The role of the fluid phase in the viscous response of bovine periodontal ligament

The mechanical response of the periodontal ligament (PDL) is complex. This tissue responds as a hyperelastic solid when pulled in tension while demonstrating a viscous behavior under compression. This intricacy is reflected in the tissue's morphology, which comprises fibers, glycosaminoglycans, a jagged interface with the surrounding porous bone and an extensive vascular network.In the present study we offer an analysis of the viscous behavior and the interplay between the fibrous matrix and its fluid phase.Cylindrical specimens comprising layers of dentine, PDL and bone were extracted from bovine first molars and affixed to a tensile-compressive loading machine. The viscous properties of the tissue were analyzed (1) by subjecting the specimens to sinusoidal displacements at various frequencies and (2) by cycling the specimens in ‘fully saturated’ and in ‘partially dry’ conditions. Both modes assisted in determining the contribution of the fluid phase to the mechanical response.It was concluded that: (1) PDL showed pseudo-plastic viscous features for cyclic compressive loading, (2) these viscous features essentially resulted from interactions between the porous matrix and unbound fluid content of the tissue. Removing the liquid from the PDL largely eliminates its damping effect in compression.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.     

Nonlinear viscoelastic characterization of bovine trabecular bone

The time-independent elastic properties of trabecular bone have been extensively investigated, and several stiffness–density relations have been proposed. Although it is recognized that trabecular bone exhibits time-dependent mechanical behaviour, a property of viscoelastic materials, the characterization of this behaviour has received limited attention. The objective of the present study was to investigate the time-dependent behaviour of bovine trabecular bone through a series of compressive creep–recovery experiments and to identify its nonlinear constitutive viscoelastic material parameters. Uniaxial compressive creep and recovery experiments at multiple loads were performed on cylindrical bovine trabecular bone samples (\(n = 19\)). Creep response was found to be significant and always comprised of recoverable and irrecoverable strains, even at low stress/strain levels. This response was also found to vary nonlinearly with applied stress. A systematic methodology was developed to separate recoverable (nonlinear viscoelastic) and irrecoverable (permanent) strains from the total experimental strain response. We found that Schapery’s nonlinear viscoelastic constitutive model describes the viscoelastic response of the trabecular bone, and parameters associated with this model were estimated from the multiple load creep–recovery (MLCR) experiments. Nonlinear viscoelastic recovery compliance was found to have a decreasing and then increasing trend with increasing stress level, indicating possible stiffening and softening behaviour of trabecular bone due to creep. The obtained parameters from MLCR tests, expressed as second-order polynomial functions of stress, showed a similar trend for all the samples, and also demonstrate stiffening–softening behaviour with increasing stress.     

Rheology of hyaluronic acid

The dynamic viscoelastic properties of hyaluronic acid solutions have been measured over the frequency range 0.02–1.67 cps. The effects of varying temperature, hyaluronic acid concentration, pH, and ionic strength on the dynamic shear moduli were studied. The solutions exhibited a sharp transition from viscous to elastic behavior as the strain frequency increased. No entanglement coupling of the hyaluronic acid molecules was evident over the concentration range 2.0–4.0 mg./ml. Solutions at pH 2.5 showed a pronounced elastic behavior relative to both higher and lower pH's. This effect was attributed to a stiffening of the hyaluronic acid molecule at this pH.     

Hydro-mechanical coupling in the periodontal ligament: A porohyperelastic finite element model

