Mesoscale simulation of blood flow in small vessels

Computational modeling of blood flow in microvessels with internal diameter 20-500 microm is a major challenge. It is because blood in such vessels behaves as a multiphase suspension of deformable particles. A continuum model of blood is not adequate if the motion of individual red blood cells in the suspension is of interest. At the same time, multiple cells, often a few thousands in number, must also be considered to account for cell-cell hydrodynamic interaction. Moreover, the red blood cells (RBCs) are highly deformable. Deformation of the cells must also be considered in the model, as it is a major determinant of many physiologically significant phenomena, such as formation of a cell-free layer, and the Fahraeus-Lindqvist effect. In this article, we present two-dimensional computational simulation of blood flow in vessels of size 20-300 microm at discharge hematocrit of 10-60%, taking into consideration the particulate nature of blood and cell deformation. The numerical model is based on the immersed boundary method, and the red blood cells are modeled as liquid capsules. A large RBC population comprising of as many as 2500 cells are simulated. Migration of the cells normal to the wall of the vessel and the formation of the cell-free layer are studied. Results on the trajectory and velocity traces of the RBCs, and their fluctuations are presented. Also presented are the results on the plug-flow velocity profile of blood, the apparent viscosity, and the Fahraeus-Lindqvist effect. The numerical results also allow us to investigate the variation of apparent blood viscosity along the cross-section of a vessel. The computational results are compared with the experimental results. To the best of our knowledge, this article presents the first simulation to simultaneously consider a large ensemble of red blood cells and the cell deformation.     

Pulsatile flow in a coronary artery using multiphase kinetic theory

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

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.     

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.     

Platelet Dynamics in Three-Dimensional Simulation of Whole Blood

A high-fidelity computational model using a 3D immersed boundary method is used to study platelet dynamics in whole blood. We focus on the 3D effects of the platelet-red blood cell (RBC) interaction on platelet margination and near-wall dynamics in a shear flow. We find that the RBC distribution in whole blood becomes naturally anisotropic and creates local clusters and cavities. A platelet can enter a cavity and use it as an express lane for a fast margination toward the wall. Once near the wall, the 3D nature of the platelet-RBC interaction results in a significant platelet movement in the transverse (vorticity) direction and leads to anisotropic platelet diffusion within the RBC-depleted zone or cell-free layer (CFL). We find that the anisotropy in platelet motion further leads to the formation of platelet clusters, even in the absence of any platelet-platelet adhesion. The transverse motion, and the size and number of the platelet clusters are observed to increase with decreasing CFL thickness. The 3D nature of the platelet-RBC collision also induces fluctuations in off-shear plane orientation and, hence, a rotational diffusion of the platelets. Although most marginated platelets are observed to tumble just outside the RBC-rich zone, platelets further inside the CFL are observed to flow with an intermittent dynamics that alters between sliding and tumbling, as a result of the off-shear plane rotational diffusion, bringing them even closer to the wall. To our knowledge, these new findings are based on the fundamentally 3D nature of the platelet-RBC interaction, and they underscore the importance of using cellular-scale 3D models of whole blood to understand platelet margination and near-wall platelet dynamics.     

Platelet Dynamics in Three-Dimensional Simulation of Whole Blood

A high-fidelity computational model using a 3D immersed boundary method is used to study platelet dynamics in whole blood. We focus on the 3D effects of the platelet-red blood cell (RBC) interaction on platelet margination and near-wall dynamics in a shear flow. We find that the RBC distribution in whole blood becomes naturally anisotropic and creates local clusters and cavities. A platelet can enter a cavity and use it as an express lane for a fast margination toward the wall. Once near the wall, the 3D nature of the platelet-RBC interaction results in a significant platelet movement in the transverse (vorticity) direction and leads to anisotropic platelet diffusion within the RBC-depleted zone or cell-free layer (CFL). We find that the anisotropy in platelet motion further leads to the formation of platelet clusters, even in the absence of any platelet-platelet adhesion. The transverse motion, and the size and number of the platelet clusters are observed to increase with decreasing CFL thickness. The 3D nature of the platelet-RBC collision also induces fluctuations in off-shear plane orientation and, hence, a rotational diffusion of the platelets. Although most marginated platelets are observed to tumble just outside the RBC-rich zone, platelets further inside the CFL are observed to flow with an intermittent dynamics that alters between sliding and tumbling, as a result of the off-shear plane rotational diffusion, bringing them even closer to the wall. To our knowledge, these new findings are based on the fundamentally 3D nature of the platelet-RBC interaction, and they underscore the importance of using cellular-scale 3D models of whole blood to understand platelet margination and near-wall platelet dynamics.     

