A transversely isotropic viscoelastic constitutive equation for brainstem undergoing finite deformation

The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was 12.7 Pa, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was 121.2 Pa, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and 1.4538 s, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.     

Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation

High-resolution architecture-based finite element models are commonly used for characterizing the mechanical behavior of cancellous bone. The vast majority of studies use homogeneous material properties to model trabecular tissue. The objectives of this study were to demonstrate that inhomogeneous finite element models that account for microcomputed tomography-measured tissue modulus variability more accurately predict the apparent stiffness of cancellous bone than homogeneous models, and to examine the sensitivity of an inhomogeneous model to the degree of tissue property variability. We tested five different material cases in finite element models of ten cancellous cubes in simulated uniaxial compression. Three of these cases were inhomogeneous and two were homogeneous. Four of these cases were unique to each specimen, and the remaining case had the same tissue modulus for all specimens. Results from all simulations were compared with measured elastic moduli from previous experiments. Tissue modulus variability for the most accurate of the three inhomogeneous models was then artificially increased to simulate the effects of non-linear CT-attenuation-modulus relationships. Uniqueness of individual models was more critical for model accuracy than level of inhomogeneity. Both homogeneous and inhomogeneous models that were unique to each specimen had at least 8% greater explanatory power for apparent modulus than models that applied the same material properties to all specimens. The explanatory power for apparent modulus of models with a tissue modulus coefficient of variation (COV) range of 21-31% was 13% greater than homogeneous models (COV=0). The results of this study indicate that inhomogenous finite element models that have tissue moduli unique to each specimen more accurately predict the elastic behavior of cancellous cubic specimens than models that have common tissue moduli between all specimens.     

Trabecular bone's mechanical properties are affected by its non-uniform mineral distribution

The bone remodeling process takes place at the surface of trabeculae and results in a non-uniform mineral distribution. This will affect the mechanical properties of cancellous bone, because the properties of bone tissue depend on its mineral content. We investigated how large this effect is by simulating several non-uniform mineral distributions in 3D finite element models of human trabecular bone and calculating the apparent stiffness of these models. In the ‘linear model’ we assumed a linear relation between mineral content and Young's modulus of the tissue. In the ‘exponential model’ we included an empirical exponential relation in the model. When the linear model was used the mineral distribution slightly changed the apparent stiffness, the difference varied between an 8% decrease and a 4% increase compared to the uniform model with the same BMD. The exponential model resulted in up to 20% increased apparent stiffness in the main load-bearing direction. A thin less mineralized surface layer (28 μm) and highly mineralized interstitial bone (mimicking mineralization resulting from anti-resorptive treatment) resulted in the highest stiffness. This could explain large reductions in fracture risk resulting from small increases in BMD. The non-uniform mineral distribution could also explain why bone tissue stiffness determined using nano-indentation is usually higher than finite element (FE)-determined stiffness. We conclude that the non-uniform mineral distribution in trabeculae does affect the mechanical properties of cancellous bone and that the tissue stiffness determined using FE-modeling could be improved by including detailed information about mineral distribution in trabeculae in the models.     

Tensile properties of the mandibular condylar cartilage

Mandibular condylar cartilage plays a crucial role in temporomandibular joint (TMJ) function, which includes facilitating articulation with the temporomandibular joint disc and reducing loads on the underlying bone. The cartilage experiences considerable tensile forces due to direct compression and shear. However, only scarce information is available about its tensile properties. The present study aims to quantify the biomechanical characteristics of the mandibular condylar cartilage to aid future three-dimensional finite element modeling and tissue engineering studies. Porcine condylar cartilage was tested under uniaxial tension in two directions, anteroposterior and mediolateral, with three regions per direction. Stress relaxation behavior was modeled using the Kelvin model and a second-order generalized Kelvin model, and collagen fiber orientation was determined by polarized light microscopy. The stress relaxation behavior of the tissue was biexponential in nature. The tissue exhibited greater stiffness in the anteroposterior direction than in the mediolateral direction as reflected by higher Young's (2.4 times), instantaneous (1.9 times), and relaxed (1.9 times) moduli. No significant differences were observed among the regional properties in either direction. The predominantly anteroposterior macroscopic fiber orientation in the fibrous zone of condylar cartilage correlated well with the biomechanical findings. The condylar cartilage appears to be less stiff and less anisotropic under tension than the anatomically and functionally related TMJ disc. The anisotropy of the condylar cartilage, as evidenced by tensile behavior and collagen fiber orientation, suggests that the shear environment of the TMJ exposes the condylar cartilage to predominantly but not exclusively anteroposterior loading.     

