A fabric-dependent fracture criterion for bone.

A fracture criterion for bone tissue is proposed. Bone material is considered to be anisotropic and its properties are described by invoking the concept of directional variation of porosity. The fracture criterion is expressed as a scalar-valued function of the stress tensor and it incorporates an orientation-dependent distribution of compressive/tensile strength. The proposed mathematical framework is applied to a numerical analysis of fracture in the proximal femur due to a fall from standing height. The risk of fracture is assessed in the context of two different porosity distributions, simulating a healthy and an osteoporotic bone.     

Densitometric, morphometric and mechanical distributions in the human proximal femur

With the prevalent use of DXA-measured BMD to assess pathologic hip fractures and its recently reported lack of reliability to predict fracture or account for efficacy of anti-resorptive therapy, it is reasonable to assess whether variations in the primary and secondary tensile and compressive trabecular microstructure can account for variations in proximal femur strength in comparison to DXA-measured BMD. To that end, microstructural and densitometric measures of trabecular bone specimens, from discrete sites within the proximal femur, were correlated with their mechanical properties. We hypothesize that accounting for regional variations in trabecular microstructure will improve predictions of proximal femur strength and stiffness compared to bone density measured by DXA. Forty-seven samples (seven donors) from seven distinct sites of human proximal femur underwent DXA and muCT imaging and mechanical testing. The results revealed significant variations in BMC, morphometric indices and mechanical properties within the proximal femur. This work has demonstrated that the mechanical performance of each sub-region is highly dependent on the corresponding trabecular microstructure. BMD measured by DXA at standard regions of interest cannot resolve the variations in trabecular density and microstructure that govern the mechanical behavior of the proximal femur. This work suggests that a quantitative Singh index that uses high resolution QCT to monitor the trabecular microstructure at specific sub-regions of the proximal femur may allow better predictions of hip fracture risk in individual patients and an improved assessment of changing bone structure in response to pharmacological interventions.     

Three-dimensional reconstruction of fracture callus morphogenesis.

Rat and mouse femur and tibia fracture calluses were collected over various time increments of healing. Serial sections were produced at spatial segments across the fracture callus. Standard histological methods and in situ hybridization to col1a1 and col2a1 mRNAs were used to define areas of cartilage and bone formation as well as tissue areas undergoing remodeling. Computer-assisted reconstructions of histological sections were used to generate three-dimensional images of the spatial morphogenesis of the fracture calluses. Endochondral bone formation occurred in an asymmetrical manner in both the femur and tibia, with cartilage tissues seen primarily proximal or distal to the fractures in the respective calluses of these bones. Remodeling of the calcified cartilage proceeded from the edges of the callus inward toward the fracture producing an inner-supporting trabecular structure over which a thin outer cortical shell forms. These data suggest that the specific developmental mechanisms that control the asymmetrical pattern of endochondral bone formation in fracture healing recapitulated the original asymmetry of development of a given bone because femur and tibia grow predominantly from their respective distal and proximal physis. These data further show that remodeling of the calcified cartilage produces a trabecular bone structure unique to fracture healing that provides the rapid regain in weight-bearing capacity to the injured bone.     

A topology optimization based model of bone adaptation

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.     

A topology optimization based model of bone adaptation

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.     

Estimation of 3D shape, internal density and mechanics of proximal femur by combining bone mineral density images with shape and density templates

Measurement of bone mineral density (BMD) by dual-energy X-ray absorptiometry (DXA) alone is only a moderate predictor of fracture risk. Finite element analysis (FEA) of bone mechanics, based on DXA images, may improve the prediction of fracture risk. We developed a method to estimate the 3D shape and density distribution of the proximal femur, using a 2D BMD image and a femur shape template. Proximal femurs of eighteen human cadavers were imaged using computed tomography and divided into two sets (N = 9 + 9). The template was created from the samples in first set by using 3D generalized Procrustes analysis and thin-plate splines. Subsequently, the template and 2D BMD image were utilized to estimate the shape and internal density distribution of the femurs in the second set. Finally, FEA was conducted based on the original and the estimated bone models to evaluate the effect of geometrical and density distributional errors on the mechanical strength. The volumetric errors induced by the estimation itself were low (<1.4%). In the estimation of bones in the second set, the mean distance difference between the estimated and the original bone surfaces was 0.80 ± 0.19 mm, suggesting feasible estimation of the femoral shape. The mean absolute error in voxel-by-voxel BMD was 120±8 mg cm?3. In FEA, the stiffness of the proximal femur differed by -7±16% between the original and estimated bones. The present method, in comparison with methods used in previous studies, improved the prediction of the geometry, the BMD distribution and the mechanical characteristics of the proximal femur. Potentially, the proposed method could ultimately improve the determination of bone fracture risk.     

