Perforation of cancellous bone trabeculae by damage-stimulated remodelling at resorption pits: a computational analysis

Loss of trabeculae in cancellous bone is often attributed to a general decline in the bone mass leading to fracture of the thin trabeculae. It has never been investigated whether trabecular perforation may have any other biomechanical mechanism. In this paper, an alternative hypothesis is proposed and tested using a computational model. Taking it as given that osteoclastic resorption is targeted to microdamage, it is hypothesised that the creation of a resorption cavity during normal bone remodelling could cause a stress-concentration in the bone tissue. If the resorption cavities were excessively deep, as is seen during osteoporosis, then this stress concentration may be sufficient to generate more microdamage so that osteoclasts "chase" newly formed damage leading to perforation. If this were true then we should find that, for a given trabecular thickness, there is a critical depth of resorption cavity such that smaller cavities refill whereas deeper cavities cause microdamage accumulation, continued osteoclast activity, and eventual trabecular perforation. Computer simulation is used to test this hypothesis. Using a remodelling stimulus calculated from both strain and damage and a simplified finite element model of a trabeculum with cavities of different sizes, it is predicted that such a critical depth of resorption cavity does indeed exist. Therefore we suggest that an increase in resorption depth relative to the thickness of trabeculae may be responsible for trabecular perforation during osteoporosis, rather than simply trabecular fracture due to insufficient strength.     

Simulation of vertebral trabecular bone loss using voxel finite element analysis

Trabecular bone loss in human vertebral bone is characterised by thinning and eventual perforation of the horizontal trabeculae. Concurrently, vertical trabeculae are completely lost with no histological evidence of significant thinning. Such bone loss results in deterioration in apparent modulus and strength of the trabecular core. In this study, a voxel-based finite element program was used to model bone loss in three specimens of human vertebral trabecular bone. Three sets of analyses were completed. In Set 1, strain adaptive resorption was modelled, whereby elements which were subject to the lowest mechanical stimulus (principal strain) were removed. In Set 2, both strain adaptive and microdamage mechanisms of bone resorption were included. Perforation of vertical trabeculae occurred due to microdamage resorption of elements with strains that exceeded a damage threshold. This resulted in collapse of the trabecular network under compression loading for two of the specimens tested. In Set 3, the damage threshold strain was gradually increased as bone loss progressed, resulting in reduced levels of microdamage resorption. This mechanism resulted in trabecular architectures in which vertical trabeculae had been perforated and which exhibited similar apparent modulus properties compared to experimental values reported in the literature. Our results indicate that strain adaptive remodelling alone does not explain the deterioration in mechanical properties that have been observed experimentally. Our results also support the hypothesis that horizontal trabeculae are lost principally by strain adaptive resorption, while vertical trabeculae may be lost due to perforation from microdamage resorption followed by rapid strain adaptive resorption of the remaining unloaded trabeculae.     

Homogenization theory and digital imaging: A basis for studying the mechanics and design principles of bone tissue

Bone tissue is a complex multilevel composite which has the ability to sense ad respond to its mechanical environment. It is believed that bone cells called osteocytes within the bone matrix sense the mechanical environment and determine whether structural alterations are needed. At present it is not known, however, how loads are transferred from the whole bone level to cells. A computational procedure combining representative volume element (RVE) based homogenization theory with digital imaging is proposed to estimate strains at various levels of bone structure. Bone tissue structural organization and RVE based analysis are briefly reviewed. The digital image based computational procedure was applied to estimate strains in individual trabeculae (first-level microstructure). Homogenization analysis of an idealized model was used to estimate strains at one level of bone structure around osteocyte lacunae (second-level trabecular microstructure). The results showed that strain at one level of bone structure is amplified to a broad range at the next microstructural level. In one case, a zeor-level tensile principal strain of 495 muE engendered strains ranging between -1000 and 7000 muE in individual trabeculae (first-level microstructure). Subsequently, a first-level tensile principal strains of 1325 muE within an inidividual trabecula engendered strains ranging between 782 and 2530 muE around osteocyte lacunae. Lacunar orientation was found to influence strains around osteocyte lacunae much more than lacunar ellipticity. In conclusion, the computational procedure combining homogenization theory with digital imaging can proveide estimates of cell level strains within whole bones. Such results may be used to bridge experimental studies of bone adaptation at the whole bone and cell culture level. (c) 1994 John Wiley & Sons, Inc.     

