Erratum to: Mechanical properties of femoral trabecular bone in dogs

After the publication of this work [1], we became aware of the fact that the frequency of the ultrasound transmitter that we used for determining the elastic moduli of the trabecular bone specimens was not correctly specified. The oscillation frequency of the ultrasound transmitter was 2 MHz (and not 100 MHz as stated in our work) while we used a sampling rate of 100 MHz. In our publication, the oscillation frequency and sampling rate were confounded. Therefore also the statement in the discussion that we might have determined elastic moduli of trabecular bone tissue rather than the elastic properties of whole specimens because we used an ultrasound frequency > 2 MHz is wrong and has to be omitted.     

Effect of boundary conditions on yield properties of human femoral trabecular bone

Trabecular bone plays an important mechanical role in bone fractures and implant stability. Homogenized nonlinear finite element (FE) analysis of whole bones can deliver improved fracture risk and implant loosening assessment. Such simulations require the knowledge of mechanical properties such as an appropriate yield behavior and criterion for trabecular bone. Identification of a complete yield surface is extremely difficult experimentally but can be achieved in silico by using micro-FE analysis on cubical trabecular volume elements. Nevertheless, the influence of the boundary conditions (BCs), which are applied to such volume elements, on the obtained yield properties remains unknown. Therefore, this study compared homogenized yield properties along 17 load cases of 126 human femoral trabecular cubic specimens computed with classical kinematic uniform BCs (KUBCs) and a new set of mixed uniform BCs, namely periodicity-compatible mixed uniform BCs (PMUBCs). In stress space, PMUBCs lead to 7–72 % lower yield stresses compared to KUBCs. The yield surfaces obtained with both KUBCs and PMUBCs demonstrate a pressure-sensitive ellipsoidal shape. A volume fraction and fabric-based quadric yield function successfully fitted the yield surfaces of both BCs with a correlation coefficient \(R^{2} \ge 0.93\). As expected, yield strains show only a weak dependency on bone volume fraction and fabric. The role of the two BCs in homogenized FE analysis of whole bones will need to be investigated and validated with experimental results at the whole bone level in future studies.     

Mechanical properties of trabecular bone. Dependency on strain rate.

The effect of strain rate (epsilon) and apparent density (rho) on stiffness (E), strength (sigma u), and ultimate strain (epsilon u) was studied in 60 human trabecular bone specimens from the proximal tibia. Testing was performed by uniaxial compression to 5% specimen strain. Six different strain rates were used: 0.0001, 0.001, 0.01, 0.1, 1, and 10 s-1. Apparent density ranged between 0.23 and 0.59 g cm-3. Linear and non-linear regression analyses using strength, stiffness and ultimate strain as dependent variables (Y) and strain rate and apparent density as independent variables were performed using the following models: Y = a rho b epsilon c, Y = rho b(a + c epsilon; Y = (a + b rho)epsilon c, Y = a rho 2 epsilon c, E = a rho 3 epsilon c. The variations of strength and stiffness were explained equally well by the linear and the power function relationship to strain rate. The exponent was 0.07 in the power function relationship between strength and strain rate and 0.05 between stiffness and strain rate. The variation of ultimate strain was explained best using a power function relationship to strain rate (exponent = 0.03). The variation of strength and stiffness was explained equally well by the linear, power function and quadratic relationship to apparent density. The cubic relationship between stiffness and apparent density showed a less good fit. Ultimate strain varied independently of apparent density.     

Mechanical properties of porcine femoral cortical bone measured by nanoindentation

This study uses a nanoindentation technique to examine variations in the local mechanical properties of porcine femoral cortical bone under hydrated conditions. Bone specimens from three age groups (6, 12 and 42 months), representing developing bone, ranging from young to mature animals, were tested on the longitudinal and transverse cross-sectional surfaces. Elastic modulus and hardness of individual lamellae within bone's microstructure: laminar bone, interstitial bone, and osteons, were measured. Both the elastic modulus and hardness increased with age. However, the magnitudes of these increases were different for each microstructural component. The longitudinal moduli were higher than the transverse moduli. Dehydrated samples were also tested to allow a comparison with hydrated samples and these resulted in higher moduli and hardness than the hydrated samples. Again, the degree of variation was different for each microstructural component. These results indicate that the developmental changes in bone have different rates of mechanical change within each microstructural component.     

