Modeling Modulus Reduction in Bovine Trabecular Bone Damaged in Compression

[+] Author and Article Information
T. L. Arthur Moore

Division of Health Sciences and Technology, Harvard Medical School–Massachusetts Institute of Technology, Cambridge, MA 02139; Orthopedic Biomechanics Laboratory, Beth Israel Deaconess Medical Center, Boston, MA 02215

L. J. Gibson

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

J Biomech Eng 123(6), 613-622 (Jun 07, 2001) (10 pages) doi:10.1115/1.1407828 History: Received August 29, 2000; Revised June 07, 2001
Copyright © 2001 by ASME
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Schematic of trabecular bone, showing different levels of structure. Individual trabeculae may have microcracks or be fractured, reducing their stiffness. A region of trabecular bone may have trabeculae with no damage, with microcracks, or with fractures. As strain increases, the damage is concentrated in a band, indicated by the darker area on the whole specimen view, within the specimen. The model described in the text considers all three scales.
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An idealized two-dimensional representation of a single trabecula. The trabecula width b is either thickness t (rod model) or length l (plate model). (a) An idealized undamaged trabecula. (b) The damaged area is outlined to contain the local damage. (c) The damaged area is represented by a new material with a reduced modulus, Ē.
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Each damaged trabecula is replaced by an intact trabecula of the same initial modulus but a new effective thickness, t. The effective thickness, t, is chosen such that the intact trabecula of thickness t has the same stiffness as the damaged trabecula of thickness t0.
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Damage location and depth through the thickness, as measured in the previous companion paper. Damage depth through the thickness is rounded to the nearest 25 percent. Schematic showing damage occurring (a) in the middle of a trabecula (50 percent depth through the thickness) and (b) at one end of a trabecula (25 percent depth through the thickness).
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Trabeculae organized into a cellular solid network: (a) Entire network, showing trabeculae with no damage (black), trabeculae with partial damage (dark gray), and fractured trabeculae with a crack or cracks through the entire thickness (light gray). (b) Fractured trabeculae have been removed from the model. (c) Partially damaged trabeculae have been reduced in thickness. (d) The thickness reduction is averaged over all the remaining trabeculae, to give an average thickness of t̄.
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Plot of modulus reduction versus density fraction reduction 11 for random removal of rodlike struts in an open-cell cellular solid (black). An estimate for the modulus reduction for random removal of platelike struts has also been included; the dashed line indicates the interpolation between the two limits (gray).
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Damage within trabecular bone specimens is localized: (a) Image of a specimen tested to −4.0 percent compressive strain, stained after testing with calcein green. (b) Schematic representation of localized damage.
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Typical stress–strain curve, loaded to a strain of −3.00 percent, then unloaded to zero strain. Initial modulus E0 and secant modulus Esec are shown.
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Comparison of model predictions with experimentally determined modulus loss. Experimental modulus loss is the secant modulus divided by the initial loading modulus. (a) Closed-cell model based on platelike trabeculae. (b) Open-cell model based on rodlike trabeculae.
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Effects of changes in input variables on the specimen reduced modulus. Closed symbols represent a specimen tested to −4.0 percent strain, with considerable damage. Black diamond represent the closed-cell model (plate); gray squares represent the open-cell model (rod) for the specimen tested to −4.0 percent. A second specimen was tested to −1.1 percent strain, with little damage. The open triangles represent the closed-cell (plate model), and the solid gray line represents the open-cell (rod) model for the specimen loaded to −1.1 percent. Input parameters varied include: (a) reduced modulus due to removal of fractured struts, Estrut_loss/E0, (b) width of the localized damage band, (c) total damaged area in the specimen, Atotal, (d) estimate of depth of damage through the thickness, tdamage/t0, (e) fraction of undamaged struts, f0, and (f ) ratio of middle damage to end damage, plotted on a log scale.
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Effects of changing the distribution of damage on predicted modulus reduction. Black diamonds represent an even distribution of areas; black circles represent all damage concentrated in fractured struts (no partial damage). The error bars cover the range of damage created by different distributions. (a) Rod model. (b) Plate model.
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General form of a beam with a segment missing. The beam may be deformed in bending (rod model) or by axial stretching (plate model).



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