Slipper Shape Research Articles

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.

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 m in a circular microchannel with diameters of 15 m. 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.

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.

Deformation of a Red Blood Cell in a Narrow Rectangular Microchannel.

The deformability of a red blood cell (RBC) is one of the most important biological parameters affecting blood flow, both in large arteries and in the microcirculation, and hence it can be used to quantify the cell state. Despite numerous studies on the mechanical properties of RBCs, including cell rigidity, much is still unknown about the relationship between deformability and the configuration of flowing cells, especially in a confined rectangular channel. Recent computer simulation techniques have successfully been used to investigate the detailed behavior of RBCs in a channel, but the dynamics of a translating RBC in a narrow rectangular microchannel have not yet been fully understood. In this study, we numerically investigated the behavior of RBCs flowing at different velocities in a narrow rectangular microchannel that mimicked a microfluidic device. The problem is characterized by the capillary number , which is the ratio between the fluid viscous force and the membrane elastic force. We found that confined RBCs in a narrow rectangular microchannel maintained a nearly unchanged biconcave shape at low , then assumed an asymmetrical slipper shape at moderate , and finally attained a symmetrical parachute shape at high . Once a RBC deformed into one of these shapes, it was maintained as the final stable configurations. Since the slipper shape was only found at moderate , measuring configurations of flowing cells will be helpful to quantify the cell state.

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.

Vesicle dynamics in a confined Poiseuille flow: from steady state to chaos.

Red blood cells (RBCs) are the major component of blood, and the flow of blood is dictated by that of RBCs. We employ vesicles, which consist of closed bilayer membranes enclosing a fluid, as a model system to study the behavior of RBCs under a confined Poiseuille flow. We extensively explore two main parameters: (i) the degree of confinement of vesicles within the channel and (ii) the flow strength. Rich and complex dynamics for vesicles are revealed, ranging from steady-state shapes (in the form of parachute and slipper shapes) to chaotic dynamics of shape. Chaos occurs through a cascade of multiple periodic oscillations of the vesicle shape. We summarize our results in a phase diagram in the parameter plane (degree of confinement and flow strength). This finding highlights the level of complexity of a flowing vesicle in the small Reynolds number where the flow is laminar in the absence of vesicles and can be rendered turbulent due to elasticity of vesicles.

Two-dimensional numerical modeling for separation of deformable cells using dielectrophoresis.

In this paper, we numerically explore the possibility of separating two groups of deformable cells, by a very small dielectrophoretic (DEP) microchip with the characteristic length of several cell diameters. A 2D two-fluid model is developed to describe the separation process, where three types of forces are considered, the aggregation force for cell-cell interaction, the deformation force for cell deformation, and the DEP force for cell dielectrophoresis. As a model validation, we calculate the levitation height of a cell subject to DEP force, and compare it with the experimental data. After that, we simulate the separation of two groups of cells with different dielectric properties at high and low frequencies, respectively. The simulation results show that the deformable cells can be separated successfully by a very small DEP microchip, according to not only their different permittivities at the high frequency, but also their different conductivities at the low frequency. In addition, both two groups of cells have a shape deformation from an original shape to a lopsided slipper shape during the separation process. It is found that the cell motion is mainly determined by the DEP force arising from the electric field, causing the cells to deviate from the centerline of microchannel. However, the cell deformation is mainly determined by the deformation force arising from the fluid flow, causing the deviated cells to undergo an asymmetric motion with the deformation of slipper shape.

Three‐dimensional numerical simulation of red blood cell motion in Poiseuille flows

SUMMARYAn immersed boundary method based on an FEM has been successfully combined with an elastic spring network model for simulating the dynamical behavior of a red blood cell (RBC) in Poiseuille flows. This elastic spring network preserves the biconcave shape of the RBC in the sense that after the removal of the body force for driving the Poiseuille flow, the RBC with its typical parachute shape in a tube does restore its biconcave resting shape. As a benchmark test, the relationship between the deformation index and the capillary number of the RBCs flowing through a narrow cylindrical tube has been validated. For the migration properties of a single cell in a slit Poiseuille flow, a slipper shape accompanied by a cell membrane tank‐treading motion is obtained for Re , and the cell mass center is away from the center line of the channel due to its asymmetric slipper shape. For the lower Re ⩽0.0137, an RBC with almost undeformed biconcave shape has a tumbling motion. A transition from tumbling to tank‐treading happens at the Reynolds number between 0.0137 and 0.03. In slit Poiseuille flow, the RBC can also exhibit a rolling motion like a wheel during the migration when the cell is released in the fluid flow with φ = π/2 and θ = π/2 (see Figure 12 for the definition of φ and θ). The lower the Reynolds number, the longer the rolling motion lasts; but the equilibrium shape and position are independent from the cell initial position in the channel. Copyright © 2014 John Wiley & Sons, Ltd.

Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow.

