Abstract
Aim of this study was the determination of transverse-isotropic elastic properties of the human femoral diaphysis and the evaluation of density-elasticity laws from literature with the acquired data. Five specimens of cortical bone were extracted from human cadaveric femora and mechanically tested via three point bending and compression tests, in order to determine the Young’s modulus in proximal-distal and in transverse direction respectively. Bone mineral density was determined via quantitative Computed Tomography for all samples and material properties were calculated according to the available regression laws from literature. Deviations of the calculated and experimentally determined Young’s moduli were analyzed. Some elasticity density laws showed good correlation to the acquired data in anterior-posterior direction, such as the law of Morgan et al. (2003) and the law of Carter & Hayes (1977). The results indicate high importance of an adequate determination of bone mineral density.
Introduction
Patient-specific finite element models help to estimate the load-bearing capacity, the reaction of bone to stresses induced by voluntary or involuntary movements and the interaction between bone and implant. Due to the non-invasive nature, finite element simulations are not only a promising concept for basic research and implant design but also for clinical orthopedics. In particular such models allow a preoperative estimation about the mechanical behavior of bone in response to loads, which may lead to an optimized patient-specific, evidence-based treatment.
The material property assignment to a finite element model is not trivial since bone is inhomogeneous. Its density is dependent on porosity and mineralization [1]. A second aspect is the anisotropy of bone. The material properties are dependent on direction [2]. The orientation changes within bone and the degree of anisotropy varies at specific anatomical regions, due to mechanical stimuli that are induced by physical movement [3, 4]. In general, latter aspects are often neglected and bone is mostly considered to be inhomogeneous but isotropic.
Quantitative CT-scans serve as a basis for the generation of finite element models and the assignment of material properties as they provide detailed geometric information and radiological density of bone. Material laws are described in literature, providing a correlation between bone density and Young’s modulus, mostly in a power regression form. However, universality of these laws is not fulfilled. Thus, the application of a specific, available law to finite element models is uncertain.
Aim of this study was the determination of transverse-isotropic Young’s moduli of the human femoral diaphysis. The acquired data was used to evaluate regression laws from literature, describing the correlation between density and Young’s moduli of bone.
Materials and methods
CT Scans
In order to determine material properties of human cortical bone experimentally, fresh male cadaveric femora were cleaned from soft tissue. The human cadavers were used and dissected in this examination under permission of the „Gesetz über das Leichen-, Bestattungs- und Friedhofswesen (Bestattungsgesetz) des Landes Schleswig-Holstein vom 04.02.2005, Abschnitt II, § 9 (Leichenöffnung, anatomisch)“. In this case it is allowed to dissect the bodies of the donators (Körperspender/in) for scientific and/or educational purposes. The anatomical appearance of the femora was normal and without deformities. The age of the donors varied from 67 to 78 years. No medical treatments were known that could have influenced bone mineralization.
The diaphyses were sawn in frozen condition. Five rectangular samples of 3 mm x 15 mm x 70 mm size were extracted from the medial and lateral proximal mid-diaphysis. The samples were scanned with a five chamber phantom in a multislice CT scanner (SOMATOM Definition AS+, Siemens Medical Solutions, Erlangen, Germany) with following baseline parameters: gantry rotation time 0.3 s, 120 kV, 400 mAs, collimation 64 x 0.6 mm, pitch 0.75. Image reconstruction was performed using a bone convolution kernel (B70 f) and a voxel size of 0.41 x 0.41 x 0.8 mm.
Some authors give advice to scan bone samples in water, in order to reduce beam hardening artifacts [5-7]. In this work the samples were scanned in air as well as in containers filled with a physiological saline solution, in order to investigate whether the solution reduces beam hardening artifacts. From this moment on, the specimens stayed in the container to keep them constantly moistened during the experimental phase.
