Flow resistance and drag forces due to multiple adherent leukocytes in postcapillary vessels.

Computational fluid dynamics was used to model flow past multiple adherent leukocytes in postcapillary size vessels. A finite-element package was used to solve the Navier-Stokes equations for low Reynolds number flow of a Newtonian fluid past spheres adhering to the wall of a cylindrical vessel. We determined the effects of sphere number, relative geometry, and spacing on the flow resistance in the vessel and the fluid flow drag force acting to sweep the sphere off the vessel wall. The computations show that when adherent leukocytes are aligned on the same side of the vessel, the drag force on each of the interacting leukocytes is less than the drag force on an isolated adherent leukocyte and can decrease by up to 50%. The magnitude of the reduction depends on the ratio of leukocyte to blood vessel diameter and distance between adherent leukocytes. However, there is an increase in the drag force when leukocytes adhere to opposite sides of the vessel wall. The increase in resistance generated by adherent leukocytes in vessels of various sizes is calculated from the computational results. The resistance increases with decreasing vessel size and is most pronounced when leukocytes adhere to opposite sides of the vessel.     

血浆层对血管壁剪应力和剪应力梯度的影响

在细小血管中,由于血细胞明显的趋轴效应,管中的血液分为两个不同的区域,即具有血细胞的核心区和邻近管壁和血浆层。应用两相分层流模型,研究在相同的流量和管径下,当核心区中的血液分别为牛顿流体和Casson流体时,不同的血浆层厚度对细小血管壁剪应力和剪应力梯度的影响。结果表明,血浆层的存在对壁剪应力和壁剪应力梯度有较大影响,当血浆层厚度仅为血管半径的1％和3％时,壁剪应力梯度分别下降约10％和20％。     

Particulate nature of blood determines macroscopic rheology: a 2-D lattice Boltzmann analysis

Historically, predicting macroscopic blood flow characteristics such as viscosity has been an empirical process due to the difficulty in rigorously including the particulate nature of blood in a mathematical representation of blood rheology. Using a two-dimensional lattice Boltzmann approach, we have simulated the flow of red blood cells in a blood vessel to estimate flow resistance at various hematocrits and vessel diameters. By including white blood cells (WBCs) in the flow, we also calculate the increase in resistance due to white cell rolling and adhesion. The model considers the blood as a suspension of particles in plasma, accounting for cell-cell and cell-wall interactions to predict macroscopic blood rheology. The model is able to reproduce the Fahraeus-Lindqvist effect, i.e., the increase in relative apparent viscosity as tube size increases, and the Fahraeus effect, i.e., tube hematocrit is lower than discharge hematocrit. In addition, the model allows direct assessment of the effect of WBCs on blood flow in the microvasculature, reproducing the dramatic increases in flow resistance as WBCs enter short capillary segments. This powerful and flexible model can be used to predict blood flow properties in any vessel geometry and with any blood composition.     

Theory for flow of Casson and Herschel-Bulkley fluids in cone-plate viscometers

Mathematical models for blood flow in cone-plate viscometer have been considered, by assuming blood as a Casson/Herschel-Bulkley fluid. Three different cases have been analyzed (i) when there is no shearing, (ii) partial shearing and (iii) full shearing. The relationships between the angular velocity and torque have been obtained for the above three cases. By assuming total shearing, the analytical expression for apparent viscosity has been obtained. Variation of apparent viscosity with yield stress, angular velocity, Casson co-efficient of viscosity, consistency index and flow behaviour index has been computed. It is observed that as the angular velocity increases, the apparent viscosity decreases for both fluids. Further, it is found that as the cone angle increases, the apparent viscosity increases. This behaviour of apparent viscosity in cone-plate viscometer is interesting and unexpected and is being reported first time.     

