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ARTICLE IN PRESS Journal of Biomechanics 40 (2007) 252–264 www.elsevier.com/locate/jbiomech www.JBiomech.com Anisotropic Poisson’s ratio and compression modulus of cortical bone determined by speckle interferometry R. Shahara,, P. Zaslanskyb, M. Barakb, A.A. Friesemc, J.D. Curreyd, S. Weinerb a Koret School of Veterinary Medicine, The Hebrew University of Jerusalem, P.O. Box 12, 76100 Rehorot, Israel b Department of Structural Biology, Weizmann Institute of Science, Israel c Department of Physics of Complex Systems, Weizmann Institute of Science, Israel d Department of Biology, University of York, UK Accepted 16 January 2006 Abstract Young’s modulus and Poisson’s ratios of 6 mm-sized cubes of equine cortical bone were measured in compression using a micromechanical loading device. Surface displacements were determined by electronic speckle pattern-correlation interferometry. This method allows for non-destructive testing of very small samples in water. Analyses of standard materials showed that the method is accurate and precise for determining both Young’s modulus and Poisson’s ratio. Material properties were determined concurrently in three orthogonal anatomic directions (axial, radial and transverse). Young’s modulus values were found to be anisotropic and consistent with values of equine cortical bone reported in the literature. Poisson’s ratios were also found to be anisotropic, but lower than those previously reported. Poisson’s ratios for the radial–transverse and transverse–radial directions were 0:15 0:02, for the axial–transverse and axial–radial directions 0:19 0:04, and for the transverse–axial and radial–axial direction 0:09 0:02 (mean7SD). Cubes located only millimetres apart had signiﬁcantly different elastic properties, showing that signiﬁcant spatial variation occurs in equine cortical bone. r 2006 Elsevier Ltd. All rights reserved. Keywords: Bone; Mechanical properties; Interferometry; Poisson’s ratio; ESPI 1. Introduction Cortical bone is anisotropic, with the elastic modulus in the axial direction being signiﬁcantly higher than in the transverse and radial directions (Reilly and Burstein, 1975, Taylor et al., 2002, Dong and Guo, 2004, Iyo et al., 2004). In fact the mechanical properties of bone are affected by many aspects of its complex structure (Weiner and Wagner, 1998), and in particular by the mineral content (Currey, 2002). Few studies have attempted to correlate structure with function at the micron to millimeter meso-scale Corresponding author. Tel.: +972 9 7433968; fax: +972 9 7488994. E-mail address: [email protected] (R. Shahar). 0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.01.021 (Zysset et al., 1999, Liu et al., 1999, Turner et al., 1999, 2000; Hengsberger et al., 2003; Enstrom, et al., 2001). Since many structural differences between (and even within) various types of cortical bone are found at that length scale, the study of the mechanical properties using millimeter-sized samples is of great importance. However testing such samples is complicated, posing many technical challenges. Various experimental methods have been reported for determining the elastic properties of cortical bone, with sample sizes ranging from several centimetres (Reilly et al., 1974, Reilly and Burstein, 1975) to single osteons with dimensions of hundreds of micrometres (Ascenzi and Bonucci, 1968). Most methods were based on loading relatively bulky bone samples in material testing machines, and recording load–displacement curves for loads such as tension, ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 compression, torsion and bending. In tension and compression experiments, the slope of the stress–strain curve within the elastic region was used to estimate the Young’s modulus of the bone. Most tests are performed by applying a controlled deformation rate, and often progress continuously until failure. Such experiments are usually used to also provide information regarding the yield point (the point at which the behaviour of the sample ceases to be linearly elastic), load and work to failure and ultimate strain. However, since these tests are destructive, often only one estimate of modulus can be obtained from each sample, thus experimental precision cannot be determined. Furthermore, since the use of large and bulky samples is required, local micro-scale variations are ignored. Several non-destructive methods are widely used for obtaining mechanical properties of small samples. Micro- and nano-indentation have been used to measure the hardness and elastic constants of cortical bone at the micro-structural level (Weiner et al., 1997, Rho et al., 1997, Zysset et al., 1999, Silva et al., 2004, Hengsberger et al., 2003; Bensamoun et al., 2004). These methods allow estimation of hardness and Young’s modulus from contact stiffness between the indenter tip and the sample. They require a highly polished sample surface, and the calculation assumes knowledge of the Poisson’s ratios of the sample. When used to measure the elastic modulus of anisotropic materials such as bone, the modulus derived from the method is an average of the anisotropic constants biased towards the modulus of the direction of testing (Rho et al., 1997). Another approach is based on the measurement of the speed at which sound travels through bone (Yoon and Katz, 1976a,b, Ashman et al., 1987, Rho et al., 1993). Although these methods are non-destructive, they are also indirect, and rely on the application