Rheology of red blood cells under flow in highly confined microchannels: I. effect of elasticity

Response to the comments on "Cellular aggregation dictates universal spreading behaviour of a whole-blood drop on a paper strip"

In 2023, we published a research article in the Journal of Colloidal and Interface Science, based on our experimental findings and substantiating scaling arguments leading to a simple theoretical insight on the effect of red blood cell (RBC) aggregation on the wicking behaviour of a finite volume of blood as it navigates through the porous passages of a paper matrix (Laha et al., 2023). Of late, we received comments from Li (2024), which offered certain suggestions regarding the possible improvement of the capillary bundle model as considered in our article for analyzing the transport of blood through the paper pores. Herein, we provide a detailed discussion on each of the points raised by Li (2024) and rationalize our views in further details in addition to the contents already provided in our concerned article.

Lattice Boltzmann simulation of deformable fluid-filled bodies: progress and perspectives

With the rapid development of studies involving droplet microfluidics, drug delivery, cell detection, and microparticle synthesis, among others, many scientists have invested significant efforts to model the flow of these fluid-filled bodies. Motivated by the intricate coupling between hydrodynamics and the interactions of fluid-filled bodies, several methods have been developed. The objective of this review is to present a compact foundation of the methods used in the literature in the context of lattice Boltzmann methods. For hydrodynamics, we focus on the lattice Boltzmann method due to its specific ability to treat time- and spatial-dependent boundary conditions and to incorporate new physical models in a computationally efficient way. We split the existing methods into two groups with regard to the interfacial boundary: fluid-structure and fluid-fluid methods. The fluid-structure methods are characterised by the coupling between fluid dynamics and mechanics of the flowing body, often used in applications involving membranes and similar flexible solid boundaries. We further divide fluid-structure-based methods into two subcategories, those which treat the fluid-structure boundary as a continuum medium and those that treat it as a discrete collection of individual springs and particles. Next, we discuss the fluid-fluid methods, particularly useful for the simulations of fluid-fluid interfaces. We focus on models for immiscible droplets and their interaction in a suspending fluid and describe benchmark tests to validate the models for fluid-filled bodies.

Yield-stress transition in suspensions of deformable droplets

Yield-stress materials, which require a sufficiently large forcing to flow, are currently ill-understood theoretically. To gain insight into their yielding transition, we study numerically the rheology of a suspension of deformable droplets in 2D. We show that the suspension displays yield-stress behavior, with droplets remaining motionless below a critical body-force. In this phase, droplets jam to form an amorphous structure, whereas they order in the flowing phase. Yielding is linked to a percolation transition in the contacts of droplet-droplet overlaps and requires strict conservation of the droplet area to exist. Close to the transition, we find strong oscillations in the droplet motion that resemble those found experimentally in confined colloidal glasses. We show that even when droplets are static, the underlying solvent moves by permeation so that the viscosity of the composite system is never truly infinite, and its value ceases to be a bulk material property of the system.

Effect of Cell Age and Membrane Rigidity on Red Blood Cell Shape in Capillary Flow

Blood flow in the microcirculatory system is crucially affected by intrinsic red blood cell (RBC) properties, such as their deformability. In the smallest vessels of this network, RBCs adapt their shapes to the flow conditions. Although it is known that the age of RBCs modifies their physical properties, such as increased cytosol viscosity and altered viscoelastic membrane properties, the evolution of their shape-adapting abilities during senescence remains unclear. In this study, we investigated the effect of RBC properties on the microcapillary in vitro flow behavior and their characteristic shapes in microfluidic channels. For this, we fractioned RBCs from healthy donors according to their age. Moreover, the membranes of fresh RBCs were chemically rigidified using diamide to study the effect of isolated graded-membrane rigidity. Our results show that a fraction of stable, asymmetric, off-centered slipper-like cells at high velocities decreases with increasing age or diamide concentration. However, while old cells form an enhanced number of stable symmetric croissants at the channel centerline, this shape class is suppressed for purely rigidified cells with diamide. Our study provides further knowledge about the distinct effects of age-related changes of intrinsic cell properties on the single-cell flow behavior of RBCs in confined flows due to inter-cellular age-related cell heterogeneity.

