Finite element modeling of trabecular bone damage

This paper presents a finite element-based, computational model for analysis of structural damage to trabecular bone tissues. A modulus reduction method was formulated from elasto-plasticity theory, and was used to account for site-specific trabecular bone tissue damage. Trabecular bone tissue damage is illustrated using a large-scale, anatomically accurate, two-dimensional, microstructural finite element model of a human thoracic vertebral body. Four models with varying specifications for damage accumulation were subjected to compressive loading and unloading cycles. The numerical results and experimental validation demonstrated that the modulus reduction method reproduced the non-linear mechanical behaviour of vertebal trabecular bone. The iterative computational approach presented provides a methodology to study trabecular bone damage, and should provide researchers with a computational approach to study bone fracture and repair and to predict vertebral fragility.     

Finite element analysis of vertebral body mechanics with a nonlinear microstructural model for the trabecular core.

In this study, a finite element model of a vertebral body was used to study the load-bearing role of the two components (shell and core) under compression. The model of the vertebral body has the characteristic kidney shape transverse cross section with concave lateral surfaces and flat superior and inferior surfaces. A nonlinear unit cell based foam model was used for the trabecular core, where nonlinearity was introduced as coupled elastoplastic beam behavior of individual trabeculae. The advantage of the foam model is that architecture and material properties are separated, thus facilitating studies of the effects of architecture on the apparent behavior. Age-related changes in the trabecular architecture were considered in order to address the effects of osteoporosis on the load-sharing behavior. Stiffness changes with age (architecture and porosity changes) for the trabecular bone model were shown to follow trends in published experimental results. Elastic analyses showed that the relative contribution of the shell to the load-bearing ability of the vertebra decreases with increasing age and lateral wall curvature. Elasto-plastic (non-linear) analyses showed that failure regions were concentrated in the upper posterior region of the vertebra in both the shell and core components. The ultimate load of the vertebral body model varied from 2800 N to 5600 N, depending on age (architecture and porosity of the trabecular core) and shell thickness. The model predictions lie within the range of experimental results. The results provide an understanding of the relative role of the core and shell in vertebral body mechanics and shed light on the yield and post-yield behavior of the vertebral body.     

Finite element thermal analysis of bone cement for joint replacements

A finite element technique was developed to investigate the thermal behavior of bone cement in joint replacement procedures. Thermal tests were designed and performed to provide the parameters in a kinetic model of bone cement exothermic polymerization. The kinetic model was then coupled with an energy balance equation using a finite element formulation to predict the temperature history and polymerization development in the bone-cement-prosthesis system. Based on the temperature history, the possibility of the thermal bone necrosis was then evaluated. As a demonstration, the effect of cement mantle thickness on the thermal behavior of the system was investigated. The temperature profiles in the bone-cement-prosthesis system have shown that the thicker the cement, the higher the peak temperature in the bone. In the 7 mm thick cement case, a peak temperature of over 55 degrees C was predicted. These high temperatures occurred in a small region near the bone/cement interface. No damage was predicted in the 3 mm and 5 mm cement mantle thickness cases. Although thermal damage was predicted in the bone for the 7 mm mantle thickness case, the amount of thermal necrosis predicted was minimal. If more cement is used in the surgical procedure, more heat will be generated and the potential for thermal bone damage may rise. The systems should be carefully selected to reduce thermal tissue damage when more cement is used. The methodology developed in this paper provides a numerical tool for the quantitative simulation of the thermal behavior of bone-cement-prosthesis designs.     

Simulation of vertebral trabecular bone loss using voxel finite element analysis

Trabecular bone loss in human vertebral bone is characterised by thinning and eventual perforation of the horizontal trabeculae. Concurrently, vertical trabeculae are completely lost with no histological evidence of significant thinning. Such bone loss results in deterioration in apparent modulus and strength of the trabecular core. In this study, a voxel-based finite element program was used to model bone loss in three specimens of human vertebral trabecular bone. Three sets of analyses were completed. In Set 1, strain adaptive resorption was modelled, whereby elements which were subject to the lowest mechanical stimulus (principal strain) were removed. In Set 2, both strain adaptive and microdamage mechanisms of bone resorption were included. Perforation of vertical trabeculae occurred due to microdamage resorption of elements with strains that exceeded a damage threshold. This resulted in collapse of the trabecular network under compression loading for two of the specimens tested. In Set 3, the damage threshold strain was gradually increased as bone loss progressed, resulting in reduced levels of microdamage resorption. This mechanism resulted in trabecular architectures in which vertical trabeculae had been perforated and which exhibited similar apparent modulus properties compared to experimental values reported in the literature. Our results indicate that strain adaptive remodelling alone does not explain the deterioration in mechanical properties that have been observed experimentally. Our results also support the hypothesis that horizontal trabeculae are lost principally by strain adaptive resorption, while vertical trabeculae may be lost due to perforation from microdamage resorption followed by rapid strain adaptive resorption of the remaining unloaded trabeculae.     

