The superposition of steady on oscillatory shear and its effect on the viscoelasticity of human blood and a blood-like model fluid

To understand the pulsatility of human blood flow in vivo, it is necessary to separately investigate (1) steady shear and oscillatory flow, and (2) the superposition of steady shear flow on oscillatory flow performed under in vitro conditions. In this study a variable steady shear rate was superimposed in parallel on oscillatory shear at a constant frequency (0.5 Hz) for human blood (45% hematocrit), and an aqueous polyacrylamide polymer solution (AP 30E, concentration 300 ppm). The effect of superposition of the above two shear flows on the viscoelasticity of blood was more pronounced for the elastic (η′') than for the viscous (η′) component of viscoelasticity. With increasing superimposed shear rate, both η′ and η′' decreased, especially at the low shear region. This behavior can be explained by the viscoelastic properties of blood and the phenomena of blood aggregation and disaggregation. Quantitatively, the dependence of the viscous component of complex viscosity on superimposed shear for both blood and polymer solution is described by a modified Carreau equation. The elastic component of complex viscosity decreased exponentially with increasing superimposed shear, and it is described by an exponential model. © 1997 Elsevier Science Ltd     

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.     

Mechanical perturbation elicits a phenotypic difference between Dictyostelium wild-type cells and cytoskeletal mutants.

To determine the specific contribution of cytoskeletal proteins to cellular viscoelasticity we performed rheological experiments with Dictyostelium discoideum wild-type cells (AX2) and mutant cells altered by homologous recombination to lack alpha-actinin (AHR), the ABP120 gelation factor (GHR), or both of these F-actin cross-linking proteins (AGHR). Oscillatory and steady flow measurements of Dictyostelium wild-type cells in a torsion pendulum showed that there is a large elastic component to the viscoelasticity of the cell pellet. Quantitative rheological measurements were performed with an electronic plate-and-cone rheometer, which allowed determination of G', the storage shear modulus, and G", the viscous loss modulus, as a function of time, frequency, and strain, respectively. Whole cell viscoelasticity depends strongly on all three parameters, and comparison of wild-type and mutant strains under identical conditions generally produced significant differences. Especially stress relaxation experiments consistently revealed a clear difference between cells that lacked alpha-actinin as compared with wild-type cells or transformants without ABP120 gelation factor, indicating that alpha-actinin plays an important role in cell elasticity. Direct observation of cells undergoing shear deformation was done by incorporating a small number of AX2 cells expressing the green fluorescent protein of Aequorea victoria and visualizing the strained cell pellet by fluorescence and phase contrast microscopy. These observations confirmed that the shear strain imposed by the rheometer does not injure the cells and that the viscoelastic response of the cell pellet is due to deformation of individual cells.     

Aggregation and disaggregation of red blood cells

The aggregation of red blood cells may be analyzed as an interaction of an adhesive surface energy and the elastic stored energy which results from deformation of the cell. The adhesive surface energy is the work required to separate a unit adhered area and is the resultant of adhesive forces due to the bridging molecules that bind the cells together and the electrostatic repulsion due to surface charge. The elastic strain energy in the case of the red blood is associated with the membrane elasticity only since the interior of the cell is liquid. The membrane elasticity is due both to bending stiffness and shear. The area expansion is small and may be neglected. These assumptions allow realistic computation of red cell shapes in rouleaux. The disaggregation of rouleaux requires an external force which must overcome the adhesive energy and also supply additional elastic energy of deformation. Depending on the geometry, the initial effect of elastic energy may tend to aid disaggregation. In a shear flow, the stresses on a suspended rouleau alternately tend to compress and to disaggregate the cells if they are free to rotate. This introduces a time dependence so that viscous effects due to the viscosity of the cell membrane, the cell cytoplasm and the external fluid may play a role in determining whether disaggregation proceeds to completion or not.     

Viscoelasticity of Human Blood

Measurements made for oscillatory flow of blood in circular tubes show that blood possesses elastic properties which make consideration of its viscous properties alone inadequate. Results are for a frequency of 10 Hz while varying the amplitude of the velocity gradient for red blood cells in plasma at concentrations ranging from 0 to 100% apparent hematocrit. For velocity gradients less than 1-2 sec^-1 both the viscous and elastic components of the shearing stress are linearly related to the gradient. For hematocrits above 20% the elastic component of the complex coefficient of viscosity increases with hematocrit approximately to the third power while the viscous component increases exponentially. Oscillatory flow measurements at very low hematocrits, when extrapolated to zero cell concentration, give the intrinsic viscosity of the average individual isolated red cell. The viscous part of this is found to be 1.7 which is compared with theoretical values from the rigid ellipsoid model for which the minimum possible value is 2.5. This difference is attributed to cell deformability. With increasing velocity gradient nonlinear properties develop. The viscous component of the complex viscosity becomes of the order of the steady flow viscosity at high gradients while the elastic component tends to decrease in inverse proportion to the gradient. Thus, the elastic component of the oscillatory stress tends to saturate, this tendency appearing at the approximate level of the yield stress.     