Vesicle formation induced by thermal fluctuations

The process of fission and vesicle formation depends on the geometry of the membrane that will split. For instance, a flat surface finds it difficult to form vesicles because of the lack of curved regions where to start the process. Here we show that vesicle formation can be promoted by temperature, by using a membrane phase field model with Gaussian curvature. We find a phase transition between fluctuating and vesiculation phases that depends on temperature, spontaneous curvature, and the ratio between bending and Gaussian moduli. We analysed the energy dynamical behaviour of these processes and found that the main driving ingredient is the Gaussian energy term, although the curvature energy term usually helps with the process as well. We also found that the chemical potential can be used to investigate the temperature of the system. Finally we address how temperature changes the condition for spontaneous vesiculation for all geometries, making it happen in a wider range of values of the Gaussian modulus.

Effect of droplet deformability on shear thinning in a cylindrical channel

Droplets suspended in fluids flowing through microchannels are often encountered in different contexts and scales, from oil extraction down to microfluidics. They are usually flexible and deform as a product of the interplay between flexibility, hydrodynamics, and interaction with confining walls. Deformability adds distinct characteristics to the nature of the flow of these droplets. We simulate deformable droplets suspended in a fluid at a high volume fraction flowing through a cylindrical wetting channel. We find a discontinuous shear thinning transition, which depends on the droplet deformability. The capillary number is the main dimensionless parameter that controls the transition. Previous results have focused on two-dimensional configurations. Here we show that, in three dimensions, even the velocity profile is different. To perform this study, we improve and extend to three dimensions a multicomponent lattice Boltzmann method which prevents the coalescence between the droplets.

Dynamics of an oscillating microbubble in a blood-like Carreau fluid

A numerical model for cavitation in blood is developed based on the Keller-Miksis equation for spherical bubble dynamics with the Carreau model to represent the non-Newtonian behavior of blood. Three different pressure waveforms driving the bubble oscillations are considered: a single-cycle Gaussian waveform causing free growth and collapse, a sinusoidal waveform continuously driving the bubble, and a multi-cycle pulse relevant to contrast-enhanced ultrasound. Parameters in the Carreau model are fit to experimental measurements of blood viscosity. In the Carreau model, the relaxation time constant is 5-6 orders of magnitude larger than the Rayleigh collapse time. As a result, non-Newtonian effects do not significantly modify the bubble dynamics but do give rise to variations in the near-field stresses as non-Newtonian behavior is observed at distances 10-100 initial bubble radii away from the bubble wall. For sinusoidal forcing, a scaling relation is found for the maximum non-Newtonian length, as well as for the shear stress, which is 3 orders of magnitude larger than the maximum bubble radius.

Activation energy and force fields during topological transitions of fluid lipid vesicles

Topological transitions of fluid lipid membranes are fundamental processes for cell life. For example, they are required for endo- and exocytosis or to enable neurotransmitters to cross the neural synapses. Here, inspired by the idea that fusion and fission proteins could have evolved in Nature in order to carry out a minimal work expenditure, we evaluate the minimal free energy pathway for the transition between two spherical large unilamellar vesicles and a dumbbell-shaped one. To address the problem, we propose and successfully use a Ginzburg-Landau type of free energy, which allows us to uniquely describe without interruption the whole, full-scale topological change. We also compute the force fields needed to overcome the involved energy barriers. The obtained forces are in excellent agreement, in terms of intensity, scale, and spatial localization with experimental data on typical fission protein systems, whereas they suggest the presence of additional features in fusion proteins.

Measuring the red blood cell shape in capillary flow using spectrally encoded flow cytometry

Red blood cells in small capillaries exhibit a wide variety of deformations that reflect their true physiological conditions at these important locations. By applying a technique for the high-speed microscopy of flowing cells, termed spectrally encoded flow cytometry (SEFC), we image the light reflected from the red blood cells in human capillaries, and propose an analytical slipper-like model for the cell morphology that can reproduce the experimental in vivo images. The results of this work would be useful for studying the unique flow conditions in these vessels, and for extracting useful clinical parameters that reflect the true physiology of the blood cells in situ.

Numerical Study on Dynamics of Blood Cell Migration and Deformation in Atherosclerotic Vessels

A phase field model is used to study the effect of atherosclerotic plaque on hemodynamics. The migration of cells in blood flows is described by a set of multiple phase field equations, which incorporate elastic energies and the interacting effects of cells. Several simulations are carried out to reveal the influences of initial velocities of blood cells, cellular elasticity and block rates of hemodynamic vessels. The results show that the cell deformation increases with the growth of the initial active velocity and block rate but with the decrease of the cellular elasticity. The atherosclerotic plaque not only affects the deformation and migration of cells but also can promote the variation in hemodynamic properties. The atherosclerotic plaque causes a burst in cell velocity, and the greater the block rate and cellular elasticity, the more dramatic the variation of instantaneous velocity. The present work demonstrates that the phase field method could be extended to reveal formation atherosclerosis at the microscopic level from the perspective of hemodynamics.

