A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

Zong Ping Luo, Kai Nan An

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff's law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone.

Original languageEnglish (US)
Pages (from-to)557-568
Number of pages12
JournalJournal of Mathematical Biology
Issue number6
StatePublished - Jun 1998


  • Apparent density
  • Bone remodeling
  • Fabric tensor
  • Theoretical model

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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