The phantom contains a solid bone mineral equivalent solution of dipotassium hydrogen phosphate (K2HPO4), absorbing X-rays similarly to bone, so that the density could be calibrated in relation to Hounsfield Units. The bone mineral density of each sample was determined, referring to the manual of the phantom manufacturer [8].
Mechanical Testing
After CT scanning the samples were tested with a universal testing machine machine (Zwick 1456, Zwick, Ulm, Germany). First a varying span three point bending test was performed (see Figure 1). At different spans a bending moment was applied to the samples and the center deflection was recorded. The span varied from L = 15 mm to L = 60 mm, in steps of 5 mm from L = 15 mm to L = 30 mm and steps of 10 mm from L = 30 mm to L = 60 mm. All tests were performed at room temperature (25°C). A preload of 5 N was applied. An incremental linear encoder (resolution 1 µm, MS30-1-LD-2; MEGATRON Elektronik AG & Co., Putzbrunn, Germany) was placed underneath the sample. A 2000 N force transducer (U2A, HBM, Darmstadt, Germany) measured the force during the test. The applied force and the deflection, represented by the vertical movement of the attached linear encoder, were recorded with DIADEM (Version 10, National Instruments, Munich, Germany). The sample was exposed to different loads, according to the specific span. In order to compensate for geometric irregularities four measurements were performed for each span. After the first measurement the sample was rotated by 180° and the same force was applied again. Afterwards the sample was inverted and the two measurements were repeated. Attaining yield strength was avoided. All forces were applied at a constant deformation rate of 0.5 mm/min. For each of the four measurements the slope was determined within the elastic section of the force-deflection curve by a regression line, so that the stability index R² was close to one (i.e. ranging from 0.983 to 0.999). From all four slopes a mean slope was calculated for each span, which enabled the determination of a mean apparent Young’s modulus according to equation 1.
This apparent Young’s modulus includes the deflection due to shear and bending forces and is used to determine the Young’s modulus by rearranging equation 2.
In this work the Young’s modulus was determined graphically. (h/L)² was plotted against (1/Eapp) for each span. From the intersection of the resulting regression line with the ordinate the flexural Young’s modulus could be determined, calculating the reciprocal of this value. The whole procedure was repeated for each sample.
After the three point bending test four smaller specimens of 3 mm x 3 mm x 7 mm size were extracted from each sample. Two of these specimens were orientated in proximal-distal direction, and the other ones were orientated in anterior-posterior direction. A compression test was performed at room temperature (25°C) with the universal material testing machine (Zwick 1456, Zwick, Ulm, Germany) (see Figure 1). A cortical bone sample was placed on the platform of the test rig with defined orientation. A preload of 10 N was applied by the moving plunger. Afterwards, the sample was exposed to a quasi-static uniaxial load, according to the orientation of the sample. A load of 600 N was applied for longitudinal test specimens in proximal-distal direction and a force of 400 N was applied for transverse test specimens in anterior-posterior direction. Attaining yield strength had to be avoided. The forces were applied at a constant deformation rate of 1 mm/min.
The applied force and the vertical deformation were recorded with DIADEM (Version 10, National Instruments, Munich, Germany). Each sample was tested eight times to compensate for geometric irregularities. After each measurement the sample was rotated by 90°. After four measurements it was inverted and again rotated after each measurement.
For the determination of Young’s moduli the vertical deformation was plotted against the applied force for each of the non-destructive measurements. The slope of the resulting deformation curve was determined within the elastic section. A regression line was drawn so that the stability index R² was close to one (i.e. R² ranged from 0.998 to 1). Considering the machine compliance, the stiffness of the sample could be calculated. Then a mean Young’s modulus of all longitudinal test specimens and all transverse test specimens respectively were determined for each sample, according to equation (3).
