Current Address: Department of Orthopaedic Surgery Rush University Medical Center Chicago, IL 60612

Numerical models commonly account for elastic inhomogeneity in cortical bone using power-law scaling relationships with various measures of tissue density, but limited experimental data exists for anatomic variation in elastic anisotropy. A recent study revealed anatomic variation in the magnitude and anisotropy of elastic constants along the entire femoral diaphysis of a single human femur (_{11}) from the mid-diaphysis toward the epiphyses (_{22}) and longitudinal (_{33}) elastic constants were not statistically significant (^{2}>0.46), as expected, but was only weakly correlated with _{33}/_{11} (^{2}=0.04) and not significantly correlated with _{33}/_{22} and _{11}/_{22}.

Elastic inhomogeneity in human cortical bone tissue is commonly accounted for by power-law scaling relationships with apparent tissue density, mineral density, or porosity (

Ultrasonic wave propagation has been used to enable non-destructive measurement of elastic constants on small specimens which can be sampled from various anatomic locations (

Whole femora were harvested from the lower extremity of nine human donors, including six females (ages 41, 59, 73, 89, 93, and 99) and three males (ages 18, 53, and 78), presenting no toxicology or bone-related pathology. All tissues were obtained post-mortem with prior donor consent following protocol approved by the Notre Dame Human Subjects Institutional Review Board. Each femur was stored in a freezer at −20°C wrapped with gauze soaked in phosphate-buffered saline. A total of 108 parallelepiped cortical bone specimens (12 specimens/donor) were prepared from each anatomic quadrant at 20, 50, and 80% of the total femur length (

Longitudinal elastic constants from the main diagonal of the reduced fourth-order stiffness tensor were measured on hydrated specimens using the pulse-transmission method for ultrasonic wave propagation as,
_{ii}_{w} is the density of DI water (_{ii}_{i}_{i}_{44}, _{55}, and _{66}) were also measured for a subset of the donors and are reported in

The nominal specimen size was approximately 5×5×5 mm, although the radial dimension was maximized within the available cortical thickness. Six specimens were removed from the analysis due to exhibiting a small radial thickness (<1.5 mm) that inhibited accurate measurement of the longitudinal and circumferential elastic constants by compromising the bulk wave assumption (

One-way analysis of variance (ANOVA) was used to examine the effect of the anatomic location (% total femur length) on elastic constant magnitude and anisotropy (JMP 8, SAS Institute Inc., Cary, NC).

Each elastic constant and anisotropy ratio exhibited statistically significant differences overall and at each given location along the length of the femoral diaphysis (_{33}>_{22}>_{11} (_{11} decreased from the mid-diaphysis toward the distal (_{22} and _{33} did not exhibit statistically significant differences (_{33}/_{11} increased (_{33}/_{22} decreased (_{11}/_{22} decreased (_{33}/_{11} increased (_{33}/_{22} did not exhibit a statistically significant difference (_{11}/_{22} decreased (

Multivariate analysis of covariance indicated a statistically significant effect of the anatomic location along the femoral diaphysis on the elastic constant magnitude and anisotropy (_{11}, _{22}, _{33}/_{11}, and _{11}/_{22}; _{33} and _{33}/_{22}), and that this effect was not confounded by the effects of anatomic quadrant, donor age, and gender which were all not statistically significant. The apparent tissue density exhibited a statistically significant effect on elastic constant magnitudes (^{2}>0.46), as expected, but was only weakly correlated with _{33}/_{11} (^{2}=0.04) and not significantly correlated with _{33}/_{22} and _{11}/_{22}.

The apparent tissue density decreased toward the distal epiphysis (

The entire dataset, including data and analysis for shear elastic constants (_{44}, _{55}, and _{66}), is available in

The results of this study confirmed trends previously reported for a single donor (_{11}) from the mid-diaphysis toward the epiphyses, since _{22} and _{33} did not change (_{11} increased _{33}/_{11} and decreased _{11}/_{22} (

Variations in intracortical porosity along the length of the femoral diaphysis were previously hypothesized to account for differences in the radial elastic constant (_{3}/_{2} (Young's moduli) with increased tissue porosity (^{2}=0.35) (_{33}/_{11} (^{2}=0.04) and not significantly correlated with _{33}/_{22} and _{11}/_{22}. Moreover, the multivariate analysis suggested that the anatomic location along the femoral diaphysis and apparent tissue density were together able to account for 56–64% of the variability in elastic constants, but only 7–33% of the variability in elastic anisotropy.

A true scalar quantity like apparent tissue density or porosity would not be expected to influence a directional property like the anisotropy ratio. However, the spatial size, morphology, orientation, and distribution of pores along the length of the femur may contribute to the observed property variation independent of the porosity volume fraction. Finite element models on micro-computed tomography reconstructions with segmented intracortical porosity (

Other structural features within the extracellular matrix may also be responsible for variations in elastic anisotropy. The apatite crystal orientation distribution was recently shown to exhibit the greatest influence on elastic anisotropy in a specimen-specific micromechanical model accounting for seven structural parameters across multiple length scales in human cortical bone (

The data gathered in this study may be useful for numerical models of the femur in order to account for anatomic variations in elastic inhomogeneity

The statistical power in this study was sufficient to meet the study objective, but was not sufficient to examine differences in the magnitude and anisotropy of elastic constants with respect to the anatomic quadrant, donor gender, and donor age. Statistical power for the effect of location along the femoral diaphysis was greater than 0.99 for _{11}, _{22}, _{33}/_{11}, and _{11}/_{22}, and greater than 0.60 for _{33} and _{33}/_{22}. Statistical power for the effects of the location along the femoral diaphysis, anatomic quadrant, and donor age on the apparent tissue density was 0.51, 0.65, and 0.85, respectively. Statistical power for the effect of the apparent tissue density on elastic constants was 1.00. All other effects exhibited statistical power less than 0.34 in the multivariate analysis.

Shear.pdf Data analysis for shear elastic constants.

This research was partially supported by the U.S. Army Medical Research and Material Command (W81XWH-06-1-0196) through the Peer-Reviewed Medical Research Program (PR054672) and the National Institutes of Health (AR052008).

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Cortical bone specimens were sectioned from the diaphysis of whole human femora at (a) 20, 50, and 80% of the total femur length and (b) from each anatomic quadrant (A = anterior, M = medial, P = posterior, L = lateral), using an orthogonal curvilinear coordinate system (1 = radial, 2 = circumferential, 3 = longitudinal).

Mean (a) elastic constants and (b) anisotropy ratios in the three orthogonal specimen axes measured along the length of the femoral diaphysis. Error bars show one standard deviation. Statistically significant differences existed between each elastic constant and anisotropy ratio at a given location along the length of the femoral diaphysis (