Effects of shear rate and suspending medium viscosity on elongation of red cells tank-treading in shear flow



Elongation measurements of red cells subjected to simple shear flow are usually performed using a single suspending medium (viscosity η0) and varying the mean shear rate equation image. Such data are often plotted versus the shear stress equation image suggesting that the elongation scales with τ. In this work, normal blood samples were tested in a rheoscope varying both η0 and equation image. The ranges of equation image were chosen to restrict the elongation of the red cells to low values where the behavior is dominated by their intrinsic properties. It was found that the elongation scales with equation image with s decreasing from two at η0 = 20 mPas to unity at η0 = 70 mPas. Above η0 = 70 mPas, the elongation is therefore essentially determined by the membrane elasticity alone. A side observation was a large variation of the elongation both intraindividually and interindividually. © 2011 International Society for Advancement of Cytometry

A standard method to measure the mechanical properties of red cells is to suspend the cells in solutions much more viscous than blood plasma and to measure their elongation when the suspension is subjected to simple shear flow (1, 2). In addition to being elongated, the membrane moves around the elongated shape thus inducing an eddy flow within the cytoplasm. The membrane motion has been termed tank-tread motion (1). The elongation is usually quantified by an elongation index equation image, where L and B denote the length and the width of the elongated cell.

Measurements of EI of different blood samples are usually performed using a constant viscosity of the suspending medium (η0) under variation of the mean shear rate equation image. Such data are often plotted versus the shear stress of the undisturbed shear flow equation image suggesting that EI scales with τ. If, on the other hand, the same blood sample was tested varying both η0 and equation image, it was shown that EI does not scale with τ. Using four viscosities between 12 and 51 mPas, EI was found to scale with equation image (3). An exponent greater unity can also be inferred from other experimental results (4, 5).

To explain this finding, the following hypothesis is put forward. (i) Besides elastic stresses, viscous stresses resist an elongation of tank-treading red cells. (ii) A variable contribution of the viscous stresses being three dimensional in the cytoplasm and two dimensional in the membrane is responsible for an exponent greater unity.

To rationalize the hypothesis, we consider the following experiment. We first measure EI with a certain set of equation image and η0. Then we decrease equation image by a certain amount and increase η0 to such an extent that EI remains constant. The elastic stresses and bending moments in the membrane remain essentially constant as EI did not change. The viscous stresses, on the other hand, decrease as the tank-tread frequency decreases with decreasing equation image (6). Therefore, the required increase of η0 is less than the reciprocal of the decrease in equation image. This in turn, gives η0 a greater weight than equation image and as a consequence an exponent greater than unity in the experimentally determined scaling law.

Of course, endowing η0 with an exponent is one of two choices. One could have set as well the exponent to equation image in which case its value would have been less than unity.

Based on the hypothesis the following prediction is made. At sufficiently large η0, the viscous contribution can be neglected against the elastic contribution, i.e., the exponent approaches unity with increasing η0. It follows that above a threshold value of η0, the elongation of tank-treading red cells is essentially determined by the membrane elasticity alone.

To test the hypothesis, this prediction was checked by repeating the previous experiment (3) with the following modifications. First, the viscosity range was extended to about 100 mPas. Second, the exponent was determined as a function of η0 instead of a single value for the whole viscosity range as previously. Third, the initial rise of the elongation curve was used instead of its middle part.

We will show that the prediction is confirmed by our experiments and give an estimate for the threshold value for η0, above which a moderate elongation of tank-treading red cells solely depends on the membrane elasticity. As a side observation, we report that the variation of the elongation is large both intraindividually and interindividually.

Materials and Methods

Dextran Solutions

The sources of the dextrans are shown in Table 1. Dextran solutions were prepared as reported previously (7). In short, dextrans of different MW were dissolved in water. The concentrations (w/v) were the same for all dextrans as controlled by refractometry (Abbe, Zeiss, Oberkochen, Germany). Electrolytes were added as a stock solution of phosphate buffered saline (PBS-Dulbecco 10×, Biochrome, Berlin, Germany). The amount of electrolytes was chosen to preserve the volume of red cells in blood plasma. To counteract red cell crenation in the small gap of the cone–plate chamber, about 1% (v/v) of a 10% (w/v) solution of human albumin in water was added. The viscosity of the final solutions was measured in a rolling ball viscometer (Anton Paar, Graz, Austria) at 23°C (Table 1).

