Predicting capillary vessel network hemodynamics in silico by machine learning.

Blood velocity and red blood cell (RBC) distribution profiles in a capillary vessel cross-section in the microcirculation are generally complex and do not follow Poiseuille's parabolic or uniform pattern. Existing imaging techniques used to map large microvascular networks in vivo do not allow a direct measurement of full 3D velocity and RBC concentration profiles, although such information is needed for accurate evaluation of the physiological variables, such as the wall shear stress (WSS) and near-wall cell-free layer (CFL), that play critical roles in blood flow regulation, disease progression, angiogenesis, and hemostasis. Theoretical network flow models, often used for hemodynamic predictions in experimentally acquired images of the microvascular network, cannot provide the full 3D profiles either. In contrast, such information can be readily obtained from high-fidelity computational models that treat blood as a suspension of deformable RBCs. These models, however, are computationally expensive and not feasible for extension to the microvascular network at large spatial scales up to an organ level. To overcome such limitations, here we present machine learning (ML) models that bypass such expensive computations but provide highly accurate and full 3D profiles of the blood velocity, RBC concentration, WSS, and CFL in every vessel in the microvascular network. The ML models, which are based on artificial neural networks and convolution-based U-net models, predict hemodynamic quantities that compare very well against the true data but reduce the prediction time by several orders. This study therefore paves the way for ML to make detailed and accurate hemodynamic predictions in spatially large microvascular networks at an organ-scale.

Hematocrit skewness along sequential bifurcations within a microfluidic network induces significant changes in downstream red blood cell partitioning.

There has been a wealth of research conducted regarding the partitioning of red blood cells (RBCs) at bifurcations within the microvasculature. In previous studies, partitioning has been characterized as either regular partitioning, in which the higher flow rate daughter channel receives a proportionally larger percentage of RBCs, or reverse partitioning, in which the opposite occurs. While there are many examples of network studies in silico, most in vitro work has been conducted using single bifurcation. When microfluidic networks have been used, the channel dimensions are typically greater than 20 µm, ignoring conditions where RBCs are highly confined. This paper presents a study of RBC partitioning in a network of sequential bifurcations with channel dimensions less than 8 µm in hydraulic diameter. The study investigated the effect of the volumetric flow rate ratio (Q*) at each bifurcation, solution hematocrit, and channel length on the erythrocyte flux ratio (N*), a measure of RBC partitioning. We report significant differences in partitioning between upstream and downstream bifurcations even when the flow rate ratio remains the same. Skewness analysis, a measure of cell distribution across the width of a vessel, strongly suggests that immediately following the first bifurcation most RBCs are skewed toward the inner channel wall, leading to preferential RBC perfusion into one daughter channel at the subsequent bifurcation even at higher downstream flow rate ratios. The skewness of RBC distribution following the first bifurcation can either manifest as enhanced regular partitioning or reverse partitioning at the succeeding branch.

Application of machine learning in predicting blood flow and red cell distribution in capillary vessel networks.

Capillary blood vessels in the body partake in the exchange of gas and nutrients with tissues. They are interconnected via multiple vascular junctions forming the microvascular network. Distributions of blood flow and red cells (RBCs) in such networks are spatially uneven and vary in time. Since they dictate the pathophysiology of tissues, their knowledge is important. Theoretical models used to obtain flow and RBC distribution in large networks have limitations as they treat each vessel as a one-dimensional segment and do not explicitly consider cell-cell and cell-vessel interactions. High-fidelity computational models that accurately model each individual RBC are computationally too expensive to predict haemodynamics in large vascular networks and over a long time. Here we investigate the applicability of machine learning (ML) techniques to predict blood flow and RBC distributions in physiologically realistic vascular networks. We acquire data from high-fidelity simulations of deformable RBC suspension flowing in the networks. With the flow and haematocrit specified at an inlet of vasculature, the ML models predict the time-averaged flow rate and RBC distributions in the entire network, time-dependent flow rate and haematocrit in each vessel and vascular bifurcation in isolation over a long time, and finally, simultaneous spatially and temporally evolving quantities through the vessel hierarchy in the networks.

A computational study of red blood cell deformability effect on hemodynamic alteration in capillary vessel networks.

