Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method

In this paper we develop a lattice Boltzmann algorithm to simulate red blood cell (RBC) behavior in shear flows. The immersed boundary method is employed to incorporate the fluid-membrane interaction between the flow field and deformable cells. The cell membrane is treated as a neo-Hookean viscoelastic material and a Morse potential is adopted to model the intercellular interaction. Utilizing the available mechanical properties of RBCs, multiple cells have been studied in shear flows using a two-dimensional approximation. These cells aggregate and form a rouleau under the action of intercellular interaction. The equilibrium configuration is related to the interaction strength. The end cells exhibit concave shapes under weak interaction and convex shapes under strong interaction. In shear flows, such a rouleau-like aggregate will rotate or be separated, depending on the relative strengths of the intercellular interaction and hydrodynamic viscous forces. These behaviors are qualitatively similar to experimental observations and show the potential of this numerical scheme for future studies of blood flow in microvessels.     

Radionuclide plethysmography for noninvasive evaluation of peripheral arterial blood flow

We validated a noninvasive radionuclide plethysmography technique to evaluate peripheral arterial blood flow during reactive hyperemia. This method, based on the measurement of blood volume variations during repetitive venous occlusions, was compared with strain-gauge venous impedance plethysmography. The technique uses 99mTc-labeled autologous red blood cells scintigraphy to determine the rate of change of forearm scintigraphic counts during venous occlusion. Thirteen subjects were simultaneously evaluated with radionuclide and impedance plethysmography. Six baseline flow measurements were performed to evaluate the reproducibility of each method. Twenty-seven serial measurements were then made to evaluate flow variation during forearm reactive hyperemia. After 30 min of recovery, resting forearm blood flows were again evaluated. Impedance and radionuclide methods showed excellent reproducibility with intraclass correlation coefficients of 0.96 and 0.93, respectively. There was also good correlation of flows between both methods during reactive hyperemia (r = 0.87). Resting flows at 30 min after reactive hyperemia were slightly lower than at baseline with both methods. We conclude that radionuclide plethysmography could be used for the noninvasive evaluation of forearm blood flow and its dynamic variations during reactive hyperemia.     

Red blood cell mechanics and capillary blood rheology

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessel walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic “tank-treading” motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

Red blood cell mechanics and capillary blood rheology.

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessels walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic "tank-treading" motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

Mechanics of blood flow

The historical development of the mechanics of blood flow can be traced from ancient times, to Leonardo da Vinci and Leonhard Euler and up to the present times with increasing biological knowledge and mathematical analysis. In the last two decades, quantitative and numerical methods have steadily given more complete and precise understanding. In the arterial system wave propagation computations based on nonlinear one-dimensional modeling have given the best representation of pulse wave propagation. In the veins, the theory of unsteady flow in collapsible tubes has recently been extensively developed. In the last decade, progress has been made in describing the blood flow at junctions, through stenoses, in bends and in capillary blood vessels. The rheological behavior of individual red blood cells has been explored. A working model consists of an elastic membrane filled with viscous fluid. This model forms a basis for understanding the viscous and viscoelastic behavior of blood.     

Non-Newtonian effects of blood flow on hemodynamics in distal vascular graft anastomoses

Non-Newtonian fluid flow in a stenosed coronary bypass is investigated numerically using the Carreau-Yasuda model for the shear thinning behavior of the blood. End-to-side coronary bypass anastomosis is considered in a simplified model geometry where the host coronary artery has a 75% severity stenosis. Different locations of the bypass graft to the stenosis and different flow rates in the graft and in the host artery are studied. Particular attention is given to the non-Newtonian effect of the blood on the primary and secondary flow patterns in the host coronary artery and the wall shear stress (WSS) distribution there. Interaction between the jet flow from the stenosed artery and the flow from the graft is simulated by solving the three-dimensional Navier-Stokes equation coupled with the non-Newtonian constitutive model. Results for the non-Newtonian flow, the Newtonian flow and the rescaled Newtonian flow are presented. Significant differences in axial velocity profiles, secondary flow streamlines and WSS between the non-Newtonian and Newtonian fluid flows are revealed. However, reasonable agreement between the non-Newtonian and the rescaled Newtonian flows is found. Results from this study support the view that the residual flow in a partially occluded coronary artery interacts with flow in the bypass graft and may have significant hemodynamic effects in the host vessel downstream of the graft. Non-Newtonian property of the blood alters the flow pattern and WSS distribution and is an important factor to be considered in simulating hemodynamic effects of blood flow in arterial bypass grafts.     

