6.1.1 Motivation
The analysis and prediction of shapes assumed by red blood cells flowing in micro-sized channels is a long-standing issue [283–285] with far-reaching consequences. For example, it is necessary for a better understanding of blood as a suspension [61,119,186,286–288] and it is required to design more efficient and cheap analysis methods, e.g. in the form of lab-on-a-chip devices [2,3,5]. Despite its long history, we are still far away from a full comprehension owing to the complex interplay between the cell membranes and the fluid.
Qualitatively, experimental studies mostly observed two different shapes [285, 289–294]:
croissants6and slippers. Example images can be seen in figure6.1. Yet, more detailed systematic experimental reports on the behavior of single RBCs are basically non-existent.
Numerically, this topic was often approached for simpler model systems such as vesicles in 2D [130, 296–302]. On the other hand, detailed investigations with realistic red blood cell models are rather scarce. Notable exceptions are three studies by the group of G. Gompper [11, 215,217]: Two of these studies found discocytes below and parachutes above a certain velocity, depending on the elastic parameters [215] and the confinement [217]. Very detailed phase diagrams as functions of velocity and confinement were presented in the third paper [11].
Yet, all these works on Poiseuille-like flows neglect or only shortly mention an important aspect of the problem, namely the influence of the initial condition. The only exception with a more detailed study regarding its influence considered a vesicle in anunboundedPoiseuille flow where a pronounced dependence on the initial position was found for higher viscosity ratios [295].
Moreover, most of the experimental and numerical studies (including references [11,215,217]) used viscosity ratios below or equal to one. This can insofar be criticized as recent works on 2D [298,300] and 3D [295] vesicles found that the dynamics can change significantly when using e.g. the physiologically more relevant value ofλ= 5[14].
6Croissants are similar to parachutes. The difference is that the latter are perfectly rotationally symmetric while the first exhibit only two symmetry planes [295].
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(a) 3D measurements
(b) Simulation results
slipper perspective slipper side croissant perspective croissant side
Fig. 6.1: (a) Top row: Confocal microscope recordings of red blood cells flowing in a 25µm×10µm channel with velocities of 0.33 mm/s (slipper) and 0.37 mm/s (croissant). (b) Bottom row: Corresponding BIM simulations in the same geometry. Reprinted from publication [Pub4] with permission from AIP Publishing.
To enable more detailed future studies, publication [Pub4] presents a new confocal mi-croscope technique developed in the group of Prof. C. Wagner which allows to record fully three-dimensional images of flowing RBCs. Croissants and slippers are observed. The author of the present thesis performed complementary BIM simulations and found slippers and metastable croissants that are very similar to the experimental recordings (see figure6.1).
Publication [Pub5] comprises the first detailed and systematic study that combines both experiments and simulations of single RBCs in microchannel flows. The experiments were conducted by members of the group of Prof. C. Wagner. In contrast to the existing literature, we useλ= 5and pay special attention to the initial condition in both methodologies, which enables us to make quantitative comparisons between the experimental and numerical results.
We observe bistability: Croissants and slippers are found to be stable at the same set of parameters over a wide range of velocities. The results of [Pub5] are shortly summarized in the following.
6.1.2 Setup
The experiments in publication [Pub5] consider the behavior of individual healthy human red blood cells in a rectangular microchannel. The channel has a width of approximately 12µm and a height of around 10µm. The viscosity of the fluid inside the cells is roughly five times higher than the viscosity of the ambient fluid. Experimental analysis of single cells is realized by using hematocrit values of less than 1 % in the input reservoir and consideration of images showing only a single cell. Furthermore, as the 3D confocal microscopy from publication [Pub4] is limited to velocities below 1 mm/s, 2D images with standard bright-field microscopy are recorded. Shapes and positions of the cells are then extracted from these images. Cells at two locations in the channel are examined: at the channel entrance and 10 mm downstream. The first location provides important information about the entry position, while the second shows the converged
6.1 Bistability of red blood cells in microchannels
shapes.
