Knowledge of the anisotropic elastic properties of osteon and osteonal lamellae provides a better understanding of various pathophysiological conditions, such as aging, osteoporosis, osteoarthritis, and other degenerative diseases. For this reason, it is important to investigate and understand the elasticity of cortical bone. We created a bidirectional micromechanical model based on inverse homogenization for predicting the elastic properties of osteon and osteonal lamellae of cortical bone. The shape, the dimensions, and the curvature of osteon and osteonal lamellae are described by appropriately chosen curvilinear coordinate systems, so that the model operates close to the real morphology of these bone components. The model was used to calculate nine orthotropic elastic constants of osteonal lamellae. The input values have the elastic properties of a single osteon. We also expressed the dependence of the elastic properties of the lamellae on the angle of orientation. To validate the model, we performed nanoindentation tests on several osteonal lamellae. We compared the experimental results with the calculated results, and there was good agreement between them. The inverted model was used to calculate the elastic properties of a single osteon, where the input values are the elastic constants of osteonal lamellae. These calculations reveal that the model can be used in both directions of homogenization, i.e., direct homogenization and also inverse homogenization. The model described here can provide either the unknown elastic properties of a single lamella from the known elastic properties at the level of a single osteon, or the unknown elastic properties of a single osteon from the known elastic properties at the level of a single lamella.
Issue Section:
Research Papers
Keywords:
Biomechanics
References
1.
Cowin
, S. C.
, and Doty
, S. B.
, 2007
, Tissue Mechanics
, Springer Science+Business Media, LLC
, New York
.10.1007/978-0-387-49985-72.
Laszlo
, M.
, 1991
, Organization of the Extracellular Matrix: A Polarization Microscopic Approach
, CRC Press
, Boca Raton, FL
.3.
Fratzl
, P.
, and Weinkamer
, R.
, 2007
, “Nature's Hierarchical Materials
,” Prog. Mater. Sci.
, 52
(8
), pp. 1263
–1334
.10.1016/j.pmatsci.2007.06.0014.
Fratzl
, P.
, Gupta
, H. S.
, Paschalis
, E. P.
, and Roschger
, P.
, 2004
, “Structure and Mechanical Quality of the Collagen-Mineral Nano-Composite in Bone
,” J. Mater. Chem.
, 14
(14
), pp. 2115
–2123
.10.1039/b402005g5.
Wagermaier
, W.
, Gupta
, H. S.
, Gourrier
, A.
, Burghammer
, M.
, Roschger
, P.
, and Fratzl
, P.
, 2006
, “Spiral Twisting of Fiber Orientation Inside Bone Lamellae
,” Biointerphases
, 1
(1
), pp. 1
–5
.10.1116/1.21783866.
Wagermaier
, W.
, Gupta
, H. S.
, Gourrier
, A.
, Paris
, O.
, Roschger
, P.
, Burghammer
, M.
, Riekel
, C.
, and Fratzl
, P.
, 2007
, “Scanning Texture Analysis of Lamellar Bone Using Microbeam Synchrotron X-ray Radiation
,” J. Appl. Crystallogr.
, 40
, pp. 115
–120
.10.1107/S00218898060448887.
Techawiboonwong
, A.
, Song
, H. K.
, Leonard
, M. B.
, and Wehrli
, F. W.
, 2008
, “Cortical Bone Water: In Vivo Quantification With Ultrashort Echo-Time MR Imaging
,” Radiology
, 248
(3
), pp. 824
–833
.10.1148/radiol.24820719958.
Weiner
, S.
, and Wagner
, H. D.
, 1998
, “The Material Bone: Structure-Mechanical Function Relations
,” Annu. Rev. Mater. Sci.
, 28
, pp. 271
–298
.10.1146/annurev.matsci.28.1.2719.
Looker
, A. C.
, Borrud
, L. G.
, Hughes
, J. P.
, Fan
, B.
, Shepherd
, J. A.
, and Sherman
, M.