Capillary blood vessels, the smallest vessels in the body, form an intricate network with constantly bifurcating, merging and winding vessels. Red blood cells (RBCs) must navigate through such complex microvascular networks in order to maintain tissue perfusion and oxygenation. Normal, healthy RBCs are extremely deformable and able to easily flow through narrow vessels. However, RBC deformability is reduced in many pathological conditions and during blood storage. The influence of reduced cell deformability on microvascular hemodynamics is not well established. Here we use a high-fidelity, 3D computational model of blood flow that retains exact geometric details of physiologically realistic microvascular networks, and deformation of every one of nearly a thousand RBCs flowing through the networks. We predict that reduced RBC deformability alters RBC trafficking with significant and heterogeneous changes in hematocrit. We quantify such changes along with RBC partitioning and lingering at vascular bifurcations, perfusion and vascular resistance, and wall shear stress. We elucidate the cellular-scale mechanisms that cause such changes. We show that such changes arise primarily due to the altered RBC dynamics at vascular bifurcations, as well as cross-stream migration. Less deformable cells tend to linger less at majority of bifurcations increasing the fraction of RBCs entering the higher flow branches. Changes in vascular resistance also seen to be heterogeneous and correlate with hematocrit changes. Furthermore, alteration in RBC dynamics is shown to cause localized changes in wall shear stress within vessels and near vascular bifurcations. Such heterogeneous and focal changes in hemodynamics may be the cause of morphological abnormalities in capillary vessel networks as observed in several diseases.

Investigation of red blood cell partitioning in an in vitro microvascular bifurcation.

There is a long history of research examining red blood cell (RBC) partitioning in microvasculature bifurcations. These studies commonly report results describing partitioning that exists as either regular partitioning, which occurs when the RBC flux ratio is greater than the bulk fluid flowrate ratio, or reverse partitioning when the RBC flux ratio is less than or equal to that of the bulk fluid flowrate. This paper presents a study of RBC partitioning in a single bifurcating microchannel with dimensions of 6 to 16 µm, investigating the effects of hematocrit, channel width, daughter channel flowrate ratio, and bifurcation angle. The erythrocyte flux ratio, N*, manifests itself as either regular or reverse partitioning, and time-dependent partitioning is much more dynamic, occurring as both regular and reverse partitioning. We report a significant reduction in the well-known sigmoidal variation of the erythrocyte flux ratio (N*) versus the volumetric flowrate ratio (Q*), partitioning behavior with increasing hematocrit in microchannels when the channel dimensions are comparable with cell size. RBCs "lingering" or jamming at the bifurcation were also observed and quantified in vitro. Results from trajectory analyses suggest that the RBC position in the feeder channel strongly affects both partitioning and lingering frequency of RBCs, with both being significantly reduced when RBCs flow on streamlines near the edge of the channel as opposed to the center of the channel. Furthermore, our experiments suggest that even at low Reynolds number, partitioning is affected by the bifurcation angle by increasing cell-cell interactions. The presented results provide further insight into RBC partitioning as well as perfusion throughout the microvasculature.

A computational study of amoeboid motility in 3D: the role of extracellular matrix geometry, cell deformability, and cell-matrix adhesion.

Amoeboid cells often migrate using pseudopods, which are membrane protrusions that grow, bifurcate, and retract dynamically, resulting in a net cell displacement. Many cells within the human body, such as immune cells, epithelial cells, and even metastatic cancer cells, can migrate using the amoeboid phenotype. Amoeboid motility is a complex and multiscale process, where cell deformation, biochemistry, and cytosolic and extracellular fluid motions are coupled. Furthermore, the extracellular matrix (ECM) provides a confined, complex, and heterogeneous environment for the cells to navigate through. Amoeboid cells can migrate without significantly remodeling the ECM using weak or no adhesion, instead utilizing their deformability and the microstructure of the ECM to gain enough traction. While a large volume of work exists on cell motility on 2D substrates, amoeboid motility is 3D in nature. Despite recent progress in modeling cellular motility in 3D, there is a lack of systematic evaluations of the role of ECM microstructure, cell deformability, and adhesion on 3D motility. To fill this knowledge gap, here we present a multiscale, multiphysics modeling study of amoeboid motility through 3D-idealized ECM. The model is a coupled fluid‒structure and coarse-grain biochemistry interaction model that accounts for large deformation of cells, pseudopod dynamics, cytoplasmic and extracellular fluid motion, stochastic dynamics of cell-ECM adhesion, and microstructural (pore-scale) geometric details of the ECM. The key finding of the study is that cell deformation and matrix porosity strongly influence amoeboid motility, while weak adhesion and microscale structural details of the ECM have secondary but subtle effects.

