Continuum finite element (FE) models of bones and bone-implant configurations are usually based on clinical CT scans. In virtually all of these models, material properties assigned to the bone elements are chosen as isotropic. It has been shown, however, that cancellous bone can be highly anisotropic and that its elastic behavior is best described as orthotropic. Material models have been proposed to derive the orthotropic elastic constants from measurements of density and a fabric tensor. The use of such relationships in FE models derived from CT scans, however, is hampered by the fact that the measurement of such a fabric tensor is not possible from clinical CT images since the resolution of such images is not good enough to resolve the trabecular micro-architecture. In this study, we explore an alternative approach that is based on the paradigm that bone adapts its micro-architecture to the loading conditions, hence that fabric and stress tensors should be aligned and correlated. With this approach, the eigenvectors and eigenvalues of the element continuum-level stress tensor are used as an estimate of the element fabric tensor, from which the orthotropic material properties then are derived. Using an iterative procedure, element orthotropic material properties and fabric tensors are updated until a converged situation is reached. The goals of this study were to investigate the feasibility and accuracy of such an iterative approach to derive orthotropic material properties for a human proximal femur and to investigate whether models derived in this way can provide more accurate results than isotropic models. Results were compared to those obtained from models of the same femurs for which the fabric was measured from micro-CT scans. It was found that the iterative approach could well estimate the orientation of the fabric principal directions. When comparing the stress/damage values in the models with material properties based on estimated and measured fabric tensors, the differences were not significant, suggesting that the material properties based on the estimated fabric tensor well reflected those based on the measured fabric tensor. Errors were less than those obtained when using isotropic models. It is concluded that this novel approach can provide a reasonable estimate of anisotropic material properties of cancellous bone. We expect that this approach can lead to more accurate results in particular for models used to study implants, which are usually anchored in highly anisotropic cancellous bone regions.
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Beaupré GS, Orr TE, Carter DR (1990) An approach for time-dependent bone modeling and remodeling–theoretical development. J Orthop Res 8:651–661
Bergmann G, Graichen F, Rohlmann A, Bender A, Heinlein B, Duda GN, Heller MO, Morlock MM (2010) Realistic loads for testing hip implants. Bio-Med Mater Eng 20:65–75
Campoli G, Weinans H, Zadpoor AA (2012) Computational load estimation of the femur. J Mech Beh Biomed Mater 10:108–119
Carter DR, Hayes WC (1976) Bone compressive strength: the influence of density and strain rate. Science 194:1174–1176
Charlebois M, Jirasek M, Zysset PK (2010) A nonlocal constitutive model for trabecular bone softening in compression. Biomech Model Mechanobiol 9:597–611
Christen P, van Rietbergen B, Lambers F, Müller R, Ito K (2012) Bone morphology allows estimation of loading history in a murine model of bone adaptation. Biomech Model Mechanobiol 11:483–492
Courtney AC, Wachtel EF, Myers ER, Hayes WC (1995) Age-related reductions in the strength of the femur tested in a fall-loading configuration. J Bone Joint Surg Am 77:387–395
Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mech Mater 4:137–147
Fyhrie DP, Carter DR (1986) A unifying principle relating stress to trabecular bone morphology. J Orthop Res 4:304–317
Harrigan TP, Mann RW (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J Mater Sci 19:761–767
Harrigan TP, Jasty M, Mann RW, Harris WH (1988) Limitations of the continuum assumption in cancellous bone. J Biomech 21:269–275
