Blood flow in small tubes: quantifying the transition to the non-continuum regime

Establishing the distribution of cerebrovascular resistance using computational fluid dynamics and 4D flow MRI

Cerebrovascular resistance (CVR) regulates blood flow in the brain, but little is known about the vascular resistances of the individual cerebral territories. We present a method to calculate these resistances and investigate how CVR varies in the hemodynamically disturbed brain. We included 48 patients with stroke/TIA (29 with symptomatic carotid stenosis). By combining flow rate (4D flow MRI) and structural computed tomography angiography (CTA) data with computational fluid dynamics (CFD) we computed the perfusion pressures out from the circle of Willis, with which CVR of the MCA, ACA, and PCA territories was estimated. 56 controls were included for comparison of total CVR (tCVR). CVR were 33.8 ± 10.5, 59.0 ± 30.6, and 77.8 ± 21.3 mmHg s/ml for the MCA, ACA, and PCA territories. We found no differences in tCVR between patients, 9.3 ± 1.9 mmHg s/ml, and controls, 9.3 ± 2.0 mmHg s/ml (p = 0.88), nor in territorial CVR in the carotid stenosis patients between ipsilateral and contralateral hemispheres. Territorial resistance associated inversely to territorial brain volume (p < 0.001). These resistances may work as reference values when modelling blood flow in the circle of Willis, and the method can be used when there is need for subject-specific analysis.

Portable platform for leukocyte extraction from blood using sheath-free microfluidic DLD

Leukocyte count is routinely performed for diagnostic purposes and is rapidly emerging as a significant biomarker for a wide array of diseases. Additionally, leukocytes have demonstrated considerable promise in novel cell-based immunotherapies. However, the direct retrieval of leukocytes from whole blood is a significant challenge due to their low abundance compared to erythrocytes. Here, we introduce a microfluidic-based platform that isolates and recovers leukocytes from diluted whole blood in a single step. Our platform utilizes a novel, sheathless method to initially sediment and focus blood cells into a dense stream while flowing through a tubing before entering the microfluidic device. A hexagonal-shaped structure, patterned at the device's inlet, directs all the blood cells against the channel's outer walls. The focused cells are then separated based on their size using the deterministic lateral displacement (DLD) microfluidic technique. We evaluated various parameters that could influence leukocyte separation, including different focusing structures (assessed both computationally and experimentally), the orientation of the tubing-chip interface, the effects of blood sample hematocrit (dilution), and flow rate. Our device demonstrated the ability to isolate leukocytes from diluted blood with a separation efficiency of 100%, a recovery rate of 76%, and a purity of 80%, while maintaining a cell viability of 98%. The device operates for over 30 min at a flow rate of 2 μL min-1. Furthermore, we developed a handheld pressure controller to drive fluid flow, enhancing the operability of our platform outside of central laboratories and enabling near-patient testing. Our platform can be integrated with downstream cell-based assays and analytical methods that require high leukocyte purity (80%), ranging from cell counting to diagnostics and cell culture applications.

A homogenized two-phase computational framework for meso- and macroscale blood flow simulations

BACKGROUND AND OBJECTIVE

Owing to the complexity of physics linked with blood flow and its associated phenomena, appropriate modeling of the multi-constituent rheology of blood is of primary importance. To this effect, various kinds of computational fluid dynamic models have been developed, each with merits and limitations. However, when additional physics like thrombosis and embolization is included within the framework of these models, computationally efficient scalable translation becomes very difficult. Therefore, this paper presents a homogenized two-phase blood flow framework with similar characteristics to a single fluid model but retains the flow resolution of a classical two-fluid model. The presented framework is validated against four different sets of experiments.

METHODS

The two-phase model of blood presented here is based on the classical diffusion-flux framework. Diffusion flux models are known to be less computationally expensive than two-fluid multiphase models since the numerical implementation resembles single-phase flow models. Diffusion flux models typically use empirical slip velocity correlations to resolve the motion between phases. However, such correlations do not exist for blood. Therefore, a modified slip velocity equation is proposed, derived rigorously from the two-fluid governing equations. An additional drag law for red blood cells (RBCs) as a function of volume fraction is evaluated using a previously published cell-resolved solver. A new hematocrit-dependent expression for lift force on RBCs is proposed. The final governing equations are discretized and solved using the open-source software OpenFOAM.

RESULTS

The framework is validated against four sets of experiments: (i) flow through a rectangular microchannel to validate RBC velocity profiles against experimental measurements and compare computed hematocrit distributions against previously reported simulation results (ii) flow through a sudden expansion microchannel for comparing experimentally obtained contours of hematocrit distributions and normalized cell-free region length obtained at different flowrates and inlet hematocrits, (iii) flow through two hyperbolic channels to evaluate model predictions of cell-free layer thickness, and (iv) flow through a microchannel that mimics crevices of a left ventricular assist device to predict hematocrit distributions observed experimentally. The simulation results exhibit good agreement with the results of all four experiments.

CONCLUSION

The computational framework presented in this paper has the advantage of resolving the multiscale physics of blood flow while still leveraging numerical techniques used for solving single-phase flows. Therefore, it becomes an excellent candidate for addressing more complicated problems related to blood flow, such as modeling mechanical entrapment of RBCs within blood clots, predicting thrombus composition, and visualizing clot embolization.

