Probabilistic failure analysis of bone using a finite element model of mineral–collagen composites

Microdamage accumulation is a major pathway for energy dissipation during the post-yield deformation of bone. In this study, a two-dimensional probabilistic finite element model of a mineral–collagen composite was developed to investigate the influence of the tissue and ultrastructural properties of bone on the evolution of microdamage from an initial defect in tension. The probabilistic failure analyses indicated that the microdamage progression would be along the plane of the initial defect when the debonding at mineral–collagen interfaces was either absent or limited in the vicinity of the defect. In this case, the formation of a linear microcrack would be facilitated. However, the microdamage progression would be scattered away from the initial defect plane if interfacial debonding takes place at a large scale. This would suggest the possible formation of diffuse damage. In addition to interfacial debonding, the sensitivity analyses indicated that the microdamage progression was also dependent on the other material and ultrastructural properties of bone. The intensity of stress concentration accompanied with microdamage progression was more sensitive to the elastic modulus of the mineral phase and the nonlinearity of the collagen phase, whereas the scattering of failure location was largely dependent on the mineral to collagen ratio and the nonlinearity of the collagen phase. The findings of this study may help understanding the post-yield behavior of bone at the ultrastructural level and shed light on the underlying mechanism of bone fractures.     

Determinants of Microdamage in Elderly Human Vertebral Trabecular Bone

Previous studies have shown that microdamage accumulates in bone as a result of physiological loading and occurs naturally in human trabecular bone. The purpose of this study was to determine the factors associated with pre-existing microdamage in human vertebral trabecular bone, namely age, architecture, hardness, mineral and organic matrix. Trabecular bone cores were collected from human L2 vertebrae (n = 53) from donors 54–95 years of age (22 men and 30 women, 1 unknown) and previous cited parameters were evaluated. Collagen cross-link content (PYD, DPD, PEN and % of collagen) was measured on surrounding trabecular bone. We found that determinants of microdamage were mostly the age of donors, architecture, mineral characteristics and mature enzymatic cross-links. Moreover, linear microcracks were mostly associated with the bone matrix characteristics whereas diffuse damage was associated with architecture. We conclude that linear and diffuse types of microdamage seemed to have different determinants, with age being critical for both types.     

The behaviour of microcracks in compact bone

This paper summarises four separate studies carried out by our group over the past number of years in the area of bone microdamage. The first study investigated the manner by which microcracks accumulate and interact with bone microstructure during fatigue testing of compact bone specimens. In a series of fatigue tests carried out at four different stress ranges between 50 and 80 MPA, crack density increased with loading cycles at a rate determined by the applied stress. Variations in the patterns of microdamage accumulation suggest that that at low stress levels, larger amounts of damage can build up without failure occurring. In a second study using a series of four-pont bending tests carried out on ovine bone samples, it was shown that bone microstructure influenced the ability of microcracks to propagate, with secondary osteons acting as barriers to crack growth. In a third study, the manner by which crack growth disrupts the canalicular processes connecting osteocytes was investigated. Analysis of individual cracks showed that disruption of the canalicular processes connecting osteocytes occurred due to shear displacement at the face of propagating microcracks, suggesting that this may play some role in the mechanism that signals bone remodelling. In a fourth in vivo study, it was shown that altering the mechanical load applied to the long bones of growing rats causes microcrack formation. In vivo microdamage was present in rats subjected to hindlimb suspension with a higher microcrack density found in the humeri than the femora. Microdamage was also found in control animals. This is the first study to demonstrate in vivo microcracks in normally loaded bones in a rat model.     

Development of a fluorescent light technique for evaluating microdamage in bone subjected to fatigue loading.