Physical properties of flowing blood

The changes of viscosity, optical reflection and electrical resistivity of blood due to flow are dependent on the orientation and deformation of red cells. From electrical point of view, it can be assumed that blood is suspension of small insulating particles (red cells) in conductive fluid (plasma) when the frequency of supplied voltage is lower than several hundreds KHz. When blood flows, red cells deform and orient in flow direction. Therefore, flowing blood shows anisotropic electrical and optical properties. In steady flow, blood resistivity longitudinal to flow decrease with flow rate, and transverse one increases. Blood flow in living body is not steady but pulsatile. We measured both longitudinal and transverse resistivity changes, optical reflection change and viscosity change of sinusoidally flowing blood in a rectangular conduit. The results are 1) during one period of sinusoidal flow the longitudinal resistivity change is opposite to that of transverse one, 2) the waveform of reflection light change is similar to that of resistance change, and 3) minimum points of both longitudinal resistivity and viscosity changes do not appear at the moment when flow is zero but are delayed. When the amplitude of sinusoidal flow is small and oscillation frequency is high, the phase difference between the zero crossing period of flow and the period of minimum change in resistivity, increases up to 90 degrees. Viscosity of blood decreases with increase of amplitude and frequency of sinusoidal flow.     

Effects of shear rate and suspending viscosity on deformation and frequency of red blood cells tank-treading in shear flows

The tank-treading rotation of red blood cells (RBCs) in shear flows has been studied extensively with experimental, analytical, and numerical methods. Even for this relatively simple system, complicated motion and deformation behaviors have been observed, and some of the underlying mechanisms are still not well understood. In this study, we attempt to advance our knowledge of the relationship among cell motion, deformation, and flow situations with a numerical model. Our simulation results agree well with experimental data, and confirm the experimental finding of the decrease in frequency/shear-rate ratio with shear rate and the increase of frequency with suspending viscosity. Moreover, based on the detailed information from our simulations, we are able to interpret the frequency dependency on shear rate and suspending viscosity using a simple two-fluid shear model. The information obtained in this study thus is useful for understanding experimental observations of RBCs in shear and other flow situations; the good agreement to experimental measurements also shows the potential usefulness of our model for providing reliable results for microscopic blood flows.     

Numerical simulations of pulsatile blood flow using a new constitutive model

In the present paper we use a new constitutive equation for whole human blood [R.G. Owens, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech. (2006), to appear] to investigate the steady, oscillatory and pulsatile flow of blood in a straight, rigid walled tube at modest Womersley numbers. Comparisons are made with the experimental results of Thurston [Elastic effects in pulsatile blood flow, Microvasc. Res. 9 (1975), 145-157] for the pressure drop per unit length against volume flow rate and oscillatory flow rate amplitude. Agreement in all cases is very good. In the presentation of the numerical and experimental results we discuss the microstructural changes in the blood that account for its rheological behaviour in this simple class of flows. In this context, the concept of an apparent complex viscosity proves to be useful.     

Ovine fetal electrocortical activity and regional cerebral blood flow

Fetuses of 12 near-term sheep were prepared for microsphere determination of cerebral blood flow. Experiments were performed 5 days postsurgery. The regional blood flows were measured in successive high (HV), low (LV) and high voltage electrocorticographic states. Comparisons were made between the observations made in the LV and averaged flanking HV cycles. Total cerebral blood flow was 95 +/- 8, 119 +/- 11 and 100 +/- 9 ml/min/100 g in HV, LV and HV, respectively. Low voltage electrocortical activity increased average cerebral blood flow by 22% (P less than 0.01). Significant changes were seen in all regions except the occipital cortex. The maximum change was observed in the thalamus in which the flows were 152 +/- 23, 243 +/- 35 and 138 +/- 20 ml/min/per 100 g tissue, respectively. The increase was 68% (P less than 0.001). The percent changes seen in the cerebrum are as follows: Frontal grey + 18%, frontal white + 22%, parietal white + 22%, temporal + 18%. A + 17% change was seen in the cord (P less than 0.03). It is concluded that in low voltage electrocortical activity all of the brain, except the occipital region, shows an increase in cerebral blood flow. This is probably secondary to a variance in cerebral activity. This preparation may be useful in localizing function in the fetal brain.     

Transient lateral transport of platelet-sized particles in flowing blood suspensions.