Three-dimensional distribution of wall shear stress and its gradient in red cell-resolved computational modeling of blood flow in in vivo-like microvascular networks.

Using a high-fidelity, 3D computational model of blood flow in microvascular networks, we provide the full 3D distribution of wall shear stress (WSS), and its gradient (WSSG), and quantify the influence of red blood cells (RBCs) on WSS and WSSG. The deformation and flow dynamics of the individual RBCs are accurately resolved in the model, while physiologically realistic microvascular networks comprised of multiple bifurcations, convergences, and tortuous vessels are considered. A strong heterogeneity in WSS and WSSG is predicted across the networks, with the highest WSS occurring in precapillary bifurcations and capillary vessels. 3D variations of WSS and WSSG are shown to occur due to both network morphology and the influence of RBCs. The RBCs increase the WSS by as much as three times compared to that when no RBCs are present, and the highest increase is observed in venules. WSSG also increases significantly, and high WSSGs occur over wider regions in the presence of RBCs. In most vessels, the circumferential component of WSSG is observed to be greater than the axial component in the presence of RBCs, while the opposite trend is observed when RBCs are not considered. These results underscore the important role of RBCs on WSS and WSSG that cannot be predicted by widely used 1D models of network blood flow. Furthermore, the subendothelium-scale variations of WSS and WSSG predicted by the present model have implications in terms of endothelial cell functions in the microvasculature.

A computational model of amoeboid cell motility in the presence of obstacles.

Locomotion of amoeboid cells is mediated by finger-like protrusions of the cell body, known as pseudopods, which grow, bifurcate, and retract in a dynamic fashion. Pseudopods are the primary mode of locomotion for many cells within the human body, such as leukocytes, embryonic cells, and metastatic cancer cells. Amoeboid motility is a complex and multiscale process, which involves bio-molecular reactions, cell deformation, and cytoplasmic and extracellular fluid motion. Additionally, cells within the human body are subject to a confined 3D environment known as the extra-cellular matrix (ECM), which resembles a fluid-filled porous medium. In this article, we present a 3D, multiphysics computational approach coupling fluid mechanics, solid mechanics, and a pattern formation model to simulate locomotion of amoeboid cells through a porous matrix composed of a viscous fluid and an array of finite-sized spherical obstacles. The model combines reaction-diffusion of activator/inhibitors, extreme deformation of the cell, pseudopod dynamics, cytoplasmic and extracellular fluid motion, and fully resolved extracellular matrix. A surface finite-element method is used to obtain the cell deformation and activator/inhibitor concentrations, while the fluid motion is solved using a combined finite-volume and spectral method. The immersed-boundary methods are used to couple the cell deformation, obstacles, and fluid. The model is able to recreate squeezing and weaving motion of cells through the matrix. We study the influence of matrix porosity, obstacle size, and cell deformability on the motility behavior. It is found that below certain values of these parameters, cell motion is completely inhibited. Phase diagrams are presented depicting such motility limits. Interesting dynamics seen in the presence of obstacles but absent in unconfined medium, such as freezing or cell arrest, probing, doubling-back, and tug-of-war are predicted. Furthermore, persistent unidirectional motion of cells that is often observed in an unconfined medium is shown to be lost in presence of obstacles, and is attributed to an alteration of the pseudopod dynamics. The same mechanism, however, allows the cell to find a new direction to penetrate further into the matrix without being stuck in one place. The results and analysis presented here show a strong coupling between cell deformability and ECM properties, and provide new fluid mechanical insights on amoeboid motility in confined medium.

Direct Numerical Simulation of Cellular-Scale Blood Flow in 3D Microvascular Networks.

