Stability of particle motions in a narrow channel flow

Single rows or two rows of identical circular cylinders spaced regularly in a narrow channel flow have been shown to be capable of steady flow provided the cylinders are located at equal lateral positions and with equal spacings in the flow direction. The stability of such steady flows of circular cylinders is studied for periodic perturbations of the particle positions, assuming that every other cylinder is equally perturbed in lateral position and spacing along the channel. This results in two rows which are not symmetrically placed. The suspending fluid is assumed to be an incompressible Newtonian fluid. It is assumed that no external forces or moments act on the cylinders and the effects of inertia forces on the motion of the fluid and the cylinders are negligible. The velocity field of the suspending fluid and the instantaneous velocities of the cylinders are computed by the finite element method. The translational velocities of the cylinders are obtained for a large number of particle positions, from which the trajectories of their relative motion are determined for various initial positions near the regular single-file and two-file arrangements. It is shown that when the initial arrangements of the cylinders are slightly perturbed from the regular (alternating) two-file flows, the trajectories of their relative motions become small closed loops around the regular two-file arrangements. In contrast, such small closed trajectories are not obtained when they start from the arrangements near the regular single-file flows or regular (symmetric) double-file flows, suggesting that these flows are unstable under the conditions examined.     

Asymmetric flows of spherical particles in a cylindrical tube

To study the rheological behavior of blood cells in various flow patterns through narrow vessels, we analyzed numerically the motion of blood cells arranged in one row or two rows in tube flow, at low Reynolds numbers. The particles are assumed to be identical rigid spheres placed periodically along the vessel axis at off-axis positions with equal spacings. The flow field of the suspending fluid in a circular cylindrical tube is analyzed by a finite element method applied to the Stokes equations, and the motion of each particle is simultaneously determined by a force-free and torque-free condition. In both cases of single- and two-file arrangements of the particles, their longitudinal and angular velocities are largely affected by the radial position and the axial spacing between neighboring particles. The apparent viscosity of the asymmetric flows is higher than that of the symmetric flow where particles are located on the tube centerline, and this is more pronounced when particles are placed farther from the tube centerline and when the axial distance between neighboring particles is reduced.     

Cyclic loading on the red cell membrane in a shear flow: a possible cause of hemolysis

The fluid force acting on single human red cells in a high shear flow was analyzed. A two-dimensional elliptical microcapsule as a model of the deformed red cells was adopted to numerically calculate the distributions of the shear forces on both sides of the cell membrane. It is theoretically shown that the cell membrane undergoes an unsteady cyclic loading under the rotational motion around the interior. The mechanism leading to blood cell trauma is examined by repeatedly loading the continuously moving cell membrane.     

Flow around cells adhered to a microvessel wall. I. Fluid stresses and forces acting on the cells

To evaluate the fluid forces acting on cells adhered to a microvessel wall, we numerically studied the flow field around adherent cells and the distribution of the stresses on their surfaces. For simplicity, the cells were modeled as rigid particles attached to a wall of a circular cylindrical tube regularly in the flow direction, in a row or two rows. It was found that not the detailed shape of the model cells but their height from the vessel wall is a key determinant of the fluid forces and torque acting on them. In both arrangements of one row and two rows, the axial spacing between neighboring adherent cells significantly affects the distributions of the stresses on them, which results in drastic variations of the fluid forces with the axial spacing and the relative positions with respect to their neighboring cells. The drag force acting on an adherent cell in the vessel was evaluated to be larger than the value in the 2D chamber flow at the same wall shear stress, mainly due to much larger variations of the pressure distribution on the cell surface in the vessel flow.     

Motion of a doublet of two cylinders in contact in a narrow channel flow.

The motion of two rigid circular cylinders in contact immersed in an incompressible Newtonian fluid in a channel is examined numerically in the zero Reynolds number limit, for various values of the cylinder radius/channel width ratio. Analyses of the time courses of the lateral position and the orientation of the doublet showed that, depending on the initial condition and the doublet/channel size ratio, the doublet exhibit one of the three types of motion: a continuous rotation in the same direction during a period, and a rotation changing its direction at every half period with a large or a small variation of the orientation.     

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%.     

The flow of sickle-cell blood in the capillaries.

