A cellular solid criterion for predicting the axial-shear failure properties of bovine trabecular bone.

In a long-term effort to develop a complete multi-axial failure criterion for human trabecular bone, the overall goal of this study was to compare the ability of a simple cellular solid mechanistic criterion versus the Tsai-Wu, Principal Strain, and von Mises phenomenological criteria--all normalized to minimize effects of interspecimen heterogeneity of strength--to predict the on-axis axial-shear failure properties of bovine trabecular bone. The Cellular Solid criterion that was developed here assumed that vertical trabeculae failed due to a linear superposition of axial compression/tension and bending stresses, induced by the apparent level axial and shear loading, respectively. Twenty-seven bovine tibial trabecular bone specimens were destructively tested on-axis without end artifacts, loaded either in combined tension-torsion (n = 10), compression-torsion (n = 11), or uniaxially (n = 6). For compression-shear, the mean (+/- S.D.) percentage errors between measured values and criterion predictions were 7.7 +/- 12.6 percent, 19.7 +/- 23.2 percent, 22.8 +/- 18.9 percent, and 82.4 +/- 64.5 percent for the Cellular Solid, Tsai-Wu, Principal Strain, and von Mises criteria, respectively; corresponding mean errors for tension-shear were -5.2 +/- 11.8 percent, 14.3 +/- 12.5 percent, 6.9 +/- 7.6 percent, and 57.7 +/- 46.3 percent. Statistical analysis indicated that the Cellular Solid criterion was the best performer for compression-shear, and performed as well as the Principal Strain criterion for tension-shear. These data should substantially improve the ability to predict axial-shear failure of dense trabecular bone. More importantly, the results firmly establish the importance of cellular solid analysis for understanding and predicting the multiaxial failure behavior of trabecular bone.     

Biaxial failure behavior of bovine tibial trabecular bone

Multiaxial failure properties of trabecular bone are important for modeling of whole bone fracture and can provide insight into structure-function relationships. There is currently no consensus on the most appropriate form of multiaxial yield criterion for trabecular bone. Using experimentally validated, high-resolution, non-linear finite element models, biaxial plain strain boundary conditions were applied to seven bovine tibial specimens. The dependence of multiaxial yield properties on volume fraction was investigated to quantify the interspecimen heterogeneity in yield stresses and strains. Two specimens were further analyzed to determine the yield properties for a wide range of biaxial strain loading conditions. The locations and quantities of tissue level yielding were compared for on-axis, transverse, and biaxial apparent level yielding to elucidate the micromechanical failure mechanisms. As reported for uniaxial loading of trabecular bone, the yield strains in multiaxial loading did not depend on volume fraction, whereas the yield stresses did. Micromechanical analysis indicated that the failure mechanisms in the on-axis and transverse loading directions were mostly independent. Consistent with this, the biaxial yield properties were best described by independent curves for on-axis and transverse loading. These findings establish that the multiaxial failure of trabecular bone is predominantly governed by the strain along the loading direction, requiring separate analytical expressions for each orthotropic axis to capture the apparent level yield behavior.     

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.     

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.     

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.     

Creep contributes to the fatigue behavior of bovine trabecular bone.

Repetitive, low-intensity loading from normal daily activities can generate fatigue damage in trabecular bone, a potential cause of spontaneous fractures of the hip and spine. Finite element models of trabecular bone (Guo et al., 1994) suggest that both creep and slow crack growth contribute to fatigue failure. In an effort to characterize these damage mechanisms experimentally, we conducted fatigue and creep tests on 85 waisted specimens of trabecular bone obtained from 76 bovine proximal tibiae. All applied stresses were normalized by the previously measured specimen modulus. Fatigue tests were conducted at room temperature; creep tests were conducted at 4, 15, 25, 37, 45, and 53 degrees C in a custom-designed apparatus. The fatigue behavior was characterized by decreasing modulus and increasing hysteresis prior to failure. The hysteresis loops progressively displaced along the strain axis, indicating that creep was also involved in the fatigue process. The creep behavior was characterized by the three classical stages of decreasing, constant, and increasing creep rates. Strong and highly significant power-law relationships were found between cycles-to-failure, time-to-failure, steady-state creep rate, and the applied loads. Creep analyses of the fatigue hysteresis loops also generated strong and highly significant power law relationships for time-to-failure and steady-state creep rate. Lastly, the products of creep rate and time-to-failure were constant for both the fatigue and creep tests and were equal to the measured failure strains, suggesting that creep plays a fundamental role in the fatigue behavior of trabecular bone. Additional analysis of the fatigue strain data suggests that creep and slow crack growth are not separate processes that dominate at high and low loads, respectively, but are present throughout all stages of fatigue.     