Pulsatile blood flow and oxygen transport past a circular cylinder

The fundamental study of blood flow past a circular cylinder filled with an oxygen source is investigated as a building block for an artificial lung. The Casson constitutive equation is used to describe the shear-thinning and yield stress properties of blood. The presence of hemoglobin is also considered. Far from the cylinder, a pulsatile blood flow in the x direction is prescribed, represented by a time periodic (sinusoidal) component superimposed on a steady velocity. The dimensionless parameters of interest for the characterization of the flow and transport are the steady Reynolds number (Re), Womersley parameter (alpha), pulsation amplitude (A), and the Schmidt number (Sc). The Hill equation is used to describe the saturation curve of hemoglobin with oxygen. Two different feed-gas mixtures were considered: pure O(2) and air. The flow and concentration fields were computed for Re=5, 10, and 40, 0< or =A< or =0.75, alpha=0.25, 0.4, and Schmidt number, Sc=1000. The Casson fluid properties result in reduced recirculations (when present) downstream of the cylinder as compared to a Newtonian fluid. These vortices oscillate in size and strength as A and alpha are varied. Hemoglobin enhances mass transport and is especially important for an air feed which is dominated by oxyhemoglobin dispersion near the cylinder. For a pure O(2) feed, oxygen transport in the plasma dominates near the cylinder. Maximum oxygen transport is achieved by operating near steady flow (small A) for both feed-gas mixtures. The time averaged Sherwood number, Sh, is found to be largely influenced by the steady Reynolds number, increasing as Re increases and decreasing with A. Little change is observed with varying alpha for the ranges investigated. The effect of pulsatility on Sh is greater at larger Re. Increasing Re aids transport, but yields a higher cylinder drag force and shear stresses on the cylinder surface which are potentially undesirable.     

The fluid shear stress distribution on the membrane of leukocytes in the microcirculation

Recent in-vivo and in-vitro evidence indicates that fluid shear stress on the membrane of leukocytes has a powerful control over several aspects of their cell function. This evidence raises a question about the magnitude of the fluid shear stress on leukocytes in the circulation. The flow of plasma on the surface of a leukocyte at a very low Reynolds number is governed by the Stokes equation for the motion of a Newtonian fluid. We numerically estimated the distribution of fluid shear stress on a leukocyte membrane in a microvessel for the cases when the leukocyte is freely suspended, as well as rolling along or attached to a microvessel wall. The results indicate that the fluid shear stress distribution on the leukocyte membrane is nonuniform with a sharp increase when the leukocyte makes membrane attachment to the microvessel wall. In a microvessel (10 microns diameter), the fluid shear stress on the membrane of a freely suspended leukocyte (8 microns diameter) is estimated to be several times larger than the wall shear stress exerted by the undisturbed Poiseuille flow, and increases on an adherent leukocyte up to ten times. High temporal stress gradients are present in freely suspended leukocytes in shear flow due to cell rotation, which are proportional to the local shear rate. In comparison, the temporal stress gradients are reduced on the membrane of leukocytes that are rolling or firmly adhered to the endothelium. High temporal gradients of shear stress are also present on the endothelial wall. At a plasma viscosity of 1 cPoise, the peak shear stresses for suspended and adherent leukocytes are of the order of 10 dyn/cm2 and 100 dyn/cm2, respectively.     

Imaging Leukocyte Adhesion to the Vascular Endothelium at High Intraluminal Pressure

Worldwide, hypertension is reported to be in approximately a quarter of the population and is the leading biomedical risk factor for mortality worldwide. In the vasculature hypertension is associated with endothelial dysfunction and increased inflammation leading to atherosclerosis and various disease states such as chronic kidney disease², stroke³ and heart failure⁴. An initial step in vascular inflammation leading to atherogenesis is the adhesion cascade which involves the rolling, tethering, adherence and subsequent transmigration of leukocytes through the endothelium. Recruitment and accumulation of leukocytes to the endothelium is mediated by an upregulation of adhesion molecules such as vascular cell adhesion molecule-1 (VCAM-1), intracellular cell adhesion molecule-1 (ICAM-1) and E-selectin as well as increases in cytokine and chemokine release and an upregulation of reactive oxygen species⁵. In vitro methods such as static adhesion assays help to determine mechanisms involved in cell-to-cell adhesion as well as the analysis of cell adhesion molecules. Methods employed in previous in vitro studies have demonstrated that acute increases in pressure on the endothelium can lead to monocyte adhesion, an upregulation of adhesion molecules and inflammatory markers⁶ however, similar to many in vitro assays, these findings have not been performed in real time under physiological flow conditions, nor with whole blood. Therefore, in vivo assays are increasingly utilised in animal models to demonstrate vascular inflammation and plaque development. Intravital microscopy is now widely used to assess leukocyte adhesion, rolling, migration and transmigration^7-9. When combining the effects of pressure on leukocyte to endothelial adhesion the in vivo studies are less extensive. One such study examines the real time effects of flow and shear on arterial growth and remodelling but inflammatory markers were only assessed via immunohistochemistry¹⁰. Here we present a model for recording leukocyte adhesion in real time in intact pressurised blood vessels using whole blood perfusion. The methodology is a modification of an ex vivo vessel chamber perfusion model⁹ which enables real-time analysis of leukocyte -endothelial adhesive interactions in intact vessels. Our modification enables the manipulation of the intraluminal pressure up to 200 mmHg allowing for study not only under physiological flow conditions but also pressure conditions. While pressure myography systems have been previously demonstrated to observe vessel wall and lumen diameter¹¹ as well as vessel contraction this is the first time demonstrating leukocyte-endothelial interactions in real time. Here we demonstrate the technique using carotid arteries harvested from rats and cannulated to a custom-made flow chamber coupled to a fluorescent microscope. The vessel chamber is equipped with a large bottom coverglass allowing a large diameter objective lens with short working distance to image the vessel. Furthermore, selected agonist and/or antagonists can be utilized to further investigate the mechanisms controlling cell adhesion. Advantages of this method over intravital microscopy include no involvement of invasive surgery and therefore a higher throughput can be obtained. This method also enables the use of localised inhibitor treatment to the desired vessel whereas intravital only enables systemic inhibitor treatment.     