Harmonic tension–compression tests at 0.1, 0.5 and 1 Hz on hydrated bovine periodontal ligament (PDL) were numerically simulated. The process was modeled by finite elements (FE) within the framework of poromechanics, with the objective of isolating the contributions of the solid- and fluid phases. The solid matrix was modeled as a porous hyperelastic material (hyperfoam) through which the incompressible fluid filling the pores flowed in accordance with the Darcy’s law. The hydro-mechanical coupling between the porous solid matrix and the fluid phase circulating through it provided an apparent time-dependent response to the PDL, whose rate of deformation depended on the permeability of the porous solid with respect to the interstitial fluid. Since the PDL was subjected to significant deformations, finite strains were taken into account and an exponential dependence of PDL permeability on void ratio – and therefore on the deformation state – was assumed. PDL constitutive parameters were identified by fitting the simulated response to the experimental data for the tests at 1 Hz. The values thus obtained were then used to simulate the tests at 0.1 and 0.5 Hz. The results of the present simulation demonstrate that a porohyperelastic model with variable permeability is able to describe the two main aspects of the PDL’s response: (1) the dependency on strain-rate—the saturated material can develop volumetric strains by only exchanging fluid and (2) the asymmetry between tension and compression, which is due to the effect of both the permeability and the elastic properties on deformation.     

Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Absence of desmin in skeletal muscle was found to induce an increase in passive stiffness. The present study aimed at developing rheological models of passive muscle to explain this stiffening. Models were elaborated by using experimental data depicting muscle viscoelastic behaviour. The experimental protocol included stepwise extension tests applied on control and desmin knockout soleus muscles from mice. Linear and non-linear models were composed of elastic and viscous elements. They were constructed with the aim at taking the presence or absence of desmin into account by simulating desmin as an elastic element. Furthermore, associated adaptation of connective tissues in absence of desmin was modelled as an additional elastic element. Differences in passive behaviour induced by absence of desmin were predicted by using a linear model and a non-linear one. The non-linear model was selected because: (1) it is able to predict experimental viscoelastic kinetics accounting for the increase in passive stiffness in muscles lacking desmin, (2) its design is consistent with morphological data, and (3) stiffness characteristics of its elements are in accordance with the literature. Finally, this modelling approach demonstrates that both absence of desmin and adaptation of connective tissue are required to explain the increase in passive stiffness in desmin knockout muscles.     

Analysis of contribution of collagen fibre component in viscoelastic behaviour of periodontal ligament using enzyme probe

The aims of this study are to observe microscopic changes in the periodontal ligament (PDL) collagen fibres after collagenase treatment, to analyse stress-relaxation behaviour of PDL specimens treated with collagenase, and to elucidate the contribution of the collagen component to the viscoelastic behaviour of the PDL. Transverse sections of rat mandibular first molars (n=24) were treated in vitro with 0, 8, 16, or 24 units of bacterial collagenase for 4h at 37 degrees C. Histological specimens were then prepared, and image analyses were done for polarised light microscopic appearances of collagen fibres. Further, stress-relaxation tests were performed for PDL specimens treated with 8 units of collagenase (n=7) and control specimens (n=7). Image analysis showed that higher concentrations of collagenase reduced greater area occupied by the PDL collagen fibres and birefringent retardation of the fibres. The amount of stress-relaxation during 600 s was 1.37 times greater in the collagenase-treated specimens than in the controls. The observed values of the stress-relaxation process were well described by a function with three exponential decay terms. The relaxation parameters of the first and second terms did not show significant differences, but those of the third term did so between the collagenase-treated and control specimens. The ratio and relaxation time of the third term for the collagenase-treated specimens were significantly less than those for the controls. These findings suggest that in the long-term relaxation component of the stress-relaxation process of the PDL the viscoelastic properties of the collagen fibres may play an important role.     

A visco-elastic model for the prediction of orthodontic tooth movement

This study presents a biomechanical model of orthodontic tooth movement. Although such models have already been presented in the literature, most of them incorporate computationally expensive finite elements (FE) methods to determine the strain distribution in the periodontal ligament (PDL). In contrast, the biomechanical model presented in this work avoids the use of FE methods. The elastic deformation of the PDL is modelled using an analytical approach, which does not require setting up a 3D model of the tooth. The duration of the lag phase is estimated using the calculated hydrostatic stresses, and bone remodelling is predicted by modelling the alveolar bone as a viscous material. To evaluate the model, some typically used motion patterns were simulated and a sensitivity analysis was carried out on the parameters. Results show that despite some shortcomings, the model is able to describe commonly used motion patterns in orthodontic tooth movement, in both single- and multi-rooted teeth.     