Oxygen tension levels and red cell velocities for the flow of sickle-cell blood in the capillaries are determined by using the Krogh model for oxygen transport and lubrication theory for the cell motion. The coupling and interaction between these arises from the red cell compliance, which is assumed to vary with the oxygen concentration. Microsieving data is used to establish an upper bound for this relationship. Calculations are carried out for a range of capillary sizes, taking into account the rightward shift of the oxyhemoglobin dissociation curve and the reduced hematocrit of sickle-cell blood, and are compared to, as a base case, the flow of normal blood under normal pressure gradient. The results indicate that under normal pressure gradients the oxygen tensions and cell velocities for sickle blood are considerably higher than for normal blood, thus acting against the tendency for cells to sickle, or significantly change their rheological properties, in the capillaries. Under reduced pressure gradients, however, the concentrations and velocities drop dramatically, adding to the likelihood of such shape or flow property changes.     

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.     

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.     

How malaria parasites reduce the deformability of infected red blood cells

The pathogenesis of malaria is largely due to stiffening of the infected red blood cells (RBCs). Contemporary understanding ascribes the loss of RBC deformability to a 10-fold increase in membrane stiffness caused by extra cross-linking in the spectrin network. Local measurements by micropipette aspiration, however, have reported only an increase of ～3-fold in the shear modulus. We believe the discrepancy stems from the rigid parasite particles inside infected cells, and have carried out numerical simulations to demonstrate this mechanism. The cell membrane is represented by a set of discrete particles connected by linearly elastic springs. The cytosol is modeled as a homogeneous Newtonian fluid, and discretized by particles as in standard smoothed particle hydrodynamics. The malaria parasite is modeled as an aggregate of particles constrained to rigid-body motion. We simulate RBC stretching tests by optical tweezers in three dimensions. The results demonstrate that the presence of a sizeable parasite greatly reduces the ability of RBCs to deform under stretching. With the solid inclusion, the observed loss of deformability can be predicted quantitatively using the local membrane elasticity measured by micropipettes.     

Blood flow: Theory,effective viscosity and effects of particle distribution

A study is made of blood flow by assuming that the blood constitutes a suspension of cells in plasma instead of a simple homogeneous fluid. A macroscopic theory governing the motion of plasma in a plasma-cell system is derived from the local volume averaging method for a system without mass transfer between the phases, and its characteristic length is much larger than the size of the cells. The equations governing the motion of the local averaged fluid quantities include one additional term in the equation of motion and two additional terms in the energy equation. These terms represent, respectively, the force exerted upon the fluid by the particles, and the rate of heat transfer and work kone upon the fluid by the particles. The theory is applied to obtain the effective viscosity as the explicit function of the volume concentration of the cells by assuming that the cells behave like rigid spherical particles with slip-collision, and the plasma is an compressible Newtonian fluid. Comparison with existing experimental results shows a good agreement. The theory is also used to obtain the effects of cell distribution upon the overall effective viscosity in a circular tube. The quantitative result shows that there is a decrease in overall effective viscosity as the concentration of cells increases toward the center of the tube, and the overall effective viscosity is smaller than the flow with evenly distributed cells.     

Motion of nonaxisymmetric red blood cells in cylindrical capillaries

We analyze theoretically the single-file flow of asymmetric red blood cells along cylindrical capillaries. Red cells in narrow capillaries are typically nonaxisymmetric, with the cell membrane moving continuously around the cell. In our analysis, cell shape and streamlines of membrane motion are prescribed. Lubrication theory is used to compute velocities and pressures in the fluid surrounding the cell. Conditions of zero lift, zero torque, zero drag, and energy conservation in the cell are imposed. Predicted tank-treading frequency, cell inclination and transverse displacement are small. Cell asymmetry and tank-treading are found to have little effect on the apparent viscosity of blood in capillaries with diameters up to 7 microns.     

Modeling of hemodynamics arising from malaria infection

We propose a numerical model of hemodynamics arising from malaria infection. This model is based on a particle method, where all the components of blood are represented by the finite number of particles. A two-dimensional spring network of membrane particles is employed for expressing the deformation of malaria infected red blood cells (IRBCs). Malaria parasite within the IRBC is modeled as a rigid object. This model is applied to the stretching of IRBCs by optical tweezers, the deformation of IRBCs in shear flow, and the occlusion of narrow channels by IRBCs. We also investigate the effects of IRBCs on the rheological property of blood in micro-channels. Our results indicate that apparent viscosity is drastically increased for the period from the ring stage and the trophozoite stage, whereas it is not altered in the early stage of infection.     

Motion and deformation of a red blood cell in a shear flow: a two-dimensional simulation of the wall effect

The motion and deformation of a single red blood cell in a simple shear flow between two parallel walls is studied theoretically. A two-dimensional deformable microcapsule is adopted as a model for the cell, which has a thin moving membrane, like a tank-tread, around the interior and is deformed into an elliptical shape with a constant area. Applying the finite element method to the Stokes equations, the tank-tread motion and deformation is determined in a stationary motion, under fluid dynamic interaction between the cell and the walls. It is shown that the motion and deformation of the microcapsule crucially depends on the channel width between the two walls. As the width decreases, the microcapsule is more elongated and the frequency of tank-tread motion decreases at a constant shear rate. In addition, the angle of inclination decreases at the low range of the viscosity ratio of internal to external fluids and increases at the high range. The results obtained are compared with experimental observations and applied to the behavior of cells under mutual interaction.     