Single-cell characterization of deformation and dynamics of mesenchymal stem cells in microfluidic systems: A computational study

Understanding the homing dynamics of individual mesenchymal stem cells (MSCs) in physiologically relevant microenvironments is crucial for improving the efficacy of MSC-based therapies for therapeutic and targeting purposes. This study investigates the passive homing behavior of individual MSCs in micropores that mimic interendothelial clefts through predictive computational simulations informed by previous microfluidic experiments. Initially, we quantified the size-dependent behavior of MSCs in micropores and elucidated the underlying mechanisms. Subsequently, we analyzed the shape deformation and traversal dynamics of each MSC. In addition, we conducted a systematic investigation to understand how the mechanical properties of MSCs impact their traversal process. We considered geometric and mechanical parameters, such as reduced cell volume, cell-to-nucleus diameter ratio, and cytoskeletal prestress states. Furthermore, we quantified the changes in the MSC traversal process and identified the quantitative limits in their response to variations in micropore length. Taken together, the computational results indicate the complex dynamic behavior of individual MSCs in the confined microflow. This finding offers an objective way to evaluate the homing ability of MSCs in an interendothelial-slit-like microenvironment.

Quantifying Shear-induced Margination and Adhesion of Platelets in Microvascular Blood Flow

Platelet margination and adhesion are two critical and closely related steps in thrombus formation. Using dissipative particle dynamics (DPD) method that seamlessly models blood cells, blood plasma, and vessel walls with functionalized surfaces, we quantify the shear-induced margination and adhesion of platelets in microvascular blood flow. The results show that the occurrence of shear-induced RBC-platelet collisions has a remarkable influence on the degree of platelet margination. We characterize the lateral motion of individual platelets by a mean square displacement analysis of platelet trajectories, and find that the wall-induced lift force and the shear-induced displacement in wall-bounded flow cause the variation in near-wall platelet distribution. We then investigate the platelet adhesive dynamics under different flow conditions, by conducting DPD simulations of blood flow in a microtube with fibrinogen-coated wall surfaces. We find that the platelet adhesion is enhanced with the increase of fibrinogen concentration level but decreased with the increase of shear rate. These results are consistent with available experimental results. In addition, we demonstrate that the adherent platelets have a negative impact on the margination dynamics of the circulating platelets, which is mainly due to the climbing effect induced by the adherent ones. Taken together, these findings provide useful insights into the platelet margination and adhesion dynamics, which may facilitate the understanding of the predominant processes governing the initial stage of thrombus formation.

Patient-Specific Inverse Modeling of In Vivo Cardiovascular Mechanics with Medical Image-Derived Kinematics as Input Data: Concepts, Methods, and Applications

Inverse modeling approaches in cardiovascular medicine are a collection of methodologies that can provide non-invasive patient-specific estimations of tissue properties, mechanical loads, and other mechanics-based risk factors using medical imaging as inputs. Its incorporation into clinical practice has the potential to improve diagnosis and treatment planning with low associated risks and costs. These methods have become available for medical applications mainly due to the continuing development of image-based kinematic techniques, the maturity of the associated theories describing cardiovascular function, and recent progress in computer science, modeling, and simulation engineering. Inverse method applications are multidisciplinary, requiring tailored solutions to the available clinical data, pathology of interest, and available computational resources. Herein, we review biomechanical modeling and simulation principles, methods of solving inverse problems, and techniques for image-based kinematic analysis. In the final section, the major advances in inverse modeling of human cardiovascular mechanics since its early development in the early 2000s are reviewed with emphasis on method-specific descriptions, results, and conclusions. We draw selected studies on healthy and diseased hearts, aortas, and pulmonary arteries achieved through the incorporation of tissue mechanics, hemodynamics, and fluid-structure interaction methods paired with patient-specific data acquired with medical imaging in inverse modeling approaches.

Scattering-driven PPG signal model

This article discusses the origin of PPG signals. Two plausible hypotheses are analyzed: the volumetric hypothesis and a model wherein the PPG is driven by the RBC aggregation process. To verify the model predictions, the PPG signals at the fingertip were measured. External pressure was applied to the fingertip, presumably reducing the blood flow. The results expressed in terms of gamma, used in pulse-oximetry, agree with the aggregation model. In addition, the oscillometric signal and the PPG signal amplitude were simultaneously measured in the fingertip. All of the experimental results favor the proposed aggregation mechanism as responsible the PPG signal.

Microfluidics Approach to the Mechanical Properties of Red Blood Cell Membrane and Their Effect on Blood Rheology

In this article, we describe the general features of red blood cell membranes and their effect on blood flow and blood rheology. We first present a basic description of membranes and move forward to red blood cell membranes’ characteristics and modeling. We later review the specific properties of red blood cells, presenting recent numerical and experimental microfluidics studies that elucidate the effect of the elastic properties of the red blood cell membrane on blood flow and hemorheology. Finally, we describe specific hemorheological pathologies directly related to the mechanical properties of red blood cells and their effect on microcirculation, reviewing microfluidic applications for the diagnosis and treatment of these diseases.

