Research Papers

Biaxial Normal Strength Behavior in the Axial-Transverse Plane for Human Trabecular Bone—Effects of Bone Volume Fraction, Microarchitecture, and Anisotropy

[+] Author and Article Information
Arnav Sanyal

Orthopaedic Biomechanics Laboratory,
Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
e-mail: arnavsanyal@berkeley.edu

Tony M. Keaveny

Orthopaedic Biomechanics Laboratory,
Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
Department of Bioengineering,
University of California, Berkeley, CA 94720
e-mail: tmk@me.berkeley.edu

1Corresponding author.

2Please address all reprint requests to Tony M. Keaveny.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 9, 2013; final manuscript received September 23, 2013; accepted manuscript posted October 14, 2013; published online November 6, 2013. Assoc. Editor: Kristen Billiar.

J Biomech Eng 135(12), 121010 (Nov 06, 2013) (9 pages) Paper No: BIO-13-1309; doi: 10.1115/1.4025679 History: Received July 09, 2013; Revised September 23, 2013; Accepted October 14, 2013

The biaxial failure behavior of the human trabecular bone, which has potential relevance both for fall and gait loading conditions, is not well understood, particularly for low-density bone, which can display considerable mechanical anisotropy. Addressing this issue, we investigated the biaxial normal strength behavior and the underlying failure mechanisms for human trabecular bone displaying a wide range of bone volume fraction (0.06–0.34) and elastic anisotropy. Micro-computed tomography (CT)-based nonlinear finite element analysis was used to simulate biaxial failure in 15 specimens (5 mm cubes), spanning the complete biaxial normal stress failure space in the axial-transverse plane. The specimens, treated as approximately transversely isotropic, were loaded in the principal material orientation. We found that the biaxial stress yield surface was well characterized by the superposition of two ellipses—one each for yield failure in the longitudinal and transverse loading directions—and the size, shape, and orientation of which depended on bone volume fraction and elastic anisotropy. However, when normalized by the uniaxial tensile and compressive strengths in the longitudinal and transverse directions, all of which depended on bone volume fraction, microarchitecture, and mechanical anisotropy, the resulting normalized biaxial strength behavior was well described by a single pair of (longitudinal and transverse) ellipses, with little interspecimen variation. Taken together, these results indicate that the role of bone volume fraction, microarchitecture, and mechanical anisotropy is mostly accounted for in determining the uniaxial strength behavior and the effect of these parameters on the axial-transverse biaxial normal strength behavior per se is minor.

