Micro-finite element simulation of trabecular-bone post-yield behaviour--effects of material model, element size and type

High-resolution local trabecular strain within trabecular structure under cyclic loading

Trabecular bone structure is a complex microstructure consisting of rods and plates, which poses challenges for its mechanical characterization. Digital image correlation (DIC) offers the possibility to characterize the strain response on the surface of trabecular bone. This study employed DIC equipped with a telecentric lens to investigate the strain state of individual trabeculae within their trabecular structure by assessing the longitudinal strain of the trabeculae at both the middle and near the edges of the trabeculae. Due to the high-resolution of the used DIC system, local surface strain of trabeculae was analyzed too. Lastly, the correlation between longitudinal trabecular strain and the orientation and slenderness of the trabeculae was investigated. The results showed that the strain magnification close to the edge of the trabeculae was higher and reached up to 8-folds the strain along the middle of the trabeculae. On the contrary, no strain magnification was found for most of the trabeculae between the longitudinal trabecular strain along the middle of the trabeculae and the globally applied strain. High-resolution full-field strain maps were obtained on the surface of trabeculae showing heterogeneous strain distribution with increasing load. No significant correlation was found between longitudinal trabecular strain and its orientation or slenderness. These findings and the applied methodology can be used to broaden our understanding of the deformation mechanisms of trabeculae within the trabecular network.

Comparison of linear and nonlinear stepwise μFE displacement predictions to digital volume correlation measurements of trabecular bone biopsies

Digital volume correlation (DVC) enables to evaluate the ability of μFE models in predicting experimental results on the mesoscale. In this study predicted displacement fields of three different linear and materially nonlinear μFE simulation methods were compared to DVC measured displacement fields at specific load steps in the elastic regime (Step_El) and after yield (Step_Ult). Five human trabecular bone biopsies from a previous study were compressed in several displacement steps until failure. At every compression step, μCT images (resolution: 36 μm) were recorded. A global DVC algorithm was applied to compute the displacement fields at all loading steps. The unloaded 3D images were then used to generate homogeneous, isotropic, linear and materially nonlinear μFE models. Three different μFE simulation methods were used: linear (L), nonlinear (NL), and nonlinear stepwise (NLS). Regarding L and NL, the boundary conditions were derived from the interpolated displacement fields at Step_El and Step_Ult, while for the NLS method nonlinear changes of the boundary conditions of the experiments were captured using the DVC displacement field of every available load step until Step_El and Step_Ult. The predicted displacement fields of all μFE simulation methods were in good agreement with the DVC measured displacement fields (individual specimens: R²>0.83 at Step_El and R²>0.59 at Step_Ult; pooled data: R²>0.97 at Step_El and R²>0.92 at Step_Ult). At Step_El, all three simulation methods showed similar intercepts, slopes, and coefficients of determination while the nonlinear μFE models improved the prediction of the displacement fields slightly in all Cartesian directions at Step_Ult (individual specimens: L: R²>0.59 and NL, NLS: R²>0.68; pooled data: L: R²>0.92 and NL, NLS: R²>0.94). Damaged/overstrained elements in L, NL, and NLS occurred at similar locations but the number of overstrained elements was overestimated when using the L simulation method. Considering the increased solving time of the nonlinear μFE models as well as the acceptable performance in displacement prediction of the linear μFE models, one can conclude that for similar use cases linear μFE models represent the best compromise between computational effort and accuracy of the displacement field predictions.

Application of phase-field fracture theories and digital volume correlation to synchrotron X-ray monitored fractures in human trabecular bone: A case study

Fracture processes of trabecular bone have been studied using various approaches over the years. However, reliable methods to analyse fracture at the single trabecula level are limited. In this study, a digital volume correlation (DVC) and a phase-field fracture model are applied and contrasted for human trabecular bone to analyse its failure under global compression at high resolution. A human trabecular bone sample was fractured in situ under synchrotron-based X-ray micro computed tomography (CT). Reconstructed CT data was then used in DVC algorithms to obtain high-resolution displacement fields in the bone at different load steps. A high-resolution specimen-specific structural mesh was discretized from the CT data and used for the phase-field simulation of the fracturing bone. The DVC analysis showed opening mode cracks as well as shear mode cracks. Strains in cracked regions were analysed. The load distribution in the trabecular structure resulted in two completely separated fracture regions in the sample body. A phenomenon that was also captured in the phase-field model. The results encourage us to believe improvements in boundary conditions and material models are worthwhile pursuing. Findings in this study support further development of a phase-field method to analyse fracture in samples with complex morphology, such as trabecular bone, and the capacity of DVC to quantify strains and slowly growing stable fractures during step-wise loading of trabecular bone.

