Blood Flow in Microvascular Networks

Morphology, repulsion, and ordering of red blood cells in viscoelastic flows under confinement

Red blood cells (RBC), the primary carriers of oxygen in the body, play a crucial role across several biomedical applications, while also being an essential model system of a deformable object in the microfluidics and soft matter fields. However, RBC behavior in viscoelastic liquids, which holds promise in enhancing microfluidic diagnostic applications, remains poorly studied. We here show that using viscoelastic polymer solutions as a suspending carrier causes changes in the clustering and shape of flowing RBC in microfluidic flows when compared to a standard Newtonian suspending liquid. Additionally, when the local RBC concentration increases to a point where hydrodynamic interactions take place, we observe the formation of equally-spaced RBC structures, resembling the viscoelasticity-driven ordered particles observed previously in the literature, thus providing the first experimental evidence of viscoelasticity-driven cell ordering. The observed RBC ordering, unaffected by polymer molecular architecture, persists as long as the surrounding medium exhibits shear-thinning, viscoelastic properties. Complementary numerical simulations reveal that viscoelasticity-induced repulsion between RBCs leads to equidistant structures, with shear-thinning modulating this effect. Our results open the way for the development of new biomedical technologies based on the use of viscoelastic liquids while also clarifying fundamental aspects related to multibody hydrodynamic interactions in viscoelastic microfluidic flows.

Confinement effect on the microcapillary flow and shape of red blood cells

The ability to change shape is essential for the proper functioning of red blood cells (RBCs) within the microvasculature. The shape of RBCs significantly influences blood flow and has been employed in microfluidic lab-on-a-chip devices, serving as a diagnostic biomarker for specific pathologies and enabling the assessment of RBC deformability. While external flow conditions, such as the vessel size and the flow velocity, are known to impact microscale RBC flow, our comprehensive understanding of how their shape-adapting ability is influenced by channel confinement in biomedical applications remains incomplete. This study explores the impact of various rectangular and square channels, each with different confinement and aspect ratios, on the in vitro RBC flow behavior and characteristic shapes. We demonstrate that rectangular microchannels, with a height similar to the RBC diameter in combination with a confinement ratio exceeding 0.9, are required to generate distinctive well-defined croissant and slipper-like RBC shapes. These shapes are characterized by their equilibrium positions in the channel cross section, and we observe a strong elongation of both stable shapes in response to the shear rate across the different channels. Less confined channel configurations lead to the emergence of unstable other shape types that display rich shape dynamics. Our work establishes an experimental framework to understand the influence of channel size on the single-cell flow behavior of RBCs, providing valuable insights for the design of biomicrofluidic single-cell analysis applications.

Erythrocyte metabolism

Our aim is to present an updated overview of the erythrocyte metabolism highlighting its richness and complexity. We have manually collected and connected the available biochemical pathways and integrated them into a functional metabolic map. The focus of this map is on the main biochemical pathways consisting of glycolysis, the pentose phosphate pathway, redox metabolism, oxygen metabolism, purine/nucleoside metabolism, and membrane transport. Other recently emerging pathways are also curated, like the methionine salvage pathway, the glyoxalase system, carnitine metabolism, and the lands cycle, as well as remnants of the carboxylic acid metabolism. An additional goal of this review is to present the dynamics of erythrocyte metabolism, providing key numbers used to perform basic quantitative analyses. By synthesizing experimental and computational data, we conclude that glycolysis, pentose phosphate pathway, and redox metabolism are the foundations of erythrocyte metabolism. Additionally, the erythrocyte can sense oxygen levels and oxidative stress adjusting its mechanics, metabolism, and function. In conclusion, fine-tuning of erythrocyte metabolism controls one of the most important biological processes, that is, oxygen loading, transport, and delivery.

Cell-free layer development and spatial organization of healthy and rigid red blood cells in a microfluidic bifurcation

Bifurcations and branches in the microcirculation dramatically affect blood flow as they determine the spatiotemporal organization of red blood cells (RBCs). Such changes in vessel geometries can further influence the formation of a cell-free layer (CFL) close to the vessel walls. Biophysical cell properties, such as their deformability, which is impaired in various diseases, are often thought to impact blood flow and affect the distribution of flowing RBCs. This study investigates the flow behavior of healthy and artificially hardened RBCs in a bifurcating microfluidic T-junction. We determine the RBC distribution across the channel width at multiple positions before and after the bifurcation. Thus, we reveal distinct focusing profiles in the feeding mother channel for rigid and healthy RBCs that dramatically impact the cell organization in the successive daughter channels. Moreover, we experimentally show how the characteristic asymmetric CFLs in the daughter vessels develop along their flow direction. Complimentary numerical simulations indicate that the buildup of the CFL is faster for healthy than for rigid RBCs. Our results provide fundamental knowledge to understand the partitioning of rigid RBC as a model of cells with pathologically impaired deformability in complex in vitro networks.

Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation

A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.

Anomalous Diffusion of Deformable Particles in a Honeycomb Network

Transport of deformable particles in a honeycomb network is studied numerically. It is shown that the particle deformability has a strong impact on their distribution in the network. For sufficiently soft particles, we observe a short memory behavior from one bifurcation to the next, and the overall behavior consists in a random partition of particles, exhibiting a diffusionlike transport. On the contrary, stiff enough particles undergo a biased distribution whereby they follow a deterministic partition at bifurcations, due to long memory. This leads to a lateral ballistic drift in the network at small concentration and anomalous superdiffusion at larger concentration, even though the network is ordered. A further increase of concentration enhances particle-particle interactions which shorten the memory effect, turning the particle anomalous diffusion into a classical diffusion. We expect the drifting and diffusive regime transition to be generic for deformable particles.

