Pseudoshear Rates Research Articles

A Microvascular Wall Shear Rate Function Derived From In Vivo Hemodynamic and Geometric Parameters in Continuously Branching Arterioles.

Conventional approaches to WSR estimation in the microcirculation involve assumptions that may result in under-/over-estimation of WSR. Therefore, our objectives were: (i) calculate WSR from RBC velocity profiles for a wide range of arteriolar diameters, (ii) provide an experimentally derived and straightforward WSR estimation function, and (iii) compare calculated to conventional WSR estimations. We characterized RBC velocity profiles in arterioles (n=39) of branching networks (21-115μm) in the rat gluteus maximus muscle (n=6). Measures included mean and maximum velocities, CFL thickness, and RBC column edge velocity, and an experiment-based WSR function was derived. CFL thickness (1-4.3μm) positively correlated with arteriolar diameter (r(2) =0.64). Results from the WSR equationwere similar to values from edge RBC velocities/CFL. Experimental WSRs (1317-4334/sec) were independent of arteriolar diameter, and were greater than pseudoshear rates (for VRatio of 1.6, 2, or diameter-dependent VRatio function) (p<0.05). A WSR equationwas derived from experimental hemodynamic parameters, and is adaptable to other velocity measurement techniques in order to obtain WSR and stress (when plasma viscosity is known). These findings provide insight on the nature of conventional WSR calculation methods in underestimating microvascular WSR values.

Process of cell‐free layer formation after arteriolar bifurcation and its physiological effects

Understanding the formation of cell‐free layer (CFL) in arteriolar networks has been of great interest, particularly for its significance in regulation of blood flow. The heterogeneous distribution of red blood cells (RBCs) modulated by the formation of CFL in arteriolar bifurcations potentially leads to alterations in tissue oxygenation (O2) and vascular tone due to varying nitric oxide (NO) production in the endothelium. Accordingly, we investigated the formation of CFL from 2 to 6 vessel‐diameter (2D–6D) downstream of arteriolar bifurcations in the rat cremaster muscle under reduced flow conditions (pseudoshear rates = 15.6 ± 2.0 s−1). We accounted for the spatiotemporal variations in the wall shear stress, velocity profile and core hematocrit along the downstream vessels by extending our previous image‐based two‐dimensional transient model, which was unprecedented in previous computational models. Our experimental results showed that an asymmetric formation of CFL width along the vessel walls was evident up to 6D from the bifurcating point. The recovery of CFL symmetry appeared to be impeded by RBC aggregation. Our numerical prediction of soluble guanylate cyclase (sGC) activity revealed that vasoactivity might not occur uniformly within an inter‐bifurcation distance, in particular under reduced flow and aggregating conditions.Support or Funding InformationNUS FRC

Recovery of cell-free layer and wall shear stress profile symmetry downstream of an arteriolar bifurcation

Unequal RBC partitioning at arteriolar bifurcations contributes to dissimilar flow developments between daughter vessels in a bifurcation. Due to the importance of the cell-free layer (CFL) and the wall shear stress (WSS) to physiological processes such as vasoregulation and gas diffusion, we investigated the effects of a bifurcation disturbance on the development of the CFL width and WSS in bifurcation daughter branches. The analysis was performed on a two-dimensional (2-D) computational model of a transverse arteriole at three different flow rates corresponding to parent branch (PB) pseudoshear rates of 60, 170 and 470s−1, while maintaining a 2-D hematocrit of about 55% in the PB. Flow symmetry was defined using the statistical similarity of the CFL and WSS distributions between the two walls of the vessel branch. In terms of the flow symmetry recovery, higher flow rates caused larger reductions in the flow symmetry indices in the MB and subsequently required longer vessel lengths for complete recovery. Lower tube hematocrits in the SB led to complete symmetry recovery for all flow rates despite the higher initial asymmetry in the SB than in the MB. Arteriolar bifurcations produce unavoidable local CFL asymmetry and the persistence of the asymmetry downstream may increase effective blood viscosity which is especially significant at higher physiological flow rates.

