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Anisotropic properties of human tibial cortical bone as measured by nanoindentation
ELSEVIER
Journal of Orthopaedic Research
Journal of Orthopaedic Research 20 (2002) 806-8 10
www .elsevier.com/locate/orthres
Anisotropic properties of human tibial cortical bone as measured by nanoindentation Z. Fan
a,
J.G. Swadener
b,c,
J.Y. Rho
a,*,
M.E. Roy
a,
G.M. Pharr
b3c
Depurrnzent of Biomedical Engineerinz, Uniaer.yity o j Memphis, Meiriplzis, TN 38152, USA Ouk RidglgP National Lahorutory, Metals and Ceramics Dirision, P. 0. Box 2008, MS-6093, Ouk Ridge, TN 37831-6093, USA Department of Muterial.? Science and Engineering, Uniutwity of Tennessee, Knoxville, TN 37996-2200, USA I'
Accepted 5 December 2001
Abstract The purpose of this study was to investigate the effects of elastic anisotropy on nanoindentation measurements in human tibial cortical bone. Nanoindentation was conducted in 12 different directions in three principal planes for both osteonic and interstitial lamellae. The experimental indentation modulus was found to vary with indentation direction and showed obvious anisotropy (oneway analysis of variance test, P < 0.0001). Because experimental indentation modulus in a specific direction is determined by all of the elastic constants of cortical bone, a complex theoretical model is required to analyze the experimental results. A recently developed analysis of indentation for the properties of anisotropic materials was used to quantitatively predict indentation modulus by using the stiffness matrix of human tibial cortical bone, which was obtained from previous ultrasound studies. After allowing for the effects of specimen preparation (dehydrated specimens in nanoindentation tests vs. moist specimens in ultrasound tests) and the structural properties of bone (different microcomponents with different mechanical properties), there were no statistically significant differences between the corrected experimental indentation modulus (M,:np)values and corresponding predicted indentation modulus (M,J values (two-tailed unpaired t-test, P > 0.5). The variation of M,,, values was found to exhibit the same trends as the corrected McTpdata. These results show that the effects of anisotropy on nanoindentation measurements can be quantitatively evaluated. 0 2002 Orthopaedic Research Society. Published by Elsevier Science Ltd. All rights reserved. Kejwordsr Anisotropy: Nanoindentation; Modulus; Bone
Introduction Cortical bone is an anisotropic material, and its mechanical properties are determined by its composition as well as microstructure. At the microstructural level, cortical bone is composed primarily of osteonic lamellae and interstitial lamellae. A typical secondary osteon is a cylinder measuring approximately 200-250 ym in diameter and is surrounded by interstitial lamellae. Osteonic lamellae and interstitial lamellae exhibit significantly different mechanical properties [12,13,21]. These differences may be due to a combination of factors, such as collagen fiber orientation, degree of mineralization, and arrangements of these materials [2,3,5,12]. Even within a whole osteon, a decline has been observed in both elastic modulus and hardncss from the center of the osteon *Corresponding author. Tel.: +1-901-678-5485; fax: +1-901-678528 I . E-muiL crtldress: [email protected] (J.Y. Rho).
outward in mature secondary osteons [14]. The complexity of cortical bone also arises from its hierarchical structural organization. The anisotropy may be partly due to the highly anisotropic structure of mineralized collagen fibrils (the basic building block of cortical bone). The fibrils are found as bundles or aligned arrays, and these can be arranged in a variety of different patterns, resulting in different mechanical properties in all three orthogonal directions. Although the patterns of lamellae are still a matter of dispute, many researchers have indicated that cortical bone has orthotropic material properties [8,10,19,20]. A better understanding of bone anisotropy can help determine the basic mechanical functions of cortical bone. In this study, the nanoindentation technique was applied to probe the mechanical properties at the microstructural level. Nanoindentation has been shown to be an effective method to probe the mechanical properties of microstructures at the micron scale with a positioning resolution of 1 pm. The lamellar units of
0736-02661021S - see front matter 0 2002 Orthopaedic Research Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 0 2 6 6 ( 0 1)OOl86-3
Z . Fun et ul. I Journul
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Orihopuediic Reseurch 20 (2002) 806-810
cortical bone are about 3-5 pm thick. The dimensions of the microstructural building blocks of osteons are well within the capabilities of nanoindentation measurement. For isotropic materials, nanoindentation can be used to obtain intrinsic material properties, such as hardness and Young's modulus. However, for anisotropic materials, the indentation modulus in a specific direction is a composite quantity that depends on all of the elastic constants [ 13,181. To our knowledge, there are no studies in the literature that quantitatively evaluate the anisotropy of cortical bone via the nanoindentation technique. Therefore, in this study a recently developed theoretical model [I 6,171, which can predict indentation modulus from the material elastic constants matrix by assuming cortical bone to have orthotropic properties, was used to calculate the predicted indentation modulus. The values of the experimental indentation modulus (Mexp)and the predicted indentation modulus (MPrc) were compared to quantitatively investigate the effects of elastic anisotropy on nanoindentation measurements in cortical bone. Materials and methods E.xperimen tul metliod5 A bone sample was obtained from the midshaft of one human tibia (52 years old, male; cause of death: sudden heart attack). After being dehydrated in a series of alcohol baths, 12 specimens (about 3 x 3 x 4 mm' in size) were cut with various surface orientations to assess the anisotropy of cortical bone. As shown in Fig. I , an orthogonal coordinate system was chosen, where directions 1, 2, and 3 were the radial, circumferential and longitudinal directions, respectively. In this figure, D l , , 0 2 2 , and D13 represent indentation directions parallel to the three principal directions. Besides the three principal directions, three indentation direcrions were chosen that were equally distributed between the principal directions at each principal plane. For example, D12,30 is the indentation direction in the transverse (1-2) plane. 30" away from the radial (1) axis. Specimens were then embedded in epoxy resin without infiltration at room temperature. The epoxy resin was used to provide support during tests. and has no effect on test results provided
1
D,, Longitudinal
807
no indents were performed close to the bonelresin boundary. After grinding the surface of the specimens with silicon carbide abrasive papers of progressively finer grit sizes (600, 800, and 1200 grit), specimens were polished on microcloths with a 0.05 pm alumina suspension. Nanoindentation was conducted in 12 dillerent directions with respect to the principal planes. All nanoindentation tests (Nano Indenter 11, MTS Systems Co., Oak Ridge, TN, USA) were performed in load control mode with a Berkovich tip. The position of each indent was carefully selected based on microscopy observations. A total of 360 indents were made in thick lamellae from five osteons and five surrounding interstitial lamellae for each orientation. Three indents were made in each target area to reduce the effects of variations within osteonic lamellae and interstitial lamellae. The indenter was loaded and unloaded three times, with two intervening periods during which the load was held constant. To minimize the elltcts of viscoelasticity and creep on property measurements, a relatively long constant load hold period was conducted before the final unloading to diminish viscoelastic deformation to a negligible rate. The second constant load hold period, performed near the end of the test at 10% of the peak load. was used to establish the rate of thermal expansion o r contraction of the testing apparatus, to correct the displacement data for thermal drift. The maximum load of the nanoindentation tests was 8 mN, with a loading/unloading rate of 400 ~ N l s ,which produced impressions in bone with depths of about 700 nm. The indentation modulus (Meyp) for each indentation was determined from the well-documented method of Oliver and Pharr [9]. In a previous ultrasound study [lo], samples from three transverse sections located at 30%, 50'1/0, and 70'%1of total length from the end of the bone in each of eight tibiae (seven men and one woman ranging in age from 45 to 68 years). Four roughly cubic samples (about 5 x 5 x 3 mm3 in size) were prepared from four quadrants (anterior, medial. posterior, and lateral) of each transverse section. This resulted a total of 96 cortical bone specimens. During the ultrasound tests, the specimens were kept moist in a saline solution.
Tlieoreticul methods In nanoindentation experiments, measurements of load and displacement are utilized to determine the contact stiffness, from which the indentation modulus can be calculated. For anisotropic materials. indentation modulus ( M ) is defined by following equation [18]:
where S is contact stillness and A , the projected area of contact, which is obtained during calibration [9]. Conical indentation of isotropic materials is well understood [4]. Anisotropic materials, for which displacements of the indented surface vary in different directions, require a more complicated analysis for the determination of indentation properties. Based on a recently developed analysis for the indentation of anisotropic malerials by a cone [ I6.171, the value of M in a certain direction is given by
UI
D t t Radial
Fig. 1. Orientation ofthe principal axes and the indentation directions. DI Dzz, and D j 3 represent indentation directions parallel to the three principal directions. D I z 1 3 is~the indentation direction in transverse (12) plane, 30" away from the radial ( I ) axis.
where u 1 / u 2(the ratio of elliptical axes of the projected area of contact) can be determined numerically; B,,. the components of the first Barnett-Lothe tensor, which are derived from the elastic stiffness matrix; a,,and u3,, the direction cosines of the angles made between the indentation direction and the principal directions: and ?, the angle used to define the displacement direction at the free surface. T o evaluate the nanoindentation experimental results, Eq. (2) was utilized to calculate the theoretical indentation modulus by using the stiffness matrix of human tibia1 cortical bone, which was determined by earlier ultrasonic velocity measurements [lo]. A pyramidal tip was used in this study. but there are no exact analytical expressions for pyramidal tip indentation. Numerical studies [6,7] have suggested that a correction factor 1.034 should be used to correct the modulus values predicted from conical tip to match the results obtained from a pyramidal tip.
