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Introduction: Ultimate strength-density relationships for bone have been reported with widely varying results. Reliable bone strength predictions are crucial for many applications that aim to assess bone failure. Bone density and bone morphology have been proposed to explain most of the variance in measured bone strength. If this holds true, it could lead to the derivation of a single ultimate strength-density-morphology relationship for all anatomical sites., Methods: All relevant literature was reviewed. Ultimate strength-density relationships derived from mechanical testing of human bone tissue were included. The reported relationships were translated to ultimate strength-apparent density relationships and normalized with respect to strain rate. Results were grouped based on bone tissue type (cancellous or cortical), anatomical site, and loading mode (tension vs. compression). When possible, the relationships were compared to existing ultimate strength-density-morphology relationships., Results: Relationships that considered bone density and morphology covered the full spectrum of eight-fold inter-study difference in reported compressive ultimate strength-density relationships for trabecular bone. This was true for studies that tested specimens in different loading direction and tissue from different anatomical sites. Sparse data was found for ultimate strength-density relationships in tension and for cortical bone properties transverse to the main loading axis of the bone., Conclusions: Ultimate strength-density-morphology relationships could explain measured strength across anatomical sites and loading directions. We recommend testing of bone specimens in other directions than along the main trabecular alignment and to include bone morphology in studies that investigate bone material properties. The lack of tensile strength data did not allow for drawing conclusions on ultimate strength-density-morphology relationships. Further studies are needed. Ideally, these studies would investigate both tensile and compressive strength-density relationships, including morphology, to close this gap and lead to more accurate evaluation of bone failure., (Copyright © 2020 The Authors. Published by Elsevier Ltd.. All rights reserved.)

An efficient solver for large-scale linear [Formula: see text] simulations was extended for nonlinear material behavior. The material model included damage-based tissue degradation and fracture. The new framework was applied to 20 trabecular biopsies with a mesh resolution of [Formula: see text]. Suitable material parameters were identified based on two biopsies by comparison with axial tension and compression experiments. The good parallel performance and low memory footprint of the solver were preserved. Excellent correlation of the maximum apparent stress was found between simulations and experiments ([Formula: see text]). The development of local damage regions was observable due to the nonlinear nature of the simulations. A novel elasticity limit was proposed based on the local damage information. The elasticity limit was found to be lower than the 0.2% yield point. Systematic differences in the yield behavior of biopsies under apparent compression and tension loading were observed. This indicates that damage distributions could lead to more insight into the failure mechanisms of trabecular bone.

Low bone strength is a major risk factor for osteoporotic fractures and is only partially determined by clinical densitometry. Accumulated micro-damage induces residual strains, degrades elastic modulus and reduces bone strength independently of bone mineral density. Histologically, overloading of bone in compression and tension leads to distinct crack size, distribution and orientation which interact during combined loading scenarios. Statistics of rheological models can describe this process and reproduce experimental stress-strain curves with an unprecedented realism, but are computationally expensive and therefore difficult to generalize to 3D. Accordingly, the aim of this work is to formulate a continuum damage model that describes the key features of bone micro-damage, namely the accumulation of residual strains, the degradation of elastic modulus and the reduction of strength in compression, tension and especially in their sequential application. The promising qualitative agreement of the model with the experiments will motivate a generalization to 3D and allow the biomechanical investigation of bones and bone-implant systems subjected to cyclic overloading in tension and/or compression., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

The trabecular structure can be assessed at the wrist or tibia via high-resolution peripheral quantitative computed tomography (HR-pQCT). Yet on this modality, the performance of the existing methods, evaluating trabecular anisotropy is usually overlooked, especially in terms of reproducibility. We thus proposed to compare the TRI routine used by SCANCO Medical AG (Brüttisellen, Switzerland), the classical mean intercept length (MIL), and the grey-level structure tensor (GST) to the mean surface length (MSL), a new method for evaluating a second-order fabric tensor based on the triangulation of the bone surface. The distal radius of 24 fresh-frozen human forearms was scanned three times via HR-pQCT protocols (61μm, 82μm nominal voxel size), dissected, and imaged via micro computed tomography (μCT) at 16μm nominal voxel size. After registering the scans, we compared for each resolution the fabric tensors, determined by the mentioned techniques for 182 trabecular regions of interest. We then evaluated the reproducibility of the fabric information measured by HR-pQCT via precision errors. On μCT, TRI and GST were respectively the best and worst surrogates for MIL μCT (MIL computed on μCT) in terms of eigenvalues and main direction of anisotropy. On HR-pQCT, however, MSL provided the best approximation of MIL μCT . Surprisingly, surface-based approaches (TRI, MSL) also proved to be more precise than both MIL and GST. Our findings confirm that MSL can reproducibly estimate MIL μCT , the current gold standard. MSL thus enables the direct mapping of the fabric-dependent material properties required in homogenised HR-pQCT-based finite element models., (Copyright © 2017 Elsevier Inc. All rights reserved.)