Erythrocyte aggregation may promote uneven spatial distribution of NO/O2 in the downstream vessel of arteriolar bifurcations

This study examined the effect of red blood cell (RBC) aggregation on nitric oxide (NO) and oxygen (O2) distributions in the downstream vessels of arteriolar bifurcations. Particular attention was paid to the inherent formation of asymmetric cell-free layer (CFL) widths in the downstream vessels and its consequential impact on the NO/O2 bioavailability after the bifurcations. A microscopic image-based two-dimensional transient model was used to predict the NO/O2 distribution by utilizing the in vivo CFL width data obtained under non-, normal- and hyper-aggregating conditions at the pseudoshear rate of 15.6±2.0s−1. In vivo experimental result showed that the asymmetry of CFL widths was enhanced by the elevation in RBC aggregation level. The model demonstrated that NO bioavailability was regulated by the dynamic fluctuation of the local CFL widths, which is corollary to its modulation of wall shear stress. Accordingly, the uneven distribution of NO/O2 was prominent at opposite sides of the arterioles up to six vessel-diameter (6D) away from the bifurcating point, and this was further enhanced by increasing the levels of RBC aggregation. Our findings suggested that RBC aggregation potentially augments both the formation of asymmetric CFL widths and its influence on the uneven distribution of NO/O2 in the downstream flow of an arteriolar bifurcation. The extended heterogeneity of NO/O2 downstream (2D–6D) also implied its potential propagation throughout the entire arteriolar microvasculature.

Effect of erythrocyte aggregation at pathological levels on NO/O2 transport in small arterioles.

This study examined the effects of red blood cell (RBC) aggregation at pathological levels on NO/O2 transport in small arterioles. Transient gas diffusion simulations were performed with in vivo cell-free layer (CFL) widths data obtained from arteriolar flows in the rat cremaster muscle. The CFL data were measured at physiological and pathological levels of aggregation under reduced flow conditions (pseudoshear rate = 31.4 ± 10.5 s-1). Our results showed that the mean peak NO concentration significantly decreased with increasing the aggregation level from non-aggregating to normal-aggregating (P < 0.05) and to hyper-aggregating (P < 0.01) conditions. In contrast, the partial O2 pressure (PO2) in pathological aggregating conditions significantly increased from those under non-aggregating (P < 0.001) and normal-aggregating (P < 0.05) conditions. Although the NO scavenging by RBCs could be impaired with a thicker CFL at higher levels of aggregation, the overall decrease in NO production due to reduction of wall shear stress with the thicker CFL dominantly limited the NO availability in tissue. On the other hand, the O2 availability in tissue increased due to the relatively high core hematocrit in the blood lumen with the thicker CFL.

Two-phase model for prediction of cell-free layer width in blood flow

This study aimed to develop a numerical model capable of predicting changes in the cell-free layer (CFL) width in narrow tubes with consideration of red blood cell aggregation effects. The model development integrates to empirical relations for relative viscosity (ratio of apparent viscosity to medium viscosity) and core viscosity measured on independent blood samples to create a continuum model that includes these two regions. The constitutive relations were derived from in vitro experiments performed with three different glass-capillary tubes (inner diameter=30, 50 and 100μm) over a wide range of pseudoshear rates (5–300s−1). The aggregation tendency of the blood samples was also varied by adding Dextran 500kDa. Our model predicted that the CFL width was strongly modulated by the relative viscosity function. Aggregation increased the width of CFL, and this effect became more pronounced at low shear rates. The CFL widths predicted in the present study at high shear conditions were in agreement with those reported in previous studies. However, unlike previous multi-particle models, our model did not require a high computing cost, and it was capable of reproducing results for a thicker CFL width at low shear conditions, depending on aggregating tendency of the blood.

Numerical simulation of blood flow with different red blood cell deformability

The present study investigated the effect of changes of red blood cell deformability on micro‐hemodynamics by using the Lattice Boltzmann Method (LBM) with Immersed Boundary Method (IBM). The blood flow of 40% stiff red blood cells and 60% normal cells was simulated and the result was compared with that of 100% normal red blood cells. The deformability of stiff cells was 50% less than normal cells. Thus, with the stiff cells, the averaged red blood cell deformability became 20% less than normal condition, which is similar to the condition of malaria‐infected blood. The simulation was conducted at 40% hematocrit and 100 s−1 pseudo shear rate. In this study, the viscosity difference between inside and outside fluids of red blood cell membrane was considered by using a flood fill method. The results showed that the effective viscosity of blood flow with stiff red blood cells significantly increased compared with that without the stiff cells. The axial migration of the stiff cells moving away from the centerline was observed during the simulation.This work was supported by NUS URC Grant R‐397‐000‐091‐112.