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The indentation modulus of an anisotropic material in any direction can be predicted by using its anisotropic elastic constants. The elastic stiffness components determined by Rho [lo] in ultrasound tests were used to calculate the predicted values (Mpre) at various orientations. It is important to note that nanoindentation experiments were performed at the microstructural level (with data obtained from osteonic and interstitial lamellae) and measured the mechanical response of the solid constituents in the vicinity of the indenter. On the other hand, ultrasound experiments were performed at the macrostructural level, which included not only osteonic and interstitial lamellae, but also lacunae and cement lines. To account for differences between the indentation modulus obtained from the osteonic and the interstitial lamellae, the value for indentation modulus derived from whole specimens might be better predicted by taking the area fraction of each tissue type, multiplying by its value of indentation modulus, and adding them [15]. This procedure implies a simple rule of mixture, which can be expressed as following equation:
where CSAF is the corrected secondary osteonal area, and is 0.388 for human tibia1 bone [15]. A void volume fraction of approximately 5.5% caused differences between the moduli determined by nanoindentation and by ultrasonic velocity measurements. To account for the effect of voids, the product ofMmlrlurr and V, (1 percentage of voids) was used to produce the adjusted indentation modulus. Because the size of indentations made exceeded the dimensions of cement lines (1-2 pm wide), cement lines were not tested in this study. In addition, for ultrasonic tests the bone specimens were kept moist in a saline solution, whereas for the nanoindentation tests, specimens were dehydrated in a series of alcohol baths. Rho and Pharr [Ill suggested that dehydration increases the indentation modulus of bovine bone by approximately 15%. In the absence of wetldry nanoindentation data for human tibia, it is reasonable to assume a similar effect here, although there are differences between bovine bone and human tibia. After the correction for dehydration, the final corrected indentation moduli (Mcxp)were produced. To compare experimental indentation modulus values for osteonic and interstitial lamellae in each of the three principal planes, one-way analysis of variance (ANOVA) tests were performed, with P < 0.01 for significance. When ANOVA revealed differences, the BonferronilDunn test (significance level: 1%) was employed to determine if specific pairwise comparisons were significantly different. The Bonferronil Duiin test was chosen for the more conservative results, since the experimental data were near normality. To compare corrected Me.? data with Mprcresults, a two-tailed unpaired t-test was used. All statistical analyses were performed with StatView 5.0.1 (SAS Institute Inc., USA). ~
Results
The average values of experimental indentation modulus for osteonic and interstitial lamellae obtained from all tests are listed in Table 1. One-way ANOVA tests indicated that there were significant differences between osteonic and interstitial lamellae in all directions (P < 0.0001). Since interstitial bone is older with higher mineral content than osteon, it would be expected to have a higher elastic modulus than osteon. Indentation modulus data in each principal plane for osteonic and interstitial lamellae were compared with one-way ANOVA test. Significant differences between interstitial lamellae moduli data existed in all three principal planes while for osteonic lamellae significant differences were only observed in the 1-3 and the 2-3 planes. In the 2-3 plane, the Bonferroni/Dunn method shows that the nanoindentation moduli are not significantly different
Table 1 Average experimental indentation modulus values (Mexp)for osteonic and interstitial lamellae Indentation directions
Experimental indentation modulus (Mcxp,GPa), average SD
+
Osteonic lamellae
Interstitial lamellae
16.6+ 1.5 17.0 2.2 25.1 2.1 15.7 i 2.3 16.0 2.0 14.9 2.0 18.4+ 1.1 18.4 f 0.9 21.8 f2.6 17.6f 1.3 1 8 . 2 f 1.4 21.9 f 2.0
19.7 1.5 18.5 1.1 27.1 f 1.7 20.4 1.8 21.1 f 1.4 18.9+ 1.4 20.0 1.4 22.4 1.6 24.8 f 1.0 20.0 i 1.4 2 l . 7 + 1.2 25.2 f I .O
+ +
*
+
* * *
0 2 2 , and 0 3 3 represent indentation directions parallel to the three principal directions. D12/3,, means the indentation direction in transverse (1-2) plane, 30" away from radial ( I ) axis. (SD: standard deviation.)
01 I,
between adjacent orientations, but as the angle between indentation directions increases, significant differences are revealed. The same pattern is observed in both osteonic and interstitial lamellae and for both the 2-3 and the 1-3 planes, suggesting orthogonal mechanical properties. To compare the corrected experimental data (Mexp) with predicted (Mpre)results, a rule of mixture (Eq. (3)) and the assumption of a 15% increase due to drying was used. In all 12 directions, the average modulus values from each osteons and its surrounding interstitial lamellae were used to calculate corrected Mexp.Table 2 shows the corrected Mexp values and predicted Mpre
Table 2 Corrected experimental (Men,,)and predicted (Mpre)indentation modulus values for various orientations Indentation direction
Corrected Meap(GPa) average f S D 14.9 i0.98 14.4 f 0.35 21.1 f 1.01 14.9 0.96 15.4* 1.46 13.9 f 1.78 15.6 f 0.89 16.7 0.57 19.0 0.95 15.3 0.48 16.3 f 0.60 19.2 i 1.57
*
+
+
Mprc(GPa) average
14.0 14.5 19.7 14.1 14.2 14.4 15.0 16.3 17.8 15.5 16.7 18.0
Corrected Mcxpdata were obtained after a rule of mixture (Eq. (3))and assuming a 15'% increase for dehydration. Predicted Mprevalues were calculated using elastic stiffness components determined by ultrasound test [lo]. (SD: standard deviation.)
corrected experimental (Mew) -predicted (Mpre)
26 24
7 I
2-221 9
12
0
15 30 45 60 75 90 angle of indentation in 2-3plane (degrees)
Fig. 2. Indentation modulus shows a statistically significant variation in the radial-longitudinal (2-3) plane. Experimental modulus (Alex,,) was determined by a rule of mixture, which considers osteonic, interstitial lamellae and void ratio; corrected Mcxpwas determined by assuming a IS%> increase due to dehydration of experimental modulus [l I]. Predicted modulus (MPrc) was calculated using elastic stiffness components determined by ultrasound tests [ I 01.
values at all orientations. Comparing these results, the predicted M,,,, values were found to exhibit the same trend as the experimental results in all orientations. Fig. 2 shows the trend in the radial-longitudinal (2-3) plane. The same pattern was also found in the 1-3 plane. There were no statistically significant differences between corrected Mexpvalues and M,,,, values (two-tailed unpaired t-test, P > 0.5).