Dynamical shapes of droplets of cyclodextrin-surfactant solutions

We present a series of experiments with droplets of aqueous cyclodextrin-surfactant solutions, in which the volume is reduced after the equilibrium spherical shape is reached. The final shape of the drop after this perturbation is found to be dependent on the concentration of inclusion complexes in the bulk of the solution. These inclusion complexes are formed by two cyclodextrin molecules and one surfactat molecule. We propose a model to describe these dynamical processes. Dipole–dipole interactions on the surface of the drop trigger a competition between water surface tension and dipole–dipole interaction energies. The results of the model reproduce the spherical and rod-like shapes found in the experiments.

Normalization of Blood Viscosity According to the Hematocrit and the Shear Rate

The rheological properties of blood depend highly on the properties of its red blood cells: concentration, membrane elasticity, and aggregation. These properties affect the viscosity of blood as well as its shear thinning behavior. Using an experimental analysis of the interface advancement of blood in a microchannel, we determine the viscosity of different samples of blood. In this work, we present two methods that successfully normalize the viscosity of blood for a single and for different donors, first according to the concentration of erythrocytes and second according to the shear rate. The proposed methodology is able to predict the health conditions of the blood samples by introducing a non-dimensional coefficient that accounts for the response to shear rate of the different donors blood samples. By means of these normalization methods, we were able to determine the differences between the red blood cells of the samples and define a range where healthy blood samples can be described by a single behavior.

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.

Red blood cell shape transitions and dynamics in time-dependent capillary flows

The dynamics of single red blood cells (RBCs) determine microvascular blood flow by adapting their shape to the flow conditions in the narrow vessels. In this study, we explore the dynamics and shape transitions of RBCs on the cellular scale under confined and unsteady flow conditions using a combination of microfluidic experiments and numerical simulations. Tracking RBCs in a comoving frame in time-dependent flows reveals that the mean transition time from the symmetric croissant to the off-centered, nonsymmetric slipper shape is significantly faster than the opposite shape transition, which exhibits pronounced cell rotations. Complementary simulations indicate that these dynamics depend on the orientation of the RBC membrane in the channel during the time-dependent flow. Moreover, we show how the tank-treading movement of slipper-shaped RBCs in combination with the narrow channel leads to oscillations of the cell's center of mass. The frequency of these oscillations depends on the cell velocity, the viscosity of the surrounding fluid, and the cytosol viscosity. These results provide a potential framework to identify and study pathological changes in RBC properties.

Blood Rheological Characterization of β-Thalassemia Trait and Iron Deficiency Anemia Using Front Microrheometry

The purpose of this work is to develop a hematocrit-independent method for the detection of beta-thalassemia trait (β-TT) and iron deficiency anemia (IDA), through the rheological characterization of whole blood samples from different donors. The results obtained herein are the basis for the development of a front microrheometry point-of-care device for the diagnosis and clinical follow-up of β-TT patients suffering hematological diseases and alterations in the morphology of the red blood cell (RBC). The viscosity is calculated as a function of the mean front velocity by detecting the sample fluid-air interface advancing through a microfluidic channel. Different viscosity curves are obtained for healthy donors, β-TT and IDA samples. A mathematical model is introduced to compare samples of distinct hematocrit, classifying the viscosity curve patterns with respect to the health condition of blood. The viscosity of the fluid at certain shear rate values varies depending on several RBC factors such as shape and size, hemoglobin (Hb) content, membrane rigidity and hematocrit concentration. Blood and plasma from healthy donors are used as reference. To validate their potential clinical value as a diagnostic tool, the viscosity results are compared to those obtained by the gold-standard method for RBC deformability evaluation, the Laser-Optical Rotational Red Cell Analyzer (LoRRCA).

Red blood cells in low Reynolds number flow: A vorticity-based characterization of shapes in two dimensions

Studies on the mechanical properties of red blood cells improve the diagnosis of some blood-related diseases. Some existing numerical methods have successfully simulated the coupling between a fluid and red blood cells. This paper introduces an alternative phase-field model formulation of two-dimensional cells that solves the vorticity and stream function that simplifies the numerical implementation. We integrate red blood cell dynamics immersed in a Poiseuille flow and reproduce previously reported morphologies (slippers or parachutes). In the case of flow in a very wide channel, we discover a new metastable shape referred to as 'anti-parachute' that evolves into a horizontal slipper centered on the channel. This sort of metastable morphology may contribute to the dynamical response of the blood.