Some viscoplastic characteristics of bovine and human cortical bone

Multiple cycle tensile creep tests were performed on human and bovine cortical bone specimens. The tests enabled total strain to be decomposed into elastic, linear viscoelastic, creep and permanent plastic components. The results indicate that a stress threshold exists; above which time dependent effects dominate material response and below which the behavior is primarily linear viscoelastic, with time effects playing only a secondary role. A constant stress above the threshold produces a constant steady state creep rate, with the magnitude of the creep rate being an exponential function of the stress magnitude. Additionally, it was found that a major portion of the inelastic strain is always recovered on unloading and that the accumulation of creep strain increases the material compliance on subsequent loadings below the threshold. These two factors suggest that a damage mechanism is responsible for the nonlinear behavior.     

Material characterization of the pig kidney in relation with the biomechanical analysis of renal trauma

The objective of this study was an investigation of the material properties of the fresh pig kidney and parametric characterization of its elastic and inelastic material behavior. The material investigation included density measurements, uniaxial as well as three-dimensional compression tests, tensile tests. and shear tests on the samples extracted from the fresh pig kidney. For comparison, density measurements on a number of soft synthetic materials were also performed. Compression tests on the radial and the tangential specimens from the cortex tissue were performed at various loading rates. Three-axial compression tests were performed on the cortex tissues placed in a compression chamber. Shear tests were performed by punching a cylinder into a slice of the cortex. Tensile tests were carried out on the outer capsule. For characterization of the material behavior, a non-linear theoretical simulation based on a two parameter Blatz model was used. For characterization of the time-dependent behavior of the pig kidney cortex, a four-parameter linear viscoelastic model was employed. From the present experimental and theoretical studies, a number of conclusions were derived: (1) The general behavior of the pig kidney cortex samples under compression showed the general non-linear features typical of the soft tissues; the stress strain diagram was composed of a very flat part at very low stress level to about 30% relative deformation which was followed by a steeply rising stiffening leading to the radial rupture of samples marked by a maximum nominal rupture strain of about 50%. (2) The uniaxial compression tests on the radial and the tangential samples from the cortex tissue showed an increase of the rupture stress with the increase in the loading rate, but a decrease in the related rupture strain. (3) The long-term uniaxial compression tests on the cortex specimens under sustained constant load showed an instantaneous deformation followed by a creep response which eventually approached an asymptote. (4) Simulation of the non-linear material behavior of the cortex tissue under uniaxial compression by the Blatz model gave two pairs of material parameters for the cortex in the radial and the tangential directions. Furthermore, fitting of the assumed four-parameter linear viscoelastic model with the experimental data resulted in the viscoelastic material parameters.     

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.     

Oscillatory shear loading of bovine periodontal ligament--a methodological study

This study examined the stress response of bovine periodontal ligament (PDL) under sinusoidal straining. The principle of the test consisted in subjecting transverse tooth, PDL and bone sections of known geometries to controlled oscillatory force application. The samples were secured to the actuator by support plates fabricated using a laser sintering technique to fit their contours to the tooth and the alveolar bone. The actuator was attached to the root slices located in the specimen's center. Hence the machine was able to push or pull the root relative to its surrounding alveolar bone. After determining an optimal distraction amplitude, the samples were cyclically loaded first in ramps and then in sinusoidal oscillations at frequencies ranging from 0.2 to 5 Hz. In the present study the following observations were made: (1) Imaging and the laser sintering technique can be used successfully to fabricate custom-made support plates for cross-sectional root-PDL-bone sections using a laser sintering technique, (2) the load-response curves were symmetric in the apical and the coronal directions, (3) both the stress response versus phase angle and the stress response versus. strain curves tended to "straighten" with increasing frequency, and (4) the phase lag between applied strain and resulting stress was small and did not differ in the intrusive and the extrusive directions. As no mechanical or time-dependent anisotropy was demonstrable in the intrusive and extrusive directions, such results may considerably simplify the development of constitutive laws for the PDL.     