Effects of myocardial sheetlet sliding on left ventricular function

Left ventricle myocardium has a complex micro-architecture, which was revealed to consist of myocyte bundles arranged in a series of laminar sheetlets. Recent imaging studies demonstrated that these sheetlets re-orientated and likely slided over each other during the deformations between systole and diastole, and that sheetlet dynamics were altered during cardiomyopathy. However, the biomechanical effect of sheetlet sliding is not well-understood, which is the focus here. We conducted finite element simulations of the left ventricle (LV) coupled with a windkessel lumped parameter model to study sheetlet sliding, based on cardiac MRI of a healthy human subject, and modifications to account for hypertrophic and dilated geometric changes during cardiomyopathy remodeling. We modeled sheetlet sliding as a reduced shear stiffness in the sheet-normal direction and observed that (1) the diastolic sheetlet orientations must depart from alignment with the LV wall plane in order for sheetlet sliding to have an effect on cardiac function, that (2) sheetlet sliding modestly aided cardiac function of the healthy and dilated hearts, in terms of ejection fraction, stroke volume, and systolic pressure generation, but its effects were amplified during hypertrophic cardiomyopathy and diminished during dilated cardiomyopathy due to both sheetlet angle configuration and geometry, and that (3) where sheetlet sliding aided cardiac function, it increased tissue stresses, particularly in the myofibre direction. We speculate that sheetlet sliding is a tissue architectural adaptation to allow easier deformations of the LV walls so that LV wall stiffness will not hinder function, and to provide a balance between function and tissue stresses. A limitation here is that sheetlet sliding is modeled as a simple reduction in shear stiffness, without consideration of micro-scale sheetlet mechanics and dynamics.

     

Biomechanical properties of the mandibular condylar cartilage and their relevance to the TMJ disc

Mandibular condylar cartilage plays a crucial role in temporomandibular joint (TMJ) function, which includes facilitating articulation with the TMJ disc, reducing loads on the underlying bone, and contributing to bone remodeling. To improve our understanding of the TMJ function in normal and pathological situations, accurate and validated three-dimensional (3-D) finite element models (FEMs) of the human TMJ may serve as valuable diagnostic tools as well as predictors of thresholds for tissue damage resulting from parafunctional activities and trauma. In this context, development of reliable biomechanical standards for condylar cartilage is crucial. Moreover, biomechanical characteristics of the native tissue are important design parameters for creating functional tissue-engineered replacements. Towards these goals, biomechanical characteristics of the condylar cartilage have been reviewed here, highlighting the structure–function correlations. Structurally, condylar cartilage, like the TMJ disc, exhibits zonal and topographical heterogeneity. Early structural investigations of the condylar cartilage have suggested that the tissue possesses a somewhat transversely isotropic orientation of collagen fibers in the fibrous zone. However, recent tensile and shear evaluations have reported a higher stiffness of the tissue in the anteroposterior direction than in the mediolateral direction, corresponding to an anisotropic fiber orientation comparable to the TMJ disc. In a few investigations, condylar cartilage under compression was found to be stiffer anteriorly than posteriorly. As with the TMJ disc, further compressive characterization is warranted. To draw inferences for human tissue using animal models, establishing stiffness–thickness correlations and regional evaluation of proteoglycan/glycosaminoglycan content may be essential. Efforts directed from the biomechanics community for the characterization of TMJ tissues will facilitate the development of reliable and accurate 3-D FEMs of the human TMJ.     