of theories of composite materials to the measurements in order to obtain estimates of the elastic constants. Few studies describe experimental determination of Poisson’s ratios of cortical bone. Reilly and Burstein (1975) assumed transverse isotropy of ﬁbrolamellar bone, and used extensometers to measure strains in two orthogonal directions concurrently. They found Poisson’s ratio values which ranged between 0.29 and 0.63. Ashman et al. (1984) reported on the use of an ultrasonic continuous wave technique, and found Poisson’s ratio values which ranged between 0.27 and 0.45. Pithioux et al. (2002) also used an ultrasonic method, and found Poisson’s ratios between 0.12 and 0.29. Despite this wide range of reported values (0.12–0.63), many studies, especially ﬁnite element analyses, often use values in the much narrower range of 0.28–0.33. Optical metrology techniques allow non-contact measurement of displacements on surfaces of samples subjected to static or dynamic mechanical loading. 253 Electronic speckle pattern-correlation interferometry (ESPI) (Jones and Wykes, 1989, Rastogi, 2001) has recently been used to determine sub-micron surface displacements on the surface of millimetre-sized tooth dentin samples loaded elastically in compression (Zaslansky et al., 2005). Displacements are directly determined from variations of laser light reﬂected from samples immersed in water. Using this technique, it is possible to perform quantitative analysis of strain on compressed samples of mineralized biological tissues such as bone and dentin, by loading them in a highprecision micro-mechanical loading device. Such measurements can be performed without damaging the sample and hence anisotropic Young’s moduli and Poisson’s ratios can be determined from multiple measurements of each sample. We report on measurements performed using a commercial ESPI system (Q300—Ettemeyer, Ulm, Germany) which has been combined with a custombuilt loading device allowing non-destructive compression tests of small cubes of cortical bone in three orthogonal directions. Our set-up allows the measurement of in-plane and out-of-plane deformation ﬁelds on surfaces of very small samples. Due to the high sensitivity of our system, experiments can be conducted non-destructively by repeatedly loading the sample within its elastic region. Measurements may be performed on wet samples, thus satisfying a basic requirement for the study of biological specimens. Measurements by this technique require load to be applied in small increments, since large deformations cause optical decorrelation that renders the measurements invalid. While the ESPI technique allows for the determination of the elastic constants from the traditional stress–strain curve, as is common in classical mechanical testing methods, it also allows for the use of a compliance-based method (‘Estimated best E’ from Zaslansky et al., 2005). The compliance method is based on the principle that each incremental loading step within the elastic region can be considered as an independent experiment in which the strain ﬁeld created within the sample and the incremental load causing it are determined. Thus, the compliance of the sample is determined repetitively and non-destructively. An interesting feature of the ESPI method is its inherent ability to concurrently measure strain along two orthogonal directions on the sample surface. This allows the derivation of Poisson’s ratios from the same experimental data on exactly the same sample under identical loading conditions. For each incremental loading step, in addition to the axial strain of the sample, the lateral strain is also determined. The negative ratio of the latter to the former can be used to estimate Poisson’s ratio. Since these loading steps can be repeated at the discretion of the investigator, the ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 254 statistical error associated with mechanical tests such as these can be reduced. This paper describes the results of optically determined measurements of strain of mechanically loaded millimetre-sized samples of equine cortical bone. These measurements were used to determine the anisotropic variations in Young’s moduli and Poisson’s ratios. 2. Materials and methods Six cubic bone samples of equine cortical bone were obtained from the cadavers of two horses. They were loaded in compression, and their surface displacements were determined using ESPI following procedures similar to those described in Zaslansky et al. (2005). These data were used to determine three Young’s moduli and three Poisson’s ratios related to the three orthotropic directions (axial, radial and transverse). 2.1. Bone sample preparation The right third metatarsal bones (MT3) were obtained from a 4-year old-male Quarter horse and a 6-year-old female Arabian horse. The cause of death of both horses was unrelated to the musculoskeletal system. All external soft tissue was meticulously removed, and 2cm thick slices were cut from the mid-diaphysis of each bone using a hand saw (Fig. 1). Transverse sections of 200–300 mm thickness were then cut from each bone slice, using a low-speed water-cooled diamond saw (South Bay Technology Inc.) (Fig. 1). These sections were ground and polished (Buehler Minimet Polisher) and examined by reﬂected light microscopy. The area of the cranial mid-diaphysis in the bones of both horses consisted almost entirely of secondary remodelled osteonal bone, with small areas of primary bone. Each bone slice was further cut and used as a source of three 2 2 2 mm cubes. The cubes were cut such that their faces were aligned with the anatomical axes of the bone: proximo-distal (axial), antero-posterior (radial) and cranio-caudal (transverse) orientations (Fig. 2). Care was taken to note and mark the orientation of the cubes, so that distinct axial, radial and transverse faces could be identiﬁed. The cubes were then stored for 2-7 days on water-saturated cotton swabs at 4 1C until testing. 