minonzio JG, Chappard C, Talmant M, Laugier P (2014) Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering study on in vitro cortical bone samples. IEEE Trans Ultrason Ferroelectr Freq Control 61(9):1478–1488

    Article  Google Scholar 

  • Foldes A, Rimon A, Keinan D, Popovtzer M (1995) Quantitative ultrasound of the tibia: a novel approach for assessment of bone status. Bone 17(4):363–367

    Article  Google Scholar 

  • Garner E, Lakes R, Lee T, Swan C, Brand R (2000) Viscoelastic dissipation in compact bone: implications for stress-induced fluid flow in bone. J Biomech Eng 122(2):166–172

    Article  Google Scholar 

  • Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis, 3rd edn. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Granke M, Grimal Q, Saïed A, Nauleau P, Peyrin F, Laugier P (2011) Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women. Bone 49(5):1020–1026

    Article  Google Scholar 

  • Granke M, Grimal Q, Parnell WJ, Raum K, Gerisch A, Peyrin F, Saïed A, Laugier P (2015) To what extent can cortical bone millimeter scale elasticity be predicted by a two phase composite model with variable porosity? Acta Biomater 12:207–215

    Article  Google Scholar 

  • Grimal Q, Haupert S, Mitton D, Vastel L, Laugier P (2009) Assessment of cortical bone elasticity and strength: mechanical testing and ultrasound provide complementary data. Med Eng Phys 31(9):1140–1147

    Article  Google Scholar 

  • Grimal Q, Raum K, Gerisch A, Laugier P (2011) A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties. Biomech Model Mechanobiol 10(6):925–937

    Article  Google Scholar 

  • Grimal Q, Rus G, Parnell WJ, Laugier P (2011) A two-parameter model of the effective elastic tensor for cortical bone. J Biomech 44(8):1621–1625

    Article  Google Scholar 

  • Haïat G (2011) Linear ultrasonic properties of cortical bone: in vitro studies. In: laugier P, Haïat G (eds) Bone Quantitative Ultrasound. Springer, Berlin

    Google Scholar 

  • Haupert S, Guérard S, Peyrin F, Mitton D, Laugier P (2014) Non destructive characterization of cortical bone micro-damage by nonlinear resonant ultrasound spectroscopy. PLoS One 9(1):e83599

    Article  Google Scholar 

  • Helgason B, Perilli E, Schileo E, Taddei F, Brynjolfsson S, Viceconti M (2008) Mathematical relationships between bone density and mechanical properties: a literature review. Clin Biomech 23(2):135–146

    Article  Google Scholar 

  • Hellmich C, Kober C, Erdmann B (2008) Micromechanics-based conversion of ct data into anisotropic elasticity tensors, applied to fe simulations of a mandible. Ann Biomed Eng 36(1):108–122

    Article  Google Scholar 

  • Hellmich C, Ulm FJ (2004) Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions? Biomech Model Mechanobiol 2:219–238

    Article  Google Scholar 

  • Iyo T, Maki Y, Sasaki N, Nakata M (2004) Anisotropic viscoelastic properties of cortical bone. J Biomech 37(9):1433–1437 0021–9290

  • Keller TS (1994) Predicting the compressive mechanical behavior of bone. J Biomech 27(9):1159–1168

    Article  Google Scholar 

  • Lakes R, Yoon HS, Katz JL (1986) Ultrasonic wave propagation and attenuation in wet bone. J Biomed Eng 8(2):143–148

    Article  Google Scholar 

  • Lakes RS (2001) Bone mechanics handbook, chap. Viscoelastic properties of cortical bone. CRC Press, Boca Raton

    Google Scholar 

  • Lakes RS, Katz JL (1979) Viscoelastic properties of wet cortical bone—II. Relaxation mechanisms. J Biomech 12(9):679–687

    Article  Google Scholar 

  • Lakshmanan S, Bodi A, Raum K (2007) Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements. IEEE Trans Ultrason Ferroelectr Freq Control 54(8):1560–1570

    Article  Google Scholar 

  • Landa M, Sedlák P, Seiner H, Heller L, Bicanová L, Šittner P, Novák V (2009) Modal resonant ultrasound spectroscopy for ferroelastics. Appl Phys A 96:557–567