We present, to our knowledge, the first direct numerical simulation of 3D cellular-scale blood flow in physiologically realistic microvascular networks. The vascular networks are designed following in vivo images and data, and are comprised of bifurcating, merging, and winding vessels. Our model resolves the large deformation and dynamics of each individual red blood cell flowing through the networks with high fidelity, while simultaneously retaining the highly complex geometric details of the vascular architecture. To our knowledge, our simulations predict several novel and unexpected phenomena. We show that heterogeneity in hemodynamic quantities, which is a hallmark of microvascular blood flow, appears both in space and time, and that the temporal heterogeneity is more severe than its spatial counterpart. The cells are observed to frequently jam at vascular bifurcations resulting in reductions in hematocrit and flow rate in the daughter and mother vessels. We find that red blood cell jamming at vascular bifurcations results in several orders-of-magnitude increase in hemodynamic resistance, and thus provides an additional mechanism of increased in vivo blood viscosity as compared to that determined in vitro. A striking result from our simulations is negative pressure-flow correlations observed in several vessels, implying a significant deviation from Poiseuille's law. Furthermore, negative correlations between vascular resistance and hematocrit are observed in various vessels, also defying a major principle of particulate suspension flow. To our knowledge, these novel findings are absent in blood flow in straight tubes, and they underscore the importance of considering realistic physiological geometry and resolved cellular interactions in modeling microvascular hemodynamics.

Flow of Red Blood Cells in Stenosed Microvessels.

A computational study is presented on the flow of deformable red blood cells in stenosed microvessels. It is observed that the Fahraeus-Lindqvist effect is significantly enhanced due to the presence of a stenosis. The apparent viscosity of blood is observed to increase by several folds when compared to non-stenosed vessels. An asymmetric distribution of the red blood cells, caused by geometric focusing in stenosed vessels, is observed to play a major role in the enhancement. The asymmetry in cell distribution also results in an asymmetry in average velocity and wall shear stress along the length of the stenosis. The discrete motion of the cells causes large time-dependent fluctuations in flow properties. The root-mean-square of flow rate fluctuations could be an order of magnitude higher than that in non-stenosed vessels. Several folds increase in Eulerian velocity fluctuation is also observed in the vicinity of the stenosis. Surprisingly, a transient flow reversal is observed upstream a stenosis but not downstream. The asymmetry and fluctuations in flow quantities and the flow reversal would not occur in absence of the cells. It is concluded that the flow physics and its physiological consequences are significantly different in micro- versus macrovascular stenosis.

Flow-Induced Damage to Blood Cells in Aortic Valve Stenosis.

Valvular hemolysis and thrombosis are common complications associated with stenotic heart valves. This study aims to determine the extent to which hemodynamics induce such traumatic events. The viscous shear stress downstream of a severely calcified bioprosthetic valve was evaluated via in vitro 2D particle image velocimetry measurements. The blood cell membrane response to the measured stresses was then quantified using 3D immersed-boundary computational simulations. The shear stress level at the boundary layer of the jet flow formed downstream of the valve orifice was observed to reach a maximum of 1000-1700 dyn/cm(2), which was beyond the threshold values reported for platelet activation (100-1000 dyn/cm(2)) and within the range of thresholds reported for red blood cell (RBC) damage (1000-2000 dyn/cm(2)). Computational simulations demonstrated that the resultant tensions at the RBC membrane surface were unlikely to cause instant rupture, but likely to lead to membrane plastic failure. The resultant tensions at the platelet surface were also calculated and the potential damage was discussed. It was concluded that although shear-induced thrombotic trauma is very likely in stenotic heart valves, instant hemolysis is unlikely and the shear-induced damage to RBCs is mostly subhemolytic.

Microparticle shape effects on margination, near-wall dynamics and adhesion in a three-dimensional simulation of red blood cell suspension.