Experimental DVC validation of heterogeneous micro finite element models applied to subchondral trabecular bone of the humeral head

Subchondral trabecular bone (STB) undergoes adaptive changes during osteoarthritic (OA) disease progression. These changes alter both the mineralization patterns and structure of bone and may contribute to variations in the mechanical properties. Similarly, when images are downsampled - as is often performed in micro finite element model (microFEM) generation - the morphological and mineralization patterns may further alter the mechanical properties due to partial volume effects. MicroFEMs accounting for material heterogeneity can account for these tissue variations, but no studies have validated these with robust full-field testing methods. As such, this study compared homogeneous and heterogeneous microFEMs to experimentally loaded trabecular bone cores from the humeral head combined with digital volume correlation (DVC). These microFEMs were used to compare apparent mechanical properties between normal and OA STB. Morphological and mineralization patterns between groups were also compared. There were no significant differences in tissue or bone mineral density between groups. The only significant differences in morphometric parameters were in trabecular thickness between groups. There were no significant differences in linear regression parameters between normal and OA STB apparent mechanical properties estimated using heterogeneous microFEMs with an element-wise bilinear elastic-plastic constitutive model. Clinical significance: Validated heterogeneous microFEMs applied to STB of the humeral head have the potential to significantly improve our understanding of mechanical variations in the bone that occur during OA progression.

Loss of longitudinal superiority marks the microarchitecture deterioration of osteoporotic cancellous bones

Osteoporosis (OP), a skeletal disease making bone mechanically deteriorate and easily fracture, is a global public health issue due to its high prevalence. It has been well recognized that besides bone loss, microarchitecture degradation plays a crucial role in the mechanical deterioration of OP bones, but the specific role of microarchitecture in OP has not been well clarified and quantified from mechanics perspective. Here, we successfully decoupled and identified the specific roles of microarchitecture, bone mass and tissue property in the failure properties of cancellous bones, through μCT-based digital modeling and finite element method simulations on bone samples from healthy and ovariectomy-induced osteoporotic mice. The results show that the microarchitecture of healthy bones exhibits longitudinal superiority in mechanical properties such as the effective stiffness, strength and toughness, which fits them well to bearing loads along their longitudinal direction. OP does not only reduce bone mass but also impair the microarchitecture topology. The former is mainly responsible for the mechanical degradation of bones in magnitude, wherever the latter accounts for the breakdown of their function-favorable anisotropy, the longitudinal superiority. Hence, we identified the microarchitecture-deterioration-induced directional mismatch between material and loading as a hazardous feature of OP and defined a longitudinal superiority index as measurement of the health status of bone microarchitecture. These findings provide useful insights and guidelines for OP diagnosis and treat assessment.

Numerical analysis of the mechanical behaviour of intact and implanted lumbar functional spinal units: Effects of loading and boundary conditions

The objective of this study was to develop an improved finite element (FE) model of a lumbar functional spinal unit (FSU) and to subsequently analyse the deviations in load transfer owing to implantation. The effects of loading and boundary conditions on load transfer in intact and implanted FSUs and its relationship with the potential risk of vertebral fracture were investigated. The FE models of L1-L5 and L3-L4 FSUs, intact and implanted, were developed using patient-specific CT-scan dataset and segmentation of cortical and cancellous bone regions. The effect of submodelling technique, as compared to artificial boundary conditions, on the elastic behaviour of lumbar spine was examined. Applied forces and moments, corresponding to physiologic movements, were used as loading conditions. Results indicated that the loading and boundary conditions considerably affect stress-strain distributions within a FSU. This study, based on an improved FE model of a vertebra, highlights the importance of using the submodelling technique to adequately evaluate the mechanical behaviour of a FSU. In the intact FSU, strains of 200-400 µε were observed in the cancellous bone of vertebral body and pedicles. High equivalent stresses of 10-25 MPa and 1-5 MPa were generated around the pars interarticularis for cortical and cancellous regions, respectively. Implantation caused reductions of 85%-92% in the range of motion for all movements. Insertion of the intervertebral cage resulted in major deviations in load transfer across a FSU for all movements. The cancellous bone around cage experienced pronounced increase in stresses of 10-15 MPa, which indicated potential risk of failure initiation in the vertebra.

Determinants of estimated failure load in the distal radius after stroke: An HR-pQCT study