Nonlinear viscoelastic characterization of bovine trabecular bone

The time-independent elastic properties of trabecular bone have been extensively investigated, and several stiffness–density relations have been proposed. Although it is recognized that trabecular bone exhibits time-dependent mechanical behaviour, a property of viscoelastic materials, the characterization of this behaviour has received limited attention. The objective of the present study was to investigate the time-dependent behaviour of bovine trabecular bone through a series of compressive creep–recovery experiments and to identify its nonlinear constitutive viscoelastic material parameters. Uniaxial compressive creep and recovery experiments at multiple loads were performed on cylindrical bovine trabecular bone samples (\(n = 19\)). Creep response was found to be significant and always comprised of recoverable and irrecoverable strains, even at low stress/strain levels. This response was also found to vary nonlinearly with applied stress. A systematic methodology was developed to separate recoverable (nonlinear viscoelastic) and irrecoverable (permanent) strains from the total experimental strain response. We found that Schapery’s nonlinear viscoelastic constitutive model describes the viscoelastic response of the trabecular bone, and parameters associated with this model were estimated from the multiple load creep–recovery (MLCR) experiments. Nonlinear viscoelastic recovery compliance was found to have a decreasing and then increasing trend with increasing stress level, indicating possible stiffening and softening behaviour of trabecular bone due to creep. The obtained parameters from MLCR tests, expressed as second-order polynomial functions of stress, showed a similar trend for all the samples, and also demonstrate stiffening–softening behaviour with increasing stress.     

Creep does not contribute to fatigue in bovine trabecular bone

In both cortical and trabecular bone loaded in fatigue, the stress-strain loops translate along the strain axis. Previous studies have suggested that this translation is the result of creep associated with the mean stress applied in the fatigue test. In this study, we measured the residual strrain (corresponding to the translation of the stress-strain loops) in fatigue tests on bovine trabecular bone and compared it to an upper bound estimate of the creep strain in each test. Our results indicate that the contribution of creep to the translation of the stress-strain loops is negligible in bovine trabecular bone. These results, combined with models for fatigue in lower density bone, suggest that that creep does not contribute to the fatigue of normal human bone. Creep may make a significant contribution to fatigue in low-density osteoporotic bone in which trabeculae have resorbed, reducing the connectivity of the trabecular structure.     

The modified super-ellipsoid yield criterion for human trabecular bone

Despite the importance of multiaxial failure of trabecular bone in many biomechanical applications, to date no complete multiaxial failure criterion for human trabecular bone has been developed. By using experimentally validated nonlinear high-resolution, micromechanical finite-element models as a surrogate for multiaxial loading experiments, we determined the three-dimensional normal strain yield surface and all combinations of the two-dimensional normal-shear strain yield envelope. High-resolution finite-element models of three human femoral neck trabecular bone specimens obtained through microcomputed tomography were used. In total, 889 multiaxial-loading cases were analyzed, requiring over 41,000 CPU hours on parallel supercomputers. Our results indicated that the multiaxial yield behavior of trabecular bone in strain space was homogeneous across the specimens and nearly isotropic. Analysis of stress-strain curves along each axis in the 3-D normal strain space indicated uncoupled yield behavior whereas substantial coupling was seen for normal-shear loading. A modified super-ellipsoid surface with only four parameters fit the normal strain yield data very well with an arithmetic error +/-SD less than -0.04 +/- 5.1%. Furthermore, the principal strains associated with normal-shear loading showed excellent agreement with the yield surface obtained for normal strain loading (arithmetic error +/- SD < 2.5 +/- 6.5%). We conclude that the four-parameter "Modified Super-Ellipsoid" yield surface presented here describes the multiaxial failure behavior of human femoral neck trabecular bone very well.     

Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone

Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples (\(\textit{n}\,{=}\,13\)) at loads corresponding to physiological strain level of 2000 \({\upmu }{\upvarepsilon }\). We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships (\(r^2\,{=}\,0.73,\ p\,{<}\,0.001\)) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV.     