We present a 3D computational modeling study of the transport of micro-scale drug carriers modeled as microparticles of different shapes (spherical, oblate, and prolate) in whole blood represented as a suspension of deformable red blood cells. The objective is to quantify the effect of microparticle shapes on their margination, near-wall dynamics and adhesion. We observe that the near-wall accumulation is highest for oblate particles of moderate aspect ratio, followed by spherical particles, and lowest for very elongated prolate particles. The result is explained using micro-scale dynamics of individual particles, and their interaction with red blood cells. We observe that the orientation of microparticles in 3D space and the frequency of their collisions with red blood cells are the key factors affecting their margination. We show that due to repeated collisions with red blood cells in the presence of a bounding wall, the axes of revolution of oblate particles align near the plane of the shear flow, but those of prolate particles shift towards the vorticity axis with a wider distribution. Such specific orientations lead to more frequent collisions and a greater lateral drift for oblate particles than microspheres, but less frequent collisions and a reduced lateral drift for elongated prolate particles, resulting in the observed differences in their near-wall accumulation. Once marginated, the particle shape has an entirely different effect on the likelihood of making particle-wall contacts. We find that marginated prolate particles, due to their alignment along the vorticity axis and large angular fluctuations, are more likely to make contacts with the wall than spherical and oblate particles. We further simulate the adhesion between flowing microparticles and the wall in the presence of red blood cells, and observe that once wall contacts are established, the likelihood of firm adhesion is greater for disk-like particles, followed by elongated prolates, and microspheres. Consequently, this study suggests that the local hemorheological conditions near the targeted sites must be taken into consideration while selecting the optimum shape of micro-scale vascular drug carriers.

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.

Hydrodynamic interaction between a platelet and an erythrocyte: effect of erythrocyte deformability, dynamics, and wall proximity.

We present three-dimensional numerical simulations of hydrodynamic interaction between a red blood cell (RBC) and a platelet in a wall-bounded shear flow. The dynamics and large deformation of the RBC are fully resolved in the simulations using a front-tracking method. The objective is to quantify the influence of tank treading and tumbling dynamics of the RBC, and the presence of a bounding wall on the deflection of platelet trajectories. We observe two types of interaction: A crossing event in which the platelet comes in close proximity to the RBC, rolls over it, and continues to move in the same direction; and a turning event in which the platelet turns away before coming close to the RBC. The crossing events occur when the initial lateral separation between the cells is above a critical separation, and the turning events occur when it is below the critical separation. The critical lateral separation is found to be higher during the tumbling motion than that during the tank treading. When the RBC is flowing closer to the wall than the platelet, the critical separation increases by several fold, implying the turning events have higher probability to occur than the crossing events. On the contrary, if the platelet is flowing closer to the wall than the RBC, the critical separation decreases by several folds, implying the crossing events are likely to occur. Based on the numerical results, we propose a mechanism of continual platelet drift from the RBC-rich region of the vessel towards the wall by a succession of turning and crossing events. The trajectory deflection in the crossing events is found to depend nonmonotonically on the initial lateral separation, unlike the monotonic trend observed in tracer particle deflection and in deformable sphere-sphere collision. This nonmonotonic trend is shown to be a consequence of the deformation of the RBC caused by the platelet upon collision. An estimation of the platelet diffusion coefficient yields values that are similar to those reported in experiments and computer simulations with multicellular suspension.

Three-dimensional numerical simulation of vesicle dynamics using a front-tracking method.

Three-dimensional numerical simulation using the front-tracking method is presented on the dynamics of a vesicle in a linear shear flow. The focus here is to elucidate the parametric dependence and the self-similarity of the vesicle dynamics, quantification of vesicle deformation, and the analysis of shape dynamics. A detailed comparison of the numerical results is made with various theoretical models and experiments. It is found that the applicability of the theoretical models is limited despite some general agreement with the simulations and experiments. The deviations between the perturbative results and the simulation results occur even in the absence of thermal noise. Specifically, we find that the vesicle dynamics does not follow a self-similar behavior in a two-parameter phase space, as proposed in a theoretical model. Rather, the dynamics is governed by three controlling parameters, namely, the excess area, viscosity ratio, and dimensionless shear rate. Additionally, we find that a linear scaling of the tank-treading angle, as proposed in the theoretical model, is possible only for nearly spherical vesicles. The breakdown of the scaling occurs at higher values of the excess area even in the absence of thermal noise. We find that the vesicle deformation saturates at large shear rates, and the asymptotic deformation matches well with a theoretical prediction for nearly spherical vesicles. The dependence of the critical viscosity ratio associated with the onset of unsteady dynamics on the vesicle excess area is in excellent agreement with the experimental observation. We show that near the transition between the tank-treading and tumbling dynamics, both the vacillating-breathing-like motion characterized by a smooth ellipsoidal shape and the trembling-like motion characterized by a highly deformed shape are possible. For the trembling-like motion, the shape is highly three-dimensional with concavities and lobes, and the vesicle deforms more in the vorticity direction than in the shear plane. A Fourier spectral analysis of the vesicle shape shows the presence of the odd harmonics and higher order modes beyond fourth order.