Trabecular bone microdamage and microstructural stresses under uniaxial compression

The balance between local remodeling and accumulation of trabecular bone microdamage is believed to play an important role in the maintenance of skeletal integrity. However, the local mechanical parameters associated with microdamage initiation are not well understood. Using histological damage labeling, micro-CT imaging, and image-based finite element analysis, regions of trabecular bone microdamage were detected and registered to estimated microstructural von Mises effective stresses and strains, maximum principal stresses and strains, and strain energy density (SED). Bovine tibial trabecular bone cores underwent a stepwise uniaxial compression routine in which specimens were micro-CT imaged following each compression step. The results indicate that the mode of trabecular failure observed by micro-CT imaging agreed well with the polarity and distribution of stresses within an individual trabecula. Analysis of on-axis subsections within specimens provided significant positive relationships between microdamage and each estimated tissue stress, strain and SED parameter. In a more localized analysis, individual microdamaged and undamaged trabeculae were extracted from specimens loaded within the elastic region and to the apparent yield point. As expected, damaged trabeculae in both groups possessed significantly higher local stresses and strains than undamaged trabeculae. The results also indicated that microdamage initiation occurred prior to apparent yield at local principal stresses in the range of 88-121 MPa for compression and 35-43 MPa for tension and local principal strains of 0.46-0.63% in compression and 0.18-0.24% in tension. These data provide an important step towards understanding factors contributing to microdamage initiation and establishing local failure criteria for normal and diseased trabecular bone.     

Mineral heterogeneity affects predictions of intratrabecular stress and strain

Knowledge of the influence of mineral variations (i.e., mineral heterogeneity) on biomechanical bone behavior at the trabecular level is limited. The aim of this study is to investigate how this material property affects the intratrabecular distributions of stress and strain in human adult trabecular bone. Two different sets of finite element (FE) models of trabecular samples were constructed; tissue stiffness was either scaled to the local degree of mineralization of bone as measured with microCT (heterogeneous) or tissue stiffness was assumed to be homogeneous. The influence of intratrabecular mineral heterogeneity was analyzed by comparing both models. Interesting effects were seen regarding intratrabecular stress and strain distributions. In the homogeneous model, the highest stresses were found at the surface with a significant decrease towards the core. Higher superficial stresses could indicate a higher predicted fracture risk in the trabeculae. In the heterogeneous model this pattern was different. A significant increase in stress with increasing distance from the trabecular surface was found followed by a significant decrease towards the core. This suggests trabecular bending during a compression. In both models a decrease in strain values from surface to core was predicted, which is consistent with trabecular bending. When mineral heterogeneity was taken into account, the predicted intratrabecular patterns of stress and strain are more consistent with the expected biomechanical behavior as based on mineral variations in trabeculae. Our findings indicate that mineral heterogeneity should not be neglected when performing biomechanical studies on topics such as the (long-term or dose dependent) effects of antiresorptive treatments.     

Interstitial fluid flow in canaliculi as a mechanical stimulus for cancellous bone remodeling: in silico validation

Cancellous bone has a dynamic 3-dimensional architecture of trabeculae, the arrangement of which is continually reorganized via bone remodeling to adapt to the mechanical environment. Osteocytes are currently believed to be the major mechanosensory cells and to regulate osteoclastic bone resorption and osteoblastic bone formation in response to mechanical stimuli. We previously developed a mathematical model of trabecular bone remodeling incorporating the possible mechanisms of cellular mechanosensing and intercellular communication in which we assumed that interstitial fluid flow activates the osteocytes to regulate bone remodeling. While the proposed model has been validated by the simulation of remodeling of a single trabecula, it remains unclear whether it can successfully represent in silico the functional adaptation of cancellous bone with its multiple trabeculae. In the present study, we demonstrated the response of cancellous bone morphology to uniaxial or bending loads using a combination of our remodeling model with the voxel finite element method. In this simulation, cancellous bone with randomly arranged trabeculae remodeled to form a well-organized architecture oriented parallel to the direction of loading, in agreement with the previous simulation results and experimental findings. These results suggested that our mathematical model for trabecular bone remodeling enables us to predict the reorganization of cancellous bone architecture from cellular activities. Furthermore, our remodeling model can represent the phenomenological law of bone transformation toward a locally uniform state of stress or strain at the trabecular level.     

