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

Rodrigo Moreno*, Örjan Smedby, Dieter H. Pahr

*Corresponding author for this work

Research output: Journal article (peer-reviewed)Journal article

17 Citations (Scopus)


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 (μ FE), which was considered as the gold standard, yielding strong correlations (R2 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 μ FE stiffness predictions.

Original languageEnglish
Pages (from-to)831-844
Number of pages14
JournalBiomechanics and Modeling in Mechanobiology
Issue number4
Publication statusPublished - 01 Aug 2016
Externally publishedYes


  • Fabric tensors
  • Mean intercept length
  • Mechanical properties
  • Stiffness tensor
  • Trabecular bone

ASJC Scopus subject areas

  • Biotechnology
  • Modeling and Simulation
  • Mechanical Engineering


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