Our simulations mirror the experiments as close as possible. Hence we also use a rectangular 12µm×10µm channel (with slightly rounded corners to prevent numerical issues, compare figure6.2) and a viscosity ratio ofλ= 5. The fluid-structure problem is solved with the periodic boundary integral method (chapters3and5). We employ the discocyte from equation (4.6) as the reference shape for the in-plane elasticity (Skalak model) and the spontaneous curvature is set to zero. The implementation of the RBC was already explained in chapters2.1and4. We use Method C as bending algorithm as we found it to be the most stable one in the present setup.
In contrast to the experiments, we have full control over the initial condition. We therefore not only vary the initial position along a line that is close to the channel’s diagonal (radial offsetrinit), but also employ three different initial shapes. These include the typical discocyte, a croissant and a slipper. Also see figure6.2for an example setup.
The velocity is varied in the experiments and simulations in a range of approximately 0.1 mm/s to 10 mm/s. This completely covers the range of physiological velocities in vessels with diameters of the order of around 10µm [6,86,303].
Fig. 6.2: Typical simulation setup used to study the behavior of a single red blood cell in a rectangular microchannel. The image shows an example when using a discocyte as the starting shape (other shapes are also used). Moreover, the cell is offset along the black arrow by an offsetrinit. The arrow at the top illustrates the position of the cam-era in the experiments. Reproduced from publication [Pub5] with permission from the Royal Society of Chemistry.
6.1.3 Summary of our key results
Experiments We first consider the experimental observations 10 mm away from the channel entrance. Figure6.3a shows the observed fraction of cell shapes as a function of the cell velocity.
Cells that were not clearly classifiable as neither croissant nor slipper were termed as “others”.
The diagram clearly shows that croissants dominate the picture at lower velocities, that croissants and slippers coexist at intermediate velocities (bistability) and that only slippers exist at higher velocities. Hence, it seems to be obvious that the velocity is the major parameter determining the shape. Yet, figure6.3b shows that the offset distribution at the channelentrancewidens with higher velocities, i.e. more cells are entering off-centered. This means that either the higher velocity or the larger offset at the entrance might be responsible for the observed fraction of shapes depicted in figure6.3a. Unfortunately, the experimental data does not allow for a definite conclusion and numerical simulations are required.
Simulations By starting with the discocyte shape in the simulations and varying the initial radial positionrinitas well as the velocity, we obtain the phase diagram shown in figure6.4a. It depicts the shapes found in the steady state. As in the experiments, croissants and slippers are observed, with the result depending on the velocity. Most importantly, we also see a pronounced dependence on the initial position. Starting further off-centered tends to produce slippers while a
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0.92 0.83 0.19 0.43 0.53 0.47 0.38 0.25 0.15 0.16 0.14 0.13
0.25 0.36 0.54 0.7 0.85 0.85 0.85 0.87
Fig. 6.3: Experimental results: (a) Observed cell shapes 10 mm away from the channel entrance as a function of cell velocity. The horizontal error bars show the standard deviation of the measured cell velocities. (b) Probability density function of the center-of-mass position of the cells at the channel entrance. The numbers on the left side depict the applied pressure drops in millibar, while the numbers on the right indicate the average cell velocity in mm/s.
The curves are offset in the vertical direction to facilitate the illustration. Reproduced from publication [Pub5] with permission from the Royal Society of Chemistry.
starting position near the center yields croissants. As both are stable, we find bistability. There is one exception, namely only croissants are stable at velocities from around 2 to 3 mm/s.
Considering the observed shapes in more details (figure6.4a), we can distinguish two different types of croissants and two different types of slippers. On the one hand, most of the croissants do not exhibit any relative movement and have two symmetry planes in the steady state (“non-TT croissants”). Yet, at higher velocities somewhat asymmetric shapes with pronounced tank-treading (TT) are found that are nevertheless very similar to croissants (see inset in figure6.4b).
On the other hand, the majority of slippers show tank-treading (“TT slippers”). At lower velocities, however, the tank-treading is suppressed and the object rotates similar to a rigid body (“tumbling”), with the shape still being reminiscent of a slipper (“non-TT slippers”). See the supplementary information of publication [Pub5] for some videos.
The dependence on the initial position also suggests a possible dependence on the initial shape.
