Relationship between ultrasonic parameters and apparent trabecular bone elastic modulus: a numerical approach

Trabecular bone patterning across the human hand

Hand bone morphology is regularly used to link particular hominin species with behaviors relevant to cognitive/technological progress. Debates about the functional significance of differing hominin hand bone morphologies tend to rely on establishing phylogenetic relationships and/or inferring behavior from epigenetic variation arising from mechanical loading and adaptive bone modeling. Most research focuses on variation in cortical bone structure, but additional information about hand function may be provided through the analysis of internal trabecular structure. While primate hand bone trabecular structure is known to vary in ways that are consistent with expected joint loading differences during manipulation and locomotion, no study exists that has documented this variation across the numerous bones of the hand. We quantify the trabecular structure in 22 bones of the human hand (early/extant modern Homo sapiens) and compare structural variation between two groups associated with post-agricultural/industrial (post-Neolithic) and foraging/hunter-gatherer (forager) subsistence strategies. We (1) establish trabecular bone volume fraction (BV/TV), modulus (E), degree of anisotropy (DA), mean trabecular thickness (Tb.Th) and spacing (Tb.Sp); (2) visualize the average distribution of site-specific BV/TV for each bone; and (3) examine if the variation in trabecular structure is consistent with expected joint loading differences among the regions of the hand and between the groups. Results indicate similar distributions of trabecular bone in both groups, with those of the forager sample presenting higher BV/TV, E, and lower DA, suggesting greater and more variable loading during manipulation. We find indications of higher loading along the ulnar side of the forager sample hand, with high site-specific BV/TV distributions among the carpals that are suggestive of high loading while the wrist moves through the 'dart-thrower's' motion. These results support the use of trabecular structure to infer behavior and have direct implications for refining our understanding of human hand evolution and fossil hominin hand use.

Quantitative Ultrasonography Measurements of the Phalanges in Adolescents: A Mixed Longitudinal Study

This study examined the effect of pubertal development on Amplitude Dependent Speed of Sound (AD-SoS), accounting for the growth in stature among adolescents. A mixed-longitudinal design with 3 assessments across a 15-mo period in 439 adolescents (girls: 215; boys: 224) aged 9-16 y was used. Bayesian multilevel models were used to describe gender-specific AD-SoS variations among participants during pubertal years. Substantial increments in AD-SoS during pubertal years were observed in both genders. AD-SoS changes were positively related to stature, and the rate of stature growth per year. Quantitative ultrasonography was sensible to describe age-related changes of bone mass during pubertal development. It seemed clinically reliable to use AD-SoS in the study of bone growth and development.

Fast decomposition of two ultrasound longitudinal waves in cancellous bone using a phase rotation parameter for bone quality assessment: Simulation study

Ultrasound signals that pass through cancellous bone may be considered to consist of two longitudinal waves, which are called fast and slow waves. Accurate decomposition of these fast and slow waves is considered to be highly beneficial in determination of the characteristics of cancellous bone. In the present study, a fast decomposition method using a wave transfer function with a phase rotation parameter was applied to received signals that have passed through bovine bone specimens with various bone volume to total volume (BV/TV) ratios in a simulation study, where the elastic finite-difference time-domain method is used and the ultrasound wave propagated parallel to the bone axes. The proposed method succeeded to decompose both fast and slow waves accurately; the normalized residual intensity was less than -19.5 dB when the specimen thickness ranged from 4 to 7 mm and the BV/TV value ranged from 0.144 to 0.226. There was a strong relationship between the phase rotation value and the BV/TV value. The ratio of the peak envelope amplitude of the decomposed fast wave to that of the slow wave increased monotonically with increasing BV/TV ratio, indicating the high performance of the proposed method in estimation of the BV/TV value in cancellous bone.