Dependence of cell-free-layer width on rheological parameters: Combining empirical data on flow separation at microvascular bifurcations with geometrical considerations

When blood flows through vessel networks, red blood cells (RBCs) are typically concentrated close to the vessel center line, creating a lubrication layer near the vessel wall. As RBCs bind oxygen, the width of this cell-free layer (CFL) impacts not only the blood rheology inside the vasculature, but also oxygen delivery to the tissues they perfuse and, hence, their function. Existing attempts to relate the width of the CFL to the rheological properties of the blood and the geometrical properties of the vessel are based on an analysis of the forces acting on RBCs suspended in the blood. However, the complexity of interactions in the blood makes this a challenging task. Here, we propose an alternative, two-step approach to derive such a functional relationship. First, we extend widely accepted empirical fits describing the minimum flow fraction needed for RBCs to enter a daughter vessel downstream of a microvascular bifurcation so that it depends not only on the diameter and discharge haematocrit of the parent vessel, but also on its average shear rate. Second, we propose a simple geometrical model for the minimum flow fraction based on the cross-sectional blood flow profile within the parent vessel upstream of the bifurcation-considering uniform, parabolic, and blunt velocity profiles-and derive the leading-order approximation to this model for small CFL widths. By equating the functional relationships obtained using these two approaches, we derive expressions relating the CFL width to the vessel diameter, discharge haematocrit, and mean shear rate. The resulting expressions are in good agreement with available in vivo data and represent a promising basis for future research.

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.

Hemorheology: the critical role of flow type in blood viscosity measurements

The crucial role of the hemorheological characteristics of blood in a range of diagnoses, treatments and drug delivery mechanisms is widely accepted. Nonetheless, the literature on blood rheology remains inconclusive and sometimes even contradictory. This is in part due to natural variance of blood samples from one study to another, but also stems from fundamental differences in the consequences of the choice of rheometric flow employed. Here, and using a detailed and accurate computational scheme, we thoroughly study the role of flow type in measurement of blood viscosity. Performing these in silico measurements, we isolate the role of flow type and geometry at different hematocrit levels. We show that flow curves obtained in pressure-driven flows relevant to laminar circulatory flows deviate greatly from ones obtained in drag flow at the same hematocrit level. Our numerical platform also allows for the yield stress to be measured under quiescent conditions and without imposing any flow for different hematocrits. We discuss the scaling of the yield stress with the hematocrit level, and show that the differences in pressure vs. drag flows stem from the Red Blood Cell (RBC) orientation at different flow rates as well as the existence of a cell free layer close to the walls.

Numerical simulation of spatiotemporal red blood cell aggregation under sinusoidal pulsatile flow

Previous studies on red blood cell (RBC) aggregation have elucidated the inverse relationship between shear rate and RBC aggregation under Poiseuille flow. However, the local parabolic rouleaux pattern in the arterial flow observed in ultrasonic imaging cannot be explained by shear rate alone. A quantitative approach is required to analyze the spatiotemporal variation in arterial pulsatile flow and the resulting RBC aggregation. In this work, a 2D RBC model was used to simulate RBC motion driven by interactional and hydrodynamic forces based on the depletion theory of the RBC mechanism. We focused on the interaction between the spatial distribution of shear rate and the dynamic motion of RBC aggregation under sinusoidal pulsatile flow. We introduced two components of shear rate, namely, the radial and axial shear rates, to understand the effect of sinusoidal pulsatile flow on RBC aggregation. The simulation results demonstrated that specific ranges of the axial shear rate and its ratio with radial shear rate strongly affected local RBC aggregation and parabolic rouleaux formation. These findings are important, as they indicate that the spatiotemporal variation in shear rate has a crucial role in the aggregate formation and local parabolic rouleaux under pulsatile flow.

Complexus cordis

The science-based study of the heart has allowed us to know its structure and function deeply, through the fragmentation and analysis of its parts, following the guidelines that so many achievements have given to us. However, at the time of reassembling those analyzed fragments, we realize that something is missing; the simply sum of the parts is not equal to everything. Thus, for decades, numerous scientists have studied novel strategies that allow us understanding, every natural phenomena from a more inclusive, open and integrative models, which closely address to interactions rather than components. In this way, we can observe how, the behavior of many variables usually transgress the conventional plane and moves towards non-linearity and fractality, making a complex tissue that will maintain its structure while thermodynamically viable. Thus, this document shows the way how, the non-linear study of complex cardiovascular dynamics, begins to give us answers to many questions that the clinical cardiologist poses every day.

Effect of cytosol viscosity on the flow behavior of red blood cell suspensions in microvessels

OBJECTIVE

The flow behavior of blood is strongly affected by red blood cell (RBC) properties, such as the viscosity ratio C between cytosol and suspending medium, which can significantly be altered in several pathologies (e.g. sickle-cell disease, malaria). The main objective of this study is to understand the effect of C on macroscopic blood flow properties such as flow resistance in microvessels, and to link it to the deformation and dynamics of single RBCs.

METHODS

We employ mesoscopic hydrodynamic simulations to investigate flow properties of RBC suspensions with different cytosol viscosities for various flow conditions in cylindrical microchannels.