Prediction of strength and strain of the proximal femur by a CT-based finite element method

Hip fractures are the most serious complication of osteoporosis and have been recognized as a major public health problem. In elderly persons, hip fractures occur as a result of increased fragility of the proximal femur due to osteoporosis. It is essential to precisely quantify the strength of the proximal femur in order to estimate the fracture risk and plan preventive interventions. CT-based finite element analysis could possibly achieve precise assessment of the strength of the proximal femur. The purpose of this study was to create a simulation model that could accurately predict the strength and surface strains of the proximal femur using a CT-based finite element method and to verify the accuracy of our model by load testing using fresh frozen cadaver specimens. Eleven right femora were collected. The axial CT scans of the proximal femora were obtained with a calibration phantom, from which the 3D finite element models were constructed. Materially nonlinear finite element analyses were performed. The yield and fracture loads were calculated, while the sites where elements failed and the distributions of the principal strains were determined. The strain gauges were attached to the proximal femoral surfaces. A quasi-static compression test of each femur was conducted. The yield loads, fracture loads and principal strains of the prediction significantly correlated with those measured (r=0.941, 0.979, 0.963). Finite element analysis showed that the solid elements and shell elements in undergoing compressive failure were at the same subcapital region as the experimental fracture site.     

Combined finite element model of human proximal femur behaviour considering remodeling and fracture

The purpose of this work was to develop a combined remodeling-to-fracture finite element model allowing for the combined simulation of human proximal femur remodeling under a given boundary conditions followed by the simulation of its fracture behaviour under quasi-static load. The combination of remodeling and fracture simulation into one unified model consists in considering that the femur properties resulting from the remodeling simulation correspond to the initial state for the fracture prediction. The remodeling model is based on a coupled strain and fatigue damage stimulus approach. The fracture model is based on continuum damage mechanics in order to predict the progressive fracturing process, which allows to predict the fracture pattern and the complete force-displacement curve under quasi-static load. To investigate the potential of the proposed unified remodeling-to-fracture model, we performed remodeling simulations on a 3D proximal femur model for a duration of 365 days followed by a side fall fracture simulation reproducing.     

Scanner influence on the mechanical response of QCT-based finite element analysis of long bones

Patient-specific QCT-based finite element (QCTFE) analyses enable highly accurate quantification of bone strength. We evaluated CT scanner influence on QCTFE models of long bones.A femur, humerus, and proximal femur without the head were scanned with K₂HPO₄ phantoms by seven CT scanners (four models) using typical clinical protocols. QCTFE models were constructed. The geometrical dimensions, as well as the QCT-values expressed in Hounsfield unit (HU) distribution was compared. Principal strains at representative regions of interest (ROIs), and maximum principal strains (associated with fracture risk) were compared. Intraclass correlation coefficients (ICCs) were calculated to evaluate strain prediction reliability for different scanners. Repeatability was examined by scanning the femur twice and comparing resulting QCTFE models.Maximum difference in geometry was 2.3%. HU histograms before phantom calibration showed wide variation between QCT scans; however, bone density histogram variability was reduced after calibration and algorithmic manipulation. Relative standard deviation (RSD) in principal strains at ROIs was <10.7%. ICC estimates between scanners were >0.9. Fracture-associated strain had 6.7%, 8.1%, and 13.3% maximum RSD for the femur, humerus, and proximal femur, respectively. The difference in maximum strain location was <2 mm. The average difference with repeat scans was 2.7%.Quantification of strain differences showed mean RSD bounded by ∼6% in ROIs. Fracture-associated strains in “regular” bones showed a mean RSD bounded by ∼8%. Strains were obtained within a ±10% difference relative to the mean; thus, in a longitudinal study only changes larger than 20% in the principal strains may be significant. ICCs indicated high reliability of QCTFE models derived from different scanners.     