Influence of anisotropic bone properties on the biomechanical behavior of the acetabular cup implant: a multiscale finite element study

Although the biomechanical behavior of the acetabular cup (AC) implant is determinant for the surgical success, it remains difficult to be assessed due to the multiscale and anisotropic nature of bone tissue. The aim of the present study was to investigate the influence of the anisotropic properties of peri-implant trabecular bone tissue on the biomechanical behavior of the AC implant at the macroscopic scale. Thirteen bovine trabecular bone samples were imaged using micro-computed tomography (μCT) with a resolution of 18 μm. The anisotropic biomechanical properties of each sample were determined at the scale of the centimeter based on a dedicated method using asymptotic homogenization. The material properties obtained with this multiscale approach were used as input data in a 3D finite element model to simulate the macroscopic mechanical behavior of the AC implant under different loading conditions. The largest stress and strain magnitudes were found around the equatorial rim and in the polar area of the AC implant. All macroscopic stiffness quantities were significantly correlated (R² > 0.85, p < 6.5 e-6) with BV/TV (bone volume/total volume). Moreover, the maximum value of the von Mises stress field was significantly correlated with BV/TV (R² > 0.61, p < 1.6 e-3) and was always found at the bone-implant interface. However, the mean value of the microscopic stress (at the scale of the trabeculae) decrease as a function of BV/TV for vertical and torsional loading and do not depend on BV/TV for horizontal loading. These results highlight the importance of the anisotropic properties of bone tissue.

Trabecular architecture in the thumb of Pan and Homo: implications for investigating hand use, loading, and hand preference in the fossil record

OBJECTIVES

Humans display an 85-95% cross-cultural right-hand bias in skilled tasks, which is considered a derived behavior because such a high frequency is not reported in wild non-human primates. Handedness is generally considered to be an evolutionary byproduct of selection for manual dexterity and augmented visuo-cognitive capabilities within the context of complex stone tool manufacture/use. Testing this hypothesis requires an understanding of when appreciable levels of right dominant behavior entered the fossil record. Because bone remodels in vivo, skeletal asymmetries are thought to reflect greater mechanical loading on the dominant side, but incomplete preservation of external morphology and ambiguities about past loading environments complicate interpretations. We test if internal trabecular bone is capable of providing additional information by analyzing the thumb of Homo sapiens and Pan.

MATERIALS AND METHODS

We assess trabecular structure at the distal head and proximal base of paired (left/right) first metacarpals using micro-CT scans of Homo sapiens (n = 14) and Pan (n = 9). Throughout each epiphysis we quantify average and local bone volume fraction (BV/TV), degree of anisotropy (DA), and elastic modulus (E) to address bone volume patterning and directional asymmetry.

RESULTS

We find a right directional asymmetry in H. sapiens consistent with population-level handedness, but also report a left directional asymmetry in Pan that may be the result of postural and/or locomotor loading.

CONCLUSION

We conclude that trabecular bone is capable of detecting right/left directional asymmetry, but suggest coupling studies of internal structure with analyses of other skeletal elements and cortical bone prior to applications in the fossil record.

Ultrasound Speed of Sound Measurements in Trabecular Bone Using the Echographic Response of a Metallic Pin

Bone quality is an important parameter in spine surgery, but its clinical assessment remains difficult. The aim of the work described here was to demonstrate in vitro the feasibility of employing quantitative ultrasound to retrieve bone mechanical properties using an echographic technique taking advantage of the presence of a metallic pin inserted in bone tissue. A metallic pin was inserted in bone tissue perpendicular to the transducer axis. The echographic response of the bone sample was determined, and the echo of the pin inserted in bone tissue and water were compared to determine speed of sound, which was compared with bone volume fraction. A 2-D finite-element model was developed to assess the effect of positioning errors. There was a significant correlation between speed of sound and bone volume fraction (R(2) = 0.6). The numerical results indicate the relative robustness of the measurement method, which could be useful to estimate bone quality intra-operatively.

Numerical Analysis of Ultrasound Backscattered Waves in Cancellous Bone Using a Finite-Difference Time-Domain Method: Isolation of the Backscattered Waves From Various Ranges of Bone Depths

Using a finite-difference time-domain method, ultrasound backscattered waves inside cancellous bone were numerically analyzed to investigate the backscatter mechanism. Two bone models with different thicknesses were modeled with artificial absorbing layers positioned at the back surfaces of the model, and an ultrasound pulse wave was transmitted toward the front surface. By calculating the difference between the simulated waveforms obtained using the two bone models, the backscattered waves from a limited range of depths in cancellous bone could be isolated. The results showed that the fast and slow longitudinal waves, which have previously been observed only in the ultrasound waveform transmitted through the bone, could be distinguished in the backscattered waveform from a deeper bone depth when transmitting the ultrasound wave parallel to the main orientation of the trabecular network. The amplitudes of the fast and slow backscattered waves were more closely correlated with the bone porosity [R2 = 0.84 and 0.66 (p < 0.001), respectively] than the amplitude of the whole (nonisolated) backscattered waves [R2 = 0.48 (p < 0.001)]. In conclusion, the nonisolated backscattered waves could be regarded as the superposition of the fast and slow waves reflected from various bone depths, returning at different times.