Beyond bone mineral density (BMD), bone quality designates the mechanical integrity of bone tissue. In vivo images based on X-ray attenuation, such as CT reconstructions, provide size, shape, and local BMD distribution and may be exploited as input for finite element analysis (FEA) to assess bone fragility. Further key input parameters of FEA are the material properties of bone tissue. This review discusses the main determinants of bone mechanical properties and emphasizes the added value, as well as the important assumptions underlying finite element analysis. Bone tissue is a sophisticated, multiscale composite material that undergoes remodeling but exhibits a rather narrow band of tissue mineralization. Mechanically, bone tissue behaves elastically under physiologic loads and yields by cracking beyond critical strain levels. Through adequate cell-orchestrated modeling, trabecular bone tunes its mechanical properties by volume fraction and fabric. With proper calibration, these mechanical properties may be incorporated in quantitative CT-based finite element analysis that has been validated extensively with ex vivo experiments and has been applied increasingly in clinical trials to assess treatment efficacy against osteoporosis.

Osteoporosis leads to bone fragility and represents a major health problem in our aging societies. Bone is a quasi-brittle hierarchical composite that exhibits damage with distinct crack morphologies in compression and tension when overloaded. A recent study reported the complex damage response of bovine compact bone under four different cyclic overloading experiments combining compression and tension. The aim of the present work is to propose a mechanistic model by which cracking bone accumulates residual strain and reduces elastic modulus in distinct compressive and tensile overloading modes. A simple rheological unit of bone with two types of cracks is formulated in the framework of continuum damage mechanics. A statistics of these rheological units is then assembled in parallel to compute the response of a macroscopic bone sample in which compressive and tensile cracks are opened, closed or propagated towards failure. The resulting constitutive model reproduces the key macroscopic features of bone tissue damage and delivers an excellent agreement with the four cyclic overloading experiments. The remarkable predictions of the model support the presence of (1) friction between the crack surfaces producing hystereses, (2) an incomplete closure of cracks leading to residual strains, (3) a bridging mechanism of collagen fibrils which failure reduces elastic modulus, and (4) two distinct classes of cracks where compressive cracks have a strong influence on tensile damage and tensile cracks have a limited impact on compressive damage. This work is expected to help improve our understanding of the bone damage mechanisms contributing to skeletal fragility and to foster a proper generalization of this damage behavior in 3D for computational analysis of bone and bone-implant systems., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

Purpose: Investigation of trabecular bone strength and compaction is important for fracture risk prediction. At 1-2% compressive strain, trabecular bone undergoes strain softening, which may lead to numerical instabilities and mesh dependency in classical local damage-plastic models. The aim of this work is to improve our continuum damage-plastic model of bone by reducing the influence of finite element mesh size under large compression., Methodology: This spurious numerical phenomenon may be circumvented by incorporating the nonlocal effect of cumulated plastic strain into the constitutive law. To this end, an over-nonlocal implicit gradient model of bone is developed and implemented into the finite element software ABAQUS using a user element subroutine. The ability of the model to detect the regions of bone failure is tested against experimental stepwise loading data of 16 human trabecular bone biopsies., Findings: The numerical outcomes of the nonlocal model revealed reduction of finite element mesh dependency compared with the local damage-plastic model. Furthermore, it helped reduce the computational costs of large-strain compression simulations., Originality: To the best of our knowledge, the proposed model is the first to predict the failure and densification of trabecular bone up to large compression independently of finite element mesh size. The current development enables the analysis of trabecular bone compaction as in osteoporotic fractures and implant migration, where large deformation of bone plays a key role., (Copyright © 2015 John Wiley & Sons, Ltd.)