Comparative analysis of bone remodelling models with respect to computerised tomography-based finite element models of bone

Subject-specific finite element models are an extensively used tool for the numerical analysis of the biomechanical behaviour of human bones. However, bone modelling is not an easy task due to the complex behaviour of bone tissue, involving non-homogeneous and anisotropic mechanical properties. Moreover, bone is a living tissue and therefore its microstructure and mechanical properties evolve with time in a known process called bone remodelling. This phenomenon has been widely studied, many being the numerical models that have been formulated to predict density distribution and its evolution in several bones. The aim of the present study is to assess the capability of a bone remodelling model to predict the bone density distribution of different types of human bone (femur, tibia and mandible) comparing the obtained results with the bone density estimated by means of computerised tomography. Good accuracy was observed for the bone remodelling predictions including the thickness of the cortical layer.     

The relationship between whole bone stiffness and strength is age and sex dependent

Accurately estimating whole bone strength is critical for identifying individuals that may benefit from prophylactic treatments aimed at reducing fracture risk. Strength is often estimated from stiffness, but it is not known whether the relationship between stiffness and strength varies with age and sex. Cadaveric proximal femurs (44 Male: 18–78 years; 40 Female: 24–95 years) and radial (36 Male: 18–89 years; 19 Female: 24–95 years) and femoral diaphyses (34 Male: 18–89 years; 19 Female: 24–95 years) were loaded to failure to evaluate how the stiffness-strength relationship varies with age and sex. Strength correlated significantly with stiffness at all sites and for both sexes, as expected. However, females exhibited significantly less strength for the proximal femur (58% difference, p < 0.001). Multivariate regressions revealed that stiffness, age and PYD were significant negative independent predictors of strength for the proximal femur (Age: M: p = 0.005, F: p < 0.001, PYD: M: p = 0.022, F: p = 0.025), radial diaphysis (Age: M = 0.055, PYD: F = 0.024), and femoral diaphysis (Age: M: p = 0.014, F: p = 0.097, PYD: M: p = 0.003, F: p = 0.091). These results indicated that older bones tended to be significantly weaker for a given stiffness than younger bones. These results suggested that human bones exhibit diminishing strength relative to stiffness with aging and with decreasing PYD. Incorporating these age- and sex-specific factors may help to improve the accuracy of strength estimates.     

Incidence and mechanical significance of pneumatization in the long bones of birds

The incidence of pneumatization in avian long bones was studied, by direct observation, in a large sample of species. Only proximal bones (humerus and femur) presented pneumatization in the sample studied. The incidence obtained was related to the variation of the maximum cortical thickness and mechanical properties, such as bending strength and flexural Young's modulus. Cortical thickness, bending strength and flexural Young's modulus were significantly lower in pneumatized bones than in marrow-filled bones. Furthermore, some congruence was found between pneumatization and systematic groups when compared. In this sense, Charadriformes was the only order studied with total absence of long bone pneumatization. Results on cortical thickness appear to be in agreement with modelling predictions previously made and with results obtained on other groups of flying vertebrates. The possible selective advantage of reduction in cortical thickness in relation to flying is suggested.     

Comparison of micro-level and continuum-level voxel models of the proximal femur

Continuum-level finite element (FE) models became standard computational tools for the evaluation of bone mechanical behavior from in vivo computed tomography scans. Such scans do not account for the anisotropy of the bone. Instead, local mechanical properties in the continuum-level FE models are assumed isotropic and are derived from bone density, using statistical relationships. Micro-FE models, on the other hand, incorporate the anisotropic structure in detail. This study aimed to quantify the effects of assumed isotropy, by comparing continuum-level voxel models of a healthy and a severely osteoporotic proximal femur with recently analyzed micro-FE models of the same bones. The micro-model element size was coarsened to generate continuum FE models with two different element sizes (0.64 and 3.04 mm) and two different density–modulus relationships found in the literature for wet and ash density. All FE models were subjected to the same boundary conditions that simulated a fall to the side, and the stress and strain distributions, model stiffness and yield load were compared. The results indicated that the stress and strain distributions could be reproduced well with the continuum models. The smallest differences between the continuum-level model and micro-level model predictions of the stiffness and yield load were obtained with the coarsest element size. Better results were obtained for both continuum-element sizes when isotropic moduli were based on ash density rather than wet density.     

