Visco-hyperelastic law for finite deformations: a frequency analysis.

Some biological tissues are repeatedly stimulated under cyclic loading, and this stimulation can be combined with large pressures, thus leading to large deformations. For such applications, visco-hyperelastic models have been proposed in the literature and used in finite-element studies. An extensively used quasi-linear model (QLVH), which assumes linear evolution equations, is compared with a nonlinear model (NLVH), which assumes a multiplicative split of the deformation gradient. The comparison is made here using sets of simulations covering a large frequency range. Lost and stored energies are computed, and the additional parameter of the NLVH model is set to two values found in the literature (NLVH-2 and NLVH-30 models). The predicted behaviour is very similar for all models at small strains, with each time constant (and corresponding viscous modulus) being associated with a damping peak and a stored-energy increase. When the strain amplitude is increased, the ratio of lost to stored energy increases for the QLVH model, but decreases for the NLVH models. The NLVH-30 model also displays a shift of the peak damping towards higher frequencies. Before reaching a steady state, all models display a decay of energy independent of the frequency, and the additional parameter of the NLVH model permits the modelling of complex types of evolution of the damping. In conclusion, this study compares the behaviour of two viscous hyper-elastic laws to allow an informed choice between them.

A non-linear viscoelastic model for the tympanic membrane.

The mechanical behavior of the tympanic membrane displays both non-linearity and viscoelasticity. Previous finite-element models of the tympanic membrane, however, have been either non-linear or viscoelastic but not both. In this study, these two features are combined in a non-linear viscoelastic model. The constitutive equation of this model is a convolution integral composed of a non-linear elastic part, represented by an Ogden hyperelastic model, and an exponential time-dependent part, represented by a Prony series. The model output is compared with the relaxation curves and hysteresis loops observed in previous measurements performed on strips of tympanic membrane. In addition, a frequency-domain analysis is performed based on the obtained material parameters, and the effect of strain rate is explored. The model presented here is suitable for modeling large deformations of the tympanic membrane for frequencies less than approximately 3 rad/s or about 0.6 Hz. These conditions correspond to the pressurization involved in tympanometry.

The role of fabric in the large strain compressive behavior of human trabecular bone.

Osteoporosis-related vertebral body fractures involve large compressive strains of trabecular bone. The small strain mechanical properties of the trabecular bone such as the elastic modulus or ultimate strength can be estimated using the volume fraction and a second order fabric tensor, but it remains unclear if similar estimations may be extended to large strain properties. Accordingly, the aim of this work is to identify the role of volume fraction and especially fabric in the large strain compressive behavior of human trabecular bone from various anatomical locations. Trabecular bone biopsies were extracted from human T12 vertebrae (n=31), distal radii (n=43), femoral head (n=44), and calcanei (n=30), scanned using microcomputed tomography to quantify bone volume fraction (BV/TV) and the fabric tensor (M), and tested either in unconfined or confined compression up to very large strains (∼70%). The mechanical parameters of the resulting stress-strain curves were analyzed using regression models to examine the respective influence of BV/TV and fabric eigenvalues. The compressive stress-strain curves demonstrated linear elasticity, yielding with hardening up to an ultimate stress, softening toward a minimum stress, and a steady rehardening followed by a rapid densification. For the pooled experiments, the average minimum stress was 1.89 ± 1.77 MPa, while the corresponding mean strain was 7.15 ± 1.84%. The minimum stress showed a weaker dependence with fabric as the elastic modulus or ultimate strength. For the confined experiments, the stress at a logarithmic strain of 1.2 was 8.08 ± 7.91 MPa, and the dissipated energy density was 5.67 ± 4.42 MPa. The latter variable was strongly related to the volume fraction (R(2)=0.83) but the correlation improved only marginally with the inclusion of fabric (R(2)=0.84). The influence of fabric on the mechanical properties of human trabecular bone decreases with increasing strain, while the role of volume fraction remains important. In particular, the ratio of the minimum versus the maximum stress, i.e., the relative amount of softening, decreases strongly with fabric, while the dissipated energy density is dominated by the volume fraction. The collected results will prove to be useful for modeling the softening and densification of the trabecular bone using the finite element method.

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.

