Recent advances on the measurement and calculation of the elastic moduli of cortical and trabecular bone: A review

In this review, recent advances on the measurement and modeling of elastic properties of cortical and trabecular bone are presented. Bone is a multifunctional material which among its other functions serves as a support for other tissues in the body. As a structural material it is stiff, strong, tough, lightweight and is adaptable. Its excellent mechanical properties are due to its complex, composite and hierarchical structure. In this paper, we outline the experimental approaches that have been used to characterize bone’s structure, composition and elastic properties at several different length scales. Then, we discuss different modeling approaches that have been employed to compute bone’s elastic moduli. We conclude by discussing the challenges and open issues in this area. Analysis of bone is of importance in orthopedics. Also, gained knowledge on bone can be used by engineers to design new bioinspired composite materials for a wide range of engineering applications.

[1]  J. Currey.  The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone. , 1988, Journal of biomechanics.

[2]  E. Olevsky,et al.  Energy absorbent natural materials and bioinspired design strategies: A review , 2010 .

[3]  R. Ritchie,et al.  TGF-beta regulates the mechanical properties and composition of bone matrix. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[4]  G. Niebur,et al.  Contrast-enhanced micro-computed tomography of fatigue microdamage accumulation in human cortical bone. , 2011, Bone.

[5]  M. Bouxsein,et al.  Biomechanics of Age-Related Fractures , 2001 .

[6]  M. Zimmerman,et al.  The acoustic properties of normal and imbedded bovine bone as measured by acoustic microscopy. , 1994, Journal of biomedical materials research.

[7]  W. Landis,et al.  Aspects of mineral structure in normally calcifying avian tendon. , 2001, Journal of structural biology.

[8]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[9]  L Kinzl,et al.  Correlation of bone mineral density with strength and microstructural parameters of cortical bone in vitro. , 2002, Bone.

[10]  K.,et al.  Differential changes in bone mineral density of the appendicular and axial skeleton with aging: relationship to spinal osteoporosis. , 1981, The Journal of clinical investigation.

[11]  S. Jackson,et al.  The morphology of bone mineral crystals , 1978, Calcified Tissue Research.

[12]  W. Hayes,et al.  Mechanical properties of trabecular bone from the proximal femur: a quantitative CT study. , 1990, Journal of computer assisted tomography.

[13]  Michael D. Morris,et al.  Chemical Microstructure of Cortical Bone Probed by Raman Transects , 1999 .

[14]  M Ding,et al.  Accuracy of cancellous bone volume fraction measured by micro-CT scanning. , 1999, Journal of biomechanics.

[15]  Steve Weiner,et al.  THE MATERIAL BONE: Structure-Mechanical Function Relations , 1998 .

[16]  G. Pharr,et al.  The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. , 1999, Journal of biomechanics.

[17]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[18]  F. Linde,et al.  Mechanical properties of trabecular bone. Dependency on strain rate. , 1991, Journal of Biomechanics.

[19]  A. Miller,et al.  Structural study of the calcifying collagen in turkey leg tendons. , 1979, Journal of molecular biology.

[20]  J. Bollerslev,et al.  Thermal stability of cortical bone collagen in relation to age in normal individuals and in individuals with osteopetrosis. , 1994, Bone.

[21]  R. Ritchie,et al.  On the Mechanistic Origins of Toughness in Bone , 2010 .

[22]  P Zioupos,et al.  The anisotropic Young's modulus of equine secondary osteones and interstitial bone determined by nanoindentation. , 2001, The Journal of experimental biology.

[23]  Klaus Klaushofer,et al.  Nucleation and growth of mineral crystals in bone studied by small-angle X-ray scattering , 1991, Calcified Tissue International.

[24]  S. Goldstein,et al.  Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. , 1999, Journal of biomechanics.

[25]  Joanna McKittrick,et al.  Anisotropy in the compressive mechanical properties of bovine cortical bone and the mineral and protein constituents. , 2011, Acta biomaterialia.