Biomechanical properties across trabeculae from the proximal femur of normal and ovariectomised sheep

The elastic behaviour of trabecular bone is a function not only of bone volume and architecture, but also of tissue material properties. Variation in tissue modulus can have a substantial effect on the biomechanical properties of trabecular bone. However, the nature of tissue property variation within a single trabecula is poorly understood. This study uses nanoindentation to determine the mechanical properties of bone tissue in individual trabeculae. Using an ovariectomised ovine model, the modulus and hardness distribution across trabeculae were measured. In both normal and ovariectomised bone, the modulus and hardness were found to increase towards the core of the trabeculae. Across the width of the trabeculae, the modulus was significantly less in the ovariectomised bone than in the control bone. However, in contrast to this hardness was found not to differ significantly between the two groups. This study provides valuable information on the variation of mechanical material properties in healthy and diseased trabecular bone tissue. The results of the current study will be useful in finite element modelling where more accurate values of trabecular bone modulus will enable the prediction of the macroscale behaviour of trabecular bone.     

Tissue stresses and strain in trabeculae of a canine proximal femur can be quantified from computer reconstructions

A quantitative assessment of bone tissue stresses and strains is essential for the understanding of failure mechanisms associated with osteoporosis, osteoarthritis, loosening of implants and cell- mediated adaptive bone-remodeling processes. According to Wolff's trajectorial hypothesis, the trabecular architecture is such that minimal tissue stresses are paired with minimal weight. This paradigm at least suggests that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture. Although bone stresses at the apparent level were determined with finite element analysis (FEA), by assuming it to be continuous, there is no data available on trabecular tissue stresses or strains of bones in situ under physiological loading conditions. The objectives of this project were to supply reasonable estimates of these quantities for the canine femur, to compare trabecular-tissue to apparent stresses, and to test Wolff's hypothesis in a quantitative sense. For that purpose, the newly developed method of large-scale micro-FEA was applied in conjunction with micro-CT structural measurements. A three-dimensional high-resolution computer reconstruction of a proximal canine femur was made using a micro-CT scanner. This was converted to a large-scale FE-model with 7.6 million elements, adequately refined to represent individual trabeculae. Using a special-purpose FE-solver, analyses were conducted for three different orthogonal hip-joint loading cases, one of which represented the stance-phase of walking. By superimposing the results, the tissue stress and strain distributions could also be calculated for other force directions. Further analyses of results were concentrated on a trabecular volume of interest (VOI) located in the center of the head. For the stance phase of walking an average tissue principal strain in the VOI of 279 strain was found, with a standard deviation of 212 microstrain. The standard deviation depended not only on the hip-force magnitude, but also on its direction. In more than 95% of the tissue volume the principal stresses and strains were in a range from zero to three times the averages, for all hip-force directions. This indicates that no single load creates even stress or strain distributions in the trabecular architecture. Nevertheless, excessive values occurred at few locations only, and the maximum tissue stress was approximately half the value reported for the tissue fatigue strength. These results thus indicate that trabecular bone tissue has a safety factor of approximately two for hip-joint loads that occur during normal activities.     

Tissue stresses and strain in trabeculae of a canine proximal femur can be quantified from computer reconstructions

A quantitative assessment of bone tissue stresses and strains is essential for the understanding of failure mechanisms associated with osteoporosis, osteoarthritis, loosening of implants and cell-mediated adaptive bone-remodeling processes. According to Wolff's trajectorial hypothesis, the trabecular architecture is such that minimal tissue stresses are paired with minimal weight. This paradigm at least suggests that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture. Although bone stresses at the apparent level were determined with finite element analysis (FEA), by assuming it to be continuous, there is no data available on trabecular tissue stresses or strains of bones in situ under physiological loading conditions. The objectives of this project were to supply reasonable estimates of these quantities for the canine femur, to compare trabecular-tissue to apparent stresses, and to test Wolff's hypothesis in a quantitative sense. For that purpose, the newly developed method of large-scale micro-FEA was applied in conjunction with micro-CT structural measurements. A three-dimensional high-resolution computer reconstruction of a proximal canine femur was made using a micro-CT scanner. This was converted to a large-scale FE-model with 7.6 million elements, adequately refined to represent individual trabeculae. Using a special-purpose FE-solver, analyses were conducted for three different orthogonal hip-joint loading cases, one of which represented the stance-phase of walking. By superimposing the results, the tissue stress and strain distributions could also be calculated for other force directions. Further analyses of results were concentrated on a trabecular volume of interest (VOI) located in the center of the head. For the stance phase of walking an average tissue principal strain in the VOI of 279 strain was found, with a standard deviation of 212 microstrain. The standard deviation depended not only on the hip-force magnitude, but also on its direction. In more than 95% of the tissue volume the principal stresses and strains were in a range from zero to three times the averages, for all hip-force directions. This indicates that no single load creates even stress or strain distributions in the trabecular architecture. Nevertheless, excessive values occurred at few locations only, and the maximum tissue stress was approximately half the value reported for the tissue fatigue strength. These results thus indicate that trabecular bone tissue has a safety factor of approximately two for hip-joint loads that occur during normal activities.     

