Tissue modulus calculated from beam theory is biased by bone size and geometry: implications for the use of three-point bending tests to determine bone tissue modulus

wbldb

van Lenthe, G. Harry¹; Voide, Romain¹; Boyd, Steven K.²; Müller, Ralph¹

Tissue modulus calculated from beam theory is biased by bone size and geometry: implications for the use of three-point bending tests to determine bone tissue modulus

Bone. October 2008;43(4):717-723

©2008 Elsevier Inc. All rights reserved.

Affiliations

¹Institute for Biomechanics, ETH Zürich, Zürich, Switzerland

²Department of Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Calgary, Canada
Abstract and keywords

Current practice to determine bone tissue modulus of murine cortical bone is to estimate it from three-point bending tests, using Euler–Bernoulli beam theory. However, murine femora are not perfect beams; hence, results can be inaccurate. Our aim was to assess the accuracy of beam theory, which we tested for two commonly used inbred strains of mice, C57BL/6 (B6) and C3H/He (C3H).

We measured the three-dimensional structure of male and female B6 and C3H femora (N = 20/group) by means of micro-computed tomography. For each femur five micro-finite element (micro-FE) models were created that simulated three-point bending tests with varying distances between the supports. Tissue modulus was calculated from beam theory using micro-FE results. The accuracy of beam theory was assessed by comparing the beam theory-derived moduli with the modulus as used in the micro-FE analyses. An additional set of fresh-frozen femora (10 B6 and 12 C3H) was biomechanically tested and subjected to the same micro-FE analyses. These combined experimental–computational analyses enabled an unbiased assessment of specimen-specific tissue modulus.

We found that by using beam theory, tissue modulus was underestimated for all femora. Femoral geometry and size had strong effects on beam theory-derived tissue moduli. Owing to their relatively thin cortex, underestimation was markedly higher for B6 than for C3H. Underestimation was dependent on support width in a strain-specific manner. From our combined experimental–computational approach we calculated tissue moduli of 12.0 ± 1.3 GPa and 13.4 ± 2.1 GPa for B6 and C3H, respectively.

We conclude that tissue moduli in murine femora are strongly underestimated when calculated from beam theory. Using image-based micro-FE analyses we could precisely quantify this underestimation. We showed that previously reported murine inbred strain-specific differences in tissue modulus are largely an effect of geometric differences, not accounted for by beam theory. We suggest a re-evaluation of the tissue properties obtained from three-point bending tests, especially in mouse genetics.

Keywords: Tissue modulus; Biomechanical testing; Finite element analysis; Mouse; _Femur
Links

DOI: 10.1016/j.bone.2008.06.008

PubMed: 18639658

WoS: 000259711600013
History

Received: 2007-12-20

Revised: 2008-06-13

Accepted: 2008-06-19

Online: 2008-06-27

Cited Works (18)

Year	Entry
1996	Rüegsegger P, Koller B, Müller R. A microtomographic system for the nondestructive evaluation of bone architecture. Calcif Tiss Int. 1996;58(1):24-29.
2006	van Lenthe GH, Stauber M, Müller R. Specimen-specific beam models for fast and accurate prediction of human trabecular bone mechanical properties. Bone. December 2006;39(6):1182-1189.
2000	Turner CH, Hsieh Y, Müller R, Bouxsein ML, Baylink DJ, Rosen CJ, Grynpas MD, Donahue LR, Beamer WG. Genetic regulation of cortical and trabecular bone strength and microstructure in inbred strains of mice. J Bone Miner Res. June 2000;15(6):1126-1131.
2001	Turner CH, Burr DB. Experimental techniques for bone mechanics. In: Cowin SC, ed. Bone Mechanics Handbook. 2nd ed. Boca Raton, FL: CRC Press; 2001:chap 7.
1999	Jacobs CR, Davis BR, Rieger CJ, Francis JJ, Saad M, Fyhrie DP. The impact of boundary conditions and mesh size on the accuracy of cancellous bone tissue modulus determination using large-scale finite-element modeling. J Biomech. November 1999;32(11):1159-1164.
1997	Jepsen KJ, Schaffler MB, Kuhn JL, Goulet RW, Bonadio J, Goldstein SA. Type I collagen mutation alters the strength and fatigue behavior of Mov13 cortical tissue. J Biomech. November–December 1997;30(11-12):1141-1147.
2000	Guo XE, Goldstein SA. Vertebral trabecular bone microscopic tissue elastic modulus and hardness do not change in ovariectomized rats. J Orthop Res. March 2000;18(2):333-336.
2004	Silva MJ, Brodt MD, Fan Z, Rho J-Y. Nanoindentation and whole-bone bending estimates of material properties in bones from the senescence accelerated mouse SAMP6. J Biomech. November 2004;37(11):1639-1646.
2005	Schriefer JL, Robling AG, Warden SJ, Fournier AJ, Mason JJ, Turner CH. A comparison of mechanical properties derived from multiple skeletal sites in mice. J Biomech. March 2005;38(3):467-475.
1999	Kabel J, van Rietbergen B, Dalstra M, Odgaard A, Huiskes R. The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone. J Biomech. 1999;32(7):673-680.
1999	Niebur GL, Yuen JC, Hsia AC, Keaveny TM. Convergence behavior of high-resolution finite element models of trabecular bone. J Biomech Eng. December 1999;121(6):629-635.
1995	van Rietbergen B, Weinans H, Huiskes R, Odgaard A. A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. J Biomech. January 1995;28(1):69-81.
2002	Boyd SK, Müller R, Zernicke RF. Mechanical and architectural bone adaptation in early stage experimental osteoarthritis. J Bone Miner Res. April 2002;17(4):687-694.
2007	Ramasamy JG, Akkus O. Local variations in the micromechanical properties of mouse femur: the involvement of collagen fiber orientation and mineralization. J Biomech. 2007;40(4):910-918.
1997	Rho J-Y, Tsui TY, Pharr GM. Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials. 1997;18(20):1325-1330.
2007	Chevalier Y, Pahr D, Allmer H, Charlebois M, Zysset P. Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation. J Biomech. 2007;40(15):3333-3340.
2000	Akhter MP, Iwaniec UT, Covey MA, Cullen DM, Kimmel DB, Recker RR. Genetic variations in bone density, histomorphometry, and strength in mice. Calcif Tiss Int. October 2000;67(4):337-344.
2002	Robling AG, Turner CH. Mechanotransduction in bone: genetic effects on mechanosensitivity in mice. Bone. November 2002;31(5):562-569.