[26]  S. Weiner,et al.  Bone structure: from ångstroms to microns , 1992, FASEB journal : official publication of the Federation of American Societies for Experimental Biology.

[27]  U. Hansen,et al.  Microcracking damage and the fracture process in relation to strain rate in human cortical bone tensile failure. , 2008, Journal of biomechanics.

[28]  E. P. Katz,et al.  Structure and function of bone collagen fibrils. , 1973, Journal of molecular biology.

[29]  R Huiskes,et al.  The Prospects of Estimating Trabecular Bone Tissue Properties from the Combination of Ultrasound, Dual‐Energy X‐Ray Absorptiometry, Microcomputed Tomography, and Microfinite Element Analysis , 2001, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[30]  M. Marko,et al.  Mineralization of collagen may occur on fibril surfaces: evidence from conventional and high-voltage electron microscopy and three-dimensional imaging. , 1996, Journal of structural biology.

[31]  P. Schneider,et al.  Ultrastructural Properties in Cortical Bone Vary Greatly in Two Inbred Strains of Mice as Assessed by Synchrotron Light Based Micro‐ and Nano‐CT , 2007, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[32]  W. Bonfield,et al.  Elastic deformation of compact bone , 1973 .

[33]  Michael F. Ashby,et al.  The mechanical efficiency of natural materials , 2004 .

[34]  R. Huiskes,et al.  A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. , 1995, Journal of biomechanics.

[35]  A. Palazoglu,et al.  Nanoscale heterogeneity promotes energy dissipation in bone. , 2007, Nature materials.

[36]  P. Thurner.  Atomic force microscopy and indentation force measurement of bone. , 2009, Wiley interdisciplinary reviews. Nanomedicine and nanobiotechnology.

[37]  T M Keaveny,et al.  The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation. , 1996, Journal of biomechanics.

[38]  S. Weiner,et al.  Bone crystal sizes: a comparison of transmission electron microscopic and X-ray diffraction line width broadening techniques. , 1994, Connective tissue research.

[39]  Masako Ito,et al.  Assessment of bone quality using micro-computed tomography (micro-CT) and synchrotron micro-CT , 2009, Journal of Bone and Mineral Metabolism.

[40]  I. Jasiuk,et al.  Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid. , 2009, Journal of biomechanical engineering.

[41]  S. Weiner,et al.  Small‐angle x‐ray scattering study of dispersed crystals from bone and tendon , 1994, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[42]  P. Price,et al.  The Size Exclusion Characteristics of Type I Collagen , 2007, Journal of Biological Chemistry.

[43]  M. Glimcher,et al.  Size and Shape of Mineralites in Young Bovine Bone Measured by Atomic Force Microscopy , 2003, Calcified Tissue International.

[44]  Iwona M Jasiuk,et al.  Multiscale modeling of elastic properties of cortical bone , 2010 .

[45]  Stephen C. Cowin,et al.  The estimated elastic constants for a single bone osteonal lamella , 2008, Biomechanics and modeling in mechanobiology.

[46]  B F McEwen,et al.  Structural relations between collagen and mineral in bone as determined by high voltage electron microscopic tomography , 1996, Microscopy research and technique.

[47]  Christian Hellmich,et al.  Mineral–collagen interactions in elasticity of bone ultrastructure – a continuum micromechanics approach , 2004 .

[48]  M. Giraud‐Guille.  Twisted plywood architecture of collagen fibrils in human compact bone osteons , 1988, Calcified Tissue International.

[49]  Alexander D. MacKerell,et al.  All-atom empirical potential for molecular modeling and dynamics studies of proteins. , 1998, The journal of physical chemistry. B.

[50]  T M Keaveny,et al.  Nonlinear behavior of trabecular bone at small strains. , 2001, Journal of biomechanical engineering.

[51]  B. Budiansky.  On the elastic moduli of some heterogeneous materials , 1965 .