Axial and Nonaxial Migration of Red Blood Cells in a Microtube

Human red blood cells (RBCs) are subjected to high viscous shear stress, especially during microcirculation, resulting in stable deformed shapes such as parachute or slipper shape. Those unique deformed RBC shapes, accompanied with axial or nonaxial migration, cannot be fully described according to traditional knowledge about lateral movement of deformable spherical particles. Although several experimental and numerical studies have investigated RBC behavior in microchannels with similar diameters as RBCs, the detailed mechanical characteristics of RBC lateral movement—in particular, regarding the relationship between stable deformed shapes, equilibrium radial RBC position, and membrane load—has not yet been fully described. Thus, we numerically investigated the behavior of single RBCs with radii of 4 μm in a circular microchannel with diameters of 15 μm. Flow was assumed to be almost inertialess. The problem was characterized by the capillary number, which is the ratio between fluid viscous force and membrane elastic force. The power (or energy dissipation) associated with membrane deformations was introduced to quantify the state of membrane loads. Simulations were performed with different capillary numbers, viscosity ratios of the internal to external fluids of RBCs, and initial RBC centroid positions. Our numerical results demonstrated that axial or nonaxial migration of RBC depended on the stable deformed RBC shapes, and the equilibrium radial position of the RBC centroid correlated well with energy expenditure associated with membrane deformations.

Capillary-size flow of human blood plasma: Revealing hidden elasticity and scale dependence

The dynamical mechanical analysis of blood generally uses models inspired by conventional flows, assuming scale-independent homogeneous flows and without considering fluid-surface boundary interactions. The present experimental study highlights the relevance of using an approach in line with physiological reality providing a strong interaction between the fluid and the boundary interface. New dynamic properties of human blood plasma are found: a finite shear elastic response (solid-like property) is identified in nearly static conditions, which also depends on the scale (being reinforced at small scales). The elastic behavior is confirmed by the induction, without heat transfer, of local hot and cold thermodynamic states evidencing a thermo-mechanical coupling in blood plasma so far known only in elastic materials. This finding opens new routes for medical diagnosis and device fabrication.

On Gaussian curvature and membrane fission

We propose a three-dimensional mathematical model to describe dynamical processes of membrane fission. The model is based on a phase field equation that includes the Gaussian curvature contribution to the bending energy. With the addition of the Gaussian curvature energy term numerical simulations agree with the predictions that tubular shapes can break down into multiple vesicles. A dispersion relation obtained with linear analysis predicts the wavelength of the instability and the number of formed vesicles. Finally, a membrane shape diagram is obtained for the different Gaussian and bending modulus, showing different shape regimes.

Vortical flow structures induced by red blood cells in capillaries

OBJECTIVE

Knowledge about the flow field of the plasma around the red blood cells in capillary flow is important for a physical understanding of blood flow and the transport of micro- and nanoparticles and molecules in the flowing plasma. We conducted an experimental study on the flow field around red blood cells in capillary flow that is complemented by simulations of vortical flow between red blood cells.

METHODS

Red blood cells were injected in a 10 × 12 µm rectangular microchannel at a low hematocrit, and the flow field around one or two cells was captured by a high-speed camera that tracked 250 nm nanoparticles in the flow field, acting as tracers.

RESULTS

While the flow field around a steady "croissant" shape is found to be similar to that of a rigid sphere, the flow field around a "slipper" shape exhibits a small vortex at the rear of the red blood cell. Even more pronounced are vortex-like structures observed in the central region between two neighboring croissants.

CONCLUSIONS

The rotation frequency of the vortices is to a good approximation, inversely proportional to the distance between the cells. Our experimental data are complemented by numerical simulations.

Collective behavior of red blood cells in confined channels

We study the flow properties of red blood cells in confined channels, when the channel width is comparable to the cell size. We focus on the case of intermediate concentrations when hydrodynamic interactions between cells play a dominant role. This regime is different to the case of low concentration in which the cells behave as hydrodynamically isolated. In this last case, the dynamic behavior is entirely controlled by the interplay between the interaction with the wall and the elastic response of the cell membrane. Our results highlight the different fluid properties when collective flow is present. The cells acquire a characteristic slipper shape, and parachute shapes are only observed at very large capillary numbers. We have characterized the spatial ordering and the layering by means of a pairwise correlation function. Focusing effects are observed at the core of the channel instead of at the lateral position typical of the single-train case. These results indicate that at these intermediate concentrations we observed at the microscale the first steps of the well-known macroscopic Fahraeus-Lindqvist effect. The rheological properties of the suspension are studied by means of the effective viscosity, with an expected shear-thinning behavior. Two main differences are obtained with respect to the single-train case. First, a large magnitude of the viscosity is obtained indicating a high resistance to flow. Secondly, the shear-thinning behavior is obtained at larger values of the capillary number respect to the single-train case. These results suggest that the phenomena of ordering in space and orientation occur at higher values of the capillary number.