Modeling of dynamic fracture and damage in two-dimensional trabecular bone microstructures using the cohesive finite element method

Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Micro-finite-element models have been used to investigate the elastic and yield properties of trabecular bone but have only seen limited application in modeling the microstructure dependent fracture of trabecular bone. In this research, dynamic fracture in two-dimensional (2D) micrographs of ovine (sheep) trabecular bone is modeled using the cohesive finite element method. For this purpose, the bone tissue is modeled as an orthotropic material with the cohesive parameters calculated from the experimental fracture properties of the human cortical bone. Crack propagation analyses are carried out in two different 2D orthogonal sections cut from a three-dimensional 8 mm diameter cylindrical trabecular bone sample. The two sections differ in microstructural features such as area fraction (ratio of the 2D space occupied by bone tissue to the total 2D space), mean trabecula thickness, and connectivity. Analyses focus on understanding the effect of the rate of loading as well as on how the rate variation interacts with the microstructural features to cause anisotropy in microdamage accumulation and in the fracture resistance. Results are analyzed in terms of the dependence of fracture energy dissipation on the microstructural features as well as in terms of the changes in damage and stresses associated with the bone architecture variation. Besides the obvious dependence of the fracture behavior on the rate of loading, it is found that the microstructure strongly influences the fracture properties. The orthogonal section with lesser area fraction, low connectivity, and higher mean trabecula thickness is more resistant to fracture than the section with high area fraction, high connectivity, and lower mean trabecula thickness. In addition, it is found that the trabecular architecture leads to inhomogeneous distribution of damage, irrespective of the symmetry in the applied loading with the fracture of the entire bone section rapidly progressing to bone fragmentation once the accumulated damage in any trabeculae reaches a critical limit.     

Fluid–structure interactions in micro-interlocked regions of the cement–bone interface

Experimental tests and computational modelling were used to explore the fluid dynamics at the trabeculae–cement interlock regions found in the tibial component of total knee replacements. A cement–bone construct of the proximal tibia was created to simulate the immediate post-operative condition. Gap distributions along nine trabeculae–cement regions ranged from 0 to 50.4 μm (mean = 12 μm). Micro-motions ranged from 0.56 to 4.7 μm with a 1 MPa compressive load to the cement. Fluid–structure analysis between the trabeculae and the cement used idealised models with parametric evaluation of loading direction, gap closing fraction (GCF), gap thickness, loading frequency and fluid viscosity. The highest fluid shear stresses (926 Pa) along the trabecular surface were found for conditions with very thin and large GCFs, much larger than reported physiological levels (～1–5 Pa). A second fluid–structure model was created with a provision for bone resorption using a constitutive model with resorption velocity proportional to fluid shear rate. A lower cut-off was used, below which bone resorption would not occur (50 s^? 1). Results showed that there was initially high shear rates (>1000 s^? 1) that diminished after initial trabecular resorption. Resorption continued in high shear rate regions, resulting in a final shape with bone left deep in the cement layer, and is consistent with morphology found in post-mortem retrievals. Small gaps between the trabecular surface and the cement in the immediate post-operative state produce fluid flow conditions that appear to be supra-physiologic; these may cause fluid-induced lysis of trabeculae in the micro-interlock regions.     

Computational determination of the critical microcrack size that causes a remodeling response in a trabecula: a feasibility study

Bone is a living tissue, which undergoes continuous renewal to repair local defects. Two separate processes, adaptation and remodeling, are involved when a defect appears. The defect produces stress concentrations that provoke regional adaptation, and is gradually repaired, first by resorption and then by deposition of new bone. Using a mathematical formulation of the adaptation mechanism in trabeculae of cancellous bone, we hypothesize that in some cases, where a microcrack is small enough relative to the dimensions of the trabecula, the adaptation response of the whole trabecula may be sufficient to regain homeostatic mechanical conditions (with no need for a remodeling process). The simulation results showed that for trabeculae with nominal length of 900 microm and nominal thickness of 80-800 microm, a microcrack with minimal length of 48 microm and minimal depth of 13% of the trabecula's thickness was required to initiate a remodeling process. A longer (100 microm) but shallower (depth of 7% of the trabecula's thickness) crack also triggered remodeling. These computational results support our hypothesis that when a microcrack small enough relative to the dimensions of the trabecula occurs, adaptation of the whole trabecula may be sufficient to regain homeostatic mechanical conditions with no need for a local remodeling process.     