A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall

A two-phase model for the flow of blood in narrow tubes is described. The model consists of a central core of suspended erythrocytes and a cell-free layer surrounding the core. It is assumed that the viscosity in the cell-free layer differs from that of plasma as a result of additional dissipation of energy near the wall caused by the red blood cell motion near the cell-free layer. A consistent system of nonlinear equations is solved numerically to estimate: (i) the effective dimensionless viscosity in the cell-free layer (beta), (ii) thickness of the cell-free layer (1-lambda) and (iii) core hematocrit (H(c)). We have taken the variation of apparent viscosity (mu(app)) and tube hematocrit with the tube diameter (D) and the discharge hematocrit (H(D)) from in vitro experimental studies [16]. The thickness of the cell-free layer computed from the model is found to be in agreement with the observations [3,21]. Sensitivity analysis has been carried out to study the behavior of the parameters 1-lambda, beta, H(c), B (bluntness of the velocity profile) and mu(app) with the variation of D and H(D).     

Red blood cells under flow show maximal ATP release for specific hematocrit

ATP release by red blood cells (RBCs) under shear stress (SS) plays a pivotal role in endothelial biochemical signaling cascades. The aim of this study is to investigate through numerical simulation how RBC spatiotemporal organization depends on flow and geometrical conditions to generate ATP patterns. Numerical simulations were conducted in a straight channel by considering both plasma and explicit presence of RBCs, their shape deformation and cell-cell interaction, and ATP release by RBCs. Two ATP release pathways through cell membrane are taken into account: pannexin 1 channel, sensitive to SS, and cystic fibrosis transmembrane conductance regulator, which responds to cell deformation. Several flow and hematocrit conditions are explored. The problem is solved by the lattice Boltzmann method. Application of SS to the RBC suspension triggers a nontrivial spatial RBC organization and ATP patterns. ATP localizes preferentially in the vicinity of the cell-free layer close to channel wall. Conditions for maximal ATP release per cell are identified, which depend on vessel size and hematocrit Ht. Increasing further Ht beyond optimum enhances the total ATP release but should degrade oxygen transport capacity, a compromise between an efficient ATP release and minimal blood dissipation. Moreover, ATP is boosted in capillaries, suggesting a vasomotor activity coordination throughout the resistance network.     

Dependence of red blood cell dynamics in microvessel bifurcations on the endothelial surface layer’s resistance to flow and compression

Red blood cells (RBCs) make up 40–45% of blood and play an important role in oxygen transport. That transport depends on the RBC distribution throughout the body, which is highly heterogeneous. That distribution, in turn, depends on how RBCs are distributed or partitioned at diverging vessel bifurcations where blood flows from one vessel into two. Several studies have used mathematical modeling to consider RBC partitioning at such bifurcations in order to produce useful insights. These studies, however, assume that the vessel wall is a flat impenetrable homogeneous surface. While this is a good first approximation, especially for larger vessels, the vessel wall is typically coated by a flexible, porous endothelial glycocalyx or endothelial surface layer (ESL) that is on the order of 0.5–1 µm thick. To better understand the possible effects of this layer on RBC partitioning, a diverging capillary bifurcation is analyzed using a flexible, two-dimensional model. In addition, the model is also used to investigate RBC deformation and RBC penetration of the ESL region when ESL properties are varied. The RBC is represented using interconnected viscoelastic elements. Stokes flow equations (viscous flow) model the surrounding fluid. The flow in the ESL is modeled using the Brinkman approximation for porous media with a corresponding hydraulic resistivity. The ESL’s resistance to compression is modeled using an osmotic pressure difference. One cell passes through the bifurcation at a time, so there are no cell–cell interactions. A range of physiologically relevant hydraulic resistivities and osmotic pressure differences are explored. Decreasing hydraulic resistivity and/or decreasing osmotic pressure differences (ESL resistance to compression) produced four behaviors: (1) RBC partitioning nonuniformity increased slightly; (2) RBC deformation decreased; (3) RBC velocity decreased relative to blood flow velocity; and (4) RBCs penetrated more deeply into the ESL. Decreasing the ESL’s resistance to flow and/or compression to pathological levels could lead to more frequent cell adhesion and clotting as well as impaired vascular regulation due to weaker ATP and nitric oxide release. Potential mechanisms that can contribute to these behaviors are also discussed.

     

Force acting on spheres adhered to a vessel wall

To evaluate the force and torque acting on leukocytes attached to the vessel wall, we numerically study the flow field around the leukocytes by using rigid spherical particles adhered to the wall of a circular cylindrical tube as a model of adherent leukocytes. The adherent particles are assumed to be placed regularly in the flow direction with equal spacings, in one row or two rows. The flow field of the suspending fluid is analyzed by a finite element method applied to the Stokes equations, and the drag force and torque acting on each particle, as well as the apparent viscosity, are evaluated as a function of the particle to tube diameter ratio and the particle arrangements. For two-row arrangements of adhered particles where neighboring particles are placed alternately on opposite sides of the vessel, the drag and the torque exerted on each particle are higher than those for single-row arrangements, for constant particle to tube diameter ratio and axial spacing between neighboring particles. This is enhanced for Larger particles and smaller axial spacings. The apparent viscosity of the flow through vessels with adhered particles is found to be significantly higher than that without adhered particles or when the particles are freely floating through the vessels.     

