Non-linear material models for tracheal smooth muscle tissue.

The aim of this study is to investigate the hyperelastic material models to describe the non-linear stress-strain behavior of tracheal smooth muscle tissue. Specifically, the goal is to validate the material model with experimental data using different finite element models and discuss the trends in stress-strain behavior of smooth muscle tissue. Both 2D and 3D finite element analyses were carried out to estimate the stress-strain behavior of the smooth muscle tissue. The results obtained indicate that the developed Ogden material model is valid and useful in explaining the stress strain behavior of tracheal smooth muscle tissue under different conditions. Finite element simulation results of the stress-strain behavior in the transverse direction are presented.

Anisotropic properties of tracheal smooth muscle tissue.

The anisotropic (directional-dependent) properties of contracting tracheal smooth muscle tissue are estimated from a computational model based on the experimental data of length-dependent stiffness. The area changes are obtained at different muscle lengths from experiments in which stimulated muscle undergoes unrestricted shortening. Then, through an interative process, the anisotropic properties are estimated by matching the area changes obtained from the finite element analysis to those derived from the experiments. The results obtained indicate that the anisotropy ratio (longitudinal stiffness to transverse stiffness) is about 4 when the smooth muscle undergoes 70% strain shortening, indicating that the transverse stiffness reduces as the longitudinal stiffness increases. It was found through a sensitivity analysis from the simulation model that the longitudinal stiffness and the in-plane shear modulus are not very sensitive as compared to major Poisson's ratio to the area changes of the muscle tissue.

Experimental investigation of Poisson's ratio as a damage parameter for bone fatigue.

The fatigue loading of bone results in the degradation of mechanical properties such as strength and stiffness. Even though several authors have investigated the relationship between the longitudinal modulus and loading cycles, the reduction in Poisson's ratio and its relationship to fatigue loading cycles have not previously been investigated. In this study, the reduction in the major Poisson's ratio and longitudinal modulus for cortical bone specimens as a result of tensile fatigue was experimentally investigated. We compared the results of the major Poisson's ratio reduction to the reduction in longitudinal modulus to determine if there was a relationship between the two. The results showed that the reduction in Poisson's ratio was about 8-22% higher than the reduction in longitudinal modulus, indicating that more microdamage accumulated transversely than longitudinally. Both the longitudinal modulus and major Poisson's ratio decreased in a logarithmic fashion with increasing loading cycles for the bone specimens tested.

Modeling fatigue damage evolution in bone.

A simple analytical model for damage evolution of bone fatigue is presented. A probabilistic method for characterizing the damage accumulation in terms of microcracks for bone fatigue was developed. The crack numerical density distributions were obtained from the Monte Carlo simulations with a Weibull distribution fit in this study. The results predicted from the present model are compared with existing experimental data and discussed. The quantitative relationship between stiffness loss, loading cycles and microdamage parameter developed in this study may be useful for fatigue life and failure stress predictions.

Fatigue data analysis of canine femurs under four-point bending.

When bone is subjected to fatigue loading, micro-cracks initiate and grow. This reduces the mechanical properties and quantitative relationships between stiffness loss and loading cycles may be derived. We developed the relationships between stiffness loss and loading cycles for whole canine femurs subjected to cyclic fatigue in four-point bending. The fatigue data from experiments followed Weibull statistics. When the stiffness loss is less than 15%, a linear relationship is best-fitted (R2 = 0.96, p < 0.0001) between the stiffness loss and loading cycles. However, when the stiffness loss is greater than 30%, a power law relationship is best-fitted (R2 = 0.97, p < 0.0001) between the stiffness loss and loading cycles. Thus, we conclude that the derived relationships between stiffness loss and loading cycles might be useful for the prediction of bone failure under cyclic bending subjected to an initial strain of 2700 microstrain.

Development of a fluorescent light technique for evaluating microdamage in bone subjected to fatigue loading.