Fast characterization of two ultrasound longitudinal waves in cancellous bone using an adaptive beamforming technique

The received signal in through-transmission ultrasound measurements of cancellous bone consists of two longitudinal waves, called the fast and slow waves. Analysis of these fast and slow waves may reveal characteristics of the cancellous bone that would be good indicators of osteoporosis. Because the two waves often overlap, decomposition of the received signal is an important problem in the characterization of bone quality. This study proposes a fast and accurate decomposition method based on the frequency domain interferometry imaging method with a modified wave transfer function that uses a phase rotation parameter. The proposed method accurately characterized the fast and slow waves in the experimental study, and the residual intensity, which was normalized with respect to the received signal intensity, was less than -20 dB over the bone specimen thickness range from 6 to 15 mm. In the simulation study, the residual intensity was less than -20 dB over the specimen thickness range from 3 to 8 mm. Decomposition of a single received signal takes only 5 s using a laptop personal computer with a single central processing unit. The proposed method has great potential to provide accurate and rapid measurements of indicators of osteoporosis in cancellous bone.

Biomechanics and mechanobiology of trabecular bone: a review

Trabecular bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. Studying the mechanical properties of trabecular bone is important, since trabecular bone is the main load bearing bone in vertebral bodies and also transfers the load from joints to the compact bone of the cortex of long bones. This review article highlights the high dependency of the mechanical properties of trabecular bone on species, age, anatomic site, loading direction, and size of the sample under consideration. In recent years, high resolution micro finite element methods have been extensively used to specifically address the mechanical properties of the trabecular bone and provide unique tools to interpret and model the mechanical testing experiments. The aims of the current work are to first review the mechanobiology of trabecular bone and then present classical and new approaches for modeling and analyzing the trabecular bone microstructure and macrostructure and corresponding mechanical properties such as elastic properties and strength.

Effects of biomechanical properties of the bone-implant interface on dental implant stability: from in silico approaches to the patient's mouth

Dental implants have become a routinely used technique in dentistry for replacing teeth. However, risks of failure are still experienced and remain difficult to anticipate. Multiscale phenomena occurring around the implant interface determine the implant outcome. The aim of this review is to provide an understanding of the biomechanical behavior of the interface between a dental implant and the region of bone adjacent to it (the bone-implant interface) as a function of the interface's environment. First, we describe the determinants of implant stability in relation to the different multiscale simulation approaches used to model the evolution of the bone-implant interface. Then, we review the various aspects of osseointegration in relation to implant stability. Next, we describe the different approaches used in the literature to measure implant stability in vitro and in vivo. Last, we review various factors affecting the evolution of the bone-implant interface properties.

Multichannel instantaneous frequency analysis of ultrasound propagating in cancellous bone

An ultrasonic pulse propagating in cancellous bone can be separated into two waves depending on the condition of the specimen. These two waves, which are called the fast wave and the slow wave, provide important information for the diagnosis of osteoporosis. The present study proposes to utilize a signal processing method that extracts the instantaneous frequency (IF) of waveforms from multiple spectral channels. The instantaneous frequency was expected to be able to show detailed time-frequency properties of ultrasonic waves being transmitted through cancellous bone. The employed method, termed the multichannel instantaneous frequency (MCIF) method, showed robustness against background noise as compared to the IF that was directly derived from the original waveform. The extracted IF revealed that the frequency of the fast wave was affected by both the propagation distance within the specimen and the bone density, independently. On the other hand, the alternation of the center frequency of the originally transmitted wave did not produce proportional changes in the extracted IF values of the fast waves, suggesting that the fast wave IF mainly reflected the thickness of the specimens. These findings may provide the possibility of obtaining a more precise diagnosis of osteoporosis.