Cell‐Free Layer Formation in Small Arterioles at Pathological Levels of Erythrocyte Aggregation

To test our hypothesis that an elevation in the aggregation level of red blood cells found in human pathological conditions will significantly enhance cell-free layer formation in small arterioles. Visualization of arteriolar blood flow in rat cremaster muscle was carried out in both normal and reduced flow conditions before and after Dextran 500 infusion to simulate physiological and pathological levels of red blood cell aggregation in humans. Both normalized mean (p < 0.0001) and SD (p < 0.002) of the layer width were significantly enhanced after hyper-aggregation induction in reduced flow conditions (mean pseudoshear rate = 57.3 ± 7.2/sec). Normalized mean and SD of the layer width generally increased with decreasing vessel radius and this effect was most pronounced with hyper-aggregation in reduced flow conditions. The threshold pseudoshear rate at which the layer formation became more pronounced when compared with non-aggregating condition was higher with hyper-aggregation (217/sec) than normal-aggregation induction (139/sec). Our findings confirmed the formation of a prominent cell-free layer in the arterioles under higher shear conditions at pathological aggregation levels and this effect became more pronounced in smaller arterioles in normalizing the layer to the vessel radius.

Effect of plasma expander viscosity on the cell free layer

The effect of low and high viscosity hemodilution with plasma expanders on the extent of the cell free layer (CFL) width was analyzed in the microcirculation of the exteriorized cremaster muscle preparation of Sprague-Dawley male rats. Anesthetized animals were subjected to 40% hemodilution by blood volume, using 5% human serum albumin (HSA) or 6% Hetastarch (hydroxyethyl starch 670 kDa). Arterioles (n=5 for each treatment) were investigated. Mean arterial pressure, heart rate, vessel flow velocity and CFL width were measured at baseline and 5, 20 and 40 min post-exchange transfusion. Blood and plasma viscosity was determined from terminal blood collections. CFL width and pseudoshear rate, diameter and flow, normalized to baseline, were significantly elevated at all post-exchange assessments. Peripheral vascular resistance decreased. The increase of the CFL width was greater with HSA by comparison with Hetastarch hemodilution (p<0.05). Hetastarch blood and plasma viscosities increased significantly compared to those of HSA (p<0.05). This study shows that CFL widths are influenced by plasma expander viscosity, a phenomenon proportional to the increase in molecular weight of the colloids in solution.

Effect of Cell-Free Layer Variation on Arteriolar Wall Shear Stress

Relationship between a cell-free layer and wall shear stress (WSS) in small arterioles has been of interest in microcirculatory research. However, influence of temporal variation in the cell-free layer width on the WSS in vivo has not been fully elucidated. In this study, we tested the hypothesis that the layer variation would increase the WSS, and this effect would be enhanced by red blood cell aggregation. The cell-free layer width in arterioles (29.5–67.1 μm ID) in rat cremaster muscles were obtained with a high-speed video camera, and the layer width data were introduced into WSS estimation. Dextran 500 was administrated to elevate the aggregation level of red blood cells to those seen in normal human blood. The variation of the layer was quantified by the variability (coefficient of variation), and its effect on WSS was studied under normal and reduced flow conditions. We found that the dextran-induced red blood cell aggregation significantly elevated the variability (p < 0.01) at low pseudoshear rates of 9.2 ± 0.6 s−1. The WSS estimated without taking account of the variability showed underestimation of its value than that of with consideration of the variability under all flow conditions, and this effect became more pronounced with increasing the variability. The variation of the cell-free layer should, therefore, be considered in the determination of the WSS particularly in the presence of red blood cell aggregation under reduced flow condition.