Time-dependent deformations in bone cells exposed to fluid flow in vitro: investigating the role of cellular deformation in fluid flow-induced signaling

Numerous experiments have shown fluid flow to be a potent stimulator of bone cells in vitro, suggesting that fluid flow is an important physical signal in bone mechanotransduction. In fluid flow experiments, bone cells are exposed to both time-dependent (e.g., oscillating or pulsing) and time-independent (e.g., steady) flow profiles. Interestingly, the signaling response of bone cells shows dependence on loading frequency and/or rate that has been postulated to be due to viscoelastic behavior. Thus, the objective of this study was to investigate the time-dependent deformations of bone cells exposed to fluid flow in vitro. Specifically, our goal was to characterize the mechanical response of bone cells exposed to oscillatory flow from 0.5 to 2.0 Hz and steady flow, since these flow profiles have previously been shown to induce different morphological and biochemical responses in vitro. By tracking cell-bound sulfate and collagen coated fluorescent beads of varying sizes, we quantified the normalized peak deformation (peak displacement normalized by the maximum peak displacement observed for all frequencies) and phase lag in bone cells exposed to 1.0 Pa oscillating flow at frequencies of 0.5-2.0 Hz. The phase lag was small (3-10 degrees ) and frequency dependent, while the normalized peak displacements decreased as a weak power law of frequency ( approximately f(-0.2)). During steady flow, the cells exhibited a nearly instantaneous deformation, followed by creep. Our results suggest that while substantial viscous deformation may occur during steady flow (compared to oscillating flow at approximately 1 Hz), bone cells behave primarily as elastic bodies when exposed to flow at frequencies associated with habitual loading.     

Viscoelastic properties of isolated collagen fibrils

Understanding the viscoelastic behavior of collagenous tissues with complex hierarchical structures requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar level. Using a microelectromechanical systems platform, in vitro coupled creep and stress relaxation tests were performed on collagen fibrils isolated from the sea cucumber dermis. Stress-strain-time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell-Weichert model. The fibrils showed an elastic modulus of 123 ± 46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell-Weichert model with a fast time response of 7 ± 2 s and a slow time response of 102 ± 5 s. The fibrillar relaxation time was smaller than literature values for tissue-level relaxation time, suggesting that tissue relaxation is dominated by noncollagenous components (e.g., proteoglycans). Each specimen was tested three times, and the only statistically significant difference found was that the elastic modulus is larger in the first test than in the subsequent two tests, indicating that viscous properties of collagen fibrils are not sensitive to the history of previous tests.     

Tensile and compressive behaviour of the bovine periodontal ligament

The mechanical response of the bovine periodontal ligament (PDL) subjected to uniaxial tension and compression is reported. Several sections normal to the longitudinal axis of bovine incisors and molars were extracted from different depths. Specimens with dimensions 10 x 5 x 2 mm including dentine, PDL and alveolar bone were obtained from these sections. Scanning electron microscopy suggested a strong similarity between the bovine PDL and the human PDL microstructure described in the literature. The prepared specimens were tested in a custom made uniaxial testing machine. They were clamped on their bone and dentine extremities and immersed in a saline solution at 37 degrees C. Stress-strain curves indicated that the PDL is characterized by a non-linear and time-dependent mechanical behaviour with the typical features of collagenous soft tissues. The curves exhibited hysteresis and preconditioning effects. The mechanical parameters evaluated in tension were maximum tangent modulus, strength, maximizer strain and strain energy density. For the molars, all these parameters increased with depth except for the apical region. For the incisors, all parameters increased with depth except ultimate strain which decreased. It was assumed that collagen fibre density and orientation were responsible for these findings.     