Interrelationship of trabecular mechanical and microstructural properties in sheep trabecular bone

The ability to evaluate fracture risk at an early time point is essential for improved prognostics as well as enhanced treatment in cases of bone loss such as from osteoporosis. Improving the diagnostic ability is inherent upon both high-resolution non-invasive imaging, and a thorough understanding of how the derived indices of structure and density relate to its true mechanical behavior. Using sheep femoral trabecular bone with a range of strength, the interrelationship of mechanical and microstructural parameters was analyzed using multi-directional mechanical testing and micro-computed tomography. Forty-five cubic trabecular bone samples were harvested from 23 adult female sheep, some of whom had received hind-limb vibratory stimuli over the course of 2 years with consequently enhanced mechanical properties. These samples were pooled into a low, medium, or high strength group for further analysis. The findings show that microCT indices that are structural in nature, e.g., structural model index (SMI) (r2=0.85, p<0.0001) is as good as more density oriented indices like bone volume/total volume (BV/TV) (r2=0.81, p<0.0001) in predicting the ultimate strength of a region of trabecular bone. Additionally, those indices more related to global changes in trabecular structure such as connectivity density (ConnD) or degree of anisotropy (DA) are less able to predict the mechanical properties of bone. Interrelationships of trabecular indices such as trabecular number (TbN), thickness (TbTh), and spacing (TbSp) provide clues as to how the trabecular bone will remodel to ultimately achieve differences in the apparent mechanical properties. For instance, the analysis showed that a loss of bone primarily affects the connectedness and overall number of trabeculae, while increased strength results in an increase of the overall thickness of trabeculae while not improving the connectedness. Certainly, the microCT indices studied are able to predict the bulk mechanical properties of a trabecular ROI well, leaving unaccounted only about 15-20% of its inherent variability. Diagnostically, this implies that future work on the early prediction of fracture risk should continue to explore the role of bone quality as the key factors or as an adjuvant to bone quantity (e.g., apparent density).     

Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue

The ability to determine trabecular bone tissue elastic and failure properties has biological and clinical importance. To date, trabecular tissue yield strains remain unknown due to experimental difficulties, and elastic moduli studies have reported controversial results. We hypothesized that the elastic and tensile and compressive yield properties of trabecular tissue are similar to those of cortical tissue. Effective tissue modulus and yield strains were calibrated for cadaveric human femoral neck specimens taken from 11 donors, using a combination of apparent-level mechanical testing and specimen-specific, high-resolution, nonlinear finite element modeling. The trabecular tissue properties were then compared to measured elastic modulus and tensile yield strain of human femoral diaphyseal cortical bone specimens obtained from a similar cohort of 34 donors. Cortical tissue properties were obtained by statistically eliminating the effects of vascular porosity. Results indicated that mean elastic modulus was 10% lower (p<0.05) for the trabecular tissue (18.0+/-2.8 GPa) than for the cortical tissue (19.9+/-1.8 GPa), and the 0.2% offset tensile yield strain was 15% lower for the trabecular tissue (0.62+/-0.04% vs. 0.73+/-0.05%, p<0.001). The tensile-compressive yield strength asymmetry for the trabecular tissue, 0.62 on average, was similar to values reported in the literature for cortical bone. We conclude that while the elastic modulus and yield strains for trabecular tissue are just slightly lower than those of cortical tissue, because of the cumulative effect of these differences, tissue strength is about 25% greater for cortical bone.     