Bone health is often compromised after stroke and the distal radius is a common site of fragility fractures. The macro- and mircoproperties of bone tissue after stroke and their clinical correlates are understudied. The objectives of the study were to use High-Resolution peripheral Quantitative Computed Tomography (HR-pQCT) to investigate the bone properties at the distal radius, and to identify the correlates of estimated failure load for the distal radius in people with chronic stroke. This was a cross-sectional study of 64 people with stroke (age: 60.8 ± 7.7 years, stroke duration: 5.7 ± 3.9 years) and 64 age- and sex-matched controls. Bilateral bone structural, densitometric, geometric and strength parameters of the distal radius were measured using HR-pQCT. The architecture, stiffness and echo intensity of the bilateral biceps brachii muscle and brachial artery blood flow were evaluated using diagnostic ultrasound. Other outcomes included the Fugl-Meyer Motor Assessment (FMA), Motor Activity Log (MAL), and Composite Spasticity Scale (CSS). The results revealed a significant side (paretic vs non-paretic for the stroke group, non-dominant vs dominant for controls) by group (stroke vs control) interaction effect for estimated failure load, cortical area, cortical thickness, trabecular number and trabecular separation, and all volumetric density parameters. Post-hoc analysis showed percent side-to-side differences in bone outcomes were greater in the stroke group than the control group, with the exception of trabecular thickness and intracortical porosity. Among the HR-pQCT variables, percent side-to-side difference in trabecular volumetric bone mineral density contributed the most to the percent side-to-side difference in estimated failure load in the stroke group (R² change = 0.334, β = 1.106). Stroke-related impairments (FMA, MAL, CSS) were found to be significant determinants of the percent side-to-side difference in estimated failure load (R² change = 0.233, β = -0.480). This was the first study to examine bone microstructure post-stroke. We found that the paretic distal radius had compromised bone structural properties and lower estimated failure load compared to the non-paretic side. Motor impairment was a determinant of estimated bone strength at the distal radius and may be a potential intervention target for improving bone health post-stroke.

Analysis of the effects of finite element type within a 3D biomechanical model of a human optic nerve head and posterior pole

BACKGROUND AND OBJECTIVE

Biomechanical stresses and strains can be simulated in the optic nerve head (ONH) using the finite element (FE) method, and various element types have been used. This study aims to investigate the effects of element type on the resulting ONH stresses and strains.

METHODS

A single eye-specific model was constructed using 3D delineations of anatomic surfaces in a high-resolution, fluorescent, 3D reconstruction of a human posterior eye, then meshed using our simple meshing algorithm at various densities using 4- and 10-noded tetrahedral elements, as well as 8- and 20-noded hexahedral elements. A mesh-free approach was used to assign heterogeneous, anisotropic, hyperelastic material properties to the lamina cribrosa, sclera and pia. The models were subjected to elevated IOP of 45 mmHg after pre-stressing from 0 to 10 mmHg, and solved in the open-source FE package Calculix; results were then interpreted in relation to computational time and simulation accuracy, using the quadratic hexahedral model as the reference standard.

RESULTS

The 10-noded tetrahedral and 20R-noded hexahedral elements exhibited similar scleral canal and laminar deformations, as well as laminar and scleral stress and strain distributions; the quadratic tetrahedral models ran significantly faster than the quadratic hexahedral models. The linear tetrahedral and hexahedral elements were stiffer compared to the quadratic element types, yielding much lower stresses and strains in the lamina cribrosa.

CONCLUSIONS

Prior studies have shown that 20-noded hexahedral elements yield the most accurate results in complex models. Results show that 10-noded tetrahedral elements yield very similar results to 20-noded hexahedral elements and so they can be used interchangeably, with significantly lower computational time. Linear element types did not yield acceptable results.

Microstructural and mechanical evaluations of region segmentation methods in classifications of osteonecrosis

Measuring the location of necrotic lesions is necessary to diagnosis of osteonecrosis. Different region segmentation methods of the femoral head were proposed to quantitatively measure necrotic lesions including Japanese Investigation Committee for Avascular Necrosis (JIC) classification and China-Japan Friendship Hospital (CJFH) classification. Biomechanical methods could bring important information to evaluate the reasonability of these classifications. In this study, microstructural and mechanical properties of trabecular bone were quantitatively analyzed according to the region segmentation methods described in these classifications. Microstructural parameters of trabecular bone were analyzed based on micro-CT scanning. Mechanical properties were measured through Nanoindentation and micro-finite element analysis. It was found that microstructural and mechanical properties of trabecular bone in the middle region was more adaptive to load bearing than the medial and lateral regions according to the CJFH classification; lesions in the middle region could bring more changes to microstructure and stress distribution. According to JIC classification, differences of microstructural and mechanical properties among the three regions were not significant. Biomechanical characteristics of trabecular bones could be better distinguished with CJFH classification.

Evaluation of bone excision effects on a human skull model-II: Finite element analysis

Patient-specific computational models are powerful tools which may assist in predicting the outcome of invasive surgery on the musculoskeletal system, and consequently help to improve therapeutic decision-making and post-operative care. Unfortunately, at present the use of personalized models that predict the effect of biopsies and full excisions is so specialized that tends to be restricted to prominent individuals, such as high-profile athletes. We have developed a finite element analysis model to determine the influence of the location of an ellipsoidal excision (14.2 mm × 11.8 mm) on the structural integrity of a human skull when exposed to impact loading, representing a free fall of an adult male from standing height. The finite element analysis model was compared to empirical data based on the drop-tower testing of three-dimensional-printed physical skull models where deformations were recorded by digital image correlation. In this bespoke example, we found that the excision site did not have a major effect on the calculated stress and strain magnitudes unless the excision was in the temporal region, where the reduction in stiffness around the excision caused failure within the neighboring area. The finite element analysis model allowed meaningful conclusions to be drawn for the implications of using such a technique based on what we know about such conditions indicating that the approach could be both clinically beneficial and also cost-effective for wider use.