Biomechanical analysis of the shear behaviour adjacent to an axially loaded implant

Good mechanical fixation of an implant to the surrounding bone is important for its longevity, and is influenced by both biological and mechanical factors. This study parametrically evaluates the mechanics of the interface with a computationally efficient analytic structural model of the shear stress field and global shear stiffness of an axially loaded implant. The utility of the analytic model was first established by validating its assumptions with a case-specific finite element model. We then used the analytic model for a sensitivity analysis of the relationship between the pattern of tissue growth and shear properties of the interface for our previously reported loaded in vivo experimental micromotion device. The bone located directly at the implant surface was found to be the most effective site for increasing interface stiffness. This suggests that the implant surface is the most desirable site for bone growth, yet is also the most mechanically challenging environment due to its maximal shear stresses. Thus, these findings support the further investigation of osteo-conductive coatings and other biological stimuli to overcome the challenging mechanics, and to promote bone growth directly at the implant surface. The model also demonstrated that the mechanical contribution to the global implant shear stiffness of a commonly observed isolated sclerotic bone rim is very limited. The results of this sensitivity analysis agree with experimental studies with the micromotion device, and with clinical studies reporting good results with osteo-conductive coatings.     

How does muscle stiffness affect the internal deformations within the soft tissue layers of the buttocks under constant loading?

Mechanical loading of soft tissues covering bony prominences can cause skeletal muscle damage, ultimately resulting in a severe pressure ulcer termed deep tissue injury (DTI). Deformation plays an important role in the aetiology of DTI. Therefore, it is essential to minimise internal muscle deformations in subjects at risk of DTI. As an example, spinal cord-injured (SCI) individuals exhibit structural changes leading to a decrease in muscle thickness and stiffness, which subsequently increase the tissue deformations. In the present study, an animal-specific finite element model, where the geometry and boundary conditions were derived from magnetic resonance images, was developed. It was used to investigate the internal deformations in the muscle, fat and skin layers of the porcine buttocks during loading. The model indicated the presence of large deformations in both the muscle and the fat layers, with maximum shear strains up to 0.65 in muscle tissue and 0.63 in fat. Furthermore, a sensitivity analysis showed that the tissue deformations depend considerably on the relative stiffness values of the different tissues. For example, a change in muscle stiffness had a large effect on the muscle deformations. A 50% decrease in stiffness caused an increase in maximum shear strain from 0.65 to 0.99, whereas a 50% increase in stiffness resulted in a decrease in maximum shear strain from 0.65 to 0.49. These results indicate the importance of restoring tissue properties after SCI, with the use of, for example, electrical stimulation, to prevent the development of DTI.     

An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment

This study deals with the viscoelastic constitutive modeling and the respective computational analysis of the human passive myocardium. We start by recapitulating the locally orthotropic inner structure of the human myocardial tissue and model the mechanical response through invariants and structure tensors associated with three orthonormal basis vectors. In accordance with recent experimental findings the ventricular myocardial tissue is assumed to be incompressible, thick-walled, orthotropic and viscoelastic. In particular, one spring element coupled with Maxwell elements in parallel endows the model with viscoelastic features such that four dashpots describe the viscous response due to matrix, fiber, sheet and fiber-sheet fragments. In order to alleviate the numerical obstacles, the strictly incompressible model is altered by decomposing the free-energy function into volumetric-isochoric elastic and isochoric-viscoelastic parts along with the multiplicative split of the deformation gradient which enables the three-field mixed finite element method. The crucial aspect of the viscoelastic formulation is linked to the rate equations of the viscous overstresses resulting from a 3-D analogy of a generalized 1-D Maxwell model. We provide algorithmic updates for second Piola–Kirchhoff stress and elasticity tensors. In the sequel, we address some numerical aspects of the constitutive model by applying it to elastic, cyclic and relaxation test data obtained from biaxial extension and triaxial shear tests whereby we assess the fitting capacity of the model. With the tissue parameters identified, we conduct (elastic and viscoelastic) finite element simulations for an ellipsoidal geometry retrieved from a human specimen.     

