Viscoelastic material properties of the myocardium and cardiac jelly in the looping chick heart

Accurate material properties of developing embryonic tissues are a crucial factor in studies of the mechanics of morphogenesis. In the present work, we characterize the viscoelastic material properties of the looping heart tube in the chick embryo through nonlinear finite element modeling and microindentation experiments. Both hysteresis and ramp-hold experiments were performed on the intact heart and isolated cardiac jelly (extracellular matrix). An inverse computational method was used to determine the constitutive relations for the myocardium and cardiac jelly. With both layers assumed to be quasilinear viscoelastic, material coefficients for an Ogden type strain-energy density function combined with Prony series of two terms or less were determined by fitting numerical results from a simplified model of a heart segment to experimental data. The experimental and modeling techniques can be applied generally for determining viscoelastic material properties of embryonic tissues.     

Surprisingly simple mechanical behavior of a complex embryonic tissue

Background

Previous studies suggest that mechanical feedback could coordinate morphogenetic events in embryos. Furthermore, embryonic tissues have complex structure and composition and undergo large deformations during morphogenesis. Hence we expect highly non-linear and loading-rate dependent tissue mechanical properties in embryos.

Methodology/Principal Findings

We used micro-aspiration to test whether a simple linear viscoelastic model was sufficient to describe the mechanical behavior of gastrula stage Xenopus laevis embryonic tissue in vivo. We tested whether these embryonic tissues change their mechanical properties in response to mechanical stimuli but found no evidence of changes in the viscoelastic properties of the tissue in response to stress or stress application rate. We used this model to test hypotheses about the pattern of force generation during electrically induced tissue contractions. The dependence of contractions on suction pressure was most consistent with apical tension, and was inconsistent with isotropic contraction. Finally, stiffer clutches generated stronger contractions, suggesting that force generation and stiffness may be coupled in the embryo.

Conclusions/Significance

The mechanical behavior of a complex, active embryonic tissue can be surprisingly well described by a simple linear viscoelastic model with power law creep compliance, even at high deformations. We found no evidence of mechanical feedback in this system. Together these results show that very simple mechanical models can be useful in describing embryo mechanics.     

Dynamics of viscoelastic spherical membranes—the balloon model of the alveolus

Double-valued pressure-volume relationships in dynamic conditions for spherical membranes, modelling the lung alveoli, were obtained at small deformations. This hysteretic behavior was considered to be produced by at least three independent mechanisms: (1) the lung parenchyma exhibits viscoelastic properties; (2) the lung surface film, independent of the tissue, exhibits viscoelastic properties and (3) the pressure acting on the inner membrane surface depends on the rate of the alveolus volume change, due to the air viscous resistance in the bronchial tree. In each case, the maximum volume change, the hysteresis loop area, the tilt angle of the hysteresis loop and the relaxation time of the system were calculated. The results show pronounced hysteresis at normal breathing due to the air viscous resistance and smaller one due to the tissue and surface viscoelastic properties. In quasistatic conditions the values of the surface viscoelasticity and the tissue viscoelasticity effects are comparable or different, depending on the concrete external conditions. Comparison with the available experimental data is discussed in detail.     

Cell poking: quantitative analysis of indentation of thick viscoelastic layers.

A recently introduced device, the cell poker, measures the force required to indent the exposed surface of a cell adherent to a rigid substratum. The cell poker has provided phenomenological information about the viscoelastic properties of several different types of cells, about mechanical changes triggered by external stimuli, and about the role of the cytoskeleton in these mechanical functions. Except in special cases, however, it has not been possible to extract quantitative estimates of viscosity and elasticity moduli from cell poker measurements. This paper presents cell poker measurements of well characterized viscoelastic polymeric materials, polydimethylsiloxanes of different degrees of polymerization, in a simple shape, a flat, thick layer, which for our purposes can be treated as a half space. Analysis of the measurements in terms of a linear viscoelasticity theory yields viscosity values for three polymer samples in agreement with those determined by measurements on a macroscopic scale. Theoretical analysis further indicates that the measured limiting static elasticity of the layers may result from the tension generated at the interface between the polymer and water. This work demonstrates the possibility of obtaining quantitative viscoelastic material properties from cell poker measurements and represents the first step in extending these quantitative studies to more complicated structures including cells.     