Bone turnover and trabecular plate survival after artificial menopause.

Considerable bone loss often occurs after menopause, particularly if menopause is induced by surgery. For perhaps two years bone formation fails to keep pace with the rapid acceleration of bone resorption that occurs after sex hormone withdrawal. The threat that this poses to the integrity of the skeleton is not clear. Because ethical constraints limit histological studies in normal women existing normal data and statistical modelling techniques were used to explore the dynamics of iliac trabecular bone after menopause. Trabeculae are breached during remodelling when the osteoclasts resorb to a depth equal to the trabecular thickness. Since holes in trabecular plates cannot normally be bridged such defects are probably permanent. Men lose 7% of their vertebral trabecular bone every 10 years; deeper than average resorption of trabeculae at the thin end of the normal range would account for it. The dramatic losses of trabecular bone that are seen in some postmenopausal women, however, require a period of imbalance between bone formation and bone resorption since this leads rapidly to generalised thinning. The statistical model suggested that an imbalance lasting only two years may account for eventual losses of up to half of the iliac trabecular bone. Further understanding is needed of what determines the amount of bone lost in the immediate postmenopause, which varies considerably among women. A simple mean is needed of identifying women who will lose bone most rapidly at the menopause. This must be suitable for use in general practice because these women should probably be offered long term hormone replacement treatment within a few months of the last menstruation.     

Poroelastic analysis of interstitial fluid flow in a single lamellar trabecula subjected to cyclic loading

Trabecula, an anatomical unit of the cancellous bone, is a porous material that consists of a lamellar bone matrix and interstitial fluid in a lacuno-canalicular porosity. The flow of interstitial fluid caused by deformation of the bone matrix is believed to initiate a mechanical response in osteocytes for bone remodeling. In order to clarify the effect of the lamellar structure of the bone matrix—i.e., variations in material properties—on the fluid flow stimuli to osteocytes embedded in trabeculae, we investigated the mechanical behavior of an individual trabecula subjected to cyclic loading based on poroelasticity. We focused on variations in the trabecular permeability and developed an analytical solution containing both transient and steady-state responses for interstitial fluid pressure in a single trabecular model represented by a multilayered two-dimensional poroelastic slab. Based on the obtained solution, we calculated the pressure and seepage velocity of the interstitial fluid in lacuno-canalicular porosity, within the single trabecula, under various permeability distributions. Poroelastic analysis showed that a heterogeneous distribution of permeability produces remarkable variations in the fluid pressure and seepage velocity in the cross section of the individual trabecula, and suggests that fluid flow stimuli to osteocytes are mostly governed by the value of permeability in the neighborhood of the trabecular surfaces if there is no difference in the average permeability in a single trabecula.     

Age-related differences in the morphology of microdamage propagation in trabecular bone

Microdamage density has been shown to increase with age in trabecular bone and is associated with decreased fracture toughness. Numerous studies of crack propagation in cortical bone have been conducted, but data in trabecular bone is lacking. In this study, propagation of severe, linear, and diffuse damage was examined in trabecular bone cores from the femoral head of younger (61.3±3.1 years) and older (75.0±3.9 years) men and women. Using a two-step mechanical testing protocol, damage was first initiated with static uniaxial compression to 0.8% strain then propagated at a normalized stress level of 0.005 to a strain endpoint of 0.8%. Coupling mechanical testing with a dual-fluorescent staining technique, the number and length/area of propagating cracks were quantified. It was found that the number of cycles to the test endpoint was substantially decreased in older compared to younger samples (younger: 77,372±15,984 cycles; older: 34,944±11,964 cycles, p=0.06). This corresponded with a greater number of severely damaged trabeculae expanding in area during the fatigue test in the older group. In the younger group, diffusely damaged trabeculae had a greater damage area, which illustrates an efficient energy dissipation mechanism. These results suggest that age-related differences in fatigue life of human trabecular bone may be due to differences in propagated microdamage morphology.     