Evaluation method
Table 1 summarizes the material laws, which were evaluated in this study [9-12]. Using the radiologically obtained bone mineral density in g/cm³ and material laws from literature the material parameters were calculated and evaluated to the measured data. Particularly, each law was investigated including the radiologically determined density of each sample. Therein radiological density was assumed to be equivalent to ash density. For laws, which originally include apparent density, a conversion factor of 0.6 was applied, according to Schileo et al. (2008), who found a constant ash density / apparent density ratio of 0.598 ± 0.004 for cortical bone [13]. The calculations were performed separately for samples scanned in air and samples immersed in physiological saline solution, in order to investigate the effect of beam hardening. The resulting Young’s moduli were compared with the experimentally determined results. Mean deviations were calculated for each law, as well as the standard deviations of single errors.
Results
The calculated mean densities of all samples are summarized in Table 2. The first column shows the sample densities derived from scans in air, the second one illustrates the results of scans with saline solution. The sample densities vary tremendously for both specific measurements. The densities of the scans in air are in general higher and more heterogeneous results are present among all samples.
The resulting Young’s moduli of both different mechanical test methods are summarized in Table 2. Differences can be observed between the three point bending and compression test method, regarding the results of single samples. The resulting Young’s moduli of the three point bending test are within a range of 16178.6 MPa to 19142.4 MPa. Due to the nature of the test method no standard deviations could be obtained. The longitudinal Young’s moduli of the compression test are in a similar range compared with the results of the bending test, varying from 17131.9 ± 3864.9 MPa to 19567.2 ± 2209.7 MPa. In summary, the bending test method resulted in slightly lower values. The transverse Young’s moduli, determined with the compression test method, are lower compared to the longitudinal ones, ranging from 6602.3 ± 443.7 MPa to 12021.6 ± 438.6 MPa. A strong diversity can be noticed among the mean values of each sample.
The resulting Young’s moduli, derived by material laws from literature are diverse. Table 3 shows all deviations of Young’s moduli involving bone mineral density from samples scanned in air and immersed in saline solution. Some material laws from literature are in good agreement to the experimentally determined compressive moduli, which is especially true for the law established by Morgan et al. (2003), resulting in mean deviations of 6.1 ± 3.2 % in air and 17.1 ± 8.3 % in saline solution. In this case the mean deviations of both investigations are even within the range of experimental standard deviations of each investigated sample. The modulus determined with the law of Carter & Hayes (1977) is also in good agreement to the experimentally determined one, resulting in mean deviations of 7.5 ± 3.4 % for samples scanned in air. Higher deviations of 32.9 ± 5.6 % can be observed for samples immersed in saline solution in this case. Application of the law established by Keller (1994) led to worst results compared to the other investigated laws. The only transverse isotropic law, indicated by Wirtz et al. (2000) is in acceptable agreement to the experimentally determined Young’s moduli for samples scanned in air.
Although the resulting flexural Young’s moduli varied from the compressive ones the application of material laws resulted in similar tendencies. The law of Morgan et al. (2003) is again in good agreement for both investigations, representing mean deviations of 7.4 ± 2.6% for samples in air and 15.8 ± 7% for those immersed in saline solution. The transverse law of Wirtz et al. (2000) was again in acceptable agreement for scans in air but did not match the results with densities of samples in saline solution. Again highest deviations could be observed applying the law of Keller (1994). The law of Carter and Hayes (1977) was not applied on the flexural test data, since it considers the strain rate, which could not be determined from the measured data of this test method.
Discussions and conclusion
In this study, Young’s moduli of five different samples were determined via a varying three point bending test and a compression test. The results of single samples were diverse for both tests. One reason is the difference in sample size. Due to the inhomogeneous character of bone locally different moduli can be observed for one specific donor, extracting smaller samples from a bigger one. Another reason may be the different mechanical nature of both tests. Ideally the results should be the same but due to different stress conditions and material alterations, deviations can be observed in practice, which prohibit a direct comparison. In compression tests for instance, friction between sample and test rig can cause a triaxial stress condition instead of a uniaxial stress field in the specimen. The accuracy may thus be decreased compared to tensile tests [14]. In a three point bending test the stress field is per se different to uniaxial tests, since compressive, tensile as well as shear stresses are existent. The stability indexes of the resulting regression lines were in general not confidable in case of the varying three point bending test. An error analysis demonstrated huge systematical errors of this test method. Therefore the results have to be regarded with suspicion.