Finite element analysis of defibrillation fields in a human torso model for ventricular defibrillation

In order to optimize defibrillation electrode systems for ventricular defibrillation thresholds (DFTs), a Finite Element Torso model was built from fast CT scans of a patient who had large cardiac dimensions (upper bound of normal) but no heart disease. Clinically used defibrillation electrode configurations, i.e. Superior Vena Cava (SVC) to Right Ventricle (RV) (SVC-RV), left pectoral Can to RV (Can-RV) and Can + SVC-RV, were analyzed. The DFTs were calculated based on 95% ventricular mass having voltage gradient> 5 V/cm and these results were also compared with clinical data. The low voltage gradient regions with voltage gradient < 5 V/cm were identified and the effect of electrode dimension and location on DFTs were also investigated for each system. A good correlation between the model results and the clinical data supports the use of Finite Element Analysis of a human torso model for optimization of defibrillation electrode systems. This correlation also indicates that the critical mass hypothesis is the primary mechanism of defibrillation. Both the FEA results and the clinical data show that Can + SVC-RV system offers the lowest voltage DFTs when compared with SVC-RV and Can-RV systems. Analysis of the effect of RV, SVC and Can electrode dimensions and locations can have an important impact on defibrillation lead designs.     

Quantification of a rat tail vertebra model for trabecular bone adaptation studies

A feedback controlled loading apparatus for the rat tail vertebra was developed to deliver precise mechanical loads to the eighth caudal vertebra (C8) via pins inserted into adjacent vertebrae. Cortical bone strains were recorded using strain gages while subjecting the C8 in four cadaveric rats to mechanical loads ranging from 25 to 100 N at 1 Hz with a sinusoidal waveform. Finite element (FE) models, based on micro computed tomography, were constructed for all four C8 for calculations of cortical and trabecular bone tissue strains. The cortical bone strains predicted by FE models agreed with strain gage measurements, thus validating the FE models. The average measured cortical bone strain during 25-100 N loading was between 298 +/- 105 and 1210 +/- 297 microstrain (muepsilon). The models predicted average trabecular bone tissue strains ranging between 135 +/- 35 and 538 +/- 138 mu epsilon in the proximal region, 77 +/- 23-307 +/- 91 muepsilon in the central region, and 155 +/- 36-621 +/- 143 muepsilon in the distal region for 25-100 N loading range. Although these average strains were compressive, it is also interesting that the trabecular bone tissue strain can range from compressive to tensile strains (-1994 to 380 mu epsilon for a 100 N load). With this novel approach that combines an animal model with computational techniques, it could be possible to establish a quantitative relationship between the microscopic stress/strain environment in trabecular bone tissue, and the biosynthetic response and gene expression of bone cells, thereby study bone adaptation.     

Visual analysis of trabecular bone structure

Acquiring image data of bone biopsies by a micro-CT scanner is today a common technique. The amount of data to be assessed is huge. The task to assess quantitative measures requires a concise visualization. We present visualization techniques that can be used interactively on state-of-the-art PCs and demonstrate how the frontier can be pushed further. A skeletonization process is applied to the image of the bone to create the central surface. After triangulation this surface can be renderd at interactive frame rates. When the surface is additionally colored by local measures (mean grey value of image data, local thickness) the overall structure and details can be recognized at the same time. This can facilitate the exploration of the biopsy and can help finding special features.     

Experimental and finite element analysis of the mouse caudal vertebrae loading model: prediction of cortical and trabecular bone adaptation