As our population ages, more individuals suffer from osteoporosis. This disease leads to impaired trabecular architecture and increased fracture risk. It is essential to understand how morphological and mechanical properties of the cancellous bone are related. Morphology-elasticity relationships based on bone volume fraction (BV/TV) and fabric anisotropy explain up to 98% of the variation in elastic properties. Yet, other morphological variables such as individual trabeculae segmentation (ITS) and trabecular bone score (TBS) could improve the stiffness predictions. A total of 743 micro-computed tomography (μCT) reconstructions of cubic trabecular bone samples extracted from femur, radius, vertebrae, and iliac crest were analyzed. Their morphology was assessed via 25 variables and their stiffness tensor (CFE) was computed from six independent load cases using micro finite element (μFE) analyses. Variance inflation factors were calculated to evaluate collinearity between morphological variables and decide upon their inclusion in morphology-elasticity relationships. The statistically admissible morphological variables were included in a multiple linear regression model of the dependent variable CFE. The contribution of each independent variable was evaluated (ANOVA). Our results show that BV/TV is the best determinant of CFE(r(2) adj  = 0.889), especially in combination with fabric anisotropy (r(2) adj  = 0.968). Including the other independent predictors hardly affected the amount of variance explained by the model (r(2) adj  = 0.975). Across all anatomical sites, BV/TV explained 87% of the variance of the bone elastic properties. Fabric anisotropy further described 10% of the bone stiffness, but the improvement in variance explanation by adding other independent factors was marginal (<1%). These findings confirm that BV/TV and fabric anisotropy are the best determinants of trabecular bone stiffness and show, against common belief, that other morphological variables do not bring any further contribution. These overall conclusions remain to be confirmed for specific bone diseases and postelastic properties., (© 2015 American Society for Bone and Mineral Research.)

Objective: Although osteopenia is frequent in spondyloarthritis (SpA), the underlying cellular mechanisms and association with other symptoms are poorly understood. This study aimed to characterize bone loss during disease progression, determine cellular alterations, and assess the contribution of inflammatory bowel disease (IBD) to bone loss in HLA-B27 transgenic rats., Methods: Bones of 2-, 6-, and 12-month-old non-transgenic, disease-free HLA-B7 and disease-associated HLA-B27 transgenic rats were examined using peripheral quantitative computed tomography, μCT, and nanoindentation. Cellular characteristics were determined by histomorphometry and ex vivo cultures. The impact of IBD was determined using [21-3 x 283-2]F1 rats, which develop arthritis and spondylitis, but not IBD., Results: HLA-B27 transgenic rats continuously lost bone mass with increasing age and had impaired bone material properties, leading to a 3-fold decrease in bone strength at 12 months of age. Bone turnover was increased in HLA-B27 transgenic rats, as evidenced by a 3-fold increase in bone formation and a 6-fold increase in bone resorption parameters. Enhanced osteoclastic markers were associated with a larger number of precursors in the bone marrow and a stronger osteoclastogenic response to RANKL or TNFα. Further, IBD-free [21-3 x 283-2]F1 rats also displayed decreased total and trabecular bone density., Conclusions: HLA-B27 transgenic rats lose an increasing amount of bone density and strength with progressing age, which is primarily mediated via increased bone remodeling in favor of bone resorption. Moreover, IBD and bone loss seem to be independent features of SpA in HLA-B27 transgenic rats., (Copyright © 2015 Elsevier Inc. All rights reserved.)

In this study, we investigated the scaling relations between trabecular bone volume fraction (BV/TV) and parameters of the trabecular microstructure at different skeletal sites. Cylindrical bone samples with a diameter of 8mm were harvested from different skeletal sites of 154 human donors in vitro: 87 from the distal radius, 59/69 from the thoracic/lumbar spine, 51 from the femoral neck, and 83 from the greater trochanter. μCT images were obtained with an isotropic spatial resolution of 26μm. BV/TV and trabecular microstructure parameters (TbN, TbTh, TbSp, scaling indices (< > and σ of α and αz), and Minkowski Functionals (Surface, Curvature, Euler)) were computed for each sample. The regression coefficient β was determined for each skeletal site as the slope of a linear fit in the double-logarithmic representations of the correlations of BV/TV versus the respective microstructure parameter. Statistically significant correlation coefficients ranging from r=0.36 to r=0.97 were observed for BV/TV versus microstructure parameters, except for Curvature and Euler. The regression coefficients β were 0.19 to 0.23 (TbN), 0.21 to 0.30 (TbTh), -0.28 to -0.24 (TbSp), 0.58 to 0.71 (Surface) and 0.12 to 0.16 (<α>), 0.07 to 0.11 (<αz>), -0.44 to -0.30 (σ(α)), and -0.39 to -0.14 (σ(αz)) at the different skeletal sites. The 95% confidence intervals of β overlapped for almost all microstructure parameters at the different skeletal sites. The scaling relations were independent of vertebral fracture status and similar for subjects aged 60-69, 70-79, and >79years. In conclusion, the bone volume fraction-microstructure scaling relations showed a rather universal character., (© 2013.)

