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The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity
ARTICLE IN PRESS
Journal of Biomechanics 37 (2004) 1281–1287
Short communication
The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity X. Neil Dong1, X. Edward Guo* Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA Accepted 10 December 2003
Abstract The objective of this study was to examine the dependence of the elastic properties of cortical bone as a transversely isotropic material on its porosity. The longitudinal Young’s modulus, transverse Young’s modulus, longitudinal shear modulus, transverse shear modulus, and longitudinal Poisson’s ratio of cortical bone were determined from eighteen groups of longitudinal and transverse specimens using tensile and torsional tests on a servo-hydraulic material testing system. These cylindrical waisted specimens of cortical bone were harvested from the middle diaphysis of three pairs of human femora. The porosity of these specimens was assessed by means of histology. Our study demonstrated that the longitudinal Young’s and shear moduli of human femoral cortical bone were significantly (po0.01) negatively correlated with the porosity of cortical bone. Conversely, the elastic properties in the transverse direction did not have statistically significant correlations with the porosity of cortical bone. As a result, the transverse elastic properties of cortical bone were less sensitive to changes in porosity than those in the longitudinal direction. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly (po0.01) negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone. r 2003 Elsevier Ltd. All rights reserved. Keywords: Cortical bone; Porosity; Transversely isotropic; Elastic moduli; Micromechanics
1. Introduction Age-related changes in the microstructure of bone tissue that reduce bone strength become increasingly important as the world-wide aging population continues to grow (Cooper et al., 1992; Cummings et al., 1990). One of the hallmarks of age-related changes in human cortical bone is the increase in porosity (Bell et al., 1999a, 2001; Crabtree et al., 2001; Jordan et al., 2000). For example, cortical bone porosity increased significantly in femoral neck specimens from patients with agerelated fractures compared to age-matched controls (Bell et al., 1999b, c). These microscopic changes in cortical bone may be partially responsible for the increased prevalence of hip fractures in the elderly (Bell *Corresponding author. Tel.: +1-212-854-6196; fax: +1-212-8548725. E-mail address: [email protected] (X. Edward Guo). 1 Current address: Bone and Joint Center, Department of Orthopaedic Surgery, Henry Ford Health System, Detroit, MI 48202, USA. 0021-9290/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.12.011
et al., 1999c). Therefore, it is important to study the dependence of the mechanical properties of bone tissue on its architecture and, specially, porosity. A few studies have demonstrated that the mechanical properties of cortical bone depend on porosity (Currey, 1988; Martin, 1993; McCalden et al., 1993; Schaffler and Burr, 1988; Wachter et al., 2002). However, these measurements are in general derived from either uniaxial or bending tests for uniaxial elastic properties such as Young’s modulus along the axis of the bone. Although it is well known that human cortical bone exhibits at least transverse isotropy in terms of its elastic properties (Burstein et al., 1976; Evans, 1976; Katz, 1980; Reilly and Burstein, 1975; Reilly et al., 1974; Yoon and Katz, 1976a, b), to our knowledge, there are no published data on the dependence of the anisotropic elastic properties of human cortical bone on porosity. The objective of current study was to experimentally determine the transversely isotropic elastic properties of human cortical bone by mechanical testing and their dependence on the histologically assessed porosity.
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2. Materials and methods 2.1. Elastic moduli of a transversely isotropic material A transversely isotropic material (e.g., cortical bone) has the same properties in all directions of the transverse plane and significantly different properties in the longitudinal direction (Lai et al., 1993; Martin et al., 1998). The elastic behavior of transversely isotropic materials can be fully characterized by the five elastic moduli (EL: longitudinal Young’s modulus; ET: transverse Young’s modulus; GL: longitudinal shear modulus; GT: transverse shear modulus; and nL: longitudinal Poisson’s ratio). These five elastic moduli can be experimentally determined by mechanical testing (Evans, 1978; Reilly and Burstein, 1975). In the present study, a group of four specimens of the same material (i.e., longitudinal tensile specimens, transverse tensile specimens, longitudinal torsional specimens and transverse torsional specimens) are needed to determine the elastic properties of a transversely isotropic material by the use of four different mechanical tests (Dong, 2002). 2.2. Specimen preparation Eighteen groups of waisted cylindrical specimens of cortical bone were obtained from three pairs of human femora from one male (64 years old) and two females (44 and 60 years old) who were free of bone diseases (Fig. 1). The grip ends of these waisted cylindrical
specimens were coated with cyanoacrylate glue and press-fit into internally threaded brass cylindrical endcaps using an alignment jig (Keaveny et al., 1994). 2.3. Mechanical testing Tensile tests were performed on the longitudinal and transverse tensile specimens using a servo-hydraulic material testing system (MTS 810, MTS System Corporation, Minneapolis, MN) at room temperature with a strain rate of 0.1%/s (Reilly and Burstein, 1975). Torsional tests were conducted on the longitudinal and transverse torsional specimens using a servohydraulic bi-axial material testing system (MTS 858, MTS Systems Corporation, Eden Prairie, MN) at room temperature with a rate of 1 /s under angle displacement control (Evans, 1978; Reilly and Burstein, 1975). 2.4. Histological measurements After mechanical testing, a histological cross-section was prepared from the central portion of a longitudinal or transverse test specimen such that the Haversian systems appeared on a transmitted light microscope (Fig. 1). The cross-section was then ground to a thickness of 200 mm with successive grits of sandpaper until the microstructural components of cortical bone were clearly shown on the microscope (Fig. 2). Consequently, the porosity of the specimen was calculated as the ratio of the area of Haversian canals and resorption
Fig. 1. Specimen preparation of cortical bone samples from the diaphysis of a human femur. First, three bone slabs with a thickness of 35 mm were sectioned from the diaphysis of the femur by a band saw. The posterior-medial quadrant of these bone slabs was then cut into three bone slices (proximal, middle, and distal) using a low-speed diamond saw. Next, the middle slice with a thickness of 20 mm was sectioned into two parallelepipeds with dimensions of 5 5 20 mm to produce a longitudinal tensile specimen and a longitudinal torsional specimen, respectively. Concurrently, the proximal and distal slices with a thickness of 5 mm were sectioned into two parallelepipeds with dimensions of 5 5 20 mm to yield a transverse tensile specimen and a transverse torsional specimen, respectively.