Leukocyte adhesion and microvessel permeability

To investigate the direct effect of leukocyte adherence to microvessel walls on microvessel permeability, we developed a method to measure changes in hydraulic conductivity (L(p)) before and after leukocyte adhesion in individually perfused venular microvessels in frog mesentery. In 19 microvessels that were initially free of leukocyte sticking or rolling along the vessel wall, control L(p) was measured first with Ringer-albumin perfusate. Blood flow was then restored in each vessel with a reduced flow rate in the range of 30-116 microm/s to facilitate leukocyte adhesion. Each vessel was recannulated in 45 min. The mean number of leukocytes adhering to the vessel wall was 237 +/- 22 leukocytes/mm(2). At the same time, L(p) increased to 4.7 +/- 0.5 times the control value. Superfusion of isoproterenol (10 microM) after leukocyte adhesion brought the increased L(p) back to 1.1 +/- 0.2 times the control in 5-10 min (n = 9). Superfusing isoproterenol before leukocyte adhesion prevented the increase in L(p) (n = 6). However, the number of leukocytes adhering to the vessel wall was not significantly affected. These results demonstrated that leukocyte adhesion caused an increase in microvessel permeability that could be prevented or restored by increasing cAMP levels in endothelial cells using isoproterenol. Thus cAMP-dependent mechanisms that regulate inflammatory agent-induced increases in permeability also modulate leukocyte adhesion-induced increases in permeability but act independently of mechanisms that regulate leukocyte adhesion to the microvessel wall. Application of ketotifen, a mast cell stabilizer, and desferrioxamine mesylate, an iron-chelating reagent, attenuated the increase in L(p) induced by leukocyte adhesion, suggesting the involvement of oxidants and the activation of mast cells in leukocyte adhesion-induced permeability increase. Furthermore, with the use of an in vivo silver stain technique, the locations of the adherent leukocytes on the microvessel wall were identified quantitatively in intact microvessels.     

Deformation mechanism of leukocyte adhering to vascular surface under steady shear flow

The adhesion of leukocytes to vascular surface is an important biomedical problem and has drawn extensive attention. In this study, we propose a compound drop model to simulate a leukocyte with a nucleus adhering to the surface of blood vessel under steady shear flow. A two-dimensional computational fluid dynamics (CFD) is conducted to determine the local distribution of pressure on the surface of the adherent model cell. By introducing the parameter of deformation index (DI), we investigate the deformation of the leukocyte and its nucleus under controlled conditions. Our numerical results show that: (i) the leukocyte is capable of deformation under external exposed flow field. The deformation index increases with initial contact angle and Reynolds number of external exposed flow. (ii) The nucleus deforms with the cell, and the deformation index of the leukocyte is greater than that of the nucleus. The leukocyte is more deformable while the nucleus is more capable of resisting external shear flow. (iii) The leukocyte and the nucleus are not able to deform infinitely with the increase of Reynolds number because the deformation index reaches a maximum. (iv) Pressure distribution confirms that there exists a region downstream of the cell, which produces high pressure to retard continuous deformation and provide a positive lift force on the cell. Meanwhile, we have measured the deformation of human leukocytes exposed to shear flow by using a flow chamber system. We found that the numerical results are well consistent with those of experiment. We conclude that the nucleus with high viscosity plays a particular role in leukocyte deformation.     

