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.     

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.     

Computational fluid dynamic studies of leukocyte adhesion effects on non-Newtonian blood flow through microvessels

The study of the effect of leukocyte adhesion on blood flow in small vessels is of primary interest to understand the resistance changes in venular microcirculation. Available computational fluid dynamic studies provide information on the effect of leukocyte adhesion when blood is considered as a homogeneous Newtonian fluid. In the present work we aim to understand the effect of leukocyte adhesion on the non-Newtonian Casson fluid flow of blood in small venules; the Casson model represents the effect of red blood cell aggregation. In our model the blood vessel is considered as a circular cylinder and the leukocyte is considered as a truncated spherical protrusion in the inner side of the blood vessel. The cases of single leukocyte adhesion and leukocyte pairs in positions aligned along the same side, and opposite sides of the vessel wall are considered. The Casson fluid parameters are chosen for cat blood and human blood and comparisons are made for the effects of leukocyte adhesion in both species. Numerical simulations demonstrated that for a Casson fluid with hematocrit of 0.4 and flow rate Q = 0.072 nl/s, a single leukocyte increases flow resistance by 5% in a 32 microns diameter and 100 microns long vessel. For a smaller vessel of 18 microns, the flow resistance increases by 15%.     

A couple stress model of blood flow in the microcirculation

A simple mathematical model depicting blood flow in the capillary is developed with an emphasis on the permeability property of the blood vessel based on Starling's hypothesis. In this study the effect of inertia has been neglected in comparison with the viscosity on the basis of the smallness of the Reynolds number of the flow in the capillary. The capillary blood vessel is approximated by a circular cylindrical tube with a permeable wall. The blood is represented by a couple stress fluid. With such an ideal model the velocity and pressure fields are determined. It is shown that an increase in the couple stress parameter increases the resistance to the flow and thereby decreases the volume rate flow. A comparison of the results with those of the Newtonian case has also been made.     

Hematocrit,viscosity and velocity distributions of aggregating and non-aggregating blood in a bifurcating microchannel

Microscale blood flow is characterised by heterogeneous distributions of hematocrit, viscosity and velocity. In microvascular bifurcations, cells are unevenly distributed between the branches, and this effect can be amplified in subsequent branches depending on a number of parameters. We propose an approach to infer hematocrit profiles of human blood flowing through a bifurcating microchannel. The influence of aggregation, induced by the addition of Dextran 2000 to the samples, is also considered. Averaged values indicate plasma skimming, particularly in the presence of red blood cell (RBC) aggregation. Using an empirical model, the hematocrit profiles are used to estimate local relative viscosity distributions. Simulations are used to predict how the non-uniform viscosity influences the velocity profiles. Comparing these data to velocity profiles of RBCs measured using particle image velocimetry provides validation of the model. It is observed that aggregation blunts velocity profiles after a long straight section of channel. Downstream of the bifurcation, skewing of the velocity profiles is detected, which is enhanced by aggregation. The proposed methodology is capable of providing hitherto unreported information on important aspects of microscale blood rheology.     

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.     

Mixed Exciton-Charge-Transfer States in Photosystem II: Stark Spectroscopy on Site-Directed Mutants

Sickle erythrocytes exhibit abnormal morphology and membrane mechanics under deoxygenated conditions due to the polymerization of hemoglobin S. We employed dissipative particle dynamics to extend a validated multiscale model of red blood cells (RBCs) to represent different sickle cell morphologies based on a simulated annealing procedure and experimental observations. We quantified cell distortion using asphericity and elliptical shape factors, and the results were consistent with a medical image analysis. We then studied the rheology and dynamics of sickle RBC suspensions under constant shear and in a tube. In shear flow, the transition from shear-thinning to shear-independent flow revealed a profound effect of cell membrane stiffening during deoxygenation, with granular RBC shapes leading to the greatest viscosity. In tube flow, the increase of flow resistance by granular RBCs was also greater than the resistance of blood flow with sickle-shape RBCs. However, no occlusion was observed in a straight tube under any conditions unless an adhesive dynamics model was explicitly incorporated into simulations that partially trapped sickle RBCs, which led to full occlusion in some cases.     

