# The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements

- 493 Downloads
- 6 Citations

## Abstract

Boundary conditions (BCs) and sample size affect the measured elastic properties of cancellous bone. Samples too small to be representative appear stiffer under kinematic uniform BCs (KUBCs) than under periodicity-compatible mixed uniform BCs (PMUBCs). To avoid those effects, we propose to determine the effective properties of trabecular bone using an embedded configuration. Cubic samples of various sizes (2.63, 5.29, 7.96, 10.58 and 15.87 mm) were cropped from \(\mu \hbox {CT}\) scans of femoral heads and vertebral bodies. They were converted into \(\mu \hbox {FE}\) models and their stiffness tensor was established via six uniaxial and shear load cases. PMUBCs- and KUBCs-based tensors were determined for each sample. “In situ” stiffness tensors were also evaluated for the embedded configuration, i.e. when the loads were transmitted to the samples via a layer of trabecular bone. The Zysset–Curnier model accounting for bone volume fraction and fabric anisotropy was fitted to those stiffness tensors, and model parameters \(\nu _{0}\) (Poisson’s ratio) \(E_{0}\) and \(\mu _{0}\) (elastic and shear moduli) were compared between sizes. BCs and sample size had little impact on \(\nu _{0}\). However, KUBCs- and PMUBCs-based \(E_{0}\) and \(\mu _{0}\), respectively, decreased and increased with growing size, though convergence was not reached even for our largest samples. Both BCs produced upper and lower bounds for the in situ values that were almost constant across samples dimensions, thus appearing as an approximation of the effective properties. PMUBCs seem also appropriate for mimicking the trabecular core, but they still underestimate its elastic properties (especially in shear) even for nearly orthotropic samples.

## Keywords

Trabecular bone Elastic properties Boundary conditions Micro-finite elements In situ Embedded configuration## Notes

### Acknowledgments

The script manager Medtool (www.dr-pahr.at) was used to generate the \(\mu \hbox {FE}\) models, start the computations and analyse the results. The authors would like to thank Prof. Bert van Rietbergen for sharing the \(\mu \hbox {CT}\) data. Karol Daszkiewicz is supported by grants from the Faculty of Civil and Environmental Engineering, Gdańsk University of Technology. The Swiss National Foundation (Grant No. 143769) and the Gebert Rüf Foundation (GRS-079/14) are also gratefully acknowledged.

### Compliance with ethical standards

### Conflict of interest

The authors have no conflict of interest to report.