Microdamage accumulation in bovine trabecular bone in uniaxial compression

In this study we investigated how microdamage accumulated with increasing compressive strain in bovine trabecular bone. We found that little damage is created in the linear elastic region, up to -0.4 percent strain. At an average strain of -0.76 percent +/-0.25 percent, the stress-strain curve became nonlinear, and peaked at -1.91 percent +/-0.55 percent strain. Microdamage increases rapidly during the peak of the stress-strain curve, and a localized band of damage formed. At strains beyond the ultimate strain, the damaged band widened and the density of damage within the band increased. Microdamage occurred as groupings of cracks; the majority of damage occurred as regions of cross-hatching. All microdamage parameters increased with increasing maximum compressive strain. We also observed exponential relationships between crack numerical density and damage (1(o) - (o)Esec/E0) and between crack length density and damage.     

Application of homogenization theory to the study of trabecular bone mechanics.

It is generally accepted that the strength and stiffness of trabecular bone is strongly affected by trabecular microstructure. It has also been hypothesized that stress induced adaptation of trabecular bone is affected by trabecular tissue level stress and/or strain. At this time, however, there is no generally accepted (or easily accomplished) technique for predicting the effect of microstructure on trabecular bone apparent stiffness and strength or estimating tissue level stress or strain. In this paper, a recently developed mechanics theory specifically designed to analyze microstructured materials, called the homogenization theory, is presented and applied to analyze trabecular bone mechanics. Using the homogenization theory it is possible to perform microstructural and continuum analyses separately and then combine them in a systematic manner. Stiffness predictions from two different microstructural models of trabecular bone show reasonable agreement with experimental results, depending on metaphyseal region, (R2 greater than 0.5 for proximal humerus specimens, R2 less than 0.5 for distal femur and proximal tibia specimens). Estimates of both microstructural strain energy density (SED) and apparent SED show that there are large differences (up to 30 times) between apparent SED (as calculated by standard continuum finite element analyses) and the maximum microstructural or tissue SED. Furthermore, a strut and spherical void microstructure gave very different estimates of maximum tissue SED for the same bone volume fraction (BV/TV). The estimates from the spherical void microstructure are between 2 and 20 times greater than the strut microstructure at 10-20% BV/TV.     

Axial-shear interaction effects on microdamage in bovine tibial trabecular bone

Because many osteoporotic fractures occur during a fall, understanding the effect of off-axis loads on initiation and propagation of microdamage in trabecular bone should provide further insight into the biomechanics of age-related fractures. Fourteen on-axis cylindrical specimens were prepared from 12 bovine tibiae. Fluorescent stains were used to label the microdamage due to a sequence of compressive and torsional damaging loads. The mean decrease in Young's modulus was over four times greater than that in the shear modulus after the compressive overload, while there was no difference between the decrease in the axial and torsional stiffnesses after the torsional overload. The total microcrack density due to compression was uniform across the radius of the cylindrical specimens, while the mean density of microcracks due to torsional overloading increased from the axis of the cylindrical specimen to the circumference. The high density of microcracks near the axis of the specimen following torsional overloading was unexpected because of the low strains. Nearly 40% of the microcracks due to torsion propagated from pre-existing microcracks caused by axial compression, indicating that existing microcracks may extend at relatively low strain if the loading mode changes. The propagating microcracks were, on average, longer than the initiating microcracks due to either compressive or torsional loading. Damage due to axial compression appears to increase the susceptibility of trabecular bone to damage propagation during subsequent torsional loads, but it has little effect on the elastic properties in shear.     

Effects of damage on the orthotropic material symmetry of bovine tibial trabecular bone

The macroscopic mechanical properties of trabecular bone can be predicted by its architecture using theoretical relationships between the elastic and architectural properties. Microdamage caused by overloading or fatigue decreases the apparent elastic moduli of trabecular bone requiring these relationships to be modified to predict the damaged elastic properties. In the case of isotropic damage, the apparent level elastic properties could be determined by multiplying all of the elastic constants by a single scalar factor. If the damage is anisotropic, the elastic constants may change by differing factors and the material coordinate system could become misaligned with the fabric coordinate system. High-resolution finite element models were used to simulate damage overloading on seven trabecular bone specimens subjected to pure shear strain in two planes. Comparison of the apparent elastic moduli of the specimens before and after damage showed that the reduction of the elastic moduli was anisotropic. This suggests that the microdamage within the specimens was inhomogeneous. However, after damage the specimens exhibited nearly orthotropic material symmetry as they did before damage. Changes in the orientation of the orthotropic material coordinate system were also small and occurred primarily in the transverse plane. Thus, while damage in trabecular bone is anisotropic, the material coordinate system remains aligned with the fabric tensor.     