, 2013
, “Total Body Bone Area, Bone Mineral Content, and Bone Mineral Density for Individuals Aged 8 Years and Over: United States, 1999–2006
,” Vital Health Stat. 11
, 2013
(253
), pp. 1
–86
.10.
Jasiuk
, I.
, and Ostoja-Starzewski
, M.
, 2004
, “Modeling of Bone at a Single Lamella Level
,” Biomech. Model. Mechanobiol.
, 3
(2
), pp. 67
–74
.10.1007/s10237-004-0048-511.
Yoon
, Y. J.
, and Cowin
, S. C.
, 2008
, “The Estimated Elastic Constants for a Single Bone Osteonal Lamella
,” Biomech. Model. Mechanobiol.
, 7
(1
), pp. 1
–11
.10.1007/s10237-006-0072-812.
Dong
, X. N.
, and Guo
, X. E.
, 2006
, “Prediction of Cortical Bone Elastic Constants by a Two-Level Micromechanical Model Using a Generalized Self-Consistent Method
,” ASME J. Biomech. Eng.
, 128
(3
), pp. 309
–316
.10.1115/1.218703913.
Fritsch
, A.
, and Hellmich
, C.
, 2007
, “Universal Microstructural Patterns in Cortical and Trabecular, Extracellular and Extravascular Bone Materials: Micromechanics-Based Prediction of Anisotropic Elasticity
,” J. Theor. Biol.
, 244
(4
), pp. 597
–620
.10.1016/j.jtbi.2006.09.01314.
Nikolov
, S.
, and Raabe
, D.
, 2008
, “Hierarchical Modeling of the Elastic Properties of Bone at Submicron Scales: The Role of Extrafibrillar Mineralization
,” Biophys. J.
, 94
(11
), pp. 4220
–4232
.10.1529/biophysj.107.12556715.
Predoi-Racila
, M.
, and Crolet
, J. M.
, 2008
, “Human Cortical Bone: The SiNuPrOs Model
,” Comput. Methods Biomech. Biomed. Eng.
, 11
(2
), pp. 169
–187
.10.1080/1025584070169514016.
Hamed
, E.
, Lee
, Y.
, and Jasiuk
, I.
, 2010
, “Multiscale Modeling of Elastic Properties of Cortical Bone
,” Acta Mech.
, 213
(1–2
), pp. 131
–154
.10.1007/s00707-010-0326-517.
Zysset
, P. K.
, and Curnier
, A.
, 1995
, “An Alternative Model for Anisotropic Elasticity Based on Fabric Tensors
,” Mech. Mater.
, 21
(4
), pp. 243
–250
.10.1016/0167-6636(95)00018-618.
Franzoso
, G.
, and Zysset
, P. K.
, 2009
, “Elastic Anisotropy of Human Cortical Bone Secondary Osteons Measured by Nanoindantation
,” ASME J. Biomech. Eng.
, 131
(2
), p
. 021001.10.1115/1.300516219.
Spiesz
, E. M.
, Reisinger
, A. G.
, Kaminsky
, W.
, Roschger
, P.
, Pahr
, D. H.
, and Zysset
, P. K.
, 2013
, “Computational and Experimental Methodology for Site-Matched Investigations of the Influence of Mineral Mass Fraction and Collagen Orientation on the Axial Indentation Modulus of Lamellar Bone
,” J. Mech. Behav. Biomed. Mater.
, 28
, pp. 195
–205
.10.1016/j.jmbbm.2013.07.00420.
Hengsberger
, S.
, Kulik
, A.
, and Zysset
, P. K.
, 2002
, “Nanoindentation Discriminates the Elastic Properties of Individual Human Bone Lamellae Under Dry and Physiological Conditions
,” Bone
, 30
(1
), pp. 178
–184
.10.1016/S8756-3282(01)00624-X21.
Lukes
, J.
, and Otahal
, S.
, 2009
, “Nanoindentation of Porcine Bone Lamellae
,” Comput. Methods Biomech. Biomed. Eng.