Evaluation of material laws from literature showed adequate regression functions in proximal-distal direction for measurements of the human femoral diaphysis. Comparing the calculated Young’s moduli with the measured data, best agreement of all measurements was observed using the regression law of Morgan et al. (2003), although it was established for trabecular bone, investigating specimens from the femoral neck [10]. However, good agreement of this law was also confirmed by Schileo et al. (2007), who evaluated finite element models of the femur, which were generated with different elasticity-density laws. In their study best prediction could be gained with the implementation of the law derived by Morgan et al. (2003) [15]. The law was established by loading the specimens along the principal bone orientation, which implies a lack of information with respect to material mapping of anisotropic finite element models. The law of Carter & Hayes (1977) was also in good agreement, especially for scans in air. However, the scans in saline solution indicate, that this law is less robust compared to that of Morgan et al. (2003). It was established for samples from the human tibial plateau. Despite that fact it was chosen for this study, since it is one of the most applied elasticity-density laws for patient-specific finite element models [11, 15]. It remains unclear why laws established with trabecular bone of different anatomical locations performed better than those established with cortical bone.
Wirtz et al. (2000) give a formulation for the transverse-isotropic material behavior of cortical bone. The law in axial load direction was derived by pooling values of two different material laws, which were established by Abendschein et al. (1970) and Lotz et al. (1991). The law in transverse direction was taken from Lotz et al. (1991) [12]. The varying experimental results of the transverse Young’s moduli determined in this study may indicate the difficulty of establishing transversally isotropic material laws. Apparently the transverse modulus is more vulnerable to patient-specific anatomical conditions and motions.
The similar tendencies, which can be observed between compressive and flexural experiments, indicate that the density affects the application of material laws in a higher degree, rather than the determination of Young’s modulus. Generally, densities of scans in air led to better agreement with established material laws, although the saline solution was considered to reduce beam hardening artifacts and thereby to improve the results. However, unexpectedly more beam hardening could be observed in the scans with saline solution, since the samples were stored in a rectangular container during scanning. Long edges of this container enforced beam hardening. Consequently, some small shadows in the rod area of the calibration phantom could be observed, which might have biased the determination of the radiological density. Thus, it is of great importance to use cylindrical containers or containers of sufficient size, in order to avoid artifacts in the regions of interest.
One further limitation of this study is the small amount of samples, reducing the statistical probability of a significant outcome. Another limitation is the rough assumption of ash density, being equal to radiological density. Schileo et al. (2008) showed underestimation of ρash in low-density regions, and overestimation of ρash in high-density regions. Obviously phantoms do not perfectly imitate bone as compositional differences exist in the mineralized phase of bone compared to the phantom. Furthermore, bone is inhomogeneous whereas phantoms are made of homogenous materials. The factor of 0.6 was derived with a calcium hydroxyapatite phantom. The application of this factor on results of a dipotassium hydrogen phosphate phantom may be questionable [13]. Furthermore, the material laws were evaluated at a specific anatomic location, i.e. the proximal mid-diaphysis of the human femur. It is known that the material behavior of bone depends on its anatomical location [10, 16]. This aspect may also affect the validity of the evaluation outlined in this study, when the results are transferred to an entire femur.
In summary, material laws from literature were evaluated by two different mechanical test methods. Best results were obtained with the law indicated by Morgan et al. (2003). Since the determination of density affected the validation of available laws most, it is of highest importance for material mapping in patient-specific finite element models. Thus, further studies will focus on best conditions for quantitative Computed Tomography. One aspect in this context will be the application of material laws on densities determined in vivo, which is essential for the generation of patient-specific finite element models. In this context, the effect of surrounding tissue and CT reconstructions will be investigated.
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