A finite element based method to determine the properties of planar soft tissue.

A finite element based method to determine the incremental elastic material properties of planar membranes was developed and evaluated. The method is applicable to tissues that exhibit inhomogeneity, geometric and material nonlinearity, and anisotropy. Markers are placed on the tissue to form a four-node quadrilateral element. The specimen is loaded to an initial reference state, then three incremental loading sets are applied and the nodal displacements recorded. One of these loadings must include shear. These data are used to solve an over-determined system of equations for the tangent stiffness matrix. The method was first verified using analytical data. Next, data obtained from a latex rubber sheet were used to evaluate experimental procedures. Finally, experiments conducted on preconditioned rat skin revealed nonlinear orthotropic behavior. The vector norm comparing the applied and calculated nodal force vectors was used to evaluate the accuracy of the solutions.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Computational study of the elastic properties of Rheum rhabarbarum tissues via surrogate models of tissue geometry

Plant petioles and stems are hierarchical cellular structures, displaying geometrical features defined at multiple length scales. One or more of the intermediate hierarchical levels consists of tissues in which the cellular distribution is quasi-random, a factor that affects the elastic properties of the tissues. The current work focuses on the finite element analysis (FEA) of the constituent tissues of the plant Rheum rhabarbarum (rhubarb). The geometric model is generated via a recently introduced method: the finite edge centroidal Voronoi tessellation (FECVT), which is capable to capture the gradients of cellularity and diversified pattern of cellular materials, as opposed to current approaches in literature. The effective stiffness of the tissues is obtained by using an accurate numerical homogenization technique via detailed finite element analysis of the models of sub-regions of the tissues. As opposed to a large-scale representative volume element (RVE), statistical volume elements (SVE) are considered in this work to model tissue microstructures that are highly random. 2D finite element analyses demonstrate that the distribution of cells in collenchyma and parenchyma tissue make them stiffer in two different directions, while the overall effect of the combined tissues results in approximately equal stiffness in both directions. The rhubarb tissues, on the other hand, are more compliant than periodic and quasi-uniform random cellular materials by a factor of up to 47% and 44%, respectively. The variations of the stiffness shows the stiffening role that cell shape, size, and graded cellular distribution play in the mechanics of the rhubarb tissue.     

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.     

Shear wave elastography can assess the in-vivo nonlinear mechanical behavior of heel-pad

This study combines non-invasive mechanical testing with finite element (FE) modelling to assess for the first time the reliability of shear wave (SW) elastography for the quantitative assessment of the in-vivo nonlinear mechanical behavior of heel-pad. The heel-pads of five volunteers were compressed using a custom-made ultrasound indentation device. Tissue deformation was assessed from B-mode ultrasound and force was measured using a load cell to calculate the force – deformation graph of the indentation test. These results were used to design subject specific FE models and to inverse engineer the tissue’s hyperelastic material coefficients and its stress – strain behavior. SW speed was measured for different levels of compression (from 0% to 50% compression). SW speed for 0% compression was used to assess the initial stiffness of heel-pad (i.e. initial shear modulus, initial Young’s modulus). Changes in SW speed with increasing compressive loading were used to quantify the tissue’s nonlinear mechanical behavior based on the theory of acoustoelasticity. Statistical analysis of results showed significant correlation between SW-based and FE-based estimations of initial stiffness, but SW underestimated initial shear modulus by 64%(±16). A linear relationship was found between the SW-based and FE-based estimations of nonlinear behavior. The results of this study indicate that SW elastography is capable of reliably assessing differences in stiffness, but the absolute values of stiffness should be used with caution. Measuring changes in SW speed for different magnitudes of compression enables the quantification of the tissue’s nonlinear behavior which can significantly enhance the diagnostic value of SW elastography.     