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.

Young's modulus of trabecular bone at the tissue level: A review

The tissue-level Young's modulus of trabecular bone is important for detailed mechanical analysis of bone and bone-implant mechanical interactions. However, the heterogeneity and small size of the trabecular struts complicate an accurate determination. Methods such as micro-mechanical testing of single trabeculae, ultrasonic testing, and nanoindentation have been used to estimate the trabecular Young's modulus. This review summarizes and classifies the trabecular Young's moduli reported in the literature. Information on species, anatomic site, and test condition of the samples has also been gathered. Advantages and disadvantages of the different methods together with recent developments are discussed, followed by some suggestions for potential improvement for future work. In summary, this review provides a thorough introduction to the approaches used for determining trabecular Young's modulus, highlights important considerations when applying these methods and summarizes the reported Young's modulus for follow-up studies on trabecular properties.

STATEMENT OF SIGNIFICANCE

The spongy trabecular bone provides mechanical support while maintaining a low weight. A correct measure of its mechanical properties at the tissue level, i.e. at a single-trabecula level, is crucial for analysis of interactions between bone and implants, necessary for understanding e.g. bone healing mechanisms. In this study, we comprehensively summarize the Young's moduli of trabecular bone estimated by currently available methods, and report their dependency on different factors. The critical review of different methods with recent updates is intended to inspire improvements in estimating trabecular Young's modulus. It is strongly suggested to report detailed information on the tested bone to enable statistical analysis in the future.

Nonlinear micro-CT based FE modeling of trabecular bone-Sensitivity of apparent response to tissue constitutive law and bone volume fraction

In this study, the sensitivity of the apparent response of trabecular bone to different constitutive models at the tissue level was investigated using finite element (FE) modeling based on micro-computed tomography (micro-CT). Trabecular bone specimens from porcine femurs were loaded under a uniaxial compression experimentally and computationally. The apparent behaviors computed using von Mises, Drucker-Prager, and Cast Iron plasticity models were compared. Secondly, the effect of bone volume fraction was studied by changing the bone volume fraction of a trabecular bone sample while keeping the same basic architecture. Also, constitutive models' parameters of the tissue were calibrated for porcine bone, and the effects of different parameters on resulting apparent response were investigated through a parametric study. The calibrated effective tissue elastic modulus of porcine trabecular bone was 10±1.2 GPa, which is in the lower range of modulus values reported in the literature for human and bovine trabecular bones (4-23.8 GPa). It was also observed that, unlike elastic modulus, yield properties of tissue could not be uniquely calibrated by fitting an apparent response from simulations to experiments under a uniaxial compression. Our results demonstrated that using these 3 tissue constitutive models had only a slight effect on the apparent response. As expected, there was a significant change in the apparent response with varying bone volume fraction. Also, both apparent modulus and maximum stress had a linear relation with bone volume fraction.

Biomechanical assessment of new surgical method instead of kyphoplasty to improve the mechanical behavior of the vertebra: Micro finite element study

AIM

To reduce post treatments of kyphoplasty, as a common treatment for osteoporotic vertebrae.

METHODS

This study suggests a new method for treating vertebrae by setting the hexagonal porous structure instead of the rigid bone cement mass in the kyphoplasty (KP). The KP procedure was performed on the fresh ovine vertebra of the level L1. Micro finite element modeling was performed based on micro computed tomography of ovine trabecular cube. The hexagonal porous structure was set on one cube instead of the bone cement mass. For the implant designing, two geometrical parameters were considered: Spacing diameter and thickness.

RESULTS

The results of micro finite element analyses indicated the improvement in the mechanical behavior of the vertebra treated by the hexagonal porous structures, as compared to those treated by vertebroplasty (VP) and KP under static loading. The improvement in the mechanical behavior of the vertebra, was observed as 54% decrease in the amount of maximum Von Misses stress (improvement of stress distribution), in trabecular cube with embedded hexagonal structure, as compared to VP and KP. This is comparable to the results of the experimental study already performed; it was shown that the improvement of mechanical behavior of the vertebra was observed as: 83% increase in the range of displacements before getting to the ultimate strength (increasing the toughness) after setting hexagonal pearls inside vertebrae. Both the material and geometry of implant influenced the amount of Von Mises stress in the structure.

CONCLUSION

The new proposed method can be offered as a substitute for the KP. The implant geometry had a more obvious effect on the amount of Von Mises stress, as compared to the implant material.

