The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone

Limit analysis of human proximal femur

A limit analysis numerical approach oriented to predict the peak/collapse load of human proximal femur, under two different loading conditions, is presented. A yield criterion of Tsai-Hu-type, expressed in principal stress space, is used to model the orthotropic bone tissues. A simplified human femur 3D model is envisaged to carry on numerical simulation of in-vitro tests borrowed from the relevant literature and to reproduce their findings. A critical discussion, together with possible future developments, is presented.

Biomechanical evaluation of total ankle arthroplasty. Part II: Influence of loading and fixation design on tibial bone-implant interaction

Finite element (FE) models to evaluate the burden placed on the interaction between total ankle arthroplasty (TAA) implants and the bone often rely on peak axial forces. However, the loading environment of the ankle is complex, and it is unclear whether peak axial forces represent a challenging scenario for the interaction between the implant and the bone. Our goal was to determine how the loads and the design of the fixation of the tibial component of TAA impact the interaction between the implant and the bone. To this end, we developed a framework that integrated robotic cadaveric simulations to determine the ankle kinematics, musculoskeletal models to determine the ankle joint loads, and FE models to evaluate the interaction between TAA and the bone. We compared the bone-implant micromotion and the risk of bone failure of three common fixation designs for the tibial component of TAA: spikes, a stem, and a keel. We found that the most critical conditions for the interaction between the implant and the bone were dependent on the specimen and the fixation design, but always involved submaximal forces and large moments. We also found that while the fixation design influenced the distribution and the peak value of bone-implant micromotion, the amount of bone at risk of failure was specimen dependent. To account for the most critical conditions for the interaction between the implant and the bone, our results support simulating multiple specimens under complex loading profiles that include multiaxial moments and span entire activity cycles.

Differing trabecular bone architecture in dinosaurs and mammals contribute to stiffness and limits on bone strain

The largest dinosaurs were enormous animals whose body mass placed massive gravitational loads on their skeleton. Previous studies investigated dinosaurian bone strength and biomechanics, but the relationships between dinosaurian trabecular bone architecture and mechanical behavior has not been studied. In this study, trabecular bone samples from the distal femur and proximal tibia of dinosaurs ranging in body mass from 23-8,000 kg were investigated. The trabecular architecture was quantified from micro-computed tomography scans and allometric scaling relationships were used to determine how the trabecular bone architectural indices changed with body mass. Trabecular bone mechanical behavior was investigated by finite element modeling. It was found that dinosaurian trabecular bone volume fraction is positively correlated with body mass similar to what is observed for extant mammalian species, while trabecular spacing, number, and connectivity density in dinosaurs is negatively correlated with body mass, exhibiting opposite behavior from extant mammals. Furthermore, it was found that trabecular bone apparent modulus is positively correlated with body mass in dinosaurian species, while no correlation was observed for mammalian species. Additionally, trabecular bone tensile and compressive principal strains were not correlated with body mass in mammalian or dinosaurian species. Trabecular bone apparent modulus was positively correlated with trabecular spacing in mammals and positively correlated with connectivity density in dinosaurs, but these differential architectural effects on trabecular bone apparent modulus limit average trabecular bone tissue strains to below 3,000 microstrain for estimated high levels of physiological loading in both mammals and dinosaurs.

Efficient materially nonlinear [Formula: see text]FE solver for simulations of trabecular bone failure

An efficient solver for large-scale linear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \hbox {FE}$$\end{document}μFE 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${36}\,{{\upmu }\hbox {m}}$$\end{document}36μm. 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 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2 > 0.97$$\end{document}R2>0.97). 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.

An explicit micro-FE approach to investigate the post-yield behaviour of trabecular bone under large deformations

Homogenised finite element (FE) analyses are able to predict osteoporosis-related bone fractures and become useful for clinical applications. The predictions of FE analyses depend on the apparent, heterogeneous, anisotropic, elastic, and yield material properties, which are typically determined by implicit micro-FE (μFE) analyses of trabecular bone. The objective of this study is to explore an explicit μFE approach to determine the apparent post-yield behaviour of trabecular bone, beyond the elastic and yield properties. The material behaviour of bone tissue was described by elasto-plasticity with a von Mises yield criterion closed by a planar cap for positive hydrostatic stresses to distinguish the post-yield behaviour in tension and compression. Two ultimate strains for tension and compression were calibrated to trigger element deletion and reproduce damage of trabecular bone. A convergence analysis was undertaken to assess the role of the mesh. Thirteen load cases using periodicity-compatible mixed uniform boundary conditions were applied to three human trabecular bone samples of increasing volume fractions. The effect of densification in large strains was explored. The convergence study revealed a strong dependence of the apparent ultimate stresses and strains on element size. An apparent quadric strength surface for trabecular bone was successfully fitted in a normalised stress space. The effect of densification was reproduced and correlated well with former experimental results. This study demonstrates the potential of the explicit FE formulation and the element deletion technique to reproduce damage in trabecular bone using μFE analyses. The proper account of the mesh sensitivity remains challenging for practical computing times.

