Errors induced by off-axis measurement of the elastic properties of bone

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.     

Trabecular shear stress in human vertebral cancellous bone: intra- and inter-individual variations

Correlation of the mean and standard deviation of trabecular stresses has been proposed as a mechanism by which a strong relationship between the apparent strength and stiffness of cancellous bone can be achieved. The current study examined whether the relationship between the mean and standard deviation of trabecular von Mises stresses can be generalized for any group of cancellous bone. Cylindrical human vertebral cancellous bone specimens were cut in the infero-superior direction from T12 of 23 individuals (inter-individual group). Thirty nine additional specimens were prepared similarly from the T4-T12 and L2-L5 vertebrae of a 63 year old male (intra-individual group). The specimens were scanned by micro-computed tomography (microCT) and trabecular von Mises stresses were calculated using finite element modeling. The expected value, standard deviation and coefficient of variation of the von Mises stress were calculated form a three-parameter Weibull function fitted to von Mises stress data from each specimen. It was found that the average and standard deviation of trabecular von Mises shear stress were: (i) correlated with each other, supporting the idea that high correlation between the apparent strength and stiffness of cancellous bone can be achieved through controlling the trabecular level shear stress variations, (ii) dependent on anatomical site and sample group, suggesting that the variation of stresses are correlated to the mean stress to different degrees between vertebrae and individuals, and (iii) dependent on bone volume fraction, consistent with the idea that shear stress is less well controlled in bones with low BV/TV. The conversion of infero-superior loading into trabecular von Mises stresses was maximum for the tissue at the junction of the thoracic and lumbar spine (T12-L1) consistent with this junction being a common site of vertebral fracture.     

Permeability study of vertebral cancellous bone using micro-computational fluid dynamics

Understanding of cancellous bone permeability is lacking despite its importance in designing tissue engineering scaffolds for bone regeneration and orthopaedic surgery that relies on infiltration of bone cement into porous cancellous bone. We employed micro-computational fluid dynamics to investigate permeability for 37 cancellous bone specimens, eliminating stringent technical requirements of bench-top testing. Microarchitectural parameters were also determined for the specimens and correlated, using uni-variate and multi-variate regression analyses, against permeability. We determined that bone surface density, trabecular pattern factor, structure model index and trabecular number are other possible predictors of permeability (with R values of 0.47, 0.44, 0.40 and 0.33), in addition to the commonly used porosity parameter (R value of 0.38). Pooling these parameters and performing multi-variate linear regression analysis improved yield the R-value of 0.50, indicating that porosity alone is a poor predictor of cancellous bone permeability and, therefore, other parameters should be included for a better and improved linear model.     

Detection of trabecular bone microdamage by micro-computed tomography

Microdamage is an important component of bone quality and affects bone remodeling. Improved techniques to assess microdamage without the need for histological sectioning would provide insight into the role of microdamage in trabecular bone strength by allowing the spatial distribution of damage within the trabecular microstructure to be measured. Nineteen cylindrical trabecular bone specimens were prepared and assigned to two groups. The specimens in group I were damaged to 3% compressive strain and labeled with BaSO(4). Group II was not loaded, but was labeled with BaSO(4). Micro-computed tomography (Micro-CT) images of the specimens were obtained at 10 microm resolution. The median intensity of the treated bone tissue was compared between groups. Thresholding was also used to measure the damaged area fraction in the micro-CT scans. The histologically measured damaged area fraction, the median CT intensity, and the micro-CT measured damaged area fraction were all higher in the loaded group than in the unloaded group, indicating that the micro-CT images could differentiate the damaged specimen group from the unloaded specimens. The histologically measured damaged area fraction was positively correlated with the micro-CT measured damaged area fraction and with the median CT intensity of the bone, indicating that the micro-CT images can detect microdamage in trabecular bone with sufficient accuracy to differentiate damage levels between samples. This technique provides a means to non-invasively assess the three-dimensional distribution of microdamage within trabecular bone test specimens and could be used to gain insight into the role of trabecular architecture in microdamage formation.     