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van Rietbergen, B., Huiskes, R., Eckstein, F., and Rüegsegger, P., 2003, “Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur,” J. Bone Miner. Res., 18(10), pp. 1781–1788. [CrossRef] [PubMed]
Verhulp, E., van Rietbergen, B., and Huiskes, R., 2008, “Load Distribution in the Healthy and Osteoporotic Human Proximal Femur During a Fall to the Side,” Bone, 42(1), pp. 30–35. [CrossRef] [PubMed]
Homminga, J., van Rietbergen, B., Lochmuller, E. M., Weinans, H., Eckstein, F., and Huiskes, R., 2004, “The Osteoporotic Vertebral Structure is Well Adapted to the Loads of Daily Life, but not to Infrequent ‘Error’ Loads,” Bone, 34(3), pp. 510–516. [CrossRef] [PubMed]
Newitt, D. C., Majumdar, S., van Rietbergen, B., von Ingersleben, G., Harris, S. T., Genant, H. K., Chesnut, C., Garnero, P., and MacDonald, B., 2002, “In Vivo Assessment of Architecture and Micro-Finite Element Analysis Derived Indices of Mechanical Properties of Trabecular Bone in the Radius,” Osteoporosis Int., 13(1), pp. 6–17. [CrossRef]
Ciarelli, T. E., Fyhrie, D. P., Schaffler, M. B., and Goldstein, S. A., 2000, “Variations in Three-Dimensional Cancellous Bone Architecture of the Proximal Femur in Female Hip Fractures and in Controls,” J. Bone Miner. Res., 15(1), pp. 32–40. [CrossRef] [PubMed]
Tanck, E., Bakker, A. D., Kregting, S., Cornelissen, B., Klein-Nulend, J., and van Rietbergen, B., 2009, “Predictive Value of Femoral Head Heterogeneity for Fracture Risk,” Bone, 44(4), pp. 590–595. [CrossRef] [PubMed]
Homminga, J., McCreadie, B. R., Ciarelli, T. E., Weinans, H., Goldstein, S. A., and Huiskes, R., 2002, “Cancellous Bone Mechanical Properties From Normals and Patients With Hip Fractures Differ on the Structure Level, Not on the Bone Hard Tissue Level,” Bone, 30(5), pp. 759–764. [CrossRef] [PubMed]
Cowin, S. C., 1986, “Fabric Dependence of an Anisotropic Strength Criterion,” Mech. Mater., 5, pp. 251–260. [CrossRef]
Keaveny, T. M., Wachtel, E. F., Zadesky, S. P., and Arramon, Y. P., 1999, “Application of the Tsai-Wu Quadratic Multiaxial Failure Criterion to Bovine Trabecular Bone,” ASME J. Biomech. Eng., 121(1), pp. 99–107. [CrossRef]
Fenech, C. M., and Keaveny, T. M., 1999, “A Cellular Solid Criterion for Predicting the Axial-Shear Failure Properties of Trabecular Bone.,” ASME J. Biomech. Eng., 121(4), pp. 414–422. [CrossRef]
Niebur, G. L., Feldstein, M. J., and Keaveny, T. M., 2002, “Biaxial Failure Behavior of Bovine Tibial Trabecular Bone,” ASME J. Biomech. Eng., 124(6), pp. 699–705. [CrossRef]
Bayraktar, H. H., Gupta, A., Kwon, R. Y., Papadopoulos, P., and Keaveny, T. M., 2004, “The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone,” ASME J. Biomech. Eng., 126(6), pp. 677–684. [CrossRef]
Rincon-Kohli, L., and Zysset, P. K., 2009, “Multi-Axial Mechanical Properties of Human Trabecular Bone,” Biomech. Model. Mechanobiol., 8(3), pp. 195–208. [CrossRef] [PubMed]
Wolfram, U., Gross, T., Pahr, D. H., Schwiedrzik, J., Wilke, H. J., and Zysset, P. K., 2012, “Fabric-Based Tsai-Wu Yield Criteria for Vertebral Trabecular Bone in Stress and Strain Space,” J. Mech. Behav. Biomed. Mater., 15, pp. 218–228. [CrossRef] [PubMed]
Bevill, G., Farhamand, F., and Keaveny, T. M., 2009, “Heterogeneity of Yield Strain in Low-Density Versus High-Density Human Trabecular Bone,” J. Biomech., 42(13), pp. 2165–2170. [CrossRef] [PubMed]
Morgan, E. F., and Keaveny, T. M., 2001, “Dependence of Yield Strain of Human Trabecular Bone on Anatomic Site,” J. Biomech., 34(5), pp. 569–577. [CrossRef] [PubMed]
Rüegsegger, P., Koller, B., and Müller, R., 1996, “A Microtomographic System for the Nondestructive Evaluation of Bone Architecture,” Calcif. Tissue Int., 58(1), pp. 24–29. [CrossRef] [PubMed]
Beck, J. D., Canfield, B. L., Haddock, S. M., Chen, T. J., Kothari, M., and Keaveny, T. M., 1997, “Three-Dimensional Imaging of Trabecular Bone Using the Computer Numerically Controlled Milling Technique,” Bone, 21(3), pp. 281–287. [CrossRef] [PubMed]
Hildebrand, T., Laib, A., Müller, R., Dequeker, J., and Rüegsegger, P., 1999, “Direct Three-Dimensional Morphometric Analysis of Human Cancellous Bone: Microstructural Data From Spine, Femur, Iliac Crest, and Calcaneus,” J. Bone Miner. Res., 14(7), pp. 1167–1174. [CrossRef] [PubMed]
van Rietbergen, B., Odgaard, A., Kabel, J., and Huiskes, R., 1996, “Direct Mechanics Assessment of Elastic Symmetries and Properties of Trabecular Bone Architecture,” J. Biomech., 29(12), pp. 1653–1657. [PubMed]
Papadopoulos, P., and Lu, J., 2001, “On the Formulation and Numerical Solution of Problems in Anisotropic Finite Plasticity,” Comput. Methods Appl. Mech. Eng., 190(37–38), pp. 4889–4910. [CrossRef]
Bevill, G., Eswaran, S. K., Gupta, A., Papadopoulos, P., and Keaveny, T. M., 2006, “Influence of Bone Volume Fraction and Architecture on Computed Large-Deformation Failure Mechanisms in Human Trabecular Bone.,” Bone, 39(6), pp. 1218–1225. [CrossRef] [PubMed]
Sanyal, A., Gupta, A., Bayraktar, H. H., Kwon, R. Y., and Keaveny, T. M., 2012, “Shear Strength Behavior of Human Trabecular Bone,” J. Biomech., 45(15), pp. 2513–2519. [CrossRef] [PubMed]