A proof-of-concept pipeline to guide evaluation of tumor tissue perfusion by dynamic contrast-agent imaging: Direct simulation and inverse tracer-kinetic procedures

Dynamic contrast-enhanced (DCE) perfusion imaging has shown great potential to non-invasively assess cancer development and its treatment by their characteristic tissue signatures. Different tracer kinetics models are being applied to estimate tissue and tumor perfusion parameters from DCE perfusion imaging. The goal of this work is to provide an in silico model-based pipeline to evaluate how these DCE imaging parameters may relate to the true tissue parameters. As histology data provides detailed microstructural but not functional parameters, this work can also help to better interpret such data. To this aim in silico vasculatures are constructed and the spread of contrast agent in the tissue is simulated. As a proof of principle we show the evaluation procedure of two tracer kinetic models from in silico contrast-agent perfusion data after a bolus injection. Representative microvascular arterial and venous trees are constructed in silico. Blood flow is computed in the different vessels. Contrast-agent input in the feeding artery, intra-vascular transport, intra-extravascular exchange and diffusion within the interstitial space are modeled. From this spatiotemporal model, intensity maps are computed leading to in silico dynamic perfusion images. Various tumor vascularizations (architecture and function) are studied and show spatiotemporal contrast imaging dynamics characteristic of in vivo tumor morphotypes. The Brix II also called 2CXM, and extended Tofts tracer-kinetics models common in DCE imaging are then applied to recover perfusion parameters that are compared with the ground truth parameters of the in silico spatiotemporal models. The results show that tumor features can be well identified for a certain permeability range. The simulation results in this work indicate that taking into account space explicitly to estimate perfusion parameters may lead to significant improvements in the perfusion interpretation of the current tracer-kinetics models.

Shear stress and intravascular pressure effects on vascular dynamics: two-phase blood flow in elastic microvessels accounting for the passive stresses

We study the steady hemodynamics in physiological elastic microvessels proposing an advanced fluid-structure interaction model. The arteriolar tissue is modeled as a two-layer fiber-reinforced hyperelastic material representing its Media and Adventitia layers. The constitutive model employed (Holzapfel et al. in J Elast 61:1-48, 2000) is parametrized via available data on stress-strain experiments for arterioles. The model is completed by simulating the blood/plasma flow in the lumen, using the thixotropic elasto-viscoplastic model in its core, and the linear Phan-Thien and Tanner viscoelastic model in its annular part. The Cell-Free Layer (CFL) and the Fåhraeus and Fåhraeus-Lindqvist effects are considered via analytical expressions based on experimental data (Giannokostas et al. in Materials (Basel) 14:367, 2021b). The coupling between tissue deformation and blood flow is achieved through the experimentally verified pressure-shear hypothesis (Pries et al. Circ Res 77:1017-1023, 1995). Our calculations confirm that the increase in the reference inner radius produces larger expansion. Also, by increasing the intraluminal pressure, the thinning of the walls is more pronounced and it may reach 40% of the initial thickness. Comparing our predictions with those in rigid-wall microtubes, we conclude that apart from the vital importance of vasodilation, there is an up to 25% reduction in wall shear stress. The passive vasodilation contributes to the decrease in the tissue stress fields and affects the hemodynamic features such as the CFL thickness, reducing the plasma layer when blood flows in vessels with elastic walls, in quantitative agreement with previous experiments. Our calculations verify the correctness of the pressure-shear hypothesis but not that of the Laplace law.

Functional implications of microvascular heterogeneity for oxygen uptake and utilization

In the vascular system, an extensive network structure provides convective and diffusive transport of oxygen to tissue. In the microcirculation, parameters describing network structure, blood flow, and oxygen transport are highly heterogeneous. This heterogeneity can strongly affect oxygen supply and organ function, including reduced oxygen uptake in the lung and decreased oxygen delivery to tissue. The causes of heterogeneity can be classified as extrinsic or intrinsic. Extrinsic heterogeneity refers to variations in oxygen demand in the systemic circulation or oxygen supply in the lungs. Intrinsic heterogeneity refers to structural heterogeneity due to stochastic growth of blood vessels and variability in flow pathways due to geometric constraints, and resulting variations in blood flow and hematocrit. Mechanisms have evolved to compensate for heterogeneity and thereby improve oxygen uptake in the lung and delivery to tissue. These mechanisms, which involve long-term structural adaptation and short-term flow regulation, depend on upstream responses conducted along vessel walls, and work to redistribute flow and maintain blood and tissue oxygenation. Mathematically, the variance of a functional quantity such as oxygen delivery that depends on two or more heterogeneous variables can be reduced if one of the underlying variables is controlled by an appropriate compensatory mechanism. Ineffective regulatory mechanisms can result in poor oxygen delivery even in the presence of adequate overall tissue perfusion. Restoration of endothelial function, and specifically conducted responses, should be considered when addressing tissue hypoxemia and organ failure in clinical settings.

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

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

Continuum microhaemodynamics modelling using inverse rheology

Modelling blood flow in microvascular networks is challenging due to the complex nature of haemorheology. Zero- and one-dimensional approaches cannot reproduce local haemodynamics, and models that consider individual red blood cells (RBCs) are prohibitively computationally expensive. Continuum approaches could provide an efficient solution, but dependence on a large parameter space and scarcity of experimental data for validation has limited their application. We describe a method to assimilate experimental RBC velocity and concentration data into a continuum numerical modelling framework. Imaging data of RBCs were acquired in a sequentially bifurcating microchannel for various flow conditions. RBC concentration distributions were evaluated and mapped into computational fluid dynamics simulations with rheology prescribed by the Quemada model. Predicted velocities were compared to particle image velocimetry data. A subset of cases was used for parameter optimisation, and the resulting model was applied to a wider data set to evaluate model efficacy. The pre-optimised model reduced errors in predicted velocity by 60% compared to assuming a Newtonian fluid, and optimisation further reduced errors by 40%. Asymmetry of RBC velocity and concentration profiles was demonstrated to play a critical role. Excluding asymmetry in the RBC concentration doubled the error, but excluding spatial distributions of shear rate had little effect. This study demonstrates that a continuum model with optimised rheological parameters can reproduce measured velocity if RBC concentration distributions are known a priori. Developing this approach for RBC transport with more network configurations has the potential to provide an efficient approach for modelling network-scale haemodynamics.

