A new computational efficient approach for trabecular bone analysis using beam models generated with skeletonized graph technique

Micro-finite element (FE) analysis is a well established technique for the evaluation of the elastic properties of trabecular bone, but is limited in its application due to the large number of elements that it requires to represent the complex internal structure of the bone. In this paper, we present an alternative FE approach that makes use of a recently developed 3D-Line Skeleton Graph Analysis (LSGA) technique to represent the complex internal structure of trabecular bone as a network of simple straight beam elements in which the beams are assigned geometrical properties of the trabeculae that they represent. Since an enormous reduction of cputime can be obtained with this beam modeling approach, ranging from approximately 1,200 to 3,600 for the problems investigated here, we think that the FE modeling technique that we introduced could potentially constitute an interesting alternative for the evaluation of the elastic mechanical properties of trabecular bone.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

A finite element beam-model for efficient simulation of large-scale porous structures

This paper presents a new method for the generation of a beam finite element (FE) model from a three-dimensional (3D) data set acquired by micro-computed tomography (micro-CT). This method differs from classical modeling of trabecular bone because it models a specific sample only and differs from conventional solid hexahedron element-based FE approaches in its computational efficiency. The stress-strain curve, characterizing global mechanical properties of a porous structure, could be well predicted (R(2)=0.92). Furthermore, validation of the method was achieved by comparing local displacements of element nodes with the displacements directly measured by time-lapsed imaging methods of failure, and these measures were in good agreement. The presented model is a first step in modeling specific samples for efficient strength analysis by FE modeling. We believe that with upcoming high-resolution in-vivo imaging methods, this approach could lead to a novel and accurate tool in the risk assessment for osteoporotic fractures.     

Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.     

Validation of an efficient method of assigning material properties in finite element analysis of pelvic bone

Bone in the pelvis is a composite material with a complex anatomical structure that is difficult to model computationally. Rather than assigning material properties to increasingly smaller elements to capture detail in three-dimensional finite element (FE) models, properties can be assigned to Gauss points within larger elements. As part of a validation process, we compared experimental and analytical results from a composite beam under four-point load to FE models with material properties assigned to refined elements and Gauss points within larger elements. Both FE models accurately predicted deformation and the analytical predictions of internal shear stress.     

Fast and accurate specimen-specific simulation of trabecular bone elastic modulus using novel beam-shell finite element models

Elastic modulus and strength of trabecular bone are negatively affected by osteoporosis and other metabolic bone diseases. Micro-computed tomography-based beam models have been presented as a fast and accurate way to determine bone competence. However, these models are not accurate for trabecular bone specimens with a high number of plate-like trabeculae. Therefore, the aim of this study was to improve this promising methodology by representing plate-like trabeculae in a way that better reflects their mechanical behavior. Using an optimized skeletonization and meshing algorithm, voxel-based models of trabecular bone samples were simplified into a complex structure of rods and plates. Rod-like and plate-like trabeculae were modeled as beam and shell elements, respectively, using local histomorphometric characteristics. To validate our model, apparent elastic modulus was determined from simulated uniaxial elastic compression of 257 cubic samples of trabecular bone (4mm×4mm×4mm; 30μm voxel size; BIOMED I project) in three orthogonal directions using the beam-shell models and using large-scale voxel models that served as the gold standard. Excellent agreement (R(2)=0.97) was found between the two, with an average CPU-time reduction factor of 49 for the beam-shell models. In contrast to earlier skeleton-based beam models, the novel beam-shell models predicted elastic modulus values equally well for structures from different skeletal sites. It allows performing detailed parametric analyses that cover the entire spectrum of trabecular bone microstructures.     