ARTICLE IN PRESS X. Neil Dong, X. Edward Guo / Journal of Biomechanics 37 (2004) 1281–1287
cavities to the total area of the cross section (Fig. 2). The lacunae and canaliculi were not considered in the present study because the lacunae and the canaliculi represented an average of only 0.80% and 1.48% of
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vascular-free cortical volume in human long bones, respectively (Evans, 1973).
2.5. Statistical analysis Statistical analyses of the experimental results were conducted using SYSTAT (SPSS Inc., Chicago, IL). Simple linear regression was used to examine the dependence of the elastic properties of cortical bone on its porosity. The relationship was considered statistically significant when the p-value was less than 0.05.
3. Results 3.1. Elastic properties of cortical bone
Fig. 2. A histological section of cortical bone indicating how to measure the porosity of cortical bone. The images of the cross section were captured by a video imaging system and transferred to a public domain NIH Image program (developed at the U.S. National Institutes of Health). The area of Haversian canals and resorption cavities was measured by manually tracing the borders of these pores. Meanwhile, the total area of the cross section was measured from the images.
During specimen preparation, a longitudinal tensile specimen from group 11 and a transverse torsional specimen from group 7 were broken (Table 1). Therefore, mechanical tests were not available for these two specimens. For the rest of specimens from all 18 groups, the elastic properties of cortical bone measured from mechanical testing were listed at full length in Table 1. A simple linear correlation analysis indicated that the elastic properties of cortical bone in the same direction were significantly (po0.05) correlated with each other (Table 2). However, the elastic properties of cortical bone in the longitudinal direction were not correlated with those in the transverse direction (Table 2).
Table 1 The transversely isotropic elastic properties of cortical bone by mechanical testing Group number
EL (GPa)
ET (GPa)
GL (GPa)
GT (GPa)
nL
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
20.66 18.12 15.82 17.86 18.84 17.88 15.68 17.98 13.83 17.38 N/Aa 15.09 13.37 17.04 15.74 15.90 16.22 15.82
10.79 10.16 7.97 9.34 10.10 11.15 8.52 9.20 12.81 9.25 9.85 6.46 9.40 10.24 8.63 8.78 10.20 9.05
5.40 5.53 5.31 5.08 5.41 5.04 4.77 5.06 3.32 4.69 3.83 4.18 3.70 5.27 4.54 4.13 5.02 4.97
3.76 3.88 3.65 3.12 2.88 3.15 N/Aa 3.05 4.36 2.75 3.03 2.23 3.34 3.32 3.62 3.35 2.76 3.50
0.39 0.34 0.34 0.32 0.39 0.38 0.39 0.40 0.32 0.41 N/Aa 0.44 0.34 0.35 0.37 0.38 0.37 0.38
Mean7SD
16.6171.83
4.7470.65
3.2870.50
0.3770.03
a
9.5571.36
Mechanical testing data were not available (N/A) for broken specimens during sample preparation.
