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Contribution of inter-site variations in architecture to trabecular bone apparent yield strains
ARTICLE IN PRESS
Journal of Biomechanics 37 (2004) 1413–1420
Contribution of inter-site variations in architecture to trabecular bone apparent yield strains Elise F. Morgana,*, Harun H. Bayraktara, Oscar C. Yehb, Sharmila Majumdarc,d, Andrew Burghardtc, Tony M. Keavenya,d a
Orthopaedic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA b Orthopaedic Research Laboratories, School of Medicine, University of California, Davis, CA 95817, USA c Department of Radiology, University of California, San Francisco, CA 94143, USA d Department of Bioengineering, University of California, Berkeley, CA 94720, USA Accepted 8 December 2003
Abstract Apparent yield strains for trabecular bone are uniform within an anatomic site but can vary across site. The overall goal of this study was to characterize the contribution of inter-site differences in trabecular architecture to corresponding variations in apparent yield strains. High-resolution, small deformation finite element analyses were used to compute apparent compressive and tensile yield strains in four sites (n=7 specimens per site): human proximal tibia, greater trochanter, femoral neck, and bovine proximal tibia. These sites display differences in compressive, but not tensile, apparent yield strains. Inter-site differences in architecture were captured implicitly in the model geometries, and these differences were isolated as the sole source of variability across sites by using identical tissue properties in all models. Thus, the effects inter-site variations in architecture on yield strain could be assessed by comparing computed yield strains across site. No inter-site differences in computed yield strains were found for either loading mode (p>0.19), indicating that, within the context of small deformations, inter-site variations in architecture do not affect apparent yield strains. However, results of ancillary analyses designed to test the validity of the small deformation assumption strongly suggested that the propensity to undergo large deformations constitutes an important contribution of architecture to inter-site variations in apparent compressive yield strains. Large deformations substantially reduced apparent compressive, but not tensile, yield strains. These findings indicate the importance of incorporating large deformation capabilities in computational analyses of trabecular bone. This may be critical when investigating the biomechanical consequences of trabecular thinning and loss. r 2004 Elsevier Ltd. All rights reserved. Keywords: Cancellous bone; Anatomic variation; Trabecular architecture; Trabecular tissue properties; Biomechanics
1. Introduction The development of effective descriptors of the mechanical behavior of trabecular bone is complicated by the large degree of mechanical heterogeneity that exists between individuals as well as between and within anatomic sites. One exception is the relative uniformity of apparent yield strains. Apparent yield strains for trabecular bone are only weakly dependent on density (Kopperdahl and Keaveny, 1998; Turner, 1989) and display intra-site variations on the order of one-tenth of *Corresponding author. Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215, USA. Tel.: +1-617-353-2791; fax: +1-617-353-5866. E-mail address: [email protected] (E.F. Morgan). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.12.037
the mean value (Morgan and Keaveny, 2001). In contrast, modulus and yield stress depend heavily on density and exhibit coefficients of variation per site of up to 70% (Linde et al., 1989; Morgan and Keaveny, 2001; Mosekilde et al., 1987). Thus, yield strain may be a more statistically powerful measure of the age-, disease-, and drug-related changes that occur in trabecular bone strength. Identifying factors such as trabecular architecture that contribute to changes in yield strain will improve this diagnostic capability and increase the potential for therapeutic interventions. While apparent yield strains are uniform within a single anatomic site, they can vary across sites (Morgan and Keaveny, 2001). These inter-site differences may be due to differences in trabecular architecture (Hildebrand et al., 1999; Ulrich et al., 1999) and tissue material
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properties (Hoffler et al., 2000). High-resolution finite element modeling provides the ability to control tissue properties, thereby isolating the effect of architecture. This technique has been used to explore contributions of architecture to apparent elastic behavior (Kabel et al., 1999; Ulrich et al., 1999). However, the contribution of architecture to yield strain may be very different, given that values of yield strain are greater than twice the applied strain at which modulus is measured. At higher applied strains, deformations within the trabecular network such as bending and buckling can become increasingly important (Bell et al., 1967; Gibson, 1985; Kopperdahl and Keaveny, 1998; Muller . et al., 1998; Pugh et al., 1973; Stolken and Kinney, 2003). The magnitude and prevalence of these deformations depends on architecture; for example, slender, rod-like trabeculae are more susceptible than are thicker, platelike trabeculae. Quantifying tissue level strains can provide mechanistic insight into the types of deformations that occur during yielding and how these deformations differ with architecture. It is important to note that most high-resolution analysis techniques assume that deformations within the trabecular network are relatively small, due to the prohibitively high computational cost of allowing for large deformations in such large models. Consequently, these analyses may not sufficiently model the full effect of architecture on apparent yield strains. Whether large deformations must be incorporated in high-resolution analyses, and whether these deformations significantly impact relationships between architecture and yield strain are open questions. The overall goal of this study was to characterize the contribution of inter-site variations in trabecular architecture to corresponding variations in apparent yield strains. A combined approach was taken whereby highresolution finite element analyses were supplemented by lattice-type (Yeh and Keaveny, 1999) finite element
analyses that included large deformation capabilities. The high-resolution analyses were used to compute apparent compressive and tensile yield strains in four anatomic sites that display differences in compressive but not tensile apparent yield strains (Morgan and Keaveny, 2001): the human proximal tibia, femoral greater trochanter, femoral neck, and the bovine proximal tibia. Inter-site differences in architecture were captured implicitly in the model geometries, and these differences were isolated as the sole source of variability across sites by using a single set of tissue properties in all models. Thus, under the assumption of small deformations, the effects inter-site variations in architecture on yield strain could be assessed by comparing computed yield strains across site. The lattice-type analyses were then used to quantify the effect on apparent yield strains of large deformations. Results of both types of finite element analyses were interpreted in the context of experimentally measured yield strains. The specific objectives of this study were to: (1) compare computed apparent yield strains across anatomic site; (2) quantify tissue strain distributions at the apparent yield point; and (3) assess the effect of large deformations on apparent yield strains.
2. Methods 2.1. High-resolution finite element analyses Seven cylindrical, on-axis specimens from each of the four anatomic sites were used (Table 1, Fig. 3); these specimens were randomly selected from larger groups used in previously published studies (Morgan and Keaveny, 2001; Niebur et al., 2000). Details of the specimen preparation and mechanical testing methods are described in these studies. Briefly, specimens were loaded to either the 0.2% offset yield point (human
Table 1 Specimen and cadaver information by anatomic site Anatomic site
n
No. donorsa (male/female)
Age
Volume fraction
Tb.Sp (mm)
Tb.Th (mm)
Tb.N (1/mm)
Greater trochanter
7
0.960b70.190
0.137b70.013
1.022b70.165
7
0.12b70.03
0.803b70.093
0.135b70.006
1.184b70.150
Femoral neck
7
0.29c70.06
0.605c70.085
0.196c70.030
1.589c70.186
Bovine proximal tibia
7
7579 62–87 62716 40–84 6375 57–72 —
0.11b70.02
Proximal tibia
7 (5/2) 5 (5/0) 7 (5/2) —
0.23d70.03
0.548c70.043
0.135b70.010
1.656c70.129
Age is reported as mean7standard deviation, followed by the range. Volume fraction, mean trabecular spacing (Tb.Sp), trabecular thickness (Tb.Th), and trabecular number (Tb.N) are reported as mean standard deviation. a Three donors (two male and one female) that provided specimens from the greater trochanter also provided specimens from the femoral neck. There was no overlap between donors for the human proximal tibia specimens and those for either the femoral neck or greater trochanter specimens. bac po0.04. bad po0.001. cad po0.04.