RESULTS

Starting from a dispersed cell configuration which approximates RBC dispersion at vessel bifurcations in the microvasculature, we find that the flow convergence and development of RBC-free layer (RBC-FL) depend only weakly on C, and require a convergence length in the range of 25D-50D, where D is channel diameter. In vessels with D ≤ 20 μ m , the final resistance of developed flow is nearly the same for C = 5 and C = 1, while for D = 40 μ m , the flow resistance for C = 5 is about 10% larger than for C = 1. The similarities and differences in flow resistance can be explained by viscosity-dependent RBC-FL thicknesses, which are associated with the viscosity-dependent dynamics of single RBCs.

CONCLUSIONS

The weak effect on the flow resistance and RBC-FL explains why RBCs can contain a high concentration of hemoglobin for efficient oxygen delivery, without a pronounced increase in the flow resistance. Furthermore, our results suggest that significant alterations in microvascular flow in various pathologies are likely not due to mere changes in cytosolic viscosity.

Spherization of red blood cells and platelet margination in COPD patients

Red blood cells (RBCs) in pathological situations undergo biochemical and conformational changes, leading to alterations in rheology involved in cardiovascular events. The shape of RBCs in volunteers and stable and exacerbated chronic obstructive pulmonary disease (COPD) patients was analyzed. The effects of RBC spherization on platelet transport (displacement in the flow field caused by their interaction with RBCs) were studied in vitro and by numerical simulations. RBC spherization was observed in COPD patients compared with volunteers. In in vitro experiments at a shear rate of 100 s^-1 , treatment of RBCs with neuraminidase induced greater sphericity, which mainly affected platelet aggregates without changing aggregate size. At 400 s^-1 , neuraminidase treatment changes both the size of the aggregates and the number of platelet aggregates. Numerical simulations indicated that RBC spherization induces an increase of the platelet mean square displacement, which is traditionally linked to the platelet diffusion coefficient. RBCs of COPD patients are more spherical than healthy volunteers. Experimentally, RBC spherization induces increased platelet transport to the wall. Additional studies are needed to understand the link between the effect of RBCs on platelet transport and the increased cardiovascular events observed in COPD patients.

Quantifying Platelet Margination in Diabetic Blood Flow

Patients with type 2 diabetes mellitus (T2DM) develop thrombotic abnormalities strongly associated with cardiovascular diseases. In addition to the changes of numerous coagulation factors such as elevated levels of thrombin and fibrinogen, the abnormal rheological effects of red blood cells (RBCs) and platelets flowing in blood are crucial in platelet adhesion and thrombus formation in T2DM. An important process contributing to the latter is the platelet margination. We employ the dissipative particle dynamics method to seamlessly model cells, plasma, and vessel walls. We perform a systematic study on RBC and platelet transport in cylindrical vessels by considering different cell shapes, sizes, and RBC deformabilities in healthy and T2DM blood, as well as variable flowrates and hematocrit. In particular, we use cellular-level RBC and platelet models with parameters derived from patient-specific data and present a sensitivity study. We find T2DM RBCs, which are less deformable compared to normal RBCs, lower the transport of platelets toward the vessel walls, whereas platelets with higher mean volume (often observed in T2DM) lead to enhanced margination. Furthermore, increasing the flowrate or hematocrit enhances platelet margination. We also investigated the effect of platelet shape and observed a nonmonotonic variation with the highest near-wall concentration corresponding to platelets with a moderate aspect ratio of 0.38. We examine the role of white blood cells (WBCs), whose count is increased notably in T2DM patients. We find that WBC rolling or WBC adhesion tends to decrease platelet margination due to hydrodynamic effects. To the best of our knowledge, such simulations of blood including all blood cells have not been performed before, and our quantitative findings can help separate the effects of hydrodynamic interactions from adhesive interactions and potentially shed light on the associated pathological processes in T2DM such as increased inflammatory response, platelet activation and adhesion, and ultimately thrombus formation.

Accounting for residence-time in blood rheology models: do we really need non-Newtonian blood flow modelling in large arteries?

Patient-specific computational fluid dynamics (CFD) is a promising tool that provides highly resolved haemodynamics information. The choice of blood rheology is an assumption in CFD models that has been subject to extensive debate. Blood is known to exhibit shear-thinning behaviour, and non-Newtonian modelling has been recommended for aneurysmal flows. Current non-Newtonian models ignore rouleaux formation, which is the key player in blood's shear-thinning behaviour. Experimental data suggest that red blood cell aggregation and rouleaux formation require notable red blood cell residence-time (RT) in a low shear rate regime. This study proposes a novel hybrid Newtonian and non-Newtonian rheology model where the shear-thinning behaviour is activated in high RT regions based on experimental data. Image-based abdominal aortic and cerebral aneurysm models are considered and highly resolved CFD simulations are performed using a minimally dissipative solver. Lagrangian particle tracking is used to define a backward particle RT measure and detect stagnant regions with increased rouleaux formation likelihood. Our novel RT-based non-Newtonian model shows a significant reduction in shear-thinning effects and provides haemodynamic results qualitatively identical and quantitatively close to the Newtonian model. Our results have important implications in patient-specific CFD modelling and suggest that non-Newtonian models should be revisited in large artery flows.