Viscoelastic shear properties of the corneal stroma

The cornea is a highly specialized transparent tissue which covers the front of the eye. It is a tough tissue responsible for refracting the light and protecting the sensitive internal contents of the eye. The biomechanical properties of the cornea are primarily derived from its extracellular matrix, the stroma. The majority of previous studies have used strip tensile and pressure inflation testing methods to determine material parameters of the corneal stroma. Since these techniques do not allow measurements of the shear properties, there is little information available on transverse shear modulus of the cornea. The primary objectives of the present study were to determine the viscoelastic behavior of the corneal stroma in shear and to investigate the effects of the compressive strain. A thorough knowledge of the shear properties is required for developing better material models for corneal biomechanics. In the present study, torsional shear experiments were conducted at different levels of compressive strain (0–30%) on porcine corneal buttons. First, the range of linear viscoelasticity was determined from strain sweep experiments. Then, frequency sweep experiments with a shear strain amplitude of 0.2% (which was within the region of linear viscoelasticity) were performed. The corneal stroma exhibited viscoelastic properties in shear. The shear storage modulus, G′, and shear loss modulus, G″, were reported as a function of tissue compression. It was found that although both of these parameters were dependent on frequency, shear strain amplitude, and compressive strain, the average shear storage and loss moduli varied from 2 to 8 kPa, and 0.3 to 1.2 kPa, respectively. Therefore, it can be concluded that the transverse shear modulus is of the same order of magnitude as the out-of-plane Young's modulus and is about three orders of magnitude lower than the in-plane Young's modulus.     

Influence of static stretching on viscoelastic properties of human tendon structures in vivo.

The purpose of this study was to investigate the influences of static stretching on the viscoelastic properties of human tendon structures in vivo. Seven male subjects performed static stretching in which the ankle was passively flexed to 35 degrees of dorsiflexion and remained stationary for 10 min. Before and after the stretching, the elongation of the tendon and aponeurosis of medial gastrocnemius muscle (MG) was directly measured by ultrasonography while the subjects performed ramp isometric plantar flexion up to the maximum voluntary contraction (MVC), followed by a ramp relaxation. The relationship between the estimated muscle force (Fm) of MG and tendon elongation (L) during the ascending phase was fitted to a linear regression, the slope of which was defined as stiffness of the tendon structures. The percentage of the area within the Fm-L loop to the area beneath the curve during the ascending phase was calculated as an index representing hysteresis. Stretching produced no significant change in MVC but significantly decreased stiffness and hysteresis from 22.9 +/- 5.8 to 20.6 +/- 4.6 N/mm and from 20.6 +/- 8.8 to 13.5 +/- 7.6%, respectively. The present results suggest that stretching decreased the viscosity of tendon structures but increased the elasticity.     

Comparison of elastic,viscoelastic and failure tensile material properties of knee ligaments and patellar tendon

The knee ligaments and patellar tendon function in concert with each other and other joint tissues, and are adapted to their specific physiological function via geometry and material properties. However, it is not well known how the viscoelastic and quasi-static material properties compare between the ligaments. The purpose of this study was to characterize and compare these material properties between the knee ligaments and patellar tendon.Dumbbell-shaped tensile test samples were cut from bovine knee ligaments (ACL, LCL, MCL, PCL) and patellar tendon (PT) and subjected to tensile testing (n = 10 per ligament type). A sinusoidal loading test was performed at 8% strain with 0.5% strain amplitude using 0.1, 0.5 and 1 Hz frequencies. Subsequently, an ultimate tensile test was performed to investigate the stress-strain characteristics.At 0.1 Hz, the phase difference between stress and strain was higher in LCL compared with ACL, PCL and PT (p < 0.05), and at 0.5 Hz that was higher in LCL compared with all other ligaments and PT (p < 0.05). PT had the longest toe-region strain (p < 0.05 compared with PCL and MCL) and MCL had the highest linear and strain-dependent modulus, and toughness (p < 0.05 compared with ACL, LCL and PT).The results indicate that LCL is more viscous than other ligaments at low-frequency loads. MCL was the stiffest and toughest, and its modulus increased most steeply at the toe-region, possibly implying a greater amount of collagen. This study improves the knowledge about elastic, viscoelastic and failure properties of the knee ligaments and PT.     

Rheology of gastric mucin exhibits a pH-dependent sol-gel transition

Gastric mucin, a high molecular weight glycoprotein, is responsible for providing the gel-forming properties and protective function of the gastric mucus layer. Bulk rheology measurements in the linear viscoelastic regime show that gastric mucin undergoes a pH-dependent sol-gel transition from a viscoelastic solution at neutral pH to a soft viscoelastic gel in acidic conditions, with the transition occurring near pH 4. In addition to pH-dependent gelation behavior in this system, further rheological studies under nonlinear deformations reveal shear thinning and an apparent yield stress in this material which are also highly influenced by pH.     