Yield behavior of bovine cancellous bone

The compressive yield strain was measured for 61 specimens of bovine cancellous bone from three distal femora. There was no significant relationship (p = 0.08, R2 = 0.051) between yield strain and the degree of trabecular orientation. There was a significant positive correlation (p less than 0.00001, R2 = 0.319) between yield strain and structural (apparent) density and significant negative correlation (p less than 0.0025, R2 = 0.145) between yield strain and bone density. Yield strain correlated best with bone solid volume fraction Vv (epsilon y = 0.592 +2- 1.446vv, R2 = 0.337). The quantity, yield strain, is highly dependent on specific definitions of the yield point and the point of zero strain. For this study the yield point was defined by a 0.0003 offset criterion, and the point of zero strain was defined as the point where the tangent at 15 percent of yield crosses zero. The results using these definitions were compared with results using yield strain values determined by other definitions of the yield point and zero strain. The correlations between yield strain and trabecular orientation, structural density and bone density changed very little for differing definitions of yield. The results suggest that yield strain in cancellous bone is isotropic or independent of textural anisotropy, so the yield behaviour may be characterized by a maximum strain yield criterion. The results also suggest that the primary mode of yield in cancellous bone is buckling of the trabeculae.     

The mechanical characteristics of cancellous bone at the upper femoral region

Mechanical behaviour of trabecular bone at the upper femoral region of human bones has been studied by compression tests on trabecular bone specimens removed from normal femora obtained at autopsy. Compression tests were performed along three different axes of loading on wet specimens and high loading rates. Femoral head specimens proved to be the strongest for any axis of loading.

Large variation in compressive strength and modulus of elasticity is seen within and between femoral bone samples. Anisotropy and differences in anisotropy for the different regions have been observed. A significant correlation between mechanical properties (σ max − E) and bone mineral content of the specimen was found.

Tests on whole bone structures demonstrate that removal of the central part of the trabecular bone at the proximal femur reduces the strength for impact loading considerably (± 50%).     

Permeability study of vertebral cancellous bone using micro-computational fluid dynamics

Understanding of cancellous bone permeability is lacking despite its importance in designing tissue engineering scaffolds for bone regeneration and orthopaedic surgery that relies on infiltration of bone cement into porous cancellous bone. We employed micro-computational fluid dynamics to investigate permeability for 37 cancellous bone specimens, eliminating stringent technical requirements of bench-top testing. Microarchitectural parameters were also determined for the specimens and correlated, using uni-variate and multi-variate regression analyses, against permeability. We determined that bone surface density, trabecular pattern factor, structure model index and trabecular number are other possible predictors of permeability (with R values of 0.47, 0.44, 0.40 and 0.33), in addition to the commonly used porosity parameter (R value of 0.38). Pooling these parameters and performing multi-variate linear regression analysis improved yield the R-value of 0.50, indicating that porosity alone is a poor predictor of cancellous bone permeability and, therefore, other parameters should be included for a better and improved linear model.     

Tensile strength of bovine trabecular bone

Data on the tensile and compressive properties of trabecular bone are needed to define input parameters and failure criteria for modeling total joint replacements. To help resolve differences in reports comparing tensile and compressive properties of trabecular bone, we have developed new methods, based on porous foam technology, for tensile testing of fresh/frozen trabecular bone specimens. Using bovine trabecular bone from an isotropic region from the proximal humerus as a model material, we measured ultimate strengths in tension and compression for two groups of 24 specimens each. The average ultimate strength in tension was 7.6 +/- 2.2 (95% C.I.) MPa and in compression was 12.4 +/- 3.2 MPa. This difference was statistically significant (p = 0.013) and was not related to density differences between the test groups (p = 0.28). Strength was related by a power-law function of the local apparent density, but, even accounting for density influences, isotropic bovine trabecular bone exhibits significantly lower strengths in tension than in compression.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Dependence of mechanical compressive strength on local variations in microarchitecture in cancellous bone of proximal human femur

Human cancellous bone is a heterogeneous material. Despite this, most of the published studies report correlations between mechanical properties and morphometric parameters averaged on the whole specimen. This work investigated whether local variations in morphometric parameters were linked to the localized failure regions of cancellous bone. Additionally, it was examined whether local values of morphometric parameters can predict the ultimate stress better than the average bone volume fraction (BV/TV). Cylindrical cancellous bone specimens extracted along the primary compressive group of human femoral heads were studied. These were microCT-imaged to assess the morphometric parameters, compressed to determine the ultimate stress, and rescanned by microCT to visualize the failure region. Failure involved slightly less than half of the free height of the specimens. Significant differences were found in the morphometric parameters calculated in the failure and in the non-failure regions. The cross-sections containing minimum BV/TV values were those most often located inside the failure region (83%, p<0.001). Regression analysis confirmed that variations in BV/TV best describe variations in ultimate stress (R2=0.84) out of the averaged morphometric parameters. The prediction of ultimate stress increased when minimum or maximum values of the morphometric parameters were taken, with the highest prediction found by considering the minimum BV/TV (R2=0.95). In conclusion, due to the heterogeneity of cancellous bone, there may exist regions characterized by a different microarchitecture, where the bone is weaker and consequently is more likely to fail. These regions mostly contain minimum values in BV/TV, which were found to predict ultimate stress better than average BV/TV.     