Influence of the shape of the micro-finite element model on the mechanical properties calculated from micro-finite element analysis

Assessing the biomechanical properties of trabecular bone is of major biological and clinical significance for the research of bone diseases, fractures and their treatments. Micro-finite element (µFE) models are becoming increasingly popular for investigating the biomechanical properties of trabecular bone. The shapes of µFE models typically include cube and cylinder. Whether there are differences between cubic and cylindrical µFE models has not yet been studied. In the present study, cubic and cylindrical µFE models of human vertebral trabecular bone were constructed. A 1% strain was prescribed to the model along the superior-inferior direction. E values were calculated from these models, and paired t-tests were performed to determine whether these were any differences between E values obtained from cubic and cylindrical models. The results demonstrated that there were no statistically significant differences in the E values between cubic and cylindrical models, and there were no significant differences in Von Mises stress distributions between the two models. These findings indicated that, to construct µFE models of vertebral trabecular bone, cubic or cylindrical models were both feasible. Choosing between the cubic or cylindrical µFE model is dependent upon the specific study design.

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.

Influence of Trabecular Bone on Peri-Implant Stress and Strain Based on Micro-CT Finite Element Modeling of Beagle Dog

The objective of this investigation is to analyze the influence of trabecular microstructure modeling on the biomechanical distribution of the implant-bone interface. Two three-dimensional finite element mandible models, one with trabecular microstructure (a refined model) and one with macrostructure (a simplified model), were built. The values of equivalent stress at the implant-bone interface in the refined model increased compared with those of the simplified model and strain on the contrary. The distributions of stress and strain were more uniform in the refined model of trabecular microstructure, in which stress and strain were mainly concentrated in trabecular bone. It was concluded that simulation of trabecular bone microstructure had a significant effect on the distribution of stress and strain at the implant-bone interface. These results suggest that trabecular structures could disperse stress and strain and serve as load buffers.

Modeling the Mechanical Consequences of Age-Related Trabecular Bone Loss by XFEM Simulation

The elderly are more likely to suffer from fracture because of age-related trabecular bone loss. Different bone loss locations and patterns have different effects on bone mechanical properties. Extended finite element method (XFEM) can simulate fracture process and was suited to investigate the effects of bone loss on trabecular bone. Age-related bone loss is indicated by trabecular thinning and loss and may occur at low-strain locations or other random sites. Accordingly, several ideal normal and aged trabecular bone models were created based on different bone loss locations and patterns; then, fracture processes from crack initiation to complete failure of these models were observed by XFEM; finally, the effects of different locations and patterns on trabecular bone were compared. Results indicated that bone loss occurring at low-strain locations was more detrimental to trabecular bone than that occurring at other random sites; meanwhile, the decrease in bone strength caused by trabecular loss was higher than that caused by trabecular thinning, and the effects of vertical trabecular loss on mechanical properties were more severe than horizontal trabecular loss. This study provided a numerical method to simulate trabecular bone fracture and distinguished different effects of the possible occurrence of bone loss locations and patterns on trabecular bone.

Inverse finite element modeling for characterization of local elastic properties in image-guided failure assessment of human trabecular bone

The local interpretation of microfinite element (μFE) simulations plays a pivotal role for studying bone structure–function relationships such as failure processes and bone remodeling.In the past μFE simulations have been successfully validated on the apparent level,however, at the tissue level validations are sparse and less promising. Furthermore,intra trabecular heterogeneity of the material properties has been shown by experimental studies. We proposed an inverse μFE algorithm that iteratively changes the tissue level Young's moduli such that the μFE simulation matches the experimental strain measurements.The algorithm is setup as a feedback loop where the modulus is iteratively adapted until the simulated strain matches the experimental strain. The experimental strain of human trabecular bone specimens was calculated from time-lapsed images that were gained by combining mechanical testing and synchrotron radiation microcomputed tomography(SRlCT). The inverse μFE algorithm was able to iterate the heterogeneous distribution of moduli such that the resulting μFE simulations matched artificially generated and experimentally measured strains.

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.

Predicting mouse vertebra strength with micro-computed tomography-derived finite element analysis

As in clinical studies, finite element analysis (FEA) developed from computed tomography (CT) images of bones are useful in pre-clinical rodent studies assessing treatment effects on vertebral body (VB) strength. Since strength predictions from microCT-derived FEAs (μFEA) have not been validated against experimental measurements of mouse VB strength, a parametric analysis exploring material and failure definitions was performed to determine whether elastic μFEAs with linear failure criteria could reasonably assess VB strength in two studies, treatment and genetic, with differences in bone volume fraction between the control and the experimental groups. VBs were scanned with a 12-μm voxel size, and voxels were directly converted to 8-node, hexahedral elements. The coefficient of determination or R (2) between predicted VB strength and experimental VB strength, as determined from compression tests, was 62.3% for the treatment study and 85.3% for the genetic study when using a homogenous tissue modulus (E t) of 18 GPa for all elements, a failure volume of 2%, and an equivalent failure strain of 0.007. The difference between prediction and measurement (that is, error) increased when lowering the failure volume to 0.1% or increasing it to 4%. Using inhomogeneous tissue density-specific moduli improved the R (2) between predicted and experimental strength when compared with uniform E t=18 GPa. Also, the optimum failure volume is higher for the inhomogeneous than for the homogeneous material definition. Regardless of model assumptions, μFEA can assess differences in murine VB strength between experimental groups when the expected difference in strength is at least 20%.