Rheology and microrheology of deformable droplet suspensions

Dense suspensions of soft colloidal particles display a broad range of physical and rheological properties which are still far from being fully understood. To elucidate the role of deformability on colloidal flow, we employ computer simulations to measure the apparent viscosity of a system of droplets of variable surface tension subjected to a pressure-driven flow. We confirm that our suspension generically undergoes discontinuous shear thinning, and determine the dependence of the onset of the discontinuity on surface tension. We find that the effective viscosity of the suspension is mainly determined by a capillary number. We present active microrheology simulations, where a single droplet is dragged through the suspension. These also show a dynamical phase transition, analogous to the one associated with discontinuous shear thinning in our interpretation. Such a transition is signalled by a discontinuity in the droplet velocity versus applied force.

The dynamics of a healthy and infected red blood cell in flow through constricted channels: A DPD simulation

Understanding the dynamics of red blood cell (RBC) motion under in silico conditions is central to the development of cost-effective diagnostic tools. Specifically, unraveling the relationship between the rheological properties and the nature of shape change in the RBC (healthy or infected) can be extremely useful. In case of malarial infection, RBC progressively loses its deformability and tends to occlude the microvessel. In the present study, detailed mesoscopic simulations are performed to investigate the deformation dynamics of an RBC in flow through a constricted channel. Specifically, the manifestation of viscous forces (through flow rates) on the passage and blockage characteristics of a healthy red blood cell (hRBC) vis-á-vis an infected red blood cell (iRBC) are investigated. A finite-sized dissipative particle dynamics framework is used to model plasma in conjunction with a discrete model for the RBC. Instantaneous wall boundary method was used to model no-slip wall boundary conditions with a good control on the near-wall density fluctuations and compressibility effects. To investigate the microvascular occlusion, the RBC motion through 2 types of constricted channels, viz, (1) a tapered microchannel and (2) a stenosed-type microchannel, were simulated. It was observed that the deformation of an infected cell was much less compared with a healthy cell, with an attendant increase in the passage time. Apart from the qualitative features, deformation indices were obtained. The deformation of hRBC was sudden, while the iRBC deformed slowly as it traversed through the constriction. For higher flow rates, both hRBC and iRBC were found to undergo severe deformation. Even under low flow rates, hRBC could easily traverse past the constricted channel. However, for sufficiently slow flow rates (eg, capillary flows), the microchannel was found to be completely blocked by the iRBC.

Classification of red blood cell shapes in flow using outlier tolerant machine learning

The manual evaluation, classification and counting of biological objects demands for an enormous expenditure of time and subjective human input may be a source of error. Investigating the shape of red blood cells (RBCs) in microcapillary Poiseuille flow, we overcome this drawback by introducing a convolutional neural regression network for an automatic, outlier tolerant shape classification. From our experiments we expect two stable geometries: the so-called 'slipper' and 'croissant' shapes depending on the prevailing flow conditions and the cell-intrinsic parameters. Whereas croissants mostly occur at low shear rates, slippers evolve at higher flow velocities. With our method, we are able to find the transition point between both 'phases' of stable shapes which is of high interest to ensuing theoretical studies and numerical simulations. Using statistically based thresholds, from our data, we obtain so-called phase diagrams which are compared to manual evaluations. Prospectively, our concept allows us to perform objective analyses of measurements for a variety of flow conditions and to receive comparable results. Moreover, the proposed procedure enables unbiased studies on the influence of drugs on flow properties of single RBCs and the resulting macroscopic change of the flow behavior of whole blood.

How viscoelastic is human blood plasma?