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3.2. Correlations between elastic moduli and porosity of cortical bone The porosity of cortical bone was calculated by averaging the porosity of four individual specimens from each group (Table 3). The porosity was listed in full length to help provide experimental data to validate future micromechanics modeling of cortical bone. Two groups (7 and 11) were excluded from further regression analysis between the elastic properties of cortical bone and its porosity because mechanical testing data were not available. The elastic properties of cortical bone in the longitudinal direction had significant correlations with its porosity. For instance, the longitudinal Young’s modulus of cortical bone was significantly (po0.01) correlated with its porosity (Fig. 3A). In addition, the shear modulus of cortical bone in the longitudinal direction was also significantly (po0.01) correlated with its porosity (Fig. 3B). Both a linear regression model and a power regression model were used to determine the correlations between elastic properties and porosity Table 2 Correlation coefficients (R) between longitudinal and transverse elastic properties of cortical bone (n=16) R EL (GPa) ET (GPa) GL (GPa) GT (GPa) a
EL (GPa)
ET (GPa)
1 0.18 0.80a 0.09
1 0.04 0.56a
GL (GPa)
(Table 4). The predictabilities of these two models were almost equivalent. For example, the power model (R2=0.69) was slightly better than the linear model (R2=0.66) in consideration of the relationship between longitudinal Young’s modulus and porosity. On the other hand, the linear model (R2=0.72) was slightly better than the power model (R2=0.67) when the relationship between longitudinal shear modulus and porosity was determined. Both the linear and power models indicated that transverse Young’s modulus (Fig. 3C) and transverse shear modulus (Fig. 3D) were not significantly (p>0.65) correlated with the porosity of cortical bone (Table 4). 3.3. The dependence of elastic anisotropy of cortical bone on its porosity The anisotropic ratios of cortical bone elasticity were significantly correlated with the porosity of cortical bone. The ratio of longitudinal to transverse Young’s moduli decreased significantly (po0.01) with the porosity of cortical bone (Fig. 4A). In addition, the ratio of longitudinal to transverse shear moduli had a significant negative correlation with the porosity of cortical bone (Fig. 4B).
GT (GPa)
4. Discussion 1 0.12
1
Significant correlations (po0.05).
The elastic modulus of cortical bone is well known to be reduced by porosity. A number of researchers have observed the significant negative correlations between
Table 3 The histologically assessed porosity of longitudinal tensile, transverse tensile, longitudinal torsional and transverse torsional test specimens of cortical bone Group number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Mean7SD
Porosity of individual specimen (%)
Group porosity (%)
Longitudinal tensile
Transverse tensile
Longitudinal torsional
Transverse torsional
4.77 9.78 4.41 3.93 5.37 7.34 12.59 5.58 13.70 4.53 10.37 12.11 18.42 8.99 7.86 11.49 13.70 7.66
4.14 4.56 10.31 6.09 5.13 9.99 14.31 6.58 19.47 9.57 9.67 8.96 11.12 8.18 6.34 13.74 10.84 11.52
5.98 5.76 5.51 7.58 7.09 10.02 9.24 6.20 12.75 9.26 13.55 10.72 16.36 8.54 10.08 12.90 9.59 10.51
6.55 7.06 5.36 5.58 5.97 4.92 19.93 9.09 8.92 12.19 14.03 7.04 10.97 8.74 9.00 14.65 12.32 8.33
8.9574.16
9.4773.87
9.5472.99
8.8673.93
5.36 6.79 6.40 5.80 5.89 8.07 14.02 6.86 13.71 8.89 11.91 9.71 14.22 8.61 8.32 13.20 11.61 9.51 9.3873.03
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Table 4 The relationships between the elastic properties and the porosity (p) of cortical bone Linear regression model EL= 0.53p+21.43 ET= 0.06p+9.06 GL= 0.19p+6.50 GT= 0.02p+3.16 nL= 0.002p+0.38
R2=0.66 R2=0.01 R2=0.72 R2=0.01 R2=0.02
p=0.0001a p=0.65 p=0.00001a p=0.76 p=0.62
Power regression model log EL= 0.30 log p+1.50 log ET=0.02 log p+0.96 log GL= 0.38 log p+1.03 log GT=0.001 log p+0.51 log nL= 0.02 log p 0.41
R2=0.69 R2=0.002 R2=0.67 R2=0.0001 R2=0.01
p=0.00001a p=0.88 p=0.0001a p=0.99 p=0.77
a
Significant correlations.
Fig. 4. The relationships between the elastic anisotropy of cortical bone and its porosity, (A) the ratio of longitudinal Young’s modulus to transverse Young’s modulus; (B) the ratio of longitudinal shear modulus to transverse shear modulus.
Fig. 3. The relationships between the elastic properties of cortical bone and its porosity, (A) longitudinal Young’s modulus: EL= 0.53p+21.43, R2=0.66; (B) longitudinal shear modulus: GL = 0.19p+6.50, R2=0.72; (C) transverse Young’s modulus: ET; (D) transverse shear modulus: GT.