2.2. Experimental set-up The experimental set-up, shown schematically in Fig. 3, consisted of a mechanical tension–compression device, a water chamber, an optical ESPI head capable of determining surface displacements and a computer which controlled the various components of the system and analysed the data. Aside from the computer, the entire set-up was enclosed in an acoustically insulated box mounted on top of a ﬂoating optical table. 2.3. Mechanical testing device The mechanical tension–compression device used for loading the bone samples was custom-designed with stainless-steel parts (SS 316) (see Fig. 4). The device consists of a sealed chamber which includes a high-grade glass window (BK-7 l/10 grade), allowing the use of aqueous solutions as a medium in which sample testing took place. The loading apparatus in the test chamber consisted of an axial motion DC motor (PI M-235.5 DG, Physik Instrumente, GmbH, Germany) capable of displacing a metal shaft in small sub-micron steps while applying substantial force (4100 N). The metal shaft acted as a movable upper anvil, pressing against samples mounted on the stationary lower anvil which was attached rigidly to the testing chamber base. The movable metal shaft was ﬁtted with an in-chamber 2 mm 2 mm Radial 2 mm Axial Transverse Fig. 2. The preparation of three 2 2 2 mm cubes from the slice shown in Fig. 1, with the orthogonal anatomic orientations preserved. Fig. 1. Removal of a 2-cm thick slice from the mid-diaphyseal area of the third metatarsal, and a thin transverse section. ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 255 DC axial motor Sample Laser Water Chamber Glass window Trigger Micro-step (PI) motor controller and computer ESPI System Fig. 3. Schematic of the entire experimental system. immersable load cell (AL311BN-6I, Sensotec, Honeywell, USA) capable of measuring loads up to 120 N in tension or compression. Force measurements were collected and stored on a computer using an A/D converter (Omega DAQP-308 PCMCIA 16-bit Analogue I/O) and further analysed by custom-written software (National Instruments Labview v 7.0 and Matlab v 6.0). 2.4. Sample mounting A thin lining layer of soft composite material (Z250, 3 M ESPE, St. Paul, MN USA) was placed over the lower, stationary anvil in the test chamber. The face of the bone cube which corresponded to the direction being tested was then placed upon it. The cube was gently pushed down onto the composite such that it was mounted in the appropriate orientation (inset in Fig. 4). Manual adjustment facilitated the alignment of the cube. The composite material was then polymerized using a hand-held light-cure device (LITEX 682, Dentamerica CA, USA). A small amount of composite was placed on the edge of the upper (travelling) anvil which was then lowered until the composite was brought into contact with the entire upper surface of the cube. While the cube was held in the desired line of compression (axial, radial or transverse) between the two anvils, the top composite was cured. This mounting procedure ensured that an intimate and stiff contact was established between both anvils and the bone cube through composite load–transfer layers. With each sample mounted and ﬁxed in place, the chamber was sealed and ﬁlled with physiologic saline. Small pieces of bone were added to the saline solution which was refrigerated at 4 1C, for at least 24 h prior to initiation of testing, in order to ensure mineral saturation of the solution. The same solution was used for all experiments. 2.5. Mechanical compression testing A small compression preload of about 10 N was applied at the beginning of each experiment, and the sample was allowed to reach near-equilibrium force readings while undergoing some initial stress-relaxation. Once a stable reading of force was reached (approximately within 200–300 s), a series of 15 compression increments was initiated. Each loading step consisted of a small incremental load, which was produced by forcing the upper anvil to move 2 mm downwards. Just before and immediately after each such loading step, force was determined by averaging 20,000 readings (taken at 80 kHz), and surface deformation was determined by laser speckle intensity measurements (ESPI, Section 2.6). Each experiment was repeated 15 times, so as to collect a large and robust data set. Then, at the end of 15 experiments, the test chamber was emptied and the cube was dismounted. The cube was then remounted in another (orthogonal) orientation, and 15 load-deformation experiments repeated. We were thus able to collect data from 15 repeated experiments of force-deformation measurements for each of the three orthogonal sample axes for each cube (therefore each cube underwent 45 experiments). Every experiment contained 15 incremental compression steps. These measurements provided the database required to determine the 3 Young’s moduli and 3 of the 6 Poisson’s ratios of the cubes of cortical bone reported here. For 2 of the cubes we were also able to calculate the three remaining Poisson’s