Cited By (20)

Year	Entry
2011	Turnbull TL, Gargac JA, Niebur GL, Roeder RK. Detection of fatigue microdamage in whole rat femora using contrast-enhanced micro-computed tomography. J Biomech. September 2, 2011;44(13):2395-2400;
2015	Jepsen KJ, Silva MJ, Vashishth D, Guo XE, van der Meulen MCH. Establishing biomechanical mechanisms in mouse models: practical guidelines for systematically evaluating phenotypic changes in the diaphyses of long bones. J Bone Miner Res. June 2015;30(6):951-966.
2013	Carretta R, Luisier B, Bernoulli D, Stüssi E, Müller R, Lorenzetti S. Novel method to analyze post-yield mechanical properties at trabecular bone tissue level. J Mech Behav Biomed Mater. 2013;20:6-18.
2021	Collins CJ, Yang B, Crenshaw TD, Ploeg H-L. Evaluation of experimental, analytical, and computational methods to determine long-bone bending stiffness. J Mech Behav Biomed Mater. March 2021;115:104253.
2011	Lorenzetti S, Carretta R, Müller R, Stüssi E. A new device and method for measuring the elastic modulus of single trabeculae. Med Eng Phys. October 2011;33(8):993-1000.
2024	Szabo E. Insights Into the Mechanical Behavior of Maturing Cortical Bone: A Porcine Model [PhD thesis]. Cleveland, OH: Case Western Reserve University; May 2024.
2012	Webb BT. Persistent Bovine Viral Diarrhea Virus Infection in the Bovine Fetus: Morphogenesis of Skeletal Lesions and Innate Immune Response [PhD thesis]. Colorado State University; 2012.
2013	Kim G. The Effects of Mineralization and Crystallinity on the Mechanical Behavior of Bone [PhD thesis]. Ithaca, NY: Cornell University; May 2013.
2010	Müller T. Bioimaging and Biomechanics of Bone Competence and Implant Stability [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2010.
2014	Carretta R. Post-Yield Mechanics and Material Composition of Single Trabeculae: A Combined Experimental and Modelling Approach [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2014.
2018	Muckatira SK. Error Estimation in Whole Bone Microstrain Measurement When Using Digital Volume Correlation [Master's thesis]. University of Illinois at Urbana-Champaign; 2018.
2012	Druchok C. The Effect of a High-Fat Diet on Bone Strain in Adult Rat Femurs [Master's thesis]. Hamilton, ON: McMaster University; October 2012.
2013	Turnbull TL. The Spatial Distribution of Fatigue Microdamage Accumulation in Cortical Bone and Factors Influencing Fracture Risk [PhD thesis]. University of Notre Dame; July 2013.
2015	Gargac J. Evaluation of Bone Healing, Damage, and Adaptation Using Computational Modeling and Image Processing Techniques [PhD thesis]. University of Notre Dame; July 2015.
2021	Zojaji M. Quantifying the Accuracy of Euler-Bernoulli and Timoshenko Beam Theories for Long Bones in Three- and Four-Point Bend Tests With Finite Element Analysis [Master's thesis]. Queen's University; September 2021.
2019	Beltejar M-JG. Understanding the Genetic Basis for Bone-Matrix Quality [PhD thesis]. University of Rochester; 2019.
2011	Kourtis LC. Computed Tomography Based Estimation of Bone Rigidity and Strength Under Combined Bending and Torsion [PhD thesis]. Stanford University; September 2011.
2011	Kristensen E. The Role of Growth Hormone in the Development of Bone Microarchitecture, Craniofacial Shape and Bone Strength [PhD thesis]. Calgary, AB: University of Calgary; February 2011.
2014	Sinder BP. Sclerostin Antibody as a Treatment for Osteogenesis Imperfecta [PhD thesis]. University of Michigan; 2014.
2016	Collins CJ. A Comprehensive Evaluation of Methods to Determine Mechanical Properties of Long Bone [PhD thesis]. University of Wisconsin – Madison; 2016.