Blood plasma has been considered a Newtonian fluid for decades. Recent experiments (Brust et al., Phys. Rev. Lett., 2013, 110) revealed that blood plasma has a pronounced viscoelastic behavior. This claim was based on purely elastic effects observed in the collapse of a thin plasma filament and the fast flow of plasma inside a contraction-expansion microchannel. However, due to the fact that plasma is a solution with very low viscosity, conventional rotational rheometers are not able to stretch the proteins effectively and thus, provide information about the viscoelastic properties of plasma. Using computational rheology and a molecular-based constitutive model, we predict accurately the rheological response of human blood plasma in strong extensional and constriction complex flows. The complete rheological characterization of plasma yields the first quantitative estimation of its viscoelastic properties in shear and extensional flows. We find that although plasma is characterized by a spectrum of ultra-short relaxation times (on the order of 10-3-10-5 s), its elastic nature dominates in flows that feature high shear and extensional rates, such as blood flow in microvessels. We show that plasma exhibits intense strain hardening when exposed to extensional deformations due to the stretch of the proteins in its bulk. In addition, using simple theoretical considerations we propose fibrinogen as the main candidate that attributes elasticity to plasma. These findings confirm that human blood plasma features bulk viscoelasticity and indicate that this non-Newtonian response should be seriously taken into consideration when examining whole blood flow.

Onsager's variational principle for the dynamics of a vesicle in a Poiseuille flow

We propose a systematic formulation of the migration behaviors of a vesicle in a Poiseuille flow based on Onsager's variational principle, which can be used to determine the most stable steady state. Our model is described by a combination of the phase field theory for the vesicle and the hydrodynamics for the flow field. The dynamics is governed by the bending elastic energy and the dissipation functional, the latter being composed of viscous dissipation of the flow field, dissipation of the bending energy of the vesicle, and the friction between the vesicle and the flow field. We performed a series of simulations on 2-dimensional systems by changing the bending elasticity of the membrane and observed 3 types of steady states, i.e., those with slipper shape, bullet shape, and snaking motion, and a quasi-steady state with zig-zag motion. We show that the transitions among these steady states can be quantitatively explained by evaluating the dissipation functional, which is determined by the competition between the friction on the vesicle surface and the viscous dissipation in the bulk flow.

Numerical-experimental observation of shape bistability of red blood cells flowing in a microchannel

Red blood cells flowing through capillaries assume a wide variety of different shapes owing to their high deformability. Predicting the realized shapes is a complex field as they are determined by the intricate interplay between the flow conditions and the membrane mechanics. In this work we construct the shape phase diagram of a single red blood cell with a physiological viscosity ratio flowing in a microchannel. We use both experimental in vitro measurements as well as 3D numerical simulations to complement the respective other one. Numerically, we have easy control over the initial starting configuration and natural access to the full 3D shape. With this information we obtain the phase diagram as a function of initial position, starting shape and cell velocity. Experimentally, we measure the occurrence frequency of the different shapes as a function of the cell velocity to construct the experimental diagram which is in good agreement with the numerical observations. Two different major shapes are found, namely croissants and slippers. Notably, both shapes show coexistence at low (<1 mm s^-1) and high velocities (>3 mm s^-1) while in-between only croissants are stable. This pronounced bistability indicates that RBC shapes are not only determined by system parameters such as flow velocity or channel size, but also strongly depend on the initial conditions.

Flow of Deformable Droplets: Discontinuous Shear Thinning and Velocity Oscillations

We study the rheology of a suspension of soft deformable droplets subjected to a pressure-driven flow. Through computer simulations, we measure the apparent viscosity as a function of droplet concentration and pressure gradient, and provide evidence of a discontinuous shear thinning behavior, which occurs at a concentration-dependent value of the forcing. We further show that this response is associated with a nonequilibrium transition between a "hard" (or less deformable) phase, which is nearly jammed and flows very slowly, and a "soft" (or more deformable) phase, which flows much more easily. The soft phase is characterized by flow-induced time dependent shape deformations and internal currents, which are virtually absent in the hard phase. Close to the transition, we find sustained oscillations in both the droplet and fluid velocities. Polydisperse systems show similar phenomenology but with a smoother transition, and less regular oscillations.

Elastic and dynamic properties of membrane phase-field models

Phase-field models have been extensively used to study interfacial phenomena, from solidification to vesicle dynamics. In this article, we analyze a phase-field model that captures the relevant physical features that characterize biological membranes. We show that the Helfrich theory of elasticity of membranes can be applied to phase-field models, allowing to derive the expressions of the stress tensor, lateral stress profile and elastic moduli. We discuss the relevance and interpretations of these magnitudes from a phase-field perspective. Taking the sharp-interface limit we show that the membrane macroscopic equilibrium equation can be derived from the equilibrium condition of the phase-field interface. We also study two dynamic models that describe the behaviour of a membrane. From the study of the relaxational behaviour of the membrane we characterize the relevant dynamics of each model, and discuss their applications.

