Subject-specific finite element models implementing a maximum principal strain criterion are able to estimate failure risk and fracture location on human femurs tested in vitro

No agreement on the choice of the failure criterion to adopt for the bone tissue can be found in the literature among the finite element studies aiming at predicting fracture risk of bones. The use of stress-based criteria seems to prevail on strain-based ones, while basic bone biomechanics suggest using strain parameters to describe failure. The aim of the present combined experimental-numerical study was to verify, using subject-specific finite element models able to accurately predict strains, if a strain-based failure criterion could identify the failure patterns of bones. Three cadaver femurs were CT-scanned and subsequently fractured in a clinically relevant single-stance loading scenario. Load-displacement curves and high-speed movies were acquired to define the failure load and the location of fracture onset, respectively. Subject-specific finite element models of the three femurs were built from CT data following a validated procedure. A maximum principal strain criterion was implemented in the finite element models, and two stress-based criteria selected for comparison. The failure loads measured were applied to the models, and the computed risks of fracture were compared to the results of the experimental tests. The proposed principal strain criterion managed to correctly identify the level of failure risk and the location of fracture onset in all the modelled specimens, while Von Mises or maximum principal stress criteria did not give significant information. A maximum principal strain criterion can thus be defined a suitable candidate for the in vivo risk factor assessment on long bones.     

Characterization of craniofacial sutures using the finite element method

Characterizing the biomechanical behavior of sutures in the human craniofacial skeleton (CFS) is essential to understand the global impact of these articulations on load transmission, but is challenging due to the complexity of their interdigitated morphology, the multidirectional loading they are exposed to and the lack of well-defined suture material properties. This study aimed to quantify the impact of morphological features, direction of loading and suture material properties on the mechanical behavior of sutures and surrounding bone in the CFS. Thirty-six idealized finite element (FE) models were developed. One additional specimen-specific FE model was developed based on the morphology obtained from a µCT scan to represent the morphological complexity inherent in CFS sutures. Outcome variables of strain energy (SE) and von Mises stress (σ_vm) were evaluated to characterize the sutures’ biomechanical behavior. Loading direction was found to impact the relationship between SE and interdigitation index and yielded varied patterns of σ_vm in both the suture and surrounding bone. Adding bone connectivity reduced suture strain energy and altered the σ_vm distribution. Incorporating transversely isotropic material properties was found to reduce SE, but had little impact on stress patterns. High-resolution µCT scanning of the suture revealed a complex morphology with areas of high and low interdigitations. The specimen specific suture model results were reflective of SE absorption and σ_vm distribution patterns consistent with the simplified FE results. Suture mechanical behavior is impacted by morphologic factors (interdigitation and connectivity), which may be optimized for regional loading within the CFS.     

The effects of misalignment during in vivo loading of bone: Techniques to detect the proximity of objects in three-dimensional models

Theories of mechanical adaptation of bone suggest that mechanical loading causes bone formation at discrete locations within bone microstructure experiencing the greatest mechanical stress/strain. Experimental testing of such theories requires in vivo loading experiments and high-resolution finite element models to determine the distribution of mechanical stresses. Finite element models of in vivo loading experiments typically assume idealized boundary conditions with applied load perfectly oriented on the bone, however small misalignments in load orientation during an in vivo experiment are unavoidable, and potentially confound the ability of finite element models to predict locations of bone formation at the scale of micrometers. Here we demonstrate two different three-dimensional spatial correlation methods to determine the effects of misalignment in load orientation on the locations of high mechanical stress/strain in the rodent tail loading model. We find that, in cancellous bone, the locations of tissue with high stress are maintained under reasonable misalignments in load orientation (p<0.01). In cortical bone, however, angular misalignments in the dorsal direction can alter the locations of high mechanical stress, but the locations of tissue with high stress are maintained under other misalignments (p<0.01). We conclude that, when using finite element models of the rodent tail loading model, small misalignments in loading orientation do not affect the predicted locations of high mechanical stress within cancellous bone.     