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specimens) or to 0.2% strain (bovine specimens) using mechanical testing protocols designed to minimize end artifacts (Keaveny et al., 1994a; Kopperdahl and Keaveny, 1998). Volume fractions were calculated using Archimedes’ Principle (Galante et al., 1970). Following mechanical testing, high-resolution threedimensional reconstructions of each specimen were obtained either using serial milling (Beck et al., 1997) or micro-computed tomography (Scanco mCT 20; Scanco Medical AG, Bassersdorf, Switzerland) at 10 and 22 mm resolution, respectively. For each specimen, the grayscale threshold was selected to match the reconstruction volume fraction with the experimentally measured value. To provide quantitative evidence of inter-site differences in trabecular architecture, standard metrics of architecture were measured from the threedimensional reconstructions (Hildebrand et al., 1999): mean trabecular thickness (Tb.Th), trabecular spacing (Tb.Sp), and trabecular number (Tb.N). These metrics were compared across site via individual analyses of variance (ANOVA). Using a voxel-based approach (Hollister et al., 1994), a 5 mm cube high-resolution finite element model was created from the central region of each specimen reconstruction. Prior to this conversion, region averaging was used to decrease the resolution to 44–66 mm in order to decrease computation time but still ensure that the voxel size was less than one-fourth of the mean trabecular thickness, as recommended for numerical convergence (Niebur et al., 1999). Materially nonlinear, high-resolution finite element analyses using an implicit incremental method were conducted to determine the apparent yield point, in each of compression and tension, as defined by the 0.2% offset criterion. The analyses incorporated material nonlinearity (i.e. tissue yielding) but not geometric nonlinearity (i.e. large deformations). Trabecular tissue was modeled as a bilinearly elastic material with a postyield modulus equal to 5% of the initial modulus (Niebur et al., 2000). Although a specimen-specific effective tissue modulus was calibrated for each specimen, the computed apparent yield strain for each loading mode was independent of the modulus due to the use of a strain-based tissue constitutive model. The tissue tensile and compressive yield strains used in all models were 0.412% and 0.825%, respectively. These values were the mean tissue yield strains from a calibration study conducted on 12 human femoral neck specimens (Bayraktar et al., 2004). To our knowledge, there are currently no other data in the literature on tissue yield strains in human trabecular bone for any anatomic site. Each analysis required 150–250 CPU h on an IBM SP2 parallel supercomputer (IBM, Armonk, NY), and a total of 56 analyses were performed. In each loading mode, computed apparent yield strains were compared across anatomic site and against
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experimental data (Chang et al., 1999; Morgan and Keaveny, 2001) using a two-factor ANOVA and Tukey post hoc test (JMP 5.0, SAS Institute, Cary, NC). Although the comparison of computed values across site was the primary comparison in this study design, the comparison against experimental values was important for two reasons. First, it provided a positive control. The single pair of tissue yield strains used in all finite element models was the pair of mean values calculated previously for trabecular tissue from the femoral neck. Agreement between computed and experimental apparent yield strains for this site would indicate that the model results can match the experimental data without accounting for intra-site variations in tissue yield properties. Second, the comparison against experimental data provided means for synthesizing the results of the high-resolution and lattice-type analysis results in order to assess the role of large deformations. The differences between computed and experimental values of yield strain and those between yield strains resulting from small and large deformation lattice-type analyses were juxtaposed to provide insight into the extent to which large deformations impact relationships between architecture and yield strain. The bovine experimental stress–strain curves were reprocessed to have a uniform definition of apparent modulus (slope at zero strain of a quadratic fit from 0% to 0.2% strain (Morgan et al., 2001). This had the effect of lowering both the compressive and tensile apparent yield strains for these data as compared to our previously published values (Chang et al., 1999; Keaveny et al., 1994b). The reprocessed mean (7SD) apparent compressive and tensile yield strains for this site were 0.79 (70.06)% and 0.61 (70.06)%, respectively. Tissue strain distributions were quantified at the apparent yield point for each simulation to provide mechanistic insight into the model predictions. At the apparent yield point, either compressive or tensile, the percentage of the total amount of tissue in the specimen that had reached either the compressive or tensile tissue level yield strain was calculated. The calculation was also performed separately for each tissue level yielding mode such that the amount of tissue that had reached the tissue compressive yield point could be compared to the amount that had reached the tissue tensile yield point. Linear regression analyses were used to assess relationships between the amount of tissue yielded and volume fraction. 2.2. Lattice-type finite element analyses To assess the contribution of large deformations to apparent yield strains, a three-dimensional lattice-type finite element model (Yeh and Keaveny, 1999) was used to simulate apparent loading of a 7.1 mm cube of trabecular bone with rod-like architecture and a volume
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fraction in the range 0.098–0.109. Each trabecula consisted of five quadratic beam elements and was assigned a unique thickness, length, and orientation typical of real trabeculae. The intra-specimen thickness distribution followed a b-distribution with a coefficient of variation of 40%, a mean of 215 mm (vertical trabeculae) or 106.6 mm (horizontal trabeculae), a minimum of 50 mm, and a maximum of 550 mm (Yeh and Keaveny, 1999). The length and orientation of each trabecula were perturbed by a random amount controlled by a ‘‘relative lattice disorder constant’’ (Jensen et al., 1990) of a=0.6. Five models, each with unique mesh geometry, were generated using these parameters. Trabecular tissue was modeled as an elastic–plastic material (E=18 GPa, v=0.3, Epost-yield=0.9 GPa) with the same tensile and compressive yield strains as used in the high-resolution finite element analyses. Monotonic, uniaxial compression and tension simulations (ABAQUS version 6.2; Abaqus, Inc., Pawtucket, RI) to 70.8% strain were performed on each model with the assumption of small deformations enforced (material nonlinearity only), and again without this assumption (material and geometric nonlinearity). These simulations were also performed using pure linear elastic tissue material behavior in order to examine the contribution of geometric nonlinearity alone. In each loading mode, resulting apparent yield strains for each type of analysis were averaged across the five models, and a paired t-test was used to detect differences in yield strains between the materially nonlinear and fully nonlinear cases.
