Skip to main content

Advertisement

Springer Nature Link
Log in

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

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

Abstract

At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Abbreviations

(x, y, z):

Orthogonal frame. Bone cross-section is assumed to be in the plane (x, y)

Z = Z(x, y):

Surface acoustic impedance value measured at point (x, y)

\({\hat Z}\) :

Mineralized matrix impedance averaged over the mesodomain matrix area

p :

Porosity

N (i) :

Number of mesodomains in each size group i = 1, . . . , 4

L (i) :

Edge size of the square cross-section of mesodomains in the plane (x, y) in each group i = 1, . . . , 4

θ :

Denotes one random realization of a mesodomain

L z :

Dimension of one mesodomain in direction z

C(θ):

Apparent stiffness tensor for one mesodomain realization

C t (θ):

Lower bound of the apparent stiffness tensor calculated with SUBC for one mesodomain realization

C d (θ):

Upper bound of the apparent stiffness tensor calculated with KUBC for one mesodomain realization

h :

Characteristic size of the finite element mesh

E, G and ν:

Apparent engineering elastic moduli, respectively, Young modulus, shear modulus and Poisson ratio

W :

Denotes any of the apparent engineering moduli

W t , W d :

Denotes any of the apparent engineering moduli obtained with either SUBC or KUBC, respectively

δ(θ), d W (θ):

Measures of the interval between the bounds of the apparent elastic behavior for one mesodomain realization

\({\hat \delta_i, \hat d_{W ;i}}\) :

Mean values of δ and d W in each quartile i = 1, . . . , 4

\({\hat W (L)}\) :

Mean of the engineering modulus W of mesodomains of size L

D W (L):

Standard deviation of the engineering modulus W of mesodomains of size L

\({\epsilon_W}\) :

Absolute uncertainty on the mean value of W

\({L = L_{{\rm RVE}} (\epsilon_W; W)}\) :

Size of RVE (Definition 2) for modulus W for a prescribed precision of \({\epsilon_W}\)

W 0;L (p):

Linear regression model for property W as a function of porosity, calculated for mesodomains of size L

References

  • Aoubiza B, Crolet JM, Meunier A (1996) On the mechanical characterization of compact bone structure using the homogenization theory. J Biomech 29(12): 1539–1547

    Google Scholar 

  • Baron C, Grimal Q, Talmant M, Laugier P (2008) Investigation of the porous network as a determinant of the overall stiffness of cortical bone: Mori-Tanaka model vs. ultrasound propagation. In: ASA, J Acoust Soc Am vol 123. Paris, p 3514

  • Baron C, Talmant M, Laugier P (2007) Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1 mhz: a finite-difference time domain study. J Acoust Soc Am 122(3): 1810

    Article  Google Scholar 

  • Cooper DM, Thomas CDL, Clement JG, Turinsky AL, Sensen CW, Hallgrimsson B (2007) Age-dependent change in the 3d structure of cortical porosity at the human femoral midshaft. Bone 40(4): 957–965

    Article  Google Scholar 

  • Cowin SC (2001) Bone mechanics handbook, vol 1, 2nd edn. CRC Press, Boca Raton

    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 J (2002) Bones: structure and mechanics. Princeton Academic Press, Princeton

    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 

  • François M, Geymonat G, Berthaud Y (1998) Determination of the symmetries of an experimentally determined stiffness tensor: application to acoustic measurements. Int J Solids Struct 35(31–32): 4091–4106

    Article  MATH  Google Scholar 

  • Grimal Q, Raum K, Gerisch A, Laugier P (2008) Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale. Comput Methods Biomech Biomed Eng 11(2): 147–157

    Article  Google Scholar 

  • Grimal Q, Raum K, Gerisch A, Laugier P (2009) About the determination of the representative volume element size in compact bone. In: Congrès Français de Mécanique. Marseille

  • Hellmich C, Barthelemy J, Dormieux L (2004) Mineral-collagen interactions in elasticity of bone ultrastructure—a continuum micromechanics approach. Eur J Mech A Solids 23(5): 783–810