About the inevitable compromise between spatial resolution and accuracy of strain measurement for bone tissue: A 3D zero-strain study

The accurate measurement of local strain is necessary to study bone mechanics and to validate micro computed tomography (µCT) based finite element (FE) models at the tissue scale. Digital volume correlation (DVC) has been used to provide a volumetric estimation of local strain in trabecular bone sample with a reasonable accuracy. However, nothing has been reported so far for µCT based analysis of cortical bone. The goal of this study was to evaluate accuracy and precision of a deformable registration method for prediction of local zero-strains in bovine cortical and trabecular bone samples. The accuracy and precision were analyzed by comparing scans virtually displaced, repeated scans without any repositioning of the sample in the scanner and repeated scans with repositioning of the samples.     

Determination of orthotropic bone elastic constants using FEA and modal analysis

Finite element models have been widely employed in an effort to quantify the stress and strain distribution around implanted prostheses and to explore the influence of these distributions on their long-term stability. In order to provide meaningful predictions, such models must contain an appropriate reflection of mechanical properties. Detailed geometrical and density information is now readily available from CT scanning. However, despite the use of phantoms, a method of determining mechanical properties (or elastic constants) from bone density has yet to be made available in a usable form.In this study, a cadaveric bone was CT scanned and its natural frequencies were measured using modal analysis. Using the geometry obtained from the CT scan data, a finite element mesh was created with the distribution of density established by matching the mass of the FE bone model with the mass of the cadaveric bone. The maximum values of the orthotropic elastic constants were then established by matching the predictions from FE modal analyses to the experimental natural frequencies, giving a maximum error of 7.8% over 4 modes of vibration. Finally, the elastic constants of the bone derived from the analyses were compared with those measured using ultrasound techniques. This produced a difference of <1% for both the maximum density and axial Young's Modulus. This study has thereby produced an orthotropic finite element model of a human femur. More importantly, however, is the implication that it is possible to create a valid FE model by simply comparing the FE results with the measured resonant frequency of the CT scanned bone.     

Femoral strength is better predicted by finite element models than QCT and DXA.

Clinicians and patients would benefit if accurate methods of predicting and monitoring bone strength in-vivo were available. A group of 51 human femurs (age range 21-93; 23 females, 28 males) were evaluated for bone density and geometry using quantitative computed tomography (QCT) and dual energy X-ray absorptiometry (DXA). Regional bone density and dimensions obtained from QCT and DXA were used to develop statistical models to predict femoral strength ex vivo. The QCT data also formed the basis of a three-dimensional finite element (FE) models to predict structural stiffness. The femurs were separated into two groups; a model training set (n = 25) was used to develop statistical models to predict ultimate load, and a test set (n = 26) was used to validate these models. The main goal of this study was to test the ability of DXA, QCT and FE techniques to predict fracture load non-invasively, in a simple load configuration which produces predominantly femoral neck fractures. The load configuration simulated the single stance phase portion of normal gait; in 87% of the specimens, clinical appearing sub-capital fractures were produced. The training/test study design provided a tool to validate that the predictive models were reliable when used on specimens with "unknown" strength characteristics. The FE method explained at least 20% more of the variance in strength than the DXA models. Planned refinements of the FE technique are expected to further improve these results. Three-dimensional FE models are a promising method for predicting fracture load, and may be useful in monitoring strength changes in vivo.     

A new approach for assigning bone material properties from CT images into finite element models

Generation of subject-specific finite element (FE) models from computed tomography (CT) datasets is of significance for application of the FE analysis to bone structures. A great challenge that remains is the automatic assignment of bone material properties from CT Hounsfield Units into finite element models. This paper proposes a new assignment approach, in which material properties are directly assigned to each integration point. Instead of modifying the dataset of FE models, the proposed approach divides the assignment procedure into two steps: generating the data file of the image intensity of a bone in a MATLAB program and reading the file into ABAQUS via user subroutines. Its accuracy has been validated by assigning the density of a bone phantom into a FE model. The proposed approach has been applied to the FE model of a sheep tibia and its applicability tested on a variety of element types. The proposed assignment approach is simple and illustrative. It can be easily modified to fit users’ situations.     

