Obtaining the shear stress versus shear rate relationship and yield stress of blood from capillary viscometry data by Tikhonov regularization

This paper describes a procedure, based on Tikhonov regularization, for extracting the shear stress versus shear rate relationship and yield stress of blood from capillary viscometry data. The relevant equations and the mathematical nature of the problem are briefly described. The procedure is then applied to three sets of capillary viscometry data of blood taken from the literature. From each data set the procedure computes the complete shear stress versus shear rate relationship and the yield stress. Since the procedure does not rely on any assumed constitutive equation, the computed rheological properties are therefore model-independent. These properties are compared against one another and against independent measurements. They are found to be in good agreement for shear stress greater than 0.1 Pa but show significant deviations for shear stress below this level. A possible way of improving this situation is discussed.     

An automated tube-type blood viscometer: validation studies

The technical complexity of previous rheometers has tended to limit the availability of blood viscosity data obtained over a wide range of shear rates. However, an automated tube-type viscometer, the Rheolog, has been developed; it employs a disposable flow assembly and less than five minutes are required to obtain blood viscosity results over a shear rate range of 1-1500 s(-1). We have carried out validation studies of the Rheolog using normal human blood and have compared these results with those obtained by cone-plate and Couette viscometers; storage time and temperature effects were also evaluated. Replicate measurements indicated mean CV levels less than 5%, and were independent of hematocrit and shear rate. Rheolog blood viscosity data agreed closely with those from other viscometers: average Rheolog differences from mean cone-plate and Couette values were -0.3% at 28% hematocrit, -1.4% at 41% hematocrit (i.e., native), and 1.0% at 56% hematocrit. Storage at room temperature up to 8 hours and at 4 degrees C up to 4 days had minimal effects whereas notable changes were observed when stored for 3 hours at 37 degrees C. Our results indicate that, within the hematocrit and shear rate limits employed herein, the Rheolog provides rapid, accurate and reproducible blood viscosity data, and suggest its usefulness for both basic science and clinical studies.     

Blood's critical Taylor number and its flow behavior at supercritical Taylor numbers

When the inner cylinder of a fluid-filled Couette viscometer is rotated rapidly, a vortical flow pattern develops when a dimensionless value referred to as the critical Taylor number (Tc) is reached. We have determined its magnitude in our viscometer for three Newtonian fluids and for blood at 37 degrees C, using the inflection point of torque/RPM vs. RPM (sudden rise in apparent viscosity). Its position was identified by least squares line fitting. Because blood was studied, the viscosity used in Tc calculation was the apparent bob shear stress/shear rate ratio at the inflection marking vortical flow onset. For glycerol-water mixtures Tc was 41.8 +/- 0.3 (N = 11), for propylene glycol 42.0 +/- 0.2 (N = 14), for silicone oil 41.8 +/- 0.2 (N = 11). For healthy blood Tc was 40.7 +/- 0.9 (N = 140). This evidence against blood's increased resistance to flow instability was accompanied by a slower rate of rise in torque both above and below Tc compared to the three Newtonian fluids. Newtonian fluids and blood both developed wavy vortical flow at a rotation rate moderately higher than Tc. Blood resisted this unstable flow behavior more than the Newtonian fluids but it also experienced a slower rate of rise in torque with increasing rotation rate above the critical Taylor number. Shear-thinning is the simplest explanation for blood's mildly altered Taylor vortex behavior; blood's resistance to flow instability is otherwise not found to be sufficient to affect its flow stability in man.     

Shear-stress dependence of dinoflagellate bioluminescence

Fluid flow stimulates bioluminescence in dinoflagellates. However, many aspects of the cellular mechanotransduction are incompletely known. The objective of our study was to formally test the hypothesis that flow-stimulated dinoflagellate bioluminescence is dependent on shear stress, signifying that organisms are responding to the applied fluid force. The dinoflagellate Lingulodinium polyedrum was exposed to steady shear using simple Couette flow in which fluid viscosity was manipulated to alter shear stress. At a constant shear rate, a higher shear stress due to increased viscosity increased both bioluminescence intensity and decay rate, supporting our hypothesis that bioluminescence is shear-stress dependent. Although the flow response of non-marine attached cells is known to be mediated through shear stress, our results indicate that suspended cells such as dinoflagellates also sense and respond to shear stress. Shear-stress dependence of flow-stimulated bioluminescence in dinoflagellates is consistent with mechanical stimulation due to direct predator handling in the context of predator-prey interactions.     