3. Results In the high-resolution finite element analyses, no differences were found between sites in computed yield
strains for either compressive (p>0.50) or tensile (p>0.19) loading (power>0.84, Fig. 1). Despite statistically significant differences in architectural metrics between sites (po0.04, Table 1), differences between sites in mean computed yield strains were less than 11% in each loading mode. Tissue level strain distributions showed considerable variability between sites and with volume fraction. At each of the compressive and tensile apparent yield points, the percentage of tissue exceeding either the compressive or tensile tissue level yield strain increased with volume fraction from less than 4% to greater than 25% (po0.001, Fig. 2A). When the apparent loading mode was tensile, nearly all tissue yielding was tensile. For compressive loading, however, the ratio of the amount of tissue that had reached the tissue compressive yield strain to the amount that had reached the tissue tensile yield strain increased with volume fraction from 0.53 to 2.7 (r2=0.84, po0.001, Fig. 2B). At low-volume fractions, regions of compressive and tensile tissue failure were often found on opposite sides of slender trabeculae (Fig. 3), indicative of localized bending. Comparison of computed yield strains to the experimental data revealed that the high-resolution finite element models did not capture the experimentally observed inter-site differences in compressive yield strains. Mean computed compressive yield strains were 21%, 30%, and 16% higher than corresponding mean experimental values for the human tibia, trochanter, and bovine tibia, respectively (po0.001, Fig. 1). In tension, by contrast, no differences were found between computed and apparent yield strains (p=0.44, power=0.73). In the lattice-type finite element models, large deformations had a substantial effect on the magnitude of the apparent compressive, but not tensile, yield
Compression
*
Apparent Yield Strain (%)
1.0
a
0.8
*
Tension
*
Experimental
1.0
FEM 0.8
a,b
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Femoral Neck
Proximal Greater Tibia Trochanter
Bovine Tibia
Femoral Neck
Proximal Greater Tibia Trochanter
Bovine Tibia
Fig. 1. Mean computed (‘‘FEM’’) and experimental compressive (left) and tensile (right) yield strains. Error bars indicate 1 SD. No inter-site differences in computed apparent yield strains were found for either loading mode, indicating that, when small deformations are assumed, inter-site variations in architecture do no contribute to variations in apparent yield strains.
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deformations caused a stiffening effect that was counteracted by tissue yielding (Fig. 4B). No difference in tensile yield strains were found between the small and large deformation cases (p=1.0, Table 2), indicating that, due to the lower tensile tissue yield strain, material nonlinearity was the dominant factor for apparent tensile loading. Neither the small deformation assumption nor the choice of tissue constitutive behavior affected the apparent modulus.
Apparent Compression
35
Proximal Tibia
Percentage of Tissue Yielded (%)
Greater Trochanter
30
Femoral Neck Bovine Proximal Tibia
25 20 15 10
4. Discussion
5 y = 0.90 + 53.42x r = 0.61
0 0.00
0.10
0.20
0.30
0.40
Volume Fraction
(A)
Apparent Compression
3.0
Proximal Tibia
Tissue Yielded in Compression Tissue Yielded in Tension
Greater Trochanter
2.5
Femoral Neck Bovine Proximal Tibia
2.0 1.5 1.0 0.5 y = 0.11 + 6.48x R = 0.84
0.0 0.00 (B)
1417
0.10
0.20
0.30
0.40
Volume Fraction
Fig. 2. (A) The total amount of tissue yielded (expressed as a percentage of the volume of tissue in the specimen) at the apparent level yield point increased with volume fraction for apparent compressive loading (po0.001). Although not shown, the trend was similar for apparent tensile loading (y=0.10+75.84x, r2=0.62, po0.001). (B) The ratio of the amount of tissue yielded in compression to that yielded in tension at the apparent compressive yield point increased with volume fraction (po0.001). At low-volume fractions, this ratio was less than one, indicating that for these specimens, more tissue yielded in tension than compression even though the apparent loading mode was compressive.
strains. Both geometric nonlinearity alone and material nonlinearity alone caused a softening effect in the apparent compressive stress–strain curve (Fig. 4A). The combined effect of geometric and material nonlinearity was even more pronounced: allowing for large deformations lowered the apparent compressive yield strain by 22% as compared to the small deformation case (po0.001, Table 2). In tension, however, large
The overall goal of this study was to characterize the contribution of inter-site variations in trabecular architecture to corresponding variations in apparent yield strains. The effects of architecture were isolated through the use of identical tissue properties in high-resolution, small deformation finite element analyses, and the resulting computed apparent yield strains were compared across site. No inter-site differences in computed yield strains were found for either tension or compression, indicating that, when small deformations are assumed, inter-site variations in architecture do not contribute to apparent yield strains. However, the lattice-type finite element results demonstrated that the small deformation assumption is not appropriate for all types of trabecular architecture. In the rod-like architecture of the lattice models, large deformations reduced the apparent compressive, but not tensile, yield strain. This was consistent with the finding that, in the anatomic sites with low trabecular thickness and high trabecular spacing, the high-resolution, small deformation analyses over-predicted apparent compressive, but not tensile, yield strains. Although this over-prediction could theoretically be due to inter-site differences in tissue yield strains, data from a previous study (Bayraktar et al., 2004) suggest that attributing the discrepancy to tissue property differences alone would require that tissue compressive yield strains differ by over 40% between the greater trochanter and femoral neck. There is no evidence in the literature to suggest that tissue yield strains vary to such a large extent within metaphyses. Therefore, these collective results suggest that the propensity to undergo large deformations constitutes an important contribution of architecture to inter-site differences in apparent compressive yield strains. They further imply that computational analyses, particularly those of failure in low-density bone, should allow for large deformations even when apparent strains are less than 1%. This may be critical for the study of age- and disease-related fragility where trabecular thinning and loss occur. The main strength of this study lies in the use of a combination of computational and experimental techniques to isolate and characterize the effect of inter-site
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Fig. 3. Renderings of the central 3–4 mm cubic portions of high-resolution finite element models of representative specimens from the (A) femoral neck, (B) human proximal tibia, (C) greater trochanter, and (D) bovine proximal tibia illustrate the inter-site variability in tissue strain distributions when the apparent compressive yield point is reached. (Red) Regions exceeding the tensile tissue yield strain; (blue) regions exceeding the compressive tissue yield strain. At lower volume fractions, only a small percentage of the tissue yielded, regions of yield were highly localized, and regions of tensile and compressive yield were often located on opposite sides of slender trabeculae. In comparison, at higher volume fractions, a larger percentage of the specimen yielded and regions yield were more evenly distributed throughout the specimen.