Mechanical significance of femoral head trabecular bone structure in Loris and Galago evaluated using micromechanical finite element models

Work on the interspecific and intraspecific variation of trabecular bone in the proximal femur of primates demonstrates important architectural variation between animals with different locomotor behaviors. This variation is thought to be related to the processes of bone adaptation whereby bone structure is optimized to the mechanical environment. Micromechanical finite element models were created for the proximal femur of the leaping Galago senegalensis and the climbing and quadrupedal Loris tardigradus by converting bone voxels from high-resolution X-ray computed tomography scans of the femoral head to eight-noded brick elements. The resulting models had approximately 1.8 million elements each. Loading conditions representing takeoff phase of a leap and more generalized load orientations were applied to the models, and the models were solved using the iterative "row-by-row" matrix-vector multiplication algorithm. The principal strain and Von Mises stress results for the leaping model were similar for both species at each load orientation. Similar hip joint reaction forces in the range of 4.9 x to 12 x body weight were calculated for both species under each loading condition, but the hip reaction values estimated for Loris were higher than predicted based on locomotor behavior. These results suggest that functional adaptation to hip joint loading may not fully explain the differences in femoral head trabecular bone structure in Galago and Loris. The finite element method represents a unique and useful tool for analyzing the functional adaptation of trabecular bone in a diversity of animals and for reconstructing locomotor behavior in extinct taxa.     

Heterogeneous linear elastic trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties

High-resolution voxel-based finite element software, such as FEEBE developed at the NCBES, is widely used for studying trabecular bone at the micro-scale. A new approach to determine heterogeneous bone tissue material properties for computational models was proposed in this study. The specimen-specific range of tissue moduli across strut width was determined from nanoindentation testing. This range was mapped directly using linear interpolation to that specimen's micro-computed tomography (microCT) grey value range as input material properties for finite element analysis. The method was applied to cuboid trabecular bone samples taken from eight, 4-year-old (skeletally mature) ovine L5 vertebrae. Before undergoing experimental uniaxial compression tests, the samples were microCT scanned and 30 microm resolution finite element models were generated. The linear elastic finite element models were compressed to 1% strain. This material property assignment method for computational models accurately reproduced the experimentally determined apparent modulus and concentrations of stress at locations of failure.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

Several stimuli are proposed in the bone remodeling theory. It is not clear, if a unique solution exists and if the result is convergent using a certain stimulus. In this study, the strain stimulus, strain energy stimulus and the von Mises stress stimulus for bone remodeling are compared and applied to a square plate model using the finite element method. In the plane stress state, the remodeling equilibrium equations are transformed into functions of only the principal strains and the graphs of these functions are drawn in a diagram using the principal strains as the variables of two coordinate axes. The equation of the sum of principal strain squared equal to a constant is a circle in the diagram. The remodeling equilibrium equation of the strain stimulus is a quadrangle fitting into the circle, the remodeling equilibrium equation of the strain energy stimulus is an ellipse and the remodeling equilibrium equation of the von Mises stress stimulus is also an ellipse close to the principal strains circle when we take the same constants in the above equations. Using the finite element method, two models are performed with the uniform initial elastic properties and with the semi-random initial distribution of the elastic properties. The principal strains as the final finite element results converge within 2% of the objective constant for all the different stimuli. The obtained Young's moduli of two models as the adaptation object are different but in equilibrium, i.e. the equilibrium solution of adaptation model is not unique. The principal strains can not be used to examine the uniqueness of solution, since two different solutions can have the same results of principal strains. Using a certain stimulus, certain initial properties and a certain iterative equation, the solution is unique in equilibrium. The results using the model in this study show also that the same results can be obtained using any of the three stimuli when a proper constant in each remodeling equilibrium equation is chosen.     

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.     