the mechanical properties of cortical bone and porosity (Currey, 1988; Evans, 1978; McCalden et al., 1993; Schaffler and Burr, 1988; Wachter et al., 2001). Particularly, a power law relationship between elastic modulus and porosity has been proposed (Currey, 1988; Schaffler and Burr, 1988). In the present study, we also
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demonstrated that the elastic and shear moduli of cortical bone in the longitudinal direction decreased significantly with the porosity of cortical bone. However, the predictive capabilities of a linear model and a power law model between elastic properties and porosity were approximately equal (Table 4). In other words, the power law model was no better than the linear model. However, no statistically significant correlations were observed between the elastic properties in the transverse direction and the porosity of cortical bone in the present study, indicating that the elastic properties of cortical bone in the transverse direction were less sensitive to changes in porosity than those in the longitudinal direction. It is possible to infer this difference by the fact that the elastic properties of cortical bone in the transverse direction were not correlated with those in the longitudinal direction (Table 2). This observation on the relationships between the longitudinal and transverse elastic properties of cortical bone was in agreement with the previous finding that the modulus in the radial or circumferential direction (i.e., transverse direction) could not be predicted from modulus in superiorinferior direction (i.e., longitudinal direction) for cortical bone (Rho et al., 1995). Furthermore, the anisotropic ratios of cortical bone elasticity were found to be significantly negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased (Fig. 4). This appeared to be contradictive with the previous studies on the elastic properties of osteoporotic and osteopetrotic bone (Ashman et al., 1985; Katz et al., 1984; Martin et al., 1998). It has been demonstrated that osteoporosis has no effect on anisotropy of cortical bone (Katz et al., 1984), but osteopetrosis greatly reduces it (Ashman et al., 1985; Katz et al., 1984). The inconsistency may indicate that metabolic bone diseases such as osteoporosis and osteopetrosis cause the changes of cortical bone in the microstructural levels and therefore influence its elastic behavior. The outcome of the present study may provide appropriate experimental data to validate micromechanics models of human femoral cortical bone. A number of researchers have developed micromechanics models of cortical bone to predict the changes in mechanical properties due to changes in the microstructural parameters such as porosity, collagen, and mineralization (Aoubiza et al., 1996; Crolet et al., 1993; Hogan, 1992; Katz, 1981; Sevostianov and Kachanov, 2000). Particularly, a finite-element micromechanics model of cortical bone predicted that the longitudinal elastic modulus of cortical bone decreased linearly with porosity (Hogan, 1992). The prediction was consistent with our results that longitudinal Young’s modulus had a significant negative correlation with the porosity of cortical bone.
One limitation of this study was that the transversely isotropic elastic properties of cortical bone were measured from multiple specimens instead of one single specimen. In order to control the effect of inhomogeneity of cortical bone (Evans and Lebow, 1951) in the present study, bone specimens in the same group were taken from the same location of the femoral shaft (Fig. 1). Another limitation was that the number of subjects (n=3) used in the present study was relatively small. In order to fully understand the dependence of transversely isotropic elasticity of cortical bone on its porosity, more subjects, including different age groups and different genders, should be included to address this tissue in the future study. In conclusion, our study demonstrated that the longitudinal Young’s modulus and shear modulus of human femoral cortical bone were significantly correlated with the porosity of cortical bone. However, the elastic properties in the transverse direction of cortical bone were not significantly correlated with the porosity of cortical bone. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone.
Acknowledgements This work was supported by Whitaker Foundation Biomedical Engineering Research Grant 97-0086, NIH/ NIAMS/ASBMR AR45832 and NSF Career BES 9875633.
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ARTICLE IN PRESS X. Neil Dong, X. Edward Guo / Journal of Biomechanics 37 (2004) 1281–1287 Bell, K.L., Loveridge, N., Reeve, J., Thomas, C.D., Feik, S.A., Clement, J.G., 2001. Super-osteons (remodeling clusters) in the cortex of the femoral shaft: influence of age and gender. Anatomical Record 264, 378–386. Burstein, A.H., Reilly, D.T., Martens, M., 1976. Aging of bone tissue: mechanical properties. Journal of Bone and Joint Surgery— American Volume 58, 82–86. Cooper, C., Campion, G., Melton, L.J.D., 1992. Hip fractures in the elderly: a world-wide projection. Osteoporosis International 2, 285–289. Crabtree, N., Loveridge, N., Parker, M., Rushton, N., Power, J., Bell, K.L., Beck, T.J., Reeve, J., 2001. Intracapsular hip fracture and the region-specific loss of cortical bone: analysis by peripheral quantitative computed tomography. Journal of Bone and Mineral Research 16, 1318–1328. Crolet, J.M., Aoubiza, B., Meunier, A., 1993. Compact bone: numerical simulation of mechanical characteristics [see comments]. Journal of Biomechanics 26, 677–687. Cummings, S.R., Rubin, S.M., Black, D., 1990. The future of hip fractures in the United States. Numbers, costs, and potential effects of postmenopausal estrogen. Clinical Orthopaedics and Related Research, 163–166. Currey, J.D., 1988. The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. Journal of Biomechanics 21, 131–139. Dong, X.N., 2002. Micromechanics of osteonal cortical bone. Ph.D. Thesis, Columbia University, New York. Evans, F.G., 1973. Mechanical Properties of Bone. Charles C Thomas Publisher, Springfield, IL. Evans, F.G., 1976. Mechanical properties and histology of cortical bone from younger and older men. Anatomical Record 185, 1–11. Evans, F.G., 1978. Relations between torsion properties and histology of adult human compact bone. Journal of Biomechanics 11, 157–165. Evans, F.G., Lebow, M., 1951. Regional differences in some of the physical properties of the human femur. Journal of Applied Physiology 3, 563–572. Hogan, H.A., 1992. Micromechanics modeling of Haversian cortical bone properties. Journal of Biomechanics 25, 549–556. Jordan, G.R., Loveridge, N., Bell, K.L., Power, J., Rushton, N., Reeve, J., 2000. Spatial clustering of remodeling osteons in the femoral neck cortex: a cause of weakness in hip fracture? Bone 26, 305–313. Katz, J.L., 1980. Anisotropy of Young’s modulus of bone. Nature 283, 106–107.