With improving clinical CT scanning technology, the accuracy of CT-based finite element (FE) models of the human skeleton may be ameliorated by an enhanced description of apparent level bone mechanical properties. Micro-finite element (μFE) modeling can be used to study the apparent elastic behavior of human cancellous bone. In this study, samples from the femur, radius and vertebral body were investigated to evaluate the predictive power of morphology-elasticity relationships and to compare them across different anatomical regions. μFE models of 701 trabecular bone cubes with a side length of 5.3 mm were analyzed using kinematic boundary conditions. Based on the FE results, four morphology-elasticity models using bone volume fraction as well as full, limited or no fabric information were calibrated for each anatomical region. The 5 parameter Zysset-Curnier model using full fabric information showed excellent predictive power with coefficients of determination ([Formula: see text]) of 0.98, 0.95 and 0.94 of the femur, radius and vertebra data, respectively, with mean total norm errors between 14 and 20%. A constant orthotropy model and a constant transverse isotropy model, where the elastic anisotropy is defined by the model parameters, yielded coefficients of determination between 0.90 and 0.98 with total norm errors between 16 and 25%. Neglecting fabric information and using an isotropic model led to [Formula: see text] between 0.73 and 0.92 with total norm errors between 38 and 49%. A comparison of the model regressions revealed minor but significant (p<0.01) differences for the fabric-elasticity model parameters calibrated for the different anatomical regions. The proposed models and identified parameters can be used in future studies to compute the apparent elastic properties of human cancellous bone for homogenized FE models.

Vertebral fractures represent a major health problem and involve the progressive collapse of trabecular bone over large compressive strains. This collapse is driven by local failure and interaction of the trabecular rod and plate elements, which translates into stress softening and densification at the material level. Current constitutive models for trabecular bone are essentially limited to infinitesimal strains. Accordingly, the aim of this work was to extend our current phenomenological model of trabecular bone (Garcia et al., 2009) for the simulation of large compressive strains by including post-yield softening and densification. A constitutive model of trabecular bone based on both volume fraction and trabecular orientation was formulated in a proper theoretical framework, implemented in commercial FE software and validated with human vertebral sections subjected to large compressive strains. As it is for infinitesimal strains, the evolution of plastic strains and damage is described by local internal variables. An isotropic softening rule was controlled by the cumulated plastic strain and a non-linear elastic spring was added to account for densification of the porous material in moderate-to-large compressive strains beyond a given threshold. To avoid convergence problems occurring as a result of softening, a consistent visco-plastic regularization approach was adopted. The experimental results for 37 vertebral sections from previous work (Dall'Ara et al., 2010) were used to validate the constitutive model for compressive loading up to 45% of the average axial deformation. This validation study showed that the model provides both qualitative predictions of damage localization on the cortex and quantitative predictions of dissipated energy (ρ(C)=0.912) of vertebral body behavior under large compressive strains. Since the evolution of the internal variables was considered in local manner, a mesh sensitivity analysis of the finite element model was conducted via two different mesh sizes and revealed that strain localization was dominated by trabecular bone heterogeneity. To our knowledge, this model is the first to simulate collapse of trabecular bone and may help improve the biomechanical understanding of several musculoskeletal conditions such as vertebral fractures or orthopedic implant migration., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Osteoporosis related vertebral fractures are an increasing clinical problem in ageing societies. The prediction of vertebral fracture load from QCT-based anatomy-specific finite element simulations could be very useful in the management of patients with osteoporosis, especially with regard to a possible fracture prevention or treatment optimisation. A key property in finite element analysis is the yield surface for the trabecular bone material. This study is aimed at identifying continuum-level yield criteria for vertebral trabecular bone using micro-finite element models subjected to uni-axial, shear, and tri-axial loading. A fabric-dependent, orthotropic Tsai-Wu yield criterion is proposed in both stress and strain spaces. Nonlinear micro-finite element models of cubic vertebral trabecular bone samples with 5.62 mm edge length were generated from μCT-scans. Kinematic boundary conditions were imposed and the specimen was loaded force controlled beyond yield in 17 different load cases (six uni-axial, three shear and eight multi-axial). The proposed yield criteria were fitted to the resulting yield data. Yield strains on-axis were significantly lower (10% in tension and 6% in compression) than in the transverse directions. Average yield strains were 0.7% in tension, 1.1% in compression, 1.0% in shear and ranged from 0.6% to 1.1% under multi-axial loading. In axial direction, maximum yield stress was 2.6 MPa in tension and 4.7 MPa in compression. Lowest shear stress was found in the transverse plane with 1.3 MPa. Multi-axial yield stresses ranged between values for uni-axial tension and compression. Yield stresses depended significantly and substantially on both volume fraction and fabric. Yield strains depended also significantly on both bone volume fraction and fabric, but only weakly on the former. The standard error of the estimate and the concordance correlation coefficient of the yield surface were 5.47% and 0.93 in strain space and 13.58% and 0.96 in stress space. The results of this study are not only consistent with experimental data from the literature but also extend the current knowledge of yield to multi-axial load cases that can hardly be realised in a biomechanical experiment. The presented yield data and criteria will help improving the prediction of vertebral ultimate load using anatomy-specific finite element models., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