, 12
(1
), pp. 177
–178
.10.1080/1025584090309146022.
Faingold
, A.
, Cohen
, S. R.
, and Wagner
, H. D.
, 2012
, “Nanoindentation of Osteonal Bone Lamellae
,” J. Mech. Behav. Biomed. Mater.
, 9
, pp. 198
–206
.10.1016/j.jmbbm.2012.01.01423.
Carnelli
, D.
, Vena
, P.
, Dao
, M.
, Ortiz
, Ch.
, and Contro
, R.
, 2013
, “Orientation and Size-Dependent Mechanical Modulation Within Individual Secondary Osteons in Cortical Bone Tissue
,” J. R. Soc. Interface
, 10
(81
), p. 20120953
.10.1098/rsif.2012.095324.
Varga
, P.
, Pacureanu
, A.
, Langer
, M.
, Suhonen
, H.
, Hesse
, B.
, Grimal
, Q.
, Cloetens
, P.
, Raum
, K.
, and Peyrin
, F.
, 2013
, “Investigation of the Three-Dimensional Orientation of Mineralized Collagen Fibrils in Human Lamellar Bone Using Synchrotron X-Ray Phase Nano-Tomography
,” Acta Biomater.
, 9
(9
), pp. 8118
–8127
.10.1016/j.actbio.2013.05.01525.
Delafargue
, A.
, and Ulm
, F.-J.
, 2004
, “Explicit Approximations of the Indentation Modulus of Elastically Orthotropic Solids for Conical Indenters
,” Int. J. Solids Struct.
, 41
(26
), pp. 7351
–7360
.10.1016/j.ijsolstr.2004.06.01926.
Korsa
, R.
, and Mares
, T.
, 2012
, “Numerical Identification of Orthotropic Coefficients of the Lamella of a Bone's Osteon
,” Bull. Appl. Mech.
, 8
(31
), pp. 45
–53
.27.
Mares
, T.
, 2006
, “Nanobiomechanical Model of Cortical Bone
,” Mechanist's Jotter, 1st ed., Czech Technical University in Prague, Prague, Czech Republic, pp. 90
–116
.28.
Goldmann
, T.
, 2006
, “Propagation of Acoustic Waves in Composite Materials and Cortical Bone
,” Ph.D. thesis, Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Mechanics, Prague, Czech Republic.29.
Orias
, A. A. E.
, 2005
, “The Relationship Between the Mechanical Anisotropy of Human Cortical Bone Tissue and Its Microstructure
,” Ph.D. thesis, University of Notre Dame, Notre Dame, IN.30.
Rho
, J. Y.
, Tsui
, T. Y.
, and Pharr
, G. M.
, 1997
, “Elastic Properties of Human Cortical and Trabecular Lamellar Bone Measured by Nanoindentation
,” Biomaterials
, 18
(20
), pp. 1325
–1330
.10.1016/S0142-9612(97)00073-231.
Korsa
, R.
, Lukes
, J.
, Sepitka
, J.
, Kytyr
, D.
, and Mares
, T.
, 2014
, “Mathematical Model of Human Osteon and Its Validation by Nanomechanical Testing of Bone Lamella
,” Comput. Methods Biomech. Biomed. Eng.
, 17
(1
), pp. 24
–25
.10.1080/10255842.2014.93107832.
Vercher
, A.
, Giner
, E.
, Arango
, C.
, Tarancon
, J. E.
, and Fuenmayor
, F. J.
, 2014
, “Homogenized Stiffness Matrices for Mineralized Collagen Fibrils and Lamellar Bone Using Unit Cell Finite Element Models
,” Biomech. Model. Mechanobiol.
, 13
(2
), pp. 437
–449
.10.1007/s10237-013-0507-y33.
Saada
, A. S.
, 1974
, Elasticity: Theory and Application
, Pergamon Press
, New York
.34.
Oliver
, W. C.
, and Pharr
, G. M.
, 1992
, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments
,” J. Mater. Res.