The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes

An analytical solution for pulsatile flow of a generalized Maxwell fluid in straight rigid tubes, with and without axial vessel motion, has been used to calculate the effect of blood viscoelasticity on velocity profiles and shear stress in flows representative of those in the large arteries. Measured bulk flow rate Q waveforms were used as starting points in the calculations for the aorta and femoral arteries, from which axial pressure gradient ▿P waves were derived that would reproduce the starting Q waves for viscoelastic flow. The ▿P waves were then used to calculate velocity profiles for both viscoelastic and purely viscous flow. For the coronary artery, published ▿P and axial vessel acceleration waveforms were used in a similar procedure to determine the separate and combined influences of viscoelasticity and vessel motion.Differences in local velocities, comparing viscous flow to viscoelastic flow, were in all cases less than about 2% of the peak local velocity. Differences in peak wall shear stress were less than about 3%.In the coronary artery, wall shear stress differences between viscous and viscoelastic flow were small, regardless of whether axial vessel motion was included. The shape of the wall shear stress waveform and its difference, however, changed dramatically between the stationary and moving vessel cases. The peaks in wall shear stress difference corresponded with large temporal gradients in the combined driving force for the flow.     

Viscoelasticity reduces the dynamic stresses and strains in the vessel wall: implications for vessel fatigue

The mechanical behavior of blood vessels is known to be viscoelastic rather than elastic. The functional role of viscoelasticity, however, has remained largely unclear. The hypothesis of this study is that viscoelasticity reduces the stresses and strains in the vessel wall, which may have a significant impact on the fatigue life of the blood vessel wall. To verify the hypothesis, the pulsatile stress in rabbit thoracic artery at physiological loading condition was investigated with a quasi-linear viscoelastic model, where the normalized stress relaxation function is assumed to be isotropic, while the stress-strain relationship is anisotropic and nonlinear. The artery was subjected to the same boundary condition, and the mechanical equilibrium equation was solved for both the viscoelastic and an elastic (which has a constant relaxation function) model. Numerical results show that, compared with purely elastic response, the viscoelastic property of arteries reduces the magnitudes and temporal variations of circumferential stress and strain. The radial wall movement is also reduced due to viscoelasticity. These findings imply that viscoelasticity may be beneficial for the fatigue life of blood vessels, which undergo millions of cyclic mechanical loadings each year of life.     

The extracellular matrix viscoelasticity as a regulator of cell and tissue dynamics

The extracellular matrix mechanical properties regulate processes in development, cancer, and fibrosis. Among the distinct mechanical properties, the vast majority of research has focused on the extracellular matrix's elasticity as the primary determinant of cell and tissue behavior. However, both cells and the extracellular matrix are not only elastic but also viscous. Despite viscoelasticity being a universal feature of living tissues, our knowledge of the influence of the extracellular matrix's viscoelasticity in cell behavior is limited. This mini-review describes some of the recent findings that have highlighted the role of the extracellular matrix's viscoelasticity in cell and tissue dynamics.     

Light transmission through blood in oscillatory flow

Measurements were made of the intensity of light transmitted through various thicknesses of normal human blood confined between two parallel plane surfaces, one fixed and the other oscillating in its own plane. When the light propagation direction is perpendicular to the direction of shear flow the transmitted intensity contains a steady component and a dominant second harmonic of the oscillation frequency. For a thin layer of blood, the steady intensity is a minimum value when the oscillation amplitude produces unit strain. The second harmonic is very small at small strains, but increases rapidly near unit strain where it is approximately in phase with the strain. For thicker layers, the effects of viscoelastic shear waves reduce the size of the second harmonic and modify its phase. Changes in light transmission are interpreted by relating the optical density of the blood to the total amount of contact between red cells. In oscillatory flow at low strains (less than 1) cell-to-cell contact is reduced by disaggregation of cell groups, and light transmission decreases. Near unit strain, disaggregation becomes complete, cell alignment occurs, and light transmission is minimized. At higher strains cell-to-cell contact is increased by formation of aligned layers of compacted cells separated by parallel plasma layers, and light transmission increases.     