Microscale poroelastic metamodel for efficient mesoscale bone remodelling simulations

Bone functional tissue adaptation is a multiaspect physiological process driven by interrelated mechanical and biological stimuli which requires the combined activity of osteoclasts and osteoblasts. In previous work, the authors developed a phenomenological mesoscale structural modelling approach capable of predicting internal structure of the femur based on daily activity loading, which relied on the iterative update of the cross-sectional areas of truss and shell elements representative of trabecular and cortical bones, respectively. The objective of this study was to introduce trabecular reorientation in the phenomenological model at limited computational cost. To this aim, a metamodel derived from poroelastic microscale continuum simulations was used to predict the functional adaptation of a simplified proximal structural femur model. Clear smooth trabecular tracts are predicted to form in the regions corresponding to the main trabecular groups identified in literature, at minimal computational cost.     

Bone adaptation to altered loading after spinal cord injury: a study of bone and muscle strength

Bone loss from the paralysed limbs after spinal cord injury (SCI) is well documented. Under physiological conditions, bones are adapted to forces which mainly emerge from muscle pull. After spinal cord injury (SCI), muscles can no longer contract voluntarily and are merely activated during spasms. Based on the Ashworth scale, previous research has suggested that these spasms may mitigate bone losses. We therefore wished to assess muscle forces after SCI with a more direct measure and compare it to measures of bone strength. We hypothesized that the bones in SCI patients would be in relation to the loss of muscle forces. Six male patients with SCI 6.4 (SD 4.3) years earlier and 6 age-matched, able-bodied control subjects were investigated. Bone scans from the right knee were obtained by pQCT. The knee extensor muscles were electrically stimulated via the femoral nerve, isometric knee extension torque was measured and patellar tendon force was estimated. Tendon force upon electrical stimulation in the SCI group was 75% lower than in the control subjects (p<0.01). Volumetric bone mineral density of the patella and of the proximal tibia epiphysis were 50% lower in the SCI group than in the control subjects (p<0.01). Cortical area was lower by 43% in the SCI patients at the proximal tibia metaphysis, and by 33% at the distal femur metaphysis. No group differences were found in volumetric cortical density. Close curvilinear relationships were found between stress and volumetric density for the tibia epiphysis (r(2)=0.90) and for the patella (r(2)=0.91). A weaker correlation with the tendon force was found for the cortical area of the proximal tibia metaphysis (r(2)=0.63), and none for the distal femur metaphysis. These data suggest that, under steady state conditions after SCI, epiphyseal bones are well adapted to the muscular forces. For the metaphysis of the long bones, such an adaptation appears to be less evident. The reason for this remains unclear.     

Variation of trabecular architecture in proximal femur of postmenopausal women

This investigation of microstructure in the human proximal femur probes the relationship between the parameters of the FRAX index of fracture risk and the parameters of bone microstructure. The specificity of fracture sites at the proximal femur raises the question of whether trabecular parameters are site-specific during post-menopause, before occurrence of fragility fracture. The donated proximal femurs of sixteen post-menopausal women in the sixth and seventh decades of life, free of metabolic pathologies and therapeutic interventions that could have altered the bone tissue, constituted the material of the study. We assessed bone mineral density of the proximal femurs by dual energy X-ray absorptiometry and then sectioned the femurs through the center of the femoral head and along the femoral neck axis. For each proximal femur, morphometry of trabeculae was conducted on the plane of the section divided into conventional regions and sub-regions consistent with the previously identified trabecular families that provide regions of relatively homogeneous microstructure. Mean trabecular width and percent bone area were calculated at such sites. Our findings indicate that each of mean trabecular width and percent bone area vary within each proximal femur independently from each other, with dependence on site. Both trabecular parameters show significant differences between pairs of sites. We speculate that a high FRAX index at the hip corresponds to a reduced percent bone area among sites that gives a more homogeneous and less site-specific quality to the proximal femur. This phenomenon may render the local tissue less able to carry out the expected mechanical function.     