Table 1. Dextrans used in this study
NameMW (kD)CompanyLot numberη0 (mPas)
  1. MW denotes the average MW. Dextran T 2000 was dialyzed and freeze-dried before use. Viscosities were measured at 23°C.

Dextran FP 6062Serva1796414.6
Dextran 200250Serva3107028.9
Dextran 500497ServaG955.9
Dextran T 20002,000Pharmacia8122104

Preparation of Red Cells

Blood was obtained on a voluntary basis from repeat donors in the local blood bank and aspirated into heparin containing vacutainers. Whole blood (1,000 μl) was pipetted into an Eppendorf vial, placed on ice for 15 min, and then centrifuged at 5,600g for 4 min. Supernatant plasma was removed for later addition. Because of the preceding cooling and rewarming during centrifugation, the buffy coat was rather solid and separated from the red cell column. Therefore, it could be aspirated with minimal loss of red cells. Finally, plasma was added to obtain a red cell concentration (v/v) of about 0.5. This suspension was used to make the final suspension in dextran.


The heart of the rheoscope is a transparent cone–plate chamber. Its walls consist of a glass coverslip as plate and a cone machined of plexiglass. On counter rotation of cone and plate, a shear flow is generated in the gap between them. The determination of the shear rate is described in Determination of the Shear Rate section of Supporting Information. The chamber is adapted to an inverted microscope (Leitz 40 ×/0.65), which can be operated in two quickly interchangeable modes. One is interference contrast optics under white light illumination. The other is bright field under illumination at 415 nm (soret band), where the absorption of hemoglobin is at a maximum.

Both cone and plate are driven by separate feedback controlled motors. Their speed is set by two potentiometers. The first potentiometer controls the sum and the second controls the difference of rotational speed of cone and plate. This way, one potentiometer controls the mean shear rate in the cone–plate chamber and the other controls the vertical position of the fluid layer between cone and plate, which is stationary with respect to the microscope.

Cells suspended in the stationary layer are observed along the gradient of the undisturbed shear flow. Single frames are recorded with a charge-coupled device monochrome camera (DMK 41BF02.H, The Imaging Source, Bremen, Germany) and stored on the hard disk of a computer. The integration time (exposure) was 1 ms.


In a two-step procedure, the red cells suspended in plasma were diluted by a factor 4×10−4 into a dextran solution. About 30 μl of this suspension was pipetted onto the cone, which was then put in place to form the final cone–plate geometry.

The suspension was first sheared clearly above the transition from red cell rotation to tank-treading for about 10 s until essentially all cells had migrated off the walls into a common focus in the middle between cone and plate. Then, starting below the transition value, the shear rate was slowly increased until all cells in view were tank-treading. This was the lowest experimental shear rate. Two further shear rates were applied through increase of the lowest value by a factor of two and four.

At each shear rate, between 10 and 20 images were taken under soret illumination (see Rheoscopy section). To avoid the same individual red cells appearing on subsequent images, the following procedure was applied: after taking an image, the height of the stationary layer (see Rheoscopy section) was changed by the respective potentiometer to move the cells already photographed out of the frame of the camera. Then, the stationary layer was relocated to immobilize new tank-treading cells in focus and the next image was taken.

After enough images had been taken, the flow was stopped, and the optic was switched to interference contrast mode to detect fine details. The focus of the microscope was moved from the surface of the cone to the surface of the plate, and the mechanical stroke (TS) was measured with an inductive probe with a precision of <1 μm. To average out variations due to rotation of the chamber, this measurement was repeated three times around the periphery. The conversion of the averaged value into a gap width is described in Determination of the Shear Rate section of Supporting Information.

Nine blood samples were studied. The experiments were performed at room temperature (23°C) and were completed within 2 h after blood withdrawal.

Evaluation of the Data

The microscopic images were evaluated with a custom-built software (see Supporting Information, Processing of the microscopic images section). In short, red cells were detected through their grey scale value by means of a threshold criterion. The principal axes of their shape were used to calculate L and B. The axis oriented within ±45° of the direction of the undisturbed flow is called L, the axis perpendicular to it is called B.