Discussion For the indentation of isotropic materials, the OliverPharr method [9] can predict intrinsic material properties, such as Young modulus, within 4%) of literature values. For anisotropic materials, the indentation modulus represents a weighted average quantity, as the formation of the contact impression involves deformation in all three principal directions. Therefore, the modulus is not easily interpreted by conventional methods. Ziv et al. [20] utilized microhardness testing to investigate the variation of hardness at various orientations of both parallel-fibered and lamellar bone. Due to its limited lateral resolution (30 pm), the variation between osteonic and interstitial lamellae was beyond the capability of microhardness testing. Consequently, their results may have a lower correlation to microstructural properties compared with the results of the current study. Akiva et al. [l] studied the anisotropic properties of unidirectional plane-parallel platelet- or ribbon-reinforced composites, which have a structure similar as collagen fibrils. They found that the elastic behavior resulting from a platelet orientation is a char-
acteristic of three-dimensional orthotropic materials. Such a material has larger elastic constants at the platelet edge-on direction. The mineral in cortical bone can be treated as platelets. Because the indentation modulus values close to the longitudinal direction are larger in the present study, this suggests that more of the crystal layers are encountered edge-on in this direction. Moreover, since the gradual rotation of the crystal layers inside collagen fibrils in osteonic lamellae produces relatively isotropic properties [19], indentation modulus data from osteons in the 1-2 plane d o not show statistically significant differences (Table 2). However, the significant differences among interstitial lamellae indicate that the crystal layers might have a different structure in the 1-2 plane, requiring further investigation. These previous studies concentrated on qualitative evaluation. To describe the anisotropy of cortical bone more precisely, it is necessary to develop a mathematical model to quantitatively evaluate bone anisotropy, to consider the effects of all of the elastic constants. The currently developed analysis was designed to fulfill this objective. As shown in this study, the variation of predicted indentation modulus values was found to be consistent with the variation of the adjusted experimental values. Following correction for dehydration and void fraction, there are no statistically significant differences between the corrected values and the predicted values. Ultrasound measurement usually includes osteonic lamellac, interstitial lamellae, porosity, cement lines, and osteocytes, whereas the nanoindentation test in this study only included osteonic and interstitial lamellae. However, the similar trend between model and nanoindentation tests suggests that osteonic and interstitial lamellae are major contributors for anisotropy of cortical bone. The matrix of cortical bone is more complex than the model presented here. To move realistically from individual lamellae to a macroscopic bone sample, different lamellar types, dominant collagen bundle directions, degree of calcification and cement lines should be considered. However, there is currently no composite material model of bone that incorporates all of these items. The present model, although accounting only for osteonic and interstitial lamellae, is shown to be sufficient to describe the anisotropy of human tibia cortical bone. In the previous ultrasound study [lo], no significant variation in elastic properties was found along the length and around the circumference of cortical bone ( P > 0.1) of human tibia, so the site-specific differences of specimens were assumed not to be a critical issue in this study. However, i t is possible that part of the variation could be due to the site-specific differences. In addition, the data obtained from one individual tibia cannot be correlated to all human tibiae. Another limitation in our study is the assumption that the structure of individual human lamella is orthotropic. Such an
Z. Fun et ul. I Journal of Orthopuedic Research 20 (2002) 806-810
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assumption is not justified by the literature [S, 19,201, which presents orthotropic properties of a mesh of lamellae with various structural directions, not even one. The present study demonstrates that the effects of anisotropy of cortical bone to nanoindentation measurement can be quantitatively evaluated. These results should be carefully interpreted because two different specimens were used to compare the experimental indentation modulus with the predicted indentation modulus, and the present model has several limitations. Nevertheless, the anisotropic indentation properties of cortical bone are sufficiently described by using the analysis of indentation of anisotropic materials.
Acknowledgements
This research was sponsored in part by the NIH45297 and NSF DMR-0076497. Instrumentation for the nanoindentation work at the Oak Ridge National Laboratory SHaRE User Facility was provided by the Division of Materials Sciences and Engineering, US Department of Energy, under Contract DE-AC05-000R22725 with UT-Battelle, LLC and through the SHaRE Program under Contract DE-AC05-760R00033 with Oak Ridge Associated Universities.
References Akiva U, Itzhak E, Wagner HD. Elastic constants of 3dimensional orthotropic composites with plateletlribbon reinforcement. Compos Sci Tech 199757:173-84. Ascenzi A, Bonucci E, Ripamonti A, Roveri N. X-ray diffraction and electron microscope study of osteons during calcification. Calcif Tiss Res 1978;25:13343. Carando S, Barbos MP, Ascenzi A, Boyde A. Orientation of collagen in human tibial and fibular shaft and possible correlation with mechanical properties. Bone 1989;lO:13942. Johnson KL. In: Contact mechanics. Cambridge: Cambridge University Press; 1985. p. 1 1 1 4 .
