Computer Methods in Applied Mechanics and Engineering
ReviewInhomogeneous, orthotropic material model for the cortical structure of long bones modelled on the basis of clinical CT or density data
Introduction
As Wolff stated in his study ”Ueber die innere Architectur der Knochen und ihre Bedeutung für die Frage von Knochenwachsthum” [About the inner architecture of bone and its meaning for the question of bone growth] [1] as early as 1870, bone or bone material is subject to continuous modification processes. It reacts to altered load situations by modifying itself. An implantation, which massively modifies the distribution of forces in the bone, can have both a positive as well as a negative effect on the modification process and on the healing of the bone segments created by a fracture.
With the aid of modern imaging techniques such as computer tomography (CT), it is possible nowadays to gain relatively precise information on the geometrical structure of a bone, varying as it does from one person to another.
In order to gain reliable results regarding the altered force flow with the finite element method, however, two further conditions must be fulfilled. On the one hand, the effects of the body weight and the muscles on, for example, the femur must be known. On the other hand, a material model is required which realistically reproduces the material constants of the bone as well as the directions they are working towards.
The CT provides an image of the highly individual internal composition of the bone material which is closely connected to the material properties of the bone via the scalar field of the density.
In order to render the calculation of the stabilisation system in medicine into a useful and reliable tool, it is necessary to resort to the CT data as they often provide the only information which can be gained from a living bone.
Many recent studies on this subject [2], [3], [4] show that it is indeed possible to derive inhomogeneous, isotropic material properties from these data.
A series of further studies, for example [5], [6], [7], shows that the material properties of bone can be represented by an orthotropic material formulation.
The aim of the present study is to combine these two factors and to develop a procedure, on the basis of the data gained from clinical CT images, which can set up an orthotropic, inhomogeneous material distribution and transfer it automatically to a finite element model.
The first section of the study presents the fundamentals of orthotropic material behaviour. The procedure developed here is divided into two parts because of the information required for an orthotropic material model, i.e. the elasticity matrix and the appertaining orthotropy directions; the second section presents the algorithm which delivers the information on the directions required for the orientation of the orthotropic symmetry planes by analysing the variation – or rather smoothness – of the density field. In the third section, a function is developed which makes the elasticity matrix dependent on the CT values. This function is parameter-dependent and is based on the relationships taken from the literature between orthotropic Young’s moduli and CT data.
Section snippets
Orthotropic material behaviour
For orthotropic material behaviour, the general strain–stress relation simplifies toin Voigt notation.
The compliance matrix in Eq. (1) is formed by three orthogonal symmetry planes which characterise the orthotropic material behaviour [9]. If the load on the orthotropic material is normal to one of these symmetry planes, then only normal strains and no shear strains occur. This means that, in the
Orthotropy directions within the bone material
If a cylindrical coordinate is introduced in the diaphysis section of a long bone, then, as can be seen in Fig. 1, a local tangential coordinate system can be established. In this coordinate system, direct relations between the absolute values of the three orthotropic Young’s moduli as well as the coordinate directions radial, tangential and axial can be found [8].
It can be seen that the direction in which acts coincides with the local axial direction and the direction in which acts
Elements of the elasticity matrices
This section describes the elements of the elasticity matrix in dependence on the CT value. The procedure is based on the experimental work of Rho et al. [12] which states relations for the orthotropic Young’s moduli in dependence on the CT value.
Experimental validation of the orthotropic material model
The first application of the material model was the simulation of the process developed for the calibration of the material parameters and for the experimental validation; the set-up can be seen in Fig. 5.
In the experiment, slices of bone from the diaphysis are subjected to a load in the radial–tangential plane. To compare the simulation and the experiment, the first tests carried out in this study focused on the global displacement and the resulting reaction force.
The next step is to
Results
In this section, first results of the three simulation models are compared with one another to show the differences in the results due to the varying material models. Subsequently, the simulation results of the global displacement (see Fig. 7) are compared with the experimental results.
Summary and outlook
This study presents an orthotropic material model on the basis of clinical CT data, suitable for FE simulations. The simulation results of FE models with orthotropic and isotropic material modelling are compared with experimental results.
The study illustrates that the orthotropic material description of the cortical bone yields considerably more realistic displacement results and should consequently be used to simulate implant-bone systems.
Future work will include validating the cortical model
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Intravoxel bone micromechanics for microCT-based finite element simulations
2013, Journal of BiomechanicsCitation Excerpt :However, they took the homogeneous values directly from the literature (Turner et al., 1990), rather than deriving them by means of spatial averaging as conducted herein. Similarly, Schneider et al. (2009) reported a stiffness overestimation by a factor of 1.6 due to use of homogeneous instead of heterogeneous elastic properties. Since both Baca et al. (2008) and Schneider et al. (2009) introduce bone material properties at the macroscopic rather than at the micron-level extracellular observation scale, we may conclude that the stiffening effect due to neglection of heterogeneous elasticity distribution is independent of the level of microstructure resolution in the Finite Element analyses.
Orientation of orthotropic material properties in a femur FE model: A method based on the principal stresses directions
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