Biomechanics and mechanobiology of trabecular bone: a review

Trabecular bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. Studying the mechanical properties of trabecular bone is important, since trabecular bone is the main load bearing bone in vertebral bodies and also transfers the load from joints to the compact bone of the cortex of long bones. This review article highlights the high dependency of the mechanical properties of trabecular bone on species, age, anatomic site, loading direction, and size of the sample under consideration. In recent years, high resolution micro finite element methods have been extensively used to specifically address the mechanical properties of the trabecular bone and provide unique tools to interpret and model the mechanical testing experiments. The aims of the current work are to first review the mechanobiology of trabecular bone and then present classical and new approaches for modeling and analyzing the trabecular bone microstructure and macrostructure and corresponding mechanical properties such as elastic properties and strength.

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.

Modeling microdamage behavior of cortical bone

Bone is a complex material which exhibits several hierarchical levels of structural organization. At the submicron-scale, the local tissue porosity gives rise to discontinuities in the bone matrix which have been shown to influence damage behavior. Computational tools to model the damage behavior of bone at different length scales are mostly based on finite element (FE) analysis, with a range of algorithms developed for this purpose. Although the local mechanical behavior of bone tissue is influenced by microstructural features such as bone canals and osteocyte lacunae, they are often not considered in FE damage models due to the high computational cost required to simulate across several length scales, i.e., from the loads applied at the organ level down to the stresses and strains around bone canals and osteocyte lacunae. Hence, the aim of the current study was twofold: First, a multilevel FE framework was developed to compute, starting from the loads applied at the whole bone scale, the local mechanical forces acting at the micrometer and submicrometer level. Second, three simple microdamage simulation procedures based on element removal were developed and applied to bone samples at the submicrometer-scale, where cortical microporosity is included. The present microdamage algorithm produced a qualitatively analogous behavior to previous experimental tests based on stepwise mechanical compression combined with in situ synchrotron radiation computed tomography. Our results demonstrate the feasibility of simulating microdamage at a physiologically relevant scale using an image-based meshing technique and multilevel FE analysis; this allows relating microdamage behavior to intracortical bone microstructure.

Potential of in vivo MRI-based nonlinear finite-element analysis for the assessment of trabecular bone post-yield properties

PURPOSE

Bone strength is the key factor impacting fracture risk. Assessment of bone strength from high-resolution (HR) images have largely relied on linear micro-finite element analysis (μFEA) even though failure always occurs beyond the yield point, which is outside the linear regime. Nonlinear μFEA may therefore be more informative in predicting failure behavior. However, existing nonlinear models applied to trabecular bone (TB) have largely been confined to micro-computed tomography (μCT) and, more recently, HR peripheral quantitative computed tomography (HR-pQCT) images, and typically have ignored evaluation of the post-yield behavior. The primary purpose of this work was threefold: (1) to provide an improved algorithm and program to assess TB yield as well as post-yield properties; (2) to explore the potential benefits of nonlinear μFEA beyond its linear counterpart; and (3) to assess the feasibility and practicality of performing nonlinear analysis on desktop computers on the basis of micro-magnetic resonance (μMR) images obtained in vivo in patients.

METHODS

A method for nonlinear μFE modeling of TB yield as well as post-yield behavior has been designed where material nonlinearity is captured by adjusting the tissue modulus iteratively according to the tissue-level effective strain obtained from linear analysis using a computationally optimized algorithm. The software allows for images at in vivo μMRI resolution as input with retention of grayscale information. Associations between axial stiffness estimated from linear analysis and yield as well as post-yield parameters from nonlinear analysis were investigated from in vivo μMR images of the distal tibia (N = 20; ages: 58-84) and radius (N = 20; ages: 50-75).

RESULTS

All simulations were completed in 1 h or less for 61 strain levels using a desktop computer (dual quad-core Xeon 3.16 GHz CPUs equipped with 40 GB of RAM). Although yield stress and ultimate stress correlated strongly (R(2) > 0.95, p < 0.001) with axial stiffness, toughness correlated moderately at the distal tibia (R(2) = 0.81, p < 0.001) and only weakly at the distal radius (R(2) = 0.34, p = 0.007). Further, toughness was found to vary by up to 16% for bone of very similar axial stiffness (<2%).

CONCLUSIONS

The work demonstrates the practicality of nonlinear μFE simulations at in vivo μMRI resolution, as well as its potential for providing additional information beyond that obtainable from linear analysis. The data suggest that a direct assessment of toughness may provide information not captured by stiffness.