Mechanical significance of femoral head trabecular bone structure in Loris and Galago evaluated using micromechanical finite element models

Work on the interspecific and intraspecific variation of trabecular bone in the proximal femur of primates demonstrates important architectural variation between animals with different locomotor behaviors. This variation is thought to be related to the processes of bone adaptation whereby bone structure is optimized to the mechanical environment. Micromechanical finite element models were created for the proximal femur of the leaping Galago senegalensis and the climbing and quadrupedal Loris tardigradus by converting bone voxels from high-resolution X-ray computed tomography scans of the femoral head to eight-noded brick elements. The resulting models had approximately 1.8 million elements each. Loading conditions representing takeoff phase of a leap and more generalized load orientations were applied to the models, and the models were solved using the iterative "row-by-row" matrix-vector multiplication algorithm. The principal strain and Von Mises stress results for the leaping model were similar for both species at each load orientation. Similar hip joint reaction forces in the range of 4.9 x to 12 x body weight were calculated for both species under each loading condition, but the hip reaction values estimated for Loris were higher than predicted based on locomotor behavior. These results suggest that functional adaptation to hip joint loading may not fully explain the differences in femoral head trabecular bone structure in Galago and Loris. The finite element method represents a unique and useful tool for analyzing the functional adaptation of trabecular bone in a diversity of animals and for reconstructing locomotor behavior in extinct taxa.     

Multi-axial mechanical properties of human trabecular bone

In the context of osteoporosis, evaluation of bone fracture risk and improved design of epiphyseal bone implants rely on accurate knowledge of the mechanical properties of trabecular bone. A multi-axial loading chamber was designed, built and applied to explore the compressive multi-axial yield and strength properties of human trabecular bone from different anatomical locations. A thorough experimental protocol was elaborated for extraction of cylindrical bone samples, assessment of their morphology by micro-computed tomography and application of different mechanical tests: torsion, uni-axial traction, uni-axial compression and multi-axial compression. A total of 128 bone samples were processed through the protocol and subjected to one of the mechanical tests up to yield and failure. The elastic data were analyzed using a tensorial fabric–elasticity relationship, while the yield and strength data were analyzed with fabric-based, conewise generalized Hill criteria. For each loading mode and more importantly for the combined results, strong relationships were demonstrated between volume fraction, fabric and the elastic, yield and strength properties of human trabecular bone. Despite the reviewed limitations, the obtained results will help improve the simulation of the damage behavior of human bones and bone-implant systems using the finite element method.     

Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images

Discrepancies in finite-element model predictions of bone strength may be attributed to the simplified modeling of bone as an isotropic structure due to the resolution limitations of clinical-level Computed Tomography (CT) data. The aim of this study is to calculate the preferential orientations of bone (the principal directions) and the extent to which bone is deposited more in one direction compared to another (degree of anisotropy). Using 100 femoral trabecular samples, the principal directions and degree of anisotropy were calculated with a Gradient Structure Tensor (GST) and a Sobel Structure Tensor (SST) using clinical-level CT. The results were compared against those calculated with the gold standard Mean-Intercept-Length (MIL) fabric tensor using micro-CT. There was no significant difference between the GST and SST in the calculation of the main principal direction (median error=28°), and the error was inversely correlated to the degree of transverse isotropy (r=−0.34, p<0.01). The degree of anisotropy measured using the structure tensors was weakly correlated with the MIL-based measurements (r=0.2, p<0.001). Combining the principal directions with the degree of anisotropy resulted in a significant increase in the correlation of the tensor distributions (r=0.79, p<0.001). Both structure tensors were robust against simulated noise, kernel sizes, and bone volume fraction. We recommend the use of the GST because of its computational efficiency and ease of implementation. This methodology has the promise to predict the structural anisotropy of bone in areas with a high degree of anisotropy, and may improve the in vivo characterization of bone.     