[52]  J. McKittrick,et al.  Elastic properties of cancellous bone in terms of elastic properties of its mineral and protein phases with application to their osteoporotic degradation , 2012 .

[53]  A. Boskey.  Bone mineral crystal size , 2003, Osteoporosis International.

[54]  Steve Weiner,et al.  Are tensile and compressive Young's moduli of compact bone different? , 2009, Journal of the mechanical behavior of biomedical materials.

[55]  D Van Dyck,et al.  Quantitative analysis of bone mineral content by x-ray microtomography. , 2003, Physiological measurement.

[56]  X Edward Guo,et al.  The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. , 2004, Journal of biomechanics.

[57]  N Guzelsu,et al.  The effects of interphase and bonding on the elastic modulus of bone: changes with age-related osteoporosis. , 2000, Medical engineering & physics.

[58]  S. Weiner,et al.  Lamellar bone: structure-function relations. , 1999, Journal of structural biology.

[59]  T. J. Hirsch,et al.  Modulus of Elasticity iof Concrete Affected by Elastic Moduli of Cement Paste Matrix and Aggregate , 1962 .

[60]  Thomas Siegmund,et al.  Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking. , 2008, Journal of biomechanics.

[61]  G. Lawes,et al.  Scanning Electron Microscopy and X-Ray Microanalysis , 1987 .

[62]  P Rüegsegger,et al.  Introduction and evaluation of a gray-value voxel conversion technique. , 2001, Journal of biomechanics.

[63]  Gary D Fullerton,et al.  Evidence that collagen and tendon have monolayer water coverage in the native state , 2006, Cell biology international.

[64]  Steve Weiner,et al.  Modelling the three-dimensional elastic constants of parallel-fibred and lamellar bone , 1998 .

[65]  S A Goldstein,et al.  Mechanical properties of human trabecular bone lamellae quantified by nanoindentation. , 1998, Technology and health care : official journal of the European Society for Engineering and Medicine.

[66]  A. V. Duin,et al.  ReaxFF: A Reactive Force Field for Hydrocarbons , 2001 .

[67]  J L Lewis,et al.  The dynamic fracture and prefracture response of compact bone by split Hopkinson bar methods. , 1975, Journal of biomechanics.

[68]  Fergal J O'Brien,et al.  Microcrack accumulation at different intervals during fatigue testing of compact bone. , 2003, Journal of biomechanics.

[69]  George Sanger,et al.  Structure and Mechanics , 1991 .

[70]  G H van Lenthe,et al.  Specimen-specific beam models for fast and accurate prediction of human trabecular bone mechanical properties. , 2006, Bone.

[71]  V. H. Jacobo,et al.  Analysis of the Architecture and Mechanical Properties of Cancellous Bone Using 2 D Voronoi Cell Based Models , 2022 .

[72]  Steven K Boyd,et al.  Automatic segmentation of cortical and trabecular compartments based on a dual threshold technique for in vivo micro-CT bone analysis. , 2007, Bone.

[73]  Masahiro Taniguchi,et al.  Atomic force microscopic studies on the structure of bovine femoral cortical bone at the collagen fibril-mineral level , 2002, Journal of materials science. Materials in medicine.

[74]  R. C. Tennyson,et al.  Dynamic viscoelastic response of bone , 1972 .

[75]  W. Bonfield,et al.  Young's modulus of compact bone. , 1974, Journal of biomechanics.

[76]  H Weinans,et al.  Detecting and tracking local changes in the tibiae of individual rats: a novel method to analyse longitudinal in vivo micro-CT data. , 2004, Bone.

[77]  T. Keller.  Predicting the compressive mechanical behavior of bone. , 1994, Journal of biomechanics.

[78]  Joel W. Ager,et al.  Fracture and Ageing in Bone: Toughness and Structural Characterization , 2006 .

[79]  Theo H Smit,et al.  Osteocyte morphology in human tibiae of different bone pathologies with different bone mineral density--is there a role for mechanosensing? , 2009, Bone.

