Viscoelastic transient of confined Red Blood Cells

The unique ability of a red blood cell to flow through extremely small microcapillaries depends on the viscoelastic properties of its membrane. Here, we study in vitro the response time upon flow startup exhibited by red blood cells confined into microchannels. We show that the characteristic transient time depends on the imposed flow strength, and that such a dependence gives access to both the effective viscosity and the elastic modulus controlling the temporal response of red cells. A simple theoretical analysis of our experimental data, validated by numerical simulations, further allows us to compute an estimate for the two-dimensional membrane viscosity of red blood cells, $\eta_{mem}^{2D}\sim 10^{-7}$ N$\cdot$s$\cdot$m$^{-1}$. By comparing our results with those from previous studies, we discuss and clarify the origin of the discrepancies found in the literature regarding the determination of $\eta_{mem}^{2D}$, and reconcile seemingly conflicting conclusions from previous works

New analytical progress in the theory of vesicles under linear flow

Vesicles are becoming a quite popular model for the study of red blood cells (RBCs). This is a free boundary problem which is rather difficult to handle theoretically. Quantitative computational approaches constitute also a challenge. In addition, with numerical studies, it is not easy to scan within a reasonable time the whole parameter space. Therefore, having quantitative analytical results is an essential advance that provides deeper understanding of observed features and can be used to accompany and possibly guide further numerical development. In this paper shape evolution equations for a vesicle in a shear flow are derived analytically with precision being cubic (which is quadratic in previous theories) with regard to the deformation of the vesicle relative to a spherical shape. The phase diagram distinguishing regions of parameters where different types of motion (tank-treading, tumbling and vacillating-breathing) are manifested is presented. This theory reveals unsuspected features: including higher order terms and harmonics (even if they are not directly excited by the shear flow) is necessary, whatever the shape is close to a sphere. Not only does this theory cure a quite large quantitative discrepancy between previous theories and recent experiments and numerical studies, but also it reveals a new phenomenon: the VB mode band in parameter space, which is believed to saturate after a moderate shear rate, exhibits a striking widening beyond a critical shear rate. The widening results from excitation of fourth order harmonic. The obtained phase diagram is in a remarkably good agreement with recent three dimensional numerical simulations based on the boundary integral formulation. Comparison of our results with experiments is systematically made.Comment: a tex file and 6 figure

3D Numerical simulations of vesicle and inextensible capsule dynamics

published in Journal of Computational PhysicsInternational audienceVesicles are locally-inextensible fluid membranes while inextensible capsules are in addition endowed with in-plane shear elasticity mimicking the cytoskeleton of red blood cells (RBCs). Boundary integral (BI) methods based on the Green's function techniques are used to describe their dynamics, that falls into the category of highly nonlinear and nonlocal dynamics. Numerical solutions raise several obstacles and challenges that strongly impact the results. Of particular complexity is (i) the membrane inextensibility, (ii) the mesh stability and (iii) numerical precisions for evaluation of the boundary integral equations. Despite intense research these questions are still a matter of debate. We regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force. In addition, the regularized kernel remains explicitly self-adjoint. The stability and precision of BI calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh, constructed by a standard refinement procedure from the main mesh. We extend the partition of unity technique to boundary integral calculation on triangular meshes: We split the calculation of the boundary integral between the original and the auxiliary mesh using a smooth weight function, which takes the distance between the source and the target as the argument and falls to zero beyond a certain cut-off distance. We provide an efficient lookup algorithm that allows us to discard most of the vertices of the auxiliary mesh lying beyond the cut-off distance from a given point without actually calculating the distances to them. The proposed algorithm offers the same treatment of near-singular integration regardless if the source and the target points belong to the same surface or not. Additional innovations are used to increase the stability and precision of the method: The bending forces are calculated by differential geometry expressions using local coordinates defined in vicinity of each vertex. The approximation of the surface in vicinity of a vertex is obtained by fitting with a second-degree polynomial of local coordinates. We solve for the Lagrange multiplier associated with membrane incompressibility using two penalization parameters per suspended entity: one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex. The proposed advancement is to vary the penalization parameters at each time step in such a way, that the global area of each membrane be conserved and the sum of squares of local strains be at minimum. This optimization is achieved by solving a linear system of rank three times the number of entities involved in the simulation. If no auxiliary mesh is used, the method reduces to steepest descent method thanks to the explicit self-adjointness of the regularized single-layer kernel in the boundary integral equation. Inextensible capsules, a model of RBC, are studied by storing the position in the reference configuration for each vertex. The elastic force is then calculated by direct variation of the elastic energy. Various nonequilibrium physical examples on vesicles and capsules will be presented and the convergence and precision tests highlighted. Overall, a good convergence is observed with numerical error inversely proportional to the number of vertices used for surface discretization, the highest order of convergence allowed by piece-wise linear interpolation of the surface