A new method using fluorescent light microscopy has been developed to visualize and evaluate bone microdamage. We report the findings of two different experiments with a common aim of comparing the fluorescent light technique to the brightfield method for quantifying microdamage in bone. In Experiment 1, 36 canine femurs were tested in four-point cyclic bending until they had lost between 5 and 43% of their stiffness. The loaded portion of the bone was stained en bloc with basic fuchsin for the presence of damage. Standard point counting techniques were used to calculate fractional damaged area (Dm.Ar = Cr.Ar/B.Ar, mm2/mm2) under brightfield and fluorescent microscopy. In Experiment 2, bone microdamage adjacent to endosseous implants, subjected to fatigue loading (150,000 cycles, 2 Hz and 37 degrees C) ex vivo was examined. The bone around the implant was either allowed to heal (adapted specimen) for 12 weeks after placement in dog mid-femoral diaphyses prior to testing or was loaded immediately to simulate non-healed bone surrounding endosseous implants (non-adapted). Crack numerical density (Cr.Dn = Cr.N/B.Ar, #/mm2), crack surface density (Cr.S.Dn = Tt.Cr.Le/B.Ar, mm/mm2) and fractional damaged area were calculated separately by both techniques in the adapted and non-adapted specimens. In both Experiments 1 and 2, significantly more microdamage was detected by the fluorescent technique than by the brightfield method. Also, there was a trend towards higher intraobserver repeatability when using the fluorescent method. These results suggest that the brightfield technique underestimates microdamage accumulation and that the fluorescent technique better represents the actual amounts of microdamage present. The results demonstrate that the fluorescent method provides an accurate and precise approach for bone microdamage evaluation, and that it improves the prediction of stiffness loss from damage accumulation.     

Nanoscale Examination of Microdamage in Sheep Cortical Bone Using Synchrotron Radiation Transmission X-Ray Microscopy

Microdamage occurs in bone through repeated and excessive loading. Accumulation of microdamage weakens bone, leading to a loss of strength, stiffness and energy dissipation in the tissue. Imaging techniques used to examine microdamage have typically been limited to the microscale. In the current study microdamage was examined at the nanoscale using transmission x-ray microscopy with an x-ray negative stain, lead-uranyl acetate. Microdamage was generated in notched and unnotched beams of sheep cortical bone (2×2×20 mm), with monotonic and fatigue loading. Bulk sections were removed from beams and stained with lead-uranyl acetate to identify microdamage. Samples were sectioned to 50 microns and imaged using transmission x-ray microscopy producing projection images of microdamage with nanoscale resolution. Staining indicated microdamage occurred in both the tensile and compressive regions. A comparison between monotonic and fatigue loading indicated a statistically significant greater amount of stain present in fatigue loaded sections. Microdamage occurred in three forms: staining to existing bone structures, cross hatch damage and a single crack extending from the notch tip. Comparison to microcomputed tomography demonstrated differences in damage morphology and total damage between the microscale and nanoscale. This method has future applications for understanding the underlying mechanisms for microdamage formation as well as three-dimensional nanoscale examination of microdamage.     

Collagen IX is indispensable for timely maturation of cartilage during fracture repair in mice.

Fracture repair recapitulates in adult organisms the sequence of cell biological events of endochondral ossification during skeletal development and growth. After initial inflammation and deposition of granulation tissue, a cartilaginous callus is formed which, subsequently, is remodeled into bone. In part, bone formation is influenced also by the properties of the extracellular matrix of the cartilaginous callus. Deletion of individual macromolecular components can alter extracellular matrix suprastructures, and hence stability and organization of mesenchymal tissues. Here, we took advantage of the collagen IX knockout mouse model to better understand the role of this collagen for organization, differentiation and maturation of a cartilaginous template during formation of new bone. Although a seemingly crucial component of cartilage fibrils is missing, collagen IX-deficient mice develop normally, but are predisposed to premature joint cartilage degeneration. However, we show here that lack of collagen IX alters the time course of callus differentiation during bone fracture healing. The maturation of cartilage matrix was delayed in collagen IX-deficient mice calli as judged by collagen X expression during the repair phase and the total amount of cartilage matrix was reduced. Entering the remodeling phase of fracture healing, Col9a1(-/-) calli retained a larger percentage of cartilage matrix than in wild type indicating also a delayed formation of new bone. We concluded that endochondral bone formation can occur in collagen IX knockout mice but is impaired under conditions of stress, such as the repair of an unfixed fractured long bone.     