The Multi-Axial Failure Response of Porcine Trabecular Skull Bone Estimated Using Microstructural Simulations

The development of a multi-axial failure criterion for trabecular skull bone has many clinical and biological implications. This failure criterion would allow for modeling of bone under daily loading scenarios that typically are multi-axial in nature. Some yield criteria have been developed to evaluate the failure of trabecular bone, but there is a little consensus among them. To help gain deeper understanding of multi-axial failure response of trabecular skull bone, we developed 30 microstructural finite element models of porous porcine skull bone and subjected them to multi-axial displacement loading simulations that spanned three-dimensional (3D) stress and strain space. High-resolution microcomputed tomography (microCT) scans of porcine trabecular bone were obtained and used to develop the meshes used for finite element simulations. In total, 376 unique multi-axial loading cases were simulated for each of the 30 microstructure models. Then, results from the total of 11,280 simulations (approximately 135,360 central processing unit-hours) were used to develop a mathematical expression, which describes the average three-dimensional yield surface in strain space. Our results indicate that the yield strain of porcine trabecular bone under multi-axial loading is nearly isotropic and despite a spread of yielding points between the 30 different microstructures, no significant relationship between the yield strain and bone volume fraction is observed. The proposed yield equation has simple format and it can be implemented into a macroscopic model for the prediction of failure of whole bones.

Assessing bone quality through mechanical properties in postmenopausal trabecular bone

BACKGROUND

The inner structure of trabecular bone is a result of structural optimization provided by remodeling processes. Changes in hormonal status related to menopause cause bone tissue loss and micro-architectural deterioration with a consequent susceptibility to fracture. Accumulation of micro-damage in bone, as a function of the rate of production and rate of repair, underlies the development of stress fractures, increasing fragility associated to age and osteoporosis, especially in transmenopausal women.

PATIENTS AND METHODS

Quasi-static and nano-dynamic mechanical characterization were undertaken in trabecular bone from femoral neck biopsies of postmenopausal women. AFM (Atomic Force Microscopy) complementary studies were performed to determine nano-roughness (SRa) and the fibrils width of collagen. Nanoindentations were used to quantify transmenopausal changes in intrinsic mechanical properties of trabecular bone: hardness (Hi), modulus of Young (Ei), complex modulus (E*), tan delta (δ), storage modulus (E') and loss modulus (E").

RESULTS

As result of the quasi-static measurements, 0.149 (0.036) GPa and 2.95 (0.73) GPa of Hi and Ei were obtained, respectively. As result of the nano-dynamic measurements, 17.94 (3.15), 0.62 (0.10), 13.79 (3.21 and 6.39 (1.28) GPa of E*, tan (δ), E' and E" were achieved, respectively. 101.07 SRa and 831.28 nm of fibrils width were additionally obtained.

CONCLUSIONS

This study poses a first approach to the measurement of bone quality in postmenopausal trabecular bone by combining quasi-static, nano-DMA analysis and tribology of dentin surface through AFM characterization.

Combining specimen-specific finite-element models and optimization in cortical-bone material characterization improves prediction accuracy in three-point bending tests

Although the beam theory is widely used for calculating material parameters in three-point bending test, it cannot accurately describe the biomechanical properties of specimens after the yield. Hence, we propose a finite element (FE) based optimization method to obtain accurate bone material parameters from three-point bending test. We tested 80 machined bovine cortical bone specimens at both longitudinal and transverse directions using three-point bending. We then adopted the beam theory and the FE-based optimization method combined with specimen-specific FE models to derive the material parameters of cortical bone. We compared data obtained using these two methods and further evaluated two groups of parameters with three-point bending simulations. Our data indicated that the FE models with material properties from the FE-based optimization method showed best agreements with experimental data for the entire force-displacement responses, including the post-yield region. Using the beam theory, the yield stresses derived from 0.0058% strain offset for the longitudinal specimen and 0.0052% strain offset for the transverse specimen are closer to those derived from the FE-based optimization method, compared to yield stresses calculated without strain offset. In brief, we conclude that the optimization FE method is more appropriate than the traditional beam theory in identifying the material parameters of cortical bone for improving prediction accuracy in three-point bending mode. Given that the beam theory remains as a popular method because of its efficiency, we further provided correction functions to adjust parameters calculated from the beam theory for accurate FE simulation.

Using Non-linear Homogenization to Improve the Performance of Macroscopic Damage Models of Trabecular Bone

Realistic macro-level finite element simulations of the mechanical behavior of trabecular bone, a cellular anisotropic material, require a suitable constitutive model; a model that incorporates the mechanical response of bone for complex loading scenarios and includes post-elastic phenomena, such as plasticity (permanent deformations) and damage (permanent stiffness reduction), which bone is likely to experience. Some such models have been developed by conducting homogenization-based multiscale finite element simulations on bone micro-structure. While homogenization has been fairly successful in the elastic regime and, to some extent, in modeling the macroscopic plastic response, it has remained a challenge with respect to modeling damage. This study uses a homogenization scheme to upscale the damage behavior from the tissue level (microscale) to the organ level (macroscale) and assesses the suitability of different damage constitutive laws. Ten cubic specimens were each subjected to 21 strain-controlled load cases for a small range of macroscopic post-elastic strains. Isotropic and anisotropic criteria were considered, density and fabric relationships were used in the formulation of the damage law, and a combined isotropic/anisotropic law with tension/compression asymmetry was formulated, based on the homogenized results, as a possible alternative to the currently used single scalar damage criterion. This computational study enhances the current knowledge on the macroscopic damage behavior of trabecular bone. By developing relationships of damage progression with bone's micro-architectural indices (density and fabric) the study also provides an aid for the creation of more precise macroscale continuum models, which are likely to improve clinical predictions.