Adams, M. F., Bayraktar, H. H., Keaveny, T. M., and Papadopoulos, P., 2004, “Ultrascalable Implicit Finite Element Analyses in Solid Mechanics With Over a Half a Billion Degrees of Freedom,” Proceedings of the ACM/IEEE SC2004: High Performance Networking and Computing.
Fitzgibbon, A. M. P., and Fisher, B., 1999, “Direct Least-Squares Fitting of Ellipses,” IEEE Trans. Pattern Anal. Mach. Intell., 21(5), pp. 476–480. [CrossRef]
Kopperdahl, D. L., and Keaveny, T. M., 1998, “Yield Strain Behavior of Trabecular Bone,” J. Biomech., 31(7), pp. 601–608. [CrossRef] [PubMed]
Chang, W. C. W., Christensen, T. M., Pinilla, T. P., and Keaveny, T. M., 1999, “Isotropy of Uniaxial Yield Strains for Bovine Trabecular Bone,” J. Orthop. Res., 17, pp. 582–585. [CrossRef] [PubMed]
Mosekilde, L., 1988, “Age-Related Changes in Vertebral Trabecular Bone Architecture—Assessed by a New Method,” Bone, 9(4), pp. 247–250. [CrossRef] [PubMed]
Thomsen, J. S., Ebbesen, E. N., and Mosekilde, L., 2002, “Age-Related Differences Between Thinning of Horizontal and Vertical Trabeculae in Human Lumbar Bone as Assessed by a New Computerized Method,” Bone, 31(1), pp. 136–142. [CrossRef] [PubMed]
Gibson, L. J., Ashby, M. F., Zhang, J., and Triantafillou, T. C., 1989, “Failure Surfaces for Cellular Materials Under Multiaxial Loads-I. Modelling,” Int. J. Mech. Sci., 31(9), pp. 635–663. [CrossRef]
Zysset, P. K., and Rincon-Kohli, L., 2006, “An Alternative Fabric-Based Yield and Failure Criterion for Trabecular Bone,” Mechanics of Bilogical Tissue, G. A.Holzapfel and R. W.Ogden, eds., Springer, New York, pp. 457–470.
Wu, R., and Stachurski, Z., 1984, “Evaluation of the Normal Stress Interaction Parameter in the Tensor Polynomial Strength Theory for Anisotropic Materials,” J. Compos. Mater., 18, pp. 456–463. [CrossRef]
Traintafillou, T. C., Zhang, J., Shercliff, T. L., Gibson, L. J., and Ashby, M. F., 1989, “Failure Surfaces for Cellular Materials Under Multiaxial Loads. II. Comparison of Models With Experiment,” Int. J. Mech. Sci., 31(9), pp. 665–678. [CrossRef]
Wang, D.-A., and Pan, J., 2006, “A Non-Quadratic Yield Function for Polymeric Foams,” Int. J. Plast., 22, pp. 434–458. [CrossRef]
Cezayirlioglu, H., Bahniuk, E., Davy, D. T., and Heiple, K. G., 1985, “Anisotropic Yield Behavior of Bone Under Combined Axial Force and Torque,” J. Biomech., 18(1), pp. 61–69. [CrossRef] [PubMed]
Gioux, G., McCormack, T. M., and Gibson, L. J., 2000, “Failure of Aluminum Foams Under Multiaxial Loads,” Int. J. Mech. Sci., 42, pp. 1097–1117. [CrossRef]
Rottler, J., and Robbins, M. O., 2001, “Yield Conditions for Deformation of Amorphous Polymer Glasses,” Phys. Rev. E, 64, p. 051801. [CrossRef]
Brezny, R., and Green, D. J., 1990, “Characterization of Edge Effects in Cellular Materials,” J. Mater. Sci., 25(11), pp. 4571–4578. [CrossRef]
Turner, C. H., and Cowin, S. C., 1988, “Errors Introduced by Off-Axis Measurements of the Elastic Properties of Bone,” J. Biomech., 110, pp. 213–214. [CrossRef]
Triantafillou, T. C., and Gibson, L. J., 1990, “Multiaxial Failure Criteria for Brittle Foams,” Int. J. Mech. Sci., 32(6), pp. 479–496. [CrossRef]
Burr, D. B., 2002, “The Contribution of the Organic Matrix to Bone's Material Properties,” Bone, 31(1), pp. 8–11. [CrossRef] [PubMed]
Milovanovic, P., Potocnik, J., Djonic, D., Nikolic, S., Zivkovic, V., Djuric, M., and Rakocevic, Z., 2012, “Age-Related Deterioration in Trabecular Bone Mechanical Properties at Material Level: Nanoindentation Study of the Femoral Neck in Women by Using AFM,” Exp. Gerontol., 47(2), pp. 154–159. [CrossRef] [PubMed]
Nawathe, S., Juillard, F., and Keaveny, T. M., 2013, “Theoretical Bounds for the Influence of Tissue-Level Ductility on the Apparent-Level Strength of Human Trabecular Bone,” J. Biomech., 46(7), pp. 1293–1299. [CrossRef] [PubMed]
Kabel, J., van Rietbergen, B., Dalstra, M., Odgaard, A., and Huiskes, R., 1999, “The Role of an Effective Isotropic Tissue Modulus in the Elastic Properties of Cancellous Bone,” J. Biomech., 32(7), pp. 673–680. [CrossRef] [PubMed]
Hou, F. J., Lang, S. M., Hoshaw, S. J., Reimann, D. A., and Fyhrie, D. P., 1998, “Human Vertebral Body Apparent and Hard Tissue Stiffness,” J. Biomech., 31(11), pp. 1009–1015. [CrossRef] [PubMed]
Cowin, S., ed., 2001, Bone Mechanics Handbook, CRC Press, Boca Raton, FL.
Gross, T., Pahr, D. H., Peyrin, F., and Zysset, P. K., 2012, “Mineral Heterogeneity has a Minor Influence on the Apparent Elastic Properties of Human Cancellous Bone: a SRμCT-Based Finite Element Study,” Comput. Methods Biomech. Biomed. Eng., 15(11), pp. 1137–1144. [CrossRef]
Easley, S. K., Jekir, M. G., Burghardt, A. J., Li, M., and Keaveny, T. M., 2010, “Contribution of the Intra-Specimen Variations in Tissue Mineralization to PTH- and Raloxifene-Induced Changes in Stiffness of Rat Vertebrae,” Bone, 46(4), pp. 1162–1169. [CrossRef] [PubMed]
Wang, C., Feng, L., and Jasiuk, I., 2009, “Scale and Boundary Conditions Effects on the Apparent Elastic Moduli of Trabecular Bone Modeled as a Periodic Cellular Solid,” ASME J. Biomech. Eng., 131(12), p. 121008. [CrossRef]


