An in-vivo measurement and analysis of viscoelastic properties of the spinal cord of cats

An in-vivo experimental technique was employed to determine the linear and nonlinear characteristics of viscoelastic properties of the spinal cord of anesthetized cats. The stress relaxation and recovery curves were reproducible in a group of cat experiments. The data of linear viscoelastic properties were used to develop a power law model with Boltzmann's convolution integral. The model was capable of predicting a prolonged stress relaxation and recovery curve. For larger deformation, the results were quantified using a nonlinear analysis of viscoelastic response of the spinal cord under the uniaxial experiment.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Force relaxation and permanent deformation of erythrocyte membrane.

Force relaxation and permanent deformation processes in erythrocyte membrane were investigated with two techniques: micropipette aspiration of a portion of a flaccid cell, and extension of a whole cell between two micropipettes. In both experiments, at surface extension ratios less than 3:1, the extent of residual membrane deformation is negligible when the time of extension is less than several minutes. However, extensions maintained longer result in significant force relaxation and permanent deformation. The magnitude of the permanent deformation is proportional to the total time period of extension and the level of the applied force. Based on these observations, a nonlinear constitutive relation for surface deformation is postulated that serially couples a hyperelastic membrane component to a linear viscous process. In contrast with the viscous dissipation of energy as heat that occurs in rapid extension of a viscoelastic solid, or in plastic flow of a material above yield, the viscous process in this case represents dissipation produced by permanent molecular reorganization through relaxation of structural membrane components. Data from these experiments determine a characteristic time constant for force relaxation, tau, which is the ratio of a surface viscosity, eta to the elastic shear modulus, mu. Because it was found that the concentration of albumin in the cell suspension strongly mediates the rate of force relaxation, values for tau of 10.1, 40.0, 62.8, and 120.7 min are measured at albumin concentrations of 0.0, 0.01, 0.1, and 1.% by weight in grams, respectively. The surface viscosity, eta, is calculated from the product of tau and mu. For albumin concentrations of 0.0, 0.01, 0.1, and 1% by weight in grams, eta is equal to 3.6, 14.8, 25.6, and 51.9 dyn s/cm, respectively.     

A mathematical model for the determination of viscoelastic behavior of brain in vivo. II. Relaxation response

In a recent experiment for determining the mechanical response of brain in vivo. a probe, inserted through scalp, skull and dura, is placed in contact with and normal to the brain, given a prescribed motion, and the time variation of corresponding force is measured. In the corresponding continuum mechanical model, brain is idealized as a linear isotropic viscoelastic solid constrained by a rigid skull. At the mating surface, the shear stress and normal displacement vanish everywhere except under the probe which exerts a local radial displacement. This model introduces effective viscoelastic moduli in shear, which is unknown, and in dilatation, which is considered known from other sources. Part II of this study is concerned with stress relaxation induced by a small step displacement of the probe. From the solution of the corresponding quasi-static boundary value problem, a nonlinear Volterra integral equation is established from which the shear stress relaxation function can be solved in terms of measured probe displacement and force. A numerical method of solution is developed.     

Shear viscoelasticity of suspensions of biological cells with viscoelastic membrane II

To discuss the relaxation phenomena of biological cell suspensions, we calculate the complex intrinsic viscosity of dispersions of spherical cells with viscoelastic membrane as a function of the frequency taking account of interfacial tension at both the interfaces of the membrane. The Maxwell model and two kinds of the three-parameter models are used to describe the viscoelasticity of the cell membrane. The results are computed mainly for the Maxwell model similarly in case of the Voigt Model (Abe, K., Takano, Y. and Sakanishi, A. Biorheology 21 405-414, 1984). The computed results of the four models, the Voigt, the Maxwell and the two kinds of the three-parameter viscoelastic models, are compared with the experimental data.     

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.     

Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties

The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.     