MicroCT-based finite element models as a tool for virtual testing of cortical bone

The aim of this study was to assess a virtual biomechanics testing approach purely based on microcomputed tomography (microCT or µCT) data, providing non-invasive methods for determining the stiffness and strength of cortical bone. Mouse femurs were µCT scanned prior to three-point-bend tests. Then microCT-based finite element models were generated with spatial variation in bone elastoplastic properties and subject-specific femur geometries. Empirical relationships of density versus Young's moduli and yield stress were used in assigning elastoplastic properties to each voxel. The microCT-based finite element modeling (µFEM) results were employed to investigate the model's accuracy through comparison with experimental tests. The correspondence of elastic stiffness and strength from the µFE analyses and tests was good. The interpretation of the derived data showed a 6.1%, 1.4%, 1.5%, and 1.6% difference between the experimental test result and µFEM output on global stiffness, nominal Young's modulus, nominal yield stress, and yield force, respectively. We conclude that virtual testing outputs could be used to predict global elastic-plastic properties and may reduce the cost, time, and number of test specimens in performing physical experiments.

Comparison of the Lag Screw Placements for the Treatment of Stable and Unstable Intertrochanteric Femoral Fractures regarding Trabecular Bone Failure

Background. In this study, the cut-out risk of Dynamic Hip Screw (DHS) was investigated in nine different positions of the lag screw for two fracture types by using Finite Element Analysis (FEA). Methods. Two types of fractures (31-A1.1 and A2.1 in AO classification) were generated in the femur model obtained from Computerized Tomography images. The DHS model was placed into the fractured femur model in nine different positions. Tip-Apex Distances were measured using SolidWorks. In FEA, the force applied to the femoral head was determined according to the maximum value being observed during walking. Results. The highest volume percentage exceeding the yield strength of trabecular bone was obtained in posterior-inferior region in both fracture types. The best placement region for the lag screw was found in the middle of both fracture types. There are compatible results between Tip-Apex Distances and the cut-out risk except for posterior-superior and superior region of 31-A2.1 fracture type. Conclusion. The position of the lag screw affects the risk of cut-out significantly. Also, Tip-Apex Distance is a good predictor of the cut-out risk. All in all, we can supposedly say that the density distribution of the trabecular bone is a more efficient factor compared to the positions of lag screw in the cut-out risk.

Nonlinear homogenisation of trabecular bone: Effect of solid phase constitutive model

Micro-finite element models have been extensively employed to evaluate the elastic properties of trabecular bone and, to a limited extent, its yield behaviour. The macroscopic stiffness tensor and yield surface are of special interest since they are essential in the prediction of bone strength and stability of implants at the whole bone level. While macroscopic elastic properties are now well understood, yield and post-yield properties are not. The aim of this study is to shed some light on what the effect of the solid phase yield criterion is on the macroscopic yield of trabecular bone for samples with different microstructure. Three samples with very different density were subjected to a large set of apparent load cases (which is important since physiological loading is complex and can have multiple components in stress or strain space) with two different solid phase yield criteria: Drucker-Prager and eccentric-ellipsoid. The study found that these two criteria led to small differences in the macroscopic yield strains for most load cases except for those that were compression-dominated; in these load cases, the yield strains for the Drucker-Prager criterion were significantly higher. Higher density samples resulted in higher differences between the two criteria. This work provides a comprehensive assessment of the effect of two different solid phase yield criteria on the macroscopic yield strains of trabecular bone, for a wide range of load cases, and for samples with different morphology.

Damage in total knee replacements from mechanical overload

The mechanical loads acting across the knee joint following total knee replacements (TKR) during activities of daily living have recently been measured using instrumented TKRs. Using a series of postmortem retrieved TKR constructs we investigated whether these mechanical loads could result in damage to the implant bone interface or supporting bone in the tibia. Eighteen cemented en bloc tibial components (0 to 22 years in service) were loaded under axial compression in increments from 1 to 10 times body weight and digital image correlation was used to measure bone strain and interface micromotion during loading and unloading. Failure was considered to occur when micromotion exceeded 150µm or compressive bone strain exceeded 7300με. The results show that all retrieved specimens had sufficient bone strength to support most activities of daily living, but ~40% would be at risk under larger physiologic loads that might occur secondary to a higher impacts such as jogging or a stumble. The tray-bone micromotion (regression model R(2)=0.48, p=0.025) was greater for donors with lower age at implantation (p=0.0092). Proximal bone strain (model R(2)=0.46, p=0.03) was greater for donors with longer time in service (p=0.021). Distal bone strain (model R(2)=0.58, p=0.005) was greater for donors with more time in service (p=0.0054) and lower peri-implant BMD (p=0.049). High mechanical overload of a single or repetitive nature may be an initiating factor in aseptic loosening of total joint arthroplasties and should be avoided in order to prolong the life of the implant.

In silico investigation of vertebroplasty as a stand-alone treatment for vertebral burst fractures

BACKGROUND

The use of percutaneous vertebroplasty as a stand-alone treatment for stable vertebral burst fractures has been investigated in vitro and in clinical studies. These studies present inconsistent results on the mechanical response of vertebroplasty-treated burst fractures. In addition, observations of the loss of sagittal alignment after vertebroplasty raise questions on the applicability of vertebroplasty for burst fractures. Therefore, the aim of this study was to investigate the mechanical stability of burst fractures after stand-alone treatment by vertebroplasty.

METHODS

Finite element simulations were performed with models generated from two laboratory-induced burst fractures in human thoracolumbar specimens. The burst fracture models were virtually injected with various cement volumes using a unipedicular or bipedicular approach. The models were subjected to four individual loads (compression, lateral bending, extension and torsion) and a multi-axial load case in the physiological range.

FINDINGS

All treated burst fractures showed improvements in stiffness and a reduction in inter-fragmentary displacements, thus potentially providing a suitable mechanical environment for fracture healing. However, large volumes of the trabecular bone (<43%), cement (<53%) and bone-cement composite (<58%) were predicted to experience strain levels exceeding the yield point. While damage was not specifically modeled, this implies a potential collapse of the treated vertebra due to local failure.

