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.     

Age-related and regional differences in the stress-strain and stress-relaxation behaviours of the rat incisor periodontal ligament

Groups of rats were killed at 2, 6, 12, and 24 months of age. From dissected left and right mandibles in each rat, three pairs of transverse sections were cut at the incisal, middle, and basal regions of the incisor. One section in each pair was loaded until failure and a stress-strain curve for the periodontal ligament (PDL) was obtained. The other section was loaded to up to 50% of the maximum shear stress as determined from the contralateral section and then kept at a constant strain for 10 min, to obtain the stress-relaxation curve at the same region of the PDL. The maximum shear stress and toughness increased with age at the incisal region and the maximum shear strain increased with age at the incisal and middle regions. The tangent modulus decreased with advancing age at the middle region. The stress-relaxation during 10 min decreased with advancing age at the incisal and basal regions, but not at the middle region. The relaxation process was well described by a sum of three exponential decay functions, reflecting the short-, medium-, and long-term relaxation components. The age-related decrease in the relaxation was mainly attributable to increases in the ratio and relaxation time of the long-term relaxation component. These results suggest that with advancing age the mechanical strength and toughness of the PDL are enhanced mostly at the incisal region and that the viscous fraction is relatively decreased at the incisal and basal regions along the long axis of the rat incisor.     

Formulation of pre-extension in a practical stress-relaxation measurement of the plant cell wall

In practical measurement of stress-relaxation, a certain length of time is required to apply the initial stress to a specimen. We refer to this process as pre-extension. In the stress-relaxation analysis of the plant cell wall which we developed, a wall specimen is stretched at a constant rate in pre-extension. The condition of pre-extension, such as the length of the pre-extension period, affects the subsequent stress-relaxation process. Thus, the stress-relaxation time spectrum calculated directly from the stress-relaxation process is an approximation. In this study, the condition during pre-extension was formulated in a stress-relaxation function and the stress-relaxation time spectrum, H(τ), was calculated as H(τ) = b·exp(−T_o/τ) −exp(−T_m/τ) / r·τ·(1−exp(−T_p/τ)) where T_o and T_m are the minimum and maximum relaxation times, τ is the relaxation time, b is the relaxation rate, r is the extension rate in pre-extension and T_p is the pre-extension time. The effect of the pre-extension condition on the spectrum is discussed.     

Viscoelastic properties of the human lumbodorsal fascia

The purpose of this study is to provide better understanding of the mechanical response of the lumbodorsal fascia to dynamic and static traction loadings. Since the fascia shows a viscoelastic behaviour, tests in which time is a variable were used, namely hysteresis and stress relaxation. Load-strain and load-time curves obtained from the hysteresis and stress-relaxation tests point out three different phenomena. First, an increase in stiffness is noticed when ligaments are successively stretched, i.e. strains produced by successive and identical loads decrease. Second, if a sufficient resting period is allowed between loadings, stiffening is reversed and strains tend to recover initial values. The third phenomenon, observed in stress-relaxation tests as time progresses, is ligament contraction in stretched and isometrically held samples. This third phenomenon may be explained by the possibility that muscle fibres capable of contracting spontaneously could be present in lumbodorsal fascia ligaments.     

The influence of humidity on the viscoelastic behaviour of human hair

The possibility of curling hair is attributed to the fact that the mechanical properties of hair fibres depend on time, temperature and humidity. In this study the dependence of the viscoelastic behaviour of human hair fibres on humidity is characterised in a bending deformation. An experimental set-up was used to perform bending relaxation measurements at different humidities. The relaxation data were fitted using a so-called "stretched exponential equation". The humidity dependence was incorporated by applying time-humidity superposition. Also, the influence of humidity on the initial E-modulus was found. The influence of humidity on the behaviour of human hair fibres could be modelled by using a general characterisation of the behaviour of human hair fibres.The general characterisation is used to predict the recovery in a human hair fibre after curling. The predicted recovery appeared to agree reasonably well with experimentally determined values.     