ratios, (see Section 5). ARTICLE IN PRESS 256 R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 Fig. 4. Mechanical tension–compression device and loading chamber, with a bone cube in place. The ﬁrst stage of sample mounting, showing the bottom anvil, with a layer of composite and a bone cube. 2.6. Optical determination of surface displacements Surface displacements of the compressed cube along and across the loading direction were used to calculate the longitudinal and lateral strains caused by each incremental load. These displacements were determined using two orthogonally aligned horizontal (X-axis) and vertical (Y-axis) speckle pattern-correlation interferometers (see detailed description in Zaslansky et al., 2005). Brieﬂy, the surface of the sample is illuminated by laser light from two symmetrical opposite angles relative to the normal to the surface. The laser illumination creates a speckled light interference ﬁeld on the surface. By imaging the speckle patterns onto a CCD detector array, variations to the interference intensity patterns can be detected and captured by a computer. Any sub-micron displacement of sub-sections on the surface will affect the optical path of the light propagating towards the ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 CCD detector array, changing the detected interference pattern. If the phase of the light of one of the beams is modulated by means of a phase shifter, the phase relations over the entire CCD ﬁeld can be determined. The interference patterns can be shown to correspond to light phase differences on small surface sub-sections, which are determined after computerized image ﬁltering and phase unwrapping (Huntley, 1998, Rastogi, 2001). The phase relations were determined for small subareas on bone sample surfaces before and after each load increment. By substraction, 2D matrices of phase differences were obtained from the X and Y interferometers, of the form: MATX ;Y ¼ Djði; jÞX ;Y (1) where i and j represent the rows and columns of the matrix respectively, and corresponds to the phase difference determined independently for each axis. Each element in the two matrices corresponds to one sub-area on the surface of the sample. A scaling relation of the form: l Djði; j ÞX ;Y (2) 4p sin y was used, in which l is the laser light wavelength (l ¼ 780 nm in our system) and y is the angle between the incident laser light and the normal to the sample surface. Our system allows detection of surface displacements with magnitudes as small as l/30 (Jones and Wykes, 1989). Thus, the discrete displacement values u and v were determined, corresponding to the components of displacements of all sub-areas within the area of interest along X and Y axes, respectively (Rastogi, 2001; Jones and Wykes, 1989). u; vði; j Þ ¼ 2.7. Determination of surface strains The uði; j Þ and vði; j Þ displacement matrices were assumed to represent the displacements of the bulk of the compressed bone samples. They were assumed to vary linearly along the rows and columns of the displacement ﬁelds in the Y- and X-directions, respectively. They could therefore be approximated as planes by least-squares regression analysis that provided continuous and differentiable displacement ﬁelds of the form: uði; j Þ ¼ ai þ bj þ c, (3) vði; j Þ ¼ di þ ej þ f . (4) For inﬁnitesimal strains, the axial normal strain (along the loading direction), the lateral normal strain (perpendicular to the loading direction) and the shear strain could then be determined according to eii ði; j Þ ¼ @uði; j Þ ¼ a, @i (5) @vði; j Þ ¼ e, @j 1 @uði; j Þ @vði; j Þ 1 eij ði; j Þ ¼ þ ¼ ðb þ d Þ. 2 @j @i 2 ejj ði; j Þ ¼ 257 (6) (7) In our experiments, each detector on the CCD covered a sub-area of approximately 12 mm 12 mm on the surface of the sample. Due to the small dimensions of the samples relative to the distance to the ESPI lens (2 mm versus 225 mm, respectively), spherical aberrations could be neglected. Therefore, the distance between adjacent rows and columns could be assumed to be 12 mm. This number was then used as a scaling factor to obtain estimates of strain from the slopes of the regression planes. 3. Data analysis Stress values were calculated as the ratio between the force measurements recorded for each load increment and the initial cross-sectional area of the corresponding sample. These stresses and the optically determined strains were used for the determination of Young’s modulus and Poisson’s ratio along each of the bone axes. In order for an experiment to be considered valid, the lateral normal strain had to be larger than the shear component by at least an order of magnitude. Otherwise it was repeated entirely, including repositioning of the sample. 3.1. Young’s modulus (E) Two complementary methods were employed to determine Young’s modulus (E) for each of the orientations of the samples (axial modulus EA, radial modulus ER, and transverse modulus ET): 3.1.1. Stress– strain curves The cumulative stress was plotted against the cumulative strain for the 15 increments in each of 15 repeated experiments, for all three compression orientations. The slopes of the stress–strain curves were then estimated using linear least-squares regression analysis. In all experiments a tight linear relationship between stress and strain was found. A typical stress–strain curve obtained from a 15-step compression experiment performed in the radial direction is shown in Fig. 5. 