Heterogeneous blood flow in microvessels with applications to nanodrug transport and mass transfer into tumor tissue

Nanodrug transport in tumor microvasculature and deposition/extravasation into tumor tissue are an important link in the nanodrug delivery process. Considering heterogeneous blood flow, such a dual process is numerically studied. The hematocrit distribution is solved by directly considering the forces experienced by the red blood cells (RBCs), i.e., the wall lift force and the random cell collision force. Using a straight microvessel as a test bed, validated computer simulations are performed to determine blood flow characteristics as well as the resulting nanodrug distribution and extravasation. The results confirm that RBCs migrate away from the vessel wall, leaving a cell-free layer (CFL). Nanodrug particles tend to preferentially accumulate in the CFL, leading to increased concentration near the endothelial surface layer. However, shear-induced NP diffusion is diminished within the CFL, causing to a much slower lateral transport rate into tumor tissue. These competing effects determine the NP deposition/extravasation rates. The present modeling framework and NP flux results provide new physical insight. The analysis can be readily extended to simulations of NP transport in blood microvessels of actual tumors.

Improved understanding of the acoustophoretic focusing of dense suspensions in a microchannel

We provide improved understanding of acoustophoretic focusing of a dense suspension (volume fraction φ>10%) in a microchannel subjected to an acoustic standing wave using a proposed theoretical model and experiments. The model is based on the theory of interacting continua and utilizes a momentum transport equation for the mixture, continuity equation, and transport equation for the solid phase. The model demonstrates the interplay between acoustic radiation and shear-induced diffusion (SID) forces that is critical in the focusing of dense suspensions. The shear-induced particle migration model of Leighton and Acrivos, coupled with the acoustic radiation force, is employed to simulate the continuum behavior of particles. In the literature, various closures for the diffusion coefficient D_{φ}^{*} are available for rigid spheres at high concentrations and nonspherical deformable particles [e.g., red blood cells (RBCs)] at low concentrations. Here we propose a closure for D_{φ}^{*} for dense suspension of RBCs and validate the proposed model with experimental data. While the available closures for D_{φ}^{*} fail to predict the acoustic focusing of a dense suspension of nonspherical deformable particles like RBCs, the predictions of the proposed model match experimental data within 15%. Both the model and experiments reveal a competition between acoustic radiation and SID forces that gives rise to an equilibrium width w^{*} of a focused stream of particles at some distance L_{eq}^{*} along the flow direction. Using different shear rates, acoustic energy densities, and particle concentrations, we show that the equilibrium width is governed by Péclet number Pe and Strouhal number Stasw^{*}=1.4(PeSt)^{-0.5} while the length required to obtain the equilibrium-focused width depends on St as L_{eq}^{*}=3.8/(St)^{0.6}. The proposed model and correlations would find significance in the design of microchannels for acoustic focusing of dense suspensions such as undiluted blood.

Modeling of Biomechanics and Biorheology of Red Blood Cells in Type 2 Diabetes Mellitus

Erythrocytes in patients with type-2 diabetes mellitus (T2DM) are associated with reduced cell deformability and elevated blood viscosity, which contribute to impaired blood flow and other pathophysiological aspects of diabetes-related vascular complications. In this study, by using a two-component red blood cell (RBC) model and systematic parameter variation, we perform detailed computational simulations to probe the alteration of the biomechanical, rheological, and dynamic behavior of T2DM RBCs in response to morphological change and membrane stiffening. First, we examine the elastic response of T2DM RBCs subject to static tensile forcing and their viscoelastic relaxation response upon release of the stretching force. Second, we investigate the membrane fluctuations of T2DM RBCs and explore the effect of cell shape on the fluctuation amplitudes. Third, we subject the T2DM RBCs to shear flow and probe the effects of cell shape and effective membrane viscosity on their tank-treading movement. In addition, we model the cell dynamic behavior in a microfluidic channel with constriction and quantify the biorheological properties of individual T2DM RBCs. Finally, we simulate T2DM RBC suspensions under shear and compare the predicted viscosity with experimental measurements. Taken together, these simulation results and their comparison with currently available experimental data are helpful in identifying a specific parametric model-the first of its kind, to our knowledge-that best describes the main hallmarks of T2DM RBCs, which can be used in future simulation studies of hematologic complications of T2DM patients.

Theory and algorithms to compute Helfrich bending forces: a review

Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.

Sub-cellular modeling of platelet transport in blood flow through microchannels with constriction

Platelet transport through arterial constrictions is one of the controlling processes influencing their adhesive functions and the formation of thrombi. We perform high-fidelity mesoscopic simulations of blood flow in microchannels with constriction, resembling arterial stenoses. The wall shear rates inside the constrictions reach levels as high as ≈8000 s(-1), similar to those encountered in moderate atherosclerotic plaques. Both red blood cells and platelets are resolved at sub-cellular resolution using the Dissipative Particle Dynamics (DPD) method. We perform a systematic study on the red blood cell and platelet transport by considering different levels of constriction, blood hematocrit and flow rates. We find that higher levels of constriction and wall shear rates lead to significantly enhanced margination of platelets, which may explain the experimental observations of enhanced post-stenosis platelet aggregation. We also observe similar margination effects for stiff particles of spherical shapes such as leukocytes. To our knowledge, such numerical simulations of dense blood through complex geometries have not been performed before, and our quantitative findings could shed new light on the associated physiological processes such as ATP release, plasma skimming, and thrombus formation.