[5] Katz JL, Meunier A. Scanning acoustic microscope studies of the elastic properties of osteons and osteon lamellae. J Biomech Eng 1993;115:543-8. [6] King RB. Elastic analysis of some punch problems for a layered medium. Int J Solids Struct l987;23: 1657-64. [7] Larson PL, Giannakopoulos SA, Choi KW, et al. Analysis of Berkovich indentations. Int J Solids Struct 1996;33:22148. [8] Liu D, Weiner S, Wagner HD. Anisotropic mechanical properties of lamellar bone using miniature cantilever bending specimens. J Biomech 1999;32:647-54. [9] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 1992;7:1564-83. [lo] Rho JY. An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone. Ultrasonics 1996;34:777-83. [l I] Rho JY, Pharr GM. Effects of drying on the mechanical properties of bovine femur measured by nanoindentation. J Mater Sci Mater Med 1999;10:485-8. [I21 Rho JY, Roy ME, Tsui TY, Pharr GM. Elastic properties of microstructural components of human bone tissue as measured by nanoindentation. J Biomed Mater Res 1999;45:48-54. [I31 Rho JY, Tsui TY, Pharr GM. Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomater 1997;18:1325-30. [I41 Rho JY, Zioupos P, Currey JD, Pharr GM. Variations in the individual thick lamellar properties within osteons by nanoindentation. Bone 1999;25:295-300. [15] Rho JY, Zioupos P, Currey JD, Pharr GM. Microstructural elasticity and regional heterogeneity in aging human bone examined by nano-indentation. J Biomech 2002;35:161-5. [I61 Swadener JG. Pharr GM. Indentation of elastically anisotropic half-spaces by cones and parabolas of revolution. Phil Mag A 2001;81:447-66. [I71 Swadener JG, Rho JY, Pharr GM. Anisotropic effects on elastic moduli measured by nanoindentation in human tibial cortical bone. J Biomed Mater Res 2001;57:108-12. [I81 Vlassak JJ, Nix WD. Indentation modulus of elastically anisotropic-half spaces. Phil Mag A 1993;67:1045-56. [I91 Weiner S, Traub W, Wagner HD. Lamellar bone: structurefunction relations. J Struct Biol 1999;126:241-55. [20] Ziv V, Wagner HD, Weiner S. Microsturcture-microhardness relations in parallel-fibered and lamellar bone. Bone 1996;18: 417-28. [21] Zysset PK, Guo XE, Hoffler CE, et al. Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J Biomech 1999;32: 1005-1 2.
Journal of Orthopaedic Research
Journal of Orthopaedic Research 20 (2002) 806-8 10
www .elsevier.com/locate/orthres
Anisotropic properties of human tibial cortical bone as measured by nanoindentation Z. Fan
a,
J.G. Swadener
b,c,
J.Y. Rho
a,*,
M.E. Roy
a,
G.M. Pharr
b3c
Depurrnzent of Biomedical Engineerinz, Uniaer.yity o j Memphis, Meiriplzis, TN 38152, USA Ouk RidglgP National Lahorutory, Metals and Ceramics Dirision, P. 0. Box 2008, MS-6093, Ouk Ridge, TN 37831-6093, USA Department of Muterial.? Science and Engineering, Uniutwity of Tennessee, Knoxville, TN 37996-2200, USA I'
Accepted 5 December 2001
Abstract The purpose of this study was to investigate the effects of elastic anisotropy on nanoindentation measurements in human tibial cortical bone. Nanoindentation was conducted in 12 different directions in three principal planes for both osteonic and interstitial lamellae. The experimental indentation modulus was found to vary with indentation direction and showed obvious anisotropy (oneway analysis of variance test, P < 0.0001). Because experimental indentation modulus in a specific direction is determined by all of the elastic constants of cortical bone, a complex theoretical model is required to analyze the experimental results. A recently developed analysis of indentation for the properties of anisotropic materials was used to quantitatively predict indentation modulus by using the stiffness matrix of human tibial cortical bone, which was obtained from previous ultrasound studies. After allowing for the effects of specimen preparation (dehydrated specimens in nanoindentation tests vs. moist specimens in ultrasound tests) and the structural properties of bone (different microcomponents with different mechanical properties), there were no statistically significant differences between the corrected experimental indentation modulus (M,:np)values and corresponding predicted indentation modulus (M,J values (two-tailed unpaired t-test, P > 0.5). The variation of M,,, values was found to exhibit the same trends as the corrected McTpdata. These results show that the effects of anisotropy on nanoindentation measurements can be quantitatively evaluated. 0 2002 Orthopaedic Research Society. Published by Elsevier Science Ltd. All rights reserved. Kejwordsr Anisotropy: Nanoindentation; Modulus; Bone
Introduction Cortical bone is an anisotropic material, and its mechanical properties are determined by its composition as well as microstructure. At the microstructural level, cortical bone is composed primarily of osteonic lamellae and interstitial lamellae. A typical secondary osteon is a cylinder measuring approximately 200-250 ym in diameter and is surrounded by interstitial lamellae. Osteonic lamellae and interstitial lamellae exhibit significantly different mechanical properties [12,13,21]. These differences may be due to a combination of factors, such as collagen fiber orientation, degree of mineralization, and arrangements of these materials [2,3,5,12]. Even within a whole osteon, a decline has been observed in both elastic modulus and hardncss from the center of the osteon *Corresponding author. Tel.: +1-901-678-5485; fax: +1-901-678528 I . E-muiL crtldress: [email protected] (J.Y. Rho).