3.1.2. Compliance method Individual compliance estimates were determined for each of 15 incremental compression loads. Strain increments for each loading step were determined by least squares regression analysis of the surface displacement values, as described in Eqs. (3) and (4) above. The ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 258 Sample Strain Stress curve, Radial compression F 16 F Stress [MPa] 14 12 10 8 Stress[MPa]= 11.6x103 Strain - 0.30 R2 = 0.996 6 4 2 0 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 6.00(strain(x105)) Middle area 6 mm Fig. 5. A typical stress–strain curve obtained from a compression experiment on cube from the Arabian horse. compliance estimate was then estimated by dividing the strain increment by its corresponding stress increment. The inverse of the median of these 15 values was considered the estimate of Young’s modulus for the experiment. In this manner 15 modulus estimates were obtained for each direction (axial, radial and transverse). For each experiment, whole body rotation and shear deformation were found to be much smaller than axial deformation, and therefore could be ignored. 3.2. Poisson’s ratio (v) In each experiment, longitudinal strains elong and lateral (orthogonal to the direction of loading) strains elat were obtained for each of the 15 incremental loads. The negative ratio between elat and elong was determined, and Poisson’s ratio was estimated as the median value of these 15 separate estimates. Since each experiment was repeated 15 times, the reported result was obtained as the average of 15 medians. For each cube, 3 different combinations of lateral/longitudinal load orientation pairs (out of 6 possible combinations) were performed. 4. Method validation Three different approaches were used to validate our methodology. First, the method described above was used to determine the elastic properties of a small 2 2 2 mm cube made of an isotropic material: Acetal polyether imide (Ultem-1000s, General Electric Plastics), whose elastic constants are precisely known. Since our experimental procedure is very different from industrial ISO and ASTM testing recommendations, these results allowed us to validate the application of our novel methodology to the testing of very small samples in water and without contact. Second, an experiment was designed to determine the effect of the non-slip conditions occurring at the anvilsample interface on the determined elastic constants. Bottom area 2 mm (a) F (b) Fig. 6. Schematic of the parallelepiped bone sample, showing the central and lower areas of interest. To this end a 2 2 6 mm rectangular parallelepiped sample of equine cortical bone was compressed along its long axis to allow calculation of Young’s modulus and Poisson’s ratio in an area distant from the area of applied loads (centre of the sample), and in an area close to loaded edge (see Fig. 6). The results were compared with those obtained in cube experiments using samples from the same horse and an identical load orientation. Third, a ﬁnite element model was created to simulate a cube compression experiment. In this model the lower nodes of the cube were ﬁxed, and the topmost nodes (area of load application) were fully restrained except along the loading direction. The modelled bone material was assigned orthotropic properties, using Young’s moduli determined here, and shear moduli reported by Ashman et al. (1987). The model was analysed with two different Poisson’s ratios: 0.3 and 0.1. The model was created with Nastran software, NFW version 2002, and consisted of 12,800 8-node brick elements, with 13,671 nodes. The composite layer above and below the cube was modelled as isotropic, with values for Young’s modulus (11 GPa) and Poisson’s ratio (0.30) obtained from the manufacturer (3 M). The results were examined to determine the level at which the lateral constraints ceased to affect the lateral strain. 5. Results Table 1 shows the calculated Young’s moduli and Poisson’s ratio results based on 10 compression experiments conducted on a 2 2 2 mm cube of Ultem-1000s. The measured Young’s moduli were very ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 259 Table 1 Results of a compression experiment performed on 2 2 2 mm cubes made of Ultems Ultem-1000s Young’s modulus ET (GPa) reported by the manufacturer Young’s modulus ET (GPa) by the compliance method (in brackets standard deviation, n ¼ 15) Young’s modulus ET (GPa) by the stressstrain curve method (in brackets standard deviation, n ¼ 15) Poisson’s ratio reported by the manufacturer Poisson’s ratio (in brackets standard deviation, n ¼ 15) 3.2 3.42 ( 0.01) 3.43 (0.01) 0.36 0.38 (0.07) close to the value supplied by the manufacturer (3.42–3.43 GPa and 3.2 GPa, respectively). The measured and manufacturer-supplied Poisson’s ratios were also very similar (0.38 and 0.36, respectively). These results indicate that our methodology is accurate (the means are in excellent agreement) and precise (small standard deviations). Table 2 shows the results of Young’s moduli obtained in a typical set of 15 experiments compressing a Quarterhorse bone cube in the radial direction. For each experiment the Young’s modulus was obtained from the slope of stress–strain curves (S–S) as well as from the inverse of the median compliance. Also shown are the mean and standard deviations for these values. Clearly the results from both methods are very similar, both with respect to the mean values and their errors. Table 3 summarizes the results obtained from compression tests of all cubes of cortical bone used in this study: 