Modulation of NO bioavailability by temporal variability of cell‐free layer width in the arterioles

The formation of a cell (erythrocyte)‐free layer near the vessel wall during blood flow is capable of attenuating NO consumption by the red blood cells and thus improves the preservation of NO in the smooth muscle for eliciting vasodilatory responses. This study utilized a transient NO transport mathematical model to predict how temporal variability exhibited by the layer width and its relevant transient physiological responses could modulate NO preservation by this diffusion barrier at high pseudoshear rates (291.5 ± 13.5 s−1). Data on cell‐free layer width was experimentally acquired from arterioles (48.4 ± 2.3 μm) of the rat cremaster muscle. We found that a time‐varying diffusion barrier could lead to a slight decrease (2% – 6%) in NO bioavailability and this effect was independent of transient changes in NO scavenging rate in the blood lumen. Conversely, transient changes in wall shear stress and accompanying NO release rate played a dominant role in reversing this decline such that an augmentation in NO bioavailability of between 3% – 15% was found. However this increase was not sufficient to induce any significant change in vascular tone. This study highlighted the importance of accounting for transient wall shear stress effect on NO release rate as mediated by the layer width temporal variability when assessing the physiological implication of the layer on NO bioavailability in arterioles.

Mathematical modelling of the cell-depleted peripheral layer in the steady flow of blood in a tube

In an earlier paper, Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327-354] used kinetic theory to derive a non-homogeneous haemorheological model and applied this to simulate the properties of steady flow of blood in a tube. By adjusting the tube haematocrit to match that of the experimental fitted curve of Pries et al. [Circ. Res. 67 (1990), 826-834] the authors showed that it was possible to quantitatively predict the apparent viscosity values presented in a later paper by Pries et al. [Am. J. Physiol. 263 (1992), 1770-1778]. In the present paper, it is the discharge haematocrit rather than the tube haematocrit that is prescribed. We further develop the predictive capacities of the original model of Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327-354] by introducing a cell-free peripheral layer next to the tube wall where, following the ideas of Sharan and Popel [Biorheology 38 (2001), 415-428], dissipation in this layer is accounted for by allowing the viscosity there to exceed that of plasma. Using both the apparent viscosity data of Pries et al. [Am. J. Physiol. 263 (1992), 1770-1778] and the relative tube haematocrit relation proposed by Sharan and Popel [Biorheology 38 (2001), 415-428], we predict the thickness of the cell-free layer and the relative viscosity in this layer. The predicted thickness of the cell-free layer as a function of both a pseudo-shear rate and the tube diameter for 45% haematocrit blood is shown to be in very close conformity with the experimental measurements of Reinke et al. [Am. J. Physiol. 253 (1987), 540-547]. With increasing discharge haematocrit the cell-free layer thickness is shown to decrease, as observed in several experimental papers [Bugliarello and Hayden, Trans. Soc. Rheol. VII (1963), 209-230, Bugliarello and Sevilla, Biorheology 7 (1970), 85-107, Soutani et al., Am. J. Physiol. 268 (1995), 1959-1965]. Our prediction of the relative viscosity in the cell-free layer shows a similar trend to that computed by Sharan and Popel [Biorheology 38 (2001), 415-428]. Finally, for sufficiently large pseudo-shear rates it is shown that the Deborah number (a non-dimensional relaxation time) may be taken to be a constant, thus greatly simplifying our haemorheological model and allowing for a partially analytic solution to the problem of steady non-homogeneous flow of blood in a tube.

Computational fluid dynamics of aggregating red blood cells in postcapillary venules

Aggregate formation of red blood cells (RBCs) in a postcapillary venular bifurcation is investigated with three-dimensional computer simulations using the Chimera grid method. Interaction energy between the RBCs is modelled by a depletion interaction theory; RBCs are modelled as rigid oblate ellipsoids. The cell–cell interactions of RBCs are strongly dependent on vessel geometry and shear rates. The experimental data on vessel geometry, pseudoshear rates, and Dextran concentration obtained in our previous in vivo RBC aggregation study in postcapillary venules of the rat spinotrapezius muscle were used to simulate RBC aggregation. The computational results were compared to the experimental results from the in vivo study. The results show that cells have a larger tendency to form an aggregate under reduced flows. Aggregate formation also depends on the angle and location of the cells before they enter the bifurcation region. Comparisons with experimental data are discussed.

