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A new multiscale micromechanical model of vertebral trabecular bones

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Abstract

A new three-dimensional (3D) multiscale micromechanical model has been suggested as adept at predicting the overall linear anisotropic mechanical properties of a vertebral trabecular bone (VTB) highly porous microstructure. A nested 3D modeling analysis framework spanning the multiscale nature of the VTB is presented herein. This hierarchical analysis framework employs the following micromechanical methods: the 3D parametric high-fidelity generalized method of cells (HFGMC) as well as the 3D sublaminate model. At the nanoscale level, the 3D HFGMC method is applied to obtain the effective elastic properties of a representative unit cell (RUC) representing the mineral collagen fibrils composite. Next at the submicron scale level, the 3D sublaminate model is used to generate the effective elastic properties of a repeated stack of multilayered lamellae demonstrating the nature of the trabeculae (bone-wall). Thirdly, at the micron scale level, the 3D HFGMC method is used again on a RUC of the highly porous VTB microstructure. The VTB-RUC geometries are taken from microcomputed tomography scans of VTB samples harvested from different vertebrae of human cadavers \((n=10)\). The predicted anisotropic overall elastic properties for native VTBs are, then, examined as a function of age and sex. The predicted results of the VTBs longitudinal Young’s modulus are compared to reported values found in the literature. The proposed 3D nested modeling analysis framework provides a good agreement with reported values of Young’s modulus of single trabeculae as well as for VTB-RUC in the literature.

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References

  • Aboudi J, Arnold SM, Bednarcyk BA (2013) Micromechanics of composite materials: a generalized multiscale analysis approach. Elsevier, Amsterdam

    Google Scholar 

  • Burstein AH, Frankel VH (1968) The viscoelastic properties of some biological materials*. Ann N Y Acad Sci 146:158–165

    Article  Google Scholar 

  • Colloca M, Blanchard R, Hellmich C, Ito K, van Rietbergen B (2014) A multiscale analytical approach for bone remodeling simulations: linking scales from collagen to trabeculae. Bone 64:303–313

  • Fratzl P (2008) Collagen: structure and mechanics. Springer, Berlin

    Book  Google Scholar 

  • Gibson LJ (1985) The mechanical behaviour of cancellous bone. J Biomech 18:317–328

    Article  Google Scholar 

  • Gilmore RS, Katz JL (1982) Elastic properties of apatites. J Mater Sci 17:1131–1141

    Article  Google Scholar 

  • Goda I, Assidi M, Belouettar S, Ganghoffer JF (2012) A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J Mech Behav Biomed Mater 16:87–108

    Article  Google Scholar 

  • Goda I, Assidi M, Ganghoffer JF (2014) A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech Model Mechanobiol 13:53–83

    Article  Google Scholar 

  • Goda I, Ganghoffer J-F (2015) 3D plastic collapse and brittle fracture surface models of trabecular bone from asymptotic homogenization method. Int J Eng Sci 87:58–82

    Article  Google Scholar 

  • Haj-Ali R (2008) Nested nonlinear multiscale frameworks for the analysis of thick-section composite materials and structures. In: Kwon YW, Ramesh Talreja DHA (eds) Multiscale modeling and simulation of composite materials and structures. Springer, US, pp 317–357

  • Haj-Ali R, Aboudi J (2009) Nonlinear micromechanical formulation of the high fidelity generalized method of cells. Int J Solids Struct 46:2577–2592

    Article  MATH  Google Scholar 

  • Haj-Ali R, Aboudi J (2013) A new and general formulation of the parametric HFGMC micromechanical method for two and three-dimensional multi-phase composites. Int J Solids Struct 50:907–919

    Article  Google Scholar 

  • Haj-Ali R, Aboudi J (2016) Integrated microplane model with the HFGMC micromechanics for nonlinear analysis of composite materials with evolving damage. Int J Solids Struct 90:129–143

    Article  Google Scholar 

  • Haj-Ali R, Zemer H, El-Hajjar R, Aboudi J (2014) Piezoresistive fiber-reinforced composites: a coupled nonlinear micromechanical–microelectrical modeling approach. Int J Solids Struct 51:491–503

    Article  Google Scholar 

  • Hamed E, Jasiuk I, Yoo A, Lee Y, Liszka T (2012) Multi-scale modelling of elastic moduli of trabecular bone. J R Soc Interface 9:1654–1673

  • Jäger I, Fratzl P (2000) Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys J 79:1737–1746

    Article  Google Scholar 

  • Johnell O, Kanis JA (2006) An estimate of the worldwide prevalence and disability associated with osteoporotic fractures. Osteoporos Int 17:1726–1733

    Article  Google Scholar 

  • Keaveny TM, Morgan E, Niebur Glen L, Yeh Oscar C (2001) Biomechanics of trabecular bone. Annu Rev Biomed Eng 3:307–333

    Article  Google Scholar 

  • Kopperdahl DL, Keaveny TM (1998) Yield strain behavior of trabecular bone. J Biomech 31:601–608