Compressive properties of trabecular bone in the distal femur

Early loosening and implant migration are two problems that lead to failures in cementless (press-fit) femoral knee components of total knee replacements. To begin to address these early failures, this study determined the anterior-posterior mechanical properties from four locations in the human distal femur. Thirty-three cylindrical specimens were removed perpendicular to the press-fit surface after the surgical cuts on 10 human cadaveric femurs (age 71.5+/-14.2 years) had been made. Compression testing was performed that utilized methods to reduce the effects of end-artifacts. The bone mineral apparent density (BMAD), apparent modulus of elasticity, yield and ultimate stress, and yield and ultimate strain were measured for 28 cylindrical specimens. The apparent modulus, yield and ultimate stress, and yield and ultimate strain each significantly differed (p<0.05) in the superior and inferior locations. Linear and power law relationships between superior and inferior mechanical properties and BMAD were determined. The inferior apparent modulus and stresses were higher than those in the superior locations. These results show that the press-fit fixation characteristics of the femoral knee component differ on the anterior shield and posterior condyles. This information will be useful in the assignment of mechanical properties in finite element models for further investigations of femoral knee components. The property-density relations also have applications for implant design and preoperative assessment of bone strength using clinically available tools.     

Mechanical properties, density and quantitative CT scan data of trabecular bone with and without metastases

Pathologic fracture of the hip due to metastatic lesions in bone is a serious problem. This study examined the effect of metastatic lesions on the material properties and quantitative computed tomography (QCT) data of trabecular bone. Twelve distal femora were obtained, four with lytic and/or blastic metastatic lesions (group L), four without lesions but from donors who died from breast, prostate, or lung cancer (group NL), and four from donors with no cancer (group NC). Each specimen was CT scanned, and 56, 15x15x15-mm cubes of trabecular bone were cut. QCT density (rho(QCT)), compressive elastic modulus (E), compressive yield and ultimate strengths (S(y) and S(u)), and ash density (rho(ash)) of each cube were determined. Regression analysis was performed between rho(ash) and E, S(y), S(u) and rho(QCT), and analysis of covariance was used to identify differences between groups. Power relationships that did not depend on group (p >/= 0.1) were found between E and rho(ash) (0.74 /= 0.94; p<0.001). rho(ash) was strongly related to rho(QCT) (r >/= 0.99; p<0.001). These results indicate that metastatic disease does not significantly impair the ability of QCT to provide an accurate and precise estimate of rho(ash) that can be used to estimate mechanical properties of trabecular bone with and without metastases.     

The three-dimensional structure of trabecular bone in the femoral head of strepsirrhine primates

It has been hypothesized for over a hundred years that trabecular bone plays an important structural role in the musculoskeletal system of animals and that it responds dynamically to applied loads through growth. The objectives of this study are to quantify the three-dimensional structure of femoral head trabecular bone in a sample of extant strepsirrhines and to relate patterns of interspecific variation to locomotor behavioral differences. The bone volume fraction (BV/TV) and fabric anisotropy of trabecular bone in the femoral heads of Cheirogaleus major, Avahi laniger, Galago senegalensis, Galago alleni, Loris tardigradus, Otolemur crassicaudatus, and Perodicticus potto were quantified in three dimensions using serial high-resolution X-ray computed tomography scan data. A volume based method was used to quantify the structural anisotropy in three cubic samples located inside the central portion of the femoral head. Significant structural differences were found between the predominantly leaping galagines and indriids and the nonleaping lorisines and cheirogaleids. The leapers in general have relatively anisotropic trabecular bone. The galagines display a unique pattern of decreasing bone volume and increasing anisotropy moving from the superior to the inferior half of the femoral head. By contrast, the nonleaping taxa possess relatively uniform and isotropic bone throughout the femoral head. The differences in femoral head trabecular structure among these taxa seem to be related to locomotor behavioral differences, reflecting variation in the use and loading of the hip joint during normal locomotion.     