Indirect determination of trabecular bone effective tissue failure properties using micro-finite element simulations

Trabecular bone strength is marked not only by the onset of local yielding, but also by post-yield behavior. To study and predict trabecular bone elastic and yield properties, micro-finite element (micro-FE) models were successfully applied. However, trabecular bone strength predictions require micro-FE models incorporating post-yield behavior of trabecular bone tissue. Due to experimental difficulties, such data is currently not available. Here we used micro-FE modeling to determine failure behavior of trabecular bone tissue indirectly, by iteratively fitting FE simulation to experimental results. Failure parameters were fitted to an isotropic plasticity model based on Hill's yield function, using materially and geometrically nonlinear micro-FE models of seven bovine trabecular bone specimens. The predictive value of the averaged effective tissue properties was subsequently tested. The results showed that compression softening had to be included on the tissue level in order to accurately describe the apparent-level behavior of the bone specimens. A sensitivity study revealed that the simulated response was less sensitive to variations in the post-yield properties of the bone tissue than variations in the elastic and yield properties. Due to fitting of the tissue properties, apparent-level behavior could be accurately reproduced for each specimen separately. Predictions based on the averaged and fixed tissue properties were less accurate, due to inter-specimen variations in the tissue properties.     

The effect of resorption cavities on bone stiffness is site dependent

Resorption cavities formed during the bone remodelling cycle change the structure and thus the mechanical properties of trabecular bone. We tested the hypotheses that bone stiffness loss due to resorption cavities depends on anatomical location, and that for identical eroded bone volumes, cavities would cause more stiffness loss than homogeneous erosion. For this purpose, we used beam–shell finite element models. This new approach was validated against voxel-based FE models. We found an excellent agreement for the elastic stiffness behaviour of individual trabeculae in axial compression (R² = 1.00) and in bending (R²>0.98), as well as for entire trabecular bone samples to which resorption cavities were digitally added (R² = 0.96, RMSE = 5.2%). After validation, this new method was used to model discrete cavities, with dimensions taken from a statistical distribution, on a dataset of 120 trabecular bone samples from three anatomical sites (4th lumbar vertebra, femoral head, iliac crest). Resorption cavities led to significant reductions in bone stiffness. The largest stiffness loss was found for samples from the 4th lumbar vertebra, the lowest for femoral head samples. For all anatomical sites, resorption cavities caused significantly more stiffness loss than homogeneous erosion did. This novel technique can be used further to evaluate the impact of resorption cavities, which are known to change in several metabolic bone diseases and due to treatment, on bone competence.     

Rib fractures under anterior–posterior dynamic loads: Experimental and finite-element study

The purpose of this study was to investigate whether using a finite-element (FE) mesh composed entirely of hexahedral elements to model cortical and trabecular bone (all-hex model) would provide more accurate simulations than those with variable thickness shell elements for cortical bone and hexahedral elements for trabecular bone (hex–shell model) in the modeling human ribs. First, quasi-static non-injurious and dynamic injurious experiments were performed using the second, fourth, and tenth human thoracic ribs to record the structural behavior and fracture tolerance of individual ribs under anterior–posterior bending loads. Then, all-hex and hex–shell FE models for the three ribs were developed using an octree-based and multi-block hex meshing approach, respectively. Material properties of cortical bone were optimized using dynamic experimental data and the hex–shell model of the fourth rib and trabecular bone properties were taken from the literature. Overall, the reaction force–displacement relationship predicted by both all-hex and hex–shell models with nodes in the offset middle-cortical surfaces compared well with those measured experimentally for all the three ribs. With the exception of fracture locations, the predictions from all-hex and offset hex–shell models of the second and fourth ribs agreed better with experimental data than those from the tenth rib models in terms of reaction force at fracture (difference <15.4%), ultimate failure displacement and time (difference <7.3%), and cortical bone strains. The hex–shell models with shell nodes in outer cortical surfaces increased static reaction forces up to 16.6%, compared to offset hex–shell models. These results indicated that both all-hex and hex–shell modeling strategies were applicable for simulating rib responses and bone fractures for the loading conditions considered, but coarse hex–shell models with constant or variable shell thickness were more computationally efficient and therefore preferred.     

A global relationship between trabecular bone morphology and homogenized elastic properties.

An alternative concept of the relationship between morphological and elastic properties of trabecular bone is presented and applied to human tissue from several anatomical locations using a digital approach. The three-dimensional morphology of trabecular bone was assessed with a microcomputed tomography system and the method of directed secants as well as the star volume procedure were used to compute mean intercept length (MIL) and average bone length (ABL) of 4 mm cubic specimens. Assuming isotropic elastic properties for the trabecular tissue, the general elastic tensors of the bone specimens were determined using the homogenization method and the closest orthotropic tensors were calculated with an optimization algorithm. The assumption of orthotropy for trabecular bone was found to improve with specimen size and hold within 6.1 percent for a 4 mm cube size. A strong global relationship (r2 = 0.95) was obtained between fabric and the orthotropic elastic tensor with a minimal set of five constants. Mean intercept length and average bone length provided an equivalent power of prediction. These results support the hypothesis that the elastic properties of human trabecular bone from an arbitrary anatomical location can be estimated from an approximation of the anisotropic morphology and a prior knowledge of tissue properties.     