Concentration profiles of 2.5 microns latex beads were measured to demonstrate lateral transport of platelet-sized objects in flows of blood suspensions; the flows had equivalent Poiseuille wall shear rates (WSRs) from 250 to 1220 s-1. Each experimental trial began with a steady flow of suspension without beads in a thin-walled capillary tube (219 microns ID; 10.2 microns SD). The tube entrance was then switched to a reservoir containing suspension of equal hematocrit, but with beads, for a short interval of flow at the same WSR. This process established a paraboloidal tongue of labeled suspension with a transient concentration gradient at its surface. The tube and contents were rapidly frozen to fix the suspended particles in flow-determined locations. Segments of frozen tube were collected at distances from the entrance corresponding to 13%, 39%, and 65% of the axial extent of the ideal paraboloidal tongue. Concentration profiles were estimated from distances measured on fluorescence microscope images of cross-cut tube segments. Experiments used tubes either 40 or 50 cm long, suspension hematocrits of 0, 15, or 40%, and bead concentrations in the range of 1.5-2.2 x 10(5)/mm3. Profiles for 0% hematocrit suspension, a dilute, single-component suspension, had features expected in normal diffusive mixing in a flow. Distinctly different profiles and more lateral transport occurred when the suspensions contained red cells; then, all profiles for 13% extent had regions of excess bead concentration near the wall. Suspension flows with 40% hematocrit exhibited the largest amount of lateral transport. A case is made that, to a first approximation, the rate of lateral transport grew linearly with WSR; however, statistical analysis showed that for 40% hematocrit, less lateral transport occurred when the WSR was 250 s-1 or 1220 s-1 than 560 s-1, thus indicating that the rate behavior is more complex.     

Numerical investigation of non-Newtonian microcirculatory blood flow in hepatic lobule

The circulation in the liver is unique at macroscopic and microscopic levels. At the macroscopic level, there is an unusual presence of portal and arterial inputs rather than a single arterial input. At the microscopic level, a series of microenvironments in the acinar system is essential in controlling the functional characteristics of hepatic parenchymal cells. Since the hemodynamics is much less studied in the multifunctional liver, an attempt is made to study the hepatic hemodynamics in a segment of a hepatic lobular structure, that is made up of high-pressure oxygenated arteriole, low-pressure nutrient-rich portal venule, fenestrated sinusoidal space and hepatic venule. Our goal is to dispel some of the myths of this complex vascular bed by means of finite volume blood flow simulation. Flow features like high-velocity gradients near the fenestrations, flow reversal and Dean vortices in the sinusoidal space are analyzed within the non-Newtonian framework. Since no distinct exact or numerical solutions are available for this complex vascular bed, the present simulated results are compared with the available clinical observations. Results revealed that the pressure plays a key role in hepatic blood flow.     

The bulk rheology of close-packed red blood cells in shear flow

A theoretical analysis is made of the dynamical behavior and bulk rheology of close-packed red blood cell suspensions subjected to simple shear flow. The model for the polyhedral cell shapes and tank-treading membrane motion developed in the companion paper (1) is used. The flow in the thin lubricating plasma layers between cells is analyzed taking into account the mechanical properties of the membrane at the corner regions of sharp membrane curvature. This leads to predictions for the apparent viscosity as a function of hematocrit and shear rate. Good agreement with experimental results is obtained at moderate and high shear rates (above 20 s-1). At lower shear rates, a rapid rise in apparent viscosity has been found experimentally, and the mechanisms leading to this behavior are examined.     

Factors affecting radioactive microsphere measurement of blood flow in pregnant guinea pigs

Comparative blood flow studies were performed in pregnant guinea pigs using radioactive microspheres to test the effects of different sphere sizes on blood flow measurements and the relationship between flows obtained intraoperatively and those performed after 5 days of recovery from anesthesia and surgery. We observed that 1.5% of the cardiac output was shunted through the microcirculation of the carcass, gut, skin and endomyometrium when 15 mu microspheres were used. Intraoperative measurements of heart rate, cardiac output and placental blood flow are significantly lower than measurements made after 5 days recovery. These reductions were ameliorated with the addition of a continuous infusion of isoproterenol and the deletion of atropine from the anesthetic.     

Collateral sources of costal and crural diaphragmatic blood flow

We measured the contribution of aortic, internal mammary, and intercostal arteries to the blood flow to the costal and crural segments of the diaphragm and other respiratory muscles in seven dogs breathing against a fixed inspiratory elastic load. We used radiolabeled microspheres to measure the blood flow with control circulation, occlusion of the aorta distal to the left subclavian artery, combined occlusion of the aorta and both internal mammary arteries, and occlusion of internal mammary arteries alone. With occlusion of the aorta distal to the left subclavian artery, blood flow to the crural diaphragm decreased from 40.3 to 23.5 ml . min-1 X 100 g-1, whereas costal flow did not change significantly (from 41.7 to 38.1 ml . min-1 . 100 g-1). Blood flows to the sternomastoid and scalene muscles (above the occlusion) increased by 200 and 340%, respectively, whereas flows to the other respiratory muscles did not change significantly. Blood flows to organs above the occlusion either remained unchanged or increased, whereas flows to those below the occlusion all decreased. When the internal mammary artery was also occluded, flows to the crural segment decreased further to 12.1 and costal flow decreased to 20.4 ml X min-1 X 100 g-1. Internal mammary arterial occlusion alone in two dogs had no effect on diaphragmatic flow. In conclusion, intercostal collateral vessels are capable of supplying a significant proportion of blood flow to both segments of the diaphragm but the costal segment is better served than the crural segment.     