Micromechanical analyses of vertebral trabecular bone based on individual trabeculae segmentation of plates and rods

Trabecular plates play an important role in determining elastic moduli of trabecular bone. However, the relative contribution of trabecular plates and rods to strength behavior is still not clear. In this study, individual trabeculae segmentation (ITS) and nonlinear finite element (FE) analyses were used to evaluate the roles of trabecular types and orientations in the failure initiation and progression in human vertebral trabecular bone. Fifteen human vertebral trabecular bone samples were imaged using micro computed tomography (μCT), and segmented using ITS into individual plates and rods by orientation (longitudinal, oblique, and transverse). Nonlinear FE analysis was conducted to perform a compression simulation for each sample up to 1% apparent strain. The apparent and relative trabecular number and tissue fraction of failed trabecular plates and rods were recorded during loading and data were stratified by trabecular orientation. More trabecular rods (both in number and tissue fraction) failed at the initiation of compression (0.1–0.2% apparent strain) while more plates failed around the apparent yield point (>0.7% apparent strain). A significant correlation between plate bone volume fraction (pBV/TV) and apparent yield strength was found (r²=0.85). From 0.3% to 1% apparent strain, significantly more longitudinal trabecular plate and transverse rod failed than other types of trabeculae. While failure initiates at rods and rods fail disproportionally to their number, plates contribute significantly to the apparent yield strength because of their larger number and tissue volume. The relative failed number and tissue fraction at apparent yield point indicate homogeneous local failure in plates and rods of different orientations.     

An algorithm for bone mechanoresponsiveness: implementation to study the effect of patient-specific cell mechanosensitivity on trabecular bone loss

The rate of bone loss is subject to considerable variation between individuals. With the 'mechanostat' model of Frost, genetic variations in bone mechanoresponsiveness are modelled by different mechanostat 'setpoints'--which may also change with age or disease. In this paper, the following setpoints are used: epsilonmin (strain below which resorption is triggered); epsilonmax (strain above which deposition occurs); omegacrit (microdamage-level above which damage-stimulated resorption occurs). To simulate decreased mechanosensitivity, epsilonmax is increased. Analyses carried out on a simplified model of a trabecula show that epsilonmax is a critical parameter: if it is higher in an individual (genetics) or increases (with age) the mass deficit each remodelling cycle increases. Furthermore, there is a value of epsilonmax above which trabecular perforation occurs, leading to rapid loss of bone mass. Maintaining bone cell mechanosensitivity could therefore be a therapeutic target for the prevention of osteoporosis.     

A cellular solid model for modulus reduction due to resorption of trabeculae in bone

In osteoporotic trabecular bone, bone loss occurs by thinning and subsequent resorption of the trabeculae. In this study, we compare the effects of density reductions from uniform thinning of struts or from removal of struts in a random, open-cell, three-dimensional Voronoi structure. The results of this study, combined with those previous studies on other regular and random structures, suggest that the modulus and strength of trabecular bone are reduced more dramatically by density losses from resorption of trabeculae than by those from uniform thinning of trabeculae.     

Age-related changes in human trabecular bone: Relationship between microstructural stress and strain and damage morphology

Accumulation of microdamage in aging and disease can cause skeletal fragility and is one of several factors contributing to osteoporotic fractures. To better understand the role of microdamage in fragility fracture, the mechanisms of bone failure must be elucidated on a tissue-level scale where interactions between bone matrix properties, the local biomechanical environment, and bone architecture are concurrently examined for their contributions to microdamage formation. A technique combining histological damage assessment of individual trabeculae with linear finite element solutions of trabecular von Mises and principal stress and strain was used to compare the damage initiation threshold between pre-menopausal (32-37 years, n=3 donors) and post-menopausal (71-80 years, n=3 donors) femoral cadaveric bone. Strong associations between damage morphology and stress and strain parameters were observed in both groups, and an age-related decrease in undamaged trabecular von Mises stress was detected. In trabeculae from younger donors, the 95% CI for von Mises stress on undamaged regions ranged from 50.7-67.9MPa, whereas in trabeculae from older donors, stresses were significantly lower (38.7-50.2, p<0.01). Local microarchitectural analysis indicated that thinner, rod-like trabeculae oriented along the loading axis are more susceptible to severe microdamage formation in older individuals, while only rod-like architecture was associated with severe damage in younger individuals. This study therefore provides insight into how damage initiation and morphology relate to local trabecular microstructure and the associated stresses and strains under loading. Furthermore, by comparison of samples from pre- and post-menopausal women, the results suggest that trabeculae from younger individuals can sustain higher stresses prior to microdamage initiation.     

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.