Orthotropic properties of cancellous bone modelled as parameterized cellular material

Abstract

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.

Keywords:

• Cancellous bone
• Equivalent microstructure
• Constitutive modelling
• Elastic properties
• Orthotropy
• Fabric

Acknowledgement

The research was supported by Polish Committee for Scientific Research (KBN) under grant no. 3T11F 00727.

Notes

Directions of the x _i axes correspond to local principal directions of mechanical orthotropy of cancellous bone and do not imply orientation of microstructure in the real space. Thus, the words ‘horizontal’ and ‘vertical’ should not be understood literally–they only refer to the so-defined local coordinate system.

The SLD tensor defined by equation (7) is in fact a square root of the tensor defined in Odgaard et al. (1997) (and thus it has a dimension of length, and not squared length). According to the author's observation, the so defined tensor is better correlated with material constants measured for the equivalent microstructures.