A new material mapping procedure for quantitative computed tomography-based, continuum finite element analyses of the vertebra

Inaccuracies in the estimation of material properties and errors in the assignment of these properties into finite element models limit the reliability, accuracy, and precision of quantitative computed tomography (QCT)-based finite element analyses of the vertebra. In this work, a new mesh-independent, material mapping procedure was developed to improve the quality of predictions of vertebral mechanical behavior from QCT-based finite element models. In this procedure, an intermediate step, called the material block model, was introduced to determine the distribution of material properties based on bone mineral density, and these properties were then mapped onto the finite element mesh. A sensitivity study was first conducted on a calibration phantom to understand the influence of the size of the material blocks on the computed bone mineral density. It was observed that varying the material block size produced only marginal changes in the predictions of mineral density. Finite element (FE) analyses were then conducted on a square column-shaped region of the vertebra and also on the entire vertebra in order to study the effect of material block size on the FE-derived outcomes. The predicted values of stiffness for the column and the vertebra decreased with decreasing block size. When these results were compared to those of a mesh convergence analysis, it was found that the influence of element size on vertebral stiffness was less than that of the material block size. This mapping procedure allows the material properties in a finite element study to be determined based on the block size required for an accurate representation of the material field, while the size of the finite elements can be selected independently and based on the required numerical accuracy of the finite element solution. The mesh-independent, material mapping procedure developed in this study could be particularly helpful in improving the accuracy of finite element analyses of vertebroplasty and spine metastases, as these analyses typically require mesh refinement at the interfaces between distinct materials. Moreover, the mapping procedure is not specific to the vertebra and could thus be applied to many other anatomic sites.     

A large-strain finite element formulation for biological tissues with application to mitral valve leaflet tissue mechanics

This paper presents a finite element formulation suitable for large-strain modeling of biological tissues and uses this formulation to implement an accurate finite element model for mitral valve leaflet tissue. First, an experimentally derived strain energy function is obtained from literature. This function is implemented in finite elements using the mixed pressure-displacement formulation. A modification is made to aid in maintaining positive definiteness of the stiffness matrix at low strains. The numerical implementation is shown to be accurate in representing the analytical model of material behavior. The mixed formulation is useful for modeling of soft biological tissues in general, and the model presented here is applicable to finite element simulation of mitral valve mechanics.     

Which axial and bending stiffnesses of posterior implants are required to design a flexible lumbar stabilization system?

Dynamic stabilization devices have been introduced to clinics as an alternative to rigid fixation. The stiffness of these devices varies widely, whereas the optimal stiffness, achieving a predefined stabilization of the spine, is unknown. This study was focused on the determination of stiffness values for posterior stabilization devices achieving a flexible, semi-flexible or rigid connection between two vertebrae.An extensively validated finite element model of a lumbar spinal segment L4-5 with an implanted posterior fixation device was used in this study. The model was exposed to pure moments of 7.5 and 20 Nm around the three principal anatomical directions, simulating flexion, extension, lateral bending and axial rotation. In parametrical studies, the influence of the axial and bending fixator stiffness on the spinal range of motion was investigated. In order to examine the validity of the computed results, an in-vitro study was carried out. In this, the influence of two posterior stabilization devices (DSS? and rigidly internal fixator) on the segmental stabilization was investigated.The finite element (FE)-model predicted that each load direction caused a pairing of stiffness relations between axial and bending stiffness. In flexion and extension, however, the bending stiffness had a neglectable effect on the segmental stabilization, compared to the axial stiffness. In contrast, lateral bending and axial rotation were influenced by both stiffness parameters. Except in axial rotation, the model predictions were in a good agreement with the determined in-vitro data. In axial rotation, the FE-model predicted a stiffer segmental behavior than it was determined in the in-vitro study.It is usually expected that high stiffness values are required for a posterior stabilization device to stiffen a spinal segment. We found that already small stiffness values were sufficient to cause a stiffening. Using these data, it may possible to develop implants for certain clinical indications.     