Radial anatomic variation of ultrasonic velocity in human cortical bone

Quantitative ultrasound techniques can be used to retrieve cortical bone quality. The aim of this study was to investigate the anatomic variations in speed of sound (SOS) in the radial direction of cortical bone tissue. SOS measurements were realized in 17 human cortical bone samples with a 3.5-MHz transverse transmission device. The radial dependence of SOS was investigated in a direction perpendicular to the periosteum. For each sample, bone porosity was measured using an X-ray micro-computed tomography device. The mean SOS was 3586 ± 255 m/s. For 16 of 17 specimens, similar radial variations in SOS were observed. In the periosteal region, SOS first decreased in the direction of the endosteum and reached a minimum value approximately in the middle of the cortical bone. SOS then increased, moving to the endosteal region. A significant negative correlation was obtained between SOS and porosity (R = -0.54, p = 0.02).

Biomechanical determinants of the stability of dental implants: influence of the bone-implant interface properties

Dental implants are now widely used for the replacement of missing teeth in fully or partially edentulous patients and for cranial reconstructions. However, risks of failure, which may have dramatic consequences, are still experienced and remain difficult to anticipate. The stability of biomaterials inserted in bone tissue depends on multiscale phenomena of biomechanical (bone-implant interlocking) and of biological (mechanotransduction) natures. The objective of this review is to provide an overview of the biomechanical behavior of the bone-dental implant interface as a function of its environment by considering in silico, ex vivo and in vivo studies including animal models as well as clinical studies. The biomechanical determinants of osseointegration phenomena are related to bone remodeling in the vicinity of the implants (adaptation of the bone structure to accommodate the presence of a biomaterial). Aspects related to the description of the interface and to its space-time multiscale nature will first be reviewed. Then, the various approaches used in the literature to measure implant stability and the bone-implant interface properties in vitro and in vivo will be described. Quantitative ultrasound methods are promising because they are cheap, non invasive and because of their lower spatial resolution around the implant compared to other biomechanical approaches.

Quantitative ultrasound and fracture risk prediction in non-osteoporotic men and women as defined by WHO criteria

This study sought to determine the association between calcaneal quantitative ultrasound (QUS) and fracture risk in individuals without osteoporosis according to the World Health Organization criteria (i.e., BMD T-score > -2.5). We found that calcaneal QUS is an independent predictor of fracture risk in women with non-osteoporotic bone mineral density (BMD).

INTRODUCTION

More than 50 % of women and 70 % of men who sustain a fragility fracture have BMD above the osteoporotic threshold (T-score > -2.5). Calcaneal QUS is associated with fracture risk. This study aimed to test the hypothesis that low calcaneal QUS is associated with increased fracture risk in individuals with non-osteoporotic BMD.

METHODS

We included 312 women and 390 men aged 62-90 years with BMD T-score > -2.5 at femoral neck. QUS was measured in broadband ultrasound attenuation (BUA) at the calcaneus using a CUBA sonometer. BMD was measured at the femoral neck (FNBMD) by dual energy X-ray absorptiometry using GE Lunar DPX-L densitometer. The incidences of any fragility fracture were ascertained by X-ray reports during the follow-up period from 1994 to 2011.

RESULTS

Of the 702 participants, 26 % of women (n = 80/312) and 14 % of men (n = 53/390) experienced at least one fragility fracture during the follow-up period. In women, after adjusting for covariates, increased risk of any fracture was significantly associated with decreased BUA (HR = 1.50; 95 % CI, 1.13-1.99). Compared with that of FNBMD, the models with BUA, in women, had greater AUC (0.71, 0.85, 0.71 for any, hip and vertebral fracture, respectively), and yielded a net reclassification improvement of 16.4 % (P = 0.009) when combined with FNBMD. In men, BUA was not significantly associated with fracture risk before and after adjustment.

CONCLUSION

These results suggest that calcaneal BUA is an independent predictor of fracture risk in women with non-osteoporotic BMD.

NUMERICAL SIMULATIONS OF CHANGE IN TRABECULAR STRUCTURE DUE TO BONE REMODELING UNDER ULTRASOUND PROPAGATION

Bone remodeling is defined as the coupling of bone formation and resorption on the bone surface. Numerical simulations of the remodeling in cancellous bone were performed to reproduce the change in the trabecular structure. Assuming that the formation/resorption in cancellous bone could be generated on the trabecular surface, where the local stress under the mechanical load was larger/smaller than the averaged stress on the surrounding surface, voxel trabecular elements in a numerical model of bovine cancellous bone were added/removed. An ultrasound continuous wave in the frequency range 0.1–1.0 MHz was applied as the mechanical load, and then, the local stress was analyzed using a finite-difference time-domain (FDTD) method. Using the remodeling simulations, both changes in the trabecular structure could be reproduced with decreasing and increasing porosity. In changes, the trabecular elements and the pore spaces became strongly oriented in the direction of ultrasound propagation. In addition, the remodeling simulations indicated that both bone formation and resorption lessened as the frequency increased.