[80]  Vinod Subramaniam,et al.  Micromechanical bending of single collagen fibrils using atomic force microscopy. , 2007, Journal of biomedical materials research. Part A.

[81]  J. Mackenzie,et al.  The Elastic Constants of a Solid containing Spherical Holes , 1950 .

[82]  E. Atkins,et al.  Scanning probe microscopy of intrafibrillar crystallites in calcified collagen , 1994 .

[83]  R. Pidaparti,et al.  Bone mineral lies mainly outside collagen fibrils: predictions of a composite model for osteonal bone. , 1996, Journal of biomechanics.

[84]  Dinesh R. Katti,et al.  Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone , 2007 .

[85]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[86]  A. Reuss,et al.  Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle . , 1929 .

[87]  K. Piekarski.  Analysis of bone as a composite material , 1973 .

[88]  P. Pankaj,et al.  Virtual trabecular bone models and their mechanical response , 2008, Proceedings of the Institution of Mechanical Engineers. Part H, Journal of engineering in medicine.

[89]  G. Fullerton,et al.  Verification of simple hydration/dehydration methods to characterize multiple water compartments on Tendon Type 1 Collagen , 2007, Cell biology international.

[90]  P Zioupos,et al.  Exploring the Effects of Hypermineralisation in Bone Tissue by Using an Extreme Biological Example , 2000, Connective tissue research.

[91]  S. Cowin,et al.  On the dependence of the elasticity and strength of cancellous bone on apparent density. , 1988, Journal of biomechanics.

[92]  S. Weiner,et al.  Rotated plywood structure of primary lamellar bone in the rat: orientations of the collagen fibril arrays. , 1997, Bone.

[93]  R. B. Ashman,et al.  Relations of mechanical properties to density and CT numbers in human bone. , 1995, Medical engineering & physics.

[94]  C. Sun,et al.  Three-Dimensional Effective Elastic Constants for Thick Laminates , 1988 .

[95]  A. Goodship,et al.  A novel technique for four-point bending of small bone samples with semi-automatic analysis. , 2003, Journal of biomechanics.

[96]  Dwayne Arola,et al.  Evaluating the elastic modulus of bone using electronic speckle pattern interferometry , 2001 .

[97]  Elliot P. Douglas,et al.  Bone structure and formation: A new perspective , 2007 .

[98]  H A Hogan,et al.  Micromechanics modeling of Haversian cortical bone properties. , 1992, Journal of biomechanics.

[99]  E. Kröner.  Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls , 1958 .

[100]  S Cusack,et al.  Determination of the elastic constants of collagen by Brillouin light scattering. , 1979, Journal of molecular biology.

[101]  J. McKittrick,et al.  Minerals Form a Continuum Phase in Mature Cancellous Bone , 2011, Calcified Tissue International.

[102]  H. Mook,et al.  Neutron diffraction studies of collagen in fully mineralized bone. , 1985, Journal of molecular biology.

[103]  K. Rajan.  Linear elastic properties of trabecular bone: a cellular solid approach , 1985 .

[104]  S A Goldstein,et al.  A comparison of the fatigue behavior of human trabecular and cortical bone tissue. , 1992, Journal of biomechanics.

[105]  R. Gilbert,et al.  Application of the multiscale FEM to the modeling of cancellous bone , 2010, Biomechanics and modeling in mechanobiology.

[106]  H Weinans,et al.  Altered tissue properties induce changes in cancellous bone architecture in aging and diseases. , 2004, Journal of biomechanics.

[107]  Markus J. Buehler,et al.  Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization , 2007 .

[108]  Tsu-Wei Chou,et al.  On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: An application to a hybrid composite☆ , 1981 .

[109]  D. Taylor.,et al.  Visualisation of three‐dimensional microcracks in compact bone , 2000, Journal of anatomy.

[110]  Ernesto Raúl Caffarena,et al.  Elastic properties, Young's modulus determination and structural stability of the tropocollagen molecule: a computational study by steered molecular dynamics. , 2005, Journal of biomechanics.