Results and Interpretation

Determination of the Exponent

Figure 1 shows the result obtained from a single blood sample. In Figure 1a, the mean values and SDs of L/B are plotted versus equation image.

Figure 1.

Example of the determination of the exponent s0. (a) L/B (mean values ± SDs) versus equation image. (b–g) Left column: determination of s0. Right column: L/B versus equation image and the linear fit. (b and c) Dx60 and Dx200. (d and e) Dx200 and Dx500. (f and g) Dx500 and Dx2,000.

The exponent was determined from the data of pairs of neighboring values of η0, i.e., from Dx60 and 200, Dx 200 and 500, and Dx500 and 2,000. The six values of L/B available for each pairing versus equation image were fitted linearly. s was increased in steps of 0.1 and covered a range large enough to encompass its final value (s0). The weighted sum of squared residuals from each of these fits was plotted versus s and fitted in an iterative four step procedure by a parabola. In the last step, the range of s within which the parabola was fitted was the minimum of the parabola of the preceding iteration ±0.4 (Figs. 1b, 1d, and 1f). The minimum of this last parabola determined the value of s0. Figures 1c, 1e, and 1g show the respective plots of L/B versus equation image together with their linear fit.

Dependence of s0 on η0

Figure 2a shows the experimental result of nine blood samples in a plot of s0 versus the average of neighboring values of η0. The data are fitted by an exponential function. The deviation of the additive constant p in the fit function from unity is smaller than the standard error of p. This means that the value of the fitted curve at infinity is not significantly different from unity.

Figure 2.

(a) s0 of all blood samples versus the average of the two viscosities for which each s0 was determined. An exponential function fitted to the data is shown as a continuous line. (b) The same data, but data points belonging to the same blood sample are connected by lines. In one blood sample, only two data points are available because the sample was not tested with Dx60.

Figure 2a further suggests that η0 = 70 mPas constitutes a threshold above which the elongation of tank-treading red cells is essentially independent of cell viscosities. This holds at least for the moderate elongations induced in this work (see Fig. 1a).

Plotting the data from Figure 2a by drawing straight lines between the three points of each blood sample, one would expect that the respective lines are essentially parallel but shifted on the s0 axis for the different blood samples. Figure 2b shows that this is not the case. The succession of the slopes of the two line segments representing each blood sample is quite irregular. Further experiments are necessary to clarify the cause of this finding.

Intraindividual Distribution of Red Cell Parameters

An interesting side observation of this study is the wide distribution of L/B in a single blood sample. Part of this variation may be due to the unavoidable oscillation of the shear rate when the rheoscope executes a full rotation. To exclude the influence of shear rate oscillations, only the variation of L/ B within single images is shown in Figure 3. The ratio of the maximum and minimum value of L/B found in each of the 1,579 images is plotted versus the mean value of the same image. The ratio extends from 1.1 to 2.4, where the small values are dominated by images containing a small number of cells as indicated by the size of the symbols.

Figure 3.

Ratio of the maximum and minimum value for L/B found in each microscopic image used in this study versus the mean value of L/B in this image. The size of the symbols depends on the number N of cells found in each image.

It is likely that distributions of membrane shear modulus and membrane viscosity are mainly responsible for this variation. The ratio of surface area to volume probably contributes to a lesser extent as the elongations remained small (see Choice of L/B and Its Range section). Finally, oscillations of the inclination angle and elongation (8) may add to the variation.

L/B < 1 is frequently observed particularly at the low and to a lesser degree at the medium shear rate applied for each dextran MW. These values could in principle result from oscillations of the inclination angle and of the elongation (8), when tank-treading red cells are caught during extreme values of these quantities. However, the subjective observation through the microscope does not show oscillations in the apparent elongation sufficient to explain the measured value of L/B in the majority of cases. Rather, it is observed on stop of flow that these cells are elliptical at rest and orient with their long axis at right angles to the flow direction at low deformations; the absence of rotation indicates that they nevertheless perform the tank-tread motion.

For small elongations, the projected shape of tank-treading red cells resembles an ellipse. For larger elongations, the leading and trailing end become pointed. However, this is not the case for all the cells. Some preserve rounded ends at elongations, where the majority have pointed ends (Fig. 4). These differences in shape are suggested to be due to different ratios of bending to shear elasticity of the membrane. A larger value would make the ends more rounded and vice versa.

Figure 4.