Transport physics and biorheology in the setting of hemostasis and thrombosis

The biophysics of blood flow can dictate the function of molecules and cells in the vasculature with consequent effects on hemostasis, thrombosis, embolism, and fibrinolysis. Flow and transport dynamics are distinct for (i) hemostasis vs. thrombosis and (ii) venous vs. arterial episodes. Intraclot transport changes dramatically the moment hemostasis is achieved or the moment a thrombus becomes fully occlusive. With platelet concentrations that are 50- to 200-fold greater than platelet-rich plasma, clots formed under flow have a different composition and structure compared with blood clotted statically in a tube. The platelet-rich, core/shell architecture is a prominent feature of self-limiting hemostatic clots formed under flow. Importantly, a critical threshold concentration of surface tissue factor is required for fibrin generation under flow. Once initiated by wall-derived tissue factor, thrombin generation and its spatial propagation within a clot can be modulated by γ'-fibrinogen incorporated into fibrin, engageability of activated factor (FIXa)/activated FVIIIa tenase within the clot, platelet-derived polyphosphate, transclot permeation, and reduction of porosity via platelet retraction. Fibrin imparts tremendous strength to a thrombus to resist embolism up to wall shear stresses of 2400 dyne cm(-2) . Extreme flows, as found in severe vessel stenosis or in mechanical assist devices, can cause von Willebrand factor self-association into massive fibers along with shear-induced platelet activation. Pathological von Willebrand factor fibers are A Disintegrin And Metalloprotease with ThromboSpondin-1 domain 13 resistant but are a substrate for fibrin generation due to FXIIa capture. Recently, microfluidic technologies have enhanced the ability to interrogate blood in the context of stenotic flows, acquired von Willebrand disease, hemophilia, traumatic bleeding, and drug action.

Hemodynamics in stenotic vessels of small diameter under steady state conditions: Effect of viscoelasticity and migration of red blood cells

BACKGROUND

In microcirculation, the non-Newtonian behavior of blood and the complexity of the microvessel network are responsible for the high flow resistance and the large reduction of the blood pressure. Red blood cell aggregation along with inward radial migration are two significant mechanisms determining the former. Yet, their impact on hemodynamics in non-straight vessels is not well understood.

OBJECTIVE

In this study, the steady state blood flow in stenotic rigid vessels is examined, employing a sophisticated non-homogeneous constitutive law. The effect of red blood cells migration on the hydrodynamics is quantified and the constitutive model's accuracy is evaluated.

METHODS

A numerical algorithm based on the two-dimensional mixed finite element method and the EVSS/SUPG technique for a stable discretization of the mass and momentum conservation equations in addition to the constitutive model is employed.

RESULTS

The numerical simulations show that a cell-depleted layer develops along the vessel wall with an almost constant thickness for slow flow conditions. This causes the reduction of the drag force and the increase of the pressure gradient as the constriction ratio decreases.

CONCLUSIONS

Viscoelastic effects in blood flow were found to be responsible for steeper decreases of tube and discharge hematocrits as decreasing function of constriction ratio.

Continuum modeling of hydrodynamic particle-particle interactions in microfluidic high-concentration suspensions

A continuum model is established for numerical studies of hydrodynamic particle-particle interactions in microfluidic high-concentration suspensions. A suspension of microparticles placed in a microfluidic channel and influenced by an external force, is described by a continuous particle-concentration field coupled to the continuity and Navier-Stokes equation for the solution. The hydrodynamic interactions are accounted for through the concentration dependence of the suspension viscosity, of the single-particle mobility, and of the momentum transfer from the particles to the suspension. The model is applied on a magnetophoretic and an acoustophoretic system, respectively, and based on the results, we illustrate three main points: (1) for relative particle-to-fluid volume fractions greater than 0.01, the hydrodynamic interaction effects become important through a decreased particle mobility and an increased suspension viscosity. (2) At these high particle concentrations, particle-induced flow rolls occur, which can lead to significant deviations of the advective particle transport relative to that of dilute suspensions. (3) Which interaction mechanism that dominates, depends on the specific flow geometry and the specific external force acting on the particles.

In vitro measurement of particle margination in the microchannel flow: effect of varying hematocrit

It has long been known that platelets undergo margination when flowing in blood vessels, such that there is an excess concentration near the vessel wall. We conduct experiments and three-dimensional boundary integral simulations of platelet-sized spherical particles in a microchannel 30 μm in height to measure the particle-concentration distribution profile and observe its margination at 10%, 20%, and 30% red blood cell hematocrit. The experiments involved adding 2.15-μm-diameter spheres into a solution of red blood cells, plasma, and water and flowing this mixture down a microfluidic channel at a wall shear rate of 1000 s(-1). Fluorescence imaging was used to determine the height and velocity of particles in the channel. Experimental results indicate that margination has largely occurred before particles travel 1 cm downstream and that hematocrit plays a role in the degree of margination. With simulations, we can track the trajectories of the particles with higher resolution. These simulations also confirm that margination from an initially uniform distribution of spheres and red blood cells occurs over the length scale of O(1 cm), with higher hematocrit showing faster margination. The results presented here, from both experiments and 3D simulations, may help explain the relationship between bleeding time in vessel trauma and red blood cell hematocrit as platelets move to a vessel wall.