Blood viscosity modelling: influence of aggregate network dynamics under transient conditions

This paper reports on a theoretical examination of the hypothesis that red blood cell network characteristics influence the mechanical properties of the fluid. For this purpose a newly developed energy-rate based blood viscosity model, which incorporates network dynamics, was used to predict the transient behaviour of blood viscosity (steady-state results of this model have been reported in Biorheology 46 (2009), 487-508). The main network characteristic examined in the present work was the inter-aggregate branch size and its relationship to the evolving aggregates. Branch size was used to define a network integrity index that accounted for the strength of the developed network. For the development and validation of the model, experiments performed with an optical shearing microscope, with different step-changes in shear rate, were utilised, as well as viscosity measurements under similar flow conditions performed in a double wall Couette instrument. The experimental data were compared with the response of the model, which incorporated the network integrity index. The results suggest that network characteristics may influence the viscosity of blood at low shear rates and exhibit good agreement with experimental observations.     

Rheology of Human Blood, near and at Zero Flow: Effects of Temperature and Hematocrit Level

Static normal human blood possesses a distinctive yield stress. When the yield stress is exceeded, the same blood has a stress-shear rate function under creeping flow conditions closely following Casson's model, which implies reversible aggregation of red cells in rouleaux and flow dominated by movement of rouleaux. The yield stress is essentially independent of temperature and its cube root varies linearly with hematocrit value. The dynamic rheological properties in the creeping flow range are such that the relative viscosity of blood to water is almost independent of temperature. Questions raised by these data are discussed, including red cell aggregation promoted by elements in the plasma.     

Blood Flow, Slip, and Viscometry

The viscosity of blood, measured by the usual viscometers in which slip is not considered, is found to be flow dependent, varying markedly with shear rate, pressure gradient, and vessel diameter in the lower ranges of these factors. The study postulates, on grounds thought reasonable, that slip may be present in blood flow, as a function of the nature of the wall surfaces, shear stress at the wall, and relative cell volume (RCV) adjacent to the wall. It presumes that blood possesses a specific, flow-independent viscosity, and determines theoretically the viscosity indications of viscometers if blood slipped in the instruments. The study shows that if the slip function is of a certain plausible form, these viscosity indications would exhibit a flow dependence of much the same pattern as the actual indications supplied by the usual viscometers. The slip postulate permits, therefore, an interpretation of the “anomalous” flow behavior of blood, dispensing with the prevailing assumption of an ad hoc variability of its viscosity with flow factors. To the extent that viscometric data for blood may be representative of other non-newtonian fluids, the slip postulate may be applicable to these fluids.     

Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta

Three non-Newtonian blood viscosity models plus the Newtonian one are analysed for a patient-specific thoracic aorta anatomical model under steady-state flow conditions via wall shear stress (WSS) distribution, non-Newtonian importance factors, blood viscosity and shear rate. All blood viscosity models yield a consistent WSS distribution pattern. The WSS magnitude, however, is influenced by the model used. WSS is found to be the lowest in the vicinity of the three arch branches and along the distal walls of the branches themselves. In this region, the local non-Newtonian importance factor and the blood viscosity are elevated, and the shear rate is low. The present study revealed that the Newtonian assumption is a good approximation at mid-and-high flow velocities, as the greater the blood flow, the higher the shear rate near the arterial wall. Furthermore, the capabilities of the applied non-Newtonian models appeared at low-flow velocities. It is concluded that, while the non-Newtonian power-law model approximates the blood viscosity and WSS calculations in a more satisfactory way than the other non-Newtonian models at low shear rates, a cautious approach is given in the use of this blood viscosity model. Finally, some preliminary transient results are presented.     

Rheology of embryonic avian blood

Shear stress, a mechanical force created by blood flow, is known to affect the developing cardiovascular system. Shear stress is a function of both shear rate and viscosity. While established techniques for measuring shear rate in embryos have been developed, the viscosity of embryonic blood has never been known but always assumed to be like adult blood. Blood is a non-Newtonian fluid, where the relationship between shear rate and shear stress is nonlinear. In this work, we analyzed the non-Newtonian behavior of embryonic chicken blood using a microviscometer and present the apparent viscosity at different hematocrits, different shear rates, and at different stages during development from 4 days (Hamburger-Hamilton stage 22) to 8 days (about Hamburger-Hamilton stage 34) of incubation. We chose the chicken embryo since it has become a common animal model for studying hemodynamics in the developing cardiovascular system. We found that the hematocrit increases with the stage of development. The viscosity of embryonic avian blood in all developmental stages studied was shear rate dependent and behaved in a non-Newtonian manner similar to that of adult blood. The range of shear rates and hematocrits at which non-Newtonian behavior was observed is, however, outside the physiological range for the larger vessels of the embryo. Under low shear stress conditions, the spherical nucleated blood cells that make up embryonic blood formed into small aggregates of cells. We found that the apparent blood viscosity decreases at a given hematocrit during embryonic development, not due to changes in protein composition of the plasma but possibly due to the changes in cellular composition of embryonic blood. This decrease in apparent viscosity was only visible at high hematocrit. At physiological values of hematocrit, embryonic blood viscosity did not change significantly with the stage of development.     