In this study, we attempt to predict cortical and trabecular bone adaptation in the mouse caudal vertebrae loading model using knowledge of bone’s local mechanical environment at the onset of loading. In a previous study, we demonstrated appreciable 25.9 and 11% increases in both trabecular and cortical bone volume density, respectively, when subjecting the fifth caudal vertebrae (C5) of C57BL/6 (B6) mice to an acute loading regime (amplitude of 8N, 3000 cycles, 10 Hz, 3 times a week for 4 weeks). We have also established a validated finite element (FE) model of the C5 vertebra using micro-computed tomography (micro-CT), which characterizes, in 3D, the micro-mechanical strains present in both cortical and trabecular compartments due to the applied loads. To investigate the relationship between load-induced bone adaptation and mechanical strains in-vivo and in-silico data sets were compared. Using data from the previous cross-sectional study, we divided cortical and trabecular compartments into 15 subregions and determined, for each region, a bone formation parameter ΔBV/BS (a cross-sectional measure of the bone volume added to cortical and trabecular surfaces following the described loading regime). Linear regression was then used to correlate mean regional values of ΔBV/BS with mean values of mechanical strains derived from the FE models which were similarly regionalized. The mechanical parameters investigated were strain energy density (SED), the orthogonal strains (e _x , e _y , e _z ) and the three shear strains (e _xy , e _yz , e _zx ). For cortical regions, regression analysis showed SED to correlate extremely well with ΔBV/BS (R ² =?0.82) and e _z (R ²?=?0.89). Furthermore, SED was found to predict expansion of the cortical shell correlating significantly with the regional percentage increases in cortical tissue volume (R ² = 0.92), cortical marrow volume (R ² =?0.91) and cortical thickness (R ² = 0.56). For trabecular regions, FE parameters were found not to correlate with load-induced trabecular bone morphology. These results indicate that load-induced cortical morphology can be predicted from population data, whereas the prediction of trabecular morphology requires subject-specific micro- architecture.     

A morphological model of vertebral trabecular bone

In their micro-structures, typical natural cellular materials such as vertebral trabecular bone have a network of doubly tapered struts, thickening near the strut joints. However, past analytical models for vertebral trabecular bone do not take account of the effect of strut taper on the mechanical properties.This paper presents an analytical cell model comprised of doubly tapered struts to predict the global mechanical properties of vertebral trabecular bone. The predicted results for male, female, and both sexes fit the experimental data well. By considering several strut taper geometries, it is shown that the horizontal Young's modulus and the horizontal uniaxial collapse stress are, in some cases, approximately 1.8- and 2.2-fold higher, respectively, than those of the uniform strut model. This finding illustrates the importance of increased trabecular thickening near the strut joints (i) for improving the accuracy of calculating the mechanical properties and (ii) for the effective treatment of aged bone using drug therapy. It also highlights the need to combine trabecular architecture measurements with information about the morphology near the strut joints.     

Finite element dynamic analysis of soft tissues using state-space model

A finite element (FE) model is employed to investigate the dynamic response of soft tissues under external excitations, particularly corresponding to the case of harmonic motion imaging. A solid 3D mixed ‘u–p’ element S8P0 is implemented to capture the near-incompressibility inherent in soft tissues. Two important aspects in structural modelling of these tissues are studied; these are the influence of viscous damping on the dynamic response and, following FE-modelling, a developed state-space formulation that valuates the efficiency of several order reduction methods. It is illustrated that the order of the mathematical model can be significantly reduced, while preserving the accuracy of the observed system dynamics. Thus, the reduced-order state-space representation of soft tissues for general dynamic analysis significantly reduces the computational cost and provides a unitary framework for the ‘forward’ simulation and ‘inverse’ estimation of soft tissues. Moreover, the results suggest that damping in soft-tissue is significant, effectively cancelling the contribution of all but the first few vibration modes.     

Convergence behavior of high-resolution finite element models of trabecular bone

The convergence behavior of finite element models depends on the size of elements used, the element polynomial order, and on the complexity of the applied loads. For high-resolution models of trabecular bone, changes in architecture and density may also be important. The goal of this study was to investigate the influence of these factors on the convergence behavior of high-resolution models of trabecular bone. Two human vertebral and two bovine tibial trabecular bone specimens were modeled at four resolutions ranging from 20 to 80 microns and subjected to both compressive and shear loading. Results indicated that convergence behavior depended on both loading mode (axial versus shear) and volume fraction of the specimen. Compared to the 20 microns resolution, the differences in apparent Young's modulus at 40 microns resolution were less than 5 percent for all specimens, and for apparent shear modulus were less than 7 percent. By contrast, differences at 80 microns resolution in apparent modulus were up to 41 percent, depending on the specimen tested and loading mode. Overall, differences in apparent properties were always less than 10 percent when the ratio of mean trabecular thickness to element size was greater than four. Use of higher order elements did not improve the results. Tissue level parameters such as maximum principal strain did not converge. Tissue level strains converged when considered relative to a threshold value, but only if the strains were evaluated at Gauss points rather than element centroids. These findings indicate that good convergence can be obtained with this modeling technique, although element size should be chosen based on factors such as loading mode, mean trabecular thickness, and the particular output parameter of interest.     