variations in architecture. The excellent match between computed and experimental apparent yield strains in both loading modes for the femoral neck—the positive control—confirmed the validity of this approach. In addition, the human specimens used for the finite element models were a subset of those that were mechanically tested, thus increasing the strength of comparisons between computed and experimental yield strains. A key assumption used in both types of computational models was that trabecular tissue has similar elastic and yield properties to cortical bone. This assumption is supported by similarities in elastic moduli (Rho et al., 1997; Turner et al., 1999) and tissue composition (Guo and Goldstein, 1997), and by the success of previous high-resolution finite element studies that have modeled trabecular tissue after cortical bone (Bayraktar et al., 2004; Niebur et al., 2000). While there remains a need for experimental investigation of trabecular tissue yield properties, the results of this study lend additional insight. The close agreement between computed and experimental apparent tensile yield strains, combined with the insensitivity of apparent tensile yield strains to the tissue compressive yield strain,
suggests that tissue tensile yield strains are relatively uniform across site and species. The picture is less clear for compression, due to the confounding effects of the small deformation assumption in the high-resolution analyses. Although we believe that the differences between computed and experimental apparent compressive yield strains for the greater trochanter and human tibia were primarily due to the small deformation assumption, the corresponding discrepancy for the bovine tibia may also be due to inter-species differences in tissue compressive yield strains. This dichotomy is supported by observations that large deformations occur in low-density whale vertebral bone, but not in higher density bovine tibial bone (Muller . et al., 1998). Furthermore, the tissue level strain distributions quantified in the current study illustrated that, even with the assumption of small deformations enforced, the deformation mechanisms during apparent compressive loading were dramatically different in the bovine tibia as compared to the trochanter and human tibia. Most notably, localized bending occurred with much greater frequency in the trochanter and human tibia than in the bovine tibia (Figs. 2, 3). Incorporating large deformations into the high-resolution analyses will enable
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Table 2 Apparent yield strains of low density trabecular bone computed from lattice-type finite element models and measured experimentally
3.5 3.0
Apparent compression
Apparent tension
n
Yield strain
n
Yield strain
5
0.6370.02a
5
0.5770.02
5
0.8170.02b
5
0.5770.02
5
0.9370.02
5
Infinite
10
0.6970.05
13
0.6170.05
15
0.7270.06
16
0.6570.05
Stress (MPa)
2.5 Computed Geometric and material nonlinearity Material nonlinearity Geometric nonlinearity
2.0 1.5 1.0
Geometric and Material Nonlinearity
Experimental Greater trochanter Proximal tibia
Material Nonlinearity
0.5
Geometric Nonlinearity Experiment
0.0 0.0
0.2
(A)
0.4
0.6
0.8
1.0
Apparent Tension
3.0 2.5 Stress (MPa)
po0.001.
Strain (%)
3.5
2.0 1.5 1.0
Geometric and Material Nonlinearity Material Nonlinearity
0.5
Geometric Nonlinearity Experiment
0.0 0.0 (B)
aab
0.2
0.4
0.6
0.8
1.0
Strain (%)
Fig. 4. Stress–strain curves for the lattice-type finite element models of low-density bone in (A) apparent compression and (B) apparent tension illustrate the effect of large deformations on the apparent yield point. The apparent yield point is indicated on each curve by the solid rectangle. The experimental data are from the greater trochanter. Mean values of apparent yield strain for all simulation cases are listed in Table 2.
computational investigations of inter-site and interspecies variations in tissue compressive yield strain. Our experimental design was unique in that it capitalized on the ability of the high-resolution models to capture architecture implicitly in the model geometry. This allowed the effects inter-site variations in architecture to be assessed by comparing computed yield
strains across site. An alternative approach, which has been used for apparent modulus and ultimate stress (Majumdar et al., 1998; Ulrich et al., 1999), would be to use statistical correlation or multiple regression techniques to develop relationships between apparent yield strain and three-dimensional structural indices such as trabecular spacing. Although this latter approach has the advantage that it could identify specific quantitative descriptors of the effect of architecture on yield strain, it relies on the assumption that the available structural indices sufficiently describe the architectural features that are most relevant for apparent yield strains. For instance, the results of the current study suggest that the relevant architectural features will be related to the propensity to undergo large deformations. In this context, measures of intra-specimen variations in architecture may be as important as mean values. The effect of intra-specimen variations in thickness on apparent modulus has been demonstrated (Yeh and Keaveny, 1999). The consequence of these variations for apparent yield strain may be particularly acute because yielding at the apparent level may initiate as local failure in a susceptible region of the specimen. The contribution of large deformations represents a possible explanation for clinical observations that small changes in bone mass can have dramatic effects on fracture risk (Meunier and Boivin, 1997; Watts et al., 1990). Focal thinning (or thickening) of trabeculae resulting from slight remodeling imbalances is expected to alter the magnitude and prevalence of large deformations. Buckling has long been proposed as a mechanistic example of how trabecular strength can be affected disproportionately to stiffness (Bell et al., 1967; Gibson, 1985; Parfitt, 1992; Pugh et al., 1973; Snyder et al., 1993; Stolken and Kinney, 2003). Although the stability analyses required to explicitly model buckling were not
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performed in this study, our results indicate that pronounced bending can have appreciable effects on compressive strength as well.
Acknowledgements Funding was provided by NIH AR43784, NIH AR41481, NSF BES-96250301, and NRAC UCB266. Human tissue was obtained from the Anatomic Gift Foundation and the National Disease Research Interchange. The authors wish to thank Jacob Pollock for his technical assistance.