3 from the 4-year-old male Quarter horse, and 3 from the 6-year-old female Arabian horse. Results of Young’s moduli determined by both stress–strain curves and compliance-median methods for the three anatomic orientations (axial, radial and transverse) are given, as well as three Poisson’s ratios (for the combination of orientations measured). Only 3 of 6 Poisson’s ratios were experimentally determined. In materials of orthotropic symmetry, the 6 different Poisson’s ratios are not independent (Cowin and Van Buskirk, 1986). It can be shown that Poisson’s ratios of orthotropic materials must satisfy the following relationships: uAT =E A ¼ uTA =E T , (8) uAR =E A ¼ uRA =E R , (9) uRT =E R ¼ uTR =E T , (10) where EA is Young’s modulus in the axial direction, ER is Young’s modulus in the radial direction, ET is Young’s modulus in the transverse direction, and uij are the 6 different Poisson’s ratios, where ij denotes the respective axial, radial or transverse direction-combinations (Cowin and Van Buskirk, 1986). The three Poisson’s ratios not determined experimentally were calculated according to Eq (8)–(10) for one cube of each horse. Table 2 Experimental results of Young’s moduli for a typical set of 15 experiments for a Quarter horse cube Experiment # S–S curve Compliance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Mean Standard deviation 9.6 10.0 10.2 10.0 10.0 10.0 10.1 10.5 10.6 10.2 9.9 10.0 10.0 10.0 9.8 10.1 0.22 9.7 9.9 10.1 9.8 9.9 9.9 9.7 10.2 10.4 10.2 9.8 9.8 9.6 10.0 9.7 9.9 0.23 Compression was in the radial direction. Moduli are shown both by the compliance method and by the stress–strain (S–S) method. By rearranging Eqs. (8)–(10) it can be seen that the ratio of orthogonal moduli is equal to the reciprocal ratio of Poisson’s ratios: E i uji ¼ , E j uij (11) where i and j are two orthogonal anatomic directions. Table 4 presents a comparison of the ratio of reciprocal Poisson’s ratios and the ratio of the corresponding Young’s moduli, in those experiments were the appropriate Poisson’s values were both measured. It can be seen that the two ratios behave roughly as predicted by Eq. (11). The measured Young’s moduli values clearly demonstrate the well known axial anisotropy of secondary osteonal bone. Interestingly, there are signiﬁcant differences in the modulus values between the 6 cubes. In both horses, one-way analysis of variance performed on the results obtained for the axial Young’s modulus of the three cubes revealed that they are signiﬁcantly different (po0.00001). Poisson’s ratios were found to be in the lower range of values reported to date for cortical bone. For strains ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 260 Table 3 Summary of results for all 6 cubes Horse Cube EA ER ET Quarter horse Cube 1 23.5 (0.3) 23.4 (0.6) 10.0 (0.5) 9.9 (0.7) 11.4 (0.7) 11.3 (0.5) Cube 2 19.3 (0.8) 19.4 (0.6) 9.9 (0.2) 10.0 (0.3) Cube 3 21.7 (0.7) 21.6 (0.5) Cube A Arabian horse uRA uAR uTA uAT uRT uTR 0.060 0.208 (0.044) — — 0.122 (0.063) 0.141 (0.031) 8.8 (0.3) 9.1 (0.3) 0.059 (0.028) 0.132 0.078 (0.027) 0.128 (0.060) — — 9.3 (0.4) 9.6 (0.5) 9.5 (0.4) 9.6 (0.5) 0.074 0.165 (0.050) 0.111 (0.046) 0.138 0.166 (0.046) 17.4 (0.4) 17.5 (0.5) 13.9 (0.4) 13.8 (0.5) 11.3 (0.5) 10.9 (0.5) 0.122 (0.018) 0.196 (0.017) 0.070 (0.017) 0.108 — — Cube B 22.1 (1.1) 21.7 (0.8) 9.9 (0.7) 9.9 (0.4) 11.1 (0.6) 11.3 (0.5) — — 0.098 0.192 (0.023) 0.150 (0.015) 0.156 (0.020) Cube C 22.6 (0.5) 22.8 (0.7) 11.5 (0.3) 11.8 (0.2) 11.1 (0.3) 11.2 (0.3) 0.102 (0.010) 0.208 0.172 (0.015) 0.166 0.124 0.244 (0.015) 0.174 E represents Young’s modulus, and u represents Poisson’s ratio. The subscripts A, R, and T represent the axial, radial and transverse directions, respectively. In each cell are shown the Young’s moduli obtained from the stress–strain curve above, with its standard deviation in brackets, the Young’s modulus obtained by the compliance method below, and its standard deviation in brackets. All means and standard deviations are based on 15 observations. Poisson’s ratios are given either as measured, or in underlined italics when calculated based on Eqs. (8)–(10). along the axis of compression a reliable signal to noise (S/N) ratio was fairly easy to obtain requiring only moderate ﬁltering of the v(i,j) displacement ﬁelds (local weighted averaging with each point averaged with its 4 nearest neighbors at a ratio of 100:40). The S/N ratio for the orthogonal transverse strain values was found to be smaller since the signal was weaker while the noise remained the same, and resulted in larger standard deviations of the estimation of Poisson’s ratio. In addition to the similarity found between the measured and manufacturer-supplied values of Poisson’s ratios for the synthetic material we tested, we decided to investigate the possibility that the low values of measured Poisson’s ratios of the bone samples might have been caused by the effect of constrained sample edges on the measured strains. Speciﬁcally, we assessed the possibility that these constraints limit the lateral expansion of the sample and yield erroneously low Poisson’s ratios. We examined the effect of constrained sample edges experimentally by comparing the Poisson ratios measured in 2 different areas of a parallelepiped bone sample. Table 5 shows the results obtained for compression of the 2 2 6 mm rectangular parallelepiped along its long axis (see Fig. 6). The sample was Table 4 Comparison of ratio of reciprocal Poisson’s ratios and the ratio of the corresponding Young’s moduli, in