Intrinsic viscoelasticity of blood cell suspensions: effects of erythrocyte deformability

The intrinsic viscoelasticity of erythrocyte suspensions holds great potential for specifying the deformability of the individual, noninteracting cells in an oscillatory shear flow field. In order to extrapolate to zero cell concentration, the complex viscoelastic modulus was measured as a function of hematocrit using 2 Hertz oscillatory flow and a shear rate of 10/sec. This was done for both normal cells and cells with severely reduced deformability when hardened with glutaraldehyde. Suspension media were blood plasma, isotonic saline, and Dextran solutions. The real parts of the complex intrinsic visco-elasticities were obtained by an extrapolation using a regression fit to Huggins' equation. For normal cells in native plasma the values ranged from 1.7 to 2, increasing to the range 2.4 to 3.1 when the plasma was diluted with isotonic saline solution. For hardened cells the value obtained was near 3.5. These results are compared with theories for suspensions of both rigid and deformable particles. Several theories for deformable particles predict an increase in intrinsic viscoelasticity with increases in the ratio of the viscosity of the interior of the particle to that of the suspending medium. This ratio controls the balance between rotational and deformational response of the cell in the flow field. The trends of these theories were observed in the measurements.     

The dynamic stress responses to area change in planar lipid bilayer membranes

The viscoelastic properties of planar phospholipid (dimyristoylphosphatidylcholine) bilayer membranes at 308 K are studied, many of them for the first time, using the nonequilibrium molecular dynamics simulation (NEMD) method for membrane area change. First, we present a unified formulation of the intrinsic three-dimensional (3D) and apparent in-plane viscoelastic moduli associated with area change based on the constitutive relations for a uniaxial system. The NEMD simulations of oscillatory area change process are then used to obtain the frequency-domain moduli. In the 4-250 GHz range, the intrinsic 3D elastic moduli of 20-27 kbar and viscous moduli of 0.2-9 kbar are found with anisotropy and monotonic frequency dispersion. In contrast, the apparent in-plane elastic moduli (1-9 kbar) are much smaller than, and the viscous moduli (2-6 kbar) comparable to, their 3D counterparts, due to the interplay between the lateral and normal relaxations. The time-domain relaxation functions, separately obtained by applying stepwise strains, can be fit by 4-6 exponential decay modes spanning subpicosecond to nanosecond timescale and are consistent with the frequency-domain results. From NEMD with varying strain amplitude, the linear constitutive model is shown to be valid up to 6 and 20% area change for the intrinsic 3D elastic and viscous responses, respectively, and up to 20% area change for the apparent in-plane viscoelasticity. Inclusion of a gramicidin A dimer (approximately 1 mol %) yields similar response properties with possibly smaller (<10%) viscous moduli. Our results agree well with available data from ultrasonic experiments, and demonstrate that the third dimension (thickness) of the planar lipid bilayer is integral to the in-plane viscoelasticity.     

血液粘弹性测量的标准化及其实验研究

提出一种关于血液粘弹性测量的校正方案.一套关于平均切变率(或测头振动频率)、抗凝剂、红细胞压积和红细胞聚集等因素对粘弹性测量值影响的实验证实了这一方案的可用性.同时,实验结果表明,尽管红细胞压积的影响十分显著,仍然可以通过定义一个新的复数型的粘弹性参数L~(?)来消除这一影响.因此,该参数对建立血液粘弹性的人群正常值范围有实际意义,实验结果还显示、测量时用EDTA作抗凝剂为宜.     

Effect of hematocrit on wall shear rate in oscillatory flow: do the elastic properties of blood play a role?

Porcine blood was used to examine the relationship between hematocrit levels and wall shear rate patterns in straight and curved artery models under fixed oscillatory flow conditions characteristic of larger arteries. It is demonstrated that porcine blood models both the viscous and elastic components of the 2 Hz complex viscosity of human blood quite accurately over a broad range of shear rates (1-1000 s-1) and hematocrits (20%-80%). For a fixed oscillatory flow waveform (Poiseuille peak shear rate = 168 s-1; mean shear rate 84 s-1), increases in hematocrit produced a decrease in the peak wall shear rate in both the straight and curved artery models and a corresponding decrease in wall shear rate reversal on the inside wall of the curved artery model. The same trends were also observed for oscillatory flows of aqueous glycerin solutions of increasing viscosity in the range of viscosity of the blood samples tested. Aqueous glycerin solutions produced wall shear rate waveforms of the same magnitude and shape as the porcine blood. This indicates that variations in the shear rate, and therefore the shear stress, were caused primarily by changes in the viscous and not the elastic properties of blood. The results suggest that simple Newtonian fluids may be sufficient for in vitro determination of the first order effects to be expected of human blood flow in large vessels having complex geometries and shear rates in or above the range of the present study.     