Printed microfluidic filter for heparinized blood

A simple lab-on-a-chip method for blood plasma separation was developed by combining stereolithographic 3D printing with inkjet printing, creating a completely sealed microfluidic device. In some approaches, one dilutes the blood sample before separation, reducing the concentration of a target analyte and increasing a contamination risk. In this work, a single drop (8 μl) of heparinized whole blood could be efficiently filtered using a capillary effect without any external driving forces and without dilution. The blood storage in heparin tubes during 24 h at 4 °C initiated the formation of small crystals that formed auto-filtration structures in the sample upon entering the 3D-printed device, with pores smaller than the red blood cells, separating plasma from the cellular content. The total filtration process took less than 10 s. The presented printed plasma filtration microfluidics fabricated with a rapid prototyping approach is a miniaturized, fast and easy-to-operate device that can be integrated into healthcare/portable systems for point-of-care diagnostics.

Front microrheology of the non-Newtonian behaviour of blood: scaling theory of erythrocyte aggregation by aging

We introduce a new framework to study the non-Newtonian behaviour of fluids at the microscale based on the analysis of front advancement. We apply this methodology to study the non-linear rheology of blood in microchannels. We carry out experiments in which the non-linear viscosity of blood samples is quantified at different haematocrits and ages. Under these conditions, blood exhibits a power-law dependence on the shear rate. In order to analyse our experimental data, we put forward a scaling theory which allows us to define an adhesion scaling number. This theory yields a scaling behaviour of the viscosity expressed as a function of the adhesion capillary number. By applying this scaling theory to samples of different ages, we are able to quantify how the characteristic adhesion energy varies as time progresses. This connection between microscopic and mesoscopic properties allows us to estimate quantitatively the change in the cell-cell adhesion energies as the sample ages.

An Investigation on the Aggregation and Rheodynamics of Human Red Blood Cells Using High Performance Computations

Studies on the haemodynamics of human circulation are clinically and scientifically important. In order to investigate the effect of deformation and aggregation of red blood cells (RBCs) in blood flow, a computational technique has been developed by coupling the interaction between the fluid and the deformable RBCs. Parallelization was carried out for the coupled code and a high speedup was achieved based on a spatial decomposition. In order to verify the code's capability of simulating RBC deformation and transport, simulations were carried out for a spherical capsule in a microchannel and multiple RBC transport in a Poiseuille flow. RBC transport in a confined tube was also carried out to simulate the peristaltic effects of microvessels. Relatively large-scale simulations were carried out of the motion of 49,512 RBCs in shear flows, which yielded a hematocrit of 45%. The large-scale feature of the simulation has enabled a macroscale verification and investigation of the overall characteristics of RBC aggregations to be carried out. The results are in excellent agreement with experimental studies and, more specifically, both the experimental and simulation results show uniform RBC distributions under high shear rates (60-100/s) whereas large aggregations were observed under a lower shear rate of 10/s.

Cell-alignment patterns in the collective migration of cells with polarized adhesion

Dictyostelium discoideum (Dd) utilizes inhomogeneities in the distribution of cell-cell adhesion molecules on cell membranes for collective cell migration. A simple example of an inhomogeneity is a front-side (leading-edge) polarization in the distribution at the early streaming stage. Experiments have shown that the polarized cell-cell adhesion induces side-by-side contact between cells [Beug et al., Nature (London) 274, 445 (1978)NATUAS0028-083610.1038/274445a0]. This result is counterintuitive, as one would expect cells to align front to front in contact with each other on the basis of front-side polarization. In this work, we theoretically examine whether front-side polarization induces side-by-side contact in collective cell migration. We construct a model for expressing cells with this polarization based on the two-dimensional cellular Potts model. By a numerical simulation with this model, we find cell-cell alignment wherein cells form lateral arrays with side-by-side contacts as observed in the experiments.

Phase-field model for collective cell migration

We construct a phase-field model for collective cell migration based on a Ginzburg-Landau free-energy formulation. We model adhesion, surface tension, repulsion, coattraction, and polarization, enabling us to follow the cells' morphologies and the effect of their membranes fluctuations on collective motion. We were able to measure the tissue surface tension as a function of the individual cell cortical tension and adhesion and identify a density threshold for cell-sheet formation.

Pressure-driven occlusive flow of a confined red blood cell

When red blood cells (RBCs) move through narrow capillaries in the microcirculation, they deform as they flow. In pathophysiological processes such as sickle cell disease and malaria, RBC motion and flow are severely restricted. To understand this threshold of occlusion, we use a combination of experiment and theory to study the motion of a single swollen RBC through a narrow glass capillary of varying inner diameter. By tracking the movement of the squeezed cell as it is driven by a controlled pressure drop, we measure the RBC velocity as a function of the pressure gradient as well as the local capillary diameter, and find that the effective blood viscosity in this regime increases with both decreasing RBC velocity and tube radius by following a power-law that depends upon the length of the confined cell. Our observations are consistent with a simple elasto-hydrodynamic model and highlight the role of lateral confinement in the occluded pressure-driven slow flow of soft confined objects.