, 7
(6
), pp. 1564
–1583
.10.1557/JMR.1992.156435.
Johnson
, K. L.
, 1970
, “The Correlation of Indentation Experiments
,” J. Mech. Phys. Solids
, 18
(2
), pp. 115
–126
.10.1016/0022-5096(70)90029-336.
Fisher-Cripps
, A. C.
, 2004
, Nanoindentation
(Mechanical Engineering Series), 2nd ed., Springer Science + Business Media
, LLC, New York
.10.1007/978-1-4757-5943-337.
Courtney
, A. C.
, Hayes
, W. C.
, and Gibson
, L. J.
, 1996
, “Age-Related Differences in Post-Yield Damage in Human Cortical Bone, Experiment and Model
,” J. Biomech.
, 29
(11
), pp. 1463
–1471
.10.1016/0021-9290(96)84542-838.
Currey
, J. D.
, 2004
, “Tensile Yield in Compact Bone Is Determined by Strain, Post-Yield Behaviour by Mineral Content
,” J. Biomech.
, 37
(4
), pp. 549
–556
.10.1016/j.jbiomech.2003.08.00839.
Carretta
, R.
, Luisier
, B.
, Bernoulli
, D.
, Stüssi
, E.
, Müller
, R.
, and Lorenzetti
, S.
, 2013
, “Novel Method to Analyze Post-Yield Mechanical Properties at Trabecular Bone Tissue Level
,” J. Mech. Behav. Biomed. Mater.
, 20
, pp. 6
–18
.10.1016/j.jmbbm.2012.12.00340.
Rho
, J. Y.
, 1996
, “An Ultrasonic Method for Measuring the Elastic Properties of Human Tibial Cortical and Cancellous Bone
,” Ultrasonic
, 34
(8
), pp. 777
–783
.10.1016/S0041-624X(96)00078-941.
Maharidge
, R.
, 1984
, “Ultrasonic Properties and Microstructure of Bovine Bone and Haversian Bovine Bone Modeling
,” Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY.42.
Taylor
, W. R.
, Roland
, E.
, Ploeg
, H.
, Hertig
, D.
, Klabunde
, R.
, Warner
, M. D.
, Hobatho
, M. C.
, Rakotomanana
, L.
, and Clift
, S. E.
, 2002
, “Determination of Orthotropic Bone Elastic Constants Using FEA and Modal Analysis
,” J. Biomech.
, 35
(6
), pp. 767
–773
.10.1016/S0021-9290(02)00022-243.
Crolet
, J. M.
, Aoubiza
, B.
, and Meunier
, A.
, 1993
, “Compact Bone: Numerical Simulation of Mechanical Characteristic
,” J. Biomech.
, 26
(6
), pp. 677
–687
.10.1016/0021-9290(93)90031-944.
Huja
, S. S.
, Beck
, F. M.
, and Thurman
, D. T.
, 2006
, “Indentation Properties of Young and Old Osteons
,” Calcif. Tissue Int.
, 78
(6
), pp. 392
–397
.10.1007/s00223-006-0025-345.
Currey
, J. D.
, 1969
, “The Relationship Between the Stiffness and the Mineral Content of Bone
,” J. Biomech.
, 2
(4
), pp. 477
–480
.10.1016/0021-9290(69)90023-246.
Hellmich
, C.
, Barthelemy
, J. F.
, and Dormieux
, L.
, 2004
, “Mineral-Collagen Interactions in Elasticity of Bone Ultrastructure-A Continuum Micromechanics Approach
,” Eur. J. Mech. A: Solids
, 23
(5
), pp. 783
–810
.10.1016/j.euromechsol.2004.05.00447.
Dong
, X. N.
, and Guo
, X. E.
, 2004
, “The Dependence of Transversely Isotropic Elasticity of Human Femoral Cortical Bone on Porosity
,” J. Biomech.
, 37
(8
), pp. 1281
–1287
.10.1016/j.jbiomech.2003.12.011Copyright © 2015 by ASME
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