Assessment of corneal viscoelasticity using elastic wave optical coherence elastography

The corneal viscoelasticity have great clinical significance, such as the early diagnosis of keratoconus. In this work, an analysis method which utilized the elastic wave velocity, frequency and energy attenuation to assess the corneal viscoelasticity is presented. Using phase‐resolved optical coherence tomography, the spatial‐temporal displacement map is derived. The phase velocity dispersion curve and center frequency are obtained by transforming the displacement map into the wavenumber‐frequency domain through the 2D fast Fourier transform (FFT). The shear modulus is calculated through Rayleigh wave equation using the phase velocity in the high frequency. The normalized energy distribution is plotted by transforming the displacement map into the spatial‐frequency domain through the 1D FFT. The energy attenuation coefficient is derived by exponential fitting to calculate the viscous modulus. Different concentrations of tissue‐mimicking phantoms and porcine corneas are imaged to validate this method, which demonstrates that the method has the capability to assess the corneal viscoelasticity.     

Elasticity-dependent response of malignant cells to viscous dissipation

The stiffness of the cellular environment controls malignant cell phenotype and proliferation. However, the effect of viscous dissipation on these parameters has not yet been investigated, in part due to the lack of in vitro cell substrates reproducing the mechanical properties of normal tissues and tumors. In this article, we use a newly reported viscoelastic polyacrylamide gel cell substrate, and we characterize the impact of viscous dissipation on three malignant cell lines: DU145 and PC3 derived from prostate and LN229 from brain. The spreading, motility and proliferation rates of these cells were analyzed on 1 kPa and 5 kPa elastic and viscoelastic gels. Surprisingly, the effect of substrate viscous dissipation on cell behavior depended on substrate stiffness for the three cell types tested. We conclude that viscoelasticity controls the spreading, proliferation and migration of malignant cells in vitro. These results highlight the critical role of viscous dissipation in the phenotype and proliferation of malignant cells, especially in stiff tumor environments.

     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals.     

The Mechanical Properties of Early Drosophila Embryos Measured by High-Speed Video Microrheology

In early development, Drosophila melanogaster embryos form a syncytium, i.e., multiplying nuclei are not yet separated by cell membranes, but are interconnected by cytoskeletal polymer networks consisting of actin and microtubules. Between division cycles 9 and 13, nuclei and cytoskeleton form a two-dimensional cortical layer. To probe the mechanical properties and dynamics of this self-organizing pre-tissue, we measured shear moduli in the embryo by high-speed video microrheology. We recorded position fluctuations of injected micron-sized fluorescent beads with kHz sampling frequencies and characterized the viscoelasticity of the embryo in different locations. Thermal fluctuations dominated over nonequilibrium activity for frequencies between 0.3 and 1000 Hz. Between the nuclear layer and the yolk, the cytoplasm was homogeneous and viscously dominated, with a viscosity three orders of magnitude higher than that of water. Within the nuclear layer we found an increase of the elastic and viscous moduli consistent with an increased microtubule density. Drug-interference experiments showed that microtubules contribute to the measured viscoelasticity inside the embryo whereas actin only plays a minor role in the regions outside of the actin caps that are closely associated with the nuclei. Measurements at different stages of the nuclear division cycle showed little variation.     

A new method for measuring the yield stress in thin layers of sedimenting blood.

A new method is presented to describe the low shear rate behavior of blood. We observed the response of a thin layer of sedimenting blood to a graded shear stress in a wedge-shaped chamber. The method allows quantitation of the degree of phase separation between red cells and plasma, and extracts the yield stress of the cell phase as a function of hematocrit. Our studies showed that the behavior of normal human blood underwent a transition from a solid-like gel to a Casson fluid. This transition began at the Casson predicted yield stress. The viscoelastic properties of blood were examined at shear stresses below the yield stress. The measured Young's elastic moduli were in good agreement with published data. The yield stress of blood showed a linear dependence on hematocrit up to 60%, and increased more rapidly at higher hematocrit.