    Article  Google Scholar 

  • Lebedev AV (2002) Method of linear prediction in the ultrasonic spectroscopy of rock. Acoust Phys 48(3):339–346

    Article  Google Scholar 

  • Leisure R, Foster K, Hightower J, Agosta D (2004) Internal friction studies by resonant ultrasound spectroscopy. Mater Sci Eng A 370:34–40

    Article  Google Scholar 

  • Les CM, Spence CA, Vance JL, Christopherson GT, Patel B, Turner AS, Divine GW, Fyhrie DP (2004) Determinants of ovine compact bone viscoelastic properties: effects of architecture, mineralization, and remodeling. Bone 35(3):729–738

    Article  Google Scholar 

  • Migliori A, Sarrao J (1997) Resonant ultrasound spectroscopy. Wiley, New York

    Google Scholar 

  • Ogi H, Nakamura N, Sato K, Hirao M, Uda S (2003) Elastic, anelastic, and piezoelectric coefficients of langasite: resonance ultrasound spectroscopy with laser-doppler interferometry. IEEE Trans Ultrason Ferroelectr Freq Control 50(5):553–560

    Article  Google Scholar 

  • Parnell WJ, Vu MB, Grimal Q, Naili S (2012) Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone. Biomech Model Mechanobiol 11:883–901

    Article  Google Scholar 

  • Parnell WJ, Grimal Q (2009) The influence of mesoscale porosity on cortical bone anisotropy. investigations via asymptotic homogenization. J R Soc Interface 6(30):97–109

    Article  Google Scholar 

  • Preininger B, Checa S, Molnar FL, Fratzl P, Duda GN, Raum K (2011) Spatial–temporal mapping of bone structural and elastic properties in a sheep model following osteotomy. Ultrasound Med Biol 37(3):474–483

    Article  Google Scholar 

  • Prevrhal S, Fuerst T, Fan B, Njeh C, Hans D, Uffmann M, Srivastav S, Genant HK (2001) Quantitative ultrasound of the tibia depends on both cortical density and thickness. Osteoporos Int 12(1):28–34

    Article  Google Scholar 

  • Raum K (2008) Microelastic imaging of bone. IEEE Trans Ultrason Ferroelectr Freq Control 55(7):1417–1431

    Article  Google Scholar 

  • Rho JY, Hobatho MC, Ashman RB (1995) Relations of mechanical properties to density and ct numbers in human bone. Med Eng Phys 17(5):347–355

    Article  Google Scholar 

  • Rho JY (1996) An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone. Ultrasonics 34(8):777–783

    Article  Google Scholar 

  • Rho JY, Zioupos P, Currey JD, Pharr GM (2002) Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nano-indentation. J Biomech 35(2):189–198

    Article  Google Scholar 

  • Ritchie RO, Buehler MJ, Hansma P (2009) Plasticity and toughness in bone. Phys Today 62(6):41–47

  • Rohrbach D, Lakshmanan S, Peyrin F, Langer M, Gerisch A, Grimal Q, Laugier P, Raum K (2012) Spatial distribution of tissue level properties in a human femoral cortical bone. J Biomech 45(13):2264–2270

  • Rohrbach D, Grimal Q, Varga P, Peyrin F, Langer M, Laugier P, Raum K (2015) Distribution of mesoscale elastic properties and mass density in the human femoral shaft. Connect Tissue Res 56(2): 120–132

    Article  Google Scholar 

  • Rudy DJ, Deuerling JM, Orías AAE, Roeder RK (2011) Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors. J Biomech 44(9):1817–1820

    Article  Google Scholar 

  • Schaffler M, Burr DB (1988) Stiffness of compact bone: effects of porosity and density. J Biomech 21(1):13–16

    Article  Google Scholar 

  • Schapery R (1975) A theory of crack initiation and growth in viscoelastic media. Int J Fract 11(1):141–159

    Article  MathSciNet  Google Scholar 

  • Schileo E, Taddei F, Malandrino A, Cristofolini L, Viceconti M (2007) Subject-specific finite element models can accurately predict strain levels in long bones. J Biomech 40(13):2982–2989