Improvements in vertebral body strength under teriparatide treatment assessed in vivo by finite element analysis: results from the EUROFORS study.

Monitoring of osteoporosis therapy based solely on DXA is insufficient to assess antifracture efficacy. Estimating bone strength as a variable closely linked to fracture risk is therefore of importance. Finite element (FE) analysis-based strength measures were used to monitor a teriparatide therapy and the associated effects on whole bone and local fracture risk. In 44 postmenopausal women with established osteoporosis participating in the EUROFORS study, FE models based on high-resolution CT (HRCT) of T(12) were evaluated after 0, 6, 12, and 24 mo of teriparatide treatment (20 microg/d). FE-based strength and stiffness calculations for three different load cases (compression, bending, and combined compression and bending) were compared with volumetric BMD (vBMD) and apparent bone volume fraction (app. BV/TV), as well as DXA-based areal BMD of the lumbar spine. Local damage of the bone tissue was also modeled. Highly significant improvements in all analyzed variables as early as 6 mo after starting teriparatide were found. After 24 mo, bone strength in compression was increased by 28.1 +/- 4.7% (SE), in bending by 28.3 +/- 4.9%, whereas app. BV/TV was increased by 54.7 +/- 8.8%, vBMD by 19.1 +/- 4.0%, and areal BMD of L(1)-L(4) by 10.2 +/- 1.2%. When comparing standardized increases, FE changes were significantly larger than those of densitometry and not significantly different from app. BV/TV. The size of regions at high risk for local failure was significantly reduced under teriparatide treatment. Treatment with teriparatide leads to bone strength increases for different loading conditions of close to 30%. FE is a suitable tool for monitoring bone anabolic treatment in groups or individual patients and offers additional information about local failure modes. FE variables showed a higher standardized response to changes than BMD measurements, but further studies are needed to show that the higher response represents a more accurate estimate of treatment-induced fracture risk reduction.

A three-dimensional elastic plastic damage constitutive law for bone tissue.

Motivated by mechanical analysis of bones and bone-implant systems, a 3D constitutive law describing the macroscopic mechanical behaviour of both cortical and trabecular bone in cyclic (not fatigue) overloads is developed. The proposed model which mathematical formulation is established within the framework of generalized standard materials accounts for three distinct material evolution modes where elastic, plastic and damage aspects are closely related. The anisotropic elasticity of bone is described by a morphology-based model and distinct damage behaviour in tension and compression by a halfspacewise generalized Hill criterion. The plastic criterion is based on the intact elastic compliance tensor. The algorithm applies three distinct projections based on the relationship between the internal variables and criteria. Their respective consistent tangent operators are presented. Numerical resolutions of several boundary value problems and a biomechanical application are presented to illustrate the potential of the constitutive model and demonstrate the expected quadratic convergence of the algorithm.

A patient-specific finite element methodology to predict damage accumulation in vertebral bodies under axial compression, sagittal flexion and combined loads.

Due to the inherent limitations of DXA, assessment of the biomechanical properties of vertebral bodies relies increasingly on CT-based finite element (FE) models, but these often use simplistic material behaviour and/or single loading cases. In this study, we applied a novel constitutive law for bone elasticity, plasticity and damage to FE models created from coarsened pQCT images of human vertebrae, and compared vertebral stiffness, strength and damage accumulation for axial compression, anterior flexion and a combination of these two cases. FE axial stiffness and strength correlated with experiments and were linearly related to flexion properties. In all loading modes, damage localised preferentially in the trabecular compartment. Damage for the combined loading was higher than cumulated damage produced by individual compression and flexion. In conclusion, this FE method predicts stiffness and strength of vertebral bodies from CT images with clinical resolution and provides insight into damage accumulation in various loading modes.

Cement distribution, volume, and compliance in vertebroplasty: some answers from an anatomy-based nonlinear finite element study.