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Kopperdahl, D.L., Keaveny, T.M., 1998. Yield strain behavior of trabecular bone. Journal of Biomechanics 31, 601–608. Linde, F., Hvid, I., Pongsoipetch, B., 1989. Energy absorptive properties of human trabecular bone specimens during axial compression. Journal of Orthopaedic Research 7, 432–439. Majumdar, S., Kothari, M., Augat, P., Newitt, D.C., Link, T.M., Lin, J.C., Lang, T., Lu, Y., Genant, H.K., 1998. High-resolution magnetic resonance imaging: three-dimensional trabecular bone architecture and biomechanical properties. Bone 22, 445–454. Meunier, P.J., Boivin, G., 1997. Bone mineral density reflects bone mass but also the degree of mineralization of bone: therapeutic implications. Bone 21, 373–377. Morgan, E.F., Keaveny, T.M., 2001. Dependence of yield strain of human trabecular bone on anatomic site. Journal of Biomechanics 34, 569–577. Morgan, E.F., Yeh, O.C., Chang, W.C., Keaveny, T.M., 2001. Nonlinear behavior of trabecular bone at small strains. Journal of Biomechanical Engineering 123, 1–9. Mosekilde, L., Mosekilde, L., Danielsen, C.C., 1987. Biomechanical competence of vertebral trabecular bone in relation to ash density and age in normal individuals. Bone 8, 79–85. . Muller, R., Gerber, S.C., Hayes, W.C., 1998. Micro-compression: a novel technique for the nondestructive assessment of local bone failure. Technology and Health Care 6, 433–444. Niebur, G.L., Feldstein, M.J., Yuen, J.C., Chen, T.J., Keaveny, T.M., 2000. High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone. Journal of Biomechanics 33, 1575–1583. Niebur, G.L., Yuen, J.C., Hsia, A.C., Keaveny, T.M., 1999. Convergence behavior of high-resolution finite element models of trabecular bone. Journal of Biomechanical Engineering 121, 629–635. Parfitt, A.M., 1992. Implications of architecture for the pathogenesis and prevention of vertebral fracture. Bone 13, S41–S47. Pugh, J.W., Rose, R.M., Radin, E.L., 1973. A structural model for the mechanical behavior of trabecular bone. Journal of Biomechanics 6, 657–670. Rho, J.Y., Tsui, T.Y., Pharr, G.M., 1997. Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18, 1325–1330. Snyder, B.D., Piazza, S., Edwards, W.T., Hayes, W.C., 1993. Role of trabecular morphology in the etiology of age-related vertebral fractures. Calcified Tissue International 53S, S14–S22. Stolken, J.S., Kinney, J.H., 2003. On the importance of geometric nonlinearity in finite-element simulations of trabecular bone failure. Bone 33, 494–504. Turner, C.H., 1989. Yield behavior of bovine cancellous bone. Journal of Biomechanical Engineering 111, 256–260. Turner, C.H., Rho, J., Takano, Y., Tsui, T.Y., Pharr, G.M., 1999. The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. Journal of Biomechanics 32, 437–441. Ulrich, D., Van Rietbergen, B., Laib, A., Rueegsegger, P., 1999. The ability of three-dimensional structural indices to reflect mechanical aspects of trabecular bone. Bone 25, 55–60. Watts, N.B., Harris, S.T., Genant, H.K., Wasnich, R.D., Miller, P.D., Jackson, R.D., Licata, A.A., Ross, P., Woodson, G.C.D., Yanover, M.J., et al., 1990. Intermittent cyclical etidronate treatment of postmenopausal osteoporosis [see comments]. New England Journal of Medicine 323, 73–79. Yeh, O.C., Keaveny, T.M., 1999. Biomechanical effects of intraspecimen variations in trabecular architecture: a three-dimensional finite element study. Bone 25, 223–228.
Journal of Biomechanics 37 (2004) 1413–1420
Contribution of inter-site variations in architecture to trabecular bone apparent yield strains Elise F. Morgana,*, Harun H. Bayraktara, Oscar C. Yehb, Sharmila Majumdarc,d, Andrew Burghardtc, Tony M. Keavenya,d a
Orthopaedic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA b Orthopaedic Research Laboratories, School of Medicine, University of California, Davis, CA 95817, USA c Department of Radiology, University of California, San Francisco, CA 94143, USA d Department of Bioengineering, University of California, Berkeley, CA 94720, USA Accepted 8 December 2003
Abstract Apparent yield strains for trabecular bone are uniform within an anatomic site but can vary across site. The overall goal of this study was to characterize the contribution of inter-site differences in trabecular architecture to corresponding variations in apparent yield strains. High-resolution, small deformation finite element analyses were used to compute apparent compressive and tensile yield strains in four sites (n=7 specimens per site): human proximal tibia, greater trochanter, femoral neck, and bovine proximal tibia. These sites display differences in compressive, but not tensile, apparent yield strains. Inter-site differences in architecture were captured implicitly in the model geometries, and these differences were isolated as the sole source of variability across sites by using identical tissue properties in all models. Thus, the effects inter-site variations in architecture on yield strain could be assessed by comparing computed yield strains across site. No inter-site differences in computed yield strains were found for either loading mode (p>0.19), indicating that, within the context of small deformations, inter-site variations in architecture do not affect apparent yield strains. However, results of ancillary analyses designed to test the validity of the small deformation assumption strongly suggested that the propensity to undergo large deformations constitutes an important contribution of architecture to inter-site variations in apparent compressive yield strains. Large deformations substantially reduced apparent compressive, but not tensile, yield strains. These findings indicate the importance of incorporating large deformation capabilities in computational analyses of trabecular bone. This may be critical when investigating the biomechanical consequences of trabecular thinning and loss. r 2004 Elsevier Ltd. All rights reserved. Keywords: Cancellous bone; Anatomic variation; Trabecular architecture; Trabecular tissue properties; Biomechanics
1. Introduction The development of effective descriptors of the mechanical behavior of trabecular bone is complicated by the large degree of mechanical heterogeneity that exists between individuals as well as between and within anatomic sites. One exception is the relative uniformity of apparent yield strains. Apparent yield strains for trabecular bone are only weakly dependent on density (Kopperdahl and Keaveny, 1998; Turner, 1989) and display intra-site variations on the order of one-tenth of *Corresponding author. Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215, USA. Tel.: +1-617-353-2791; fax: +1-617-353-5866. E-mail address: [email protected] (E.F. Morgan). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.12.037
the mean value (Morgan and Keaveny, 2001). In contrast, modulus and yield stress depend heavily on density and exhibit coefficients of variation per site of up to 70% (Linde et al., 1989; Morgan and Keaveny, 2001; Mosekilde et al., 1987). Thus, yield strain may be a more statistically powerful measure of the age-, disease-, and drug-related changes that occur in trabecular bone strength. Identifying factors such as trabecular architecture that contribute to changes in yield strain will improve this diagnostic capability and increase the potential for therapeutic interventions. While apparent yield strains are uniform within a single anatomic site, they can vary across sites (Morgan and Keaveny, 2001). These inter-site differences may be due to differences in trabecular architecture (Hildebrand et al., 1999; Ulrich et al., 1999) and tissue material
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properties (Hoffler et al., 2000). High-resolution finite element modeling provides the ability to control tissue properties, thereby isolating the effect of architecture. This technique has been used to explore contributions of architecture to apparent elastic behavior (Kabel et al., 1999; Ulrich et al., 1999). However, the contribution of architecture to yield strain may be very different, given that values of yield strain are greater than twice the applied strain at which modulus is measured. At higher applied strains, deformations within the trabecular network such as bending and buckling can become increasingly important (Bell et al., 1967; Gibson, 1985; Kopperdahl and Keaveny, 1998; Muller . et al., 1998; Pugh et al., 1973; Stolken and Kinney, 2003). The magnitude and prevalence of these deformations depends on architecture; for example, slender, rod-like trabeculae are more susceptible than are thicker, platelike trabeculae. Quantifying tissue level strains can provide mechanistic insight into the types of deformations that occur during yielding and how these deformations differ with architecture. It is important to note that most high-resolution analysis techniques assume that deformations within the trabecular network are relatively small, due to the prohibitively high computational cost of allowing for large deformations in such large models. Consequently, these analyses may not sufficiently model the full effect of architecture on apparent yield strains. Whether large deformations must be incorporated in high-resolution analyses, and whether these deformations significantly impact relationships between architecture and yield strain are open questions. The overall goal of this study was to characterize the contribution of inter-site variations in trabecular architecture to corresponding variations in apparent yield strains. A combined approach was taken whereby highresolution finite element analyses were supplemented by lattice-type (Yeh and Keaveny, 1999) finite element
analyses that included large deformation capabilities. The high-resolution analyses were used to compute apparent compressive and tensile yield strains in four anatomic sites that display differences in compressive but not tensile apparent yield strains (Morgan and Keaveny, 2001): the human proximal tibia, femoral greater trochanter, femoral neck, and the bovine proximal tibia. Inter-site differences in architecture were captured implicitly in the model geometries, and these differences were isolated as the sole source of variability across sites by using a single set of tissue properties in all models. Thus, under the assumption of small deformations, the effects inter-site variations in architecture on yield strain could be assessed by comparing computed yield strains across site. The lattice-type analyses were then used to quantify the effect on apparent yield strains of large deformations. Results of both types of finite element analyses were interpreted in the context of experimentally measured yield strains. The specific objectives of this study were to: (1) compare computed apparent yield strains across anatomic site; (2) quantify tissue strain distributions at the apparent yield point; and (3) assess the effect of large deformations on apparent yield strains.