A structural model for the mechanical behavior of trabecular bone

A quantitative model is developed for trabecular bone by approximating the trabecular geometry with a hypothetical network of compact bone. For the region immediately beneath the articular cartilage in the distal end of the femur, finite element analyses were performed with a high speed computer, assuming a physiological static load. The results indicate that bending and buckling of trabeculae are considerable in any elastic deformation of the bone; that fatigue fracture in some fraction of suitably oriented trabeculae is inevitable in normal ambulation; and that the stiffness varies considerably with lateral position across the subchondral plate. The latter depends totally on trabecular arrangement and may play a role in joint function and degeneration. The adjustments necessary to bring the gross stiffness into agreement with experiment imply that the intertrabecular soft tissues are of no consequence to the mechanical properties and that the compact bone of which trabeculae are made is probably not as stiff as cortical bone.     

A three-dimensional finite element model for arterial clamping

Clamp induced injuries of the arterial wall may determine the outcome of surgical procedures. Thus, it is important to investigate the underlying mechanical effects. We present a three-dimensional finite element model, which allows the study of the mechanical response of an artery-treated as a two-layer tube-during arterial clamping. The important residual stresses, which are associated with the load-free configuration of the artery, are also considered. In particular, the finite element analysis of the deformation process of a clamped artery and the associated stress distribution is presented. Within the clamping area a zone of axial tensile peak-stresses was identified, which (may) cause intimal and medial injury. This is an additional injury mechanism, which clearly differs from the commonly assumed wall damage occurring due to compression between the jaws of the clamp. The proposed numerical model provides essential insights into the mechanics of the clamping procedure and the associated injury mechanisms. It allows detailed parameter studies on a virtual clamped artery, which can not be performed with other methodologies. This approach has the potential to identify the most appropriate clamps for certain types of arteries and to guide optimal clamp design.     

Static and dynamic finite element analyses of an idealized structural model of vertebral trabecular bone.

An idealized three-dimensional finite element model of a rodlike trabecular bone structure was developed to study its static and dynamic responses under compressive loading, considering the effects of bone marrow and apparent density. Static analysis of the model predicted hydraulic stiffening of trabecular bone due to the presence of bone marrow. The predicted power equation relating trabecular bone apparent elastic modulus to its apparent density was in good agreement with those of the reported experimental investigations. The ratio of the maximum stress in the trabecular bone tissue to its apparent stress had a high value, decreasing with increasing bone apparent density. Frequency analyses of the model predicted higher natural frequencies for the bone without marrow than those for the bone with marrow. Adding a mass relatively large compared to that of bone rendered a single-degree-of-freedom response. In this case, the resonant frequency was higher for the bone with marrow than that for the bone without marrow. The predicted vibrational measurement of apparent modulus was in good agreement with that of the static measurement, suggesting vibrational testing as a method for nondestructive measurement of trabecular bone elastic moduli.     

A nonlocal constitutive model for trabecular bone softening in compression

Using the three-dimensional morphological data provided by computed tomography, finite element (FE) models can be generated and used to compute the stiffness and strength of whole bones. Three-dimensional constitutive laws capturing the main features of bone mechanical behavior can be developed and implemented into FE software to enable simulations on complex bone structures. For this purpose, a constitutive law is proposed, which captures the compressive behavior of trabecular bone as a porous material with accumulation of irreversible strain and loss of stiffness beyond its yield point and softening beyond its ultimate point. To account for these features, a constitutive law based on damage coupled with hardening anisotropic elastoplasticity is formulated using density and fabric-based tensors. To prevent mesh dependence of the solution, a nonlocal averaging technique is adopted. The law has been implemented into a FE software and some simple simulations are first presented to illustrate its behavior. Finally, examples dealing with compression of vertebral bodies clearly show the impact of softening on the localization of the inelastic process.     

An adaptation model for trabecular bone at different mechanical levels

Background  

Bone has the ability to adapt to mechanical usage or other biophysical stimuli in terms of its mass and architecture, indicating that a certain mechanism exists for monitoring mechanical usage and controlling the bone's adaptation behaviors. There are four zones describing different bone adaptation behaviors: the disuse, adaptation, overload, and pathologic overload zones. In different zones, the changes of bone mass, as calculated by the difference between the amount of bone formed and what is resorbed, should be different.     