outward in mature secondary osteons [14]. The complexity of cortical bone also arises from its hierarchical structural organization. The anisotropy may be partly due to the highly anisotropic structure of mineralized collagen fibrils (the basic building block of cortical bone). The fibrils are found as bundles or aligned arrays, and these can be arranged in a variety of different patterns, resulting in different mechanical properties in all three orthogonal directions. Although the patterns of lamellae are still a matter of dispute, many researchers have indicated that cortical bone has orthotropic material properties [8,10,19,20]. A better understanding of bone anisotropy can help determine the basic mechanical functions of cortical bone. In this study, the nanoindentation technique was applied to probe the mechanical properties at the microstructural level. Nanoindentation has been shown to be an effective method to probe the mechanical properties of microstructures at the micron scale with a positioning resolution of 1 pm. The lamellar units of
0736-02661021S - see front matter 0 2002 Orthopaedic Research Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 0 2 6 6 ( 0 1)OOl86-3
Z . Fun et ul. I Journul
of
Orihopuediic Reseurch 20 (2002) 806-810
cortical bone are about 3-5 pm thick. The dimensions of the microstructural building blocks of osteons are well within the capabilities of nanoindentation measurement. For isotropic materials, nanoindentation can be used to obtain intrinsic material properties, such as hardness and Young's modulus. However, for anisotropic materials, the indentation modulus in a specific direction is a composite quantity that depends on all of the elastic constants [ 13,181. To our knowledge, there are no studies in the literature that quantitatively evaluate the anisotropy of cortical bone via the nanoindentation technique. Therefore, in this study a recently developed theoretical model [I 6,171, which can predict indentation modulus from the material elastic constants matrix by assuming cortical bone to have orthotropic properties, was used to calculate the predicted indentation modulus. The values of the experimental indentation modulus (Mexp)and the predicted indentation modulus (MPrc) were compared to quantitatively investigate the effects of elastic anisotropy on nanoindentation measurements in cortical bone. Materials and methods E.xperimen tul metliod5 A bone sample was obtained from the midshaft of one human tibia (52 years old, male; cause of death: sudden heart attack). After being dehydrated in a series of alcohol baths, 12 specimens (about 3 x 3 x 4 mm' in size) were cut with various surface orientations to assess the anisotropy of cortical bone. As shown in Fig. I , an orthogonal coordinate system was chosen, where directions 1, 2, and 3 were the radial, circumferential and longitudinal directions, respectively. In this figure, D l , , 0 2 2 , and D13 represent indentation directions parallel to the three principal directions. Besides the three principal directions, three indentation direcrions were chosen that were equally distributed between the principal directions at each principal plane. For example, D12,30 is the indentation direction in the transverse (1-2) plane. 30" away from the radial (1) axis. Specimens were then embedded in epoxy resin without infiltration at room temperature. The epoxy resin was used to provide support during tests. and has no effect on test results provided
1
D,, Longitudinal
807
no indents were performed close to the bonelresin boundary. After grinding the surface of the specimens with silicon carbide abrasive papers of progressively finer grit sizes (600, 800, and 1200 grit), specimens were polished on microcloths with a 0.05 pm alumina suspension. Nanoindentation was conducted in 12 dillerent directions with respect to the principal planes. All nanoindentation tests (Nano Indenter 11, MTS Systems Co., Oak Ridge, TN, USA) were performed in load control mode with a Berkovich tip. The position of each indent was carefully selected based on microscopy observations. A total of 360 indents were made in thick lamellae from five osteons and five surrounding interstitial lamellae for each orientation. Three indents were made in each target area to reduce the effects of variations within osteonic lamellae and interstitial lamellae. The indenter was loaded and unloaded three times, with two intervening periods during which the load was held constant. To minimize the elltcts of viscoelasticity and creep on property measurements, a relatively long constant load hold period was conducted before the final unloading to diminish viscoelastic deformation to a negligible rate. The second constant load hold period, performed near the end of the test at 10% of the peak load. was used to establish the rate of thermal expansion o r contraction of the testing apparatus, to correct the displacement data for thermal drift. The maximum load of the nanoindentation tests was 8 mN, with a loading/unloading rate of 400 ~ N l s ,which produced impressions in bone with depths of about 700 nm. The indentation modulus (Meyp) for each indentation was determined from the well-documented method of Oliver and Pharr [9]. In a previous ultrasound study [lo], samples from three transverse sections located at 30%, 50'1/0, and 70'%1of total length from the end of the bone in each of eight tibiae (seven men and one woman ranging in age from 45 to 68 years). Four roughly cubic samples (about 5 x 5 x 3 mm3 in size) were prepared from four quadrants (anterior, medial. posterior, and lateral) of each transverse section. This resulted a total of 96 cortical bone specimens. During the ultrasound tests, the specimens were kept moist in a saline solution.
Tlieoreticul methods In nanoindentation experiments, measurements of load and displacement are utilized to determine the contact stiffness, from which the indentation modulus can be calculated. For anisotropic materials. indentation modulus ( M ) is defined by following equation [18]:
where S is contact stillness and A , the projected area of contact, which is obtained during calibration [9]. Conical indentation of isotropic materials is well understood [4]. Anisotropic materials, for which displacements of the indented surface vary in different directions, require a more complicated analysis for the determination of indentation properties. Based on a recently developed analysis for the indentation of anisotropic malerials by a cone [ I6.171, the value of M in a certain direction is given by
UI
D t t Radial
Fig. 1. Orientation ofthe principal axes and the indentation directions. DI Dzz, and D j 3 represent indentation directions parallel to the three principal directions. D I z 1 3 is~the indentation direction in transverse (12) plane, 30" away from the radial ( I ) axis.
where u 1 / u 2(the ratio of elliptical axes of the projected area of contact) can be determined numerically; B,,. the components of the first Barnett-Lothe tensor, which are derived from the elastic stiffness matrix; a,,and u3,, the direction cosines of the angles made between the indentation direction and the principal directions: and ?, the angle used to define the displacement direction at the free surface. T o evaluate the nanoindentation experimental results, Eq. (2) was utilized to calculate the theoretical indentation modulus by using the stiffness matrix of human tibia1 cortical bone, which was determined by earlier ultrasonic velocity measurements [lo]. A pyramidal tip was used in this study. but there are no exact analytical expressions for pyramidal tip indentation. Numerical studies [6,7] have suggested that a correction factor 1.034 should be used to correct the modulus values predicted from conical tip to match the results obtained from a pyramidal tip.