An analysis of the unconfined compression of articular cartilage

Analytical solutions have been obtained for the internal deformation and fluid-flow fields and the externally observable creep, stress relaxation, and constant strain-rate behaviors which occur during the unconfined compression of a cylindrical specimen of a fluid-filled, porous, elastic solid, such as articular cartilage, between smooth, impermeable plates. Instantaneously, the "biphasic" continuum deforms without change in volume and behaves like an incompressible elastic solid of the same shear modulus. Radial fluid flow then allows the internal fluid pressure to equilibrate with the external environment. The equilibrium response is controlled by the Young's modulus and Poisson's ratio of the solid matrix.     

Tissue changes in the rabbit periodontal ligament during orthodontic tooth movement

In this (semi) quantitative animal study the reaction of the periodontal ligament (PDL) to experimental tooth movement is described. To this end, rabbit first incisors were moved sideways with helical torsion springs for periods varying from 3-24 hours. The initial force of the springs was 50 gf. The histomorphology of the PDL was studied in 5 microns thick plastic sections. Comparison with control animals and animals wearing passive springs showed that tooth movement leads to an increased trauma in the PDL within only a few hours. This trauma is characterized by hyalinization, tears and ruptures in the fibres and blood vessels, and by the presence of extravascular erythrocytes and pyknosis. Tissue damage significantly increased with time. After 24 hours of tooth movement, the PDL fibers are compressed or stretched in 68% of the sections and the blood vessels in the PDL are compressed or stretched in 62% of the sections. Even in the controls, more than 15% of the sections displayed slightly stretched or compressed fibers, and about 10% showed slightly compressed or stretched blood vessels. This indicates that some damage is regularly present in a normally functioning PDL. Increases in the percentage of sections with blood vessel compression are found in all groups wearing passive springs, especially after 6 hours. A high concordancy in compression and tension patterns of blood vessels and fibers is present in 83% of the sections. Pyknotic cells are practically confined to areas with compressed PDL fibers in rabbits wearing active springs. Extravascular erythrocytes were found in sections with all types of fiber patterns. A significant majority of extravascular erythrocytes, however, was found in areas with compressed fibers.     

Growth and Stress-Relaxation Parameters for the Cell Wall of Normal and 10 Dwarf Barley Strains

Mechanical properties of the cell wall of 2- and 3-day-old barleycoleoptiles of normal and 10 dwarf strains were measured toexplain their different growth rates. Six parameters for mechanicalproperties (minimum and maximum stress relaxation time, relaxationrate, residual stress, strain ratio and the apparent Young'smodulus) were obtained by use of a stress-relaxation technique. The minimum stress-relaxation time and relaxation rate on day2, when the coleoptiles of all the strains had their maximumgrowth rates, were correlated significantly with the growthrates (r=0.79 and 0.72). When coleoptile growth in all the strainswas slowing down, the apparent Young's modulus was correlatedto the final coleoptile length. These results suggest (1) that when the coleoptile cell is activelygrowing, the minimum stress-relaxation time and relaxation rateof the cell wall affect the growth rate and (2) that the apparentYoung's modulus limits the final coleoptile length of the barleystrains examined. (Received October 13, 1983; Accepted April 16, 1984)     

Correlation between mechanical properties and wall composition of the canine superior vena cava

The mechanical properties (modulus of elasticity and stress-relaxation) of different venous segments of the canine superior vena cava were determined as well as the composition of the vessel wall by means of physical, biochemical and histological methods. It was found that the wall of the vena cava was structurally and mechanically a function of the metric position with respect to the right heart: the modulus of elasticity increased, the stress-relaxation decreased, the concentration of hydroxyproline, collagen and elastin increased and the amount of muscle fibres decreased with increasing distal distance from the right heart. A significant linear correlation coefficient was observed between the modulus of elasticity and the structural wall components. The data presented show the axial heterogeneity and the dependency of the mechanical properties upon the venous vessel wall composition.     

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.     