Precision of nanoindentation protocols for measurement of viscoelasticity in cortical and trabecular bone

Nanoindentation has recently gained attention as a characterization technique for mechanical properties of biological tissues, such as bone, on the sub-micron level. However, optimal methods to characterize viscoelastic properties of bones are yet to be established. This study aimed to compare the time-dependent viscoelastic properties of bone tissue obtained with different nanoindentation methods. Bovine cortical and trabecular bone samples (n=8) from the distal femur and proximal tibia were dehydrated, embedded and polished. The material properties determined using nanoindentation were hardness and reduced modulus, as well as time-dependent parameters based on creep, loading-rate, dissipated energy and semi-dynamic testing under load control. Each loading protocol was repeated 160 times and the reproducibility was assessed based on the coefficient of variation (CV). Additionally, three well-characterized polymers were tested and CV values were calculated for reference.The employed methods were able to characterize time-dependent viscoelastic properties of bone. However, their reproducibility varied highly (CV 9–40%). The creep constant increased with increasing dwell time. The reproducibility was best with a 30 s creep period (CV 18%). The dissipated energy was stable after three repeated load cycles, and the reproducibility improved with each cycle (CV 23%). The viscoelastic properties determined with semi-dynamic test increased with increase in frequency. These measurements were most reproducible at high frequencies (CV 9–10%). Our results indicate that several methods are feasible for the determination of viscoelastic properties of bone material. The high frequency semi-dynamic test showed the highest precision within the tested nanoindentation protocols.     

Rheological properties of the human lumbar spine ligaments

The purpose of this study is to provide a better understanding of the rheological properties of the lumbar spinal ligaments under subfailure physiological loads. Non-destructive tests including an hysteresis experiment, stress-relaxation and stepwise load-relaxation tests were used to investigate the time-dependent properties of the interspinous-supraspinous ligament complex. Using a reduced relaxation function, the viscoelastic behaviour over the experimental time-scale was described by a linear function of the logarithm of time. Internal damping of ligament substance dissipates about 36% of the mechanical energy applied during physiological loading. Local elastic stiffness is found to be two to four times global stiffness of the bone-ligament-bone complex. These physical parameters (stiffness, energy dissipation, hysteresis, relaxation, etc) can be used to improve computer models of the lumbar spinal column.     

Viscoelastic characterization of peripapillary sclera: material properties by quadrant in rabbit and monkey eyes

In this report we characterize the viscoelastic material properties of peripapillary sclera from the four quadrants surrounding the optic nerve head in both rabbit and monkey eyes. Scleral tensile specimens harvested from each quadrant were subjected to uniaxial stress relaxation and tensile ramp to failure tests. Linear viscoelastic theory, coupled with a spectral reduced relaxation function, was employed to characterize the viscoelastic properties of the tissues. We detected no differences in the stress-strain curves of specimens from the four quadrants surrounding the optic nerve head (ONH) below a strain of 4 percent in either the rabbit or monkey. While the peripapillary sclera from monkey eyes is significantly stiffer (both instantaneously and in equilibrium) and relaxes more slowly than that from rabbits, we detected no differences in the viscoelastic material properties (tested at strains of 0-1 percent) of sclera from the four quadrants surrounding the ONH within either species group.     

Viscoelastic properties of gerbil tympanic membrane at very low frequencies

The tympanic membrane transfers sound waves in the ear canal to mechanical vibrations in the middle ear and cochlea. Good estimates of the mechanical properties of the tympanic membrane are important to obtain realistic models. Up till now, only limited resources about tympanic membrane viscoelastic properties are available in the literature. This study aimed to quantify the viscoelastic properties of gerbil tympanic membrane. Step indentations were applied with a custom indenter on four fresh, intact tympanic membranes and the resulting force relaxation was measured. The reduced relaxation functions were then fitted with two viscoelastic model representations: a 5-parameter Maxwell model and a model with a continuous relaxation spectrum. The average relaxation function is described by an initial rapid decrease of 6.5% with characteristic time 0.77 s, followed by a long term decrease with characteristic time 46 s that gradually tends stable till a total relaxation of 15%. The relaxation curves in the time domain were transformed to complex moduli in the frequency domain. It was found that these transformations yield information on strain-rate dependence only from quasi-static to the very lowest acoustic frequencies. Finally, relaxation and hysteresis were simulated in a finite element model with viscoelastic material properties.     

Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone

Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples (\(\textit{n}\,{=}\,13\)) at loads corresponding to physiological strain level of 2000 \({\upmu }{\upvarepsilon }\). We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships (\(r^2\,{=}\,0.73,\ p\,{<}\,0.001\)) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV.     

The Physical and Linear Viscoelastic Properties of Fresh Wet Foams Based on Egg White Proteins and Selected Hydrocolloids

The aim of this work was to evaluate the physicochemical properties of fresh foams based on egg white proteins, xanthan gum and gum Arabic. The distributions of the size of gas bubbles suspended in liquid were determined, as well as density and volume fraction of gas phase of the generated foams. Additionally, the viscoelastic properties in the linear range were measured, and the results were analyzed with the use of the fractional Zener model. It was shown, that foam supplementation with hydrocolloids considerably decreased their volume fraction of gas phase in comparison to pure egg white protein-based foams. Application of gum Arabic did not cause an increase in the size of foam bubbles when compared to pure white egg foam, whereas application of xanthan gum significantly decreased the size of the bubbles. Application of the fractional Zener model allowed to determine the relaxation times, their intensity in analyzed suspensions and also equilibrium module (G _e ). The increase in the concentration of xanthan gum resulted in the prolongation of the relaxation time and increased its intensity. Gum Arabic, when added, weakened the viscoelastic properties of the mixture as a viscoelastic solid.     

Understanding of the Viscoelastic Response of the Human Corneal Stroma Induced by Riboflavin/UV-A Cross-Linking at the Nano Level

Purpose

To investigate the viscoelastic changes of the human cornea induced by riboflavin/UV-A cross-linking using Atomic Force Microscopy (AFM) at the nano level.

Methods

Seven eye bank donor corneas were investigated, after gently removing the epithelium, using a commercial AFM in the force spectroscopy mode. Silicon cantilevers with tip radius of 10 nm and spring elastic constants between 26- and 86-N/m were used to probe the viscoelastic properties of the anterior stroma up to 3 µm indentation depth. Five specimens were tested before and after riboflavin/UV-A cross-linking; the other two specimens were chemically cross-linked using glutaraldehyde 2.5% solution and used as controls. The Young’s modulus (E) and the hysteresis (H) of the corneal stroma were quantified as a function of the application load and scan rate.

Results

The Young’s modulus increased by a mean of 1.1-1.5 times after riboflavin/UV-A cross-linking (P<0.05). A higher increase of E, by a mean of 1.5-2.6 times, was found in chemically cross-linked specimens using glutaraldehyde 2.5% (P<0.05). The hysteresis decreased, by a mean of 0.9-1.5 times, in all specimens after riboflavin/UV-A cross-linking (P<0.05). A substantial decrease of H, ranging between 2.6 and 3.5 times with respect to baseline values, was observed in glutaraldehyde-treated corneas (P<0.05).

Conclusions

The present study provides the first evidence that riboflavin/UV-A cross-linking induces changes of the viscoelastic properties of the cornea at the scale of stromal molecular interactions.     

Elastic response, buckling, and instability of microtubules under radial indentation

We tested the mechanical properties of single microtubules by lateral indentation with the tip of an atomic force microscope. Indentations up to approximately 3.6 nm, i.e., 15% of the microtubule diameter, resulted in an approximately linear elastic response, and indentations were reversible without hysteresis. At an indentation force of around 0.3 nN we observed an instability corresponding to an approximately 1-nm indentation step in the taxol-stabilized microtubules, which could be due to partial or complete rupture of a relatively small number of lateral or axial tubulin-tubulin bonds. These indentations were reversible with hysteresis when the tip was retracted and no trace of damage was observed in subsequent high-resolution images. Higher forces caused substantial damage to the microtubules, which either led to depolymerization or, occasionally, to slowly reannealing holes in the microtubule wall. We modeled the experimental results using finite-element methods and find that the simple assumption of a homogeneous isotropic material, albeit structured with the characteristic protofilament corrugations, is sufficient to explain the linear elastic response of microtubules.     