Theoretical modeling of filtration of blood cell suspensions

A theoretical model of filtration of suspensions containing red blood cells (RBCs) and white blood cells (WBCs) has been developed. Equations are written for the pressure drop, the filtration flow and the fractions of filter pores containing RBCs (alpha) and WBCs (alpha*). Because the relative resistances (ratios of resistance of cell to resistance of suspending fluid) of RBCs (beta) and WBCs (beta*) through the filter pore are greater than one, the transit of these cells (especially WBCs) through the filter is slower than that of suspending fluid; this leads to values of alpha and alpha* higher than those simply expected from the hematocrit and leukocrit, respectively, in the entering and exiting suspensions. In the absence of pore plugging by the cells (steady flow), the pressure drop can be computed from alpha, alpha*, beta and beta*. In order to model unsteady flow, differential equations are written to include pore plugging and the subsequent unplugging by the rising filtration pressure at a constant flow. By specifying the fractions of entering RBCs (epsilon) and WBCs (epsilon*) which would plug the pores and the rate at which the plugged pores would unplug in response to pressure rise (epsilon u), as well as the fractions of entering RBCs (epsilon p) and WBCs (epsilon p*) that would plug the pores permanently, theoretical pressure-time curves can be generated by numerical integration, and the results fit the experimental data well. From such fitting of theoretical curve to experimental data, information can be deduced for epsilon, epsilon*, epsilon u, epsilon p and epsilon* p.     

PEG-albumin supraplasma expansion is due to increased vessel wall shear stress induced by blood viscosity shear thinning

We studied the extreme hemodilution to a hematocrit of 11% induced by three plasma expanders: polyethylene glycol (PEG)-conjugated albumin (PEG-Alb), 6% 70-kDa dextran, and 6% 500-kDa dextran. The experimental component of our study relied on microelectrodes and cardiac output to measure both the rheological properties of plasma-expander blood mixtures and nitric oxide (NO) bioavailability in vessel walls. The modeling component consisted of an analysis of the distribution of wall shear stress (WSS) in the microvessels. Our experiments demonstrated that plasma expansion with PEG-Alb caused a state of supraperfusion with cardiac output 40% above baseline, significantly increased NO vessel wall bioavailability, and lowered peripheral vascular resistance. We attributed this behavior to the shear thinning nature of blood and PEG-Alb mixtures. To substantiate this hypothesis, we developed a mathematical model of non-Newtonian blood flow in a vessel. Our model used the Quemada rheological constitutive relationship to express blood viscosity in terms of both hematocrit and shear rate. The model revealed that the net effect of the hemodilution induced by relatively low-viscosity shear thinning PEG-Alb plasma expanders is to reduce overall blood viscosity and to increase the WSS, thus intensifying endothelial NO production. These changes act synergistically, significantly increasing cardiac output and perfusion due to lowered overall peripheral vascular resistance.     

Factors influencing distribution of cerebral gas embolism

The results of a series of experiments in baboons centrally cooled to a brain temperature of 18 °C have been described. These studies were characterized by 50% reduction in blood flow to cerebellum after cooling, relative increase in distribution to cerebellar cortex as opposed to white matter, and reduced cerebral oxygen extraction.The physical factors of importance are the viscosity of the blood, chiefly the effect to hematocrit, surface tension, the surface tension coefficient between gas and blood, the radius of the vessel, aud the velocity of the bubble.     

The pressure-flow relation in resting rat skeletal muscle perfused with pure erythrocyte suspensions

