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Research Papers

Elastic Properties of Human Osteon and Osteonal Lamella Computed by a Bidirectional Micromechanical Model and Validated by Nanoindentation

[+] Author and Article Information
Radim Korsa

Department of Mechanics, Biomechanics
and Mechatronics,
Czech Technical University in Prague,
Technicka 4,
Prague 166 07, Czech Republic
e-mail: radim.korsa@gmail.com

Jaroslav Lukes

Department of Mechanics, Biomechanics
and Mechatronics,
Czech Technical University in Prague,
Technicka 4,
Prague 166 07, Czech Republic
e-mail: jaroslav.lukes@fs.cvut.cz

Josef Sepitka

Department of Mechanics, Biomechanics
and Mechatronics,
Czech Technical University in Prague,
Technicka 4,
Prague 166 07, Czech Republic
e-mail: Josef.Sepitka@fs.cvut.cz

Tomas Mares

Department of Mechanics, Biomechanics
and Mechatronics,
Czech Technical University in Prague,
Technicka 4,
Prague 166 07, Czech Republic
e-mail: tomas.mares@fs.cvut.cz

1Corresponding author.

Manuscript received October 16, 2014; final manuscript received April 13, 2015; published online June 9, 2015. Assoc. Editor: David Corr.

J Biomech Eng 137(8), 081002 (Aug 01, 2015) (11 pages) Paper No: BIO-14-1516; doi: 10.1115/1.4030407 History: Received October 16, 2014; Revised April 13, 2015; Online June 09, 2015

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.

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Figures

Grahic Jump Location
Fig. 1

Schema of bone tissue showing the hierarchical organization of the components

Grahic Jump Location
Fig. 2

Frames of reference of an osteon and osteonal lamella. αn are the orientation angles of osteonal lamellae, and k is the number of osteonal lamellae, from the innermost to the outermost

Grahic Jump Location
Fig. 3

Frames of reference of an osteonal lamella

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Fig. 4

A sample embedded in epoxy resin with marked perpendicular planes. TriboIndenter's optical was used to obtain pictures of three analyzed planes. The squares demarcate indented areas of the single lamellae. There are continuous lamellae, which were tested, in the transversal and frontal planes. The lamellae in the sagittal plane do not connect lamellae from the transversal and frontal plane.

Grahic Jump Location
Fig. 5

Schematics of elastic–plastic expansion of a spherical core [35] within a single bone lamella. We took into account an extreme case of the thinnest lamella lmin = 7.9 μm and the maximum indentation contact depth hc max = 0.2 μm. The measured volume of material underneath the probe can be approximated by a hemisphere with a radius of cmax = 2.8 μm.

Grahic Jump Location
Fig. 6

Pictures of the topography of the indented lamellae obtained by in situ SPM. Asterisks indicate where the indents were performed, and black strikeout points were excluded from the overall analysis because of poor contact between the tip and the surface of the material at the beginning of loading.

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