Deformation mechanism of leukocyte adhering to vascular surface under steady shear flow

~~Deformation mechanism of leukocyte adhering to vascular surface under steady shear flow~~     

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.     

Red blood cells initiate leukocyte rolling in postcapillary expansions: a lattice Boltzmann analysis

Leukocyte rolling on the vascular endothelium requires initial contact between leukocytes circulating in the blood and the vessel wall. Although specific adhesion mechanisms are involved in leukocyte-endothelium interactions, adhesion patterns in vivo suggest other rheological mechanisms also play a role. Previous studies have proposed that the abundance of leukocyte rolling in postcapillary venules is due to interactions between red blood cells (RBCs) and leukocytes as they enter postcapillary expansions, but the details of the fluid dynamics have not been elucidated. We have analyzed the interactions of red and white blood cells as they flow from a capillary into a postcapillary venule using a lattice Boltzmann approach. This technique provides the complete solution of the flow field and quantification of the particle-particle forces in a relevant geometry. Our results show that capillary-postcapillary venule diameter ratio, RBC configuration, and RBC shape are critical determinants of the initiation of cell rolling in postcapillary venules. The model predicts that an optimal configuration of the trailing red blood cells is required to drive the white blood cell to the wall.     

Oscillatory flow of blood in a stenosed artery

In order to understand the abnormal flow conditions of blood in a locally constricted blood vessel, the analytical results are obtained for the oscillatory flow of blood which behaves as a Newtonian fluid. It is here assumed that the surface roughness is cosine-shaped and the maximum height of the roughness is very small compared with the radius of the unconstricted tube. Numerical solutions are presented for the instantaneous flow rate, resistive impedance, wall shear stress and phase lag.     

Unsteady Boundary Layer Flow and Heat Transfer of a Casson Fluid past an Oscillating Vertical Plate with Newtonian Heating

In this paper, the heat transfer effect on the unsteady boundary layer flow of a Casson fluid past an infinite oscillating vertical plate with Newtonian heating is investigated. The governing equations are transformed to a systems of linear partial differential equations using appropriate non-dimensional variables. The resulting equations are solved analytically by using the Laplace transform method and the expressions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and reduce to some well-known solutions for Newtonian fluids. Numerical results for velocity, temperature, skin friction and Nusselt number are shown in various graphs and discussed for embedded flow parameters. It is found that velocity decreases as Casson parameters increases and thermal boundary layer thickness increases with increasing Newtonian heating parameter.     

Pressure and flow in the umbilical cord

A fluid dynamic study of blood flow within the umbilical vessels of the human maternal-fetal circulatory system is considered. It is found that the umbilical coiling index (UCI) is unable to distinguish between cords of significantly varying pressure and flow characteristics, which are typically determined by the vessel curvature, torsion and length. Larger scale geometric non-uniformities superposed over the inherent coiling, including cords exhibiting width and/or local UCI variations as well as loose true knots, typically produce a small effect on the total pressure drop. Crucially, this implies that a helical geometry of mean coiling may be used to determine the steady vessel pressure drop through a more complex cord. The presence of vessel constriction, however, drastically increases the steady pressure drop and alters the flow profile. For pulsatile-flow within the arteries, the steady pressure approximates the time-averaged value with high accuracy over a wide range of cords. Furthermore, the relative peak systolic pressure measured over the period is virtually constant and approximately 25% below the equivalent straight-pipe value for a large range of non-straight vessels. Interestingly, this suggests that the presence of vessel helicity dampens extreme pressures within the arterial cycle and may provide another possible evolutionary benefit to the coiled structure of the cord.     

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.     