μFEA successfully exhibits higher stresses and strains in microdamaged regions of whole vertebrae

Micro-finite element (μFE) modeling has shown promise in evaluating the structural integrity of trabecular bone. Histologic microcrack analyses have been compared to μFE models of trabecular bone cores to demonstrate the potential of this technique. To date this has not been achieved in whole bone structures, and comparisons of histologic microcrack and μFE results have been limited due to challenges in alignment of 2D sections with 3D data sets. The goal of this study was to ascertain if image registration can facilitate determination of a relationship between stresses and strains generated from μFE models of whole vertebrae and histologically identified microdamage. μFE models of three whole vertebrae, stained sequentially with calcein and fuchsin, were generated with accurate integration of element sets representing the histologic sections based on volumetric image registration. Displacement boundary conditions were applied to the μFE models based on registration of loaded and unloaded μCT images. Histologically labeled damaged regions were found to have significantly higher von Mises stresses and principle strains in the μFE models, as compared to undamaged regions. This work provides a new robust method for generating and histologically validating μFE models of whole bones that can represent trabecular damage resulting from complex physiologic loading.

Micro-CT finite element model and experimental validation of trabecular bone damage and fracture

Most micro-CT finite element modeling of human trabecular bone has focused on linear and non-linear analysis to evaluate bone failure properties. However, prediction of the apparent failure properties of trabecular bone specimens under compressive load, including the damage initiation and its progressive propagation until complete bone failure into consideration, is still lacking. In the present work, an isotropic micro-CT FE model at bone tissue level coupled to a damage law was developed in order to simulate the failure of human trabecular bone specimens under quasi-static compressive load and predict the apparent stress and strain. The element deletion technique was applied in order to simulate the progressive fracturing process of bone tissue. To prevent mesh-dependence that generally affects the damage propagation rate, regularization technique was applied in the current work. The model was validated with experimental results performed on twenty-three human trabecular specimens. In addition, a sensitivity analysis was performed to investigate the impact of the model factors' sensitivities on the predicted ultimate stress and strain of the trabecular specimens. It was found that the predicted failure properties agreed very well with the experimental ones.

Theoretical bounds for the influence of tissue-level ductility on the apparent-level strength of human trabecular bone

The role of tissue-level post-yield behavior on the apparent-level strength of trabecular bone is a potentially important aspect of bone quality. To gain insight into this issue, we compared the apparent-level strength of trabecular bone for the hypothetical cases of fully brittle versus fully ductile failure behavior of the trabecular tissue. Twenty human cadaver trabecular bone specimens (5mm cube; BV/TV=6-36%) were scanned with micro-CT to create 3D finite element models (22-micron element size). For each model, apparent-level strength was computed assuming either fully brittle (fracture with no tissue ductility) or fully ductile (yield with no tissue fracture) tissue-level behaviors. We found that the apparent-level ultimate strength for the brittle behavior was only about half the value of the apparent-level 0.2%-offset yield strength for the ductile behavior, and the ratio of these brittle to ductile strengths was almost constant (mean±SD=0.56±0.02; n=20; R(2)=0.99 between the two measures). As a result of this small variation, although the ratio of brittle to ductile strengths was positively correlated with the bone volume fraction (R(2)=0.44, p=0.01) and structure model index (SMI, R(2)=0.58, p<0.01), these effects were small. Mechanistically, the fully ductile behavior resulted in a much higher apparent-level strength because in this case about 16-fold more tissue was required to fail than for the fully brittle behavior; also, there was more tensile- than compressive-mode of failure at the tissue level for the fully brittle behavior. We conclude that, in theory, the apparent-level strength behavior of human trabecular bone can vary appreciably depending on whether the tissue fails in a fully ductile versus fully brittle manner, and this effect is largely constant despite appreciable variations in bone volume fraction and microarchitecture.

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.

Validation of composite finite elements efficiently simulating elasticity of trabecular bone

Patient-specific analyses of the mechanical properties of bones become increasingly important for the management of patients with osteoporosis. The potential of composite finite elements (CFEs), a novel FE technique, to assess the apparent stiffness of vertebral trabecular bone is investigated in this study. Segmented volumes of cylindrical specimens of trabecular bone are compared to measured volumes. Elasticity under uniaxial loading conditions is simulated; apparent stiffnesses are compared to experimentally determined values. Computational efficiency is assessed and recommendations for simulation parameters are given. Validating apparent uniaxial stiffnesses results in concordance correlation coefficients 0.69 ≤ r(c) ≤ 0.92 for resolutions finer than 168 μm, and an average error of 5.8% between experimental and numerical results at 24 μm resolution. As an application, the code was used to compute local, macroscopic stiffness tensors for the trabecular structure of a lumbar vertebra. The presented technique allows for computing stiffness using smooth FE meshes at resolutions that are well achievable in peripheral high resolution quantitative CT. Therefore, CFEs could be a valuable tool for the patient-specific assessment of bone stiffness.