Cross-talk between red blood cells and plasma influences blood flow and omics phenotypes in severe COVID-19

Coronavirus disease 2019 (COVID-19) is caused by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) and can affect multiple organs, among which is the circulatory system. Inflammation and mortality risk markers were previously detected in COVID-19 plasma and red blood cells (RBCs) metabolic and proteomic profiles. Additionally, biophysical properties, such as deformability, were found to be changed during the infection. Based on such data, we aim to better characterize RBC functions in COVID-19. We evaluate the flow properties of RBCs in severe COVID-19 patients admitted to the intensive care unit by using microfluidic techniques and automated methods, including artificial neural networks, for an unbiased RBC analysis. We find strong flow and RBC shape impairment in COVID-19 samples and demonstrate that such changes are reversible upon suspension of COVID-19 RBCs in healthy plasma. Vice versa, healthy RBCs resemble COVID-19 RBCs when suspended in COVID-19 plasma. Proteomics and metabolomics analyses allow us to detect the effect of plasma exchanges on both plasma and RBCs and demonstrate a new role of RBCs in maintaining plasma equilibria at the expense of their flow properties. Our findings provide a framework for further investigations of clinical relevance for therapies against COVID-19 and possibly other infectious diseases.

Fluid and protein exchange in microvascular networks: Importance of modelling heterogeneity in geometrical and biophysical properties

KEY POINTS

Microvascular network architecture defines coupling of fluid and protein exchange. Network arrangements markedly reduce capillary hydrostatic pressures and resting fluid movement at the same time as increasing the capacity for change The presence of vascular remodelling or angiogenesis puts constraints of network behaviour The sites of fluid and protein exchange can be segregated to different portions of the network Although there is a net filtration of fluid from a network of exchange vessels, there are specific areas where fluid moves into the circulation (reabsorption) and, when protein is moving into tissue, the amount is insufficient under basal conditions to result in changes in oncotic pressure.

ABSTRACT

Integration of functional results obtained across scales, from chemical signalling to the whole organism, is a daunting task requiring the marriage of experimental data with mathematical modelling. In the present study, a novel coupled computational fluid dynamics model is developed incorporating fluid and protein transport using measurements in an in vivo frog (Rana pipiens) mesenteric microvascular network. The influences of network architecture and exchange are explored systematically under the common assumptions of structurally and functionally identical microvessels (Homogeneous Scenario) or microvessels classified by position in flow (Class Uniform Scenario), which are compared with realistic microvascular network components (Heterogeneous Scenario). The model incorporates ten quantities that vary within a microvessel; pressure boundary conditions are calibrated against experimental measurements. The Homogeneous Scenario standard model showed that assuming a single 'typical' capillary hides the influence of vessels arranged into a network architecture, where capillary hydrostatic pressures (pT ) are reduced, resulting in both a nonuniform distribution of blood flow and reduced volume flow rate (Jf,T ). In the Class Uniform Scenario pT was further attenuated to produce a ∼60% reduction in Jf,T . Finally, the Heterogeneous Scenario, incorporating measures of individual vessel surface area, demonstrates additional lowering of pT from inlet values favouring a >70% reduction of Jf,T in the face of a ∼120% increase in protein movement into the tissues relative to the Homogeneous Scenario. Beyond the impacts of network architecture, an unanticipated finding was the influence of a blind-end microvessel on model convergence, indicating a profound influence of the largely unexplored dynamics of vascular remodelling on tissue perfusion.

Association between erythrocyte dynamics and vessel remodelling in developmental vascular networks

Sprouting angiogenesis is an essential vascularization mechanism consisting of sprouting and remodelling. The remodelling phase is driven by rearrangements of endothelial cells (ECs) within the post-sprouting vascular plexus. Prior work has uncovered how ECs polarize and migrate in response to flow-induced wall shear stress (WSS). However, the question of how the presence of erythrocytes (widely known as red blood cells (RBCs)) and their impact on haemodynamics affect vascular remodelling remains unanswered. Here, we devise a computational framework to model cellular blood flow in developmental mouse retina. We demonstrate a previously unreported highly heterogeneous distribution of RBCs in primitive vasculature. Furthermore, we report a strong association between vessel regression and RBC hypoperfusion, and identify plasma skimming as the driving mechanism. Live imaging in a developmental zebrafish model confirms this association. Taken together, our results indicate that RBC dynamics are fundamental to establishing the regional WSS differences driving vascular remodelling via their ability to modulate effective viscosity.

The capillary fascicle in skeletal muscle: Structural and functional physiology of RBC distribution in capillary networks

KEY POINTS

The capillary module, consisting of parallel capillaries from arteriole to venule, is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres. Using intravital video microscopy of resting extensor digitorum longus muscle in rats, we demonstrated the capillary fascicle as a series of interconnected modules forming continuous columns that align naturally with the dimensions of the muscle fascicle. We observed structural heterogeneity for module topology, and functional heterogeneity in space and time for capillary-red blood cell (RBC) haemodynamics within a module and between modules. We found that module RBC haemodynamics were independent of module resistance, providing direct evidence for microvascular flow regulation at the level of the capillary module. The capillary fascicle is an updated paradigm for characterizing blood flow and RBC distribution in skeletal muscle capillary networks.

ABSTRACT

Capillary networks are the fundamental site of oxygen exchange in the microcirculation. The capillary module (CM), consisting of parallel capillaries from terminal arteriole (TA) to post-capillary venule (PCV), is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres, requiring co-ordination from numerous modules. It has previously been recognized that TAs and PCVs interact with multiple CMs, creating interconnected networks. Using label-free intravital video microscopy of resting extensor digitorum longus muscle in rats, we found that these networks form continuous columns of linked CMs spanning thousands of microns, herein denoted as the capillary fascicle (CF); this structure aligns naturally with the dimensions of the muscle fascicle. We measured capillary-red blood cell (RBC) haemodynamics and module topology (n = 9 networks, 327 modules, 1491 capillary segments). The average module had length 481 μm, width 157 μm and 9.51 parallel capillaries. We observed structural heterogeneity for CM topology, and functional heterogeneity in space and time for capillary-RBC haemodynamics within a module and between modules. There was no correlation between capillary RBC velocity and lineal density. A passive inverse relationship between module length and haemodynamics was remarkably absent, providing direct evidence for microvascular flow regulation at the level of the CM. In summary, the CF is an updated paradigm for characterizing RBC distribution in skeletal muscle, and strengthens the theory of capillary networks as major contributors to the signal that regulates capillary perfusion.