INTERPRETATION

To improve the primary stability and to prevent the collapse of treated burst fractures, the use of posterior instrumentation is suggested as an adjunct to vertebroplasty.

Bone density and anisotropy affect periprosthetic cement and bone stresses after anatomical glenoid replacement: A micro finite element analysis

Glenoid loosening is still a main complication for shoulder arthroplasty. We hypothesize that cement and bone stresses potentially leading to fixation failure are related not only to glenohumeral conformity, fixation design or eccentric loading, but also to bone volume fraction, cortical thickness and degree of anisotropy in the glenoid. In this study, periprosthetic bone and cement stresses were computed with micro finite element models of the replaced glenoid depicting realistic bone microstructure. These models were used to quantify potential effects of bone microstructural parameters under loading conditions simulating different levels of glenohumeral conformity and eccentric loading simulating glenohumeral instability. Results show that peak cement stresses were achieved near the cement-bone interface in all loading schemes. Higher stresses within trabecular bone tissue and cement mantle were obtained within specimens of lower bone volume fraction and in regions of low anisotropy, increasing with decreasing glenohumeral conformity and reaching their maxima below the keeled design when the load is shifted superiorly. Our analyses confirm the combined influences of eccentric load shifts with reduced bone volume fraction and anisotropy on increasing periprosthetic stresses. They finally suggest that improving fixation of glenoid replacements must reduce internal cement and bone tissue stresses, in particular in glenoids of low bone density and heterogeneity.

The quartic piecewise-linear criterion for the multiaxial yield behavior of human trabecular bone

Prior multiaxial strength studies on trabecular bone have either not addressed large variations in bone volume fraction and microarchitecture, or have not addressed the full range of multiaxial stress states. Addressing these limitations, we utilized micro-computed tomography (lCT) based nonlinear finite element analysis to investigate the complete 3D multiaxial failure behavior of ten specimens (5mm cube) of human trabecular bone, taken from three anatomic sites and spanning a wide range of bone volume fraction (0.09–0.36),mechanical anisotropy (range of E3/E1¼3.0–12.0), and microarchitecture. We found that most of the observed variation in multiaxial strength behavior could be accounted for by normalizing the multiaxial strength by specimen-specific values of uniaxial strength (tension,compression in the longitudinal and transverse directions). Scatter between specimens was reduced further when the normalized multiaxial strength was described in strain space.The resulting multiaxial failure envelope in this normalized-strain space had a rectangular boxlike shape for normal–normal loading and either a rhomboidal box like shape or a triangular shape for normal-shear loading, depending on the loading direction. The finite element data were well described by a single quartic yield criterion in the 6D normalized strain space combined with a piecewise linear yield criterion in two planes for normalshear loading (mean error SD: 4.660.8% for the finite element data versus the criterion).This multiaxial yield criterion in normalized-strain space can be used to describe the complete 3D multiaxial failure behavior of human trabecular bone across a wide range of bone volume fraction, mechanical anisotropy, and microarchitecture.

Effect of boundary conditions on yield properties of human femoral trabecular bone

Trabecular bone plays an important mechanical role in bone fractures and implant stability. Homogenized nonlinear finite element (FE) analysis of whole bones can deliver improved fracture risk and implant loosening assessment. Such simulations require the knowledge of mechanical properties such as an appropriate yield behavior and criterion for trabecular bone. Identification of a complete yield surface is extremely difficult experimentally but can be achieved in silico by using micro-FE analysis on cubical trabecular volume elements. Nevertheless, the influence of the boundary conditions (BCs), which are applied to such volume elements, on the obtained yield properties remains unknown. Therefore, this study compared homogenized yield properties along 17 load cases of 126 human femoral trabecular cubic specimens computed with classical kinematic uniform BCs (KUBCs) and a new set of mixed uniform BCs, namely periodicity-compatible mixed uniform BCs (PMUBCs). In stress space, PMUBCs lead to 7-72 % lower yield stresses compared to KUBCs. The yield surfaces obtained with both KUBCs and PMUBCs demonstrate a pressure-sensitive ellipsoidal shape. A volume fraction and fabric-based quadric yield function successfully fitted the yield surfaces of both BCs with a correlation coefficient [Formula: see text]. As expected, yield strains show only a weak dependency on bone volume fraction and fabric. The role of the two BCs in homogenized FE analysis of whole bones will need to be investigated and validated with experimental results at the whole bone level in future studies.

Numerical Methodology to Evaluate the Effects of Bone Density and Cement Augmentation on Fixation Stiffness of Bone-Anchoring Devices

Bone quality is one of the reported factors influencing the success of bone anchors in arthroscopic repairs of torn rotator cuffs at the shoulder. This work was aimed at developing refined numerical methods to investigate how bone quality can influence the fixation stiffness of bone anchors. To do that bone biopsies were scanned at 26-μm resolution with a high-resolution microcomputer tomography (micro-CT) scanner and their images were processed for virtual implantation of a typical design of bone anchor. These were converted to microfinite element (μFE) and homogenized classical FE models, and analyses were performed to simulate pulling on the bone anchor with and without cement augmentation. Quantification of structural stiffness for each implanted specimen was then computed, as well as stress distributions within the bone structures, and related to the bone volume fraction of the specimens. Results show that the classical method is excellently correlated to structural predictions of the more refined μFE method, despite the qualitative differences in local stresses in the bone surrounding the implant. Predictions from additional loading cases suggest that structural fixation stiffness in various directions is related to apparent bone density of the surrounding bone. Augmentation of anchoring with bone cement stiffens the fixation and alters these relations. This work showed the usability of homogenized FE (hFE) in the evaluation of bone anchor fixation and will be used to develop new methodologies for virtual investigations leading to optimized repairs of rotator cuff and glenoid Bankart lesions.