2. Methods 2.1. High-resolution finite element analyses Seven cylindrical, on-axis specimens from each of the four anatomic sites were used (Table 1, Fig. 3); these specimens were randomly selected from larger groups used in previously published studies (Morgan and Keaveny, 2001; Niebur et al., 2000). Details of the specimen preparation and mechanical testing methods are described in these studies. Briefly, specimens were loaded to either the 0.2% offset yield point (human
Table 1 Specimen and cadaver information by anatomic site Anatomic site
n
No. donorsa (male/female)
Age
Volume fraction
Tb.Sp (mm)
Tb.Th (mm)
Tb.N (1/mm)
Greater trochanter
7
0.960b70.190
0.137b70.013
1.022b70.165
7
0.12b70.03
0.803b70.093
0.135b70.006
1.184b70.150
Femoral neck
7
0.29c70.06
0.605c70.085
0.196c70.030
1.589c70.186
Bovine proximal tibia
7
7579 62–87 62716 40–84 6375 57–72 —
0.11b70.02
Proximal tibia
7 (5/2) 5 (5/0) 7 (5/2) —
0.23d70.03
0.548c70.043
0.135b70.010
1.656c70.129
Age is reported as mean7standard deviation, followed by the range. Volume fraction, mean trabecular spacing (Tb.Sp), trabecular thickness (Tb.Th), and trabecular number (Tb.N) are reported as mean standard deviation. a Three donors (two male and one female) that provided specimens from the greater trochanter also provided specimens from the femoral neck. There was no overlap between donors for the human proximal tibia specimens and those for either the femoral neck or greater trochanter specimens. bac po0.04. bad po0.001. cad po0.04.
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specimens) or to 0.2% strain (bovine specimens) using mechanical testing protocols designed to minimize end artifacts (Keaveny et al., 1994a; Kopperdahl and Keaveny, 1998). Volume fractions were calculated using Archimedes’ Principle (Galante et al., 1970). Following mechanical testing, high-resolution threedimensional reconstructions of each specimen were obtained either using serial milling (Beck et al., 1997) or micro-computed tomography (Scanco mCT 20; Scanco Medical AG, Bassersdorf, Switzerland) at 10 and 22 mm resolution, respectively. For each specimen, the grayscale threshold was selected to match the reconstruction volume fraction with the experimentally measured value. To provide quantitative evidence of inter-site differences in trabecular architecture, standard metrics of architecture were measured from the threedimensional reconstructions (Hildebrand et al., 1999): mean trabecular thickness (Tb.Th), trabecular spacing (Tb.Sp), and trabecular number (Tb.N). These metrics were compared across site via individual analyses of variance (ANOVA). Using a voxel-based approach (Hollister et al., 1994), a 5 mm cube high-resolution finite element model was created from the central region of each specimen reconstruction. Prior to this conversion, region averaging was used to decrease the resolution to 44–66 mm in order to decrease computation time but still ensure that the voxel size was less than one-fourth of the mean trabecular thickness, as recommended for numerical convergence (Niebur et al., 1999). Materially nonlinear, high-resolution finite element analyses using an implicit incremental method were conducted to determine the apparent yield point, in each of compression and tension, as defined by the 0.2% offset criterion. The analyses incorporated material nonlinearity (i.e. tissue yielding) but not geometric nonlinearity (i.e. large deformations). Trabecular tissue was modeled as a bilinearly elastic material with a postyield modulus equal to 5% of the initial modulus (Niebur et al., 2000). Although a specimen-specific effective tissue modulus was calibrated for each specimen, the computed apparent yield strain for each loading mode was independent of the modulus due to the use of a strain-based tissue constitutive model. The tissue tensile and compressive yield strains used in all models were 0.412% and 0.825%, respectively. These values were the mean tissue yield strains from a calibration study conducted on 12 human femoral neck specimens (Bayraktar et al., 2004). To our knowledge, there are currently no other data in the literature on tissue yield strains in human trabecular bone for any anatomic site. Each analysis required 150–250 CPU h on an IBM SP2 parallel supercomputer (IBM, Armonk, NY), and a total of 56 analyses were performed. In each loading mode, computed apparent yield strains were compared across anatomic site and against
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experimental data (Chang et al., 1999; Morgan and Keaveny, 2001) using a two-factor ANOVA and Tukey post hoc test (JMP 5.0, SAS Institute, Cary, NC). Although the comparison of computed values across site was the primary comparison in this study design, the comparison against experimental values was important for two reasons. First, it provided a positive control. The single pair of tissue yield strains used in all finite element models was the pair of mean values calculated previously for trabecular tissue from the femoral neck. Agreement between computed and experimental apparent yield strains for this site would indicate that the model results can match the experimental data without accounting for intra-site variations in tissue yield properties. Second, the comparison against experimental data provided means for synthesizing the results of the high-resolution and lattice-type analysis results in order to assess the role of large deformations. The differences between computed and experimental values of yield strain and those between yield strains resulting from small and large deformation lattice-type analyses were juxtaposed to provide insight into the extent to which large deformations impact relationships between architecture and yield strain. The bovine experimental stress–strain curves were reprocessed to have a uniform definition of apparent modulus (slope at zero strain of a quadratic fit from 0% to 0.2% strain (Morgan et al., 2001). This had the effect of lowering both the compressive and tensile apparent yield strains for these data as compared to our previously published values (Chang et al., 1999; Keaveny et al., 1994b). The reprocessed mean (7SD) apparent compressive and tensile yield strains for this site were 0.79 (70.06)% and 0.61 (70.06)%, respectively. Tissue strain distributions were quantified at the apparent yield point for each simulation to provide mechanistic insight into the model predictions. At the apparent yield point, either compressive or tensile, the percentage of the total amount of tissue in the specimen that had reached either the compressive or tensile tissue level yield strain was calculated. The calculation was also performed separately for each tissue level yielding mode such that the amount of tissue that had reached the tissue compressive yield point could be compared to the amount that had reached the tissue tensile yield point. Linear regression analyses were used to assess relationships between the amount of tissue yielded and volume fraction. 2.2. Lattice-type finite element analyses To assess the contribution of large deformations to apparent yield strains, a three-dimensional lattice-type finite element model (Yeh and Keaveny, 1999) was used to simulate apparent loading of a 7.1 mm cube of trabecular bone with rod-like architecture and a volume
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fraction in the range 0.098–0.109. Each trabecula consisted of five quadratic beam elements and was assigned a unique thickness, length, and orientation typical of real trabeculae. The intra-specimen thickness distribution followed a b-distribution with a coefficient of variation of 40%, a mean of 215 mm (vertical trabeculae) or 106.6 mm (horizontal trabeculae), a minimum of 50 mm, and a maximum of 550 mm (Yeh and Keaveny, 1999). The length and orientation of each trabecula were perturbed by a random amount controlled by a ‘‘relative lattice disorder constant’’ (Jensen et al., 1990) of a=0.6. Five models, each with unique mesh geometry, were generated using these parameters. Trabecular tissue was modeled as an elastic–plastic material (E=18 GPa, v=0.3, Epost-yield=0.9 GPa) with the same tensile and compressive yield strains as used in the high-resolution finite element analyses. Monotonic, uniaxial compression and tension simulations (ABAQUS version 6.2; Abaqus, Inc., Pawtucket, RI) to 70.8% strain were performed on each model with the assumption of small deformations enforced (material nonlinearity only), and again without this assumption (material and geometric nonlinearity). These simulations were also performed using pure linear elastic tissue material behavior in order to examine the contribution of geometric nonlinearity alone. In each loading mode, resulting apparent yield strains for each type of analysis were averaged across the five models, and a paired t-test was used to detect differences in yield strains between the materially nonlinear and fully nonlinear cases.