Viscous elements have little impact on measured passive length-tension properties of human gastrocnemius muscle-tendon units in vivo

Several studies have measured the elastic properties of a single human muscle-tendon unit in vivo. However the viscoelastic behavior of single human muscles has not been characterized. In this study, we adapted QLV theory to model the viscoelastic behavior of human gastrocnemius muscle-tendon units in vivo. We also determined the influence of viscoelasticity on passive length-tension properties of human gastrocnemius muscle-tendon units. Eight subjects participated in the experiment, which consisted of two parts. First, the stress relaxation response of human gastrocnemius muscle-tendon units was determined at a range of knee and ankle angles. Subsequently, passive ankle torque and ankle angle were collected during cyclic dorsiflexion and plantarflexion at a range of knee angles. Viscous parameters were determined by fitting the stress relaxation experiment data with a two-term exponential function, and elastic parameters were estimated by fitting the QLV model and viscous parameters to the cyclic experiment data. The model fitted the experimental data well at slow speeds (RMSE: 1.7 ± 0.5N) and at fast speeds (RMSE: 1.9 ± 0.2N). Muscle-tendon units demonstrated a large amount of stress relaxation. Nonetheless, viscoelastic passive length-tension curves estimated with the QLV model were similar to elastic passive length-tension curves obtained using a model that ignored viscosity. There was little difference in the elastic passive length-tension curves at different loading rates. We conclude that (a) the QLV model can be used to quantify viscoelastic behaviors of relaxed human gastrocnemius muscle-tendon units in vivo, and (b) over the range of velocities we examined, the velocity of loading has little effect on the passive length-tension properties of human gastrocnemius muscle-tendon units.     

Effects of tissue preservation on murine bone mechanical properties

Murine bone specimens are used extensively in skeletal research to assess the effects of environmental, physiologic and pathologic factors on their mechanical properties. Given the destructive nature of mechanical testing, it is normally performed as a terminal procedure, where specimens must be preserved without affecting their mechanical properties. To this end, we aimed to study the effects of tissue preservation (freezing and formalin fixation) on the elastic and viscoelastic mechanical properties of murine femur and vertebrae. A total of 120 femurs and 180 vertebral bodies (L3–L5) underwent non-destructive cyclic loading to assess their viscoelastic properties followed by mono-cyclic loading to failure to assess their elastic properties. All specimens underwent re-hydration in 0.9% saline for 30 min prior to mechanical testing. Analysis indicated that stiffness, modulus of elasticity, yield load, yield strength, ultimate load and ultimate strength of frozen and formalin-fixed femurs and vertebrae were not different from fresh specimens. Cyclic loading of both femurs and vertebrae indicated that loss, storage and dynamic moduli were not affected by freezing. However, formalin fixation altered their viscoelastic properties. Our findings suggest that freezing and formalin fixation over a 2-week period do not alter the elastic mechanical properties of murine femurs and vertebrae, provided that specimens are re-hydrated for at least half an hour prior to testing. However, formalin fixation weakened the viscoelastic properties of murine bone by reducing its ability to dissipate viscous energy. Future studies should address the long-term effects of both formalin fixation and freezing on the mechanical properties of murine bone.     

Stress-controlled viscoelastic tensile response of bovine cornea

The viscoelastic response of bovine corneas was characterized using in vitro load-controlled uniaxial tension experiments. Specifically, two types of tests were employed: cycled ramp tests over a range of loading rates and creep tests over a range of hold stresses. Multiple replicates of each were used to quantify natural variability as well as mean trends. A preconditioning protocol was used to obtain a unique reference state before testing and to overcome the effects of non-physiological loading. A quasi-linear viscoelastic model incorporating a representation of the microstructure of the cornea was compared to the experimental results. For low stresses and moderate durations this model compares favorably, but overall the material displays non-linearities that cannot be represented within the quasi-linear framework.