those experiments were the appropriate Poisson’s values were measured Bone cube Ratio of measured reciprocal Poisson’s ratios Ratio of corresponding measured Young’s moduli Quarter horse cube 1 Quarter horse cube 2 Arabian horse cube A Arabian horse cube B uTR/uRT ¼ 1.16 uTA/uAT ¼ 0.60 uRA/uAR ¼ 0.62 uRT/uTR ¼ 1.06 ER/ET ¼ 1.14 ET/ER ¼ 0.46 ET/ER ¼ 0.80 ET/ER ¼ 1.20 divided into 3 areas of interest: proximal third, central third and distal third. Results are presented for both the central and distal 2 2 mm areas and are quite similar, falling within the range of variation shown to exist in this bone within millimeter-range zones. These results show that the end constrains only affect the measurements slightly. We also tested edge effects and the validity of the measured Poisson’s ratios using ﬁnite element analysis ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 261 Table 5 Results of compression of a rectangular parallelepiped along its long axis (transverse direction) Region of interest Young’s modulus ET (Gpa) by the compliance method (in brackets standard deviation, n ¼ 15) Young’s modulus ET (Gpa) by the S–S curve method (in brackets standard deviation, n ¼ 15) Poisson’s ratio uTR (in brackets standard deviation, n ¼ 15) Central area (n ¼ 15) Bottom area (n ¼ 15) 10.7 (0.48) 9.1 (0.57) 10.9 (0.41) 9.1 (0.79) 0.142 (0.03) 0.120 (0.02) The displacements of the transverse-radial surface were measured. The measurements were performed both in the central area and the bottom area (see Fig. 6). simulating compression of 2 2 mm bone cubes with Poisson’s ratio of 0.1 and 0.3, respectively, and with 0.5 mm thick composite layers between the cube and both loading anvils. Boundary conditions simulating the no-slip contact between the anvils and the sample were set. The distributions of the computed strains along a central longitudinal line of nodes, in the compression (y-) direction and lateral (x-, orthogonal to the compression) direction for Poisson’s ratios of 0.3 and 0.1 are shown in Figs. 7a and b, respectively. It can be seen that the central area, coincident with the area on the sample’s surface from which experimental data were obtained, exhibits a consistent strain in both the direction of load application and the direction orthogonal to it. The average absolute values of the ratio between the lateral and axial strains are 0.303 and 0.107 for simulations using Poisson’s ratios of 0.3 and 0.1, respectively. These results suggest that the direct Poisson ratio measurements obtained by our experiments represent the true properties of the tested material, and are not biased by the no-slip conditions between the anvils and the composite layer. 6. Discussion This study shows that Young’s moduli and Poisson’s ratios can be determined by direct observations of surface displacements in millimeter-sized samples of secondary osteonal cortical bone loaded in compression under water. All bone samples showed much higher modulus in the axial direction (EA) than in the radial and transverse directions (ER and ET), as expected. Furthermore, the radial and transverse moduli were quite similar. This is consistent with values obtained by others using macroscopic specimens (Reilly and Burstein, 1975, Taylor et al., 2002, Dong and Guo, 2004, Iyo et al., 2004). We calculated Young’s moduli from our experimental data by two independent methods (the stress–strain and compliance methods). As can be seen in Table 3, the methods yield nearly identical results. Yet an important difference exists between these two methods. The S–S method uses a broad range of stresses and strains, whereas the compliance method is based on measurements made with very small stress increments (o1 MPa). Our results show that minute deformation can be reliably determined with our ESPI-based experimental system. Low stress increments result in very small strain increments, thus allowing experimental approximation of the local derivatives of the stress–strain curve. Furthermore, our ability to measure accurately very small strains allows us to conduct elastic experiments in which damage does not occur in the sample. Additionally, with the S–S curve slope method only one modulus estimate is obtained for each experiment, whereas with the compliance method a large number of statistically independent measurements are obtained. This has the advantage of providing multiple results for each sample, allowing improved statistics. Furthermore, these features might have application for studying other properties, such as rateand load-dependant variations of the elastic constants. We found signiﬁcant local variations in Young’s moduli to occur within a range of 1–2 mm of the equine cortical bone we tested. Axial Young’s moduli varied between 17.4 and 23.6 GPa, whereas radial and transverse Young’s moduli varied between 8.8 and 13.9 GPa. Similar variation was previously reported by Ashman et al. (1984). We have shown that the variations found in small adjacent cubes do not arise from lack of precision or lack of accuracy of our method. We suggest that in both horses, structural differences in the meso-scale, such as regions of secondary osteons versus interstitial bone, variable pore distributions, and/or local differences in mineral content have a major impact on local stiffness. In fact, when we remounted some of our samples and repeated the experiments, we obtained extremely similar results (data not shown). The broad range of results is however similar to results previously reported in the