Computer aided stress analysis of long bones utilizing computed tomography

A computer aided analysis method has been developed which utilizes computed tomography (CT) and a finite element (FE) computer program to determine the stress-displacement pattern in a long bone section. The CT data file provides the geometry, the apparent density and the elastic properties for the three-dimensional FE model. A developed pre-processor generates the FE model of a human diaphyseal tibia section which is then analyzed by the SAP IV finite element program. The results obtained are sorted and displayed by a developed post-processor and compared with stresses and deformations from the literature. The model generation method was verified by applying it to a model of simple geometry and boundary conditions, then comparing the results with the analytical solution of the same problem. The convergence behavior of nodal displacements was tested as a function of mesh refinement. This method provides an automatic, versatile, non-invasive and accurate tool of long bone modeling for finite element stress analysis.     

The use of strain energy as a convergence criterion in the finite element modelling of bone and the effect of model geometry on stress convergence

Keyak and Skinner¹ state that the adequacy of a finite element mesh may be verified by confirming that convergence of strain energy has been achieved. The paper concludes by stating that convergence of strain energy does not ensure that a particular mesh is adequate for producing accurate stress/strain results, and that only qualitative results can be achieved from 3D models of bone. This conclusion is challenged and the effect of element size on stress convergence is clarified. The influence of element size on stress convergence in regions of surface irregularity is studied and current methods are discussed that facilitate convergence.     

Localized damage in vertebral bone is most detrimental in regions of high strain energy density

It was hypothesized that damage to bone tissue would be most detrimental to the structural integrity of the vertebral body if it occurred in regions with high strain energy density, and not necessarily in regions of high or low trabecular bone apparent density, or in a particular anatomic location. The reduction in stiffness due to localized damage was computed in 16 finite element models of 10-mm-thick human vertebral sections. Statistical analyses were performed to determine which characteristic at the damage location--strain energy density, apparent density, or anatomic location--best predicted the corresponding stiffness reduction. There was a strong positive correlation between regional strain energy density and structural stiffness reduction in all 16 vertebral sections for damage in the trabecular centrum (p < 0.05, r2 = 0.43-0.93). By contrast, regional apparent density showed a significant negative correlation to stiffness reduction in only four of the sixteen bones (p < 0.05, r2 = 0.47-0.58). While damage in different anatomic locations did lead to different reductions in stiffness (p < 0.0001, ANOVA), no single location was consistently the most critical location for damage. Thus, knowledge of the characteristics of bone that determine strain energy density distributions can provide an understanding of how damage reduces whole bone mechanical properties. A patient-specific finite element model displaying a map of strain energy density can help optimize surgical planning and reinforcement of bone in individuals with high fracture risk.     

Mechanics of linear microcracking in trabecular bone

Microcracking in trabecular bone is responsible both for the mechanical degradation and remodeling of the trabecular bone tissue. Recent results on trabecular bone mechanics have demonstrated that bone tissue microarchitecture, tissue elastic heterogeneity and tissue-level mechanical anisotropy all should be considered to obtain detailed information on the mechanical stress state. The present study investigated the influence of tissue microarchitecture, tissue heterogeneity in elasticity and material separation properties and tissue-level anisotropy on the microcrack formation process. Microscale bone models were executed with the extended finite element method. It was demonstrated that anisotropy and heterogeneity of the bone tissue contribute significantly to bone tissue toughness and the resistance of trabecular bone to microcrack formation. The compressive strain to microcrack initiation was computed to increase by a factor of four from an assumed homogeneous isotropic tissue to an assumed anisotropic heterogenous tissue.     