Effects of shear rate, confinement, and particle parameters on margination in blood flow

The effects of flow and particle properties on margination of particles in red blood cell (RBC) suspensions is investigated using direct numerical simulation (DNS) of cellar blood flow. We focus on margination of particles in the flow of moderately dense suspensions of RBCs. We hypothesize that margination rate in nondilute suspensions is mainly driven by the RBC-enhanced diffusion of marginating particles in the RBC-filled region. We derive a scaling law for margination length in a straight channel. Margination length increases cubically with channel height and is independent of shear rate. We verify this scaling law for margination length by DNS of flowing RBCs and marginating particles. We also show that rigidity and size both lead to particle margination with rigidity having a more significant effect compared to size within the range of parameters in this study.

Modeling microcirculatory blood flow: current state and future perspectives

Microvascular blood flow determines a number of important physiological processes of an organism in health and disease. Therefore, a detailed understanding of microvascular blood flow would significantly advance biophysical and biomedical research and its applications. Current developments in modeling of microcirculatory blood flow already allow to go beyond available experimental measurements and have a large potential to elucidate blood flow behavior in normal and diseased microvascular networks. There exist detailed models of blood flow on a single cell level as well as simplified models of the flow through microcirculatory networks, which are reviewed and discussed here. The combination of these models provides promising prospects for better understanding of blood flow behavior and transport properties locally as well as globally within large microvascular networks.

Inflow/Outflow Boundary Conditions for Particle-Based Blood Flow Simulations: Application to Arterial Bifurcations and Trees

When blood flows through a bifurcation, red blood cells (RBCs) travel into side branches at different hematocrit levels, and it is even possible that all RBCs enter into one branch only, leading to a complete separation of plasma and RBCs. To quantify this phenomenon via particle-based mesoscopic simulations, we developed a general framework for open boundary conditions in multiphase flows that is effective even for high hematocrit levels. The inflow at the inlet is duplicated from a fully developed flow generated in a pilot simulation with periodic boundary conditions. The outflow is controlled by adaptive forces to maintain the flow rate and velocity gradient at fixed values, while the particles leaving the arteriole at the outlet are removed from the system. Upon validation of this approach, we performed systematic 3D simulations to study plasma skimming in arterioles of diameters 20 to 32 microns. For a flow rate ratio 6:1 at the branches, we observed the “all-or-nothing” phenomenon with plasma only entering the low flow rate branch. We then simulated blood-plasma separation in arteriolar bifurcations with different bifurcation angles and same diameter of the daughter branches. Our simulations predict a significant increase in RBC flux through the main daughter branch as the bifurcation angle is increased. Finally, we demonstrated the effectiveness of the new methodology in simulations of blood flow in vessels with multiple inlets and outlets, constructed using an angiogenesis model.

Blood tests, which provide a wealth of information on the state of human health, are often performed on cell-free samples. Therefore, blood-plasma separation needs to be achieved. A simple but effective solution for isolating plasma from blood utilizes capillary bifurcations. In a particle-based simulation study of plasma skimming in capillary bifurcations, the blood flow properties such as velocity and pressure fields differ drastically at the inlet and outlet regions. Therefore, a new open (non-periodic) boundary is required. In this paper, we have developed and validated a general parallel framework for open boundary conditions. This is a non-trivial enabling technology that could be used in all open boundary systems and all particle-based Lagrangian simulations. We performed systematic 3D simulations of blood flow in arteriolar bifurcations and elucidated the biophysical mechanism of blood-plasma separation as well as quantified the effects of branch size and bifurcation angle on cell separation efficiency, which have not been addressed before. We also demonstrated the applicability of the methodology in arterial trees with multiple inlets and outlets.

Hemodynamics in the microcirculation and in microfluidics

Hemodynamics in microcirculation is important for hemorheology and several types of circulatory disease. Although hemodynamics research has a long history, the field continues to expand due to recent advancements in numerical and experimental techniques at the micro-and nano-scales. In this paper, we review recent computational and experimental studies of blood flow in microcirculation and microfluidics. We first focus on the computational studies of red blood cell (RBC) dynamics, from the single cellular level to mesoscopic multiple cellular flows, followed by a review of recent computational adhesion models for white blood cells, platelets, and malaria-infected RBCs, in which the cell adhesion to the vascular wall is essential for cellular function. Recent developments in optical microscopy have enabled the observation of flowing blood cells in microfluidics. Experimental particle image velocimetry and particle tracking velocimetry techniques are described in this article. Advancements in micro total analysis system technologies have facilitated flowing cell separation with microfluidic devices, which can be used for biomedical applications, such as a diagnostic tool for breast cancer or large intestinal tumors. In this paper, cell-separation techniques are reviewed for microfluidic devices, emphasizing recent advances and the potential of this fast-evolving research field in the near future.