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The indentation modulus of an anisotropic material in any direction can be predicted by using its anisotropic elastic constants. The elastic stiffness components determined by Rho [lo] in ultrasound tests were used to calculate the predicted values (Mpre) at various orientations. It is important to note that nanoindentation experiments were performed at the microstructural level (with data obtained from osteonic and interstitial lamellae) and measured the mechanical response of the solid constituents in the vicinity of the indenter. On the other hand, ultrasound experiments were performed at the macrostructural level, which included not only osteonic and interstitial lamellae, but also lacunae and cement lines. To account for differences between the indentation modulus obtained from the osteonic and the interstitial lamellae, the value for indentation modulus derived from whole specimens might be better predicted by taking the area fraction of each tissue type, multiplying by its value of indentation modulus, and adding them [15]. This procedure implies a simple rule of mixture, which can be expressed as following equation:
where CSAF is the corrected secondary osteonal area, and is 0.388 for human tibia1 bone [15]. A void volume fraction of approximately 5.5% caused differences between the moduli determined by nanoindentation and by ultrasonic velocity measurements. To account for the effect of voids, the product ofMmlrlurr and V, (1 percentage of voids) was used to produce the adjusted indentation modulus. Because the size of indentations made exceeded the dimensions of cement lines (1-2 pm wide), cement lines were not tested in this study. In addition, for ultrasonic tests the bone specimens were kept moist in a saline solution, whereas for the nanoindentation tests, specimens were dehydrated in a series of alcohol baths. Rho and Pharr [Ill suggested that dehydration increases the indentation modulus of bovine bone by approximately 15%. In the absence of wetldry nanoindentation data for human tibia, it is reasonable to assume a similar effect here, although there are differences between bovine bone and human tibia. After the correction for dehydration, the final corrected indentation moduli (Mcxp)were produced. To compare experimental indentation modulus values for osteonic and interstitial lamellae in each of the three principal planes, one-way analysis of variance (ANOVA) tests were performed, with P < 0.01 for significance. When ANOVA revealed differences, the BonferronilDunn test (significance level: 1%) was employed to determine if specific pairwise comparisons were significantly different. The Bonferronil Duiin test was chosen for the more conservative results, since the experimental data were near normality. To compare corrected Me.? data with Mprcresults, a two-tailed unpaired t-test was used. All statistical analyses were performed with StatView 5.0.1 (SAS Institute Inc., USA). ~
Results
The average values of experimental indentation modulus for osteonic and interstitial lamellae obtained from all tests are listed in Table 1. One-way ANOVA tests indicated that there were significant differences between osteonic and interstitial lamellae in all directions (P < 0.0001). Since interstitial bone is older with higher mineral content than osteon, it would be expected to have a higher elastic modulus than osteon. Indentation modulus data in each principal plane for osteonic and interstitial lamellae were compared with one-way ANOVA test. Significant differences between interstitial lamellae moduli data existed in all three principal planes while for osteonic lamellae significant differences were only observed in the 1-3 and the 2-3 planes. In the 2-3 plane, the Bonferroni/Dunn method shows that the nanoindentation moduli are not significantly different
Table 1 Average experimental indentation modulus values (Mexp)for osteonic and interstitial lamellae Indentation directions
Experimental indentation modulus (Mcxp,GPa), average SD
+
Osteonic lamellae
Interstitial lamellae
16.6+ 1.5 17.0 2.2 25.1 2.1 15.7 i 2.3 16.0 2.0 14.9 2.0 18.4+ 1.1 18.4 f 0.9 21.8 f2.6 17.6f 1.3 1 8 . 2 f 1.4 21.9 f 2.0
19.7 1.5 18.5 1.1 27.1 f 1.7 20.4 1.8 21.1 f 1.4 18.9+ 1.4 20.0 1.4 22.4 1.6 24.8 f 1.0 20.0 i 1.4 2 l . 7 + 1.2 25.2 f I .O
+ +
*
+
* * *
0 2 2 , and 0 3 3 represent indentation directions parallel to the three principal directions. D12/3,, means the indentation direction in transverse (1-2) plane, 30" away from radial ( I ) axis. (SD: standard deviation.)
01 I,
between adjacent orientations, but as the angle between indentation directions increases, significant differences are revealed. The same pattern is observed in both osteonic and interstitial lamellae and for both the 2-3 and the 1-3 planes, suggesting orthogonal mechanical properties. To compare the corrected experimental data (Mexp) with predicted (Mpre)results, a rule of mixture (Eq. (3)) and the assumption of a 15% increase due to drying was used. In all 12 directions, the average modulus values from each osteons and its surrounding interstitial lamellae were used to calculate corrected Mexp.Table 2 shows the corrected Mexp values and predicted Mpre
Table 2 Corrected experimental (Men,,)and predicted (Mpre)indentation modulus values for various orientations Indentation direction
Corrected Meap(GPa) average f S D 14.9 i0.98 14.4 f 0.35 21.1 f 1.01 14.9 0.96 15.4* 1.46 13.9 f 1.78 15.6 f 0.89 16.7 0.57 19.0 0.95 15.3 0.48 16.3 f 0.60 19.2 i 1.57
*
+
+
Mprc(GPa) average
14.0 14.5 19.7 14.1 14.2 14.4 15.0 16.3 17.8 15.5 16.7 18.0
Corrected Mcxpdata were obtained after a rule of mixture (Eq. (3))and assuming a 15'% increase for dehydration. Predicted Mprevalues were calculated using elastic stiffness components determined by ultrasound test [lo]. (SD: standard deviation.)
