Biomedical Engineering Reference
In-Depth Information
1 Introduction
Computer simulations become more and more important for endoprosthetic inves-
tigations of bones. Therefore, a realistic material modeling is required to ensure a
reliable prediction of the inner mechanical stresses. Bones generally consist of can-
cellous bone surrounded by a thin layer of dense compact bone resulting in location-
dependent material properties. The modelling of the microstructure in detail is com-
putational out of reach nowadays. A pointwise homogenization of the stochastic
and heterogeneous microstructure would be beneficial. Thus, a constitutive law is
required that can predict the inhomogeneous elasticity depending on the local bone
density and microstructure.
Direct mechanical measurements for example are performed by Ashmann et al.
[
1
,
2
], Rho et al. [
3
,
4
], Dalstra et al. [
5
] and different regression equations are
proposed. In the mid nineties a new idea was investigated. Real microstructures based
on high resolution CT images are converted into virtual models that could be studied
by FEM simulation. Such studies are performed by Müller [
6
], Ulrich [
7
], Pahr and
Zysset [
8
]. Different issues raise by dealing with the continuummechanics approach.
Ulrich et al. investigated the influence of meshing and element formulation. Pahr and
Zysset compared several sets of boundary conditions regarding the accuracy of the
obtained stiffness of human cancellous bone specimens.
Since the procedure of calculating the anisotropic stiffness matrix seems to be
clear, it lacks of estimating a corresponding isotropic constitutive law. The theory
of micromechanics and homogenization points out to distinguish between apparent
and effective estimates. As a general rule, an apparent estimate is obtained since the
window size is limited. However, a convergence study allows the prediction of an
effective estimate by increasing the window size stepwise (cp. Kanit [
9
,
10
]).
Notwithstanding that the FEMsolution is an approximation by nature, an apparent
estimate should be expected generally due to use of boundary conditions.
This work presents a study of the different influences and proposes a procedure to
calculate effective moduli. Methods are presented to determine the “effectiveness”
of the solution. Plenty different, but stochastically equivalent structures are needed
to study the influences entirely. An algorithm is applied to generate an unlimited
number of varying representative volume elements (RVE).
2 Material and Method
2.1 Generation of Stochastic RVE
Cancellous bone can roughly be seen as a two-phase material consisting of the bone
tissue reinforcement and the interstitial bone marrow matrix. Generally, the effective
elastic properties of such materials are depending on the respective volume fractions,
the elastic properties of both materials and the structural composition.