Distinct roles of red-blood-cell-derived and wall-derived mechanisms in metabolic regulation of blood flow

OBJECTIVE

A theoretical model is used to analyze combinations of RBC-derived and wall-derived (RBC-independent) mechanisms for metabolic blood flow regulation, with regard to their oxygen transport properties.

METHODS

Heterogeneous microvascular network structures are derived from observations in rat mesentery and hamster cremaster. The effectiveness of metabolic blood flow regulation using combinations of RBC-dependent and RBC-independent mechanisms is simulated in these networks under conditions of reduced oxygen delivery and increased oxygen demand.

RESULTS

Metabolic regulation by a wall-derived mechanism results in higher predicted total blood flow rate and number of flowing vessels, and lower tissue hypoxic fraction, than regulation by combinations of RBC-derived and wall-derived signals. However, a combination of RBC-derived and wall-derived signals results in a higher predicted median tissue P_O2 than either mechanism acting alone.

CONCLUSIONS

Model results suggest complementary roles for RBC-derived and wall-derived mechanisms of metabolic flow regulation, with the wall-derived mechanism responsible for avoiding hypoxia, and the RBC-derived mechanism responsible for maintaining P_O2 levels high enough for optimal tissue function.

Simulation of oxygen transport and estimation of tissue perfusion in extensive microvascular networks: Application to cerebral cortex

Advanced imaging techniques have made available extensive three-dimensional microvascular network structures. Simulation of oxygen transport by such networks requires information on blood flow rates and oxygen levels in vessels crossing boundaries of the imaged region, which is difficult to obtain experimentally. Here, a computational method is presented for estimating blood flow rates, oxygen levels, tissue perfusion and oxygen extraction, based on incomplete boundary conditions. Flow rates in all segments are estimated using a previously published method. Vessels crossing the region boundary are classified as arterioles, capillaries or venules. Oxygen levels in inflowing capillaries are assigned based on values in outflowing capillaries, and similarly for venules. Convective and diffusive oxygen transport is simulated. Contributions of each vessel to perfusion are computed in proportion to the decline in oxygen concentration along that vessel. For a vascular network in the mouse cerebral cortex, predicted tissue oxygen levels show a broad distribution, with 99% of tissue in the range of 20 to 80 mmHg under reference conditions, and steep gradients near arterioles. Perfusion and extraction estimates are consistent with experimental values. A 30% reduction in perfusion or a 30% increase in oxygen demand, relative to reference levels, is predicted to result in tissue hypoxia.

Simulation of a detoxifying organ function: Focus on hemodynamics modeling and convection-reaction numerical simulation in microcirculatory networks

When modeling a detoxifying organ function, an important component is the impact of flow on the metabolism of a compound of interest carried by the blood. We here study the effects of red blood cells (such as the Fahraeus-Lindqvist effect and plasma skimming) on blood flow in typical microcirculatory components such as tubes, bifurcations and entire networks, with particular emphasis on the liver as important representative of detoxifying organs. In one of the plasma skimming models, under certain conditions, oscillations between states are found and analyzed in a methodical study to identify their causes and influencing parameters. The flow solution obtained is then used to define the velocity at which a compound would be transported. A convection-reaction equation is studied to simulate the transport of a compound in blood and its uptake by the surrounding cells. Different types of signal sharpness have to be handled depending on the application to address different temporal compound concentration profiles. To permit executing the studied models numerically stable and accurate, we here extend existing transport schemes to handle converging bifurcations, and more generally multi-furcations. We study the accuracy of different numerical schemes as well as the effect of reactions and of the network itself on the bolus shape. Even though this study is guided by applications in liver micro-architecture, the proposed methodology is general and can readily be applied to other capillary network geometries, hence to other organs or to bioengineered network designs.

Effects of impaired microvascular flow regulation on metabolism-perfusion matching and organ function

Impaired tissue oxygen delivery is a major cause of organ damage and failure in critically ill patients, which can occur even when systemic parameters, including cardiac output and arterial hemoglobin saturation, are close to normal. This review addresses oxygen transport mechanisms at the microcirculatory scale, and how hypoxia may occur in spite of adequate convective oxygen supply. The structure of the microcirculation is intrinsically heterogeneous, with wide variations in vessel diameters and flow pathway lengths, and consequently also in blood flow rates and oxygen levels. The dynamic processes of structural adaptation and flow regulation continually adjust microvessel diameters to compensate for heterogeneity, redistributing flow according to metabolic needs to ensure adequate tissue oxygenation. A key role in flow regulation is played by conducted responses, which are generated and propagated by endothelial cells and signal upstream arterioles to dilate in response to local hypoxia. Several pathophysiological conditions can impair local flow regulation, causing hypoxia and tissue damage leading to organ failure. Therapeutic measures targeted to systemic parameters may not address or may even worsen tissue oxygenation at the microvascular level. Restoration of tissue oxygenation in critically ill patients may depend on restoration of endothelial cell function, including conducted responses.