Unloading oxygen in a capillary vessel under a pathological condition

In this work, we study theoretically the unloading of oxygen from a hemoglobin molecule to the wall of a typical capillary vessel, considering that the hemoglobin under pathological conditions, obeys the rheological Maxwell model. Based on recent experimental evidences in hypertension, we consider that the red blood cells (RBCs) are composed by a single continuous medium in contrast with the classical particulate or discrete RBC models, which are only valid under normal physiological conditions. The analysis considers the hemodynamic interactions between the plasma and the hemoglobin, both circulating in a long horizontal capillary. We apply numerical and analytical methods to obtain the main fluid-dynamic characteristics for both fluids in the limit of low Reynolds and Womersley numbers. A diffusion boundary layer formulation for the oxygen transport in the combined plasma-hemoglobin core region is presented. The main aspects derived are the time and spatial evolution of the membrane. The hemoglobin and plasma velocities and the pressure distributions are shown. For the oxygen unloading the results are the oxy-hemoglobin saturation, the oxygen flux and the oxygen concentration in the cell-free plasma layer. The volume fraction of red blood cells and the Strouhal number have a great influence on the hemodynamic interactions.     

Analysis of early embryonic great-vessel microcirculation in zebrafish using high-speed confocal μPIV

In the developing cardiovascular system, hemodynamic vascular loading is critical for angiogenesis and cardiovascular adaptation. Normal zebrafish embryos with transgenically-labeled endothelial and red blood cells provide an excellent in vivo model for studying the fluid-flow induced vascular loading. To characterize the developmental hemodynamics of early embryonic great-vessel microcirculation in the zebrafish embryo, two complementary studies (experimental and numerical) are presented. Quantitative comparison of the wall shear stress (WSS) at the first aortic arch (AA1) of wild-type zebrafish embryos during two consecutive developmental stages is presented, using time-resolved confocal micro-particle image velocimetry (μPIV). Analysis showed that there was significant WSS difference between 32 and 48 h post-fertilization (hpf) wild-type embryos, which correlates with normal arch morphogenesis. The vascular distensibility of the arch wall at systole and the acceleration/deceleration rates of time-lapse phase-averaged streamwise blood flow curves were also analyzed. To estimate the influence of a novel intermittent red-blood cell (RBC) loading on the endothelium, a numerical two-phase, volume of fluid (VOF) flow model was further developed with realistic in vivo conditions. These studies showed that near-wall effects and cell clustering increased WSS augmentation at a minimum of 15% when the distance of RBC from arch vessel wall was less than 3 μm or when RBC cell-to-cell distance was less than 3 μm. When compared to a smooth wall, the WSS augmentation increased by a factor of ~1.4 due to the roughness of the wall created by the endothelial cell profile. These results quantitatively highlight the contribution of individual RBC flow patterns on endothelial WSS in great-vessel microcirculation and will benefit the quantitative understanding of mechanotransduction in embryonic great vessel biology, including arteriovenous malformations (AVM).     

Influence of erythrocyte aggregation on leukocyte margination in postcapillary venules of rat mesentery

The role of erythrocyte (red blood cell; RBC) aggregation in affecting leukocyte (white blood cell; WBC) margination in postcapillary venules of the mesentery (rat) was explored by direct intravital microscopy. Optical techniques were refined and applied to relate the light-scattering properties of RBCs to obtain a quantitative index of aggregate size (G), which, under idealized conditions, represents the number of RBCs per aggregate. WBC margination, defined as the radial migration of WBCs to the venular wall and their subsequent rolling along the endothelium, was measured as the percentage of the potentially maximal WBC volumetric flux within the microvessel lumen (F(WBC)(*)). In normal blood, F(WBC)(*) increased exponentially fourfold, and G increased from 1 to 1.15 as wall shear rates () were reduced from a steady-state value of approximately 600 to <100 s(-1) by proximal occlusion with a blunt microprobe. Enhancement of aggregation by infusion (iv) of dextran 500 (428 kDa), to attain a systemic concentration of 3 g/100 ml, resulted in a four- and sevenfold increase in G and F(WBC)(*), respectively, as was reduced below 100 s(-1). Inhibition of RBC aggregation by infusion of dextran 40 (37.5 kDa) caused F(WBC)(*) to fall to one-half of its steady-state level for < 100 s(-1). Thus it appears that the well-known increase of WBC margination with reductions in is strongly dependent on the occurrence of RBC aggregation. Increasing the extent of RBC aggregation during reductions in also increased the firm adhesion of WBCs to the endothelium because of an enhanced probability of contact between leukocytes and the postcapillary venular wall.     