A phenomenological model for predicting fatigue life in bovine trabecular bone

Cyclic loading of bone during daily activities can lead to fatigue degradation and increased risk of fracture in both the young and elderly population. Damage processes under cyclic loading in trabecular bone result in the reduction of the elastic modulus and accumulation of residual strain. These effects increase with increasing stress levels, leading to a progressive reduction in fatigue life. The present work analyzes the effect of stress and strain variation on the above damage processes in bovine trabecular bone, and develops a phenomenological model relating fatigue life to the imposed stress level. The elastic modulus reduction of the bone specimens was observed to depend on the maximum compressive strain, while the rate of residual strain accumulation was a function of the stress level. A model was developed for the upper and lower bounds of bone elastic modulus reduction with increasing number of cycles, at each stress range. The experimental observations were described well by the model. The model predicted the bounds of the fatigue life with change in fatigue stress. The decrease in the fatigue life with increasing stress was related to corresponding increases in the residual strain accumulation rates at the elevated stress levels. The model shows the validity of fatigue predictions from relatively few cyclic experiments, by combining trends observed in the monotonic and the cyclic tests. The model also presents a relatively simple procedure for predicting the endurance limit for bovine trabecular bone specimens.     

Modeling modulus reduction in bovine trabecular bone damaged in compression.

Loading bone beyond its yield point creates microdamage, leading to reduction in stiffness. Previously, we related microdamage accumulation to changes in mechanical properties. Here, we develop a model that predicts stiffness loss based on the presence of microdamage. Modeling is done at three levels: (1) a single trabecula, (2) a cellular solid consisting of intact, damaged, and fractured trabeculae, and (3) a specimen with a localized damage band. Predictions of a reduced modulus agree well with experimental measured modulus reductions of post-yield compression of bovine trabecular bone. The predicted reduced modulus is relatively insensitive to changes in the input parameters.     

Effect of trabecular curvature on the stiffness of trabecular bone

Simplified structural models of trabecular bone have been used to model various forms of trabecular variability. The structural effects of variability of direction, length and thickness of the trabeculae have been studied using 'lattice-type' finite element models. However, many of the trabeculae are not perfectly straight, and have a small degree of curvature. The objective of this study is to quantify the influence of small curvatures of the trabeculae on the effective modulus of trabecular bone, in the principal material direction. An analytical analysis of the effect of curvature on a single trabecula is performed, utilizing the concept of cellular-solid models. Closed-form expressions are derived for the effect of curvature on the flexibility in the principal material direction. For comparison, expressions are derived for the flexibility of a straight oblique element, representing angular variability. A quantitative comparison is presented, which is dependent on the thickness of the trabeculae. It was found that small curvatures have a large effect on the stiffness of the trabecular structure. This effect is largest for thin trabeculae, and decreases for thick trabeculae. The stiffness of the trabecular structure can be reduced by a factor of up to four for thin trabeculae and up to two for thick trabeculae, even for small curvatures. The flexibility of curved elements is found to be larger than the flexibility of oblique elements with similar eccentricities. Thus it seems that curvature might play a role in determining the effective modulus of trabecular bone.     

Indirect determination of trabecular bone effective tissue failure properties using micro-finite element simulations

Trabecular bone strength is marked not only by the onset of local yielding, but also by post-yield behavior. To study and predict trabecular bone elastic and yield properties, micro-finite element (micro-FE) models were successfully applied. However, trabecular bone strength predictions require micro-FE models incorporating post-yield behavior of trabecular bone tissue. Due to experimental difficulties, such data is currently not available. Here we used micro-FE modeling to determine failure behavior of trabecular bone tissue indirectly, by iteratively fitting FE simulation to experimental results. Failure parameters were fitted to an isotropic plasticity model based on Hill's yield function, using materially and geometrically nonlinear micro-FE models of seven bovine trabecular bone specimens. The predictive value of the averaged effective tissue properties was subsequently tested. The results showed that compression softening had to be included on the tissue level in order to accurately describe the apparent-level behavior of the bone specimens. A sensitivity study revealed that the simulated response was less sensitive to variations in the post-yield properties of the bone tissue than variations in the elastic and yield properties. Due to fitting of the tissue properties, apparent-level behavior could be accurately reproduced for each specimen separately. Predictions based on the averaged and fixed tissue properties were less accurate, due to inter-specimen variations in the tissue properties.