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.     

Blood flow in the capillary bed

From arteries to veins, the blood has to go through the ‘capillary’ blood vessels. These blood vessels are so small that often their diameter is smaller than that of the red blood cells. Intimate interactions occur, therefore, between the blood cells and the blood vessels.

A general survey of recent works on capillary blood flow is given in this article. Some details are presented for two problems: the problem of deformation of the flexible red blood cells, their motion in the capillary blood vessels, and the pressure drop due to the red cell blood vessel interaction; and the problem of the flow of plasma ‘bolus’ between neighboring red cells.

The solution supplies many details about the microcirculation phenomenon. Taken together, a method is offered for the calculation of pressure drop in the capillary as a function of various physical parameters: the red cell volume per unit blood volume, (hematocrit), the ratio of the cell diameter to the blood vessel diameter, the ratio of the length of the blood vessel to its length, the volume of individual red cells, and a parameter relating the cell membrane elasticity, plasma viscosity and the cell velocity.     

Red blood cell velocity and oxygen tension measurement in cerebral microvessels by double-wavelength photoexcitation.

Because the regulation of microcirculation in the cerebral cortex cannot be analyzed without measuring the blood flow dynamics and oxygen concentration in cerebral microvessels, we developed a fluorescence and phosphorescence system for estimating red blood cell velocity and oxygen tension in cerebral microcirculation noninvasively and continuously with high spatial resolution. Using red blood cells labeled with fluorescent isothiocyanate to visualize red cell distribution and using the oxygen quenching of Pd-meso-tetra-(4-carboxyphenyl)-porphyrin phosphorescence to measure oxygen tension enabled simultaneous measurement of blood velocity and oxygen tension. We examined how the measurement accuracy was affected by the spatial resolution and by the excitation laser light passing through the targeted microvessel and exciting the oxygen probe dye in the tissue beneath it. Focusing the excitation light into the microvessel stabilized the phosphorescence lifetime at each spatial resolution; moreover, it greatly reduced phosphorescence from the brain tissue. Animal experiments involving acute hemorrhagic shock demonstrated the feasibility of our system by showing that the changes in venular velocity and oxygen tension are synchronized to the change in mean arterial pressure. Our system measures the red cell velocity and oxygen concentration in the cerebral microcirculation by using the differences in luminescence and wavelength between fluorescence and phosphorescence, making it possible to easily acquire information about cerebral microcirculatory distribution and oxygen tension simultaneously.     

Analysis of red blood cell motion through cylindrical micropores: effects of cell properties.

Filtration through micropores is frequently used to assess red blood cell deformability, but the dependence of pore transit time on cell properties is not well understood. A theoretical model is used to simulate red cell motion through cylindrical micropores with diameters of 3.6, 5, and 6.3 microns, and 11-microns length, at driving pressures of 100-1000 dyn/cm2. Cells are assumed to have axial symmetry and to conserve surface area during deformation. Effects of membrane shear viscosity and elasticity are included, but bending resistance is neglected. A time-dependent lubrication equation describing the motion of the suspending fluid is solved, together with the equations for membrane equilibrium, using a finite difference method. Predicted transit times are consistent with previous experimental observations. Time taken for cells to enter pores represents more than one-half of the transit time. Predicted transit time increases with increasing membrane viscosity and with increasing cell volume. It is relatively insensitive to changes in internal viscosity and to changes in membrane elasticity except in the narrowest pores at low driving pressures. Elevating suspending medium viscosity does not increase sensitivity of transit time to membrane properties. Thus filterability of red cells is sensitively dependent on their resistance to transient deformations, which may be a key determinant of resistance to blood flow in the microcirculation.     

Three-dimensional,unsteady simulation of alveolar respiration

A novel macroscopic gas transport model, derived from fundamental engineering principles, is used to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations, mimicking the single-breath technique for measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of distribution and the lung diffusing capacity. Several RBC distributions are considered, namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC distribution effect on the lung diffusing capacity. The effect of red cell movement is not considered in the present study because CO does not equilibrate with capillary blood within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion is expected to be only marginal.     