Effect of erythrocyte aggregation and flow rate on temporal variation of cell‐free layer width in arterioles

The cell‐free layer (CFL) is a dynamic feature of microcirculatory blood flow and its variability is commonly described in terms of standard deviation (SD). However, CFL width in an arteriole follows a non‐Gaussian distribution imposed by asymmetrical nature of its physical environment, which implies that more detailed statistical analyses are required to better interpret CFL characteristics. In this study, rat cremaster muscle was used to visualize arteriolar flows and erythrocyte aggregability was raised by dextran infusion to levels seen in normal human blood. The effects of aggregation and flow reduction on CFL characteristics in arterioles (ID 20‐50 μm) were investigated. A positive relation was found between mean CFL widths and their corresponding SD and was predominantly enhanced by flow reduction. Both normalized mean and SD of CFL width increased significantly at low pseudoshear rates (17.1 ± 6.1 s−1), especially in the presence of aggregation. The variability of the CFL width was further analyzed in terms of its frequency and magnitude of deviations from its mean value, extending either towards the vessel wall or into the erythrocyte core. There was no significant effect of either flow condition or aggregation on the frequency of CFL deviations, whereas the magnitude of deviations significantly increased after flow reduction and dextran infusion (P &lt; 0.05), especially into the erythrocyte core.

Temporal and spatial variations of cell-free layer width in arterioles

Separation of red blood cells and plasma in microcirculatory vessels produces a cell-free layer at the wall. This layer may be an important determinant of blood viscosity and wall shear stress in arterioles, where most of the hydraulic pressure loss in the circulatory system occurs and flow regulatory mechanisms are prominent. With the use of a newly developed method, the width of the cell-free layer was rapidly and repeatedly determined in arterioles (10- to 50-microm inner diameter) in the rat cremaster muscle at normal arterial pressure. The temporal variation of the cell-free layer width was non-Gaussian, but calculated mean and median values differed by <0.2 microm. The correlation length of the temporal variations downstream (an indication of mixing) was approximately 30 microm and was independent of pseudoshear rate (ratio of mean velocity to vessel diameter) and of vessel diameter. The cell-free layer width was significantly different on opposite sides of the vessel and inversely related. Increasing red blood cell aggregability reduced this inverse relation but had no effect on correlation length. In the diameter range studied, the mean width of the cell-free layer increased from 0.8 to 3.1 microm and temporal variations increased from 30% to 70% of the mean width. Increased aggregability did not alter either relationship. In summary, the cell-free layer width in arterioles is diameter dependent and shows substantial non-Gaussian temporal variations. The temporal variations increase as diameter increases and are inversely related on opposite sides of the vessel.

Erythrocyte margination and sedimentation in skeletal muscle venules.

Previous studies in skeletal muscle of the dog and cat have shown that venous vascular resistance changes inversely with blood flow and may be due mainly to red blood cell aggregation, a phenomenon present in these species. To determine whether red blood cell axial migration and sedimentation contribute to this effect, we viewed either vertically or horizontally oriented venules of the rat spinotrapezius muscle with a horizontally oriented microscope during acute arterial pressure reduction. With normal (nonaggregating) rat blood, reduction of arterial pressure did not significantly change the relative diameter of the red blood cell column with respect to the venular wall. After induction of red blood cell aggregation in the rat by infusion of Dextran 500, red blood cell column diameter decreased up to 35% at low pseudoshear rates (below approximately 5 s(-1)); the magnitude was independent of venular orientation. In vertically oriented venules, the plasma layer was symmetrical, whereas in horizontally oriented venules, the plasma layer formed near the upper wall. We conclude that, although red blood cell axial migration and sedimentation develop in vivo, they occur only for larger flow reductions than are needed to elicit changes in venous resistance.

Effect of erythrocyte aggregation on velocity profiles in venules.