A new method using fluorescent light microscopy has been developed to visualize and evaluate bone microdamage. We report the findings of two different experiments with a common aim of comparing the fluorescent light technique to the brightfield method for quantifying microdamage in bone. In Experiment 1, 36 canine femurs were tested in four-point cyclic bending until they had lost between 5 and 43% of their stiffness. The loaded portion of the bone was stained en bloc with basic fuchsin for the presence of damage. Standard point counting techniques were used to calculate fractional damaged area (Dm.Ar = Cr.Ar/B.Ar, mm2/mm2) under brightfield and fluorescent microscopy. In Experiment 2, bone microdamage adjacent to endosseous implants, subjected to fatigue loading (150,000 cycles, 2 Hz and 37 degrees C) ex vivo was examined. The bone around the implant was either allowed to heal (adapted specimen) for 12 weeks after placement in dog mid-femoral diaphyses prior to testing or was loaded immediately to simulate non-healed bone surrounding endosseous implants (non-adapted). Crack numerical density (Cr.Dn = Cr.N/B.Ar, #/mm2), crack surface density (Cr.S.Dn = Tt.Cr.Le/B.Ar, mm/mm2) and fractional damaged area were calculated separately by both techniques in the adapted and non-adapted specimens. In both Experiments 1 and 2, significantly more microdamage was detected by the fluorescent technique than by the brightfield method. Also, there was a trend towards higher intraobserver repeatability when using the fluorescent method. These results suggest that the brightfield technique underestimates microdamage accumulation and that the fluorescent technique better represents the actual amounts of microdamage present. The results demonstrate that the fluorescent method provides an accurate and precise approach for bone microdamage evaluation, and that it improves the prediction of stiffness loss from damage accumulation.

Does microdamage accumulation affect the mechanical properties of bone?

It has never been demonstrated that microcrack accumulation in bone leads to impaired mechanical properties. We hypothesized that microdamage accumulation is positively and linearly correlated with a reduction in bone's elastic modulus. We also tested the hypothesis that damage accumulates more rapidly in tensile cortices, but crack growth is greater in compressive cortices. Canine femurs (n = 26) were tested in four-point cyclic bending under load control until they had lost between 5 and 43% of their stiffness. Ten femurs were used as nonloaded controls. The loaded portion of the bone was stained en bloc with basic fuchsin to detect the presence of microdamage. The number of stained microcracks, their lengths and the area of damaged bone were measured under the microscope. Crack numerical density, surface density, mean crack length, and the percentage of damaged area were calculated. Significant microdamage accumulation was not detected until the bone had lost 15% of its elastic modulus. The relationship between crack density and stiffness loss was approximately quadratic, but the relationship between damaged area and stiffness loss was linear. There were significantly more microcracks in tensile cortices, but on average cracks were significantly longer in compressive cortices. We conclude that microcrack accumulation impairs the mechanical properties of bone by reducing its elastic modulus. We also conclude that damage accumulates more rapidly in tensile cortices, but crack growth is greater in compressive cortices.

En bloc staining of bone under load does not improve dye diffusion into microcracks.

Microdamage accumulation in bone has been implicated in the pathogenesis of some bone fractures, and in implant loosening. Standard techniques for staining microcracks may not allow all cracks to be stained. We tested the hypothesis that crack closure in bone cortices after removal of a bending load may prevent diffusion of stain to sites of microcrack nucleation. Following cyclic loading, 26 canine femurs were divided into a group stained en bloc while applying a four point bending load, and another group stained without an applied load. No differences in number or length of microcracks were observed, indicating that crack closure does not prevent diffusion of stain to the crack location. Staining under load is unnecessary.

Stress analysis of the human temporomandibular joint.

Stress analysis of the human temporomandibular joint (TMJ) consisting of mandibular disc, condyle and fossa-eminence complex during normal sagittal jaw closure was performed using non-linear finite element analysis (FEA). The geometry of the TMJ was obtained from magnetic resonance imaging (MRI). The tissue proportion was measured from a cadaver TMJ. Contact surfaces were defined to represent the interaction between the mandibular disc and the condyle, and between the mandibular disc and the fossa-eminence complex so that finite sliding was allowed between contact bodies. Stresses in the TMJ components (disc, condyle and fossa-eminence complex), and forces in capsular ligaments were obtained. The results demonstrated that, with the given condylar displacement, the stress in the condyle was dominantly compressive and in the fossa-eminence complex was dominantly tensile. The cancellous bone was shielded by the shell shaped cortical bone from the external loading. The results illustrate the stress distributions in the TMJ during a normal jaw closure.