Example of two red cells having about the same elongation but different shape. Left: pointed ends. Right: round ends. The width of both images corresponds to 20 μm.


Choice of L/B and Its Range

When shear rates are increased beyond the values used in this work, the continuing elongation curves level off because of the limited ratio of surface area and volume of red cells. The initial part of the curve, on the other hand, can be considered to be essentially independent of red cell geometric properties and to represent the intrinsic properties of membrane and cytoplasm.

In this work, the initial part of the curve was used because we wanted to probe the intrinsic properties alone and not a mixture of intrinsic and geometric properties. In addition, L/B was chosen instead of equation image because it shows a slightly more linear increase with equation image at the moderate elongations employed in the experiments.

Interindividual Distribution of Red Cell Parameters

The threshold viscosity of 70 mPas (see Dependence of s0 on η0 section) is below the maximum value of η0 used in this study. Therefore, the data obtained for η0 = 104 mPas can be considered as a pure measure of the elastic properties of the red cell membrane. The data obtained for η0 = 14.6 mPas, on the other hand, are influenced by both elastic and viscous properties of the red cell, as indicated by the value of s0.

The slope of a regression line through the three data points available for each dextran MW (see Fig. 1a) was taken to characterize the respective intrinsic properties. In Figure 5, the slopes for Dx60 are plotted versus those for Dx2,000. The correlation between the two quantities is low. This indicates a large influence of the viscous properties of the red cell on its elongation in the shear field.

Figure 5.

Slopes of the curves L/B versus equation image (e.g., see Fig. 1a) for Dx60 versus Dx2,000. Each data point is for one blood sample.

The range of both sets of data is unexpectedly large. The ratio between maximum and minimum value is 1.9 for Dx60 and 1.6 for Dx2,000. As all blood samples were from healthy persons donating blood at the blood bank, it can be assumed that a ratio of 1.5 or more between maximum and minimum values of average intrinsic membrane properties does not noticeably influence the perfusion in vivo. This finding corroborates earlier results in an animal model, where the survival of artificially stiffened red cells was studied (9). It also sheds doubt on the relevance of minor changes found ectacytometrically in clinical studies.

Comparison to Red Cell Models

To our knowledge, only few mathematical models exist in which the ratio of cytoplasmic viscosity and η0 is in the same range as in real elongation experiments.

In one of these models (10), the membrane viscosity is taken into account. Elongation data are presented for η0 = 12.5, 20, and 100 mPas. The shape of the published curves only allows a pointwise determination of s. Choosing L/B = 1.2, an exponent of 3.7 is obtained between 12.5 and 20 mPas and a value of 1.42 between 20 and 100 mPas. These values are clearly above the experimentally determined curve in Figure 2a.

According to the hypothesis put forward in Introduction section, the discrepancy should be caused by an overemphasis of the viscous versus the elastic contribution to the resistance against deformation in the model cell. This assumed inadequacy of the model might be responsible for another discrepancy. The model cell requires equation image at η0 = 100 mPas to achieve L/B = 1.8, whereas on average in the present study equation image suffices.

In two other models, the membrane viscosity is neglected. In both of them EI depends exactly on τ when η0 ranges from 10 to 50 mPas (11) or from 50 to 100 mPas (12), respectively. The second finding does not contradict our result because the mean of the two values of η0 is above the threshold of η0 = 70 mPas (see Dependence of s0 on η0 section). The other model is in contradiction to our results because dependence on τ is equivalent to s0 = 1. Even when the membrane viscosity is neglected, accounting for the cytoplasmic viscosity alone should result in s0 > 1 as it was shown that the energy dissipation in the cytoplasm constitutes an appreciable portion of the total dissipation in the tank-treading red cell (5, 13–15).


The authors thank the staff of the blood bank, Universitätsklinikum Aachen, for cooperation in obtaining the blood samples, the staff of the mechanical workshop, Universitätsklinikum Aachen, for design and construction of the rheoscope, Dr. Thomas Herold, RWTH-Aachen, for repair of the control unit of the rheoscope, Dr. Hiroshi Noguchi, Institute for solid State Physics, University of Tokyo, for communicating the raw data of his Figure 9a (10), and Dipl. Ing. M. Laumen, RWTH-Aachen, for the opportunity to perform the viscometric measurements in his laboratory.