Magnetic resonance microscopy determined velocity and hematocrit distributions in a Couette viscometer

Magnetic resonance microscopy is used to non-invasively measure the radial velocity distribution in Couette flow of erythrocyte suspensions of varying aggregation behavior at a nominal shear rate of 2.20 s(-1) in a 1 mm gap. Suspensions of red blood cells in albumin-saline, plasma and 1.48% Dextran added plasma at average hematocrits near 0.40 are studied, providing a range of aggregation ability. The spatial distribution of the red blood cell volume fraction, hematocrit, is calculated from the velocity distribution. The hematocrit profiles provide direct measure of the thickness of the aggregation and shear rate dependent red blood cell depletion at the Couette surfaces. At the nominal shear rate studied hematocrit distributions for the red blood cells in plasma show a depletion zone near the inner Couette wall but not the outer wall. The red blood cells in plasma with Dextran show cell depletion regions of approximately 100 mum at both the inner and outer Couette surfaces, with greater depletion at the inner wall, but approach the normal blood hematocrit distribution with a doubling of shear rate due to decreased aggregation. The material response of the blood is spatially dependent with the shear rate and the hematocrit distribution non-uniform across the gap.     

Pulsatile flow in a coronary artery using multiphase kinetic theory

Pulsatile flow in a model of a right coronary artery (RCA) was previously modeled as a single-phase fluid and as a two-phase fluid using experimental rheological data for blood as a function of hematocrit and shear rate. Here we present a multiphase kinetic theory model which has been shown to compute correctly the viscosity of red blood cells (RBCs) and their migration away from vessel walls: the Fahraeus–Lindqvist effect. The computed RBC viscosity decreases with shear rate and vessel size, consistent with measurements. The pulsatile computations were performed using a typical cardiac waveform until a limit cycle was well established. The RBC volume fractions, shear stresses, shear stress gradients, granular temperatures, viscosities, and phase velocities varied with time and position during each cardiac cycle. Steady-state computations were also performed and were found to compare well with time-averaged transient results. The wall shear stress and wall shear stress gradients (both spatial and temporal) were found to be highest on the inside area of maximum curvature. Potential atherosclerosis sites are identified using these computational results.     

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.     

Measurement of erythrocyte orientation in flow by spin labeling III--erythrocyte orientation and rheological conditions

The measurements of erythrocyte orientation, obtained through a spin labeling technique, are compared with a phenomenological model. Several rheological conditions are varied: hematocrit, suspending medium viscosity, blood age, artificial reversible aggregation. We found that the onset of orientation is very sensitive to any variation of these conditions, and that its measurement would be a good method to assess erythrocyte deformability. A critical shear rate for the orientation process is then determined and compared to the corresponding parameter obtained from viscosity measurements of identical suspensions. A close qualitative relationship is found between the two sets of values of the critical shear rate.     

Flow behaviour of a POSS biopolymer solution

A non-biodegradable polyhedral oligomeric silsesquioxane (POSS) nanocomposite biopolymer has been developed for fabrication of medical devices and for tissue engineering human organs. The polymer in solution, containing 2 wt% of POSS, has been synthesized, characterized and investigated to determine its key rheological properties. Thus, the variation of shear stress and viscosity as a function of shear rate has been determined at ambient temperature to estimate yield stress and the index of pseudoplasticity, respectively. The temperature dependence of viscosity and the effect of ageing on the viscosity of the polymer have also been investigated. Results are compared with those of a conventional polycarbonate urethane (PCU) polymer solution. The POSS-PCU polymer solution shows near-Newtonian behaviour in the shear rate range to 1000 s(-1), having an apparent viscosity of approximately 3000 mPa s and a pseudoplasticity index of 0.90, decreasing slightly as the polymer solution is aged over 9 months. The temperature dependence of viscosity of the POSS polymer is extremely low and does not change with ageing but the yield strength increases from 2.7 Pa to 8.3 Pa.     

Temperature-dependent threshold shear stress of red blood cell aggregation

Red blood cell (RBC) aggregation is becoming an important hemorheological parameter, which exhibits a unique temperature dependence. However, further investigation is still required for understanding the temperature-dependent characteristics of hemorheology that includes RBC aggregation. In the present study, blood samples were examined at 3, 10, 20, 30, and 37 °C. When the temperature decreases, the whole-blood and plasma viscosities increase, whereas the aggregation indices (AI, M, and b) yield contrary results. Since these contradictory results are known to arise from an increase in the plasma viscosity as the temperature decreases, aggregation indices that were corrected for plasma viscosity were examined. The corrected indices showed mixed results with the variation of the temperature. However, the threshold shear rate and the threshold shear stress increased as the temperature decreased, which is a trend that agrees with that of the blood viscosity. As the temperature decreases, RBC aggregates become more resistant to hydrodynamic dispersion and the corresponding threshold shear stress increases as does the blood viscosity. Therefore, the threshold shear stress may help to better clarify the mechanics of RBC aggregation under both physiological and pathological conditions.     