In spite of numerous investigations of erythrocyte rheology, there is limited information about the influence of erythrocyte suspensions on whole organ pressure-flow relationships. In this study, we present whole organ pressure-flow curves for resting vasodilated gracilis muscle of the rat, in which the microanatomy and vessel properties have been determined previously. For pure erythrocyte suspensions from donor rats, the organ resistance increases only mildly with perfusion time (less than a 5% shift over a one-hour perfusion time), while in contrast, erythrocyte suspensions containing leukocytes show an increases of resistance near 100% over a period of 25 min. Variation in pressure-flow curves in the muscle at the same arterial hematocrit between different rats is less than 15%. The pressure-flow relation for pure erythrocyte suspensions depends on hematocrit. Shear thinning is exhibited at high hematocrits, while Newtonian behavior is approached at arterial hematocrits below 15%. The whole organ apparent viscosity for pure erythrocyte suspensions (normalized by cell-free plasma resistance) is a non-linear function of hematocrit; at physiological pressures, it reaches values comparable to those of apparent viscosities measured in rotational viscometers or in in vitro tube flow (diameters greater than 0.8 mm). The apparent viscosities estimated from the whole organ experiments tend to be higher than those measured in straight tubes under in vitro conditions. The pressure-flow curves for pure erythrocyte suspensions are shifted towards lower pressures than the curves for mixed suspensions of erythrocytes at the same hematocrit and with leukocytes at physiological cell counts. These acute experiments show that pure erythrocyte suspensions yield highly reproducible resistances in the skeletal muscle microcirculation with dilated arterioles. Relative apparent viscosities measured in vivo are higher than those measured in straight glass tubes of comparable dimesions.     

Rheological effects of red blood cell aggregation in the venous network: a review of recent studies

It has long been recognized that understanding the rheological properties of blood is essential to a full understanding of the function of the circulatory system. Given the difficulty of obtaining carefully controlled measurements in vivo, most of our current concepts of the flow behavior of blood in vivo are based on its properties in vitro. Studies of blood rheology in rotational and tube viscometers have defined the basic properties of blood and pointed to certain features that may be especially significant for understanding in vivo function. At the same time, differences between in vivo and in vitro systems combined with the complex rheological properties of blood make it difficult to predict in vivo blood rheology from in vitro studies. We have investigated certain flow properties of blood in vivo, using the venular network of skeletal muscle as our model system. In the presence of red blood cell aggregation, venous velocity profiles become blunted from the parabolic as in Poiseuille flow, as pseudo-shear rate (= mean fluid velocity/vessel diameter) is decreased from approximately 100 s(-1) to 5 s(-1). At control flow rates, the short distance between venular junctions does not appear to permit significant axial migration and red cell depletion of the peripheral fluid layer before additional red cells and aggregates are infused from a feeding tributary. Formation of a cell-free plasma layer at the vessel wall and sedimentation in vivo are evident only at very low pseudo-shear rates (<5 s(-1)). These findings may explain in large part observations in whole organs of increased venous resistance with reduction of blood flow.     

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.     

A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes

A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.     

Time-dependent rheological behaviour of blood flow at low shear in narrow horizontal tubes

Magnitude and time-dependence of the effects of red cell aggregation and sedimentation on the rheology of human blood were studied during low shear (tau W 2.5 to 92 mPa) flow through horizontal tubes (ID 25 to 105 microns). Immediately following reduction of perfusion pressure to a low value the red cell concentration near the tube walls decreases as a result of red cell aggregation. This is associated with a transient increase of centerline velocity. Simultaneously, sedimentation begins to occur and eventually leads to the formation of a cell-free supernatant plasma layer. Time-course and extent of this sedimentation process are strongly affected by wall shear stress variation, particularly in the larger tubes. At the lower shear stresses, centerline velocity decreases (flow resistance increases) with time following the initial acceleration period, due to sedimentation of red cells. This is followed by a further increase of resistance caused by the elevation of hematocrit occurring because of the reduction of cell/plasma velocity ratio. The time dependence of blood rheological behaviour under these flow conditions is interpreted to reflect the net effect of the partially counteracting phenomena of sedimentation and red cell aggregation.     

Effects of sedimentation of small red blood cell aggregates on blood flow in narrow horizontal tubes

The flow properties of aggregating red cell suspensions flowing at low flow rates through horizontal tubes are analyzed using a theoretical model. The effects of sedimentation of small aggregates, which will be formed at comparatively high flow rates, on the relative apparent viscosity are considered. In the case in which a large number of small aggregates are formed in a suspension flowing through a horizontal tube, it seems that red cells are transported as a concentrated suspension through the bottom part of the tube because of sedimentation of aggregates. A two-layer flow model is used for the distribution of red cells. It consists of plasma in the upper part and a concentrated red cell suspension in the bottom part of the tube divided by a smooth and horizontal interface. It is assumed that the suspension is a Newtonian fluid whose viscosity increases exponentially with hematocrit. The velocity distribution, the relative apparent viscosity and the flux of red cells are calculated as functions of width of plasma layer for a different discharge hematocrit. The theoretical results are compared with the results obtained from experimental data. The relative apparent viscosity increases rapidly with an increasing degree of sedimentation over a wide range of plasma layer widths.     