Simulation of a Single Red Blood Cell Flowing Through a Microvessel Stenosis Using Dissipative Particle Dynamics

The motion and deformation of a single red blood cell flowing through a microvessel stenosis was investigated employing dissipative particle dynamics (DPD) method. The numerical model considers plasma, cytoplasm, the RBC membrane and the microvessel walls, in which a three dimensional coarse-grained spring network model of RBC’s membrane was used to simulate the deformation of the RBC. The suspending plasma was modelled as an incompressible Newtonian fluid and the vessel walls were regarded as rigid body. The body force exerted on the free DPD particles was used to drive the flow. A modified bounce-back boundary condition was enforced on the membrane to guarantee the impenetrability. Adhesion of the cell to the stenosis vessel surface was mediated by the interactions between receptors and ligands. Firstly, the motion of a single RBC in a microfluidic channel was simulated and the results were found in agreement with the experimental data cited by [1]. Then the mechanical behavior of the RBC in the microvessel stenosis was studied. The effects of the bending rigidity of membrane, the size of the stenosis and the driven body force on the deformation and motion of red blood cell were discussed.     

A model of a radially expanding and contracting lymphangion

The lymphatic system is an extensive vascular network featuring valves and contractile walls that pump interstitial fluid and plasma proteins back to the main circulation. Immune function also relies on the lymphatic system's ability to transport white blood cells. Failure to drain and pump this excess fluid results in edema characterized by fluid retention and swelling of limbs. It is, therefore, important to understand the mechanisms of fluid transport and pumping of lymphatic vessels. Unfortunately, there are very few studies in this area, most of which assume Poiseuille flow conditions. In vivo observations reveal that these vessels contract strongly, with diameter changes of the order of magnitude of the diameter itself over a cycle that lasts typically 2-3s. The radial velocity of the contracting vessel is on the order of the axial fluid velocity, suggesting that modeling flow in these vessels with a Poiseuille model is inappropriate. In this paper, we describe a model of a radially expanding and contracting lymphatic vessel and investigate the validity of assuming Poiseuille flow to estimate wall shear stress, which is presumably important for lymphatic endothelial cell mechanotransduction. Three different wall motions, periodic sinusoidal, skewed sinusoidal and physiologic wall motions, were investigated with steady and unsteady parabolic inlet velocities. Despite high radial velocities resulting from the wall motion, wall shear stress values were within 4% of quasi-static Poiseuille values. Therefore, Poiseuille flow is valid for the estimation of wall shear stress for the majority of the lymphangion contractile cycle.     

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.     

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.     

Possible role of contact following in the generation of coherent motion of Dictyostelium cells

After aggregation by chemotaxis, cells of the cellular slime mold Dictyostelium discoideum form a multicellular structure and show coherent motion such as vortices. Here, we present a mathematical model to explain both aggregation and coherent motion of cells in two-dimensional space. The model incorporates chemotactic response of cells and the cell's property, called "contact following", to follow the other cells with which they are in contact. Analytical study and computer simulation using the model show that with contact following, cells form circular clusters within which cell rotation occurs. Unidirectional cell motion in a long belt of cells is another type of solution of the model. Besides, contact following has an effect to accelerate cell cluster merging. By considering the mechanism of cell movement, possible explanations of contact following are proposed.     

Fluid Dynamics of Ventricular Filling in the Embryonic Heart

The vertebrate embryonic heart first forms as a valveless tube that pumps blood using waves of contraction. As the heart develops, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells that drive the flow, are necessary for correct chamber and valve morphogenesis. In this article, computational fluid dynamics was used to explore how spatial distributions of the normal forces acting on the heart wall change as the endocardial cushions grow and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-moving boundary problem of the cardiac wall driving the motion of the blood in a simplified model of a two-dimensional heart. The normal forces acting on the heart walls increased during the period of one atrial contraction because inertial forces are negligible and the ventricular walls must be stretched during filling. Furthermore, the force required to fill the ventricle increased as the stiffness of the ventricular wall was increased. Increased endocardial cushion height also drastically increased the force necessary to contract the ventricle. Finally, flow in the moving boundary model was compared to flow through immobile rigid chambers, and the forces acting normal to the walls were substantially different.