A computational model of drug delivery through microcirculation to compare different tumor treatments

Starting from the fundamental laws of filtration and transport in biological tissues, we develop a computational model to capture the interplay between blood perfusion, fluid exchange with the interstitial volume, mass transport in the capillary bed, through the capillary walls and into the surrounding tissue. These phenomena are accounted at the microscale level, where capillaries and interstitial volume are viewed as two separate regions. The capillaries are described as a network of vessels carrying blood flow. We apply the model to study drug delivery to tumors. The model can be adapted to compare various treatment options. In particular, we consider delivery using drug bolus injection and nanoparticle injection into the blood stream. The computational approach is suitable for a systematic quantification of the treatment performance, enabling the analysis of interstitial drug concentration levels, metabolization rates and cell surviving fractions. Our study suggests that for the treatment based on bolus injection, the drug dose is not optimally delivered to the tumor interstitial volume. Using nanoparticles as intermediate drug carriers overrides the shortcomings of the previous delivery approach. This work shows that the proposed theoretical and computational framework represents a promising tool to compare the efficacy of different cancer treatments.

Self-assembled polymersomes formed by symmetric, asymmetric and side-chain-tethered coil-rod-coil triblock copolymers

Self-assembly behaviors of coil-rod-coil copolymers in selective solvents are explored by dissipative particle dynamics. The morphological phase diagram as a function of rod length and coil length shows five distinct types of aggregates, including spherical micelles, worm-like micelles, disk-like aggregates, honeycomb structures, and polymersomes. Small polymersomes are formed at rather poor alignment associated with monolayered rod domains. Some of the rods are even lying perpendicular to the radial direction. For symmetric copolymers (CmRxCm), the condition of vesicle formation is restricted to short coil and rod lengths. To favor the formation of CRC-polymersomes, two architecture modifications are adopted. One is to increase the coil length asymmetrically to be CmRxCn, where n > m. The other one is to tether a T-block onto the middle of the rod-block as Cm(RxTy)Cm copolymers. For those CRC-polymersomes, structural, transport, and mechanical properties of the vesicular membrane are determined, including membrane thickness, area density of coil blocks, order parameter, solvent permeability, frequency of flip-flop, membrane tension, and stretching and bending moduli. The influences of the coil length (n) and tethered block length (y) on membrane properties are examined. Finally, the mechanism of membrane fusion between CRC-polymersomes is investigated. The fusion process involves four stages and in the contact region the rods lying perpendicular to the radial direction of the polymersome play the key role. The encounter of two vesicles may result in a fused, hemifused, or non-fused polymersome. The final fate is determined by the competition between membrane tension and the steric barrier of the coil corona. The fusion outcome may change if the tension is altered by manipulating the lumen pressure.

Three-dimensional dynamics of oblate and prolate capsules in shear flow

We study computationally the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia for a range of dimensionless shear rates. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane, and its equatorial plane is initially tilted out of the plane of shear. We find, at low shear rates, the well-accepted tumbling motion is not always stable for both oblate and prolate capsules. For an oblate capsule, the dominant stable modes for increasing dimensionless shear rate are as follows: rolling with the equatorial plane staying in the plane of shear, precessing following Jeffery's orbit [Proc. R. Soc. London A 102, 161 (1922)], and tumbling. Interestingly, the order of modes is reversed for a prolate capsule: tumbling, precessing, and rolling with increasing dimensionless shear rate. At transitional regimes, we find the stable motion of a capsule can depend on its initial titled angle, even at the same shear rate. At high dimensionless shear rates, a spheroidal capsule undergoes a complicated oscillating-swinging motion: Its major axis oscillates about the plane of shear in addition to the swinging about a mean angle with flow direction found previously, and the amplitudes of both oscillations decrease when increasing the dimensionless shear rate towards a steady tank treading motion asymptotically. We summarize the results in phase diagrams and discuss the reorientation of both oblate and prolate capsules in a wide range of dimensionless shear rates.

Computational biorheology of human blood flow in health and disease

Hematologic disorders arising from infectious diseases, hereditary factors and environmental influences can lead to, and can be influenced by, significant changes in the shape, mechanical and physical properties of red blood cells (RBCs), and the biorheology of blood flow. Hence, modeling of hematologic disorders should take into account the multiphase nature of blood flow, especially in arterioles and capillaries. We present here an overview of a general computational framework based on dissipative particle dynamics (DPD) which has broad applicability in cell biophysics with implications for diagnostics, therapeutics and drug efficacy assessments for a wide variety of human diseases. This computational approach, validated by independent experimental results, is capable of modeling the biorheology of whole blood and its individual components during blood flow so as to investigate cell mechanistic processes in health and disease. DPD is a Lagrangian method that can be derived from systematic coarse-graining of molecular dynamics but can scale efficiently up to arterioles and can also be used to model RBCs down to the spectrin level. We start from experimental measurements of a single RBC to extract the relevant biophysical parameters, using single-cell measurements involving such methods as optical tweezers, atomic force microscopy and micropipette aspiration, and cell-population experiments involving microfluidic devices. We then use these validated RBC models to predict the biorheological behavior of whole blood in healthy or pathological states, and compare the simulations with experimental results involving apparent viscosity and other relevant parameters. While the approach discussed here is sufficiently general to address a broad spectrum of hematologic disorders including certain types of cancer, this paper specifically deals with results obtained using this computational framework for blood flow in malaria and sickle cell anemia.