A comparison of cartilage stress-relaxation models in unconfined compression: QLV and stretched exponential in combination with fluid flow

Cartilage exhibits nonlinear viscoelastic behaviour. Various models have been proposed to explain cartilage stress relaxation, but it is unclear whether explicit modelling of fluid flow in unconfined compression is needed. This study compared Fung's quasi-linear viscoelastic (QLV) model with a stretched-exponential model of cartilage stress relaxation and examined each of these models both alone and in combination with a fluid-flow model in unconfined compression. Cartilage explants were harvested from bovine calf patellofemoral joints and equilibrated in tissue culture for 5 days before stress-relaxation testing in unconfined compression at 5% nominal strain. The stretched exponential models fit as well as the QLV models. Furthermore, the average stretched exponential relaxation time determined by this model lies within the range of experimentally measured relaxation times for extracted proteoglycan aggregates, consistent with the hypothesis that the stretched exponential model represents polymeric mechanisms of cartilage viscoelasticity.     

Dynamic response of immature bovine articular cartilage in tension and compression, and nonlinear viscoelastic modeling of the tensile response

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with R2 = 0.970+/-0.019 at 1% strain and R2 = 0.992+/-0.007 at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at 10 Hz was approximately 2.3 times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was approximately 24. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of 16.7+/-7.4 deg versus 53.5+/-12.8 deg at 10(-3) Hz). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies (R2 = 0.908+/-0.100). The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.     

The viscoelastic standard nonlinear solid model: predicting the response of the lumbar intervertebral disk to low-frequency vibrations

Due to the mathematical complexity of current musculoskeletal spine models, there is a need for computationally efficient models of the intervertebral disk (IVD). The aim of this study is to develop a mathematical model that will adequately describe the motion of the IVD under axial cyclic loading as well as maintain computational efficiency for use in future musculoskeletal spine models. Several studies have successfully modeled the creep characteristics of the IVD using the three-parameter viscoelastic standard linear solid (SLS) model. However, when the SLS model is subjected to cyclic loading, it underestimates the load relaxation, the cyclic modulus, and the hysteresis of the human lumbar IVD. A viscoelastic standard nonlinear solid (SNS) model was used to predict the response of the human lumbar IVD subjected to low-frequency vibration. Nonlinear behavior of the SNS model was simulated by a strain-dependent elastic modulus on the SLS model. Parameters of the SNS model were estimated from experimental load deformation and stress-relaxation curves obtained from the literature. The SNS model was able to predict the cyclic modulus of the IVD at frequencies of 0.01 Hz, 0.1 Hz, and 1 Hz. Furthermore, the SNS model was able to quantitatively predict the load relaxation at a frequency of 0.01 Hz. However, model performance was unsatisfactory when predicting load relaxation and hysteresis at higher frequencies (0.1 Hz and 1 Hz). The SLS model of the lumbar IVD may require strain-dependent elastic and viscous behavior to represent the dynamic response to compressive strain.     

The influence of elastic modulus and thickness on the release of the soft-fouling green alga Ulva linza (syn. Enteromorpha linza) from poly(dimethylsiloxane) (PDMS) model networks

The effect of modulus and film thickness on the release of adhered spores and sporelings (young plants) of the green fouling alga Ulva (syn. Enteromorpha) was investigated. PDMS elastomers of constant thickness (100 microm) but different elastic moduli were prepared by varying cross-link density with functional silicone oligomers with degrees of polymerization ranging from 18-830. This provided a 50-fold range of modulus values between 0.2 and 9.4 MPa. Three PDMS coatings of different thicknesses were tested at constant elastic modulus (0.8 MPa). The data revealed no significant increase in percentage spore removal except at the lowest modulus of 0.2 MPa although sporelings released more readily at all but the highest modulus. The influence of coating thickness was also greater for the release of sporelings compared to spores. The release data are discussed in the light of fracture mechanics models that have been applied to hard fouling. New concepts appertaining to the release of soft fouling organisms are proposed, which take into account the deformation in the adhesive base of the adherand and deformation of the PDMS film.     

