Biomechanics and Modeling in Mechanobiology

, Volume 15, Issue 4, pp 831–844 | Cite as

Prediction of apparent trabecular bone stiffness through fourth-order fabric tensors

  • Rodrigo MorenoEmail author
  • Örjan Smedby
  • Dieter H. Pahr
Original Paper


The apparent stiffness tensor is an important mechanical parameter for characterizing trabecular bone. Previous studies have modeled this parameter as a function of mechanical properties of the tissue, bone density, and a second-order fabric tensor, which encodes both anisotropy and orientation of trabecular bone. Although these models yield strong correlations between observed and predicted stiffness tensors, there is still space for reducing accuracy errors. In this paper, we propose a model that uses fourth-order instead of second-order fabric tensors. First, the totally symmetric part of the stiffness tensor is assumed proportional to the fourth-order fabric tensor in the logarithmic scale. Second, the asymmetric part of the stiffness tensor is derived from relationships among components of the harmonic tensor decomposition of the stiffness tensor. The mean intercept length (MIL), generalized MIL (GMIL), and fourth-order global structure tensor were computed from images acquired through microcomputed tomography of 264 specimens of the femur. The predicted tensors were compared to the stiffness tensors computed by using the micro-finite element method (\(\upmu \)FE), which was considered as the gold standard, yielding strong correlations (\(R^2\) above 0.962). The GMIL tensor yielded the best results among the tested fabric tensors. The Frobenius error, geodesic error, and the error of the norm were reduced by applying the proposed model by 3.75, 0.07, and 3.16 %, respectively, compared to the model by Zysset and Curnier (Mech Mater 21(4):243–250, 1995) with the second-order MIL tensor. From the results, fourth-order fabric tensors are a good alternative to the more expensive \(\upmu \)FE stiffness predictions.


Fabric tensors Stiffness tensor Mechanical properties Trabecular bone Mean intercept length 



This research has been supported by the Swedish Research Council (VR), Grant Nos. 2012-3512 and 2014-6153, and the Swedish Heart-Lung Foundation (HLF), Grant No. 2011-0376.