3. Results In the high-resolution finite element analyses, no differences were found between sites in computed yield
strains for either compressive (p>0.50) or tensile (p>0.19) loading (power>0.84, Fig. 1). Despite statistically significant differences in architectural metrics between sites (po0.04, Table 1), differences between sites in mean computed yield strains were less than 11% in each loading mode. Tissue level strain distributions showed considerable variability between sites and with volume fraction. At each of the compressive and tensile apparent yield points, the percentage of tissue exceeding either the compressive or tensile tissue level yield strain increased with volume fraction from less than 4% to greater than 25% (po0.001, Fig. 2A). When the apparent loading mode was tensile, nearly all tissue yielding was tensile. For compressive loading, however, the ratio of the amount of tissue that had reached the tissue compressive yield strain to the amount that had reached the tissue tensile yield strain increased with volume fraction from 0.53 to 2.7 (r2=0.84, po0.001, Fig. 2B). At low-volume fractions, regions of compressive and tensile tissue failure were often found on opposite sides of slender trabeculae (Fig. 3), indicative of localized bending. Comparison of computed yield strains to the experimental data revealed that the high-resolution finite element models did not capture the experimentally observed inter-site differences in compressive yield strains. Mean computed compressive yield strains were 21%, 30%, and 16% higher than corresponding mean experimental values for the human tibia, trochanter, and bovine tibia, respectively (po0.001, Fig. 1). In tension, by contrast, no differences were found between computed and apparent yield strains (p=0.44, power=0.73). In the lattice-type finite element models, large deformations had a substantial effect on the magnitude of the apparent compressive, but not tensile, yield
Compression
*
Apparent Yield Strain (%)
1.0
a
0.8
*
Tension
*
Experimental
1.0
FEM 0.8
a,b
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Femoral Neck
Proximal Greater Tibia Trochanter
Bovine Tibia
Femoral Neck
Proximal Greater Tibia Trochanter
Bovine Tibia
Fig. 1. Mean computed (‘‘FEM’’) and experimental compressive (left) and tensile (right) yield strains. Error bars indicate 1 SD. No inter-site differences in computed apparent yield strains were found for either loading mode, indicating that, when small deformations are assumed, inter-site variations in architecture do no contribute to variations in apparent yield strains.
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deformations caused a stiffening effect that was counteracted by tissue yielding (Fig. 4B). No difference in tensile yield strains were found between the small and large deformation cases (p=1.0, Table 2), indicating that, due to the lower tensile tissue yield strain, material nonlinearity was the dominant factor for apparent tensile loading. Neither the small deformation assumption nor the choice of tissue constitutive behavior affected the apparent modulus.
Apparent Compression
35
Proximal Tibia
Percentage of Tissue Yielded (%)
Greater Trochanter
30
Femoral Neck Bovine Proximal Tibia
25 20 15 10
4. Discussion
5 y = 0.90 + 53.42x r = 0.61
0 0.00
0.10
0.20
0.30
0.40
Volume Fraction
(A)
Apparent Compression
3.0
Proximal Tibia
Tissue Yielded in Compression Tissue Yielded in Tension
Greater Trochanter
2.5
Femoral Neck Bovine Proximal Tibia
2.0 1.5 1.0 0.5 y = 0.11 + 6.48x R = 0.84
0.0 0.00 (B)
1417
0.10
0.20
0.30
0.40
Volume Fraction
Fig. 2. (A) The total amount of tissue yielded (expressed as a percentage of the volume of tissue in the specimen) at the apparent level yield point increased with volume fraction for apparent compressive loading (po0.001). Although not shown, the trend was similar for apparent tensile loading (y=0.10+75.84x, r2=0.62, po0.001). (B) The ratio of the amount of tissue yielded in compression to that yielded in tension at the apparent compressive yield point increased with volume fraction (po0.001). At low-volume fractions, this ratio was less than one, indicating that for these specimens, more tissue yielded in tension than compression even though the apparent loading mode was compressive.