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.     

人体股骨弹性模量与骨矿含量的关系

选取正常男性成人股骨18段,制成5-20mm测试块202个,用Ⅱ型γ线骨矿分析仪测得骨矿含量面密度分别为1.324-1.901g/cm2,再用WE-10(A)型液压式万能试验机和千分表测得测试块受纵向压应力时所承受的压力及相应的高度变化并计算其弹性模量。结果显示：骨矿含量在1.802g/cm2以下时,弹性模量及力学性能与骨矿含量呈正相关关系,但超出此范围后,骨矿含量的增加可能会使骨的力学性能下降。     

Quantification of trabecular bone microdamage using the virtual internal bond model and the individual trabeculae segmentation technique

Trabecular bone microdamage significantly influences the skeletal integrity and bone remodelling process. In this paper a novel constitutive model, called the virtual internal bond model (VIB), was adopted for simulating the damage behaviour of bone tissue. A unique 3D image analysis technique, named individual trabeculae segmentation, was used to analyse the effects of microarchitectures on the damage behaviours of trabecular bone. We demonstrated that the process of initiation and accumulation of microdamage in trabecular bone samples can be captured by the VIB-embedded finite-element method simulation without a separate fracture criterion. Our simulation results showed that the microdamage can occur at as early as about 0.2–0.4% apparent strain, and a large volume of microdamage was accumulated around the apparent yield strain. In addition we found that the plate-like trabeculae, especially the longitudinal ones, take crucial roles in the microdamage behaviours of trabecular bone.     

Low bone mass may not be the only cause of skeletal fragility in osteoporosis

Skeletal fragility in postmenopausal osteoporosis is not due solely to reduction in bone mass. This fact explains some of the overlap in bone mineral measurements observed between patients who are fracturing and age- and sex-matched normals who are not. Changes in skeletal architecture and bone remodeling occur with age which can account for some of the fragility. These changes are exaggerated in patients with postmenopausal osteoporosis who are suffering spine fractures. Three abnormalities have been described by histomorphometric methods which can account for skeletal fragility out of proportion to the degree of bone loss. They are: (i) loss of trabecular connectivity such that vertical weight-bearing bars lose their cross-attachments with each other, thus becoming susceptible to buckling; (ii) inefficient and prolonged microdamage repair due to periods of pause in the formation phase of remodeling; and (iii) accumulation of unrepaired microdamage in unremodeled bone tissue in the central part of trabeculae due to reduced osteon wall thickness coupled with maintenance of trabecular thickness. Recognition of these abnormalities should broaden our approach to the study of skeletal fragility in the syndrome of postmenopausal osteoporosis.     

Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model

Mineralized collagen fibrils are the basic building blocks of bone tissue at the supramolecular level. Several disease states, manipulation of the expression of specific proteins involved in biomineralization, and treatment with different agents alter the extent of mineralization as well as the morphology of mineral crystals which in turn affect the mechanical function of bone tissue. An experimental assessment of mineralized fibers' mechanical properties is challenged by their small size, leaving analytical and computational models as a viable alternative for investigation of the fibril-level mechanical properties. In the current study the variation of the elastic stiffness tensor of mineralized collagen fibrils with changing mineral volume fraction and mineral aspect ratios was predicted via a micromechanical model. The partitioning of applied stresses between mineral and collagen phases is also predicted for normal and shear loading of fibrils. Model predictions resulted in transversely isotropic collagen fibrils in which the modulus along the longer axis of the fibril was the greatest. All the elastic moduli increased with increasing mineral volume fraction whereas Poisson's ratios decreased with the exception of v12 (=v21). The partitioning of applied stresses were such that the stresses acting on mineral crystals were about 1.5, 15, and 3 times greater than collagen stresses when fibrils were loaded transversely, longitudinally, and in shear, respectively. In the overall the predictions were such that: (a) greatest modulus along longer axis; (b) the greatest mineral/collagen stress ratio along the longer axis of collagen fibers (i.e., greatest relief of stresses acting on collagen); and (c) minimal lateral contraction when fibers are loaded along the longer axis. Overall, the pattern of mineralization as put forth in this model predicts a superior mechanical function along the longer axis of collagen fibers, the direction which is more likely to experience greater stresses.     

Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength

There is an ongoing discussion on how bone strength could be explained from its internal structure and composition. Reviewing recent experimental and molecular dynamics studies, we here propose a new vision on bone material failure: mutual ductile sliding of hydroxyapatite mineral crystals along layered water films is followed by rupture of collagen crosslinks. In order to cast this vision into a mathematical form, a multiscale continuum micromechanics theory for upscaling of elastoplastic properties is developed, based on the concept of concentration and influence tensors for eigenstressed microheterogeneous materials. The model reflects bone's hierarchical organization, in terms of representative volume elements for cortical bone, for extravascular and extracellular bone material, for mineralized fibrils and the extrafibrillar space, and for wet collagen. In order to get access to the stress states at the interfaces between crystals, the extrafibrillar mineral is resolved into an infinite amount of cylindrical material phases oriented in all directions in space. The multiscale micromechanics model is shown to be able to satisfactorily predict the strength characteristics of different bones from different species, on the basis of their mineral/collagen content, their intercrystalline, intermolecular, lacunar, and vascular porosities, and the elastic and strength properties of hydroxyapatite and (molecular) collagen.     

Physical Aspects of Fruit Growth: STRESS DISTRIBUTION AROUND LENTICELS

The skin around a lenticel on a soft fruit has been modelled as a thin elastic plate with a rigid circular inclusion and applied tensile loads at the edges. A solution for the stress distribution in the skin has then been found using the linear theory of elasticity. From that solution the severity of the stress concentration and the location and form of initial cuticular failure have been deduced, the latter two being in broad agreement with observed crack initiation in the cuticle of grapes.     

A dynamic pattern of mechanical stimulation promotes ossification in avian embryonic long bones

We have performed a set of finite element analyses of embryonic chick hindlimb skeletal rudiments at several time points during development, around the time of initial bone formation. Using optical projection tomography, we created anatomically accurate rudiment and muscle morphologies for each stage. The changes in pattern and magnitude of biophysical stimuli (such as stress, strain, hydrostatic pressure and fluid flow) were computed, and were found to transform as bone formation proceeded in the rudiment. For each biophysical stimulus, a single concentration of the stimulus was predicted at the mid-diaphysis some time before initial bone formation begins. Then, several hours before ossification, two concentrations of the stimuli were predicted distal and proximal to the prospective bone collar. Once bone formation had begun, high concentrations of the stimuli were maintained proximal and distal to the bone collar. We propose the hypothesis that patterns of biophysical stimuli resulting from mechanical loading due to muscle contractions initiate and propagate ossification in avian embryonic long bones, whereby a region of the perichondrium experiences a period of time under high cyclic stimulus levels, promoting chondrocyte hypertrophy at the core and prompting bone collar formation some hours later.     

Post-yield nanomechanics of human cortical bone in compression using synchrotron X-ray scattering techniques

The ultrastructural response to applied loads governs the post-yield deformation and failure behavior of bone, and is correlated with bone fragility fractures. Combining a novel progressive loading protocol and synchrotron X-ray scattering techniques, this study investigated the correlation of the local deformation (i.e., internal strains of the mineral and collagen phases) with the bulk mechanical behavior of bone. The results indicated that the internal strains of the longitudinally oriented collagen fibrils and mineral crystals increased almost linearly with respect to the macroscopic strain prior to yielding, but markedly decreased first and then gradually leveled off after yielding. Similar changes were also observed in the applied stress before and after yielding of bone. However, the collagen to mineral strain ratio remained nearly constant throughout the loading process. In addition, the internal strains of longitudinal mineral and collagen phases did not exhibit a linear relationship with either the modulus loss or the plastic deformation of bulk bone tissue. Finally, the time-dependent response of local deformation in the mineral phase was observed after yielding. Based on the results, we speculate that the mineral crystals and collagen fibrils aligned with the loading axis only partially explain the post-yield deformation, suggesting that shear deformation involving obliquely oriented crystals and fibrils (off axis) is dominant mechanism of yielding for human cortical bone in compression.     