Dynamic tracking of microvascular hemoglobin content for continuous perfusion monitoring in the intensive care unit: pilot feasibility study

Purpose: There is a need for bedside methods to monitor oxygen delivery in the microcirculation. Near-infrared spectroscopy commonly measures tissue oxygen saturation, but does not reflect the time-dependent variability of microvascular hemoglobin content (MHC) that attempts to match oxygen supply with demand. The objective of this study is to determine the feasibility of MHC monitoring in critically ill patients using high-resolution near-infrared spectroscopy to assess perfusion in the peripheral microcirculation. Methods: Prospective observational cohort of 36 patients admitted within 48 h at a tertiary intensive care unit. Perfusion was measured on the quadriceps, biceps, and/or deltoid, using the temporal change in optical density at the isosbestic wavelength of hemoglobin (798 nm). Continuous wavelet transform was applied to the hemoglobin signal to delineate frequency ranges corresponding to physiological oscillations in the cardiovascular system. Results: 31/36 patients had adequate signal quality for analysis, most commonly affected by motion artifacts. MHC signal demonstrates inter-subject heterogeneity in the cohort, indicated by different patterns of variability and frequency composition. Signal characteristics were concordant between muscle groups in the same patient, and correlated with systemic hemoglobin levels and oxygen saturation. Signal power was lower for patients receiving vasopressors, but not correlated with mean arterial pressure. Mechanical ventilation directly impacts MHC in peripheral tissue. Conclusion: MHC can be measured continuously in the ICU with high-resolution near-infrared spectroscopy, and reflects the dynamic variability of hemoglobin distribution in the microcirculation. Results suggest this novel hemodynamic metric should be further evaluated for diagnosing microvascular dysfunction and monitoring peripheral perfusion.

Local vs. Global Blood Flow Modulation in Artificial Microvascular Networks: Effects on Red Blood Cell Distribution and Partitioning

Our understanding of cerebral blood flow (CBF) regulation during functional activation is still limited. Alongside with the accepted role of smooth muscle cells in controlling the arteriolar diameter, a new hypothesis has been recently formulated suggesting that CBF may be modulated by capillary diameter changes mediated by pericytes. In this study, we developed in vitro microvascular network models featuring a valve enabling the dilation of a specific micro-channel. This allowed us to investigate the non-uniform red blood cell (RBC) partitioning at microvascular bifurcations (phase separation) and the hematocrit distribution at rest and for two scenarios modeling capillary and arteriolar dilation. RBC partitioning showed similar phase separation behavior during baseline and activation. Results indicated that the RBCs at diverging bifurcations generally enter the high-flow branch (classical partitioning). Inverse behavior (reverse partitioning) was observed for skewed hematocrit profiles in the parent vessel of bifurcations, especially for high RBC velocity (i.e., arteriolar activation). Moreover, results revealed that a local capillary dilation, as it may be mediated in vivo by pericytes, led to a localized increase of RBC flow and a heterogeneous hematocrit redistribution within the whole network. In case of a global increase of the blood flow, as it may be achieved by dilating an arteriole, a homogeneous increase of RBC flow was observed in the whole network and the RBCs were concentrated along preferential pathways. In conclusion, overall increase of RBC flow could be obtained by arteriolar and capillary dilation, but only capillary dilation was found to alter the perfusion locally and heterogeneously.

The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex

The functioning of cerebral cortex depends on adequate tissue oxygenation. MRI-based techniques allow estimation of blood oxygen levels, tissue perfusion, and oxygen consumption rate (CMRO₂), but do not directly measure partial pressure of oxygen (PO₂) in tissue. To address the estimation of tissue PO₂, the oxygen mass transfer coefficient (KTO₂) is here defined as the CMRO₂ divided by the difference in spatially averaged PO₂ between blood and tissue, and is estimated by analyzing Krogh-cylinder type models. Resistance to radial diffusion of oxygen from microvessels to tissue is distributed within vessels and in the extravascular tissue. The value of KTO₂ is shown to depend strongly on vascular length density and also on microvessel tube hematocrits and diameters, but to be insensitive to blood flow rate and to transient changes in flow or oxygen consumption. Estimated values of KTO₂ are higher than implied by previous studies, implying smaller declines in PO₂ from blood to tissue. Average tissue PO₂ can be estimated from MRI-based measurements as average blood PO₂ minus the product of KTO₂ and CMRO₂. For oxygen consumption rates and vascular densities typical of mouse cortex, the predicted difference between average blood and tissue PO₂ is about 10 mmHg.

Spatio-temporal investigation of doxorubicin in a 3D heterogeneous tumor microenvironment

Doxorubicin (Adriamycin) is a type of chemotherapy drugs using to treat diseases such as breast cancer, bladder cancer, Kaposi's sarcoma, and lymphoma. Additionally, it can be first prescribed to reduce tumor size. The ratio of killed cells is varied depending on the clinical dosage regimen. Hence, the exact dosage of the drug must be administered to prevent the toxicity that could impair the immune system or leading to heart failure. In the present study, a 3D heterogeneous geometry with a solid tumor and healthy tissue is modeled for the drug delivery investigation. At the first stage, the physical properties of tumor microenvironment are obtained. Then, a five-compartmental model is used to evaluate the free, bound and internalized drug via the convection-diffusion-reaction (CDR) equation. Results are shown that a percent increase of 37.5% and 47.1% for the 75 mg m^-2 to 50 mg m^-2 in the AUC of bound drug and free drug concentration, respectively. The free and bound drugs have the same trend in time showing an apex at the earliest time of injection and then drops to the lowest amount about 9 hours after treatment. Moreover, the internalized drug has a different trend in time. It increases and reaches a constant amount of drug concentration in the cells. Besides, the fraction of surviving cells is also evaluated for both tumor and healthy tissues showing a 88.62% and 97.76% of surviving cells with 50 mg m^-2 of doxorubicin after the treatment, respectively. This model is developed to predict the heterogenous distribution of doxorubicin in three different drug concentrations for patient-specific drug treatment. This model could be used for different drugs to show the rate of perfusion and the ability to kill cancerous cells regarding their different doses and toxicity effects.