  1. Advani SG, Tucker CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31(8):751–784CrossRefGoogle Scholar
  2. Auffray N, Kolev B, Petitot M (2014) On anisotropic polynomial relations for the elasticity tensor. J Elast 115(1):77–103MathSciNetCrossRefzbMATHGoogle Scholar
  3. Backus G (1970) A geometrical picture of anisotropic elastic tensors. Rev Geophys 8(3):633–671CrossRefGoogle Scholar
  4. Boehler JP (ed) (1987) Applications of tensor functions in solid mechanics. Springer, ViennazbMATHGoogle Scholar
  5. Campanella A, Tonon ML (1994) A note on the Cauchy relations. Meccanica 29(1):105–108CrossRefzbMATHGoogle Scholar
  6. Cowin S (1985) The relationship between the elasticity tensor and the fabric tensor. Mech Mater 4(2):137–147CrossRefGoogle Scholar
  7. Gibson L (1985) The mechanical behaviour of cancellous bone. J Biomech 18(5):317–328MathSciNetCrossRefGoogle Scholar
  8. 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
  9. 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(14):2356–2362CrossRefGoogle Scholar
  10. Horn BKP (1984) Proc IEEE. Extended Gaussian images 72(12):1671–1686Google Scholar
  11. Huynh DQ (2009) Metrics for 3D rotations: comparison and analysis. J Math Imaging Vis 35(2):155–164MathSciNetCrossRefGoogle Scholar
  12. Kanatani KI (1984) Distribution of directional data and fabric tensors. Int J Eng Sci 22(2):149–164MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kim G, Cole JH, Boskey AL, Baker SP, van der Meulen MC (2014) Reduced tissue-level stiffness and mineralization in osteoporotic cancellous bone. Calcif Tissue Int 95(2):125–131CrossRefGoogle Scholar
  14. Larsson D, Luisier B, Kersh ME, Dall’Ara E, Zysset PK, Pandy MG, Pahr DH (2014) Assessment of transverse isotropy in clinical-level CT images of trabecular bone using the gradient structure tensor. Ann Biomed Eng 42(5):950–959CrossRefGoogle Scholar
  15. Lekadir K, Hazrati-Marangalou J, Hoogendoorn C, Taylor Z, van Rietbergen B, Frangi AF (2015) Statistical estimation of femur micro-architecture using optimal shape and density predictors. J Biomech 48(4):598–603CrossRefGoogle Scholar
  16. Mehrabadi MM, Cowin SC (1990) Eigentensors of linear anisotropic elastic materials. Q J Mech Appl Math 43(1):15–41MathSciNetCrossRefzbMATHGoogle Scholar
  17. Moakher M (2002) Means and averaging in the group of rotations. SIAM J Matrix Anal Appl 24(1):1–16MathSciNetCrossRefzbMATHGoogle Scholar
  18. Moakher M (2005) A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J Matrix Anal Appl 26(3):735–747MathSciNetCrossRefzbMATHGoogle Scholar
  19. Moakher M (2008) Fourth-order cartesian tensors: old and new facts, notions and applications. Q J Mech Appl Math 61(2):181–203MathSciNetCrossRefzbMATHGoogle Scholar
  20. Moakher M, Norris AN (2006) The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J Elast 85(3):215–263MathSciNetCrossRefzbMATHGoogle Scholar
  21. Moreno R, Smedby Ö (2014) Volume-based fabric tensors through Lattice-Boltzmann simulations. In: Proceedings of the international conference on pattern recognit (ICPR), pp 3179–3184Google Scholar
  22. Moreno R, Borga M, Smedby Ö (2012) Generalizing the mean intercept length tensor for gray-level images. Med Phys 39(7):4599–4612Google Scholar
  23. Moreno R, Borga M, Smedby Ö (2014) Techniques for computing fabric tensors: a review. In: Westin CF, Vilanova A, Burgeth B (eds) Visualization and processing of tensors and higher order descriptors for multi-valued data. Springer, Berlin, pp 271–292Google Scholar
  24. Moreno R, Borga M, Klintström E, Brismar T, Smedby Ö (2015) Anisotropy estimation of trabecular bone in gray-scale: Comparison between cone beam and micro computed tomography data. In: Tavares JM, Natal Jorge R (eds) Developments in medical image processing and computational vision. Springer, Berlin, pp 207–220Google Scholar
  25. Pahr DH, Zysset PK (2008) Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech Model Mechanobiol 7(6):463–476CrossRefGoogle Scholar
  26. Tabor Z, Rokita E (2007) Quantifying anisotropy of trabecular bone from gray-level images. Bone 40(4):966–972CrossRefGoogle Scholar
  27. Tjhia CK, Odvina CV, Rao DS, Stover SM, Wang X, Fyhrie DP (2011) Mechanical property and tissue mineral density differences among severely suppressed bone turnover (SSBT) patients, osteoporotic patients, and normal subjects. Bone 49(6):1279–1289CrossRefGoogle Scholar
  28. Whitehouse WJ (1974) The quantitative morphology of anisotropic trabecular bone. J Microsc 101(2):153–168CrossRefGoogle Scholar
  29. Yang G, Kabel J, Van Rietbergen B, Odgaard A, Huiskes R, Cown SC (1999) The anisotropic Hooke’s law for cancellous bone and wood. J Elast 53(2):125–146CrossRefzbMATHGoogle Scholar
  30. Zysset PK (2003) A review of morphology–elasticity relationships in human trabecular bone: theories and experiments. J Biomech 36(10):1469–1485CrossRefGoogle Scholar
  31. Zysset P, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21(4):243–250CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Technology and HealthKTH Royal Institute of TechnologyHuddingeSweden
  2. 2.Institute for Lightweight Design and Structural BiomechanicsTechnical University of ViennaViennaAustria

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