Dynamics of blood flow: modeling of Fåhraeus and Fåhraeus–Lindqvist effects using a shear-induced red blood cell migration model

Blood flow in micro capillaries of diameter approximately 15–500 μm is accompanied with a lower tube hematocrit level and lower apparent viscosity as the diameter decreases. These effects are termed the Fåhraeus and Fåhraeus–Lindqvist effects, respectively. Both effects are linked to axial accumulation of red blood cells. In the present investigation, we extend previous works using a shear-induced model for the migration of red blood cells and adopt a model for blood viscosity that accounts for the suspending medium viscosity and local hematocrit level. For fully developed hematocrit profiles (i.e., independent of axial location), the diffusion fluxes due to particle collision frequency and viscosity gradients are of equal magnitude and opposite directions. The ratio of the diffusion coefficients for the two fluxes affects both the Fåhraeus and Fåhraeus–Lindqvist effects and is found related to the capillary diameter and discharge hematocrit using a well-known data-fit correlation for apparent blood viscosity. The velocity and hematocrit profiles were determined numerically as functions of radial coordinate, tube diameter, and discharge hematocrit. The velocity profile determined numerically is consistent with the derived analytical expression and the results are in good agreement with published numerical results and experimental data for hematocrit ratio and hematocrit and velocity profiles.     

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.     

Effects of red blood cell aggregation on myocardial hematocrit gradient using two approaches to increase aggregation

The normal transmyocardial tissue hematocrit distribution (i.e., subepicardial greater than subendocardial) is known to be affected by red blood cell (RBC) aggregation. Prior studies employing the use of infused large macromolecules to increase erythrocyte aggregation are complicated by both increased plasma viscosity and dilution of plasma. Using a new technique to specifically alter the aggregation behavior by covalent attachment of Pluronic F-98 to the surface of the RBC, we have determined the effects of only enhanced aggregation (i.e., Pluronic F-98-coated RBCs) versus enhanced aggregation with increased plasma viscosity (i.e., an addition of 500 kDa dextran) on myocardial tissue hematocrit in rapidly frozen guinea pig hearts. Although both approaches equally increased aggregation, tissue hematocrit profiles differed markedly: 1) when Pluronic F-98-coated cells were used, the normal transmyocardial gradient was abolished, and 2) when dextran was added, the hematocrit remained at subepicardial levels for about one-half the thickness of the myocardium and then rapidly decreased to the control level in the subendocardial layer. Our results indicate that myocardial hematocrit profiles are sensitive to both RBC aggregation and to changes of plasma viscosity associated with increased RBC aggregation. Furthermore, they suggest the need for additional studies to explore the mechanisms affecting RBC distribution in three-dimensional vascular beds.     

Microvascular blood viscosity in vivo and the endothelial surface layer

The apparent viscosity of blood in glass tubes declines with decreasing diameter (F?hraeus-Lindqvist effect) and exhibits a distinctive minimum at 6-7 microm. However, flow resistance in vivo in small vessels is substantially higher than predicted by in vitro viscosity data. The presence of a thick endothelial surface layer (ESL) has been proposed as the primary cause for this discrepancy. Here, a physical model is proposed for microvascular flow resistance as a function of vessel diameter and hematocrit in vivo; it combines in vitro blood viscosity with effects of a diameter-dependent ESL. The model was developed on the basis of flow distributions observed in three microvascular networks in the rat mesentery with 392, 546, and 383 vessel segments, for which vessel diameters, network architecture, flow velocity, and hematocrit were determined by intravital microscopy. A previously described hemodynamic simulation was used to predict the distributions of flow and hematocrit from the assumed model for effective blood viscosity. The dependence of ESL thickness on vessel diameter was estimated by minimizing deviations of predicted values for velocities, flow directions, and hematocrits from measured data. Optimal results were obtained with a layer thickness of approximately 0.8-1 microm for 10- to 40-microm-diameter vessels and declined strongly for smaller diameters, with an additional hematocrit-dependent impact on flow resistance exhibiting a maximum for approximately 10-microm-diameter vessels. These results show that flow resistance in vivo can be explained by in vitro blood viscosity and the presence of an ESL and indicate the rheologically effective thickness of the ESL in microvessels.     

Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes

Hematocrit distribution and red blood cell aggregation are the major determinants of blood flow in narrow tubes at low flow rates. It has been observed experimentally that in microcirculation the hematocrit distribution is not uniform. This nonuniformity may result from plasma skimming and cell screening effects and also from red cell sedimentation. The goal of the present study is to understand the effect of nonaxisymmetric hematocrit distribution on the flow of human and cat blood in small blood vessels of the microcirculation. Blood vessels are modeled as circular cylindrical tubes. Human blood is described by Quemada's rheological model, in which local viscosity is a function of both the local hematocrit and a structural parameter that is related to the size of red blood cell aggregates. Cat blood is described by Casson's model. Eccentric hematocrit distribution is considered such that the axis of the cylindrical core region of red cell suspension is parallel to the axis of the blood vessel but not coincident. The problem is solved numerically by using finite element method. The calculations predict nonaxisymmetric distribution of velocity and shear stress in the blood vessel and the increase of apparent viscosity with increasing eccentricity of the core.     

Effects of flowing RBCs on adhesion of a circulating tumor cell in microvessels

Adhesion of circulating tumor cells (CTCs) to the microvessel wall largely depends on the blood hydrodynamic conditions, one of which is the blood viscosity. Since blood is a non-Newtonian fluid, whose viscosity increases with hematocrit, in the microvessels at low shear rate. In this study, the effects of hematocrit, vessel size, flow rate and red blood cell (RBC) aggregation on adhesion of a CTC in the microvessels were numerically investigated using dissipative particle dynamics. The membrane of cells was represented by a spring-based network connected by elastic springs to characterize its deformation. RBC aggregation was modeled by a Morse potential function based on depletion-mediated assumption, and the adhesion of the CTC to the vessel wall was achieved by the interactions between receptors and ligands at the CTC and those at the endothelial cells forming the vessel wall. The results demonstrated that in the microvessel of \(15\,\upmu \hbox {m}\) diameter, the CTC has an increasing probability of adhesion with the hematocrit due to a growing wall-directed force, resulting in a larger number of receptor–ligand bonds formed on the cell surface. However, with the increase in microvessel size, an enhanced lift force at higher hematocrit detaches the initial adherent CTC quickly. If the microvessel is comparable to the CTC in diameter, CTC adhesion is independent of Hct. In addition, the velocity of CTC is larger than the average blood flow velocity in smaller microvessels and the relative velocity of CTC decreases with the increase in microvessel size. An increased blood flow resistance in the presence of CTC was also found. Moreover, it was found that the large deformation induced by high flow rate and the presence of aggregation promote the adhesion of CTC.     

Linear and nonlinear analyses of pulsatile blood flow in a cylindrical tube

A non-Newtonian shear-thinning constitutive relation is proposed to study pulsatile flow of whole blood in a cylindrical tube. The constitutive relation, which satisfies the principle of material frame indifference, is derived from viscometric data obtained from whole blood over a range of hematocrits. Assuming axisymmetric flow in a rigid cylindrical tube of constant diameter, a second-order, nonlinear partial differential equation governing the axial velocity component is obtained. Imposing a periodic pressure gradient, the governing equation was solved numerically using finite difference methods over a range of Stokes values and hematocrits. For a forcing frequency of 1 Hz, results are presented over tube diameters ranging between 0.1 and 2 cm and over hematocrits ranging between 10 and 80%. For a given hematocrit, velocity profiles predicted for the non-Newtonian model under sinusoidal forcing reveal attenuated volume flow rate and enhanced vorticity transport over the tube cross-section relative to a Newtonian fluid having a viscosity corresponding to the high shear-rate limit. For moderate to high Stokes numbers, consistent with flow in large arteries, our results revealed a viscosity distribution that was nearly time invariant. An analytic solution was obtained for a fluid having arbitrarily prescribed radially varying, temporally invariant viscosity and density distributions under arbitrary periodic pressure forcing. Close agreement was observed between our numerical and analytical results when the imposed viscosity distribution was chosen to approximate the time-averaged viscosity distribution predicted by the shear-thinning non-Newtonian model. For St > or approximately= 100, the disparity between our results and those of a Newtonian fluid of constant viscosity grows with a decreasing ratio of the DC to AC components of the pressure-gradient amplitude below 50%. In particular, for any purely oscillatory pressure-gradient (vanishing DC component), the Womersley solution is a particularly poor predictor of the amplitude and phase of wall shear rate for over half of the flow cycle. Under such circumstances, the analytical models presented here provide a simple and accurate means of estimating instantaneous wall shear rate, knowing only the pressure gradient and hematocrit.     