    Article  Google Scholar 

  • Schileo E, Dall‘Ara E, Taddei F, Malandrino A, Schotkamp T, Baleani M, Viceconti M (2008) An accurate estimation of bone density improves the accuracy of subject-specific finite element models. J Biomech 41(11):2483–2491

    Article  Google Scholar 

  • Seiner H, Sedlak P, Bodnarova L, Kruisova A, Landa M, de Pablos A, Belmonte M (2012) Sensitivity of the resonant ultrasound spectroscopy to weak gradients of elastic properties. J Acoust Soc Am 131(5):3775–3785

    Article  Google Scholar 

  • Seiner H, Sedlák P, Koller M, Landa M, Ramírez C, Osendi M, Belmonte M (2013) Anisotropic elastic moduli and internal friction of graphene nanoplatelets/silicon nitride composites. Compos Sci Technol 75:93–97

    Article  Google Scholar 

  • Sievänen H, Cheng S, Ollikainen S, Uusi-Rasi K (2001) Ultrasound velocity and cortical bone characteristics in vivo. Osteoporos Int 12(5):399–405

    Article  Google Scholar 

  • Snyder SM, Schneider E (1991) Estimation of mechanical properties of cortical bone by computed tomography. J Orthop Res 9(3): 422–431

    Article  Google Scholar 

  • Talmant M, Foiret J, Minonzio JG (2010) Bone Quantitative ultrasound, chap. Guided Waves in Cortical Bone. Springer, New York

    Google Scholar 

  • Ulrich TJ, McCall KR, Guyer RA (2002) Determination of elastic moduli of rock samples using resonant ultrasound spectroscopy. J Acoust Soc Am 111(4):1667–1674

    Article  Google Scholar 

  • Yang X, Muthukumaran P, DasDe S, Teoh SH, Choi H, Lim SK, Lee T (2013) Positive alterations of viscoelastic and geometric properties in ovariectomized rat femurs with concurrent administration of ibandronate and PTH. Bone 52(1):308–317

    Article  Google Scholar 

  • Zebaze RMD, Ghasem-Zadeh A, Bohte A, Iuliano-Burns S, Mirams M, Price RI, Mackie EJ, Seeman E (2010) Intracortical remodelling and porosity in the distal radius and post-mortem femurs of women: a cross-sectional study. Lancet 375(9727):1729–1736

    Article  Google Scholar 

Download references

Acknowledgments

This work has been conducted within the European Associated Laboratory ‘Ultrasound Based Assessment of Bone’ (ULAB), a cooperation of centers in Paris (France) and Kiel and Berlin (Germany), funded by CNRS (France). This work was funded by the Agence Nationale pour la Recherche under a Grant No. ANR-13-BS09-0006-01, the Elsbeth-Bonhoff foundation, project \(\sharp \)36 ‘Verbesserung der Abschätzung der Knochenbruchfestigkeit am proximalen Femur durch multitimodale Bestimmung von festigkeitsrelevanten Knochenmaterialeigenschaften’ and was supported by the Deutsche Forschungsgemeinschaft (SPP 1420 grant Ra1380/7-3, Ra1380/9-1). JS is grateful for the support from the BSRT. We acknowledge Robert Wendlandt from UKSH Lübeck for the collection of the samples.

Conflict of interest

The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, INSERM UMR-S 1146, CNRS UMR 7371, LIB 15, Rue de l’Ecole de Medecine, 75006, Paris, France

    Simon Bernard, Pascal Laugier & Quentin Grimal

  2. Julius-Wolff-Institute & Berlin Brandenburg School for Regenerative Therapies, Charité-Universitätsmedizin Berlin, Berlin, Germany

    Joannes Schneider, Peter Varga & Kay Raum

Authors
  1. Simon Bernard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joannes Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pascal Laugier
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kay Raum
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Quentin Grimal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quentin Grimal.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (xls 24 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernard, S., Schneider, J., Varga, P. et al. Elasticity–density and viscoelasticity–density relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements. Biomech Model Mechanobiol 15, 97–109 (2016). https://doi.org/10.1007/s10237-015-0689-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-015-0689-6

Keywords

Search

Navigation