Phase diagram and breathing dynamics of a single red blood cell and a biconcave capsule in dilute shear flow.

We present phase diagrams of the single red blood cell and biconcave capsule dynamics in dilute suspension using three-dimensional numerical simulations. The computational geometry replicates an in vitro linear shear flow apparatus. Our model includes all essential properties of the cell membrane, namely, the resistance against shear deformation, area dilatation, and bending, as well as the viscosity difference between the cell interior and suspending fluids. By considering a wide range of shear rate and interior-to-exterior fluid viscosity ratio, it is shown that the cell dynamics is often more complex than the well-known tank-treading, tumbling, and swinging motion and is characterized by an extreme variation of the cell shape. As a result, it is often difficult to clearly establish whether the cell is swinging or tumbling. Identifying such complex shape dynamics, termed here as "breathing" dynamics, is the focus of this article. During the breathing motion at moderate bending rigidity, the cell either completely aligns with the flow direction and the membrane folds inward, forming two cusps, or it undergoes large swinging motion while deep, craterlike dimples periodically emerge and disappear. At lower bending rigidity, the breathing motion occurs over a wider range of shear rates, and is often characterized by the emergence of a quad-concave shape. The effect of the breathing dynamics on the tank-treading-to-tumbling transition is illustrated by detailed phase diagrams which appear to be more complex and richer than those of vesicles. In a remarkable departure from the vesicle dynamics, and from the classical theory of nondeformable cells, we find that there exists a critical viscosity ratio below which the transition is independent of the viscosity ratio, and dependent on shear rate only. Further, unlike the reduced-order models, the present simulations do not predict any intermittent dynamics of the red blood cells.

Tank-treading and tumbling frequencies of capsules and red blood cells.

This study is motivated in part by the discrepancy that exists in the literature with regard to the dependence of the tank-treading frequency of red blood cells on the shear rate and suspending medium viscosity. Here we consider three-dimensional numerical simulations of deformable capsules of initially spherical and oblate spheroidal shapes and biconcave discoid representing the red blood cell resting shape. By considering a much broader range of the viscosity ratio (ratio of capsule or cell interior to suspending fluid viscosity), shear rate, and aspect ratio (ratio of minor to major axes) than that considered in the previous experiments, we find several new characteristics of the tank-treading and tumbling frequencies that have not been reported earlier. These new characteristics are the result of the large shape deformation and the coupling between shape and angular oscillations of the capsules or cells. For the spherical and oblate spheroidal capsules, the tank-treading frequency shows a nonmonotonic trend that is characterized by an initial decrease leading to a minimum followed by an increase with increasing viscosity ratio. For red blood cells, we find two regimes of the viscosity dependence of the tank-treading frequency: an exponential regime in which the tank-treading frequency decreases at a slower rate with increasing viscosity ratio, and a logarithmic range in which it decreases at a much faster rate. While this trend agrees well with different theoretical models of shape-preserving capsules, it was not evident in previous experimental results. When the shear rate dependence is considered, the tank-treading frequency of red blood cells and capsules of highly elongated initial shapes exhibits a nonmonotonic trend that is characterized by an initial increase leading to a maximum followed by a sharp decrease with decreasing shear rate. This anomalous behavior of the tank-treading frequency is shown to be due to a breathing-like dynamics of the capsule or cell that is characterized by a repeated emergence and absence of deep, crater-like dimples, and a large swinging motion. We further observe that the tumbling frequency exhibits a decreasing trend with increasing viscosity ratio that is in contrast to the theoretical result for the shape-preserving capsules and is due to the periodic deformation and preferential alignment of the capsules in the extensional quadrant of the flow.

Rheology of a dilute suspension of liquid-filled elastic capsules.