The limitations of canine trabecular bone as a model for human: a biomechanical study

Distal canine femurs were sectioned into 8 mm cubic specimens. Orthogonal compression tests were performed to preyield in two or three directions and to failure in a third. Apparent density and ash weight density were measured for a subset of specimens. The results were compared to the human distal femur results of Ciarelli et al. (Transactions of the 32nd Annual Meeting of the Orthopaedic Research Society, Vol. 11, p. 42, 1986). Quantitative similarities existed in the fraction of components comprising the trabecular tissue of the two species. Qualitative similarities were seen in the positional and anisotropic variation of the mechanical properties, and also in the form and strength of the relationships between the mean modulus and bone density, ultimate stress and density, and ultimate stress and modulus. However, significantly different regression equations resulted for the mean modulus-density, and ultimate stress modulus relationships, indicating that for the same density, canine trabecular bone displays a lower modulus than human, and may achieve greater compressive strains before failure.     

Development of a three-dimensional finite element model of a human tibia using experimental modal analysis.

The modal analysis of a human tibia consisted of characterizing its dynamic behavior by determining natural frequency, damping ratio and mode shapes. Two methods were used to perform the modal analysis: (1) a finite element method (structural model); (2) an experimental modal analysis (modal model). The experimental modal model was used to optimize the structural model. After optimization, differences in results between the two models were found to be due only to mechanical properties and mass distribution. The influences of boundary conditions and geometric properties (such as inertia and length) were eliminated by the finite element model itself. The percent relative error between the two methods was approximately 3%, corresponding to the standard deviation of the measured frequencies. For the frequency range considered, the mode shapes were bending modes in two different vibration planes (latero-medial and sagittal), with a slight torsion effect due to the twisted geometry of the tibia.     

Artificial composite bone as a model of human trabecular bone: the implant-bone interface

The use of artificial bones in implant testing has become popular due to their low variability and ready availability. However, friction coefficients, which are critical to load transfer in uncemented implants, have rarely been compared between human and artificial bone, particularly for wet and dry conditions. In this study, the static and dynamic friction coefficients for four commercially used titanium surfaces (polished, Al(2)O(3) blasted, plasma sprayed, beaded) acting on the trabecular component of artificial bones (Sawbones) were compared to those for human trabecular bone. Artificial bones were tested in dry and wet conditions and normal interface stress was varied (0.25, 0.5, 1.0MPa). Friction coefficients were mostly lower for artificial bones than real bone. In particular, static friction coefficients for the dry polished surface were 20% of those for real bone and 42-61% for the dry beaded surface, with statistical significance (alpha<0.05). Less marked differences were observed for dynamic friction coefficients. Significant but non-systematic effects of normal stress or wet/dry condition on friction coefficients were observed within each surface type. These results indicate that the use of artificial bone models for pre-clinical implant testing that rely on interface load transfer with trabecular bone for mechanical integrity can be particularly sensitive to surface finish and lubrication conditions.     

Finite element evaluation of the AIA shear specimen for bone

An elastic-plastic finite element analysis is performed on the AIA shear specimen to evaluate its effectiveness to yield ultimate shear strength values. The effect of geometry, material properties, and yield criteria are discussed in the light of applications to human femoral cortical bone. Specimen dimensions are noted as follows: W, width, D, hole diameter and H, distance between holes. As the H/D ratio increases the stress distribution tends more toward pure shear at the same time the overshoot in the shear distribution increases. An H/D ratio equal to 1.2-1.5 is optimal. The H/W parameter does not affect the overshoot noticeably but it does slightly affect the purity of shear. The material parameters do affect the performance of the shear specimen. However, the effect of the material parameters are far more pronounced in the anisotropic case than it is in the isotropic case. In the isotropic case, the Young modulus does not affect the overshoot. The increase in Poisson's ratio does slightly decrease the overshoot. For the anisotropic case, the increase in the ratio of shear modulus to Young modulus in the transverse direction (G/E2) results in an increase in the overshoot (in the shear distribution). The increase in the ratio of the Young modulus in the transverse direction to that of the axial direction (E2/E1) also results in an increase in the overshoot. Creating a notch at the top of the hole is shown to have the effect of decreasing the overshoot. Its effect on the purity of the shear is rather slight. It is found that plasticity is initiated at the sides of the two holes where the tensile normal stresses are maximum. The plastic region first expands around the perimeter of the hole then radially outward; and finally, it expands into the significant region. If the W/H parameter is less than 5, a sizable portion of the width of the specimen around the hole can go plastic with the significant region still being in the elastic state. Such a situation can cause tearing of the specimen across the width. A W/H ratio of 6 or more can prevent that danger. It is also found that the onset of plasticity brings about higher overshoot and higher purity of shear. The notched shear specimen performs better in actual tests and is more reliable in producing shear failures. The shear strength results obtained from AIA shear tests tend to confirm those shear strength results obtained from torsion tests.