Numerical Simulation of Blood Flow Through Microvascular Capillary Networks

A numerical method is implemented for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille’s law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for a tree-like capillary network consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree.     

Two-dimensional lattice Boltzmann study of red blood cell motion through microvascular bifurcation: cell deformability and suspending viscosity effects

Red blood cell (RBC) motion and trajectories in bifurcated microvessels are simulated using a two-dimensional immersed boundary-lattice Boltzmann method (IB-LBM). A RBC is modeled as a capsule with viscous interior fluid enclosed by a flexible membrane. For the symmetric bifurcation model employed, the critical offset position in the mother branch, which separates the RBC flux toward the two branches, has been calculated. The RBC flux and the hematocrit partitioning between the two daughter branches have also been studied. Effects of the flow-rate ratio, cell deformability and suspending viscosity have been examined. Simulation results indicate that increased cell rigidity and suspending viscosity have counter effects on cell trajectory through a bifurcation: the cell trajectory shifts toward the low flow-rate branch for less deformable cells, and toward the high flow-rate branch for more viscous plasma. These results imply that a higher cell rigidity would reduce the regular phase separation of hematocrit and plasma skimming processes in microcirculation, while an increased viscosity has the opposite effect. This has implications for relevant studies in fundamental biology and biomedical applications.     

Rheological properties of the blood influencing selectin-mediated adhesion of flowing leukocytes

We studied how the rheological properties of blood influenced capture and rolling adhesion of leukocytes as well as their margination in the bloodstream. When citrated, fluorescently labeled blood was perfused through glass capillaries coated with P-selectin, leukocytes formed numerous rolling attachments. The number of attached leukocytes increased as the hematocrit was increased between 10% and 30% and was essentially constant from 30% to 50%. In EDTA-treated blood, adhesion was absent, and the flux of marginated cells varied little with increasing hematocrit. However, the velocity of marginated leukocytes increased monotonically, whereas the volumetric flow rate was constant, implying that the flow velocity profile became blunted and wall shear rate increased. Thus increasing hematocrit promoted attachment for a given total flow rate, without increasing margination, even though wall shear rate and blood viscosity increased. Blood was diluted to 20% hematocrit with plasma, 40-kDa dextran (to reduce red blood cell aggregation), or 500-kDa dextran (to enhance aggregation). Increasing aggregation correlated with increasing leukocyte adhesion and with more slow-flowing leukocytes near the wall. Thus flowing erythrocytes promote leukocyte adhesion, either by causing margination of leukocytes or by initiating and stabilizing attachments that follow.     

Detection of red cell aggregation by low shear rate viscometry in whole blood with elevated plasma viscosity

The viscosity of whole blood measured at low shear rates is determined partly by shear resistance of the red cell aggregates present, stronger aggregation increasing the viscosity in the absence of other changes. Effects of cell deformability can confound interpretation and comparison in terms of aggregation, however, particularly when the plasma viscosity is high. We illustrate the problem with a comparison of hematocrit-adjusted blood from type 1 diabetes patients and controls in which it is found the apparent and relative viscosities at a true shear rate of 0.20 s-1 are lower in the patient samples than age matched controls, in spite of reports that aggregation is increased in such populations. Because the plasma viscosities of the patients were higher on average than controls, we performed a series of experiments to examine the effect of plasma protein concentration and viscosity on normal blood viscosity. Dilution or concentration by ultrafiltration of autologous plasma and viscosity measurements at low shear on constant hematocrit red cell suspensions showed (a) suspension viscosity at 0.25 and 3 s-1 increased monotonically with plasma protein concentration and viscosity but (b) the relative viscosity increased, in concert with the microscopic aggregation grade, up to a viscosity of approximately 1.25 mPa-s but above this the value the relative viscosity no longer increased as the degree of aggregation increased in concentrated plasmas. It is suggested that in order to reduce cell deformation effects in hyperviscous pathological plasmas, patient and control plasmas should be systematically diluted before hematocrit is adjusted and rheological measurements are made. True shear rates should be calculated. Comparison of relative viscosities at low true shear rates appears to allow the effects of red cell aggregation to be distinguished by variable shear rate viscometry in clinical blood samples.     