Pulsatile blood flow and gas exchange across a cylindrical fiber array

The pulsatile blood flow and gas transport of oxygen and carbon dioxide through a cylindrical array of microfibers are numerically simulated. Blood is modeled as a homogeneous Casson fluid, and hemoglobin molecules in blood are assumed to be in local equilibrium with oxygen and carbon dioxide. It is shown that flow pulsatility enhances gas transport and the amount of gas exchange is sensitive to the blood flow field across the fibers. The steady Sherwood number dependence on Reynolds number was shown to have a linear relation consistent with experimental findings. For most cases, an enhancement in gas transport is accompanied with an increase in flow resistance. Maximum local shear stress is provided as a possible indicator of thrombosis, and the computed shear stress is shown to be below the threshold value for thrombosis formation for all cases evaluated.     

CFD analysis of pulsatile blood flow in an atherosclerotic human artery with eccentric plaques

Atherosclerosis is a slow vascular degeneration. It thickens the internal walls of a blood vessel locally depositing an atherosclerotic plaque. Such reduced lumen increases the resistance to blood flow. Plaques can be punctual (eccentric, here considered) or circumferential (symmetrical). Stenoses do not have a typical shape: we hypothesised here a reference geometry (trapezium) with its possible evolutions (semi-ellipse, triangle). Two criteria (Equivalent Area and Equivalent Dimensions) were then defined to compare the results among the 35 case studies numerically analysed with a Computational Fluid Dynamics code (Comsol Multiphysics 3.3). Blood was considered a Cassonian fluid with modified viscosity equation. The artery was cylindrical, rigid and straight, interested by a pulsatile blood flow. Among the variables: shape and dimensions of the stenoses; number of stenoses (single or coupled pathologies); mutual locations (3 possibilities). The main results were that the length of the consequent flow disturbance is due to the stenotic shape and height; blood flow recirculation, downstream of the pathology, is due to the slope of the stenotic walls; and the peak velocities depend on the shape and height of stenosis. The differences from case to case diminish in diastole.     

Leukocytes are recruited through the bronchial circulation to the lung in a spontaneously hypertensive rat model of COPD

Chronic obstructive pulmonary disease (COPD) kills approximately 2.8 million people each year, and more than 80% of COPD cases can be attributed to smoking. Leukocytes recruited to the lung contribute to COPD pathology by releasing reactive oxygen metabolites and proteolytic enzymes. In this work, we investigated where leukocytes enter the lung in the early stages of COPD in order to better understand their effect as a contributor to the development of COPD. We simultaneously evaluated the parenchyma and airways for neutrophil accumulation, as well as increases in the adhesion molecules and chemokines that cause leukocyte recruitment in the early stages of tobacco smoke induced lung disease. We found neutrophil accumulation and increased expression of adhesion molecules and chemokines in the bronchial blood vessels that correlated with the accumulation of leukocytes recovered from the lung. The expression of adhesion molecules and chemokines in other vascular beds did not correlate with leukocytes recovered in bronchoalveolar lavage fluid (BALF). These data strongly suggest leukocytes are recruited in large measure through the bronchial circulation in response to tobacco smoke. Our findings have important implications for understanding the etiology of COPD and suggest that pharmaceuticals designed to reduce leukocyte recruitment through the bronchial circulation may be a potential therapy to treat COPD.     

Patterns of constriction produced by vasoactive agents

The patterns of vasoconstriction produced by local infusions of constrictor agents and neurogenic stimuli are unique and varied. Although vasoconstrictors or neurogenic stimuli may produce similar increases in total resistance to blood flow, the effects on consecutive vascular segments may differ dramatically. Vasoconstrictors may affect primarily small vessels, large vessels, or a combination of both. The constrictor response may be restricted to precapillary vessels or may recruit both pre- and postcapillary vessels. The baroreceptors elicit a pattern of vasoconstriction distinct from that produced by electrical stimulation of a vasomotor nerve. Prearteriolar and venous resistance may contribute more than arterioles to increases in total vascular resistance produced by local infusions of vasoconstrictor agents or nerve stimulation. The constriction of large vessels also affects fluid filtration, vascular capacity, and the distribution of blood flow between shunt and exchange vessels. The waning of the resistance increase that occurs during prolonged infusions of vasoconstrictors varies, in part, as a function of the vessel segments that participate in the vasoconstrictor response. Large vessels participate in vasoconstrictor responses triggered by stimuli that impose a severe stress on the circulation. In contrast, small vessels participate primarily in normal vascular adjustments required to maintain blood pressure at the set point.