Affine versus non-affine deformation in soft biological tissues, measured by the reorientation and stretching of collagen fibres through the thickness of compressed porcine skin

Skin is a complex three-dimensional structure of cells, collagen fibres and other proteins. However most mechanical analyses treat skin as a two-dimensional membrane, neglecting the through thickness structure. In this paper we investigate through thickness reorientation of collagen fibres. The mode of deformation of skin is also considered. For modelling purposes deformation is usually assumed to be affine. This assumption was tested by constructing a simple geometrical, affine deformation model to predict the through thickness reorientation of collagen fibres, from their initial through thickness angle and the measured deformations of skin samples during compression. The measured reorientation of collagen fibres was found to be very variable, however the average reorientations were consistent with the predictions of the model, with the inclusion of a systematic error. The variation in the reorientation of individual fibres can be explained by the variations in the structure at a micrometre scale. The systematic deviation of reorientations from the model predictions can be explained by a non-affine relationship between the collagen fibres and ground substance at a micrometre scale. However, non-affine deformations at a micrometre scale caused by irregularities of structure are likely to average out at a millimetre scale, because at this level material is evenly distributed.     

Cartilage stress-relaxation proceeds slower at higher compressive strains

Articular cartilage is the connective tissue which covers bone surfaces and deforms during in vivo activity. Previous research has investigated flow-dependent cartilage viscoelasticity, but relatively few studies have investigated flow-independent mechanisms. This study investigated polymer dynamics as an explanation for the molecular basis of flow-independent cartilage viscoelasticity. Polymer dynamics predicts that stress-relaxation will proceed more slowly at higher volumetric concentrations of polymer. Stress-relaxation tests were performed on cartilage samples after precompression to different strain levels. Precompression increases the volumetric concentration of cartilage biopolymers, and polymer dynamics predicts an increase in relaxation time constant. Stress-relaxation was slower for greater precompression. There was a significant correlation between the stress-relaxation time constant and cartilage volumetric concentration. Estimates of the flow-dependent timescale suggest that flow-dependent relaxation occurs on a longer timescale than presently observed. These results are consistent with polymer dynamics as a mechanism of cartilage viscoelasticity.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

On a three–dimensional model for the description of the passive characteristics of skeletal muscle tissue

In this work, a three–dimensional model was developed to describe the passive mechanical behaviour of anisotropic skeletal muscle tissue. To validate the model, orientation–dependent axial (\(0^\circ\), \(45^\circ\), \(90^\circ\)) and semi–confined compression experiments (mode I, II, III) were performed on soleus muscle tissue from rabbits. In the latter experiments, specimen deformation is prescribed in the loading direction and prevented in an additional spatial direction, fibre compression at \(0^\circ\) (mode I), fibre elongation at \(90^\circ\) (mode II) and a neutral state of the fibres at \(90^\circ\) where their length is kept constant (mode III). Overall, the model can adequately describe the mechanical behaviour with a relatively small number of model parameters. The stiffest tissue response during orientation–dependent axial compression (\(-\,7.7\,\pm \,1.3\) kPa) occurs when the fibres are oriented perpendicular to the loading direction (\(90^\circ\)) and are thus stretched during loading. Semi–confined compression experiments yielded the stiffest tissue (\(-\,36.7\,\pm \,11.2\) kPa) in mode II when the muscle fibres are stretched. The extensive data set collected in this study allows to study the different error measures depending on the deformation state or the combination of deformation states.

     

Anharmonic compression of the glitter lattice

A previous report has described the crystal structure of glitter, which is a dense 3-,4-connected net composed of ethylenic columns that run parallel to the c-axis of the unit cell. Such a structure invites speculation as to its relative stiffness along that axis. A semiempirical expression due to Cohen was used in a previous communication to estimate its zero-pressure bulk modulus. This estimate exceeds that of any known material at 440 GPa. Further, by treating the ethylenic units as harmonic springs, a correction was computed for the elastic deformation of the carbon-carbon double bonds along the c-axis. This correction is on the order of 300 GPa for deformations of the double bonds of approximately 0.1 angstroms. The present communication treats the ethylenic units along the c-axis of glitter as anharmonic springs obeying a Morse potential and a Morse's law force. Within the anharmonic approximation, at modest bond length deformations, x', the bulk modulus at pressure of the glitter lattice exceeds 1 TPa.