ASSESSMENT OF OSTEOPOROTIC FEMORAL FRACTURE RISK: FINITE ELEMENT METHOD AS A POTENTIAL REPLACEMENT FOR CURRENT CLINICAL TECHNIQUES

Femoral fracture risk prediction is a necessary step preceding effective pharmacological intervention or pre-operative planning. Current clinical methods for fracture risk prediction rely on 2D imaging methods and have limited predictive value. Researchers are therefore trying to find improved methods for fracture prediction. During last few decades, many studies have focused on integration of 3D imaging techniques and the finite element (FE) method to improve the accuracy of fracture assessment techniques. In this paper, we review the recent advances in FE and other techniques for predicting the risk of femoral fractures. Based on a number of selected studies, the different steps that are involved in generation of patient-specific FE models are reviewed with particular emphasis on the fracture criteria. The inaccuracies that might arise due to the imperfections of the involved steps are also discussed. It is concluded that compared to image- and geometry-based techniques, FE is a more promising approach for prediction of fracture loads. However, certain technological advancements in FE modeling protocols are required before FE modeling can be recruited in clinical settings.

The sensitivity of nonlinear computational models of trabecular bone to tissue level constitutive model

Microarchitectural finite element models have become a key tool in the analysis of trabecular bone. Robust, accurate, and validated constitutive models would enhance confidence in predictive applications of these models and in their usefulness as accurate assays of tissue properties. Human trabecular bone specimens from the femoral neck (n = 3), greater trochanter (n = 6), and lumbar vertebra (n = 1) of eight different donors were scanned by μ-CT and converted to voxel-based finite element models. Unconfined uniaxial compression and shear loading were simulated for each of three different constitutive models: a principal strain-based model, Drucker-Lode, and Drucker-Prager. The latter was applied with both infinitesimal and finite kinematics. Apparent yield strains exhibited minimal dependence on the constitutive model, differing by at most 16.1%, with the kinematic formulation being influential in compression loading. At the tissue level, the quantities and locations of yielded tissue were insensitive to the constitutive model, with the exception of the Drucker-Lode model, suggesting that correlation of microdamage with computational models does not improve the ability to discriminate between constitutive laws. Taken together, it is unlikely that a tissue constitutive model can be fully validated from apparent-level experiments alone, as the calculations are too insensitive to identify differences in the outcomes. Rather, any asymmetric criterion with a valid yield surface will likely be suitable for most trabecular bone models.

Maximum principal strain as a criterion for prediction of orthodontic mini-implants failure in subject-specific finite element models

OBJECTIVE

To investigate the most reliable stress or strain parameters in subject-specific finite element (FE) models to predict success or failure of orthodontic mini-implants (OMIs).

MATERIALS AND METHODS

Subject-specific FE analysis was applied to 28 OMIs used for anchorage. Each model was developed using two computed tomography data sets, the first taken before OMI placement and the second taken immediately after placement. Of the 28 OMIs, 6 failed during the first 5 months, and 22 were successful. The bone compartment was divided into four zones in the FE models, and peak stress and strain parameters were calculated for each. Logistic regression of the failure (vs success) of OMIs on the stress and strain parameters in the models was conducted to verify the ability of these parameters to predict OMI failure.

RESULTS

Failure was significantly dependent on principal strain parameters rather than stress parameters. Peak maximum principal strain in the bone 0.5 to 1 mm from the OMI surface was the best predictor of failure (R(2) = 0.8151).

CONCLUSIONS

We propose the use of the maximum principal strain as a criterion for predicting OMI failure in FE models.

Biaxial normal strength behavior in the axial-transverse plane for human trabecular bone--effects of bone volume fraction, microarchitecture, and anisotropy

The biaxial failure behavior of the human trabecular bone, which has potential relevance both for fall and gait loading conditions, is not well understood, particularly for low-density bone, which can display considerable mechanical anisotropy. Addressing this issue, we investigated the biaxial normal strength behavior and the underlying failure mechanisms for human trabecular bone displaying a wide range of bone volume fraction (0.06-0.34) and elastic anisotropy. Micro-computed tomography (CT)-based nonlinear finite element analysis was used to simulate biaxial failure in 15 specimens (5 mm cubes), spanning the complete biaxial normal stress failure space in the axial-transverse plane. The specimens, treated as approximately transversely isotropic, were loaded in the principal material orientation. We found that the biaxial stress yield surface was well characterized by the superposition of two ellipses--one each for yield failure in the longitudinal and transverse loading directions--and the size, shape, and orientation of which depended on bone volume fraction and elastic anisotropy. However, when normalized by the uniaxial tensile and compressive strengths in the longitudinal and transverse directions, all of which depended on bone volume fraction, microarchitecture, and mechanical anisotropy, the resulting normalized biaxial strength behavior was well described by a single pair of (longitudinal and transverse) ellipses, with little interspecimen variation. Taken together, these results indicate that the role of bone volume fraction, microarchitecture, and mechanical anisotropy is mostly accounted for in determining the uniaxial strength behavior and the effect of these parameters on the axial-transverse biaxial normal strength behavior per se is minor.