3D Sugar Printing of Networks Mimicking the Vasculature

The vasculature plays a central role as the highway of the body, through which nutrients and oxygen as well as biochemical factors and signals are distributed by blood flow. Therefore, understanding the flow and distribution of particles inside the vasculature is valuable both in healthy and disease-associated networks. By creating models that mimic the microvasculature fundamental knowledge can be obtained about these parameters. However, microfabrication of such models remains a challenging goal. In this paper we demonstrate a promising 3D sugar printing method that is capable of recapitulating the vascular network geometry with a vessel diameter range of 1 mm down to 150 µm. For this work a dedicated 3D printing setup was built that is capable of accurately printing the sugar glass material with control over fibre diameter and shape. By casting of printed sugar glass networks in PDMS and dissolving the sugar glass, perfusable networks with circular cross-sectional channels are obtained. Using particle image velocimetry, analysis of the flow behaviour was conducted showing a Poisseuille flow profile inside the network and validating the quality of the printing process.

Resonances in the response of fluidic networks inherent to the cooperation between elasticity and bifurcations

A global response function (GRF) of an elastic network is introduced as a generalization of the response function (RF) of a rigid network, relating the average flow along the network with the pressure difference at its extremes. The GRF can be used to explore the frequency behaviour of a fluid confined in a tree-like symmetric elastic network in which vessels bifurcate into identical vessels. We study such dynamic response for elastic vessel networks containing viscous fluids. We find that the bifurcation structure, inherent to tree-like networks, qualitatively changes the dynamic response of a single elastic vessel, and gives resonances at certain frequencies. This implies that the average flow throughout the network could be enhanced if the pulsatile forcing at the network's inlet were imposed at the resonant frequencies. The resonant behaviour comes from the cooperation between the bifurcation structure and the elasticity of the network, since the GRF has no resonances either for a single elastic vessel or for a rigid network. We have found that resonances shift to high frequencies as the system becomes more rigid. We have studied two different symmetric tree-like network morphologies and found that, while many features are independent of network morphology, particular details of the response are morphology dependent. Our results could have applications to some biophysical networks, for which the morphology could be approximated to a tree-like symmetric structure and a constant pressure at the outlet. The GRF for these networks is a characteristic of the system fluid-network, being independent of the dynamic flow (or pressure) at the network's inlet. It might therefore represent a good quantity to differentiate healthy vasculatures from those with a medical condition. Our results could also be experimentally relevant in the design of networks engraved in microdevices, since the limit of the rigid case is almost impossible to attain with the materials used in microfluidics and the condition of constant pressure at the outlet is often given by the atmospheric pressure.

Dynamic Changes in Microvascular Flow Conductivity and Perfusion After Myocardial Infarction Shown by Image-Based Modeling

Background Microcirculation is a decisive factor in tissue reperfusion inadequacy following myocardial infarction ( MI ). Nonetheless, experimental assessment of blood flow in microcirculation remains a bottleneck. We sought to model blood flow properties in coronary microcirculation at different time points after MI and to compare them with healthy conditions to obtain insights into alterations in cardiac tissue perfusion. Methods and Results We developed an image-based modeling framework that permitted feeding a continuum flow model with anatomical data previously obtained from the pig coronary microvasculature to calculate physiologically meaningful permeability tensors. The tensors encompassed the microvascular conductivity and were also used to estimate the arteriole-venule drop in pressure and myocardial blood flow. Our results indicate that the tensors increased in a bimodal pattern at infarcted areas on days 1 and 7 after MI while a nonphysiological decrease in arteriole-venule drop in pressure was observed; contrary, the tensors and the arteriole-venule drop in pressure on day 3 after MI , and in remote areas, were closer to values for healthy tissue. Myocardial blood flow calculated using the condition-dependent arteriole-venule drop in pressure decreased in infarcted areas. Last, we simulated specific modes of vascular remodeling, such as vasodilation, vasoconstriction, or pruning, and quantified their distinct impact on microvascular conductivity. Conclusions Our study unravels time- and region-dependent alterations of tissue perfusion related to the structural changes occurring in the coronary microvasculature due to MI . It also paves the way for conducting simulations in new therapeutic interventions in MI and for image-based microvascular modeling by applying continuum flow models in other biomedical scenarios.

Quantitative measurements of cerebral blood flow with near-infrared spectroscopy

We propose a new near-infrared spectroscopy (NIRS) method for quantitative measurements of cerebral blood flow (CBF). Because this method uses concepts of coherent hemodynamics spectroscopy (CHS), we identify this new method with the acronym NIRS-CHS. We tested this method on the prefrontal cortex of six healthy human subjects during mean arterial pressure (MAP) transients induced by the rapid deflation of pneumatic thigh cuffs. A comparison of CBF dynamics measured with NIRS-CHS and with diffuse correlation spectroscopy (DCS) showed a good agreement for characteristic times of the CBF transient. We also report absolute measurements of baseline CBF with NIRS-CHS (69 ± 6 ml/100g/min over the six subjects). NIRS-CHS can provide more accurate measurements of CBF with respect to previously reported NIRS surrogates of CBF.

Spatiotemporal distribution modeling of PET tracer uptake in solid tumors

OBJECTIVE

Distribution of PET tracer uptake is elaborately modeled via a general equation used for solute transport modeling. This model can be used to incorporate various transport parameters of a solid tumor such as hydraulic conductivity of the microvessel wall, transvascular permeability as well as interstitial space parameters. This is especially significant because tracer delivery and drug delivery to solid tumors are determined by similar underlying tumor transport phenomena, and quantifying the former can enable enhanced prediction of the latter.

METHODS

We focused on the commonly utilized FDG PET tracer. First, based on a mathematical model of angiogenesis, the capillary network of a solid tumor and normal tissues around it were generated. The coupling mathematical method, which simultaneously solves for blood flow in the capillary network as well as fluid flow in the interstitium, is used to calculate pressure and velocity distributions. Subsequently, a comprehensive spatiotemporal distribution model (SDM) is applied to accurately model distribution of PET tracer uptake, specifically FDG in this work, within solid tumors.

RESULTS

The different transport mechanisms, namely convention and diffusion from vessel to tissue and in tissue, are elaborately calculated across the domain of interest and effect of each parameter on tracer distribution is investigated. The results show the convection terms to have negligible effect on tracer transport and the SDM can be solved after eliminating these terms.