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.     

Separation of red blood cells at high volume concentration under low centrifugal accelerations

The separation process of blood and RBC suspensions in a hematocrit range between 0.3-0.7 was investigated with a centrifuge allowed to run at low accelerations (100 xg-1000 xg). The position of the interface between the supernatant of plasma and the RBC column was continuously recorded by a new optoelectronic measuring system. The separation process could be mathematically described by an exponential decrease of the cell column approaching a final packing. At a given centrifugal acceleration the time constant is influenced by hematocrit, aggregation, deformation and plasma viscosity. The final packing depends linearly on the starting hematocrit (0.3-0.7) and can be used as a measure of deformability.     

Erythrocyte distribution in arterial blood flow: IV. Simplification of the proposed equations

In a previous paper (Bull. Math. Biophysics,29, 565–574, 1967) the author developed equations to represent velocity and hematocrit profiles in quasi-Poiseuille flow of blood. It was assumed that energy dissipation was minimized and that the viscosity depended on hematocrit and shear rate according to the Casson formula. These equations are simplified considerably, placed in a form more suitable for numerical solution and shown to depend on a single dimensionless parameter. Typicalin vivo values for this parameter are calculated.     

Visualization study of motion and deformation of red blood cells in a microchannel with straight,divergent and convergent sections

The size of red blood cells (RBC) is on the same order as the diameter of microvascular vessels. Therefore, blood should be regarded as a two-phase flow system of RBCs suspended in plasma rather than a continuous medium of microcirculation. It is of great physiological and pathological significance to investigate the effects of deformation and aggregation of RBCs on microcirculation. In this study, a visualization experiment was conducted to study the microcirculatory behavior of RBCs in suspension. Motion and deformation of RBCs in a microfluidic chip with straight, divergent, and convergent microchannel sections have been captured by microscope and high-speed camera. Meanwhile, deformation and movement of RBCs were investigated under different viscosity, hematocrit, and flow rate in this system. For low velocity and viscosity, RBCs behaved in their normal biconcave disc shape and their motion was found as a flipping motion: they not only deformed their shapes along the flow direction, but also rolled and rotated themselves. RBCs were also found to aggregate, forming rouleaux at very low flow rate and viscosity. However, for high velocity and viscosity, RBCs deformed obviously under the shear stress. They elongated along the flow direction and performed a tank-treading motion.     

In silico biophysics and hemorheology of blood hyperviscosity syndrome

Hyperviscosity syndrome (HVS) is characterized by an increase of the blood viscosity by up to seven times the normal blood viscosity, resulting in disturbances to the circulation in the vasculature system. HVS is commonly associated with an increase of large plasma proteins and abnormalities in the properties of red blood cells, such as cell interactions, cell stiffness, and increased hematocrit. Here, we perform a systematic study of the effect of each biophysical factor on the viscosity of blood by employing the dissipative particle dynamic method. Our in silico platform enables manipulation of each parameter in isolation, providing a unique scheme to quantify and accurately investigate the role of each factor in increasing the blood viscosity. To study the effect of these four factors independently, each factor was elevated more than its values for a healthy blood while the other factors remained constant, and viscosity measurement was performed for different hematocrits and flow rates. Although all four factors were found to increase the overall blood viscosity, these increases were highly dependent on the hematocrit and the flow rates imposed. The effect of cell aggregation and cell concentration on blood viscosity were predominantly observed at low shear rates, in contrast to the more magnified role of cell rigidity and plasma viscosity at high shear rates. Additionally, cell-related factors increase the whole blood viscosity at high hematocrits compared with the relative role of plasma-related factors at lower hematocrits. Our results, mapped onto the flow rates and hematocrits along the circulatory system, provide a correlation to underpinning mechanisms for HVS findings in different blood vessels.     