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.     

A compartmental model for oxygen transport in brain microcirculation in the presence of blood substitutes

A compartmental model is developed for oxygen (O(2)) transport in brain microcirculation in the presence of blood substitutes (hemoglobin-based oxygen carriers). The cerebrovascular bed is represented as a series of vascular compartments, on the basis of diameters, surrounded by a tissue compartment. A mixture of red blood cells (RBC) and plasma/extracellular hemoglobin solution flows through the vascular bed from the arterioles through the capillaries to the venules. Oxygen is transported by convection in the vascular compartments and by diffusion in the surrounding tissue where it is utilized. Intravascular resistance and the diffusive loss of oxygen from the arterioles to the tissue are incorporated in the model. The model predicts that most of the O(2) transport occurs at the level of capillaries. Results computed from the present model in the presence of hemoglobin-based oxygen carriers are consistent with those obtained from the earlier validated model (Sharan et al., 1989, 1998a) on oxygen transport in brain circulation in the absence of extracellular hemoglobin. We have found that: (a) precapillary PO(2) gradients increase as PO(2) in the arterial blood increases, P(50 p) (oxygen tension at 50% saturation of hemoglobin with O(2) in plasma) decreases, i.e. O(2) affinity of the extracellular hemoglobin is increased, the flow rate of the mixture decreases, hematocrit decreases at constant flow, metabolic rate increases, and intravascular transport resistance in the arterioles is neglected; (b) precapillary PO(2) gradients are not sensitive to (i) intracapillary transport resistance, (ii) cooperativity (n(p)) of hemoglobin with oxygen in plasma, (iii) hemoglobin concentration in the plasma and (iv) hematocrit when accounting for viscosity variation in the flow; (c) tissue PO(2) is not sensitive to the variation of intravascular transport resistance in the arterioles. We also found that tissue PO(2) is a non-monotonic function of the Hill coefficient n(p) for the extracellular hemoglobin with a maximum occurring when n(p) equals the blood Hill coefficient. The results of the computations give estimates of the magnitudes of the increases in tissue PO(2) as arterial PO(2) increases,P(50 p) increases, flow rate increases, hematocrit increases, hemoglobin concentration in the plasma increases, metabolic rate decreases, the capillary mass transfer coefficient increases or the intracapillary transport resistance decreases.     

The effects of chronic long-term intermittent hypobaric hypoxia on blood rheology parameters

The effect of chronic long-term intermittent hypobaric hypoxia (CLTIHH) on blood rheology is not completely investigated. We designed this study to determine the effect of CLTIHH on blood rheology parameters. Present study was performed in 16 male Spraque-Dawley rats that divided into CLTIHH and Control groups. To obtain CLTIHH, rats were placed in a hypobaric chamber (430 mmHg; 5 hours/day, 5 days/week, 5 weeks). The control rats stayed in the same environment as the CLTIHH rats but they breathed room air. In the blood samples aspirated from the heart, hematocrit, whole blood viscosity, plasma viscosity, plasma fibrinogen concentration, erythrocyte rigidity index and oxygen delivery index were determined. The whole blood viscosity, plasma viscosity, hematocrit and fibrinogen concentration values in the CLTIHH group were found to be higher than those of the control group. However, no significant difference was found in erythrocyte rigidity index and oxygen delivery index between the groups. Our results suggested that CLTIHH elevated whole blood viscosity by increasing plasma viscosity, fibrinogen concentration and hematocrit value without effecting the erythrocyte deformability. Hence, CLTIHH that may occur in intermittent high altitude exposure and some severe obstructive sleep apnea (OSA) patients may be responsible for hemorheologic changes in those subjects.