STUDY DESIGN: The biomechanics of vertebral bodies augmented with real distributions of cement were investigated using nonlinear finite element (FE) analysis. OBJECTIVES: To compare stiffness, strength, and stress transfer of augmented versus nonaugmented osteoporotic vertebral bodies under compressive loading. Specifically, to examine how cement distribution, volume, and compliance affect these biomechanical variables. SUMMARY OF BACKGROUND DATA: Previous FE studies suggested that vertebroplasty might alter vertebral stress transfer, leading to adjacent vertebral failure. However, no FE study so far accounted for real cement distributions and bone damage accumulation. METHODS: Twelve vertebral bodies scanned with high-resolution pQCT and tested in compression were augmented with various volumes of cements and scanned again. Nonaugmented and augmented pQCT datasets were converted to FE models, with bone properties modeled with an elastic, plastic and damage constitutive law that was previously calibrated for the nonaugmented models. The cement-bone composite was modeled with a rule of mixture. The nonaugmented and augmented FE models were subjected to compression and their stiffness, strength, and stress map calculated for different cement compliances. RESULTS: Cement distribution dominated the stiffening and strengthening effects of augmentation. Models with cement connecting either the superior or inferior endplate (S/I fillings) were only up to 2 times stiffer than the nonaugmented models with minimal strengthening, whereas those with cement connecting both endplates (S + I fillings) were 1 to 8 times stiffer and 1 to 12 times stronger. Stress increases above and below the cement, which was higher for the S + I cases and was significantly reduced by increasing cement compliance. CONCLUSION: The developed FE approach, which accounts for real cement distributions and bone damage accumulation, provides a refined insight into the mechanics of augmented vertebral bodies. In particular, augmentation with compliant cement bridging both endplates would reduce stress transfer while providing sufficient strengthening.

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

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

Compressive fatigue behavior of human vertebral trabecular bone.

Damage accumulation under compressive fatigue loading is believed to contribute significantly to non-traumatic, age-related vertebral fractures in the human spine. Only few studies have explored trabecular bone fatigue behavior under compressive loading and none examined the influence of trabecular architecture on fatigue life. In this study, trabecular bone samples of human lumbar and thoracic vertebrae (4 donors from age 29 to 86, n=29) were scanned with a microCT system prior to compressive fatigue testing to determine morphology-mechanical relationships for this relevant loading mode. Inspired from previous fabric-based relationships for elastic properties and quasi-static strength of trabecular bone, a simple power relationship between volume fraction, fabric eigenvalue, applied stress and the number of cycles to failure is proposed. The experimental results demonstrate a high correlation for this relationship (R2=0.95) and detect a significant contribution of the degree of anisotropy towards prediction of fatigue life. Step-wise regression for total and residual strains at failure suggested a weak, but significant correlation with volume fraction. From the obtained results, we conclude that the applied stress normalized by volume fraction and axial fabric eigenvalue can estimate fatigue life of human vertebral trabecular bone in axial compressive loading.

Nonlinear tensile properties of bovine articular cartilage and their variation with age and depth.

Tensile stiffness of articular cartilage is much greater than its compressive stiffness and plays an essential role even in compressive properties by increasing transient fluid pressures during physiological loading. Recent studies of nonlinear properties of articular cartilage in compression revealed several physiologically pertinent nonlinear behaviors, all of which required that cartilage tensile stiffness increase significantly with stretch. We therefore performed sequences of uniaxial tension tests on fresh bovine articular cartilage slices using a protocol that allowed several hours to attain equilibrium and measured longitudinal and transverse tissue strain. By testing bovine cartilage from different ages (6 months to 6 years) we found that equilibrium and transient tensile modulus increased significantly with maturation and age, from 0 to 15 MPa at equilibrium and from 10 to 28 MPa transiently. Our results indicate that cartilage stiffens with age in a manner similar to other highly hydrated connective tissues, possibly due to age-dependent content of enzymatic and nonenzymatic collagen cross links. The long relaxation period used in our tests (5-10 hours) was necessary in order to attain equilibrium and avoid a very significant overestimation of equilibrium modulus that occurs when much shorter times are used (15-30 minutes). We also found that equilibrium and transient tensile modulus increased nonlinearly when cartilage is stretched from 0 to 10% strain without any previous tare load. Although our results estimate a nonlinear increase in tensile stiffness with stretch that is an order of magnitude lower than that required to predict nonlinear properties in compression, they are in agreement with previous results from other uniaxial tension tests of collagenous materials. We therefore speculate that biaxial tensile moduli may be much higher and thereby more compatible with observed nonlinear compressive properties.