Red blood cell motions in high-hematocrit blood flowing through a stenosed microchannel

We investigated the behavior of red blood cells (RBCs) in a microchannel with stenosis using a confocal micro-PTV system. Individual trajectories of RBCs in a concentrated suspension of up to 20% hematocrit (Hct) were measured successfully. Results indicated that the trajectories of healthy RBCs became asymmetric before and after the stenosis, while the trajectories of tracer particles in pure water were almost symmetric. The asymmetry was greater in 10% Hct than in 20% Hct. We also investigated the effect of deformability of RBCs on the cell-free layer thickness by hardening RBCs using a glutaraldehyde treatment. The results indicated that deformability is the key factor in the asymmetry of cell-free layer thickness. Therefore, the motions of RBCs are influenced strongly by the Hct, the deformability, and the channel geometry. These results give fundamental knowledge for a better understanding of blood flow in microcirculation and biomedical microdevices.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

Red cell distribution and the recruitment of pulmonary diffusing capacity.

The distribution of red blood cells in alveolar capillaries is typically nonuniform, as shown by intravital microscopy and in alveolar tissue fixed in situ. To determine the effects of red cell distribution on pulmonary diffusive gas transport, we computed the uptake of CO across a two-dimensional geometric capillary model containing a variable number of red blood cells. Red blood cells are spaced uniformly, randomly, or clustered without overlap within the capillary. Total CO diffusing capacity (DLCO) and membrane diffusing capacity (DmCO) are calculated by a finite-element method. Results show that distribution of red blood cells at a fixed hematocrit greatly affects capillary CO uptake. At any given average capillary red cell density, the uniform distribution of red blood cells yields the highest DmCO and DLCO, whereas the clustered distribution yields the lowest values. Random nonuniform distribution of red blood cells within a single capillary segment reduces diffusive CO uptake by up to 30%. Nonuniform distribution of red blood cells among separate capillary segments can reduce diffusive CO uptake by >50%. This analysis demonstrates that pulmonary microvascular recruitment for gas exchange does not depend solely on the number of patent capillaries or the hematocrit; simple redistribution of red blood cells within capillaries can potentially account for 50% of the observed physiological recruitment of DLCO from rest to exercise.     

Finite Element Model of Oxygen Transport for the Design of Geometrically Complex Microfluidic Devices Used in Biological Studies

Red blood cells play a crucial role in the local regulation of oxygen supply in the microcirculation through the oxygen dependent release of ATP. Since red blood cells serve as an oxygen sensor for the circulatory system, the dynamics of ATP release determine the effectiveness of red blood cells to relate the oxygen levels to the vessels. Previous work has focused on the feasibility of developing a microfluidic system to measure the dynamics of ATP release. The objective was to determine if a steep oxygen gradient could be developed in the channel to cause a rapid decrease in hemoglobin oxygen saturation in order to measure the corresponding levels of ATP released from the red blood cells. In the present study, oxygen transport simulations were used to optimize the geometric design parameters for a similar system which is easier to fabricate. The system is composed of a microfluidic device stacked on top of a large, gas impermeable flow channel with a hole to allow gas exchange. The microfluidic device is fabricated using soft lithography in polydimethyl-siloxane, an oxygen permeable material. Our objective is twofold: (1) optimize the parameters of our system and (2) develop a method to assess the oxygen distribution in complex 3D microfluidic device geometries. 3D simulations of oxygen transport were performed to simulate oxygen distribution throughout the device. The simulations demonstrate that microfluidic device geometry plays a critical role in molecule exchange, for instance, changing the orientation of the short wide microfluidic channel results in a 97.17% increase in oxygen exchange. Since microfluidic devices have become a more prominent tool in biological studies, understanding the transport of oxygen and other biological molecules in microfluidic devices is critical for maintaining a physiologically relevant environment. We have also demonstrated a method to assess oxygen levels in geometrically complex microfluidic devices.     

Cell-free layer and wall shear stress variation in microvessels

In this study, we simulated multiple red blood cells flowing through straight microvessels with the immersed-boundary lattice-Boltzmann model to examine the shear stress variation on the microvessel surface and its relation to the properties of cell-free layer. Significant variation in shear stress has been observed due to the irregular configuration of blood cells flowing near the microvessel wall. A low shear stress is typically found at locations where there is a cell flowing close to the wall, and a large shear stress at locations with a relatively wide gap between cell and wall. This relationship between the shear stress magnitude and the distance between cell and wall has been attributed to the reverse pressure difference developed between the front and rear sides of a cell flowing near the vessel wall. We further studied the effects of several hemodynamic factors on the variation of shear stress, including the cell deformability, the flow rate, and the aggregation among red blood cells. These simulations show that the shear stress variation is less profound in situations with wider cell-free layers, since the reverse pressure difference around the edge cells is less evident, and the influence of this pressure difference on wall shear stress becomes weaker. This study also demonstrates the complexity of the flow field in the gap between cell and wall. More precise experimental techniques are required accurately measure such shear stress variation in microcirculation.     