Fracture and material degradation properties of cortical bone under accelerated stress.

The fracture stress and material property degradation of bovine cortical bone specimens were investigated experimentally under accelerated cyclic tensile stress testing. The fracture stress of a typical specimen was found from a static tensile test, and the cyclic loading/unloading was calculated as a percentage of this fracture stress. The results of accelerated cyclic stress tests were compared to monotonically increased static tests to determine if loading/unloading has an effect on the damage mechanism in bone. It was found that fracture stress of the bone increases due to accelerated stress cycling whereas the modulus decreases in a logarithmic fashion with increasing cyclic stress.

Cancellous bone architecture: advantages of nonorthogonal trabecular alignment under multidirectional joint loading.

Wolff proposed that trabeculae align at 90 degrees angles (orthogonal). However, nonorthogonal alignment of trabeculae has been observed near many joints, including the proximal femur. We propose that nonorthogonal alignment is an adaptation to multidirectional joint loads. When the loading direction does not correspond with the trabecular alignment, warping or shear coupling occurs leading to large shear strains within the cancellous structure. Using a simplified continuum model for trabecular bone, we demonstrate that shear coupling caused by multidirectional joint loads is reduced 33-75% when trabeculae are aligned 60 degrees from one another (as is observed in regions of the proximal femur), as opposed to 90 degrees from one another (as was predicted by Wolff). The results suggest that an optimal cancellous structure may appear differently under multidirectional joint loads than the 'trajectorial' organization proposed by Wolff, which was based upon assumptions drawn from unidirectional loading.

A uniform strain criterion for trabecular bone adaptation: do continuum-level strain gradients drive adaptation?

In this paper, it is postulated that the apparent density of trabecular bone adapts so that continuum-level strains within the bone are uniform and, as a consequence, spatial strain gradients within the bone/marrow continuum are minimized. The feasibility of a uniform strain criterion was tested using computational finite-element analysis of the proximal femur. We demonstrated that (1) this criterion produced a realistic apparent density distribution in the proximal femur, (2) the solutions for apparent density were convergent and unique, (3) predicted apparent densities compared well to experimental measurements, and (4) strain gradients within the bone/marrow continuum were reduced substantially. Thus, a possible goal of trabecular bone adaptation may be the reduction of strain gradients within the bone/marrow continuum. Osteocytes within the bone tissue and bone cells on the surface of a trabeculum are mechanosensitive and play a role in bone adaptation. In addition, the bone marrow is rich in osteoprogenitor cells near the bone surface that are mechanosensitive. Strain gradients within bone/marrow continuum cause pressure gradients in the marrow, causing extracellular fluid flow which could stimulate osteoprogenitor cells and contribute to bone adaptation.

A viscoelastic material model to represent smooth muscle shortening.

The mechanical properties of a contracting smooth muscle can be changed by changing its length. A viscoelastic material model was developed to predict the length-dependent stiffness changes when a constrained muscle is allowed to shorten under a constant external force. Three-dimensional finite element simulations were carried out to estimate the stiffness changes and compared to available experimental data. A good agreement was found indicating that the viscoelastic material model developed gives a valid representation of the length dependent stiffness changes of a smooth muscle. Sensitivity analysis was carried out to determine the relative effects of material constants in the model on the length dependent stiffness.

Bone stiffness changes due to microdamage under different loadings.

Stiffness changes due to microdamage in the longitudinal and cross-sectional directions in a dog bone model under different loadings were investigated using three-dimensional finite element analysis. Stiffness changes and severity of both longitudinal and cross-sectional type microcracks were estimated between the damaged and undamaged bone under four-point bending, torsion and tension. Finite element simulation results indicated that longitudinal damage was more severe than cross-sectional damage under axial tension and bending, and the opposite was true for torsional loading. However, for axial tension, the stiffness change due to cross-sectional microcracks remained constant.

Bone mineral lies mainly outside collagen fibrils: predictions of a composite model for osteonal bone.