The importance of intrinsic damage properties to bone fragility: a finite element study

As the average age of the population has increased, the incidence of age-related bone fracture has also increased. While some of the increase of fracture incidence with age is related to loss of bone mass, a significant part of the risk is unexplained and may be caused by changes in intrinsic material properties of the hard tissue. This investigation focused on understanding how changes to the intrinsic damage properties affect bone fragility. We hypothesized that the intrinsic (μm) damage properties of bone tissue strongly and nonlinearly affect mechanical behavior at the apparent (whole tissue, cm) level. The importance of intrinsic properties on the apparent level behavior of trabecular bone tissue was investigated using voxel based finite element analysis. Trabecular bone cores from human T12 vertebrae were scanned using microcomputed tomography (μCT) and the images used to build nonlinear finite element models. Isotropic and initially homogenous material properties were used for all elements. The elastic modulus (E(i)) of individual elements was reduced with a secant damage rule relating only principal tensile tissue strain to modulus damage. Apparent level resistance to fracture as a function of changes in the intrinsic damage properties was measured using the mechanical energy to failure per unit volume (apparent toughness modulus, W(a)) and the apparent yield strength (σ(ay), calculated using the 0.2% offset). Intrinsic damage properties had a profound nonlinear effect on the apparent tissue level mechanical response. Intrinsic level failure occurs prior to apparent yield strength (σ(ay)). Apparent yield strength (σ(ay)) and toughness vary strongly (1200% and 400%, respectively) with relatively small changes in the intrinsic damage behavior. The range of apparent maximum stresses predicted by the models was consistent with those measured experimentally for these trabecular bone cores from the experimental axial compressive loading (experimental: σ(max) = 3.0-4.3 MPa; modeling: σ(max) = 2-16 MPa). This finding differs significantly from previous studies based on nondamaging intrinsic material models. Further observations were that this intrinsic damage model reproduced important experimental apparent level behaviors including softening after peak load, microdamage accumulation before apparent yield (0.2% offset), unload softening, and sensitivity of the apparent level mechanical properties to variability of the intrinsic properties.

Finite element analysis and computed tomography based structural rigidity analysis of rat tibia with simulated lytic defects

Finite element analysis (FEA), CT based structural rigidity analysis (CTRA) and mechanical testing is performed to assess the compressive failure load of rat tibia with simulated lytic defects. Twenty rat tibia were randomly assigned to four equal groups (n=5). Three of the groups included a simulated defect at various locations: anterior bone surface (Group 1), posterior bone surface (Group 2) and through bone defect (Group 3). The fourth group was a control group with no defect (Group 4). Microcomputed tomography was used to assess bone structural rigidity properties and to provide 3D model data for generation of the finite element models for each specimen. Compressive failure load calculated using CT derived rigidity parameters (FCTRA) was well correlated to failure load recorded in mechanical testing (R(2)=0.96). The relationships between mechanical testing failure load and the axial rigidity (R(2)=0.61), bending rigidity (R(2)=0.71) and FEA calculated failure loads, considering bone as an elastic isotropic (R(2)=0.75) and elastic transversely isotropic (R(2)=0.90) are also well correlated. CTRA stress, calculated adjacent to the defect, were also shown to be well correlated with yield stresses calculated using the minimum density at the weakest cross section (R(2)=0.72). No statistically significant relationship between apparent density and mechanical testing failure load was found (P=0.37). In summary, the results of this study indicate that CTRA analysis of bone strength correlates well with both FEA and results obtained from compression experiments. In addition there exist a good correlation between structural rigidity parameters and experimental failure loads. In contrast, there was no correlation between average bone density and failure load.

Towards patient-specific material modeling of trabecular bone post-yield behavior

Bone diseases such as osteoporosis are one of the main causes of bone fracture and often result in hospitalization and long recovery periods. Researchers are aiming to develop new tools that consider the multiple determinants acting at the different scales of bone, and which can be used to clinically estimate patient-specific fracture risk and also assess the efficacy of new therapies. The main step towards this goal is a deep understanding of the bone organ, and is achieved by modeling the complexity of the structure and the high variability of the mechanical outcome. This review uses a hierarchical approach to evaluate bone mechanics at the macroscale, microscale, and nanoscale levels and the interactions between scales. The first section analyzes the experimental evidence of bone mechanics in the elastic and inelastic regions, microdamage generation, and post-yield toughening mechanisms from the organ level to the ultrastructural level. On the basis of these observations, the second section provides an overview of the constitutive models available to describe bone mechanics and predict patient-specific outcomes. Overall, the role of the hierarchical structure of bone and the interplay between each level is highlighted, and their effect is evaluated in terms of modeling biological variability and patient specificity.

DAMAGE INITIATION AND PROPAGATION UNTIL FAILURE OF CEMENT–BONE INTERFACE BY THE ELEMENT-FAILURE METHOD

Aseptic loosening of cemented hip implants frequently indicates failure by shear or tension of the cement–bone interface. An element-failure algorithm is proposed for the simulation of the mechanical behavior of the cement–bone interface until failure, under tensile and shear loading. To validate the model proposed, analysis on the cement–bone interface properties (elastic modulus, strength and thickness) and failure criteria, as well as mesh convergence analysis, were performed. Numerical results showed that the proposed element-failure method is able to determine the initiation and progression of interface failure in cemented hip replacements.