  • Maquer G, Musy SN, Wandel J, Gross T, Zysset PK (2015) Bone volume fraction and fabric anisotropy are better determinants of trabecular bone stiffness than other morphological variables. J Bone Miner Res 30:1000–1008

    Article  Google Scholar 

  • McDonald K, Little J, Pearcy M, Adam C (2010) Development of a multi-scale finite element model of the osteoporotic lumbar vertebral body for the investigation of apparent level vertebra mechanics and micro-level trabecular mechanics. Med Eng Phys 32:653–661

  • McElhaney J, Byars EF (1965) Dynamic response of biological materials. American Society of Mechanical Engineers, New York

    Google Scholar 

  • Pecknold DA, Haj-Ali R (1993) Integrated micromechanical/structural analysis of laminated composites. In: Hyer MW (ed) Mechanics of composite materials-nonlinear effects. Proceedings of the 1st joint mechanics meeting of SES/ASME/ASCE Charlottesville, pp 197–206

  • Peterson DR, Bronzino JD (2008) Mechanics of hard tissue. In: Katz JL (ed) Biomechanics: principles and applications. CRS Press, Boca Raton

    Google Scholar 

  • Rao RD, Singrakhia MD (2003) Painful osteoporotic vertebral fracture. J Bone Joint Surg Am 85:2010–2022

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Rho J-Y, Roy ME, Tsui TY, Pharr GM (1999) Elastic properties of microstructural components of human bone tissue as measured by nanoindentation. J Biomed Mater Res 45:48–54

    Article  Google Scholar 

  • Rho J-Y, Tsui TY, Pharr GM (1997) Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18:1325–1330

    Article  Google Scholar 

  • Rho JY, Ashman RB, Turner CH (1993) Young’s modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J Biomech 26:111–119

    Article  Google Scholar 

  • Roy ME, Rho J-Y, Tsui TY, Evans ND, Pharr GM (1999) Mechanical and morphological variation of the human lumbar vertebral cortical and trabecular bone. J Biomed Mater Res 44:191–197

    Article  Google Scholar 

  • Schwiedrzik J, Raghavan R, Bürki A, LeNader V, Wolfram U, Michler J, Zysset P (2014) In situ micropillar compression reveals superior strength and ductility but an absence of damage in lamellar bone. Nat Mater 13:740–747

    Article  Google Scholar 

  • van Rietbergen B, Weinans H, Huiskes R, Odgaard A (1995) A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. J Biomech 28:69–81

    Article  Google Scholar 

  • Weiner S, Traub W, Wagner HD (1999) Lamellar bone: structure—function relations. J Struct Biol 126:241–255

  • Wolfram U, Wilke H-J, Zysset PK (2010a) Rehydration of vertebral trabecular bone: influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level. Bone 46:348–354

    Article  Google Scholar 

  • Wolfram U, Wilke H-J, Zysset PK (2010b) Valid \(\mu \) finite element models of vertebral trabecular bone can be obtained using tissue properties measured with nanoindentation under wet conditions. J Biomech 43:1731–1737

    Article  Google Scholar 

  • Wolfram U, Wilke H-J, Zysset PK (2011) Damage accumulation in vertebral trabecular bone depends on loading mode and direction. J Biomech 44:1164–1169

  • Yang G, Kabel J, Van Rietbergen B, Odgaard A, Huiskes R, Cown SC (1998) The anisotropic Hooke’s Law for cancellous bone and wood. J Elasticity 53:125–146

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Acknowledgements

This research was supported in part by the Israeli Ministry of Science, Technology and Space under Grant No. 3-12961. The first author gratefully acknowledges the support of the Nathan Cummings Chair of Mechanics.

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Authors and Affiliations

  1. Faculty of Engineering, Tel-Aviv University, 6997801, Tel Aviv, Israel

    Rami Haj-Ali, Eyass Massarwa & Jacob Aboudi

  2. Department of Spine Surgery III, IRCCS Istituto Ortopedico Galeazzi, Milan, Italy

    Fabio Galbusera

  3. Mechanical, Process and Energy Engineering, Heriot-Watt University, Edinburgh, UK

    Uwe Wolfram

  4. Institute for Orthopedic Research and Biomechanics, Ulm University, Helmholtzstrasse. 14, 89081, Ulm, Germany

    Hans-Joachim Wilke

Authors
  1. Rami Haj-Ali
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  2. Eyass Massarwa
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  3. Jacob Aboudi
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  4. Fabio Galbusera
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  5. Uwe Wolfram
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  6. Hans-Joachim Wilke
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Correspondence to Rami Haj-Ali.

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Haj-Ali, R., Massarwa, E., Aboudi, J. et al. A new multiscale micromechanical model of vertebral trabecular bones. Biomech Model Mechanobiol 16, 933–946 (2017). https://doi.org/10.1007/s10237-016-0862-6

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  • DOI: https://doi.org/10.1007/s10237-016-0862-6

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