Grahic Jump Location
Fig. 1

Definition of the chronological yield point. This graph depicts the stress-normalized strain responses in the longitudinal (solid line) and transverse (dotted line) directions for a single specimen loaded biaxially in the longitudinal and transverse directions. The normalized strains occur at the same instant in time for both responses. For this loading, the ratio of maximum applied strain in the longitudinal to transverse direction was 0.73, and yielding first occurred along the transverse direction, which defined the chronological yield point.

Grahic Jump Location
Fig. 2

The longitudinal (solid) and transverse (open) yield points and the respective yield ellipses for one specimen. Each ellipse is represented by five parameters, the major and minor diameters (a and b, respectively), the coordinates of the center (h,k), and the angle of tilt of the major axis with respect to the horizontal (ϕ). The shaded region bounded by the two intersecting ellipses defines the elastic region.

Grahic Jump Location
Fig. 3

A dual-ellipse fit to the normalized longitudinal and transverse yield strength data pooled from all specimens. The yield strength in each quadrant was normalized by the respective specimen-specific uniaxial strengths.

Grahic Jump Location
Fig. 4

(a) Dual-ellipse fit, (b) single-ellipse fit, and (c) quartic super-ellipse fit to the (same) pooled normalized chronological yield strength data

Grahic Jump Location
Fig. 5

Comparison of dual-ellipse fit (solid) and a quadratic fit (dotted) to the pooled normalized “equivalent” yield strength data

Grahic Jump Location
Fig. 6

Variation of biaxial strength with elastic anisotropy at a constant bone volume fraction (BV/TV) of 0.09, 0.18, and 0.20 for two biaxial loading cases: longitudinal compression and transverse tension in a ratio of 5:1 (dotted line); and longitudinal tension and transverse compression, also in a ratio of 5:1 (solid line). For longitudinal compression and transverse tension, biaxial strength was always defined by the transverse direction. However, for longitudinal tension and transverse compression, the biaxial strength was defined by the longitudinal direction up to an elastic anisotropy of ∼6, beyond which the biaxial strength was defined by the transverse direction, leading to a change in the relation between biaxial strength and elastic anisotropy.

Grahic Jump Location
Fig. 7

Distribution of yielded tissue at the biaxial yield point in thin (∼0.45 mm) longitudinal slices taken from three 5 mm cube specimens subjected to two biaxial loading cases (top row: longitudinal compression and transverse tension in a ratio of about 5:1; bottom row: longitudinal tension and transverse compression, also in a ratio of about 5:1). The percentage value denotes the proportion (percentage) of total tissue yielded in the overall cube specimen at the biaxial yield point. Red regions denote tissue yielded in tension and blue regions denote tissue yielded in compression. BV/TV = bone volume fraction; EA = elastic anisotropy. (The reader is referred to the online version of this article (doi:10.1115/1.4025679) for interpretation of the references to color.)



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