Fast tool for evaluation of iliac crest tissue elastic properties using the reduced-basis methods

Computationally expensive finite element (FE) methods are generally used for indirect evaluation of tissue mechanical properties of trabecular specimens, which is vital for fracture risk prediction in the elderly. This work presents the application of reduced-basis (RB) methods for rapid evaluation of simulation results. Three cylindrical transiliac crest specimens (diameter: 7.5 mm, length: 10-12 mm) were obtained from healthy subjects (20 year-old, 22 year-old, and 24 year-old females) and scanned using microcomputed tomography imaging. Cubic samples of dimensions 5×5×5 mm(3) were extracted from the core of the cylindrical specimens for FE analysis. Subsequently, a FE solution library (test space) was constructed for each of the specimens by varying the material property parameters: tissue elastic modulus and Poisson's ratio, to develop RB algorithms. The computational speed gain obtained by the RB methods and their accuracy relative to the FE analysis were evaluated. Speed gains greater than 4000 times, were obtained for all three specimens for a loss in accuracy of less than 1% in the maxima of von-Mises stress with respect to the FE-based value. The computational time decreased from more than 6 h to less than 18 s. RB algorithms can be successfully utilized for real-time reliable evaluation of trabecular bone elastic properties.     

Finite element analysis of ramming in Ovis canadensis

The energy produced during the ramming of bighorn sheep (Ovis canadensis) would be expected to result in undesirable stresses in their frontal skull, which in turn would cause brain injury; yet, this animal seems to suffer no ill effects. In general, horn is made of an α-keratin sheath covering a bone. Despite volumes of data on the ramming behavior of Ovis canadensis, the extent to which structural components of horn and horn-associated structure or tissue absorb the impact energy generated by the ramming event is still unknown. This study investigates the hypothesis that there is a mechanical relationship present among the ramming event, the structural constituents of the horn, and the horn-associated structure. The three-dimensional complex structure of the bighorn sheep horn was successfully constructed and modeled using a computed tomography (CT) scan and finite element (FE) method, respectively. Three different three-dimensional quasi-static models, including a horn model with trabecular bone, a horn model with compact bone that instead of trabecular bone, and a horn model with trabecular bone as well as frontal sinuses, were studied. FE simulations were used to compare distributions of principal stress in the horn and the frontal sinuses and the strain energy under quasi-static loading conditions. It was noticed that strain energy due to elastic deformation of the complex structure of horn modeled with trabecular bone and with trabecular bone and frontal sinus was different. In addition, trabecular bone in the horn distributes the stresses over a larger volume, suggesting a mechanical link between the structural constituents and the ramming event. This phenomenon was elucidated through the principal stress distribution in the structure. This study will help designers in choosing appropriate material combinations for the successful design of protective structures against a similar impact.     

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.     

Adaptive diffuse domain approach for calculating mechanically induced deformation of trabecular bone

Remodelling of trabecular bone is essentially affected by the mechanical load of the trabeculae. Mathematical modelling and simulation of the remodelling process have to include time-consuming calculations of the displacement field within the complex trabecular structure under loading. We present an adaptive diffuse domain approach for calculating the elastic bone deformation based on micro computer tomogram data of real trabecular bone structures and compared it with a conventional voxel-based finite element method. In addition to allowing for higher computational efficiency, the adaptive approach is characterised by a very smooth representation of the bone surface, which suggests that this approach would be suitable as a basis for future simulations of bone resorption and formation processes within the trabecular structure.     