Measurement of microstructural strain in cortical bone

It is well known that mechanical factors affect bone remodeling such that increased mechanical demand results in net bone formation, whereas decreased demand results in net bone resorption. Current theories suggest that bone modeling and remodeling is controlled at the cellular level through signals mediated by osteocytes. The objective of this study was to investigate how macroscopically applied bone strains similar in magnitude to those that occur in vivo are manifest at the microscopic level in the bone matrix. Using a digital image correlation strain measurement technique, experimentally determined bone matrix strains around osteocyte lacuna resulting from macroscopic strains of approximately 2,000 microstrain (0.2%) reach levels of over 30,000 microstrain (3%) over fifteen times greater than the applied macroscopic strain. Strain patterns were highly heterogeneous and in some locations similar to observed microdamage around osteocyte lacuna indicating the resulting strains may represent the precursors to microdamage. This information may lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Microdamage Caused by Fatigue Loading in Human Cancellous Bone: Relationship to Reductions in Bone Biomechanical Performance

Vertebral fractures associated with osteoporosis are often the result of tissue damage accumulated over time. Microscopic tissue damage (microdamage) generated in vivo is believed to be a mechanically relevant aspect of bone quality that may contribute to fracture risk. Although the presence of microdamage in bone tissue has been documented, the relationship between loading, microdamage accumulation and mechanical failure is not well understood. The aim of the current study was to determine how microdamage accumulates in human vertebral cancellous bone subjected to cyclic fatigue loading. Cancellous bone cores (n = 32) from the third lumbar vertebra of 16 donors (10 male, 6 female, age 76±8.8, mean ± SD) were subjected to compressive cyclic loading at σ/E₀ = 0.0035 (where σ is stress and E₀ is the initial Young’s modulus). Cyclic loading was suspended before failure at one of seven different amounts of loading and specimens were stained for microdamage using lead uranyl acetate. Damage volume fraction (DV/BV) varied from 0.8±0.5% (no loading) to 3.4±2.1% (fatigue-loaded to complete failure) and was linearly related to the reductions in Young’s modulus caused by fatigue loading (r² = 0.60, p<0.01). The relationship between reductions in Young’s modulus and proportion of fatigue life was nonlinear and suggests that most microdamage generation occurs late in fatigue loading, during the tertiary phase. Our results indicate that human vertebral cancellous bone tissue with a DV/BV of 1.5% is expected to have, on average, a Young’s modulus 31% lower than the same tissue without microdamage and is able to withstand 92% fewer cycles before failure than the same tissue without microdamage. Hence, even small amounts of microscopic tissue damage in human vertebral cancellous bone may have large effects on subsequent biomechanical performance.     

Changes in the Mechanical Properties and Composition of Bone during Microdamage Repair

Under normal conditions, loading activities result in microdamage in the living skeleton, which is repaired by bone remodeling. However, microdamage accumulation can affect the mechanical properties of bone and increase the risk of fracture. This study aimed to determine the effect of microdamage on the mechanical properties and composition of bone. Fourteen male goats aged 28 months were used in the present study. Cortical bone screws were placed in the tibiae to induce microdamage around the implant. The goats were euthanized, and 3 bone segments with the screws in each goat were removed at 0 days, 21 days, 4 months, and 8 months after implantation. The bone segments were used for observing microdamage and bone remodeling, as well as nanoindentation and bone composition, separately. Two regions were measured: the first region (R1), located 1.5 mm from the interface between the screw hole and bone; and the second region (R2), located>1.5 mm from the bone-screw interface. Both diffuse and linear microdamage decreased significantly with increasing time after surgery, with the diffuse microdamage disappearing after 8 months. Thus, screw implantation results in increased bone remodeling either in the proximal or distal cortical bone, which repairs the microdamage. Moreover, bone hardness and elastic modulus decreased with microdamage repair, especially in the proximal bone tissue. Bone composition changed greatly during the production and repair of microdamage, especially for the C (Carbon) and Ca (Calcium) in the R1 region. In conclusion, the presence of mechanical microdamage accelerates bone remodeling either in the proximal or distal cortical bone. The bone hardness and elastic modulus decreased with microdamage repair, with the micromechanical properties being restored on complete repair of the microdamage. Changes in bone composition may contribute to changes in bone mechanical properties.     