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Katz, J.L., 1981. Composite material models for cortical bone. Mechanical Properties of Bone. ASME, New York, pp. 171–184. Katz, J.L., Yoon, H.S., Lipson, S., Maharidge, R., Meunier, A., Christel, P., 1984. The effects of remodeling on the elastic properties of bone. Calcified Tissue International 36 (suppl 1), S31–36. Keaveny, T.M., Guo, X.E., Wachtel, E.F., McMahon, T.A., Hayes, W.C., 1994. Trabecular bone exhibits fully linear elastic behavior and yields at low strains. Journal of Biomechanics 27, 1127–1136. Lai, W.M., Rubin, D., Krempl, E., 1993. Introduction to Continuum Mechanics. Pergamon Press, New York. Martin, B., 1993. Aging and strength of bone as a structural material. Calcified Tissue International 53, S34–39; discussion S39–40. Martin, R.B., Burr, D.B., Sharkey, N.A., 1998. Skeletal Tissue Mechanics. Springer, New York. McCalden, R.W., McGeough, J.A., Barker, M.B., Court-Brown, C.M., 1993. Age-related changes in the tensile properties of cortical bone. The relative importance of changes in porosity, mineralization, and microstructure. Journal of Bone and Joint Surgery— American Volume 75, 1193–1205. Reilly, D.T., Burstein, A.H., 1975. The elastic and ultimate properties of compact bone tissue. Journal of Biomechanics 8, 393–405. Reilly, D.T., Burstein, A.H., Frankel, V.H., 1974. The elastic modulus for bone. Journal of Biomechanics 7, 271–275. Rho, J.Y., Hobatho, M.C., Ashman, R.B., 1995. Relations of mechanical properties to density and CT numbers in human bone. Medical Engineering & Physics 17, 347–355. Schaffler, M.B., Burr, D.B., 1988. Stiffness of compact bone: effects of porosity and density. Journal of Biomechanics 21, 13–16. Sevostianov, I., Kachanov, M., 2000. Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone. Journal of Biomechanics 33, 881–888. Wachter, N.J., Augat, P., Krischak, G.D., Mentzel, M., Kinzl, L., Claes, L., 2001. Prediction of cortical bone porosity in vitro by microcomputed tomography. Calcified Tissue International 68, 38–42. Wachter, N.J., Krischak, G.D., Mentzel, M., Sarkar, M.R., Ebinger, T., Kinzl, L., Claes, L., Augat, P., 2002. Correlation of bone mineral density with strength and microstructural parameters of cortical bone in vitro. Bone 31, 90–95. Yoon, H.S., Katz, J.L., 1976a. Ultrasonic wave propagation in human cortical bone—I. Theoretical considerations for hexagonal symmetry. Journal of Biomechanics 9, 407–412. Yoon, H.S., Katz, J.L., 1976b. Ultrasonic wave propagation in human cortical bone—II. Measurements of elastic properties and microhardness. Journal of Biomechanics 9, 459–464.
Journal of Biomechanics 37 (2004) 1281–1287
Short communication
The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity X. Neil Dong1, X. Edward Guo* Bone Bioengineering Laboratory, Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA Accepted 10 December 2003
Abstract The objective of this study was to examine the dependence of the elastic properties of cortical bone as a transversely isotropic material on its porosity. The longitudinal Young’s modulus, transverse Young’s modulus, longitudinal shear modulus, transverse shear modulus, and longitudinal Poisson’s ratio of cortical bone were determined from eighteen groups of longitudinal and transverse specimens using tensile and torsional tests on a servo-hydraulic material testing system. These cylindrical waisted specimens of cortical bone were harvested from the middle diaphysis of three pairs of human femora. The porosity of these specimens was assessed by means of histology. Our study demonstrated that the longitudinal Young’s and shear moduli of human femoral cortical bone were significantly (po0.01) negatively correlated with the porosity of cortical bone. Conversely, the elastic properties in the transverse direction did not have statistically significant correlations with the porosity of cortical bone. As a result, the transverse elastic properties of cortical bone were less sensitive to changes in porosity than those in the longitudinal direction. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly (po0.01) negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone. r 2003 Elsevier Ltd. All rights reserved. Keywords: Cortical bone; Porosity; Transversely isotropic; Elastic moduli; Micromechanics
1. Introduction Age-related changes in the microstructure of bone tissue that reduce bone strength become increasingly important as the world-wide aging population continues to grow (Cooper et al., 1992; Cummings et al., 1990). One of the hallmarks of age-related changes in human cortical bone is the increase in porosity (Bell et al., 1999a, 2001; Crabtree et al., 2001; Jordan et al., 2000). For example, cortical bone porosity increased significantly in femoral neck specimens from patients with agerelated fractures compared to age-matched controls (Bell et al., 1999b, c). These microscopic changes in cortical bone may be partially responsible for the increased prevalence of hip fractures in the elderly (Bell *Corresponding author. Tel.: +1-212-854-6196; fax: +1-212-8548725. E-mail address: [email protected] (X. Edward Guo). 1 Current address: Bone and Joint Center, Department of Orthopaedic Surgery, Henry Ford Health System, Detroit, MI 48202, USA. 0021-9290/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.12.011
et al., 1999c). Therefore, it is important to study the dependence of the mechanical properties of bone tissue on its architecture and, specially, porosity. A few studies have demonstrated that the mechanical properties of cortical bone depend on porosity (Currey, 1988; Martin, 1993; McCalden et al., 1993; Schaffler and Burr, 1988; Wachter et al., 2002). However, these measurements are in general derived from either uniaxial or bending tests for uniaxial elastic properties such as Young’s modulus along the axis of the bone. Although it is well known that human cortical bone exhibits at least transverse isotropy in terms of its elastic properties (Burstein et al., 1976; Evans, 1976; Katz, 1980; Reilly and Burstein, 1975; Reilly et al., 1974; Yoon and Katz, 1976a, b), to our knowledge, there are no published data on the dependence of the anisotropic elastic properties of human cortical bone on porosity. The objective of current study was to experimentally determine the transversely isotropic elastic properties of human cortical bone by mechanical testing and their dependence on the histologically assessed porosity.