We propose that the elastic properties of osteonal bone can be modeled accurately as a simple fiber-reinforced composite, provided that accurate properties for the mineral and collagen phases of the ultrastructure are available. Off-axis stiffness coefficients were measured in anterior quadrant of canine femora at 10 degrees increments from longitudinal to transverse direction using an acoustic microscope. The resolution of these measurements was about 60 microns or less than the radius of one osteon. The bone specimens were subsequently demineralized and the off-axis measurements were repeated to determine the elasticity of bone collagen. Bone collagen fibrils were not principally aligned along the long axis of the bone, but demonstrated an alignment that was 30 degrees from the long axis. A simple composite model was developed based on the experimental data. The model that best fit experimental data assumed that (1) bone collagen was aligned 30 degrees from the long axis of the bone, (2) 75% of mineral crystals reside outside of collagen fibrils, and (3) mineral crystals outside of collagen fibrils have their c-axis in the longitudinal direction.

The anisotropy of osteonal bone and its ultrastructural implications.

The anisotropic elastic symmetry of osteonal bone reflects the ultrastructural organization of collagen fibrils and mineral crystals within the osteons as well as the lamellar microstructure. Until recently, reported values for bone's anisotropic elastic properties were limited in their interpretation by poor precision and resolution of measurement techniques. Here, we report measurements of bone anisotropy using high precision acoustic microscopy. The elastic properties of canine femoral bone specimens, taken from 23 femora, were measured at 10 degrees increments from the long axis of the bone. Half of the bone specimens subsequently were demineralized in EDTA solution, the other half were decollagenized in sodium hypochlorite solution, and the acoustic measurements were repeated. We found the elastic symmetry of osteonal bone deviates significantly from orthotropic theory supporting the hypothesis that the lamellar microstructure forms a "rotated plywood" (Weiner and Traub, FASEB J 6:879-885; 1992). The principal orientation of bone mineral was along the long axis of the bone, while bone collagen appeared to be aligned at a 30 degrees angle to the long axis. The misalignment between the mineral and the collagen suggests that (1) a substantial percentage of the mineral is extrafibrillar, and (2) the alignment of extrafibrillar mineral is governed by external influences, e.g., mechanical stresses.

Fracture toughness determination of dental materials by laboratory testing and finite element models.

This study assessed the effectiveness of finite element analysis in predicting the stress intensity factor (KIC) for three types of dental materials: a glass ionomer, a dental amalgam, and a composite resin. Laboratory tests were conducted on small single-edge notch specimens loaded in three-point bending to determine values for fracture toughness (KQ). Using the dimensions measured for each laboratory specimen, a J integral approach was employed to calculate KIC using finite element analysis. Both two-dimensional plane strain and three-dimensional models were used in determining KIC for each specimen, and these values were compared to the KQ values obtained from laboratory tests. The results indicated that no significant differences existed between laboratory results and those obtained from both two- and three-dimensional finite element models (P > .85). For the three-dimensional model, values for KIC were found to vary across the specimen thickness, with the values at the center of the specimen closely paralleling those obtained from the two-dimensional plane strain model. It was concluded that the two-dimensional plane strain J integral technique was as effective as the three-dimensional technique in calculating values for KIC.

Elastic and fracture properties of dental direct filling materials.

Five dental direct filling materials were tested in tension and compression in order to define their stress-strain behaviour for both types of applied stress. In addition, fracture toughness was determined from three-point bending tests. Linear and non-linear stress-strain behaviour was observed, and the response was specific for each material and also to the type of applied stress. Fracture resistance was also found to be material specific, which was related to differences in composition.

Collagen fiber orientation and geometry effects on the mechanical properties of secondary osteons.

The effects of collagen fiber orientation and osteon geometry on the mechanical properties of secondary osteons under axial compression/tension and combined loadings (compression, bending and torsion) were investigated using a composite-beam finite-element model. Three cross-sectional shapes of secondary osteons were studied to show the effect of geometry. The results of stiffness are presented using the tension and compression properties for each lamella. The model shows that the mechanical properties of osteons are enhanced in bending and torsion when collagen fibers are oriented within 30 degrees of the loading axis. Osteons with alternating lamellar orientation are not well adapted to resist torsional moments, but alternate collagen fiber orientation has virtually no effect on the bending stiffness of osteons. Fiber orientation affects the mechanical properties less significantly when osteons are non-circular. Collagen fiber orientation and osteon geometry interact to determine the mechanical behavior of the osteon, and may act in a compensatory manner in the adaptive process.