    Article  MATH  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 

  • Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11: 357

    Article  MATH  Google Scholar 

  • Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6): 813–841

    Article  MathSciNet  Google Scholar 

  • Jeulin D (2001) Caractérisation morphologique et modèles de structures aléatoires. In: Homogénéisation en mécanique des matériaux, vol 1. Hermès, Paris, pp 95–132

  • Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13–14): 3647–3679

    Article  MATH  Google Scholar 

  • Kanit T, N’Guyen F, Forest S, Jeulin D, Reed M, Singleton S (2006) Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput Methods Appl Mech Eng 195(33–36): 3960–3982

    Article  MATH  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 

  • Martin RB, Burr DB, Sharkey NA (1998) Skeletal tissue mechanics, vol 1, 1st edn. Springer, New York

    Google Scholar 

  • Ostoja-Starzewski M (2008) Microstructural randomness and scaling in mechanics of materials. Chapman and Hall, CRC, London

    MATH  Google Scholar 

  • Pahr D, Zysset P (2008) Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech Model Mechanobiol 7(6): 463–476

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Raum K, Cleveland R, Peyrin F, Laugier P (2006) Derivation of elastic stiffness from site-matched mineral density and acoustic impedance maps. Phys Med Biol 51(3): 747–758

    Article  Google Scholar 

  • Raum K, Reisshauer J, Brandt J (2004) Frequency and resolution dependence of the anisotropic impedance estimation in cortical bone using time-resolved scanning acoustic microscopy. J Biomed Mater Res A 71(3): 430–438

    Article  Google Scholar 

  • Ren ZY, Zheng QS (2002) A quantitative study of minimum sizes of representative volume elements of cubic polycrystals—numerical experiments. J Mech Phys Solids 50(4): 881–893

    Article  MATH  Google Scholar 

  • Sab K (1992) On the homogenization and the simulation of random materials. Eur J Mech A Solids 11: 585–607

    MathSciNet  MATH  Google Scholar 

  • Saied A, Raum K, Leguerney I, Laugier P (2008) Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone. Bone 43(1): 187–194

    Article  Google Scholar 

  • Saporta G (1990) Probabilités, analyses des données et statistiques. Technip, Paris

    Google Scholar 

  • Sevostianov I, Kachanov M (2000) Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone. J Biomech 33(7): 881–888

    Article  Google Scholar 

  • Soize C (2008) Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probab Eng Mech 23(2–3): 307–323

    Article  Google Scholar 

  • Torquato S (2002) Random heterogeneous materials. Interdisciplinary applied mathematics; mechanics and materials. Springer, New York

    Google Scholar 

  • Zohdi T, Wriggers P (2005) Introduction to computational micromechanics. Lecture notes in applied computational mechanics. Springer, Berlin

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UPMC Univ Paris 06, UMR 7623, Laboratoire d’Imagerie Paramétrique, 75005, Paris, France

    Quentin Grimal & Pascal Laugier

  2. CNRS, UMR 7623, Laboratoire d’Imagerie Paramétrique, 75005, Paris, France

    Quentin Grimal & Pascal Laugier

  3. 15, Rue de l’école de médecine, 75006, Paris, France

    Quentin Grimal

  4. Julius Wolff Institute, Berlin-Brandenburg School for Regenerative Therapies, Charité-Universitätsmedizin Berlin, Augustenburger Platz 1, 13353, Berlin, Germany

    Kay Raum

  5. Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Alf Gerisch

Authors
  1. Quentin Grimal
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kay Raum
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Alf Gerisch
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Pascal Laugier
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Quentin Grimal.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grimal, Q., Raum, K., Gerisch, A. et al. A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties. Biomech Model Mechanobiol 10, 925–937 (2011). https://doi.org/10.1007/s10237-010-0284-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-010-0284-9

Keywords

Profiles

  1. Alf Gerisch View author profile