CONCLUSION

The proposed framework of spatiotemporal modeling for PET tracers can be utilized to comprehensively assess the impact of various parameters on the spatiotemporal distribution of PET tracers.

Microfluidic cell fragmentation for mechanical phenotyping of cancer cells

Circulating tumor cells (CTCs) shed from the primary tumor undergo significant fragmentation in the microvasculature, and very few escape to instigate metastases. Inspired by this in vivo behavior of CTCs, we report a microfluidic method to phenotype cancer cells based on their ability to arrest and fragment at a micropillar-based bifurcation. We find that in addition to cancer cell size, mechanical properties determine fragmentability. We observe that highly metastatic prostate cancer cells are more resistant to fragmentation than weakly metastatic cells, providing the first indication that metastatic CTCs can escape rupture and potentially initiate secondary tumors. Our method may thus be useful in identifying phenotypes that succumb to or escape mechanical trauma in microcirculation.

Structure-based algorithms for microvessel classification

Objective

Recent developments in high‐resolution imaging techniques have enabled digital reconstruction of three‐dimensional sections of microvascular networks down to the capillary scale. To better interpret these large data sets, our goal is to distinguish branching trees of arterioles and venules from capillaries.

Methods

Two novel algorithms are presented for classifying vessels in microvascular anatomical data sets without requiring flow information. The algorithms are compared with a classification based on observed flow directions (considered the gold standard), and with an existing resistance‐based method that relies only on structural data.

Results

The first algorithm, developed for networks with one arteriolar and one venular tree, performs well in identifying arterioles and venules and is robust to parameter changes, but incorrectly labels a significant number of capillaries as arterioles or venules. The second algorithm, developed for networks with multiple inlets and outlets, correctly identifies more arterioles and venules, but is more sensitive to parameter changes.

Conclusions

The algorithms presented here can be used to classify microvessels in large microvascular data sets lacking flow information. This provides a basis for analyzing the distinct geometrical properties and modelling the functional behavior of arterioles, capillaries, and venules.

Grading fluorescein angiograms in malarial retinopathy

BACKGROUND

Malarial retinopathy is an important finding in Plasmodium falciparum cerebral malaria, since it strengthens diagnostic accuracy, predicts clinical outcome and appears to parallel cerebral disease processes. Several angiographic features of malarial retinopathy have been described, but observations in different populations can only be reliably compared if consistent methodology is used to capture and grade retinal images. Currently no grading scheme exists for fluorescein angiographic features of malarial retinopathy.

METHODS

A grading scheme for fluorescein angiographic images was devised based on consensus opinion of clinicians and researchers experienced in malarial retinopathy in children and adults. Dual grading were performed with adjudication of admission fluorescein images from a large cohort of children with cerebral malaria.

RESULTS

A grading scheme is described and standard images are provided to facilitate future grading studies. Inter-grader agreement was >70 % for most variables. Intravascular filling defects are difficult to grade and tended to have lower inter-grader agreement (>57 %) compared to other features.

CONCLUSIONS

This grading scheme provides a consistent way to describe retinal vascular damage in paediatric cerebral malaria, and can facilitate comparisons of angiographic features of malarial retinopathy between different patient groups, and analysis against clinical outcomes. Inter-grader agreement is reasonable for the majority of angiographic signs. Dual grading with expert adjudication should be used to maximize accuracy.

The effects of transit time heterogeneity on brain oxygenation during rest and functional activation

The interpretation of regional blood flow and blood oxygenation changes during functional activation has evolved from the concept of 'neurovascular coupling', and hence the regulation of arteriolar tone to meet metabolic demands. The efficacy of oxygen extraction was recently shown to depend on the heterogeneity of capillary flow patterns downstream. Existing compartment models of the relation between tissue metabolism, blood flow, and blood oxygenation, however, typically assume homogenous microvascular flow patterns. To take capillary flow heterogeneity into account, we modeled the effect of capillary transit time heterogeneity (CTH) on the 'oxygen conductance' used in compartment models. We show that the incorporation of realistic reductions in CTH during functional hyperemia improves model fits to dynamic blood flow and oxygenation changes acquired during functional activation in a literature animal study. Our results support earlier observations that oxygen diffusion properties seemingly change during various physiologic stimuli, and posit that this phenomenon is related to parallel changes in capillary flow patterns. Furthermore, our results suggest that CTH must be taken into account when inferring brain metabolism from changes in blood flow- or blood oxygenation-based signals .

Using a classic paper by Robin Fahraeus and Torsten Lindqvist to teach basic hemorheology

"The viscosity of the blood in narrow capillary tubes" by Robin Fåhraeus and Torsten Lindqvist (Am J Physiol 96: 562-568, 1931) can be a valuable opportunity for teaching basic hemorheological principles in undergraduate cardiovascular physiology. This classic paper demonstrates that a progressive decline in apparent viscosity occurs when blood flows through glass capillary tubes of diminishing radius, which was later designated as the "Fåhraeus-Lindqvist effect." Subsequent studies have shown that apparent viscosity continues to decline at diameters that correspond to the arteriolar segments of the systemic vascular tree, where the majority of the total peripheral resistance resides and is actively regulated in vivo. The Fåhraeus-Lindqvist effect thus reduces microvascular resistance, thereby maintaining local tissue perfusion at a relatively lower blood pressure. The paper by Fåhraeus and Lindqvist can be used as a platform for a plenary discussion of these concepts as well as of the relationships among hematocrit, vessel diameter, red blood cell deformability, and resistance to blood flow and how these factors may affect the work of the heart.

Angiogenesis: an adaptive dynamic biological patterning problem

Formation of functionally adequate vascular networks by angiogenesis presents a problem in biological patterning. Generated without predetermined spatial patterns, networks must develop hierarchical tree-like structures for efficient convective transport over large distances, combined with dense space-filling meshes for short diffusion distances to every point in the tissue. Moreover, networks must be capable of restructuring in response to changing functional demands without interruption of blood flow. Here, theoretical simulations based on experimental data are used to demonstrate that this patterning problem can be solved through over-abundant stochastic generation of vessels in response to a growth factor generated in hypoxic tissue regions, in parallel with refinement by structural adaptation and pruning. Essential biological mechanisms for generation of adequate and efficient vascular patterns are identified and impairments in vascular properties resulting from defects in these mechanisms are predicted. The results provide a framework for understanding vascular network formation in normal or pathological conditions and for predicting effects of therapies targeting angiogenesis.