The Force at the Tip--Modelling Tension and Proliferation in Sprouting Angiogenesis

Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art.

Sprouting angiogenesis—a process by which new blood vessels grow from existing ones—is an ubiquitous phenomenon in health and disease of higher organisms, playing a crucial role in organogenesis, wound healing, inflammation, as well as on the onset and progression of over 50 different diseases such as cancer, rheumatoid arthritis and diabetes. Mathematical models have the ability to suggest relevant hypotheses with respect to the mechanisms of cell movement and rearrangement within growing vessel sprouts. The inclusion of both biochemical and mechanical processes in a mathematical model of sprouting angiogenesis permits to describe sprout extension as a function of the forces exerted by the cells in the tissue. It also allows to question the regulation of biochemical processes by mechanical forces and vice-versa. In this work we present a compact model of sprouting angiogenesis that includes the mechanical characteristics of the vessel and the tissue. We use this model to suggest the mechanism for the regulation of proliferation within sprout formation. We conclude that the tip cell has the role of creating a tension in the cells that follow its lead. On those first cells of the stalk, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. The modelling strategy used, deemed phase-field, permits to describe the evolution of the shape of different domains in complex systems. It is focused on the movement of the interfaces between the domains, and not on an exhaustive description of the transport properties within each domain. For this reason, it requires a reduced number of parameters, and has been used extensively in modelling other biological phenomena such as tumor growth. The coupling of mechanical and biochemical processes in a compact mathematical model of angiogenesis will enable the study of lumen formation and aneurisms in the near future. Also, this framework will allow the study of the action of flow in vessel remodelling, since local forces can readily be coupled with cell movement to obtain the final vessel morphology.

Multiple scale model for cell migration in monolayers: Elastic mismatch between cells enhances motility

We propose a multiscale model for monolayer of motile cells that comprise normal and cancer cells. In the model, the two types of cells have identical properties except for their elasticity; cancer cells are softer and normal cells are stiffer. The goal is to isolate the role of elasticity mismatch on the migration potential of cancer cells in the absence of other contributions that are present in real cells. The methodology is based on a phase-field description where each cell is modeled as a highly-deformable self-propelled droplet. We simulated two types of nearly confluent monolayers. One contains a single cancer cell in a layer of normal cells and the other contains normal cells only. The simulation results demonstrate that elasticity mismatch alone is sufficient to increase the motility of the cancer cell significantly. Further, the trajectory of the cancer cell is decorated by several speed “bursts” where the cancer cell quickly relaxes from a largely deformed shape and consequently increases its translational motion. The increased motility and the amplitude and frequency of the bursts are in qualitative agreement with recent experiments.

Rheology of red blood cells under flow in highly confined microchannels. II. Effect of focusing and confinement

We study the focusing of red blood cells and vesicles in pressure-driven flows in highly confined microchannels (10-30 μm), identifying the control parameters that dictate the cell distribution along the channel. Our results show that an increase in the flow velocity leads to a sharper cell distribution in a lateral position of the channel. This position depends on the channel width, with cells flowing at outer (closer to the walls) positions in thicker channels. We also study the relevance of the object shape, exploring the different behaviour of red blood cells and different vesicles. We also analyze the implications of these phenomena in the cell suspension rheology, highlighting the crucial role of the wall confinement in the rheological properties of the suspension.

Phase-field theories for mathematical modeling of biological membranes

Biological membranes are complex structures whose mechanics are usually described at a mesoscopic level, such as the Helfrich bending theory. In this article, we present the phase-field methods, a useful tool for studying complex membrane problems which can be applied to very different phenomena. We start with an overview of the general theory of elasticity, paying special attention to its derivation from a molecular scale. We then study the particular case of membrane elasticity, explicitly obtaining the Helfrich bending energy. Within the framework of this theory, we derive a phase-field model for biological membranes and explore its physical basis and interpretation in terms of membrane elasticity. We finally explain three examples of applications of these methods to membrane related problems. First, the case of vesicle pearling and tubulation, when lipidic vesicles are exposed to the presence of hydrophobic polymers that anchor to the membrane, inducing a shape instability. Finally, we study the behavior of red blood cells while flowing in narrow microchannels, focusing on the importance of membrane elasticity to the cell flow capabilities.