A semi-empirical model of apparent blood viscosity as a function of vessel diameter and discharge hematocrit

A semi-empirical model is developed to describe the dependence of apparent viscosity of blood on vessel diameter (2.7 to 500 microns) and vessel discharge hematocrit (5% to 60%). The blood flow is modeled as a cell-rich core and a cell-free marginal layer in the larger vessels and an axial-train in the smaller vessels. Laminar (Poiseuille) flow is assumed in all cases. An equation is derived in which apparent viscosity is a function of vessel diameter, core viscosity, and width of marginal layer. This is then complemented by empirical equations in which core viscosity varies exponentially with discharge hematocrit while the width of marginal layer varies linearly with discharge hematocrit. The model correlates well with several sets of experimental data and behaves according to the Fahraeus-Lindqvist effect. Predicted apparent viscosity tends to the expected finite value for large vessel diameters. Dependence of apparent viscosity on vessel diameter is realistically smooth in the whole diameter range.     

Dynamics of blood flow: modeling of the Fåhræus–Lindqvist effect

To model the Fåhræus–Lindqvist effect, Haynes’ marginal zone theory is used, following previous works, i.e., a core layer of uniform red blood cells (RBCs) is assumed to be surrounded by an annular plasma layer in which no RBCs are present. A simplified trial-and-error solution procedure is provided to determine the size of the core region and the hematocrit level in that zone in addition to the apparent viscosity, given the (upstream) large vessel hematocrit level and the average hematocrit level in the (downstream) small vessel. To test the model, a set of experimental data is selected to provide not only apparent viscosity data but also the average hematocrit levels in small tubes of different diameters. The results are found to support Haynes’ marginal theory, with no fitting parameters used in the computations. Viscous dissipation is determined. The use of the mechanical energy balance is found to lead to results that are consistent with those based on the momentum balance, while leaving the average hematocrit level undetermined and required by either experimental data or an additional equation based on further theoretical work. The present analysis is used to model bifurcation using published empirical correlations quantifying the Fåhræus effect and phase separation. The model equations are extended to microvascular networks with repeated bifurcations.     

肝郁脾虚证模型大鼠血流变及TXB_2、PGFla的变化（英文）

目的：探查中医肝郁脾虚证模型的血流变及相关调节因子的状态。方法：采用慢性束缚应激＋过度疲劳＋饮食失节法建立大鼠肝郁脾虚证模型,测定大鼠造模三周、自然恢复一周时的血流变和血浆TXB2、PGF1a。结果：与正常组相比,模型组大鼠造模三周150/s、38/s、10/s、5/s切变率下的全血粘度、还原粘度均显著升高（P〈0.001）,红细胞聚集指数显著降低（P〈0.001）,红细胞压积显著升高（P〈0.01）,红细胞变形指数无显著性差异（P〉0.05）;血浆TXB2显著升高（P〈0.001）,6-keto-PGF1a显著降低（P〈0.05）,TXB2/PGF1a显著升高（P〈0.01）;模型组大鼠第四周150/s、38/s、10/s、5/s切变率下的全血粘度、还原粘度仍显著升高（P〈0.001或P〈0.01）;红细胞聚集指数显著降低（P〈0.001）;红细胞压积与变形指数无显著性差异（P〉0.05）;血浆TXB2和TXB2/PGF1a显著降低（P〈0.05）,6-keto-PGF1a显著升高（P〈0.05）。结论：肝郁脾虚证大鼠存在血液高粘和血栓易形成状态,恢复期血液高粘同时伴有扩血管因素的加强。提示肝郁脾虚证有血流变的异常和血浆TXB2-PGI2的平衡失调,主要涉及到血小板和血浆因素的参与。