Phase velocity and attenuation predictions of waves in cancellous bone using an iterative effective medium approximation

The quantitative determination of wave dispersion and attenuation in bone is an open research area as the factors responsible for ultrasound absorption and scattering in composite biological tissues have not been completely explained. In this study, we use the iterative effective medium approximation (IEMA) proposed in [1] so as to calculate phase velocity and attenuation in media with properties similar to those of cancellous bones. Calculations are performed for a frequency range of 0.4-0.8 MHz and for different inclusions' volume concentrations and sizes. Our numerical results are compared with previous experimental findings so as to assess the effectiveness of IEMA. It was made clear that attenuation and phase velocity estimations could provide supplementary information for cancellous bone characterization.

Microarchitecture and bone quality in the human calcaneus: local variations of fabric anisotropy

The local variability of microarchitecture of human trabecular calcaneus bone is investigated using high-resolution micro-computed tomography (µCT) scanning. The fabric tensor is employed as the measure of the microarchitecture of the pore structure of a porous medium. It is hypothesized that a fabric tensor-dependent poroelastic ultrasound approach will more effectively predict the data variance than will porosity alone. The specific aims of the present study are as follows: (1) to quantify the morphology and local anisotropy of the calcaneus microarchitecture with respect to anatomical directions; (2) to determine the interdependence, or lack thereof, of microarchitecture parameters, fabric, and volumetric bone mineral density (vBMD); and (3) to determine the relative ability of vBMD and fabric measurements in evaluating the variance in ultrasound wave velocity measurements along orthogonal directions in the human calcaneus. Our results show that the microarchitecture in the analyzed regions of human calcanei is anisotropic, with a preferred alignment along the posterior-anterior direction. Strong correlation was found between most scalar architectural parameters and vBMD. However, no statistical correlation was found between vBMD and the fabric components, the measures of the pore microstructure orientation. Therefore, among the parameters usually considered for cancellous bone (ie, classic histomorphometric parameters such as porosity, trabecular thickness, number and separation), only fabric components explain the data variance that cannot be explained by vBMD, a global mass measurement, which lacks the sensitivity and selectivity to distinguish osteoporotic from healthy subjects because it is insensitive to directional changes in bone architecture. This study demonstrates that a multidirectional, fabric-dependent poroelastic ultrasound approach has the capability of characterizing anisotropic bone properties (bone quality) beyond bone mass, and could help to better understand anisotropic changes in bone architecture using ultrasound.

Comparison of phase velocity in trabecular bone mimicking-phantoms by time domain numerical (EFIT) and analytical multiple scattering approaches

The corrected Waterman-Truell model and the Elastodynamic Finite Integration Technique were used to analyze the ultrasonic wave dispersion in trabecular bones mimicking phantoms. A simple two-phase model of the trabecular bone is assumed; the trabeculae structure and the bone marrow. The phase velocity for frequencies within the range from 400kHz to 800kHz were computed for different scatterer arrays varying their dimensions and number. The theoretical and numerical results were compared to experimental published data, obtained from a mimicking phantom composed by a periodic array of nylon shreds (trabeculae array) immersed in a water tank. Our results showed an excellent consistency when compared to experimental data. The negative dispersions of -8.48m/s/MHz and -9.16m/s/MHz were computed by the multiple scattering method and the numerical approach, respectively, where the latter is closer to the experimental dispersion of -12.09m/s/MHz. Similar result has been reported in the literature, where the dispersion predicted by the Generalized Self-Consistent Method [J. Acoust. Soc. Am. 124 (2008) 4047] is -9.96m/s/MHz.