TRANSVERSE ISOTROPIC ELASTIC PROPERTIES OF VERTEBRAL TRABECULAR BONE MATRIX MEASURED USING MICROINDENTATION UNDER DRY CONDITIONS (EFFECTS OF AGE, GENDER, AND VERTEBRAL LEVEL)

The mechanical properties of bone extracellular matrix have become of increasing interest for the understanding of vertebral fracture risk. Depth-sensing indentation techniques allow the measurement of directional elastic properties of trabecular bone ex vivo with a high spatial resolution. Transverse isotropic elastic properties of vertebral trabecular bone obtained from two orthogonal directions were investigated using microindentation under dry conditions focusing on the influence of microanatomical location, age, gender, vertebral level, and anatomic direction on these properties. Biopsies were obtained from 104 human vertebrae (T1–L3) with a median age of 65 (21–94) years. Significantly, higher indentation moduli were found for indentations on axial than on transverse cross-sections of trabeculae (p < 0.01). Indentation moduli in the core were 1.05 to 1.12 times higher than in the periphery (p < 0.01). No difference in stiffness could be detected between males and females (p > 0.05) and different ages (p > 0.5). Vertebral level showed a weak correlation (p = 0.073, r2 ≈ 0.17). These results provide insights in the transverse isotropic properties of trabecular bone matrix related to age, gender, microanatomical location, and anatomic direction for a broad spectrum of human vertebrae.

Implementation of asymmetric yielding in case-specific finite element models improves the prediction of femoral fractures

Although asymmetric yielding in bone is widely shown in experimental studies, previous case-specific non-linear finite element (FE) studies have mainly adopted material behaviour using the Von Mises yield criterion (VMYC), assuming equal bone strength in tension and compression. In this study, it was verified that asymmetric yielding in FE models can be captured using the Drucker-Prager yield criterion (DPYC), and can provide better results than simulations using the VMYC. A sensitivity analysis on parameters defining the DPYC (i.e. the degree of yield asymmetry and the yield stress settings) was performed, focusing on the effect on bone failure. In this study, the implementation of a larger degree of yield asymmetry improved the prediction of the fracture location; variations in the yield stress mainly affected the predicted failure force. We conclude that the implementation of asymmetric yielding in case-specific FE models improves the prediction of femoral bone strength.

Multiscale modelling and nonlinear finite element analysis as clinical tools for the assessment of fracture risk

The risk of osteoporotic fractures is currently estimated based on an assessment of bone mass as measured by dual-energy X-ray absorptiometry. However, patient-specific finite element (FE) simulations that include information from multiple scales have the potential to allow more accurate prognosis. In the past, FE models of bone were limited either in resolution or to the linearization of the mechanical behaviour. Now, nonlinear, high-resolution simulations including the bone microstructure have been made possible by recent advances in simulation methods, computer infrastructure and imaging, allowing the implementation of multiscale modelling schemes. For example, the mechanical loads generated in the musculoskeletal system define the boundary conditions for organ-level, continuum-based FE models, whose nonlinear material properties are derived from microstructural information. Similarly microstructure models include tissue-level information such as the dynamic behaviour of collagen by modifying the model's constitutive law. This multiscale approach to modelling the mechanics of bone allows a more accurate characterization of bone fracture behaviour. Furthermore, such models could also include the effects of ageing, osteoporosis and drug treatment. Here we present the current state of the art for multiscale modelling and assess its potential to better predict an individual's risk of fracture in a clinical setting.

Valid micro finite element models of vertebral trabecular bone can be obtained using tissue properties measured with nanoindentation under wet conditions

Osteoporosis-related vertebral fractures represent a major public health problem. Anatomy-specific CT-based finite element (FE) simulations could help in identifying which vertebrae have the highest risk of fracture and thus help to decide upon the need for vertebroplasty or other surgical intervention. Continuum level FE simulations require effective macroscopic material properties of the vertebra. Micro finite element (microFE) models can be used to circumvent the difficult experiments that are necessary to determine these effective properties. From a quantitative point of view, these microFE models depend critically on the chosen trabecular tissue properties. The question remains whether linear elastic microFE models of vertebral trabecular bone with and without specimen-specific tissue properties yield similar results as non-destructive macroscopic experiments under moist conditions. microFE models were set up from microCT scans with specimen-specific or average tissue moduli measured by nanoindentation under dry and wet testing conditions. Non-destructive macroscopic mechanical compression, tension and torsion tests were performed. Experimentally obtained and simulated apparent stiffnesses were compared. No significant difference was found when comparing microFE simulations with wet tissue properties and experiments for tension, compression and torsion (p>0.05). Concordance correlation coefficients were high for tension and compression (r(c)(wet)>or=0.96,p<0.05) but moderate for torsion (r(c)(wet)=0.81,p<0.05). The agreement between simulation and experiment was confirmed by Bland-Altman plots which showed mean differences <or=10MPa. Surprisingly, the agreement between simulation and experiment was not reduced by using an average tissue modulus. The results indicate that valid microFE models can be set up using average tissue properties obtained under wet indentation conditions.

Rehydration of vertebral trabecular bone: influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level

For understanding the fracture risk of vertebral bodies the macroscopic mechanical properties of the cancellous core are of major interest. Due to the hierarchical nature of bone, these depend in turn on the micromechanical properties of bone extracellular matrix which is at least linear elastic transverse isotropic. The experimental determination of local elastic properties of bone ex vivo necessitates a high spatial resolution which can be provided by depth-sensing indentation techniques. Using microindentation, this study investigated the effects of rehydration on the transverse isotropic elastic properties of vertebral trabecular bone matrix obtained from two orthogonal directions with a view to microanatomical location, age, gender, vertebral level and anatomic direction in a conjoint statistics. Biopsies were gained from 104 human vertebrae (T1-L3) with a median age of 65 years (21-94). Wet elastic moduli were 29% lower (p<0.05) than dry elastic moduli. For wet indentation the ratio of mean elastic moduli tested in axial to those tested in transverse indentation direction were 1.13 to 1.23 times higher than for dry indentation. The ratio of elastic moduli tested in the core to those tested in the periphery of trabeculae was 1.05 to 1.16 times higher when testing wet. Age and gender did not show any influence on the elastic moduli for wet and dry measurements. The correlation between vertebral level and elastic moduli became weaker after rehydration (p(wet)<0.09, r(wet)(2)=0.14) and (p(dry)<0.01, r(wet)(2)=0.38). Elastic and dissipated energies were similarly affected by rehydration compared to the elastic modulus. No significant difference in the energies could be found for gender (p>0.05). Significant differences in the energies were found for age (p<0.05) after rehydration. Qualitative and quantitative insights into the transverse isotropic elastic properties of trabecular bone matrix under two testing conditions over a broad spectrum of vertebrae could be given. This study could help to further improve understanding of the mechanical properties of vertebral trabecular bone.