Rheology of a dilute suspension of liquid-filled elastic capsules in linear shear flow is studied by three-dimensional numerical simulations using a front-tracking method. This study is motivated by a recent discovery that a suspension of viscous vesicles exhibits a shear viscosity minimum when the vesicles undergo an unsteady vacillating-breathing dynamics at the threshold of a transition between the tank-treading and tumbling motions. Here we consider capsules of spherical resting shape for which only a steady tank-treading motion is observed. A comprehensive analysis of the suspension rheology is presented over a broad range of viscosity ratio (ratio of internal-to-external fluid viscosity), shear rate (or, capillary number), and capsule surface-area dilatation. We find a result that the capsule suspension exhibits a shear viscosity minimum at moderate values of the viscosity ratio, and high capillary numbers, even when the capsules are in a steady tank-treading motion. It is further observed that the shear viscosity minimum exists for capsules with area-dilating membranes but not for those with nearly incompressible membranes. Nontrivial results are also observed for the normal stress differences which are shown to decrease with increasing capillary number at high viscosity ratios. Such nontrivial results neither can be predicted by the small-deformation theory nor can be explained by the capsule geometry alone. Physical mechanisms underlying these results are studied by decomposing the particle stress tensor into a contribution due to the elastic stresses in the capsule membrane and a contribution due to the viscosity differences between the internal and suspending fluids. It is shown that the elastic contribution is shear-thinning, but the viscous contribution is shear thickening. The coupling between the capsule geometry and the elastic and viscous contributions is analyzed to explain the observed trends in the bulk rheology.

Dynamics of nonspherical capsules in shear flow.

Three-dimensional numerical simulations using a front-tracking method are presented on the dynamics of oblate shape capsules in linear shear flow by considering a broad range of viscosity contrast (ratio of internal-to-external fluid viscosity), shear rate (or capillary number), and aspect ratio. We focus specifically on the coupling between the shape deformation and orientation dynamics of capsules, and show how this coupling influences the transition from the tank-treading to tumbling motion. At low capillary numbers, three distinct modes of motion are identified: a swinging or oscillatory (OS) mode at a low viscosity contrast in which the inclination angle theta(t) oscillates but always remains positive; a vacillating-breathing (VB) mode at a moderate viscosity contrast in which theta(t) periodically becomes positive and negative, but a full tumbling does not occur; and a pure tumbling mode (TU) at a higher viscosity contrast. At higher capillary numbers, three types of transient motions occur, in addition to the OS and TU modes, during which the capsule switches from one mode to the other as (i) VB to OS, (ii) TU to VB to OS, and (iii) TU to VB. Phase diagrams showing various regimes of capsule dynamics are presented. For all modes of motion (OS, VB, and TU), a large-amplitude oscillation in capsule shape and a strong coupling between the shape deformation and orientation dynamics are observed. It is shown that the coupling between the shape deformation and orientation is the strongest in the VB mode, and hence at a moderate viscosity contrast, for which the amplitude of shape deformation reaches its maximum. The numerical results are compared with the theories of Keller and Skalak, and Skotheim and Secomb. Significant departures from the two theories are discussed and related to the strong coupling between the shape deformation, inclination, and transition dynamics.

Three-dimensional computational modeling of multiple deformable cells flowing in microvessels.

Three-dimensional (3D) computational modeling and simulation are presented on the motion of a large number of deformable cells in microchannels. The methodology is based on an immersed boundary method, and the cells are modeled as liquid-filled elastic capsules. The model retains two important features of the blood flow in the microcirculation, that is, the particulate nature of blood and deformation of the erythrocytes. The tank-treading and tumbling motion and the lateral migration, as observed for erythrocytes in dilute suspension, are briefly discussed. We then present results on the motion of multiple cells in semidense suspension and study how their collective dynamics leads to various physiologically relevant processes such as the development of the cell-free layer and the Fahraeus-Lindqvist effect. We analyze the 3D trajectory and velocity fluctuations of individual cell in the suspension and the plug-flow velocity profile as functions of the cell deformability, hematocrit, and vessel size. The numerical results allow us to directly obtain various microrheological data, such as the width of the cell-free layer, and the variation in the apparent blood viscosity and hematocrit over the vessel cross section. We then use these results to calculate the core and plasma-layer viscosity and show that the two-phase (or core-annular) model of blood flow in microvessels underpredicts the blood velocity obtained in the simulations by as much as 40%. Based on a posteriori analysis of the simulation data, we develop a three-layer model of blood flow by taking into consideration the smooth variation in viscosity and hematocrit across the interface of the cell-free layer and the core. We then show that the blood velocity predicted by the three-layer model agrees very well with that obtained from the simulations.