Active nitric oxide produced in the red cell under hypoxic conditions by deoxyhemoglobin-mediated nitrite reduction

Recent studies have generated a great deal of interest in a possible role for red blood cells in the transport of nitric oxide (NO) to the microcirculation and the vascular effect of this nitric oxide in facilitating the flow of blood through the microcirculation. Many questions have, however, been raised regarding such a mechanism. We have instead identified a completely new mechanism to explain the role of red cells in the delivery of NO to the microcirculation. This new mechanism results in the production of NO in the microcirculation where it is needed. Nitrite produced when NO reacts with oxygen in arterial blood is reutilized in the arterioles when the partial pressure of oxygen decreases and the deoxygenated hemoglobin formed reduces the nitrite regenerating NO. Nitrite reduction by hemoglobin results in a major fraction of the NO generated retained in the intermediate state where NO is bound to Hb(III) and in equilibrium with the nitrosonium cation bound to Hb(II). This pool of NO, unlike Hb(II)NO, is weakly bound and can be released from the heme. The instability of Hb(III)NO in oxygen and its displacement when flushed with argon requires that reliable determinations of red blood cell NO must be performed on freshly lysed samples without permitting the sample to be oxygenated. In fresh blood samples Hb(III)NO accounts for 75% of the red cell NO with appreciably higher values in venous blood than arterial blood. These findings confirm that nitrite reduction at reduced oxygen pressures is a major source for red cell NO. The formation and potential release from the red cell of this NO could have a major impact in regulating the flow of blood through the microcirculation.     

Validation and application of an automated rheoscope for measuring red blood cell deformability distributions in different species

BACKGROUND: The deformability of red blood cells (RBCs) is of great importance for the conservation of oxygen delivery in the microcirculation. Even a small fraction of rigid cells is considered to harm the exchange of respiratory gases. Techniques that measure RBC deformability often provide an indication of the mean deformability. It may not be possible, however, to assess whether this mean value is reduced by the presence of a small rigid cell fraction or by a slight overall reduction in RBC deformability. A technique that provides a deformability distribution would be of great value to study diseases that are marked by subpopulations with a reduced deformability. METHODS: This paper describes a rheoscope system that uses advanced image analysis techniques to quickly quantify the deformability of many individual cells in shear flow, in order to find the RBC-deformability distribution. Since variations in the shear stress are responsible for variations in cell elongation, and hence introduce an additional spread in the cell deformability distribution, we first determined the spread caused by instrumental error. We then utilized the technique to investigate the relation between cell deformability and cell size of single blood samples of different species (human, pig, rat and rabbit). RESULTS: The spread caused by instrumental error was small compared to the actual RBC-deformability spread in blood samples. The deformability distribution of human and pig cells are alike although their cell sizes are different. Rat and rabbit cells show comparable deformability and size distributions. With this technique no correlation was found between cell deformability and cell size in animal RBCs. In the human sample a minor correlation was found between cell deformability and cell size. CONCLUSIONS: The automated rheoscope enables us to study the mechanical properties of RBCs more thoroughly by their deformability distribution. These deformability distributions are hardly influenced by the technique or by cell size.     

Short term starvation-induced changes in the kinetic parameters of rat red cell L-alanine and glycine uptake.

Na(+)-dependent L-Alanine and Glycine uptake by rat red blood cells were best fit to a common model of two transport components, saturable transport and diffusion. 24 hours of food deprivation provoked statistically significant increases of the Km and Vmax red cells L-Alanine uptake, whereas the diffusion constant did not change in response to starvation. The Glycine uptake kinetics poorly follows the L-Alanine pattern and no significant response to starvation can be outlined. The physiological meaning of these adaptations has to be related to short term food deprivation regulation, independent of protein synthesis in the erythrocytes. Such mechanisms could be important to account for the previously described changes in the distribution patterns of amino acids between the blood plasma and blood cell compartments in response to short term starvation.     

Computer modeling of spatial-time distribution of metabolite concentrations in phantoms of biological objects by example of rat brain pial

A method for quantitative assessment of the metabolite concentrations in digital phantoms of the biological objects has been proposed. The objects under consideration are expected to have a complex geometry of diffusive sources. We present an algorithm for calculation of non-steady state gradients of the diffusing compounds under their constant border concentration. The algorithm is based on the analytical solutions for a centrally symmetric spherical source. Spatial-time distribution of metabolites is considered by the example of glucose diffusion around blood vessels in the rat brain pial. It was shown that glucose level in the tissue was increased after glycine treatment and under condition of other biological system parameters to be unchanged.     