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2. Materials and methods 2.1. Elastic moduli of a transversely isotropic material A transversely isotropic material (e.g., cortical bone) has the same properties in all directions of the transverse plane and significantly different properties in the longitudinal direction (Lai et al., 1993; Martin et al., 1998). The elastic behavior of transversely isotropic materials can be fully characterized by the five elastic moduli (EL: longitudinal Young’s modulus; ET: transverse Young’s modulus; GL: longitudinal shear modulus; GT: transverse shear modulus; and nL: longitudinal Poisson’s ratio). These five elastic moduli can be experimentally determined by mechanical testing (Evans, 1978; Reilly and Burstein, 1975). In the present study, a group of four specimens of the same material (i.e., longitudinal tensile specimens, transverse tensile specimens, longitudinal torsional specimens and transverse torsional specimens) are needed to determine the elastic properties of a transversely isotropic material by the use of four different mechanical tests (Dong, 2002). 2.2. Specimen preparation Eighteen groups of waisted cylindrical specimens of cortical bone were obtained from three pairs of human femora from one male (64 years old) and two females (44 and 60 years old) who were free of bone diseases (Fig. 1). The grip ends of these waisted cylindrical
specimens were coated with cyanoacrylate glue and press-fit into internally threaded brass cylindrical endcaps using an alignment jig (Keaveny et al., 1994). 2.3. Mechanical testing Tensile tests were performed on the longitudinal and transverse tensile specimens using a servo-hydraulic material testing system (MTS 810, MTS System Corporation, Minneapolis, MN) at room temperature with a strain rate of 0.1%/s (Reilly and Burstein, 1975). Torsional tests were conducted on the longitudinal and transverse torsional specimens using a servohydraulic bi-axial material testing system (MTS 858, MTS Systems Corporation, Eden Prairie, MN) at room temperature with a rate of 1 /s under angle displacement control (Evans, 1978; Reilly and Burstein, 1975). 2.4. Histological measurements After mechanical testing, a histological cross-section was prepared from the central portion of a longitudinal or transverse test specimen such that the Haversian systems appeared on a transmitted light microscope (Fig. 1). The cross-section was then ground to a thickness of 200 mm with successive grits of sandpaper until the microstructural components of cortical bone were clearly shown on the microscope (Fig. 2). Consequently, the porosity of the specimen was calculated as the ratio of the area of Haversian canals and resorption
Fig. 1. Specimen preparation of cortical bone samples from the diaphysis of a human femur. First, three bone slabs with a thickness of 35 mm were sectioned from the diaphysis of the femur by a band saw. The posterior-medial quadrant of these bone slabs was then cut into three bone slices (proximal, middle, and distal) using a low-speed diamond saw. Next, the middle slice with a thickness of 20 mm was sectioned into two parallelepipeds with dimensions of 5 5 20 mm to produce a longitudinal tensile specimen and a longitudinal torsional specimen, respectively. Concurrently, the proximal and distal slices with a thickness of 5 mm were sectioned into two parallelepipeds with dimensions of 5 5 20 mm to yield a transverse tensile specimen and a transverse torsional specimen, respectively.
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cavities to the total area of the cross section (Fig. 2). The lacunae and canaliculi were not considered in the present study because the lacunae and the canaliculi represented an average of only 0.80% and 1.48% of
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vascular-free cortical volume in human long bones, respectively (Evans, 1973).
2.5. Statistical analysis Statistical analyses of the experimental results were conducted using SYSTAT (SPSS Inc., Chicago, IL). Simple linear regression was used to examine the dependence of the elastic properties of cortical bone on its porosity. The relationship was considered statistically significant when the p-value was less than 0.05.
3. Results 3.1. Elastic properties of cortical bone
Fig. 2. A histological section of cortical bone indicating how to measure the porosity of cortical bone. The images of the cross section were captured by a video imaging system and transferred to a public domain NIH Image program (developed at the U.S. National Institutes of Health). The area of Haversian canals and resorption cavities was measured by manually tracing the borders of these pores. Meanwhile, the total area of the cross section was measured from the images.
