Determination of orthotropic bone elastic constants using FEA and modal analysis

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Abstract

Finite element models have been widely employed in an effort to quantify the stress and strain distribution around implanted prostheses and to explore the influence of these distributions on their long-term stability. In order to provide meaningful predictions, such models must contain an appropriate reflection of mechanical properties. Detailed geometrical and density information is now readily available from CT scanning. However, despite the use of phantoms, a method of determining mechanical properties (or elastic constants) from bone density has yet to be made available in a usable form.

In this study, a cadaveric bone was CT scanned and its natural frequencies were measured using modal analysis. Using the geometry obtained from the CT scan data, a finite element mesh was created with the distribution of density established by matching the mass of the FE bone model with the mass of the cadaveric bone. The maximum values of the orthotropic elastic constants were then established by matching the predictions from FE modal analyses to the experimental natural frequencies, giving a maximum error of 7.8% over 4 modes of vibration. Finally, the elastic constants of the bone derived from the analyses were compared with those measured using ultrasound techniques. This produced a difference of <1% for both the maximum density and axial Young's Modulus. This study has thereby produced an orthotropic finite element model of a human femur. More importantly, however, is the implication that it is possible to create a valid FE model by simply comparing the FE results with the measured resonant frequency of the CT scanned bone.

Introduction

Material constants of an appropriate accuracy are an essential pre-requisite for any successful finite element (FE) analysis. FE models of bone have been widely employed to explore the long-term effects of the implantation of hip prostheses. These explorations have used FE approaches together with algorithms to predict bone remodelling (e.g. Orr et al., 1990; van Rietbergen et al., 1993; Smolinski and Rubash, 1992; Skinner et al., 1994; Huiskes, 1995; Weinans and Sumner, 1997). Determination of the geometry and distribution of elastic constants incorporated into such FE models therefore needs to be performed before the models can be used with any confidence. This validation is commonly achieved by comparing some aspect of experimental model behaviour with corresponding FE predictions: Strain gauge measurements (e.g. Little et al., 1986; Dalstra et al., 1995) and modal analysis (e.g. Hight et al., 1980; Khalil et al., 1981; Couteau et al., 1998a) have both been used in this capacity.

Strain gauges are capable of measuring average values of strain over the gauge length of the device and can thus provide a basis for comparison with numerical predictions, but only at particular locations on the bone surface. In contrast to the local nature of the validation in strain gauge studies, modal analysis offers the possibility of a global validation for an entire structure. The effect of a variation in the Young's Modulus of the cortical and cancellous bone of a human tibia has been explored by Hobatho et al. (1991) using an isotropic materials model. Using CT scan supplied geometry, values of Young's modulus for the two different types of bone were varied within the range of magnitudes from the literature. It was found that very good agreement between FE predictions and experimental values of natural frequency could be obtained in this way. Unfortunately, the distribution of density for the actual bone tested was not incorporated into the model. Couteau et al. (1998b) compared experiments and numerical results for the first 4 natural frequencies of a human femur. Errors of up to 8.1% were reported.

From the literature, it is well known that modal analysis is capable of validating existing FE models of structures with complex materials distributions (Couteau et al (1998a), Couteau et al (1998b)) but these techniques have never been used to derive the properties of bone. The material constants of bone are widely recognised as being better modelled as orthotropic rather than isotropic (Rho, 1996). The aim of this study therefore was to both validate our orthotropic FE model and determine the quality of this validation using the above techniques. In addition, this study aimed to establish whether modal analysis is capable of determining the distribution of material constants within bone structures.

Section snippets

Methodology

A fresh, frozen, human cadaveric femur was prepared by removing all external soft tissues and then weighed. The bone was mounted horizontally by its metaphyses using soft elastic straps (resonant frequency <10 Hz, mass 2 g) in order to simulate free–free boundary conditions during vibration. A unidirectional piezoelectric accelerometer (Brüel&Kjaer 4375, weight 2.3 g) was fixed to the surface of the bone using beeswax. The structure was then excited by hitting the femoral head normal to its

Results

The natural frequencies of the intact bone were measured and are shown in Table 3. These frequencies were similar to Couteau et al. (1998b), whose study showed 301.6 Hz for the resonant frequency (cf. 285 Hz in this study). The mass of the cadaveric bone was found to be 622 g, and therefore the corrected relationship for effective density based upon the mass of this particular bone wasρeff=0.464HU+1000(kg/m3).

An FE bone mass of 622 g was predicted for a maximum effective density of 1940 kg/m3, when

Discussion

This study has performed an FE and experimental modal analysis and compared the determined elastic constants with ultrasound results. It has been shown that modal analysis can be used, together with FE models incorporating CT scan data, to determine the distribution of elastic constants throughout a long bone.

This study has assumed that the FE results should be matched to the fundamental (resonant) frequency of the bone. Modal frequencies have an increased error with larger accelerometer

Acknowledgements

Useful discussions with Dr. H. Bereiter (Raetisches Kantons- und Regionalspital, Abteilung Orthopaedie, Chur, Switzerland) and Dr. H. Jacob (Universitaetsklinik Balgrist, Zurich, Switzerland) are gratefully acknowledged. Thanks are also due to Dr. Y. Deger of Sulzer Innotec AG, Winterthur, Switzerland, who determined the Modal Assurance Criterion values. The funding for this study was provided by Sulzer Orthopedics AG, Winterthur, Switzerland.

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