Hazrati Marangalou J, Ito K, van Rietbergen B (2012) A new approach to determine the accuracy of morphology-elasticity relationships in continuum FE analyses of human proximal femur. J Biomech 45:2884–2892
Hazrati Marangalou J, Ito K, Cataldi M, Taddei F, van Rietbergen B (2013) A novel approach to estimate trabecular bone anisotropy using a database approach. J Biomech 46:2356–2362
Hazrati Marangalou J, Eckstein F, Kuhn V, Ito K, Cataldi M, Taddei F, Rietbergen B (2014a) Locally measured microstructural parameters are better associated with vertebral strength than whole bone density. Osteoporos Int 25:1285–1296
Hazrati Marangalou J, Ito K, Taddei F, van Rietbergen B (2014b) Inter-individual variability of bone density and morphology distribution in the proximal femur and T12 vertebra. Bone 60:213–220
Heller MO, Bergmann G, Kassi JP, Claes L, Haas NP, Duda GN (2005) Determination of muscle loading at the hip joint for use in pre-clinical testing. J Biomech 38:1155–1163
Jacobs CR, Simo JC, Beaupre GS, Carter DR (1997) Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations. J Biomech 30:603–613
Keyak JH (2001) Improved prediction of proximal femoral fracture load using nonlinear finite element models. Med Eng Phys 23:165–173
Lepore F, Brun C, Yi-Yu C, Ming-Chang C, Dutton RA, Hayashi KM, Luders E, Lopez OL, Aizenstein HJ, Toga AW, Becker JT, Thompson PM (2008) Generalized tensor-based morphometry of HIV/AIDS using multivariate statistics on deformation tensors. IEEE Trans Med Imaging 27:129–141
Morgan EF, Bayraktar HH, Keaveny TM (2003) Trabecular bone modulus-density relationships depend on anatomic site. J Biomech 36:897–904
Odgaard A (1997) Three-dimensional methods for quantification of cancellous bone architecture. Bone 20:315–328
Odgaard A, Kabel J, van Rietbergen B, Dalstra M, Huiskes R (1997) Fabric and elastic principal directions of cancellous bone are closely related. J Biomech 30:487–495
Pahr DH, Zysset PK (2009) A comparison of enhanced continuum FE with micro FE models of human vertebral bodies. J Biomech 42: 455–462
Rice JC, Cowin SC, Bowman JA (1988) On the dependence of the elasticity and strength of cancellous bone on apparent density. J Biomech 21:155–168
Schwartzman A, Dougherty RF, Taylor JE (2010) Group comparison of eigenvalues and eigenvectors of diffusion tensors. J Am Stat Assoc 105:588–599
Taddei F, Schileo E, Helgason B, Cristofolini L, Viceconti M (2007) The material mapping strategy influences the accuracy of CT-based finite element models of bones: an evaluation against experimental measurements. Med Eng Phys 29:973–979
Turner CH (1992) On Wolff’s law of trabecular architecture. J Biomech 25:1–9
Vander Sloten J, Van der Perre G (1989) Trabecular structure compared to stress trajectories in the proximal femur and the calcaneus. J Biomed Eng 11:203–208
Verhulp E, van Rietbergen B, Huiskes R (2008) Load distribution in the healthy and osteoporotic human proximal femur during a fall to the side. Bone 42:30–35
Whitcher B, Wisco JJ, Hadjikhani N, Tuch DS (2007) Statistical group comparison of diffusion tensors via multivariate hypothesis testing. Magn Reson Med 57:1065–1074
Yang G, Kabel J, Van Rietbergen B, Odgaard A, Huiskes R, Cown S (1998) The anisotropic Hooke’s law for cancellous bone and wood. J Elast 53:125–146
Zysset PK, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21:243–250
Zysset PK, Goulet RW, Hollister SJ (1998) A global relationship between trabecular bone morphology and homogenized elastic properties. J Biomech Eng 120:640–646
Funding from the European Union for the osteoporotic virtual physiological human project (VPHOP FP7-ICT2008-223865) is gratefully acknowledged.
Conflict of interest
Bert van Rietbergen is a consultant for Scanco Medical AG.
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Hazrati Marangalou, J., Ito, K. & van Rietbergen, B. A novel approach to estimate trabecular bone anisotropy from stress tensors. Biomech Model Mechanobiol 14, 39–48 (2015). https://doi.org/10.1007/s10237-014-0584-6
- Trabecular bone
- Fabric tensor
- Finite element analysis