An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory

The quasi-linear viscoelastic (QLV) theory proposed by Fung (1972) has been frequently used to model the nonlinear time- and history-dependent viscoelastic behavior of many soft tissues. It is common to use five constants to describe the instantaneous elastic response (constants A and B) and reduced relaxation function (constants C, tau 1, and tau 2) on experiments with finite ramp times followed by stress relaxation to equilibrium. However, a limitation is that the theory is based on a step change in strain which is not possible to perform experimentally. Accounting for this limitation may result in regression algorithms that converge poorly and yield nonunique solutions with highly variable constants, especially for long ramp times (Kwan et al. 1993). The goal of the present study was to introduce an improved approach to obtain the constants for QLV theory that converges to a unique solution with minimal variability. Six goat femur-medial collateral ligament-tibia complexes were subjected to a uniaxial tension test (ramp time of 18.4 s) followed by one hour of stress relaxation. The convoluted QLV constitutive equation was simultaneously curve-fit to the ramping and relaxation portions of the data (r2 > 0.99). Confidence intervals of the constants were generated from a bootstrapping analysis and revealed that constants were distributed within 1% of their median values. For validation, the determined constants were used to predict peak stresses from a separate cyclic stress relaxation test with averaged errors across all specimens measuring less than 6.3 +/- 6.0% of the experimental values. For comparison, an analysis that assumed an instantaneous ramp time was also performed and the constants obtained for the two approaches were compared. Significant differences were observed for constants B, C, tau 1, and tau 2, with tau 1 differing by an order of magnitude. By taking into account the ramping phase of the experiment, the approach allows for viscoelastic properties to be determined independent of the strain rate applied. Thus, the results obtained from different laboratories and from different tissues may be compared.     

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.     

Required test duration for group comparisons in ligament viscoelasticity: a statistical approach

The goal of this study was to determine the duration of time that ligaments from a study group need to be loaded in order to adequately determine their collective viscoelastic behavior. Rat ligaments were subjected either to creep or stress relaxation for 1,000 s or stress relaxation for 10,000 s to compare estimates of viscoelastic behavior for different test durations. Stresses versus time (relaxation) or strains versus time (creep) were fit with power law models (tbeta where beta is the rate of creep or relaxation on a log-log scale). Time intervals were separated by logarithmic decade and analyzed using a Random Coefficients approach to compute residual specimen error as a function of the number of decades of data analyzed. Standard Regression was also used for comparison. Results show that by testing for 相似文献   

Prediction of internal pressure profile of compression bandages using stress relaxation parameters

The efficacy of compression therapy using compression bandages is highly dependent on the level of compression applied and the sustenance of the pressure during the course of treatment. This study attempts to predict the pressure profile generated by compression bandages using constitutive equations describing relaxation behavior of viscoelastic materials. It is observed that this pressure profile is highly correlated with the stress relaxation behavior of the bandage. To model the pressure profile, the stress relaxation behavior of compression bandages was studied and modeled using three mechanical models: the Maxwell model, the standard linear solid model and the two-component Maxwell model with a nonlinear spring. It was observed that the models with more component values explained the experimental relaxation curves better. The parameters used for modelling relaxation behavior were used to describe the pressure profile, which is significantly dependent on the longitudinal stress relaxation behavior of the bandage, using the modified Laplace's law equation. This approach thus helps in evaluating the bandage performance with time during compression therapy as novel wound care management.     

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Membrane viscoelasticity.

In this paper, we develop a theory for viscoelastic behavior of large membrane deformations and apply the analysis to the relaxation of projections produced by small micropipette aspiration of red cell discocytes. We show that this relaxation is dominated by the membrane viscosity and that the cytoplasmic and extracellular fluid flow have negligible influence on the relaxation time and can be neglected. From preliminary data, we estimate the total membrane "viscosity" when the membrane material behaves in an elastic solid manner. The total membrane viscosity is calculated to be 10(-3) dyn-s/cm, which is a surface viscosity that is about three orders of magnitude greater than the surface viscosity of lipid membrane components (as determined by "fluidity" measurements). It is apparent that the lipid bilayer contributes little to the fluid dynamic behavior of the whole plasma membrane and that a structural matrix dominates the viscous dissipation. However, we show that viscous flow in the membrane is not responsible for the temporal dependence of the isotropic membrane tension required to produce lysis and that the previous estimates of Rand, Katchalsky, et al., for "viscosity" are six to eight orders of magnitude too large.     

Pulse nuclear magnetic resonance measurements of water exchange across the erythrocyte membrane employing a low Mn concentration.