Experiments to determine the time dependent material properties of the periodontal ligament]

The periodontal ligament is a tissue that attaches the tooth (root) to its alveolar socket, and thus plays an important role in the regulation of tooth movements. Detailed knowledge of the material properties of the periodontal ligament is therefore essential to an understanding of tooth reaction to forces applied during orthodontic treatment. A knowledge of material parameters can also be used in simulations of long-term tooth movements with the aim of improving orthodontic treatment. To this end, this study investigated time-dependent material properties, namely the hysteresis behaviour of the periodontal ligament under constant-velocity loading, the influence of loading velocity on the hysteresis, and its failure under constant loading. Specimens obtained from pigs were used for testing purposes, and the experiments were conducted in a special test setup using a material testing device. The material behaviour of the periodontal ligament was shown to be viscoelastic, and the elastic parameters of material behaviour were also determined. Under constant-velocity loading, material behaviour showed a nonlinear course of the stress-strain curve, also known as hysteresis. When loading was repeated several times, the maximum stress of the hysteresis decreased with each cycle. Determination of the deflection of the specimen at different velocities showed maximum stress to be dependent on the loading rate. The measured stress-strain curves were approximated by bilinear behaviour, permitting the use of finite element calculations. Also investigated was the failure behaviour of the periodontal ligament, which revealed tissue rupture to be inconstant.     

Characterization of structure and properties of bone by spectral measure method

Novel mathematical method called spectral measure method (SMM) is developed for characterization of bone structure and indirect estimation of bone properties. The spectral measure method is based on an inverse homogenization technique which allows to derive information about the structure of composite material from measured effective electric or viscoelastic properties. The mechanical properties and ability to withstand fracture depend on the structural organization of bone as a hierarchical composite. Information about the bone structural parameters is contained in the spectral measure in the Stieltjes integral representation of the effective properties. The method is based on constructing the spectral measure either by calculating it directly from micro-CT images or using measurements of electric or viscoelastic properties over a frequency range. In the present paper, we generalize the Stieltjes representation to the viscoelastic case and show how bone microstructure, in particular, bone volume or porosity, can be characterized by the spectral function calculated using measurements of complex permittivity or viscoelastic modulus. For validation purposes, we numerically simulated measured data using micro-CT images of cancellous bone. Recovered values of bone porosity are in excellent agreement with true porosity estimated from the micro-CT images. We also discuss another application of this method, which allows to estimate properties difficult to measure directly. The spectral measure method based on the derived Stieltjes representation for viscoelastic composites, has a potential for non-invasive characterization of bone structure using electric or mechanical measurements. The method is applicable to sea ice, porous rock, and other composite materials.     

Comparison of balloon and transducer catheters for estimating lung elasticity.

Pleural pressure is usually estimated with a balloon catheter (BC) positioned in the middle third of the esophagus. An alternate method, which avoids potential inaccuracies associated with changes in balloon volume, is a catheter-mounted transducer (CMT) system. To assess the accuracy of a CMT system in defining the elastic properties of the lungs, we compared the static pressure-volume (PV) properties of the lungs measured sequentially with CMT and BC systems in six healthy subjects each on two occasions, relating static transpulmonary pressure (Pst,L) to lung volume during interrupted exhalations from total lung capacity (TLC). PV data were fitted with an exponential function (least-squares method), and the exponent (k) was used to define the shape of the PV curve; position was defined by Pst,L at TLC and at 90 and 60% TLC. These data were examined for agreement (paired t test) and repeatability (coefficient of repeatability). No significant differences were demonstrated: k was 0.10 +/- 0.02 and 0.11 +/- 0.03 (SD) and Pst,L at 60% TLC was 8.27 +/- 2.09 and 8.37 +/- 1.63 cmH2O for the CMT and BC systems, respectively. The coefficient of repeatability for each parameter was not significantly different but was consistently less with the BC, suggesting better repeatability. We conclude that a CMT system is an acceptable alternative to a BC system for defining the elastic properties of lungs.     

Mechanoptical transducer for quantifying activity in small insect muscles

Contraction strength, frequency, rate, relaxation rate, as well as transient and static muscle lengths were quantified for onion fly (Delia antiqua) oviducal muscle to demonstrate the application of a newly developed, sensitive, rapidly responding, stable and linear mechanoptical transducer system. Contractile patterns were also differentiated for the complex array of muscle segments in the oviduct. Computer-assisted analysis of analogue records showed that within-train contraction strength varied inversely as a function of contraction frequency in a train. Also, a tonic component, assumed to be a function of contraction frequency and the viscoelastic properties of the tissue, was superimposed on trains of phasic, longitudinal contractions. The transducer system described in this report provides opportunities to quantify contraction phenomena occurring at intervals approaching 1 ms in small (nom. 1 mm) tissue samples with resolution in the order of 1 μg of force and 10 μm of displacement.