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

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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

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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.
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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.
2018	Zimmermann Y. Développement d'une modélisation par éléments finis pour caractérisation non destructive de la biomécanique osseuse à partir d'images MICRO-CT [Master's thesis]. École polytechnique de Montréal; December 2018.
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.