literature (Currey, 2002, Cowin, 2001). We conﬁrmed the accuracy of our measurements and validity of the results by performing compression experiments on small cubes of Ultems (see also Zaslansky et al., 2005). Measurements made using a protocol similar to that used for bone yielded values for both Young’s modulus and Poisson’s ratio which were ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 262 x- and y-strains along a central line of the cube in the direction of compression 0.00006 0.00004 0.00002 0 strain 0 0.5 1 1.5 2 2.5 3 -0.00002 -0.00004 x - strain -0.00006 y - strain -0.00008 -0.0001 -0.00012 position of element (mm) (a) x- and y-strains along a central line of the cube in the direction of compression 0.00004 0.00002 0 0 0.5 1 1.5 2 2.5 3 strain -0.00002 x- strain -0.00004 y- strain -0.00006 -0.00008 -0.0001 -0.00012 (b) position of element (mm) Fig. 7. Finite element analysis results, showing the longitudinal (y) and lateral (x) strains along a central line of the cube in the direction of compression for bone cubes with Poisson’s ratio of 0.3 (a) and 0.1 (b). very similar to the values reported by the manufacturers, as can be seen in Table 1. The results obtained here for Poisson’s ratios of cortical bone are however lower than those usually reported. In fact, previous studies reported a very large range of values of Poisson’s ratios in bone, from 0.12 to 0.63 (Pithioux et al, 2002; Reilly and Burstein, 1975). The values found in our series of experiments fall within a narrower range. The low values obtained by our experiments for Poisson’s ratios in the RA and TA directions (0.07–0.124) are particularly striking. In a material with orthotropic symmetry the Poisson’s ratio values should be orientation-dependent. As can be seen from Eq. (11), the relationship between the different Poisson’s ratios is related to the associated Young’s moduli. Table 3 shows that when both uRA and uAR or uTA and uAT were measured in the same bone cube, they were different. Table 4 demonstrates that the ratio between these reciprocal Poisson’s ratios and the ratio between the corresponding Young’s moduli are similar. On the other hand when uRT and uTR were both measured in the same bone cube they were quite similar, as were the corresponding radial and transverse moduli. These ﬁndings strongly support the notion that these are indeed representative values for Poisson’s ratios of secondary osteonal bone. The large differences between many of our measurements of Poisson’s ratio and those often used in the literature led us to investigate the possibility that the non-slip condition between the anvils of the ARTICLE IN PRESS R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 micro-mechanical loading device and the bone samples could cause the measured lateral strain to be misleadingly low. Several validation studies were conducted to evaluate this scenario. A compression experiment conducted on a 6 2 2 mm rectangular parallelepiped bone sample along its long axis yielded similar results for Young’s modulus and Poisson’s ratio when data were obtained from the central 2 2 region (far from the conﬁned edges) and from a 2 2 region near the loaded edge. Furthermore, analysis of a ﬁnite element model of compression of a cube, with boundary conditions simulating the non-slip edge effect arising in the experiments of this study showed that in the entire region of interest on the face of the cube the ratio of strain across the direction of loading to the strain at the same point along the direction of loading yields the true Poisson’s ratio. We therefore conclude that the values of Poisson’s ratio reported here are accurate. Osteonal bone is the dominant bone type of adult equine bone (Mason et al., 1995); however it is not known how this bone behaves in the meso-scale when loaded under compression. Individual osteons have different mineral contents, and indeed mineral content varies continuously throughout their structure. Mineral content also varies between Haversian and interosteonal areas (Currey, 2002). Hence different stiffness values characterize different sites within the bone, and it is not clear how the load is distributed among these sites. We plan to further modify our system in order to achieve even greater resolution that will allow the measurement of displacements in the meso-scale of 50–500 mm. It will then be possible to determine local displacement variations in regions such as newly formed osteons with low mineral content and regions of older, more mineralized osteons. The variation of local displacements within individual regions (and the resulting strain variations) can then be compared. The results reported here demonstrate that the method described can yield quantitative measurements of surface displacements of small cubes of bone loaded in compression while in a water environment. Furthermore, these results can be used to calculate the strains, and through them values for Young’s moduli and Poisson’s ratios. Measurements are performed without contact with the sample, and can easily be repeated many times on the same sample, since the method is non-destructive, allowing truly elastic measurements. In conclusion, we described the determination of Young’s moduli and Poisson’s ratios using a novel, noncontact optical method. Small bone samples were tested non-destructively in an aqueous environment, and measurements of displacements were performed in three orthogonal directions. 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