A reconciliation of local and global models for bone remodeling through optimization theory

Remodeling rules with either a global or a local mathematical form have been proposed for load-bearing bones in the literature. In the local models, the bone architecture (shape, density) is related to the strains/energies sensed at any point in the bone, while in the global models, a criterion believed to be applicable to the whole bone is used. In the present paper, a local remodeling rule with a strain "error" form is derived as the necessary condition for the optimum of a global remodeling criterion, suggesting that many of the local error-driven remodeling rules may have corresponding global optimization-based criteria. The global criterion proposed in the present study is a trade-off between the cost of metabolic growth and use, mathematically represented by the mass, and the cost of failure, mathematically represented by the total strain energy. The proposed global criterion is shown to be related to the optimality criteria methods of structural optimization by the equivalence of the model solution and the fully stressed solution for statically determinate structures. In related work, the global criterion is applied to simulate the strength recovery in bones with screw holes left behind after removal of fracture fixation plates. The results predicted by the model are shown to be in good agreement with experimental results, leading to the conclusion that load-bearing bones are structures with optimal shape and property for their function.     

A model of bone adaptation as an optimization process

The internal bone adaptation of the proximal femur is considered. A three-dimensional finite element model of the proximal femur is used. The bone remodeling in this work is numerically described by an evolutionary remodeling scheme with anisotropic material parameters and time-dependent loading. The memory of past loading is included in the model to account for the delay in the bone response from the load changes. The remodeling rate equation is derived from the structural optimization task of maximizing the stiffness in each time step. Additional information can be extracted from the optimization process; the remodeling equilibrium parameter where no apposition or resorption takes place, is defined as the element optimality conditions and the optimal design is used as an initial design for the onset of the remodeling simulation. Two examples of bone adaptation resulting from load changes are given, and the irreversible nature of the model is illustrated.     

Use of a statistical model of the whole femur in a large scale,multi-model study of femoral neck fracture risk

Interpatient variability is often overlooked in orthopaedic computational studies due to the substantial challenges involved in sourcing and generating large numbers of bone models. A statistical model of the whole femur incorporating both geometric and material property variation was developed as a potential solution to this problem. The statistical model was constructed using principal component analysis, applied to 21 individual computer tomography scans. To test the ability of the statistical model to generate realistic, unique, finite element (FE) femur models it was used as a source of 1000 femurs to drive a study on femoral neck fracture risk. The study simulated the impact of an oblique fall to the side, a scenario known to account for a large proportion of hip fractures in the elderly and have a lower fracture load than alternative loading approaches. FE model generation, application of subject specific loading and boundary conditions, FE processing and post processing of the solutions were completed automatically. The generated models were within the bounds of the training data used to create the statistical model with a high mesh quality, able to be used directly by the FE solver without remeshing. The results indicated that 28 of the 1000 femurs were at highest risk of fracture. Closer analysis revealed the percentage of cortical bone in the proximal femur to be a crucial differentiator between the failed and non-failed groups. The likely fracture location was indicated to be intertrochantic. Comparison to previous computational, clinical and experimental work revealed support for these findings.     

A Femur-Implant Model for the Prediction of Bone Remodeling Behavior Induced by Cementless Stem

Bone remodeling simulation is an effective tool for the prediction of long-term effect of implant on the bone tissue, as well as the selection of an appropriate implant in terms of architecture and material. In this paper, a finite element model of proximal femur was developed to simulate the structures of internal trabecular and cortical bones by incorporating quantitative bone functional adaptation theory with finite element analysis. Cementless stems made of titanium, two types of Functionally Graded Material (FGM) and flexible ‘iso-elastic’ material as comparison were implanted in the structure of proximal femur respectively to simulate the bone remodeling behaviors of host bone. The distributions of bone density, von Mises stress, and interface shear stress were obtained. All the prosthetic stems had effects on the bone remodeling behaviors of proximal femur, but the degrees of stress shielding were different. The amount of bone loss caused by titanium implant was in agreement with the clinical observation. The FGM stems caused less bone loss than that of the titanium stem, in which FGM I stem (titanium richer at the top to more HAP/Col towards the bottom) could relieve stress shielding effectively, and the interface shear stresses were more evenly distributed in the model with FGM I stem in comparison with those in the models with FGM II (titanium and bioglass) and titanium stems. The numerical simulations in the present study provided theoretical basis for FGM as an appropriate material of femoral implant from a biomechanical point of view. The next steps are to fabricate FGM stem and to conduct animal experiments to investigate the effects of FGM stem on the remodeling behaviors using animal model.