A recent whole organ study in cat skeletal muscle showed that the increase in venous resistance seen at reduced arterial pressures is nearly abolished when the muscle is perfused with a nonaggregating red blood cell suspension. To explore a possible underlying mechanism, we tested the hypothesis that red blood cell aggregation alters flow patterns in vivo and leads to blunted red blood cell velocity profiles at reduced shear rates. With the use of fluorescently labeled red blood cells in tracer quantities and a video system equipped with a gated image intensifier, we obtained velocity profiles in venous microvessels (45-75 microm) of rat spinotrapezius muscle at centerline velocities between 0.3 and 14 mm/s (pseudoshear rates 3-120 s(-1)) under normal (nonaggregating) conditions and after induction of red blood cell aggregation with Dextran 500. Profiles are nearly parabolic (Poiseuille flow) over this flow rate range in the absence of aggregation. When aggregation is present, profiles are parabolic at high shear rates and become significantly blunted at pseudoshear rates of 40 s(-1) and below. These results indicate a possible mechanism for increased venous resistance at reduced flows.

Laminar flow of power law fluid foods in concentric annuli

The flow of power law fluids for isothermal, laminar flow in concentric annuli was investigated experimentally and numerically. Similar to the results of Russel & Christiansen (1974, Indust. Engng Chem. Process Des. Develop. 13 , 391–398) it was observed that plots of the natural logarithm of a reduced shear stress at the outer wall of the annulus vs the natural logarithm of a reduced pseudo-shear rate were identical to those for tube flow. However, in contrast to the findings of Russel & Christiansen (1974), the y intercepts and the slopes of these plots were found to be equal to In K′ and n, respectively. Frederickson & Bird's (1958, Indust. Engng Chem. 50 , 347–352) solution was found to be in perfect aggreement with the numerical solution in every case. The Hanks & Larsen (1979, Indust. Engng Chem. Fundamentals, 18 , 33–35) equation also agreed very well with the numerical solution. The Russel & Christiansen (1974, Indust. Engng Chem. Process Des. Develop., 13 , 391–398) method gave an absolute mean error of 3.6% against the numerical results for pressure drop.

Cell-free plasma layer in cerebral microvessels.

Two diameters of vessel and red cell column in cerebral microvessels (> 29.8 microns in diameter) of cat were measured together with red cell velocity, using a two fluorescent tracer method. A fluorescein isothiocyanate (FITC)-labeled red cell was adopted as a flow tracer to measure the cell velocity with a dual window technique. Based on the fluorescence image, the red cell column diameter was measured. Plasma was stained with rhodamine-B isothiocyanate (RITC)-labeled dextran to measure the vessel diameter. The thickness of the cell-free plasma layer could be determined from the difference of the two diameters. The obtained thickness of the cell-free layer was not described by a simple function of vessel diameter or red cell velocity; it was dependent on the pseudo shear rate defined by the ratio of cell velocity to vessel radius. The layer thickness increased with a decrease in the pseudo shear rate.

Effects of aggregation on the flow properties of red blood cell suspensions in narrow vertical tubes.

The flow properties of aggregating red cell suspensions flowing at low rates through vertical tubes with diameters from 30 microns to 150 microns are analyzed using a theoretical model. Unidirectional flow is assumed, and the distributions of velocity and red cell concentration are assumed to be axisymmetric. A three-layer approximation is used for the distribution of red cells, with a cylindrical central core of aggregated red cells moving with uniform velocity, a cell-free marginal layer near the tube wall, and an annular region located between the core and the marginal layer containing suspended non-aggregating red cells. This suspension is assumed to behave approximately as a Newtonian fluid whose viscosity increases exponentially with red cell concentration. Physical arguments concerning the mechanics of red cell attachment to, and detachment from the aggregated core lead to a kinetic equation for core formation. From this kinetic equation and the equation for conservation of red cell volume flux, a relationship between core radius and pressure gradient is obtained. Then the relative viscosity is calculated as a function of pseudo-shear rate. At low flow rates, it is shown that the relative viscosity decreases with decreasing flow and that the dependence of relative viscosity on shear rates is more pronounced in larger tubes. It is also found that the relative viscosity decreases with increasing aggregation tendency of suspension. These theoretical predictions are in good qualitative and quantitative agreement with experimental results.