At the tissue level, the local material properties of human cancellous bone are heterogeneous due to constant remodelling. Since standard high-resolution computed tomography scanning methods are unable to capture this heterogeneity in detail, local differences in mineralisation are normally not incorporated in computational models. To investigate the effects of heterogeneous mineral distribution on the apparent elastic properties, 40 cancellous bone samples from the human femoral neck were scanned by means of synchrotron radiation microcomputed tomography (SRμCT). SRμCT-based micromechanical finite element models that accounted for mineral heterogeneity were compared with homogeneous models. Evaluation of the apparent stiffness tensor of both model types revealed that homogeneous models led to a minor but significant (p < 0.05) overestimation of the elastic properties of heterogeneous models by 2.18 ± 1.89%. Variation of modelling parameters did not affect the overestimation to a great extent. It was concluded that the heterogeneous mineralisation has only a minor influence on the apparent elastic properties of human cancellous bone.

In this study, the homogenized anisotropic elastic properties of single bone lamellae are computed using a finite element unit cell method. The resulting stiffness tensor is utilized to calculate indentation moduli for multiple indentation directions in the lamella plane which are then related to nanoindentation experiments. The model accounts for different fibril orientation patterns in the lamellae--the twisted and orthogonal plywood pattern, a 5-sublayer pattern and an X-ray diffraction-based pattern. Three-dimensional sectional views of each pattern facilitate the comparison to transmission electron (TEM) images of real lamella cuts. The model results indicate, that the 5-sublayer- and the X-ray diffraction-based patterns cause the lamellae to have a stiffness maximum between 0° and 45° to the osteon axis. Their in-plane stiffness characteristics are qualitatively matching the experimental findings that report a higher stiffness in the osteon axis than in the circumferential direction. In contrast, lamellae owning the orthogonal or twisted plywood fibril orientation patterns have no preferred stiffness alignment. This work shows that the variety of fibril orientation patterns leads to qualitative and quantitative differences in the lamella elastic mechanical behavior. The study is a step toward a deeper understanding of the structure-mechanical function relationship of bone lamellae.

Although obesity traditionally has been considered a condition of low risk for osteoporosis, this classic view has recently been questioned. The aim of this study was to assess bone microarchitecture and turnover in a mouse model of high-fat diet-induced obesity. Seven-week-old male C57BL/6J mice (n = 18) were randomized into 3 diet groups. One third (n = 6) received a low-fat diet for 24 weeks, one third was kept on an extended high-fat diet (eHF), and the remaining was switched from low-fat to high-fat chow 3 weeks before sacrifice (sHF). Serum levels of insulin, leptin, adiponectin, osteocalcin, and cross-linked telopeptides of type I collagen (CTX) were measured. In addition, bone microarchitecture was analyzed by micro-computed tomography; and lumbar spine bone density was assessed by dual-energy x-ray absorptiometry. The CTX, body weight, insulin, and leptin were significantly elevated in obese animals (sHF: +48%, +24%, +265%, and +102%; eHF: +43%, +52%, +761%, and +292%). The CTX, body weight, insulin, and leptin showed a negative correlation with bone density and bone volume. Interestingly, short-term high-fat chow caused similar bone loss as extended high-fat feeding. Bone volume was decreased by 12% in sHF and 19% in eHF. Bone mineral density was 25% (sHF) and 27% (eHF) lower when compared with control mice on low-fat diet. As assessed by the structure model index, bone microarchitecture changed from plate- to rod-like appearance upon high-fat challenge. Trabecular and cortical thickness remained unaffected. Short-term and extended high-fat diet-induced obesity caused significant bone loss in male C57BL/6J mice mainly because of resorptive changes in trabecular architecture., (© 2011 Elsevier Inc. All rights reserved.)