Shear degradation as a probe of microalgal exopolymer structure and rheological properties

The bulk theological properties of exopolymers produced by three species of microalgae are destroyed by shear stress. The properties are drag reduction in capillary pressure flow and low shear rate viscosity. As such, shear stress constitutes an experimental probe into the macromolecular structure which effects bulk Theological properties. Native and sheared exopolymer solutions were subjected to analysis by electrophoresis, size exclusion chromatography, hydrolysis, dialysis, and reducing end-group analysis. The evidence indicates that shearing did not break the glycoside backbone of these exopolymers, rather shearing disrupted subtle interactions between copolymers. The interactions necessary for bulk rheological properties are likely at the quaternary level of macromolecular organization, specifically weak aggregations.     

Point perturbation analysis of experimental data

A new method of data analysis is proposed. The method is based on discrete perturbation of experimental data points, which is used to probe the metric of the parameter hyperspace. Perturbation-induced fluctuations in the residual values are analysed by discrete Fourier transform to yield the autocorrelation function and a relaxation length for each experimental point. This parameter provides a quantitative measure of correlation and hence nonrandomness of residuals. The method is applied to the analysis of measurements of the shear viscosity of a 2,6-lutidine/water mixture near the critical point, and to the oxygen and carbon monoxide binding reactions to human hemoglobin. Relaxation profiles are constructed for several experimental data sets. Departure from random behavior in the residuals is discussed in connection with the theoretical interpretations of the phenomenon under consideration.     

Shear stress distribution in arterial tree models, generated by constrained constructive optimization.

Models of arterial trees are generated by the algorithm of Constrained Constructive Optimization (CCO). Straight cylindrical, binary branching tubes are arranged in an optimized fashion so as to convey blood to the terminal sites of the tree, which are distributed over a predefined area, representing the tissue to be perfused. All terminal segments supply equal flows at a unique terminal pressure, and the radii of parent and daughter segments are related via a bifurcation law. The connective structure and geometry of the model are optimized according to a target function such as total intravascular volume. The shear rate between blood and the vessel walls is computed in each segment and a new method is presented for rescaling a given CCO tree to a desired value of shear rate in the root segment. The effect of viscosity varying with shear rate is evaluated and a new method is presented for rescaling a CCO-tree segment by segment to consistent values of radii and variable viscosity. Shear stress is evaluated for its deviation from being proportional to shear rate and then subjected to various types of analyses. Usually both, shear stress and its variability, are found to be larger in the smaller than in the larger segments of the CCO-model trees. However, it is shown how the shear-stress distribution can be reshuffled between small and large segments when rescaling a CCO tree to obey a different bifurcation law, while its whole geometry remains unchanged and all boundary conditions remain fulfilled. The selection of optimization target is found to drastically affect shear-stress variability within bifurcations, which reaches a distinct minimum if the model is optimized according to intravascular volume. Finally, a rank-analysis of shear stress within each bifurcation shows that only two out of six possible rank patterns actually occur: the parent segment always experiences medium shear stress while minimum shear stress resides mostly in the larger, less frequently in the smaller daughter.     

Adaptive regulation of wall shear stress optimizing vascular tree function

The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm²), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.     

Elasto-thixotropic properties of bronchial mucus and polymer analogs. I. Experimental results

The non-newtonian viscous and elasto-thixotropic properties of native and lyophilized pathological bronchial mucus and of polymer solutions (3% and 6% PIB in decalin) used as mucus analogs were analyzed using a cone-plate Carri-Med rheometer and a Couette viscoelastometer that we have specifically developed for measuring the rheological properties of bronchial mucus in clinical practice. The master curves obtained for apparent viscosity under steady conditions as a function of shear rates (gamma: 2.6 X 10(-3) to 6.9 X 10(1) sec-1) were fairly similar, whatever the apparatus used. Under transient conditions, at low shear rate (gamma less than 1.4 sec-1), PIB and mucus exhibited a typical viscoelastic behavior: the shear stress increased slightly up to a steady-state value. At higher gamma, a transitory overshoot of sigma characteristic of the elastothixotropic systems appeared. Such a behavior can be interpreted as resulting from structural changes such as formation and rupture of the three-dimensional network present in bronchial mucus as in polymer solutions.