The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates

The compressive properties of human cancellous bone of the distal intracondylar femur in its wet condition were determined. Specimens were obtained from six cadaveric femora and were tested at a strain rate of 0.002, 0.10 and 9.16 sec⁻¹. It was found that the compressive strength decreases with an increasing vertical distance from the joint. The highest compressive strength level was recorded in the posterior medial condyle. Correlations among the mechanical properties, the bulk specimen density and the bone mineral content yield (i) highly significant correlations between the compressive strength and the elastic modulus (ii) highly significant correlations between the compressive strength or the modulus of elasticity and the bulk specimen density (iii) a doubtful correlation between the compressive strength and the bone mineral content. All recorded graphs of the impact loaded specimens displayed several well defined stress peaks, unlike the graphs recorded at low loading rates. It can be concluded that upon impact loading the localized trabecular failure which is associated with each peak, does not affect the spongy bone's stress capacity in a detrimental way.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Fatigue of bovine trabecular bone

Fatigue loading of bone, from the activities of daily living in the elderly, or from prolonged exercise in the young, can lead to increased risk of fracture. Elderly patients with osteoporosis are particularly prone to fragility fractures of the vertebrae, where load is carried primarily by trabecular bone. In this study, specimens of bovine trabecular bone were loaded in compressive fatigue at four different normalized stresses to one of six maximum strains. The resulting change in modulus and residual strain accumulation were measured over the life of the fatigue test. The number of cycles to reach a given maximum compressive strain increased with decreasing normalized stress. Modulus reduction and specimen residual strain increased with increasing maximum compressive strain, but few differences were observed between specimens loaded to the same maximum strain at different normalized stresses.     

Fatigue microdamage in bovine trabecular bone

Microdamage, in the form of small cracks, may accumulate in trabecular bone loaded in fatigue. Specimens of bovine trabecular bone were loaded in compressive fatigue at one of four normalized stresses and loading was stopped after the specimens reached one of six maximum strains. Microdamage was identified using a fluorochrome staining technique, and microdamage parameters, including the number of damaged trabeculae and the damaged area fraction, were measured. No microdamage was observed during loading to strains below the yield strain; at higher strains, all microdamage parameters increased with increasing maximum compressive strain. Few significant differences were observed in the type or amount of microdamage accumulation between specimens loaded to the same maximum strain at different normalized stresses; however, more trabecular fractures were observed at high numbers of cycles, which corresponded to low normalized stresses.     

The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens.

The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens was studied by non-destructive uniaxial compression to 0.4% strain using cylindrical specimens with different sizes and length-to-diameter ratios, and by comparing cubic and cylindrical specimens with the same cross-sectional area. Both the length and the cross-sectional area of the specimen had a highly significant influence on the mechanical behaviour (p less than 0.0001). Within the actual range of length (2.75-11.0 mm) the normalized stiffness (Young's modulus) was related nearly linearly to the specimen length. This dependency on specimen length is suggested to be caused mainly by structural disintegrity of the trabecular specimens near the surface. The normalized stiffness (Young's modulus) was also positively correlated to the cross-sectional area. This dependency on cross-sectional area is probably due to friction-induced stress inhomogeneity at the platen-specimen interface. A cube with side length 6.5 mm or a cylindrical specimen with 7.5 mm diameter and 6.5 mm length are suggested as standard specimens for comparative studies on trabecular bone mechanics.     