Osteoporosis-related vertebral body fractures involve large compressive strains of trabecular bone. The small strain mechanical properties of the trabecular bone such as the elastic modulus or ultimate strength can be estimated using the volume fraction and a second order fabric tensor, but it remains unclear if similar estimations may be extended to large strain properties. Accordingly, the aim of this work is to identify the role of volume fraction and especially fabric in the large strain compressive behavior of human trabecular bone from various anatomical locations. Trabecular bone biopsies were extracted from human T12 vertebrae (n=31), distal radii (n=43), femoral head (n=44), and calcanei (n=30), scanned using microcomputed tomography to quantify bone volume fraction (BV/TV) and the fabric tensor (M), and tested either in unconfined or confined compression up to very large strains (∼70%). The mechanical parameters of the resulting stress-strain curves were analyzed using regression models to examine the respective influence of BV/TV and fabric eigenvalues. The compressive stress-strain curves demonstrated linear elasticity, yielding with hardening up to an ultimate stress, softening toward a minimum stress, and a steady rehardening followed by a rapid densification. For the pooled experiments, the average minimum stress was 1.89 ± 1.77 MPa, while the corresponding mean strain was 7.15 ± 1.84%. The minimum stress showed a weaker dependence with fabric as the elastic modulus or ultimate strength. For the confined experiments, the stress at a logarithmic strain of 1.2 was 8.08 ± 7.91 MPa, and the dissipated energy density was 5.67 ± 4.42 MPa. The latter variable was strongly related to the volume fraction (R(2)=0.83) but the correlation improved only marginally with the inclusion of fabric (R(2)=0.84). The influence of fabric on the mechanical properties of human trabecular bone decreases with increasing strain, while the role of volume fraction remains important. In particular, the ratio of the minimum versus the maximum stress, i.e., the relative amount of softening, decreases strongly with fabric, while the dissipated energy density is dominated by the volume fraction. The collected results will prove to be useful for modeling the softening and densification of the trabecular bone using the finite element method.

The key parameters determining the elastic properties of an unidirectional mineralized bone fibril-array decomposed in two further hierarchical levels are investigated using mean field methods. Modeling of the elastic properties of mineralized micro- and nanostructures requires accurate information about the underlying topology and the constituents' material properties. These input data are still afflicted by great uncertainties and their influence on computed elastic constants of a bone fibril-array remains unclear. In this work, mean field methods are applied to model mineralized fibrils, the extra-fibrillar matrix and the resulting fibril-array. The isotropic or transverse isotropic elastic constants of these constituents are computed as a function of degree of mineralization, mineral distribution between fibrils and extra-fibrillar matrix, collagen stiffness and fibril volume fraction. The linear sensitivity of the elastic constants was assessed at a default set of the above parameters. The strain ratios between the constituents as well as the axial and transverse indentation moduli of the fibril-array were calculated for comparison with experiments. Results indicate that the degree of mineralization and the collagen stiffness dominate fibril-array elasticity. Interestingly, the stiffness of the extra-fibrillar matrix has a strong influence on transverse and shear moduli of the fibril-array. The axial strain of the intra-fibrillar mineral platelets is 30-90% of the applied fibril strain, depending on mineralization and collagen stiffness. The fibril-to-fibril-array strain ratio is essentially ~1. This study provides an improved insight in the parameters, which govern the fibril-array stiffness of mineralized tissues such as bone.