Analysis of superimposed ultrasonic guided waves in long bones by the joint approximate diagonalization of eigen-matrices algorithm

The parameters of ultrasonic guided waves (GWs) are very sensitive to mechanical and structural changes in long cortical bones. However, it is a challenge to obtain the group velocity and other parameters of GWs because of the presence of mixed multiple modes. This paper proposes a blind identification algorithm using the joint approximate diagonalization of eigen-matrices (JADE) and applies it to the separation of superimposed GWs in long bones. For the simulation case, the velocity of the single mode was calculated after separation. A strong agreement was obtained between the estimated velocity and the theoretical expectation. For the experiments in bovine long bones, by using the calculated velocity and a theoretical model, the cortical thickness (CTh) was obtained. For comparison with the JADE approach, an adaptive Gaussian chirplet time-frequency (ACGTF) method was also used to estimate the CTh. The results showed that the mean error of the CTh acquired by the JADE approach was 4.3%, which was smaller than that of the ACGTF method (13.6%). This suggested that the JADE algorithm may be used to separate the superimposed GWs and that the JADE algorithm could potentially be used to evaluate long bones.

Numerical investigation of ultrasound refraction caused by oblique orientation of trabecular network in cancellous bone

Ultrasound propagation through cancellous bone can be greatly affected by the trabecular structure. In the present study, the ultrasound propagation for the oblique orientation of the trabecular network was numerically investigated using 3-D finite-difference time-domain (FDTD) simulations. The models of cancellous bone were reconstructed from X-ray microcomputed tomographic (μCT) images of a bovine bone. Cancellous bone models with various orientations of the trabecular network were realized by cutting the μCT images rotated from 0 to 90°. Ultrasound waveforms propagating through these cancellous bone models were simulated while changing the receiving position. The refraction of the ultrasound wave for the oblique angle of the main orientation was investigated on the basis of the variation in the arrival time and peak amplitude. As the propagation direction approached the direction parallel to the main orientation, the arrival time of the first peak became less and the peak amplitude became smaller. This means that the wave of the first peak, which corresponded to a fast wave, propagated in the direction perpendicular to the main orientation. In addition, a strong correlation between the first-peak amplitude and the arrival time was observed in the porosity range of 0.68 to 0.85, in which the slope of the amplitude with respect to time increased linearly with porosity.

Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: an in vitro study

The influence of cancellous bone microstructure on the ultrasonic wave propagation of fast and slow waves was experimentally investigated. Four spherical cancellous bone specimens extracted from two bovine femora were prepared for the estimation of acoustical and structural anisotropies of cancellous bone. In vitro measurements were performed using a PVDF transducer (excited by a single sinusoidal wave at 1 MHz) by rotating the spherical specimens. In addition, the mean intercept length (MIL) and bone volume fraction (BV/TV) were estimated by X-ray micro-computed tomography. Separation of the fast and slow waves was clearly observed in two specimens. The fast wave speed was strongly dependent on the wave propagation direction, with the maximum speed along the main trabecular direction. The fast wave speed increased with the MIL. The slow wave speed, however, was almost constant. The fast wave speeds were statistically higher, and their amplitudes were statistically lower in the case of wave separation than in that of wave overlap.

Determination of the heterogeneous anisotropic elastic properties of human femoral bone: from nanoscopic to organ scale

Cortical bone is a multiscale composite material. Its elastic properties are anisotropic and heterogeneous across its cross-section, due to endosteal bone resorption which might affect bone strength. The aim of this paper was to describe a homogenization method leading to the estimation of the variation of the elastic coefficients across the bone cross-section and along the bone longitudinal axis. The method uses the spatial variations of bone porosity and of the degree of mineralization of the bone matrix (DMB) obtained from the analysis of 3-D synchrotron micro-computed tomography images. For all three scales considered (the foam (100 nm), the ultrastructure (5 microm) and the mesoscale (500 microm)), the elastic coefficients were determined using the Eshelby's inclusion problem. DMB values were used at the scale of the foam. Collagen was introduced at the scale of the ultrastructure and bone porosity was introduced at the mesoscale. The pores were considered as parallel cylinders oriented along the bone axis. Each elastic coefficient was computed for different regions of interest, allowing an estimation of its variations across the bone cross-section and along the bone longitudinal axis. The method was applied to a human femoral neck bone specimen, which is a site of osteoporotic fracture. The computed elastic coefficients for cortical bone were in good agreement with experimental results, but some discrepancies were obtained in the endosteal part (trabecular bone). These results highlight the importance of accounting for the heterogeneity of cortical bone properties across bone cross-section and along bone longitudinal axis.