Heterogeneous glycation of cancellous bone and its association with bone quality and fragility

Non-enzymatic glycation (NEG) and enzymatic biochemical processes create crosslinks that modify the extracellular matrix (ECM) and affect the turnover of bone tissue. Because NEG affects turnover and turnover at the local level affects microarchitecture and formation and removal of microdamage, we hypothesized that NEG in cancellous bone is heterogeneous and accounts partly for the contribution of microarchitecture and microdamage on bone fragility. Human trabecular bone cores from 23 donors were subjected to compression tests. Mechanically tested cores as well as an additional 19 cores were stained with lead-uranyl acetate and imaged to determine microarchitecture and measure microdamage. Post-yield mechanical properties were measured and damaged trabeculae were extracted from a subset of specimens and characterized for the morphology of induced microdamage. Tested specimens and extracted trabeculae were quantified for enzymatic and non-enzymatic crosslink content using a colorimetric assay and Ultra-high Performance Liquid Chromatography (UPLC). Results show that an increase in enzymatic crosslinks was beneficial for bone where they were associated with increased toughness and decreased microdamage. Conversely, bone with increased NEG required less strain to reach failure and were less tough. NEG heterogeneously modified trabecular microarchitecture where high amounts of NEG crosslinks were found in trabecular rods and with the mechanically deleterious form of microdamage (linear microcracks). The extent of NEG in tibial cancellous bone was the dominant predictor of bone fragility and was associated with changes in microarchitecture and microdamage.     

Preparation of on-axis cylindrical trabecular bone specimens using micro-CT imaging

The Orientation of trabecular bone specimens for mechanical testing must be carefully controlled. A method for accurately preparing on-axis cylindrical specimens using high-resolution micro-CT imaging was developed. Sixteen cylindrical specimens were prepared from eight bovine tibiae. High-resolution finite element models were generated from micro-CT images of parallelepipeds and used to determine the principal material coordinate system of each parallelepiped. A cylindrical specimen was then machined with a diamond coring bit. The resulting specimens were scanned again to evaluate the orientation. The average deviation between the principal fabric orientation and the longitudinal axis of the cylindrical specimen was only 4.70 +/- 3.11 degrees.     

Microtensile measurements of single trabeculae stiffness in human femur

In this paper, the authors perform microtensile tests of single trabeculae excised from a human femur head. One of the main issues of this work is to establish some experimental procedures for preparing and testing the specimens. The use of a well-characterized microtensile apparatus allows for a low intraspecimen dispersion of the measured stiffness. Tensile/compressive tests were chosen because they appear less sensitive to errors in the cross-sectional area measurements with respect to bending tests. By these considerations, some tensile/compressive tests of plate-like trabecular specimens have been carried out. Typical stiffness values are 74.2+/-0.7Nmm(-1) for tensile tests, and 58.9+/-0.6Nmm(-1) for compressive test. Another compressive test performed on a shorter specimen yielded a stiffness value of 148.3+/-5.3Nmm(-1). The maximum applied load was about 0.5N. Rough measurements of specimens sizes yielded a Young's modulus value ranging from 1.41 to 1.89GPa.     

Specimen size effect in the volumetric shrinkage of cancellous bone measured at two levels of dehydration

Water is commonly removed from bone to study its effect on mechanical behaviour; however, dehydration also alters the bone structure. To make matters worse, measuring structural changes in cancellous bone is complicated by a number of factors. Therefore, the goals of this study were to address these issues by (1) comparing Archimedes' method and a helium pycnometer as methods for measuring cancellous bone volume; (2) measuring the apparent dimensional and volumetric tissue shrinkage of cancellous bone at two levels of dehydration; and, (3) identifying whether a size effect exists in cancellous bone shrinkage. Cylindrical specimens (3, 5 and 8.3 mm diameters) of cancellous bone were taken from the distal bovine femur. The apparent dimensions of each cylindrical specimen were measured in a fully hydrated state (HYD), after drying at room temperature (AIR), and after oven drying at 105 degrees C (OVEN). Tissue volume measurements for those three hydration states were obtained using both a helium pycnometer and Archimedes' method. Aluminium foams, which mimic the cancellous structure, were used as controls. The results suggest that the helium pycnometer and Archimedes' method yield identical results in the HYD and AIR states, but that Archimedes' method under-predicts the nominal OVEN volume by incorporating the collagen-apatite porosity. A distinct size effect on volumetric shrinkage is observed (p<0.025) using the pycnometer in both AIR and OVEN states. Apparent dimensional shrinkage (2% and 7%) at the two dehydration levels is much smaller than the measured volumetric tissue shrinkage (16% and 29%), which results in a reduced dehydrated bone volume fraction.