Using the three-dimensional morphological data provided by computed tomography, finite element (FE) models can be generated and used to compute the stiffness and strength of whole bones. Three-dimensional constitutive laws capturing the main features of bone mechanical behavior can be developed and implemented into FE software to enable simulations on complex bone structures. For this purpose, a constitutive law is proposed, which captures the compressive behavior of trabecular bone as a porous material with accumulation of irreversible strain and loss of stiffness beyond its yield point and softening beyond its ultimate point. To account for these features, a constitutive law based on damage coupled with hardening anisotropic elastoplasticity is formulated using density and fabric-based tensors. To prevent mesh dependence of the solution, a nonlocal averaging technique is adopted. The law has been implemented into a FE software and some simple simulations are first presented to illustrate its behavior. Finally, examples dealing with compression of vertebral bodies clearly show the impact of softening on the localization of the inelastic process.

Classical finite element (FE) models can estimate vertebral stiffness and strength with much lower computational costs than muFE analyses, but the accuracy of these models rely on calibrated material properties that are not necessarily consistent with experimental results. In general, trabecular bone material properties are scaled with computer tomography (CT) density alone, without accounting for local variations in anisotropy or micro-architecture. Moreover, the cortex is often omitted or assigned with a constant thickness. In this work, voxel FE models, as well as surface-based homogenized FE models with topologically-conformed geometry and assigned with experimentally validated properties for bone, were developed from a series of 12 specimens tested up to failure. The effects of changing from a digital mesh to a smooth mesh, including a cortex of variable thickness and/or including heterogeneous trabecular fabric, were investigated. In each case, FE predictions of vertebral stiffness and strength were compared with the experimental gold-standard, and changes in elastic strain energy density and damage distributions were reported. The results showed that a smooth mesh effectively removed zones of artificial damage locations occurring in the ragged edges of the digital mesh. Adding an explicit cortex stiffened and strengthened the models, unloading the trabecular centrum while increasing the correlations to experimental stiffness and strength. Further addition of heterogeneous fabric improved the correlations to stiffness (R(2)=0.72) and strength (R(2)=0.89) and moved the damage locations closer to the vertebral endplates, following the local trabecular orientations. It was furthermore demonstrated that predictions of vertebral stiffness and strength of homogenized FE models with topologically-conformed cortical shell and heterogeneous trabecular fabric correlated well with experimental measurements, after assigning purely experimental data for bone without further calibration of material laws at the macroscale of bone. This study successfully demonstrated the limitations of current classical FE methods and provided valuable insights into the damage mechanisms of vertebral bodies.

In the context of osteoporosis, evaluation of bone fracture risk and improved design of epiphyseal bone implants rely on accurate knowledge of the mechanical properties of trabecular bone. A multi-axial loading chamber was designed, built and applied to explore the compressive multi-axial yield and strength properties of human trabecular bone from different anatomical locations. A thorough experimental protocol was elaborated for extraction of cylindrical bone samples, assessment of their morphology by micro-computed tomography and application of different mechanical tests: torsion, uni-axial traction, uni-axial compression and multi-axial compression. A total of 128 bone samples were processed through the protocol and subjected to one of the mechanical tests up to yield and failure. The elastic data were analyzed using a tensorial fabric-elasticity relationship, while the yield and strength data were analyzed with fabric-based, conewise generalized Hill criteria. For each loading mode and more importantly for the combined results, strong relationships were demonstrated between volume fraction, fabric and the elastic, yield and strength properties of human trabecular bone. Despite the reviewed limitations, the obtained results will help improve the simulation of the damage behavior of human bones and bone-implant systems using the finite element method.

Motivated by mechanical analysis of bones and bone-implant systems, a 3D constitutive law describing the macroscopic mechanical behaviour of both cortical and trabecular bone in cyclic (not fatigue) overloads is developed. The proposed model which mathematical formulation is established within the framework of generalized standard materials accounts for three distinct material evolution modes where elastic, plastic and damage aspects are closely related. The anisotropic elasticity of bone is described by a morphology-based model and distinct damage behaviour in tension and compression by a halfspacewise generalized Hill criterion. The plastic criterion is based on the intact elastic compliance tensor. The algorithm applies three distinct projections based on the relationship between the internal variables and criteria. Their respective consistent tangent operators are presented. Numerical resolutions of several boundary value problems and a biomechanical application are presented to illustrate the potential of the constitutive model and demonstrate the expected quadratic convergence of the algorithm.