## References

- Bevill G, Easley SK, Keaveny TM (2007) Side-artifact errors in yield strength and elastic modulus for human trabecular bone and their dependence on bone volume fraction and anatomic site. J Biomech 40(15):3381–3388CrossRefGoogle Scholar
- Blöß T, Welsch M (2015) RVE procedure for estimating the elastic properties of inhomogeneous microstructures such as bone tissue. Biomedical Technol 74:1–17Google Scholar
- Böhm HJ (2016) A short introduction to basic aspects of continuum micromechanics, CDL–FMD Report 3–1998, TU Wien, Vienna. http://www.ilsb.tuwien.ac.at/links/downloads/ilsbrep206.pdf
- Choi K, Kuhn JL, Ciarelli MJ, Goldstein SA (1990) The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J Biomech 23(11):1103–1113CrossRefGoogle Scholar
- Eringen AC (1999) Microcontinuum field theories I: foundations and solids. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Flaig C (2012) A highly scalable memory efficient multigrid solver for \(\mu \)-finite element analyses. Doctoral dissertation, Eidgenössische Technische Hochschule (ETH) ZürichGoogle Scholar
- Gross T, Pahr DH, Zysset PK (2013) Morphology-elasticity relationships using decreasing fabric information of human trabecular bone from three major anatomical locations. Biomech Model Mechanobiol 12(4):793–800CrossRefGoogle Scholar
- Hadji P, Klein S, Gothe H, Häussler B, Kless Th, Schmidt T, Steinle Th, Verheyen F, Linder R (2013) The epidemiology of osteoporosis-Bone Evaluation Study (BEST): an analysis of routine health insurance data. Dtsch Arztebl Int 110(4):52–7Google Scholar
- Harrigan TP, Mann RW (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J Mater Sci 19(3):761–767CrossRefGoogle Scholar
- Harrison NM, McHugh PE (2010) Comparison of trabecular bone behavior in core and whole bone samples using high-resolution modeling of a vertebral body. Biomech Model Mechanobiol 9(4):469–480CrossRefGoogle Scholar
- Hazanov S, Amieur M (1995) On overall properties of elastic heterogeneous bodies smaller than the representative volume. Int J Eng Sci 33(9):1289–1301CrossRefzbMATHGoogle Scholar
- Hazanov S, Huet C (1994) Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J Mech Phys Solids 42(12):1995–2011MathSciNetCrossRefzbMATHGoogle Scholar
- Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11:127–140MathSciNetCrossRefzbMATHGoogle Scholar
- Hollister SJ, Kikuchi N (1992) A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput Mech 10:73–95CrossRefzbMATHGoogle Scholar
- Hollister SJ, Fyhrie DP, Jepsen KJ, Goldstein SA (1991) Application of homogenization theory to the study of trabecular bone mechanics. J Biomech 24(9):825–839CrossRefGoogle Scholar
- Hütter G (2016) Application of a microstrain continuum to size effects in bending and torsion of foams. Int J Eng Sci 101:81–91CrossRefGoogle Scholar
- Jiang M, Alzebdeh K, Jasiuk I, Ostoja-Starzewski M (2001) Scale and boundary conditions effects in elastic properties of random composites. Acta Mech 148:63–78CrossRefzbMATHGoogle Scholar
- Keaveny TM, McClung MR, Genant HK, Zanchetta JR, Kendler D, Brown JP, Goemaere S, Recknor C, Brandi ML, Eastell R, Kopperdahl DL, Engelke K, Fuerst T, Radcliffe HS, Libanati C (2014) Femoral and vertebral strength improvements in postmenopausal women with osteoporosis treated with denosumab. J Bone Miner Res 29(1):158–165CrossRefGoogle Scholar
- Lakes RS (1983) Size effects and micromechanics of a porous solid. J Mater Sci 18:2572–2580CrossRefGoogle Scholar
- Latypova A, Maquer G, Elankumaran K, Pahr D, Zysset P, Pioletti DP, Terrier A (2016) Identification of elastic properties of human patellae using micro-finite element analysis. J Biomech. doi: 10.1016/j.jbiomech.2016.07.031 Google Scholar
- Lochmüller EM, Pöschl K, Würstlin L, Matsuura M, Müller R, Link TM, Eckstein F (2008) Does thoracic or lumbar spine bone architecture predict vertebral failure strength more accurately than density? Osteoporos Int 19(4):537–545CrossRefGoogle Scholar
- Maquer G, Musy SN, Wandel J, Gross T, Zysset PK (2015) Bone volume fraction and fabric anisotropy are better determinants of trabecular bone stiffness than other morphological variables. J Bone Miner Res 30(6):1000–1008CrossRefGoogle Scholar
- Maquer G, Bürki A, Nuss K, Zysset PK, Tannast M (2016) Head-neck osteoplasty has minor effect on the strength of an ovine Cam-FAI model: in vitro and finite element analyses. Clin Orthop Relat Res. doi: 10.1007/s11999-016-5024-8
- Marangalou JH, Ito K, Cataldi M, Taddei F, van Rietbergen B (2013) A novel approach to estimate trabecular bone anisotropy using a database approach. J Biomech 46(14):2356–2362CrossRefGoogle Scholar
- Ostoja-Starzewski M (1998) Random field models of heterogeneous materials. Int J Solids Struct 35(19):2429–2455CrossRefzbMATHGoogle Scholar
- Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132CrossRefGoogle Scholar
- Pahr DH, Zysset P (2008) Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech Model Mechanobiol 7(6):463–476CrossRefGoogle Scholar
- Panyasantisuk J, Pahr DH, Gross T, Zysset PK (2015) Comparison of mixed and kinematic uniform boundary conditions in homogenized elasticity of femoral trabecular bone using microfinite element analyses. J Biomech Eng 137(1):011002CrossRefGoogle Scholar
- Pecullan S, Gibiansky LV, Torquato S (1999) Scale effects on the elastic behavior of periodic and hierarchical two-dimensional composites. J Mech Phys Solids 47:1509–1542MathSciNetCrossRefzbMATHGoogle Scholar
- Pistoia W, Van Rietbergen B, Lochmüller E-M, Lill C, Eckstein F, Rüegsegger P (2002) Estimation of distal radius failure load with micro-finite element analysis models based on three-dimensional peripheral quantitative computed tomography images. Bone 30(6):842–848CrossRefGoogle Scholar
- Riedler TW, Calvard S (1978) Picture thresholding using an iterative selection method. IEEE Trans Systems Man Cybern 8(8):630–632CrossRefGoogle Scholar
- Synek A, Chevalier Y, Baumbach SF, Pahr DH (2015) The influence of bone density and anisotropy in finite element models of distal radius fracture osteosynthesis: evaluations and comparison to experiments. J Biomech 48(15):4116–4123CrossRefGoogle Scholar
- Syroka-Korol E, Tejchman J, Mróz Z (2013) FE calculations of a deterministic and statistical size effect in concrete under bending within stochastic elasto-plasticity and non-local softening. Eng Struct 48:205–219CrossRefGoogle Scholar
- Un K, Bevill G, Keaveny TM (2006) The effects of side-artifacts on the elastic modulus of trabecular bone. J Biomech 39(11):1955–1963CrossRefGoogle Scholar
- van Rietbergen B, Weinans H, Huiskes R, Odgaard A (1995) A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. J Biomech 28(1):69–81CrossRefGoogle Scholar
- Viceconti M (2015) Biomechanics-based in silico medicine: the manifesto of a new science. J Biomech 48(2):193–194CrossRefGoogle Scholar
- Wang C, Feng L, Jasiuk I (2009) Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid. J Biomech Eng 131(12):121008CrossRefGoogle Scholar
- Wheel MA, Frame JC, Riches PE (2015) Is smaller always stiffer? On size effects in supposedly generalized continua. Int J Solids Struct 67–68:84–92CrossRefGoogle Scholar
- Whitehouse WJ (1974) The quantitative morphology of anisotropic trabecular bone. J Microsc 101(2):153–168CrossRefGoogle Scholar
- Zysset PK (2003) A review of morphology-elasticity relationships in human trabecular bone: theories and experiments. J Biomech 36(10):1469–1485CrossRefGoogle Scholar
- Zysset P, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21(4):243–250CrossRefGoogle Scholar
- Zysset P, Goulet RW, Hollister SJ (1998) A global relationship between trabecular bone morphology and homogenized elastic properties. J Biomech Eng 120(5):640–646CrossRefGoogle Scholar
- Zysset PK, Dall’Ara E, Varga P, Pahr DH (2013) Finite element analysis for prediction of bone strength. BoneKEy reports, 2Google Scholar