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, Ca, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, λ, and (iii) the reduced volume, ν=[V/(4/3)π]/(A/4π)3/2 (A and V are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Numerical simulation of the transient shape of the red blood cell in microcapillary flow

The transient shape of a red blood cell (RBC) in a microcapillary flow is simulated under different initial conditions, including various axis orientations and centroid locations, using the LBM-DLM/FD method, which is derived from the lattice Boltzmann method and the distributed Lagrange multiplier/fictitious domain method. Although the terminal velocity is not sensitive to the initial configuration, the evolution of the velocity and the shape are determined by the initial conditions. The parachute and the slipper shape are the most probable shapes for a deformed RBC in the flow. An RBC with an initial axis orientation of 90 degrees exhibits a more complicated deformation. RBCs have a tendency to move to the centerline of a tube if an offset between the RBC centroid and the centerline exists. Our numerical results are validated by experiments, and some details beyond the experiment are provided.

On the problem of slipper shapes of red blood cells in the microvasculature

Red blood cells (RBC) are known to exhibit non symmetric (slipper) shapes in the microvasculature. Vesicles have been recently used as a model for RBC and numerical simulations proved the existence of slipper shapes under Poiseuille flow (both in unconfined and confined geometry). However, in our recent numerical simulations the transition from symmetric (parachute) shape to the slipper one was found to take place upon decreasing the flow strength, while experiments on RBCs showed the contrary. In this work we show that if the viscosity contrast (ratio between the internal over external fluid viscosities) is different from unity, as is the case with RBCs, the transition from parachute to slipper shape occurs upon increasing the flow strength, in agreement with experiments. We provide the phase diagram of shapes in the microcirculation. The slipper is found to have a higher speed than the parachute (for the same parameters), that we believe to be the basic reason for its prevailing in the microvasculature. We provide a simple geometrical picture that explains the slipper flow efficiency over the parachute one. Finally, we show that there exists in parameter space regions of co-existence of slipper/parachute shapes and suggest simple experimental protocols to test these findings. The coexistence of shapes seems to be supported by experiments, though a systematic experimental study is lacking. A potential application of this work is to guide designing flow-based experiments in order to link the shape of RBCs to pathologies affecting cell deformability, such as sickle cell diseases, malaria, and those affecting blood hematocrit, as in polycythemia vera disease.

Symmetry breaking of vesicle shapes in Poiseuille flow

Vesicle behavior under unbounded axial Poiseuille flow is studied analytically. Our study reveals subtle features of the dynamics. It is established that there exists a stable off-centerline steady-state solution for low enough flow strength. This solution appears as a symmetry-breaking bifurcation upon lowering the flow strength and includes slipper shapes, which are characteristic of red blood cells in the microvasculature. A stable axisymmetric solution exists for any flow strength provided the excess area is small enough. It is shown that the mechanism of the symmetry breaking depends on the geometry of the flow: The bifurcation is subcritical in axial Poiseuille flow and supercritical in planar flow.

Dynamics of Red Blood Cells and Vesicles in Microchannels of Oscillating Width

We have studied the dynamics of red blood cells and fluid lipid vesicles in hydrodynamic flow fields created by microchannels with periodically varying channels width. For red blood cells we find a transition from a regime with oscillating tilt angle and fixed shape to a regime with oscillating shape with increasing flow velocity. We have determined the crossover to occur at a critical ratio of channel width L_y and red blood cell velocity v_m approx. 10⊥-3s. These oscillations are superposed by shape transitions from a discocyte to a slipper shape at low velocities and a slipper to parachute transition at high flow velocities.Ref.:H. Noguchi, G. Gompper, L. Schmid, A. Wixforth, T. Franke, EPL (Europhysics Letters) 2010, 89, 28002S. Braunmueller, L. Schmid, T. Franke J.Phys. Condens. Matt. accepted

Deformation and clustering of red blood cells in microcapillary flows

The shape changes and clustering of red blood cells (RBCs) under flow in cylindrical microcapillaries are studied using a triangulated surface model for the membrane and a particle-based mesoscopic simulation technique for the embedding fluid. As the flow velocity increases, the RBCs make a transition from a discocyte shape at low velocities to a parachute shape at high velocities; close to the critical flow velocity, the RBC can also be found in a transient slipper shape. The transition and critical flow velocity are examined for various capillary diameters and RBC volume fractions (hematocrit HT). At high flow velocities and low hematocrits, the parachute-shaped RBCs can be found in clusters which are hydrodynamically stabilized. Here, the formation of a fluid vortex between neighboring cells, called bolus, develops which keeps the cells at a preferred distance. Decreasing the flow velocity towards the critical velocity, we observe an increasing frequency of drastic RBC shape fluctuations to slipper-shaped RBCs that can result in cluster breakup. These clusters resemble those seen in experiments using optical microscopy.

Why Do Red Blood Cells Have Asymmetric Shapes Even in a Symmetric Flow?

Understanding why red blood cells (RBCs) move with an asymmetric shape (slipperlike shape) in small blood vessels is a long-standing puzzle in blood circulatory research. By considering a vesicle (a model system for RBCs), we discovered that the slipper shape results from a loss in stability of the symmetric shape. It is shown that the adoption of a slipper shape causes a significant decrease in the velocity difference between the cell and the imposed flow, thus providing higher flow efficiency for RBCs. Higher membrane rigidity leads to a dramatic change in the slipper morphology, thus offering a potential diagnostic tool for cell pathologies.