strains. Both geometric nonlinearity alone and material nonlinearity alone caused a softening effect in the apparent compressive stress–strain curve (Fig. 4A). The combined effect of geometric and material nonlinearity was even more pronounced: allowing for large deformations lowered the apparent compressive yield strain by 22% as compared to the small deformation case (po0.001, Table 2). In tension, however, large
The overall goal of this study was to characterize the contribution of inter-site variations in trabecular architecture to corresponding variations in apparent yield strains. The effects of architecture were isolated through the use of identical tissue properties in high-resolution, small deformation finite element analyses, and the resulting computed apparent yield strains were compared across site. No inter-site differences in computed yield strains were found for either tension or compression, indicating that, when small deformations are assumed, inter-site variations in architecture do not contribute to apparent yield strains. However, the lattice-type finite element results demonstrated that the small deformation assumption is not appropriate for all types of trabecular architecture. In the rod-like architecture of the lattice models, large deformations reduced the apparent compressive, but not tensile, yield strain. This was consistent with the finding that, in the anatomic sites with low trabecular thickness and high trabecular spacing, the high-resolution, small deformation analyses over-predicted apparent compressive, but not tensile, yield strains. Although this over-prediction could theoretically be due to inter-site differences in tissue yield strains, data from a previous study (Bayraktar et al., 2004) suggest that attributing the discrepancy to tissue property differences alone would require that tissue compressive yield strains differ by over 40% between the greater trochanter and femoral neck. There is no evidence in the literature to suggest that tissue yield strains vary to such a large extent within metaphyses. Therefore, these collective results suggest that the propensity to undergo large deformations constitutes an important contribution of architecture to inter-site differences in apparent compressive yield strains. They further imply that computational analyses, particularly those of failure in low-density bone, should allow for large deformations even when apparent strains are less than 1%. This may be critical for the study of age- and disease-related fragility where trabecular thinning and loss occur. The main strength of this study lies in the use of a combination of computational and experimental techniques to isolate and characterize the effect of inter-site
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Fig. 3. Renderings of the central 3–4 mm cubic portions of high-resolution finite element models of representative specimens from the (A) femoral neck, (B) human proximal tibia, (C) greater trochanter, and (D) bovine proximal tibia illustrate the inter-site variability in tissue strain distributions when the apparent compressive yield point is reached. (Red) Regions exceeding the tensile tissue yield strain; (blue) regions exceeding the compressive tissue yield strain. At lower volume fractions, only a small percentage of the tissue yielded, regions of yield were highly localized, and regions of tensile and compressive yield were often located on opposite sides of slender trabeculae. In comparison, at higher volume fractions, a larger percentage of the specimen yielded and regions yield were more evenly distributed throughout the specimen.
variations in architecture. The excellent match between computed and experimental apparent yield strains in both loading modes for the femoral neck—the positive control—confirmed the validity of this approach. In addition, the human specimens used for the finite element models were a subset of those that were mechanically tested, thus increasing the strength of comparisons between computed and experimental yield strains. A key assumption used in both types of computational models was that trabecular tissue has similar elastic and yield properties to cortical bone. This assumption is supported by similarities in elastic moduli (Rho et al., 1997; Turner et al., 1999) and tissue composition (Guo and Goldstein, 1997), and by the success of previous high-resolution finite element studies that have modeled trabecular tissue after cortical bone (Bayraktar et al., 2004; Niebur et al., 2000). While there remains a need for experimental investigation of trabecular tissue yield properties, the results of this study lend additional insight. The close agreement between computed and experimental apparent tensile yield strains, combined with the insensitivity of apparent tensile yield strains to the tissue compressive yield strain,
suggests that tissue tensile yield strains are relatively uniform across site and species. The picture is less clear for compression, due to the confounding effects of the small deformation assumption in the high-resolution analyses. Although we believe that the differences between computed and experimental apparent compressive yield strains for the greater trochanter and human tibia were primarily due to the small deformation assumption, the corresponding discrepancy for the bovine tibia may also be due to inter-species differences in tissue compressive yield strains. This dichotomy is supported by observations that large deformations occur in low-density whale vertebral bone, but not in higher density bovine tibial bone (Muller . et al., 1998). Furthermore, the tissue level strain distributions quantified in the current study illustrated that, even with the assumption of small deformations enforced, the deformation mechanisms during apparent compressive loading were dramatically different in the bovine tibia as compared to the trochanter and human tibia. Most notably, localized bending occurred with much greater frequency in the trochanter and human tibia than in the bovine tibia (Figs. 2, 3). Incorporating large deformations into the high-resolution analyses will enable
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Apparent Compression
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Table 2 Apparent yield strains of low density trabecular bone computed from lattice-type finite element models and measured experimentally
3.5 3.0
Apparent compression
Apparent tension
n
Yield strain
n
Yield strain
5
0.6370.02a
5
0.5770.02
5
0.8170.02b
5
0.5770.02
5
0.9370.02
5
Infinite
10
0.6970.05
13
0.6170.05
15
0.7270.06
16
0.6570.05
Stress (MPa)
2.5 Computed Geometric and material nonlinearity Material nonlinearity Geometric nonlinearity
2.0 1.5 1.0
Geometric and Material Nonlinearity
Experimental Greater trochanter Proximal tibia
Material Nonlinearity
0.5
Geometric Nonlinearity Experiment
0.0 0.0
0.2
(A)
0.4
0.6
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3.0 2.5 Stress (MPa)
po0.001.
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0.5
Geometric Nonlinearity Experiment
0.0 0.0 (B)
aab
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Fig. 4. Stress–strain curves for the lattice-type finite element models of low-density bone in (A) apparent compression and (B) apparent tension illustrate the effect of large deformations on the apparent yield point. The apparent yield point is indicated on each curve by the solid rectangle. The experimental data are from the greater trochanter. Mean values of apparent yield strain for all simulation cases are listed in Table 2.
computational investigations of inter-site and interspecies variations in tissue compressive yield strain. Our experimental design was unique in that it capitalized on the ability of the high-resolution models to capture architecture implicitly in the model geometry. This allowed the effects inter-site variations in architecture to be assessed by comparing computed yield
strains across site. An alternative approach, which has been used for apparent modulus and ultimate stress (Majumdar et al., 1998; Ulrich et al., 1999), would be to use statistical correlation or multiple regression techniques to develop relationships between apparent yield strain and three-dimensional structural indices such as trabecular spacing. Although this latter approach has the advantage that it could identify specific quantitative descriptors of the effect of architecture on yield strain, it relies on the assumption that the available structural indices sufficiently describe the architectural features that are most relevant for apparent yield strains. For instance, the results of the current study suggest that the relevant architectural features will be related to the propensity to undergo large deformations. In this context, measures of intra-specimen variations in architecture may be as important as mean values. The effect of intra-specimen variations in thickness on apparent modulus has been demonstrated (Yeh and Keaveny, 1999). The consequence of these variations for apparent yield strain may be particularly acute because yielding at the apparent level may initiate as local failure in a susceptible region of the specimen. The contribution of large deformations represents a possible explanation for clinical observations that small changes in bone mass can have dramatic effects on fracture risk (Meunier and Boivin, 1997; Watts et al., 1990). Focal thinning (or thickening) of trabeculae resulting from slight remodeling imbalances is expected to alter the magnitude and prevalence of large deformations. Buckling has long been proposed as a mechanistic example of how trabecular strength can be affected disproportionately to stiffness (Bell et al., 1967; Gibson, 1985; Parfitt, 1992; Pugh et al., 1973; Snyder et al., 1993; Stolken and Kinney, 2003). Although the stability analyses required to explicitly model buckling were not
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performed in this study, our results indicate that pronounced bending can have appreciable effects on compressive strength as well.
Acknowledgements Funding was provided by NIH AR43784, NIH AR41481, NSF BES-96250301, and NRAC UCB266. Human tissue was obtained from the Anatomic Gift Foundation and the National Disease Research Interchange. The authors wish to thank Jacob Pollock for his technical assistance.
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