The blood vessels provide an efficient system for transport of substances to all parts of the body. They are capable of growing or regressing during development, in response to changing functional needs, and in disease states. This is achieved by structural adaptation, i.e. changes in the diameters and other characteristics of existing vessels, and by angiogenesis, i.e. growth of new blood vessels. Here, we address the question: How do the processes of structural adaptation and angiogenesis lead to the formation of organized vessel networks that can supply the changing needs of the tissue? We carried out theoretical simulations of network growth and adaptation, including vessel blood flows, oxygen transport to tissue, and the generation of a growth factor in low-oxygen regions, which stimulates angiogenesis by sprouting from existing vessels. We showed that the processes of over-abundant random angiogenesis together with structural adaptation including pruning of redundant vessels can generate adequate and efficient vessel networks that are capable of continuously adapting to changing tissue needs. Our work provides insight into the biological mechanisms that are essential for formation and maintenance of functional vessel networks, and may lead to new strategies for controlling blood vessel formation in diseases.

Topology and hemodynamics of the cortical cerebrovascular system

The cerebrovascular system continuously delivers oxygen and energy substrates to the brain, which is one of the organs with the highest basal energy requirement in mammals. Discontinuities in the delivery lead to fatal consequences for the brain tissue. A detailed understanding of the structure of the cerebrovascular system is important for a multitude of (patho-)physiological cerebral processes and many noninvasive functional imaging methods rely on a signal that originates from the vasculature. Furthermore, neurodegenerative diseases often involve the cerebrovascular system and could contribute to neuronal loss. In this review, we focus on the cortical vascular system. In the first part, we present the current knowledge of the vascular anatomy. This is followed by a theory of topology and its application to vascular biology. We then discuss possible interactions between cerebral blood flow and vascular topology, before summarizing the existing body of the literature on quantitative cerebrovascular topology.

'Null method' determination of drug biophase concentration

PK/PD modeling is enhanced by improvements in the accuracy of its metrics. For PK/PD modeling of drugs and biologics that interact with enzymes or receptors, the equilibrium constant of the interaction can provide critical insight. Methodologies such as radioliogand binding and isolated tissue preparations can provide estimates of the equilibrium constants (as the dissociation constant, K value) for drugs and endogenous ligands that interact with specific enzymes and receptors. However, an impediment to further precision for PK/PD modeling is that it remains a problem to convert the concentration of drug in bulk solution (A) into an estimate of receptor occupation, since A is not necessarily the concentration (C) of drug in the biophase that yields fractional binding from the law of mass action, viz., C/(C + K). In most experimental studies A is much larger than K, so the use of administered instead of biophase concentration gives fractional occupancies very close to unity. We here provide a simple way to obtain an estimate of the factor that converts the total drug concentration into the biophase concentration in isolated tissue preparation. Our approach is an extension of the now classic 'null method' introduced and applied by Furchgott to determination of drug-receptor dissociation constants.

MEK5 is activated by shear stress, activates ERK5 and induces KLF4 to modulate TNF responses in human dermal microvascular endothelial cells

OBJECTIVE

ECs lining arteries respond to LSS by suppressing pro-inflammatory changes, in part through the activation of MEK5, ERK5 and induction of KLF4. We examined if this anti-inflammatory pathway operates in human ECs lining microvessels, the principal site of inflammatory responses.

METHODS

We used immunofluorescence microscopy of human skin to assess ERK5 activation and KLF4 expression in HDMECs in situ. We applied LSS to or overexpressed MEK5/CA in cultured HDMECs and assessed gene expression by microarrays and qRT-PCR and protein expression by Western blotting. We assessed effects of MEK5/CA on TNF responses using qRT-PCR, FACS and measurements of HDMEC monolayer electrical resistance. We used siRNA knockdown to assess the role of ERK5 and KLF4 in these responses.

RESULTS

ERK5 phosphorylation and KLF4 expression is observed in HDMECs in situ. LSS activates ERK5 and induces KLF4 in cultured HDMECs. MEK5/CA-transduced HDMECs show activated ERK5 and increased KLF4, thrombomodulin, eNOS, and ICAM-1 expression. MEK5 induction of KLF4 is mediated by ERK5. MEK5/CA-transduced HDMECs are less responsive to TNF, an effect partly mediated by KLF4.

CONCLUSIONS

MEK5 activation by LSS inhibits inflammatory responses in microvascular ECs, in part through ERK5-dependent induction of KLF4.

Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations

Movement, deformation, and partitioning of mammalian red blood cells (RBCs) in diverging microvessel bifurcations are simulated using a two-dimensional, flexible-particle model. A set of viscoelastic elements represents the RBC membrane and the cytoplasm. Motion of isolated cells is considered, neglecting cell-to-cell interactions. Center-of-mass trajectories deviate from background flow streamlines due to migration of flexible cells towards the mother vessel centerline upstream of the bifurcation and due to flow perturbations caused by cell obstruction in the neighborhood of the bifurcation. RBC partitioning in the bifurcation is predicted by determining the RBC fraction entering each branch, for a given partition of total flow and for a given upstream distribution of RBCs. Typically, RBCs preferentially enter the higher-flow branch, leading to unequal discharge hematocrits in the downstream branches. This effect is increased by migration toward the centerline but decreased by the effects of obstruction. It is stronger for flexible cells than for rigid circular particles of corresponding size, and decreases with increasing parent vessel diameter. For unequally sized daughter vessels, partitioning is asymmetric, with RBCs tending to enter the smaller vessel. Partitioning is not significantly affected by branching angles. Model predictions are consistent with previous experimental results.