Numerical Simulation of Callus Healing for Optimization of Fracture Fixation Stiffness

The stiffness of fracture fixation devices together with musculoskeletal loading defines the mechanical environment within a long bone fracture, and can be quantified by the interfragmentary movement. In vivo results suggested that this can have acceleratory or inhibitory influences, depending on direction and magnitude of motion, indicating that some complications in fracture treatment could be avoided by optimizing the fixation stiffness. However, general statements are difficult to make due to the limited number of experimental findings. The aim of this study was therefore to numerically investigate healing outcomes under various combinations of shear and axial fixation stiffness, and to detect the optimal configuration. A calibrated and established numerical model was used to predict fracture healing for numerous combinations of axial and shear fixation stiffness under physiological, superimposed, axial compressive and translational shear loading in sheep. Characteristic maps of healing outcome versus fixation stiffness (axial and shear) were created. The results suggest that delayed healing of 3 mm transversal fracture gaps will occur for highly flexible or very rigid axial fixation, which was corroborated by in vivo findings. The optimal fixation stiffness for ovine long bone fractures was predicted to be 1000–2500 N/mm in the axial and >300 N/mm in the shear direction. In summary, an optimized, moderate axial stiffness together with certain shear stiffness enhances fracture healing processes. The negative influence of one improper stiffness can be compensated by adjustment of the stiffness in the other direction.     

Trabecular shear stress in human vertebral cancellous bone: intra- and inter-individual variations

Correlation of the mean and standard deviation of trabecular stresses has been proposed as a mechanism by which a strong relationship between the apparent strength and stiffness of cancellous bone can be achieved. The current study examined whether the relationship between the mean and standard deviation of trabecular von Mises stresses can be generalized for any group of cancellous bone. Cylindrical human vertebral cancellous bone specimens were cut in the infero-superior direction from T12 of 23 individuals (inter-individual group). Thirty nine additional specimens were prepared similarly from the T4-T12 and L2-L5 vertebrae of a 63 year old male (intra-individual group). The specimens were scanned by micro-computed tomography (microCT) and trabecular von Mises stresses were calculated using finite element modeling. The expected value, standard deviation and coefficient of variation of the von Mises stress were calculated form a three-parameter Weibull function fitted to von Mises stress data from each specimen. It was found that the average and standard deviation of trabecular von Mises shear stress were: (i) correlated with each other, supporting the idea that high correlation between the apparent strength and stiffness of cancellous bone can be achieved through controlling the trabecular level shear stress variations, (ii) dependent on anatomical site and sample group, suggesting that the variation of stresses are correlated to the mean stress to different degrees between vertebrae and individuals, and (iii) dependent on bone volume fraction, consistent with the idea that shear stress is less well controlled in bones with low BV/TV. The conversion of infero-superior loading into trabecular von Mises stresses was maximum for the tissue at the junction of the thoracic and lumbar spine (T12-L1) consistent with this junction being a common site of vertebral fracture.     

Differential effects of pre-tension on shear wave propagation in elastic media with different boundary conditions as measured by magnetic resonance elastography and finite element modeling

Magnetic resonance elastography (MRE) can non-invasively determine material stiffness based on the propagating shear wavelength. Shear wave propagation in a finite homogenous isotropic material can be affected by multiple factors. In this study we examined the effects of pre-tension and frequency on MRE shear measurements of gel phantoms with different boundary conditions, frequencies, and geometries. Results from MRE measurements were compared to wave motion theory in elastic solids and qualitatively to a finite element (FE) model. Results indicated that boundary conditions, geometry and pre-tension are important factors to be considered when performing MRE tests on a finite material, and that FE modeling can help explore how the shear wave propagation is affected under various boundary conditions and axial stresses, among other potential factors.