This work introduces a novel method of automating the process of patient-specific finite element (FE) model development using a mapped mesh technique. The objective is to map a predefined mesh (template) of high quality directly onto a new bony surface (target) definition, thereby yielding a similar mesh with minimal user interaction. To bring the template mesh into correspondence with the target surface, a deformable registration technique based on the FE method has been adopted. The procedure has been made hierarchical allowing several levels of mesh refinement to be used, thus reducing the time required to achieve a solution. Our initial efforts have focused on the phalanx bones of the human hand. Mesh quality metrics, such as element volume and distortion were evaluated. Furthermore, the distance between the target surface and the final mapped mesh were measured. The results have satisfactorily proven the applicability of the proposed method.

High-resolution finite element models of trabecular bone can be used to study trabecular structure-function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653-1657, 1996) lead to "apparent" rather than to "effective" elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385-394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young's moduli, shear moduli, and Poisson's ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as "periodicity compatible" mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths > or =5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness.

Osteoporosis leads to an increased risk of bone fracture. While bone density and architecture can be assessed in vivo with increasing accuracy using CT and MRI, their relationship with the critical mechanical properties at various anatomical sites remain unclear. The objective of this study was to quantify the quasi-static compressive mechanical properties of human trabecular bone among different skeletal sites and compare their relationships with bone volume fraction and a measure of microstructural anisotropy called fabric. Over 600 trabecular bone samples from six skeletal sites were assessed by microCT and tested in uniaxial compression. Bone volume fraction correlated positively with elastic modulus, yield stress, ultimate stress, and the relationships depended strongly on skeletal site. The account of fabric improved these correlations substantially, especially when the data of all sites were pooled together, but the fabric-mechanical property relationships remained somewhat distinct among the anatomical sites. The study confirms that, beyond volume fraction, fabric plays an important role in determining the mechanical properties of trabecular bone and should be exploited in mechanical analysis of clinically relevant sites of the human skeleton.

Measuring the microscopic mechanical properties of bone tissue is important in support of understanding the etiology and pathogenesis of many bone diseases. Knowledge about these properties provides a context for estimating the local mechanical environment of bone related cells thait coordinate the adaptation to loads experienced at the whole organ level. The objective of this study was to determine the effects of experimental testing parameters on nanoindentation measures of lamellar-level bone mechanical properties. Specifically, we examined the effect of specimen preparation condition, indentation depth, repetitive loading, time delay, and displacement rate. The nanoindentation experiments produced measures of lamellar elastic moduli for human cortical bone (average value of 17.7 +/- 4.0 GPa for osteons and 19.3 +/- 4.7 GPa for interstitial bone tissue). In addition, the hardness measurements produced results consistent with data in the literature (average 0.52 +/- 0.15 GPa for osteons and 0.59 +/- 0.20 GPa for interstitial bone tissue). Consistent modulus values can be obtained from a 500-nm-deep indent. The results also indicated that the moduli and hardnesses of the dry specimens are significantly greater (22.6% and 56.9%, respectively) than those of the wet and wet and embedded specimens. The latter two groups were not different. The moduli obtained at a 5-nm/s loading rate were significantly lower than the values at the 10- and 20-nm/s loading rates while the 10- and 20-nm/s rates were not significantly different. The hardness measurements showed similar rate-dependent results. The preliminary results indicated that interstitial bone tissue has significantly higher modulus and hardness than osteonal bone tissue. In addition, a significant correlation between hardness and elastic modulus was observed.

In the perspective of predicting mechanical from morphological properties of human trabecular bone, the theoretical and experimental relationships between volume fraction, fabric and elastic properties were reviewed.Five data sets of human trabecular bone and two data sets of idealized cells were obtained from various investigators and analyzed statistically with one isotropic and four anisotropic models. For each model, multiple linear regressions were performed to fit the components of both the compliance and the stiffness tensors using volume fraction and in some cases fabric. The adjusted coefficients of determination of the regressions and the average relative errors of the reported versus the predicted tensor norms were calculated. The three anisotropic models that implied a log transformation of the data showed the best results. Excluding the idealized cell data, the adjusted coefficients of determination of these models ranged from 0.80 to 0.95 for the compliance and from 0.80 to 0.94 for the stiffness tensors, while the average relative errors varied between 16% and 55% for the compliance and between 25% and 62% for the stiffness data. The use of volume fraction alone in the isotropic model decreased the adjusted coefficients of determination by 0.03-0.25 and increased the average relative errors by 5-27%. This review confirms the potential of morphology-elasticity relationships for estimation of elastic properties of human trabecular bone using peripheral quantitative computed tomography or magnetic resonance imaging, but emphasizes the need for standardized measurements of mechanical properties at both continuum and tissue level.