A simple, precise, and rapid pulse nuclear magnetic resonance technique for measuring the rate of water exchange across the erythrocyte membrane is presented. The technique is based upon the nonlinear fit of Carr-Purcell-Meiboom-Gill (CPMG) transverse relaxation time data of blood doped with 1.7 mM MnCl2 to the general two-compartment exchange condition. Previous approaches using CPMG data required high MnCl2 concentrations (25-53 nM), shown in this work to induce systematic errors ranging from 35 to 45%. At 23 degrees C the average residence time of a water molecule inside the erythrocyte (tau a) is 21.0 +/- 0.6 ms (SE). The Arrhenius plot for water exchange is linear over the range of 3 degrees - 37 degrees C and th Arrhenius activation energy is 4.79 +/- 0.03 kcal (SE). This value does not differ significantly from the energy required for bulk water flow. Results are compared with previous determinations, and sources of systematic error in tau a and the activation energy are evaluated.     

A finite nonlinear hyper-viscoelastic model for soft biological tissues

Soft tissues exhibit highly nonlinear rate and time-dependent stress-strain behaviour. Strain and strain rate dependencies are often modelled using a hyperelastic model and a discrete (standard linear solid) or continuous spectrum (quasi-linear) viscoelastic model, respectively. However, these models are unable to properly capture the materials characteristics because hyperelastic models are unsuited for time-dependent events, whereas the common viscoelastic models are insufficient for the nonlinear and finite strain viscoelastic tissue responses. The convolution integral based models can demonstrate a finite viscoelastic response; however, their derivations are not consistent with the laws of thermodynamics. The aim of this work was to develop a three-dimensional finite hyper-viscoelastic model for soft tissues using a thermodynamically consistent approach. In addition, a nonlinear function, dependent on strain and strain rate, was adopted to capture the nonlinear variation of viscosity during a loading process. To demonstrate the efficacy and versatility of this approach, the model was used to recreate the experimental results performed on different types of soft tissues. In all the cases, the simulation results were well matched ($R^2⩾0.99$) with the experimental data.     

A precision method to measure average viscoelastic parameters of erythrocyte populations

Suspensions of erythrocytes in media of low conductivity are subjected to homogenous high frequency electric fields (1 MHz, approximately 10 to 40 kV/m). The resulting transient deformation of the cells is measured by laser light diffraction. Employing a viscoelastic model of the erythrocyte membrane, relative values of membrane shear modulus and response time can currently be determined to within 7% or better. With a measurement time of one minute the average values of some 10(5) cells can be obtained. As a test of the method, osmotic swelling and deflation of the cells and crosslinking their membrane skeleton by diamide are used to alter the viscoelastic properties of the erythrocytes.     

Viscoelastic properties of transformed cells: role in tumor cell progression and metastasis formation

The micropipette aspiration technique was used to investigate the deformation properties of a panel of nontransformed and transformed rat fibroblasts derived from the same normal cell line. In this method, a step negative pressure is applied to the cell via a micropipette and the aspiration distance into the pipette as a function of time is determined using video techniques. A standard solid viscoelastic model was then used to analyze the viscoelastic properties of the cell. From these results, it is concluded that a direct correlation exists between an increase in deformability and progression of the transformed phenotype from a nontumorigenic cell line into a tumorigenic, metastatic cell line.     

Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity

BACKGROUND: Quasilinear viscoelasticity (QLV) theory has been widely and successfully used to describe the time-dependent response of connective tissues. Difficulties remain, however, particularly in material parameter estimation and sensitivities. In this study, we introduce a new alternative: the fractional order viscoelasticity (FOV) theory, which uses a fractional order integral to describe the relaxation response. FOV implies a fractal-like tissue structure, reflecting the hierarchical arrangement of collagenous tissues. METHOD OF APPROACH: A one-dimensional (I-D) FOV reduced relaxation function was developed, replacing the QLV "box-spectrum" function with a fractional relaxation function. A direct-fit, global optimization method was used to estimate material parameters from stress relaxation tests on aortic valve tissue. RESULTS: We found that for the aortic heart valve, FOV had similar accuracy and better parameter sensitivity than QLV, particularly for the long time constant (tau2). The mean (n = 5) fractional order was 0.29, indicating that the viscoelastic response of the tissue was strongly fractal-like. RESULTS SUMMARY: mean QLV parameters were C = 0.079, tau1 = 0.004, tau2 = 76, and mean FOV parameters were beta = 0.29, tau = 0.076, and rho = 1.84. CONCLUSIONS: FOV can provide valuable new insights into tissue viscoelastic behavior Determining the fractional order can provide a new and sensitive quantitative measure for tissue comparison.