Numerical simulation of two-phase non-Newtonian blood flow with fluid-structure interaction in aortic dissection

The behavior of blood cells and vessel compliance significantly influence hemodynamic parameters, which are closely related to the development of aortic dissection. Here the two-phase non-Newtonian model and the fluid-structure interaction (FSI) method are coupled to simulate blood flow in a patient-specific dissected aorta. Moreover, three-element Windkessel model is applied to reproduce physiological pressure waves. Important hemodynamic indicators, such as the spatial distribution of red blood cells (RBCs) and vessel wall displacement, which greatly influence the hemodynamic characteristics are analyzed. Results show that the proximal false lumen near the entry tear appears to be a vortex zone with a relatively lower volume fraction of RBCs, a low time-averaged wall shear stress (TAWSS) and a high oscillatory shear index (OSI), providing a suitable physical environment for the formation of atherosclerosis. The highest TAWSS is located in the narrow area of the distal true lumen which might cause further dilation. TAWSS distributions in the FSI model and the rigid wall model show similar trend, while there is a significant difference for the OSI distributions. We suggest that an integrated model is essential to simulate blood flow in a more realistic physiological environment with the ultimate aim of guiding clinical treatment.     

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.     

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.     

Multiphase hemodynamic simulation of pulsatile flow in a coronary artery

A multiphase transient non-Newtonian three-dimensional (3-D) computational fluid dynamics (CFD) simulation has been performed for pulsatile hemodynamics in an idealized curved section of a human coronary artery. We present the first prediction, to the authors' knowledge, of particulate buildup on the inside curvature using the multiphase theory of dense suspension hemodynamics. In this study, the particulates are red blood cells (RBCs). The location of RBC buildup on the inside curvature correlates with lower wall shear stress (WSS) relative to the outside curvature. These predictions provide insight into how blood-borne particulates interact with artery walls and hence, have relevance for understanding atherogenesis since clinical observations show that atherosclerotic plaques generally form on the inside curvatures of arteries. The buildup of RBCs on the inside curvature is driven by the secondary flow and higher residence times. The higher viscosity in the central portion of the curved vessel tends to block their flow, causing them to migrate preferentially through the boundary layer. The reason for this is the nearly neutrally buoyant nature of the dense two-phase hemodynamic flow. The two-phase non-Newtonian viscosity model predicts greater shear thinning than the single-phase non-Newtonian model. Consequently, the secondary flow induced in the curvature is weaker. The waveforms for computed hemodynamic parameters, such as hematocrit, WSS, and viscosity, follow the prescribed inlet velocity waveforms. The lower oscillatory WSS produced on the inside curvature has implications for understanding thickening of the intimal layer.     

An integrated fluid–structure interaction and thrombosis model for type B aortic dissection

False lumen thrombosis (FLT) in type B aortic dissection has been associated with the progression of dissection and treatment outcome. Existing computational models mostly assume rigid wall behavior which ignores the effect of flap motion on flow and thrombus formation within the FL. In this study, we have combined a fully coupled fluid–structure interaction (FSI) approach with a shear-driven thrombosis model described by a series of convection–diffusion reaction equations. The integrated FSI-thrombosis model has been applied to an idealized dissection geometry to investigate the interaction between vessel wall motion and growing thrombus. Our simulation results show that wall compliance and flap motion can influence the progression of FLT. The main difference between the rigid and FSI models is the continuous development of vortices near the tears caused by drastic flap motion up to 4.45 mm. Flap-induced high shear stress and shear rates around tears help to transport activated platelets further to the neighboring region, thus speeding up thrombus formation during the accelerated phase in the FSI models. Reducing flap mobility by increasing the Young’s modulus of the flap slows down the thrombus growth. Compared to the rigid model, the predicted thrombus volume is 25% larger using the FSI-thrombosis model with a relatively mobile flap. Furthermore, our FSI-thrombosis model can capture the gradual effect of thrombus growth on the flow field, leading to flow obstruction in the FL, increased blood viscosity and reduced flap motion. This model is a step closer toward simulating realistic thrombus growth in aortic dissection, by taking into account the effect of intimal flap and vessel wall motion.

     

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.     

Blood flow in small curved tubes

Blood flow in small curved tubes is modeled by the two-fluid model where a relatively cell-free fluid layer envelops a fluid core of higher viscosity. The parameters in the model are successfully curve fitted to experimental data for straight tubes. The curved tube equations are then solved by perturbation theory. It was found that curvature in general lowers the tube resistance, but increases the shear stress near the inside wall.