Tendon tissue microdamage and the limits of intrinsic repair

The transmission of mechanical muscle force to bone for musculoskeletal stability and movement is one of the most important functions of tendon. The load-bearing tendon core is composed of highly aligned collagen-rich fascicles interspersed with stromal cells (tenocytes). Despite being built to bear very high mechanical stresses, supra-physiological/repetitive mechanical overloading leads to tendon microdamage in fascicles, and potentially to tendon disease and rupture. To date, it is unclear to what extent intrinsic healing mechanisms of the tendon core compartment can repair microdamage. In the present study, we investigated the healing capacity of the tendon core compartment in an ex vivo tissue explant model. To do so, we isolated rat tail tendon fascicles, damaged them by applying a single stretch to various degrees of sub-rupture damage and longitudinally assessed downstream functional and structural changes over a period of several days. Functional damage was assessed by changes in the elastic modulus of the material stress-strain curves, and biological viability of the resident tenocytes. Structural damage was quantified using a fluorescent collagen hybridizing peptide (CHP) to label mechanically disrupted collagen structures. While we observed functional mechanical damage for strains above 2% of the initial fascicle length, structural collagen damage was only detectable for 6% strain and beyond. Minimally loaded/damaged fascicles (2–4% strain) progressively lost elastic modulus over the course of tissue culture, despite their collagen structures remaining intact with high degree of maintained cell viability. In contrast, more severely overloaded fascicles (6–8% strain) with damage at the molecular/collagen level showed no further loss of the elastic modulus but markedly decreased cell viability. Surprisingly, in these heavily damaged fascicles the elastic modulus partially recovered, an effect also seen in further experiments on devitalized fascicles, implying the possibility of a non-cellular but matrix-driven mechanism of molecular repair. Overall, our findings indicate that the tendon core has very little capacity for self-repair of microdamage. We conclude that stromal tenocytes likely do not play a major role in anabolic repair of tendon matrix microdamage, but rather mediate catabolic matrix breakdown and communication with extrinsic cells that are able to effect tissue repair.     

Theoretical analysis of alendronate and risedronate effects on canine vertebral remodeling and microdamage

Bisphosphonates suppress bone remodeling activity, increase bone volume, and significantly reduce fracture risk in individuals with osteoporosis and other metabolic bone diseases. The objectives of the current study were to develop a mathematical model that simulates control and 1 year experimental results following bisphosphonate treatment (alendronate or risedronate) in the canine fourth lumbar vertebral body, validate the model by comparing simulation predictions to 3 year experimental results, and then use the model to predict potential long term effects of bisphosphonates on remodeling and microdamage accumulation. To investigate the effects of bisphosphonates on bone volume and microdamage, a mechanistic biological model was modified from previous versions to simulate remodeling in a representative volume of vertebral trabecular bone in dogs treated with various doses of alendronate or risedronate, including doses equivalent to those used for treatment of post-menopausal osteoporosis in humans. Bisphosphonates were assumed to affect remodeling by suppressing basic multicellular unit activation and reducing resorption area. Model simulation results for trabecular bone volume fraction, microdamage, and activation frequency following 1 year of bisphosphonate treatment are consistent with experimental measurements. The model predicts that trabecular bone volume initially increases rapidly with 1 year of bisphosphonate treatment, and continues to slowly rise between 1 and 3 years of treatment. The model also predicts that microdamage initially increases rapidly, 0.5–1.5-fold for alendronate or risedronate during the first year of treatment, and reaches its maximum value by 2.5 years before trending downward for all dosages. The model developed in this study suggests that increasing bone volume fraction with long term bisphosphonate treatment may sufficiently reduce strain and damage formation rate so that microdamage does not accumulate above that which is initiated in the first two years of treatment.