Simulation of the behaviour of the L1 vertebra for different material properties and loading conditions

Three-dimensional finite element models of the thoracolumbar junction (T12–L2) and isolated L1 vertebra were developed to investigate the role of material properties and loading conditions on vertebral stresses and strains to predict fracture risk. The geometry of the vertebrae was obtained from computed tomography images. The isolated vertebra model included an L1 vertebra loaded through polymethylmethacrylate plates located at the top and bottom of the vertebra, and the segment model included T12 to L2 vertebrae and seven ligaments, fibrous intervertebral discs and facet joints. Each model was examined with both homogeneous and spatially varying bone tissue properties. Stresses and strains were compared for uniform compression and flexion. Including material heterogeneity remarkably reduced the stiffness of the isolated L1 vertebra and increased the magnitudes of the minimum principal strains and stresses in the mid-transverse section. The stress and strain distributions further changed when physiological loading was applied to the L1 vertebra. In the segment models, including heterogeneous material properties increased the magnitude of the minimum principal strain by 158% in the centre of the mid-transverse section. Overall, the inclusion of heterogeneity and physiological loading increased the magnitude of the strains up to 346% in flexion and 273% in compression.     

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

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

Predicting bone remodeling around tissue- and bone-level dental implants used in reduced bone width

The objective of this study was to predict time-dependent bone remodeling around tissue- and bone-level dental implants used in patients with reduced bone width. The remodeling of bone around titanium tissue-level, and titanium and titanium–zirconium alloy bone-level implants was studied under 100 N oblique load for one month by implementing the Stanford theory into three-dimensional finite element models. Maximum principal stress, minimum principal stress, and strain energy density in peri-implant bone and displacement in x- and y- axes of the implant were evaluated. Maximum and minimum principal stresses around tissue-level implant were higher than bone-level implants and both bone-level implants experienced comparable stresses. Total strain energy density in bone around titanium implants slightly decreased during the first two weeks of loading followed by a recovery, and the titanium–zirconium implant showed minor changes in the axial plane. Total strain energy density changes in the loading and contralateral sides were higher in tissue-level implant than other implants in the cortical bone at the horizontal plane. The displacement values of the implants were almost constant over time. Tissue-level implants were associated with higher stresses than bone-level implants. The time-dependent biomechanical outcome of titanium–zirconium alloy bone-level implant was comparable to the titanium implant.     

The combination of photogrammetry and finite elements for a fine grained functional analysis of anatomical structures

Summary A new method of functional morphological analysis is presented. Combining stereophotogrammetry with the finite element technique, a new approach, permits a three-dimensional numerical stress analysis of arbitrarily shaped bodies to be performed. The stereophotogrammetric method which originated for three-dimensional calculations in the study of surfaces in land surveying is well suited for the determination of the nodal co-ordinates required for the finite element method, an engineering technique developed for behavioural analysis of solids and fluids responding to external forces. This approach was tested in a study of the functional morphology of the bill of an African wading bird, the shoebill Balaeniceps rex. A few findings of that study are given here in order to demonstrate the method. Advantages of the finite element method compared with other techniques for stress analysis of anatomical structures are also discussed. The method presents exciting possibilities for predicting displacement and stress responses more accurately and in much greater detail. The scope of this powerful computerized stress analysis technique is greatly enhanced with the introduction of stereophotogrammetry for determining the three-dimensional co-ordinates of complex anatomical structures. With the finite element method, the properties of the bone structure can be modelled as they occur in the life of the animal. This is not possible with physical models. Furthermore, rare specimens can be analysed non-destructively.     

Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation

The aim of this paper is to develop a multiscale hierarchical hybrid model based on finite element analysis and neural network computation to link mesoscopic scale (trabecular network level) and macroscopic (whole bone level) to simulate the process of bone remodelling. As whole bone simulation, including the 3D reconstruction of trabecular level bone, is time consuming, finite element calculation is only performed at the macroscopic level, whilst trained neural networks are employed as numerical substitutes for the finite element code needed for the mesoscale prediction. The bone mechanical properties are updated at the macroscopic scale depending on the morphological and mechanical adaptation at the mesoscopic scale computed by the trained neural network. The digital image-based modelling technique using μ-CT and voxel finite element analysis is used to capture volume elements representativeof 2 mm³ at the mesoscale level of the femoral head. The input data for the artificial neural network are a set of bone material parameters, boundary conditions and the applied stress. The output data are the updated bone properties and some trabecular bone factors. The current approach is the first model, to our knowledge, that incorporates both finite element analysis and neural network computation to rapidly simulate multilevel bone adaptation.