Hydrodynamic pairing of soft particles in a confined flow

The mechanism of hydrodynamics-induced pairing of soft particles, namely closed bilayer membranes (vesicles, a model system for red blood cells) and drops, is studied numerically with a special attention paid to the role of the confinement (the particles are within two rigid walls). This study unveils the complexity of the pairing mechanism due to hydrodynamic interactions. We find both for vesicles and for drops that two particles attract each other and form a stable pair at weak confinement if their initial separation is below a certain value. If the initial separation is beyond that distance, the particles repel each other and adopt a longer stable interdistance. This means that for the same confinement we have (at least) two stable branches. To which branch a pair of particles relaxes with time depends only on the initial configuration. An unstable branch is found between these two stable branches. At a critical confinement the stable branch corresponding to the shortest interdistance merges with the unstable branch in the form of a saddle-node bifurcation. At this critical confinement we have a finite jump from a solution corresponding to the continuation of the unbounded case to a solution which is induced by the presence of walls. The results are summarized in a phase diagram, which proves to be of a complex nature. The fact that both vesicles and drops have the same qualitative phase diagram points to the existence of a universal behavior, highlighting the fact that with regard to pairing the details of mechanical properties of the deformable particles are unimportant. This offers an interesting perspective for simple analytical modeling

Blood crystal: emergent order of red blood cells under wall-confined shear flow

Driven or active suspensions can display fascinating collective behavior, where coherent motions or structures arise on a scale much larger than that of the constituent particles. Here, we report experiments and numerical simulations revealing that red blood cells (RBCs) assemble into regular patterns in a confined shear flow. The order is of pure hydrodynamic and inertialess origin, and emerges from a subtle interplay between (i) hydrodynamic repulsion by the bounding walls which drives deformable cells towards the channel mid-plane and (ii) intercellular hydrodynamic interactions which can be attractive or repulsive depending on cell-cell separation. Various crystal-like structures arise depending on RBC concentration and confinement. Hardened RBCs in experiments and rigid particles in simulations remain disordered under the same conditions where deformable RBCs form regular patterns, highlighting the intimate link between particle deformability and the emergence of order. The difference in structuring ability of healthy (deformable) and diseased (stiff) RBCs creates a flow signature potentially exploitable for diagnosis of blood pathologies

Crawling in a fluid

There is increasing evidence that mammalian cells not only crawl on substrates but can also swim in fluids. To elucidate the mechanisms of the onset of motility of cells in suspension, a model which couples actin and myosin kinetics to fluid flow is proposed and solved for a spherical shape. The swimming speed is extracted in terms of key parameters. We analytically find super- and subcritical bifurcations from a non-motile to a motile state and also spontaneous polarity oscillations that arise from a Hopf bifurcation. Relaxing the spherical assumption, the obtained shapes show appealing trends

Viscoelastic transient of confined Red Blood Cells

The unique ability of a red blood cell to flow through extremely small microcapillaries depends on the viscoelastic properties of its membrane. Here, we study in vitro the response time upon flow startup exhibited by red blood cells confined into microchannels. We show that the characteristic transient time depends on the imposed flow strength, and that such a dependence gives access to both the effective viscosity and the elastic modulus controlling the temporal response of red cells. A simple theoretical analysis of our experimental data, validated by numerical simulations, further allows us to compute an estimate for the two-dimensional membrane viscosity of red blood cells, $\eta_{mem}^{2D}\sim 10^{-7}$ N$\cdot$s$\cdot$m$^{-1}$. By comparing our results with those from previous studies, we discuss and clarify the origin of the discrepancies found in the literature regarding the determination of $\eta_{mem}^{2D}$, and reconcile seemingly conflicting conclusions from previous works

Erythrocyte-erythrocyte aggregation dynamics under shear flow

Red blood cells (RBCs) -- erythrocytes -- suspended in plasma tend to aggregate and form rouleaux. During aggregation the first stage consists in the formation of RBC doublets [Blood cells, molecules, and diseases 25, 339 (1999)]. While aggregates are normally dissociated by moderate flow stresses, under some pathological conditions the aggregation becomes irreversible, which leads to high blood viscosity and vessel occlusion. We perform here two-dimensional simulations to study the doublet dynamics under shear flow in different conditions and its impact on rheology. We sum up our results on the dynamics of doublet in a rich phase diagram in the parameter space (flow strength, adhesion energy) showing four different types of doublet configurations and dynamics. We find that membrane tank-treading plays an important role in doublet disaggregation, in agreement with experiments on RBCs. A remarkable feature found here is that when a single cell performs tumbling (by increasing vesicle internal viscosity) the doublet formed due to adhesion (even very weak) remains stable even under a very strong shear rate. It is seen in this regime that an increase of shear rate induces an adaptation of the doublet conformation allowing the aggregate to resist cell-cell detachment. We show that the normalized effective viscosity of doublet suspension increases significantly with the adhesion energy, a fact which should affect blood perfusion in microcirculation.Comment: 14page