During specimen preparation, a longitudinal tensile specimen from group 11 and a transverse torsional specimen from group 7 were broken (Table 1). Therefore, mechanical tests were not available for these two specimens. For the rest of specimens from all 18 groups, the elastic properties of cortical bone measured from mechanical testing were listed at full length in Table 1. A simple linear correlation analysis indicated that the elastic properties of cortical bone in the same direction were significantly (po0.05) correlated with each other (Table 2). However, the elastic properties of cortical bone in the longitudinal direction were not correlated with those in the transverse direction (Table 2).
Table 1 The transversely isotropic elastic properties of cortical bone by mechanical testing Group number
EL (GPa)
ET (GPa)
GL (GPa)
GT (GPa)
nL
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
20.66 18.12 15.82 17.86 18.84 17.88 15.68 17.98 13.83 17.38 N/Aa 15.09 13.37 17.04 15.74 15.90 16.22 15.82
10.79 10.16 7.97 9.34 10.10 11.15 8.52 9.20 12.81 9.25 9.85 6.46 9.40 10.24 8.63 8.78 10.20 9.05
5.40 5.53 5.31 5.08 5.41 5.04 4.77 5.06 3.32 4.69 3.83 4.18 3.70 5.27 4.54 4.13 5.02 4.97
3.76 3.88 3.65 3.12 2.88 3.15 N/Aa 3.05 4.36 2.75 3.03 2.23 3.34 3.32 3.62 3.35 2.76 3.50
0.39 0.34 0.34 0.32 0.39 0.38 0.39 0.40 0.32 0.41 N/Aa 0.44 0.34 0.35 0.37 0.38 0.37 0.38
Mean7SD
16.6171.83
4.7470.65
3.2870.50
0.3770.03
a
9.5571.36
Mechanical testing data were not available (N/A) for broken specimens during sample preparation.
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3.2. Correlations between elastic moduli and porosity of cortical bone The porosity of cortical bone was calculated by averaging the porosity of four individual specimens from each group (Table 3). The porosity was listed in full length to help provide experimental data to validate future micromechanics modeling of cortical bone. Two groups (7 and 11) were excluded from further regression analysis between the elastic properties of cortical bone and its porosity because mechanical testing data were not available. The elastic properties of cortical bone in the longitudinal direction had significant correlations with its porosity. For instance, the longitudinal Young’s modulus of cortical bone was significantly (po0.01) correlated with its porosity (Fig. 3A). In addition, the shear modulus of cortical bone in the longitudinal direction was also significantly (po0.01) correlated with its porosity (Fig. 3B). Both a linear regression model and a power regression model were used to determine the correlations between elastic properties and porosity Table 2 Correlation coefficients (R) between longitudinal and transverse elastic properties of cortical bone (n=16) R EL (GPa) ET (GPa) GL (GPa) GT (GPa) a
EL (GPa)
ET (GPa)
1 0.18 0.80a 0.09
1 0.04 0.56a
GL (GPa)
(Table 4). The predictabilities of these two models were almost equivalent. For example, the power model (R2=0.69) was slightly better than the linear model (R2=0.66) in consideration of the relationship between longitudinal Young’s modulus and porosity. On the other hand, the linear model (R2=0.72) was slightly better than the power model (R2=0.67) when the relationship between longitudinal shear modulus and porosity was determined. Both the linear and power models indicated that transverse Young’s modulus (Fig. 3C) and transverse shear modulus (Fig. 3D) were not significantly (p>0.65) correlated with the porosity of cortical bone (Table 4). 3.3. The dependence of elastic anisotropy of cortical bone on its porosity The anisotropic ratios of cortical bone elasticity were significantly correlated with the porosity of cortical bone. The ratio of longitudinal to transverse Young’s moduli decreased significantly (po0.01) with the porosity of cortical bone (Fig. 4A). In addition, the ratio of longitudinal to transverse shear moduli had a significant negative correlation with the porosity of cortical bone (Fig. 4B).
GT (GPa)
4. Discussion 1 0.12
1
Significant correlations (po0.05).
The elastic modulus of cortical bone is well known to be reduced by porosity. A number of researchers have observed the significant negative correlations between
Table 3 The histologically assessed porosity of longitudinal tensile, transverse tensile, longitudinal torsional and transverse torsional test specimens of cortical bone Group number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Mean7SD
Porosity of individual specimen (%)
Group porosity (%)
Longitudinal tensile
Transverse tensile
Longitudinal torsional
Transverse torsional
4.77 9.78 4.41 3.93 5.37 7.34 12.59 5.58 13.70 4.53 10.37 12.11 18.42 8.99 7.86 11.49 13.70 7.66
4.14 4.56 10.31 6.09 5.13 9.99 14.31 6.58 19.47 9.57 9.67 8.96 11.12 8.18 6.34 13.74 10.84 11.52
5.98 5.76 5.51 7.58 7.09 10.02 9.24 6.20 12.75 9.26 13.55 10.72 16.36 8.54 10.08 12.90 9.59 10.51
6.55 7.06 5.36 5.58 5.97 4.92 19.93 9.09 8.92 12.19 14.03 7.04 10.97 8.74 9.00 14.65 12.32 8.33
8.9574.16
9.4773.87
9.5472.99
8.8673.93
5.36 6.79 6.40 5.80 5.89 8.07 14.02 6.86 13.71 8.89 11.91 9.71 14.22 8.61 8.32 13.20 11.61 9.51 9.3873.03
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Table 4 The relationships between the elastic properties and the porosity (p) of cortical bone Linear regression model EL= 0.53p+21.43 ET= 0.06p+9.06 GL= 0.19p+6.50 GT= 0.02p+3.16 nL= 0.002p+0.38
R2=0.66 R2=0.01 R2=0.72 R2=0.01 R2=0.02
p=0.0001a p=0.65 p=0.00001a p=0.76 p=0.62
Power regression model log EL= 0.30 log p+1.50 log ET=0.02 log p+0.96 log GL= 0.38 log p+1.03 log GT=0.001 log p+0.51 log nL= 0.02 log p 0.41
R2=0.69 R2=0.002 R2=0.67 R2=0.0001 R2=0.01
p=0.00001a p=0.88 p=0.0001a p=0.99 p=0.77
a
Significant correlations.