Improving the local solution accuracy of large-scale digital image-based finite element analyses

Digital image-based finite element modeling (DIBFEM) has become a widely utilized approach for efficiently meshing complex biological structures such as trabecular bone. While DIBFEM can provide accurate predictions of apparent mechanical properties, its application to simulate local phenomena such as tissue failure or adaptation has been limited by high local solution errors at digital model boundaries. Furthermore, refinement of digital meshes does not necessarily reduce local maximum errors. The purpose of this study was to evaluate the potential to reduce local mean and maximum solution errors in digital meshes using a post-processing filtration method. The effectiveness of a three-dimensional, boundary-specific filtering algorithm was found to be mesh size dependent. Mean absolute and maximum errors were reduced for meshes with more than five elements through the diameter of a cantilever beam considered representative of a single trabecula. Furthermore, mesh refinement consistently decreased errors for filtered solutions but not necessarily for non-filtered solutions. Models with more than five elements through the beam diameter yielded absolute mean errors of less than 15% for both Von Mises stress and maximum principal strain. When applied to a high-resolution model of trabecular bone microstructure, boundary filtering produced a more continuous solution distribution and reduced the predicted maximum stress by 30%. Boundary-specific filtering provides a simple means of improving local solution accuracy while retaining the model generation and numerical storage efficiency of the DIBFEM technique.     

Development and validation of a subject-specific finite element model of a human clavicle

This study aimed to develop and validate a finite element (FE) model of a human clavicle which can predict the structural response and bone fractures under both axial compression and anterior–posterior three-point bending loads. Quasi-static non-injurious axial compression and three-point bending tests were first conducted on a male clavicle followed by a dynamic three-point bending test to fracture. Then, two types of FE models of the clavicle were developed using bone material properties which were set to vary with the computed tomography image density of the bone. A volumetric solid FE model comprised solely of hexahedral elements was first developed. A solid-shell FE model was then created which modelled the trabecular bone as hexahedral elements and the cortical bone as quadrilateral shell elements. Finally, simulations were carried out using these models to evaluate the influence of variations in cortical thickness, mesh density, bone material properties and modelling approach on the biomechanical responses of the clavicle, compared with experimental data. The FE results indicate that the inclusion of density-based bone material properties can provide a more accurate reproduction of the force–displacement response and bone fracture timing than a model with uniform bone material properties. Inclusion of a variable cortical thickness distribution also slightly improves the ability of the model to predict the experimental response. The methods developed in this study will be useful for creating subject-specific FE models to better understand the biomechanics and injury mechanism of the clavicle.     

The accuracy of digital image-based finite element models.

Digital image-based finite element meshing is an alternative approach to time-consuming conventional meshing techniques for generating realistic three-dimensional (3D) models of complex structures. Although not limited to biological applications, digital image-based modeling has been used to generate structure-specific (i.e., non-generic) models of whole bones and trabecular bone microstructures. However, questions remain regarding the solution accuracy provided by the digital meshing approach, particularly at model or material boundaries. The purpose of this study was to compare the accuracy of digital and conventional smooth boundary models based on theoretical solutions for a two-dimensional (2D) compression plate and a 3D circular cantilever beam. For both the plate and beam analyses, the predicted solution at digital model boundaries was characterized by local oscillations, which produced potentially high errors within individual boundary elements. Significantly, however, the digital model boundary solution oscillated approximately about the theoretical solution. A marked improvement in solution accuracy was therefore achieved by considering average results within a region composed of several elements. Absolute errors for Von Mises stress averaged over the beam cross section, for example, converged to less than 4 percent, and the predicted free-end displacement of the cantilever beam was within 1 percent of the theoretical solution. Analyses at several beam orientations and mesh resolutions suggested a minimum discretization of three to four digital finite elements through the beam cross section to avoid high numerical stiffening errors under bending.     

Magnetic resonance microscopy for the quantitative analysis of trabecular bone architecture

A major concern for long-term spaceflight is the effect of microgravity on bone structure and mass as a loss of cortical and trabecular bone volume and density, both of which can lead to decreased bone strength and an increased risk of bone fracture. Detailed analysis of the three-dimensional structure of trabecular bone, and its relation to bone strength has become feasible only recently using high-resolution 3D imaging techniques. In particular, magnetic resonance microscopy (MRM) has proved to be particularly useful for the ex vivo evaluation of the complex architecture of trabecular bone. In this study, we describe the use of two different MRM-based methods for the quantitative evaluation of the three-dimensional structure of trabecular bone explants and for the prediction of their biomechanical properties. The in vivo application of such methods is also discussed.