corrected experimental (Mew) -predicted (Mpre)
26 24
7 I
2-221 9
12
0
15 30 45 60 75 90 angle of indentation in 2-3plane (degrees)
Fig. 2. Indentation modulus shows a statistically significant variation in the radial-longitudinal (2-3) plane. Experimental modulus (Alex,,) was determined by a rule of mixture, which considers osteonic, interstitial lamellae and void ratio; corrected Mcxpwas determined by assuming a IS%> increase due to dehydration of experimental modulus [l I]. Predicted modulus (MPrc) was calculated using elastic stiffness components determined by ultrasound tests [ I 01.
values at all orientations. Comparing these results, the predicted M,,,, values were found to exhibit the same trend as the experimental results in all orientations. Fig. 2 shows the trend in the radial-longitudinal (2-3) plane. The same pattern was also found in the 1-3 plane. There were no statistically significant differences between corrected Mexpvalues and M,,,, values (two-tailed unpaired t-test, P > 0.5).
Discussion For the indentation of isotropic materials, the OliverPharr method [9] can predict intrinsic material properties, such as Young modulus, within 4%) of literature values. For anisotropic materials, the indentation modulus represents a weighted average quantity, as the formation of the contact impression involves deformation in all three principal directions. Therefore, the modulus is not easily interpreted by conventional methods. Ziv et al. [20] utilized microhardness testing to investigate the variation of hardness at various orientations of both parallel-fibered and lamellar bone. Due to its limited lateral resolution (30 pm), the variation between osteonic and interstitial lamellae was beyond the capability of microhardness testing. Consequently, their results may have a lower correlation to microstructural properties compared with the results of the current study. Akiva et al. [l] studied the anisotropic properties of unidirectional plane-parallel platelet- or ribbon-reinforced composites, which have a structure similar as collagen fibrils. They found that the elastic behavior resulting from a platelet orientation is a char-
acteristic of three-dimensional orthotropic materials. Such a material has larger elastic constants at the platelet edge-on direction. The mineral in cortical bone can be treated as platelets. Because the indentation modulus values close to the longitudinal direction are larger in the present study, this suggests that more of the crystal layers are encountered edge-on in this direction. Moreover, since the gradual rotation of the crystal layers inside collagen fibrils in osteonic lamellae produces relatively isotropic properties [19], indentation modulus data from osteons in the 1-2 plane d o not show statistically significant differences (Table 2). However, the significant differences among interstitial lamellae indicate that the crystal layers might have a different structure in the 1-2 plane, requiring further investigation. These previous studies concentrated on qualitative evaluation. To describe the anisotropy of cortical bone more precisely, it is necessary to develop a mathematical model to quantitatively evaluate bone anisotropy, to consider the effects of all of the elastic constants. The currently developed analysis was designed to fulfill this objective. As shown in this study, the variation of predicted indentation modulus values was found to be consistent with the variation of the adjusted experimental values. Following correction for dehydration and void fraction, there are no statistically significant differences between the corrected values and the predicted values. Ultrasound measurement usually includes osteonic lamellac, interstitial lamellae, porosity, cement lines, and osteocytes, whereas the nanoindentation test in this study only included osteonic and interstitial lamellae. However, the similar trend between model and nanoindentation tests suggests that osteonic and interstitial lamellae are major contributors for anisotropy of cortical bone. The matrix of cortical bone is more complex than the model presented here. To move realistically from individual lamellae to a macroscopic bone sample, different lamellar types, dominant collagen bundle directions, degree of calcification and cement lines should be considered. However, there is currently no composite material model of bone that incorporates all of these items. The present model, although accounting only for osteonic and interstitial lamellae, is shown to be sufficient to describe the anisotropy of human tibia cortical bone. In the previous ultrasound study [lo], no significant variation in elastic properties was found along the length and around the circumference of cortical bone ( P > 0.1) of human tibia, so the site-specific differences of specimens were assumed not to be a critical issue in this study. However, i t is possible that part of the variation could be due to the site-specific differences. In addition, the data obtained from one individual tibia cannot be correlated to all human tibiae. Another limitation in our study is the assumption that the structure of individual human lamella is orthotropic. Such an
Z. Fun et ul. I Journal of Orthopuedic Research 20 (2002) 806-810
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assumption is not justified by the literature [S, 19,201, which presents orthotropic properties of a mesh of lamellae with various structural directions, not even one. The present study demonstrates that the effects of anisotropy of cortical bone to nanoindentation measurement can be quantitatively evaluated. These results should be carefully interpreted because two different specimens were used to compare the experimental indentation modulus with the predicted indentation modulus, and the present model has several limitations. Nevertheless, the anisotropic indentation properties of cortical bone are sufficiently described by using the analysis of indentation of anisotropic materials.
Acknowledgements
This research was sponsored in part by the NIH45297 and NSF DMR-0076497. Instrumentation for the nanoindentation work at the Oak Ridge National Laboratory SHaRE User Facility was provided by the Division of Materials Sciences and Engineering, US Department of Energy, under Contract DE-AC05-000R22725 with UT-Battelle, LLC and through the SHaRE Program under Contract DE-AC05-760R00033 with Oak Ridge Associated Universities.
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