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.     

Comparison of blood rheological models for physiological flow simulation

The present study investigates the flow effects that different blood constitutive equations induce when employed in numerical simulations in the framework of computational hemodynamics. In accord with experimental studies on the rheological behavior of blood, three blood constitutive equations namely the Casson, Power-Law and Quemada models were used for simulating the shear flow behavior of blood. The case studied is the flow in a channel with a moving part of the boundary and was selected because it reproduces the flow phenomena occurring in realistic arterial conditions. Flow simulation for every model is carried out assuming the same flow rate at the inlet of the channel and different Strouhal numbers reflecting different intensities of the boundary movement. Results show that the modeling of blood as non-Newtonian fluid has marked qualitative and quantitative effects on both the flow field and the wall shear stress whereas comparison of the different models shows good agreement between the flow effects by the Casson and Quemada models.     

Dynamical clustering of red blood cells in capillary vessels

We have modeled the dynamics of a 3-D system consisting of red blood cells (RBCs), plasma and capillary walls using a discrete-particle approach. The blood cells and capillary walls are composed of a mesh of particles interacting with harmonic forces between nearest neighbors. We employ classical mechanics to mimic the elastic properties of RBCs with a biconcave disk composed of a mesh of spring-like particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, translational and angular momenta. Realistic behavior of blood cells is modeled by considering RBCs and plasma flowing through capillaries of various shapes. Three types of vessels are employed: a pipe with a choking point, a curved vessel and bifurcating capillaries. There is a strong tendency to produce RBC clusters in capillaries. The choking points and other irregularities in geometry influence both the flow and RBC shapes, considerably increasing the clotting effect. We also discuss other clotting factors coming from the physical properties of blood, such as the viscosity of the plasma and the elasticity of the RBCs. Modeling has been carried out with adequate resolution by using 1 to 10 million particles. Discrete particle simulations open a new pathway for modeling the dynamics of complex, viscoelastic fluids at the microscale, where both liquid and solid phases are treated with discrete particles. Figure A snapshot from fluid particle simulation of RBCs flowing along a curved capillary. The red color corresponds to the highest velocity. We can observe aggregation of RBCs at places with the most stagnant plasma flow.     

Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity

A computational fluid dynamics (CFD) approach was presented to model the blood flows in the carotid bifurcation and the brain arteries under altered gravity. Physical models required for CFD simulation were introduced including a model for arterial wall motion due to fluid-wall interactions, a shear thinning fluid model of blood, a vascular bed model for outflow boundary conditions, and a model for autoregulation mechanism. The three-dimensional unsteady incompressible Navier-Stokes equations coupled with these models were solved iteratively using the pseudocompressibility method and dual time stepping. Gravity source terms were added to the Navier-Stokes equations to take the effect of gravity into account. For the treatment of complex geometry, a chimera overset grid technique was adopted to obtain connectivity between arterial branches. For code validation, computed results were compared with experimental data for both steady-state and time-dependent flows. This computational approach was then applied to blood flows through a realistic carotid bifurcation and two Circle of Willis models, one using an idealized geometry and the other using an anatomical data set. A three-dimensional Circle of Willis configuration was reconstructed from subject-specific magnetic resonance images using an image segmentation method. Through the numerical simulation of blood flow in two model problems, namely, the carotid bifurcation and the brain arteries, it was observed that the altered gravity has considerable effects on arterial contraction/dilatation and consequent changes in flow conditions.     

Theory to Predict Shear Stress on Cells in Turbulent Blood Flow

Shear stress on blood cells and platelets transported in a turbulent flow dictates the fate and biological activity of these cells. We present a theoretical link between energy dissipation in turbulent flows to the shear stress that cells experience and show that for the case of physiological turbulent blood flow: (a) the Newtonian assumption is valid, (b) turbulent eddies are universal for the most complex of blood flow problems, and (c) shear stress distribution on turbulent blood flows is possibly universal. Further we resolve a long standing inconsistency in hemolysis between laminar and turbulent flow using the theoretical framework. This work demonstrates that energy dissipation as opposed to bulk shear stress in laminar or turbulent blood flow dictates local mechanical environment of blood cells and platelets universally.