Vertebral fragility and structural redundancy

The mechanisms of age-related vertebral fragility remain unclear, but may be related to the degree of "structural redundancy" of the vertebra; ie, its ability to safely redistribute stress internally after local trabecular failure from an isolated mechanical overload. To better understand this issue, we performed biomechanical testing and nonlinear micro-CT-based finite element analysis on 12 elderly human thoracic ninth vertebral bodies (age 76.9 ± 10.8 years). After experimentally overloading the vertebrae to measure strength, we used nonlinear finite element analysis to estimate the amount of failed tissue and understand the failure mechanisms. We found that the amount of failed tissue per unit bone mass decreased with decreasing bone volume fraction (r(2)  = 0.66, p < 0.01). Thus, for the weak vertebrae with low bone volume fraction, overall failure of the vertebra occurred after failure of just a tiny proportion of the bone tissue (<5%). This small proportion of failed tissue had two sources: the existence of fewer vertically oriented load paths to which load could be redistributed from failed trabeculae; and the vulnerability of the trabeculae in these few load paths to undergo bending-type failure mechanisms, which further weaken the bone. Taken together, these characteristics suggest that diminished structural redundancy may be an important aspect of age-related vertebral fragility: vertebrae with low bone volume fraction are highly susceptible to collapse because so few trabeculae are available for load redistribution if the external loads cause any trabeculae to fail.

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.

MICRO-FINITE ELEMENT ANALYSIS OF TRABECULAR BONE YIELD BEHAVIOR — EFFECTS OF TISSUE NONLINEAR MATERIAL PROPERTIES

Bone tissue material nonlinearity and large deformations within the trabecular network are important for the characterization of failure behavior of trabecular bone at both the apparent and tissue levels. Micro-finite element analysis (μFEA) is a useful tool for determining the mechanical properties of trabecular bone due to certain experimental difficulties. The aim of this study was to determine the effects of bone tissue nonlinear material properties on the apparent- and tissue-level mechanical parameters of trabecular bone using μFEA. A bilinear tissue constitutive model was proposed to describe the bone tissue material nonlinearity. Two trabecular specimens with different micro-architectures were taken as examples. The effects of four parameters, i.e., tissue Young's modulus, tissue yield strain in tension, tissue yield strain in compression, and post-yield modulus on the apparent yield stress/strain, tissue von Mises stress distribution, the amount of tissue elements yielded in compression and tension under compressive and tensile loading conditions were obtained using nine cases for different values of those parameters by totally 36 nonlinear μFEA. These data may provide a reference for more sophisticated evaluations of bone strength and the related fracture risk.

Heavy ion irradiation and unloading effects on mouse lumbar vertebral microarchitecture, mechanical properties and tissue stresses

Astronauts are exposed to both musculoskeletal disuse and heavy ion radiation in space. Disuse alters the magnitude and direction of forces placed upon the skeleton causing bone remodeling, while energy deposited by ionizing radiation causes free radical formation and can lead to DNA strand breaks and oxidative damage to tissues. Radiation and disuse each result in a net loss of mineralized tissue in the adult, although the combined effects, subsequent consequences for mechanical properties and potential for recovery may differ. First, we examined how a high dose (2 Gy) of heavy ion radiation ((56)Fe) causes loss of mineralized tissue in the lumbar vertebrae of skeletally mature (4 months old), male, C57BL/6 mice using microcomputed tomography and determined the influence of structural changes on mechanical properties using whole bone compression tests and finite element analyses. Next, we tested if a low dose (0.5 Gy) of heavy particle radiation prevents skeletal recovery from a 14-day period of hindlimb unloading. Irradiation with a high dose of (56)Fe (2 Gy) caused bone loss (-14%) in the cancellous-rich centrum of the fourth lumbar vertebra (L4) 1 month later, increased trabecular stresses (+27%), increased the propensity for trabecular buckling and shifted stresses to the cortex. As expected, hindlimb unloading (14 days) alone adversely affected microarchitectural and mechanical stiffness of lumbar vertebrae, although the reduction in yield force was not statistically significant (-17%). Irradiation with a low dose of (56)Fe (0.5 Gy) did not affect vertebrae in normally loaded mice, but significantly reduced compressive yield force in vertebrae of unloaded mice relative to sham-irradiated controls (-24%). Irradiation did not impair the recovery of trabecular bone volume fraction that occurs after hindlimb unloaded mice are released to ambulate normally, although microarchitectural differences persisted 28 days later (96% increase in ratio of rod- to plate-like trabeculae). In summary, (56)Fe irradiation (0.5 Gy) of unloaded mice contributed to a reduction in compressive strength and partially prevented recovery of cancellous microarchitecture from adaptive responses of lumbar vertebrae to skeletal unloading. Thus, irradiation with heavy ions may accelerate or worsen the loss of skeletal integrity triggered by musculoskeletal disuse.