Heterogeneity of yield strain in low-density versus high-density human trabecular bone

Understanding the off-axis behavior of trabecular yield strains may lend unique insight into the etiology of fractures since yield strains provide measures of failure independent of elastic behavior. We sought to address anisotropy of trabecular yield strains while accounting for variations in both density and anatomic site and to determine the mechanisms governing this behavior. Cylindrical specimens were cored from vertebral bodies (n=22, BV/TV=0.11+/-0.02) and femoral necks (n=28, BV/TV=0.22+/-0.06) with the principal trabecular orientation either aligned along the cylinder axis (on-axis, n=22) or at an oblique angle of 15 degrees or 45 degrees (off-axis, n=28). Each specimen was scanned with micro-CT, mechanically compressed to failure, and analysed with nonlinear micro-CT-based finite element analysis. Yield strains depended on anatomic site (p=0.03, ANOVA), and the effect of off-axis loading was different for the two sites (p=0.04)-yield strains increased for off-axis loading of the vertebral bone (p=0.04), but were isotropic for the femoral bone (p=0.66). With sites pooled together, yield strains were positively correlated with BV/TV for on-axis loading (R(2)=58%, p<0.0001), but no such correlation existed for off-axis loading (p=0.79). Analysis of the modulus-BV/TV and strength-BV/TV relationships indicated that, for the femoral bone, the reduction in strength associated with off-axis loading was greater than that for modulus, while the opposite trend occurred for the vertebral bone. The micro-FE analyses indicated that these trends were due to different failure mechanisms for the two types of bone and the different loading modes. Taken together, these results provide unique insight into the failure behavior of human trabecular bone and highlight the need for a multiaxial failure criterion that accounts for anatomic site and bone volume fraction.

Cancellous bone lamellae strongly affect microcrack propagation and apparent mechanical properties: separation of patients with osteoporotic fracture from normal controls using a 2D nonlinear finite element method (biomechanical stereology)

Biomechanical stereology is proposed as a two-dimensional (2D) finite element (FE) method to estimate the ability of bone tissue to sustain damage and to separate patients with osteoporotic fracture from normal controls. Briefly, 2D nonlinear compact tension FE models were created from quantitative back scattered electron images taken of iliac crest bone specimens collected from the individuals with or without osteoporotic fracture history. The effects of bone mineral microstructure on predicted bone fracture toughness and microcrack propagation were examined. The 2D FE models were used as surrogates for the real bone tissues. The calculated microcrack propagation results and bone mechanical properties were examined as surrogates for measurements from mechanical testing of actual specimens. The results for the 2D FE simulation separated patients with osteoporotic fracture from normal controls even though only the variability in tissue mineral microstructure was used to build the models. The models were deliberately created to ignore all differences in mean mineralization. Hence, the current results support the following hypotheses: (1) that material heterogeneity is important to the separation of patients with osteoporotic fracture from normal controls; and (2) that 2D nonlinear finite element modeling can produce surrogate mechanical parameters that separate patients with fracture from normal controls.

Modeling of Dynamic Fracture and Damage in Two-Dimensional Trabecular Bone Microstructures Using the Cohesive Finite Element Method

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

Effects of intracortical porosity on fracture toughness in aging human bone: a microCT-based cohesive finite element study

The extent to which increased intracortical porosity affects the fracture properties of aging and osteoporotic bone is unknown. Here, we report the development and application of a microcomputed tomography based finite element approach that allows determining the effects of intracortical porosity on bone fracture by blocking all other age-related changes in bone. Previously tested compact tension specimens from human tibiae were scanned using microcomputed tomography and converted to finite element meshes containing three-dimensional cohesive finite elements in the direction of the crack growth. Simulations were run incorporating age-related increase in intracortical porosity but keeping cohesive parameters representing other age-related effects constant. Additional simulations were performed with reduced cohesive parameters. The results showed a 6% decrease in initiation toughness and a 62% decrease in propagation toughness with a 4% increase in porosity. The reduction in toughnesses became even more pronounced when other age-related effects in addition to porosity were introduced. The initiation and propagation toughness decreased by 51% and 83%, respectively, with the combined effect of 4% increase in porosity and decrease in the cohesive properties reflecting other age-related changes in bone. These results show that intracortical porosity is a significant contributor to the fracture toughness of the cortical bone and that the combination of computational modeling with advanced imaging improves the prediction of the fracture properties of the aged and the osteoporotic cortical bone.

[Elbow dysplasia in the dog: finite element analysis]

For young active dogs of large, fast-growing breeds, diseases of the elbow represent an increasing important disorder. Genetic predisposition, overweight and joint overload have been proposed as possible causes of elbow dysplasia. In this study, the influence of various biomechanical parameters on load transfer in healthy and pathological dog elbows has been analysed by means of a two-dimensional finite element model. Pathological changes in the elbow structure, such as altered material properties or asynchronous bone growth, have a distinct influence on the contact pressure in the joint articulation, internal bone deformation and stresses in the bones. The results obtained support empirical observations made during years of experience and offer explanations for clinical findings that are not yet well understood.