Microfluidic Study of Enhanced Deposition of Sickle Cells at Acute Corners

Sickle cell anemia is a blood disorder, known to affect the microcirculation and is characterized by painful vaso-occlusive crises in deep tissues. During the last three decades, many scenarios based on the enhanced adhesive properties of the membrane of sickle red blood cells have been proposed, all related to a final decrease in vessels lumen by cells accumulation on the vascular walls. Up to now, none of these scenarios considered the possible role played by the geometry of the flow on deposition. The question of the exact locations of occlusive events at the microcirculatory scale remains open. Here, using microfluidic devices where both geometry and oxygen levels can be controlled, we show that the flow of a suspension of sickle red blood cells around an acute corner of a triangular pillar or of a bifurcation, leads to the enhanced deposition and aggregation of cells. Thanks to our devices, we follow the growth of these aggregates in time and show that their length does not depend on oxygenation levels; instead, we find that their morphology changes dramatically to filamentous structures when using autologous plasma as a suspending fluid. We finally discuss the possible role played by such aggregates in vaso-occlusive events.     

Water diffusion permeability of human erythrocytes studied by a pulsed gradient NMR technique

Water diffusion permeability of human erythrocytes has been measured by NMR using a pulsed magnetic field gradient technique. The measurement of exchange rates was based on restricted diffusion of water molecules within red blood cells. This method avoids addition of paramagnetic ions, such as Mn²⁺ and is used in vivo.The mean lifetime of water inside human erythrocytes was found to be 17 ms at 24°C. A sulfhydryl reagent, known to inhibit water osmotic permeability, reduced significantly water diffusion across the red cell membrane.     

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.     

Red Cell Membrane Permeability Deduced from Bulk Diffusion Coefficients

The permeability coefficients of dog red cell membrane to tritiated water and to a series of[¹⁴C]amides have been deduced from bulk diffusion measurements through a "tissue" composed of packed red cells. Red cells were packed by centrifugation inside polyethylene tubing. The red cell column was pulsed at one end with radiolabeled solute and diffusion was allowed to proceed for several hours. The distribution of radioactivity along the red cell column was measured by sequential slicing and counting, and the diffusion coefficient was determined by a simple plotting technique, assuming a one-dimensional diffusional model. In order to derive the red cell membrane permeability coefficient from the bulk diffusion coefficient, the red cells were assumed to be packed in a regular manner approximating closely spaced parallelopipeds. The local steady-state diffusional flux was idealized as a one-dimensional intracellular pathway in parallel with a one-dimensional extracellular pathway with solute exchange occurring within the series pathway and between the pathways. The diffusion coefficients in the intracellular and extracellular pathways were estimated from bulk diffusion measurements through concentrated hemoglobin solutions and plasma, respectively; while the volume of the extracellular pathway was determined using radiolabeled sucrose. The membrane permeability coefficients were in satisfactory agreement with the data of Sha'afi, R. I., C. M. Gary-Bobo, and A. K. Solomon (1971. J. Gen. Physiol. 58:238) obtained by a rapid-reaction technique. The method is simple and particularly well suited for rapidly permeating solutes.     

Water diffusion permeability of human erythrocytes studied by a pulsed gradient NMR technique.

Water diffusion permeability of human erythrocytes has been measured by NMR using a pulsed magnetic field gradient technique. The measurement of exchange rates was based on restricted diffusion of water molecules within red blood cells. This method avoids addition of paramagnetic ions, such as Mn2+, and is used in vivo. The mean lifetime of water insed human erythrocytes was found to be 17 ms at 24 degrees C. A sulfhydryl reagent, known to inhibit water osmotic permeability, reduced significantly water diffusion across the red cell membrane.     

Model of Red Blood Cell Rotation in the Flow toward a Cell Sizing Orifice: Application to Volume Distribution

The rotation of human red blood cells (RBC) as they flow in the shear field established by a Coulter type orifice is modeled. This model, based on hydrodynamics of ellipsoid rotation in laminar creeping flow, is used to calculate the probability of the cells entering the orifice with a specific orientation. The electrical resistance change produced by a cell passing through the orifice of an electronic cell volume detector is the product of an orientation-dependent shape factor and the cell volume. This paper presents a method to calculate the shape factor probability distribution which can be used to predict its effect on the cell volume distribution. Experimental results confirm the theoretical prediction that the right skewness of resistance change distributions is in part a result of the nonspherical shape of red cells.