We therefore show in figure6.4b the corresponding phase diagram when starting with a slipper.7 Compared to the first diagram, the “area” of croissants is reduced, i.e. slippers are found at smaller values of the initial radial offset. Apart from this, no qualitative differences are found.
Especially the croissant-only region from 2 to 3 mm/s still exists.
Comparison Comparing the experimental (figure6.3a) and numerical (figure6.4) diagrams qualitatively, one can see that both are very similar: Croissants dominate at lower and slippers at higher velocities, with both coexisting at intermediate values. Making a quantitative comparison, however, is not immediately possible. Rather, we first need to use the results from the simulations
7In publication [Pub5] we also start with a croissant and rotated discocytes. The corresponding results are omitted
6.1 Bistability of red blood cells in microchannels
(a) Init. as discocyte
0 2 4 6 8 10
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non-TT Croissant TT Croissant non-TT Slipper TT Slipper
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(b) Init. as slipper
0 2 4 6 8 10
0 1 2 3 4
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Fig. 6.4: Numerical results: Red blood cell shapes in the steady state as a function of velocityuand initial radial offset rinit. The top axis shows the results in terms of the capillary numberCaB:=µuRRBC2 /κB, whereµ=1.2 mPa s is the ambient fluid viscosity andRRBC=4µm the large radius of the equilibrium discocyte shape. Each mark indicates one simulation, and the lines and shaded areas are guides to the eyes. Above the dashed line, the cell would overlap with the channel walls. The cell was started as a discocyte in figure (a) and as a slipper in figure (b). The inset in the right figure illustrates a tank-treading croissant. Reproduced from publication [Pub5] with permission from the Royal Society of Chemistry.
and make a prediction regarding the expected number of cell shapes. The idea to do this is the following. The simulations (figure 6.4) yield a critical value rtrans for the initial radial position below which croissants and above which slippers occur. Extracting the fraction of cells from figure6.3b that enter the channel in the experiments with an offset8below this particular valuertransgives a certain valueφ. Hence, the simulations predict that a fractionφof the cells should become croissants. φcan therefore be directly compared with the experimental findings from figure6.3a.
The result of this procedure can be seen in figure 6.5 where we show the prediction for both numerical phase diagrams (i.e. started with discocyte and slipper) in comparison with the experimental observations. Obviously, the prediction using the discocyte as starting shape shows somewhat more notable deviations. On the other hand, starting with the slipper gives very good agreement. This suggests that the cells are entering the channel in the experiments with rather asymmetric shapes because a slipper is less symmetric than a discocyte. Indeed, most of the cells observed at the channel entrance cannot be clearly classified (i.e. they are “others”).
Conclusion Due to the very good agreement, we can conclude that the physical models used in the simulations (chapter2) can properly describe the dynamics of RBCs in microchannel flows.
It also means that the chosen algorithms and their implementation is appropriate (chapters4 and5). Moreover, our experimental and numerical results clearly show that the initial condition has a decisive influence on the shapes assumed by red blood cells. This constitutes a fundamental finding. It also suggests that croissants and slippers occurring in-vivo are not just transients but rather intrinsically stable shapes. For more analyses and information we point the reader to publication [Pub5].
8Actually, we additionally take into account that the initial radial offset in the simulations is approximately along the diagonal while we only have the projection of the offset in the experiments.
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0 200 400 600
Capillary numberCaB
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8
1 (a)
Cell velocity[mm/s]
Fractionofcroissantsφ
Init. as discocyte Experiment
0 200 400 600
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0 0.2 0.4 0.6 0.8
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Cell velocity[mm/s] Init. as slipper Experiment
Fig. 6.5: Comparison between experimental and numerical results. The black dashed line is identical to the experi-mental findings depicted in figure6.3a. The other points and lines show the fraction of croissantsφas predicted from the simulations and as explained in the main text. We show the standard deviation of the measured cell velocities as horizontal error bars, while the vertical error bars indicate the uncertainty in the prediction (see publication [Pub5]).
Figure (a) shows the result when starting the simulations with the discocyte shape, figure (b) when starting with the slipper shape. Reproduced from publication [Pub5] with permission from the Royal Society of Chemistry.