A micro-computed tomography study of the trabecular bone structure in the femoral head

The goal of this study was to characterize the trabecular microarchitecture of the femoral head using micro-computed tomography (ICT). Femoral head specimens were obtained from subjects following total hip replacement. Cylindrical cores from the specimens were scanned to obtain 3-D images with an isotropic resolution of 26 Im. Bone structural parameters were evaluated on a per millimeter basis: relative bone volume (BV/TV), trabecular number (Tb.N), thickness (Tb.Th) and separation (Tb.Sp), structure model index (SMI), and connectivity (Conn.D). The ICT data show that the first two millimeters, starting at the joint surface, are characterized by more plate-like trabeculae, and are significantly denser than the underlying trabecular bone. Regional differences in the trabecular architecture reveal that the superior pole has significantly higher BV/TV, Tb.N and Tb.Th values, with lower Tb.Sp compared to the inferior and side poles. Because subchondral bone is essential in the load attenuation of joints, the difference in bone structure between the subchondral and trabecular bone might arise from the different functions each have within joint-forming bones. The denser trabecular structure of the superior pole as compared to the inferior pole can be interpreted as a functional adaptation to higher loading in this area.     

Mechanical strength of tibial trabecular bone evaluated by X-ray computed tomography

The prospects for the use of quantitative computed tomography (QCT) for evaluation of mechanical properties of tibial trabecular bone were investigated. Computed tomography (CT) data from the proximal tibial epi- and metaphysis of six human cadaver knees were correlated with mechanical data obtained from compression tests and penetration strength measurements. In addition CT and intraoperative penetration data were compared in 20 patients. If spatial agreement between CT and mechanical measurement sites is optimized, close correlations are found between the relative linear attenuation coefficient determined by CT and the ultimate strength (r = 0.84), the yield strength (r = 0.85), the elastic modulus (r = 0.78), the ultimate energy absorption (r = 0.83), the yield energy absorption (r = 0.81), and the penetration strength (r = 0.82). It is concluded that these correlations are sufficient to make QCT a valuable tool for non-invasive evaluation of the spatial distribution of bone properties in several clinical applications.     

Predicting changes in mechanical properties of trabecular bone by adaptive remodeling

Because changes in the mechanical properties of bone are closely related to trabecular bone remodeling, methods that consider the temporal morphological changes induced by adaptive remodeling of trabecular bone are needed to estimate long-term fracture risk and bone quality in osteoporosis. We simulated bone remodeling using simplified and pig trabecular bone models and estimated the morphology of healthy and osteoporotic cases. We then displayed the fracture risk of the remodeled models based on a cumulative histogram from high stress. The histogram showed more elements had higher stresses in the osteoporosis model, indicating that the osteoporosis model had a greater risk.     

Physical and mechanical properties of calf lumbosacral trabecular bone.

The physical and mechanical properties of calf lumbar and sacral trabecular bone were determined and compared with those of human trabecular bone. The mean tissue density (1.66 +/- 0.12 g cm-3), equivalent mineral density (169 +/- 36 mg cm-3), apparent density (453 +/- 89 mg cm-3), ash density (194 +/- 59 mg cm-3), ash content (0.6 +/- 0.05%), compressive strength (7.1 +/- 3.0 MPa) and compressive modulus (173 +/- 97 MPa) of calf trabecular bone are similar to those of young human. There were moderate, positive linear correlations between apparent density and equivalent mineral density, ash density, and compressive strength; and between compressive strength and equivalent mineral density (R2 ranging from 0.35 to 0.48, p less than 0.001). Apparent density, ash density, and equivalent mineral density did not differ significantly in different regions. In contrast to humans, the compressive strength increased from posterior, near the facet, to the anterior vertebral body. These comparisons of physical and mechanical properties, as well as anatomical comparisons by others, indicate that the calf spine is a good model of the young non-osteoporotic human spine and thus useful for the testing of spinal instrumentation.