Fig. 4. The relationships between the elastic anisotropy of cortical bone and its porosity, (A) the ratio of longitudinal Young’s modulus to transverse Young’s modulus; (B) the ratio of longitudinal shear modulus to transverse shear modulus.
Fig. 3. The relationships between the elastic properties of cortical bone and its porosity, (A) longitudinal Young’s modulus: EL= 0.53p+21.43, R2=0.66; (B) longitudinal shear modulus: GL = 0.19p+6.50, R2=0.72; (C) transverse Young’s modulus: ET; (D) transverse shear modulus: GT.
the mechanical properties of cortical bone and porosity (Currey, 1988; Evans, 1978; McCalden et al., 1993; Schaffler and Burr, 1988; Wachter et al., 2001). Particularly, a power law relationship between elastic modulus and porosity has been proposed (Currey, 1988; Schaffler and Burr, 1988). In the present study, we also
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demonstrated that the elastic and shear moduli of cortical bone in the longitudinal direction decreased significantly with the porosity of cortical bone. However, the predictive capabilities of a linear model and a power law model between elastic properties and porosity were approximately equal (Table 4). In other words, the power law model was no better than the linear model. However, no statistically significant correlations were observed between the elastic properties in the transverse direction and the porosity of cortical bone in the present study, indicating that the elastic properties of cortical bone in the transverse direction were less sensitive to changes in porosity than those in the longitudinal direction. It is possible to infer this difference by the fact that the elastic properties of cortical bone in the transverse direction were not correlated with those in the longitudinal direction (Table 2). This observation on the relationships between the longitudinal and transverse elastic properties of cortical bone was in agreement with the previous finding that the modulus in the radial or circumferential direction (i.e., transverse direction) could not be predicted from modulus in superiorinferior direction (i.e., longitudinal direction) for cortical bone (Rho et al., 1995). Furthermore, the anisotropic ratios of cortical bone elasticity were found to be significantly negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased (Fig. 4). This appeared to be contradictive with the previous studies on the elastic properties of osteoporotic and osteopetrotic bone (Ashman et al., 1985; Katz et al., 1984; Martin et al., 1998). It has been demonstrated that osteoporosis has no effect on anisotropy of cortical bone (Katz et al., 1984), but osteopetrosis greatly reduces it (Ashman et al., 1985; Katz et al., 1984). The inconsistency may indicate that metabolic bone diseases such as osteoporosis and osteopetrosis cause the changes of cortical bone in the microstructural levels and therefore influence its elastic behavior. The outcome of the present study may provide appropriate experimental data to validate micromechanics models of human femoral cortical bone. A number of researchers have developed micromechanics models of cortical bone to predict the changes in mechanical properties due to changes in the microstructural parameters such as porosity, collagen, and mineralization (Aoubiza et al., 1996; Crolet et al., 1993; Hogan, 1992; Katz, 1981; Sevostianov and Kachanov, 2000). Particularly, a finite-element micromechanics model of cortical bone predicted that the longitudinal elastic modulus of cortical bone decreased linearly with porosity (Hogan, 1992). The prediction was consistent with our results that longitudinal Young’s modulus had a significant negative correlation with the porosity of cortical bone.
One limitation of this study was that the transversely isotropic elastic properties of cortical bone were measured from multiple specimens instead of one single specimen. In order to control the effect of inhomogeneity of cortical bone (Evans and Lebow, 1951) in the present study, bone specimens in the same group were taken from the same location of the femoral shaft (Fig. 1). Another limitation was that the number of subjects (n=3) used in the present study was relatively small. In order to fully understand the dependence of transversely isotropic elasticity of cortical bone on its porosity, more subjects, including different age groups and different genders, should be included to address this tissue in the future study. In conclusion, our study demonstrated that the longitudinal Young’s modulus and shear modulus of human femoral cortical bone were significantly correlated with the porosity of cortical bone. However, the elastic properties in the transverse direction of cortical bone were not significantly correlated with the porosity of cortical bone. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone.
Acknowledgements This work was supported by Whitaker Foundation Biomedical Engineering Research Grant 97-0086, NIH/ NIAMS/ASBMR AR45832 and NSF Career BES 9875633.
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