Biomech.Model.Mechanobiol. 7, 463-476, 2008
D.H. Pahr, P.K. Zysset
Institute of Lightweight Design and Structural Biomechanics,
TU Wien, Vienna, Austria
High-resolution finite element models of trabecular bone can be used to study
trabecular structure-function relationships, elasticity, multiaxial strength,
and tissue remodelling in more detail than experiments.
Beside effects of the model size, scan/analysis resolution, segmentation
process, etc., the type of the applied boundary conditions (BCs) have a strong
influence on the predicted elastic properties.
Appropriate BCs have to be applied on hexahedral digital finite element models
in order to obtain effective elastic properties.
Homogeneous displacement BCs as proposed by Van Rietbergen et al.
(J Biomech 29(12):1653-1657, 1996) lead to "apparent" rather than to
"effective" elastic properties.
This study provides some answers concerning such differences by comparing
various BC types (uniform displacement, mixed BCs, periodic BCs), different
volume element definitions (original and mirrored models), and several bone
volume fractions (BVTV ranging from 6.5 to 37.6%).
First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385-394, 1998)
are theoretically extended to shear loading of a porous media. Second, six
human bone samples are analyzed, their orthotropic Young's moduli, shear
moduli, and Poisson's ratios computed and compared.
It is found that the proposed mixed BCs give exactly the same effective elastic
properties as periodic BCs if a periodic and orthotropic micro-structured
material is used and thus denoted as "periodicity compatible" mixed uniform BCs
As bone samples were shown to be nearly orthotropic for volume element side
lengths ≤ 5 mm the proposed mixed BCs turn out to be the best choice because
they give again essentially the same overall elastic properties as periodic
For bone samples of smaller dimensions (< 5 mm) with a strong anisotropy
(beyond orthotropy) uniform displacement BCs remain applicable but they can
significantly overestimate the effective stiffness.