Dynamic Deformation and Recovery Response of Red Blood Cells to a Cyclically Reversing Shear Flow: Effects of Frequency of Cyclically Reversing Shear Flow and Shear Stress Level

INTRODUCTION

The deformability of red blood cells (RBCs) under shear How has been studied during the last 30 years to understand the biomechanical capability of RBCs in circulatory dynamics. The RBCs with a mean diameter of S µm subjected to a randomly varying shear stress in the cardiac chamber and larger vessels must flexibly alter their shape to pass through 3- to 4-µm-diameter capillaries of the microcirculation (1) as oxygen carriers. The ability of RBCs to deform and maintain the oxygen-carrying capability under a randomly varying shear flow is of great interest from the standpoint of rheological and functional behavior of RBCs in the cardiovascular system.

Previous studies have uncovered various findings related to the deformability of RBCs in a steady, uniform shear-flow field. These included the ellipsoidal shape in a high shear flow (2), the rotation of the RBC membrane, called tanktreading motion, the effect of shear-stress amplitude on elongation, the effect of shear rate on the rotational speed of tank-treading motion (3-5), the effect of internal and external viscosity ratio and membrane stiffness and hematocrit on deformability (6,7). Theoretical models have also been published (8-13) that describe the deformation mechanism of RBCs in the uniform shear flow. Many experimental studies have been performed to understand hemolysis (14) with respect to a specific uniform shear-stress level and exposure time (15-17). Consequently, the mathematical relations between the uniform shear-stress level and hemolysis of RBCs have also been reported (18,19). In addition, works by Nevaril and Velker (20,21) on morphological changes and/or decrease in cell deformability caused by various levels of shear stress suggested that affected blood cells might clog up the microvessels, leading to tissue infarction, or might be filtered by the spleen function (22). In addition, fluctuating shear How under a normal physiological condition (peak shear stress <3.0 Pa) has also been addressed (23,24).

The study of RBCs has also extended to the analysis and design of cardiovascular devices such as heart-lung bypass systems to minimize mechanical effects that cause hemolysis. In relation to hemolysis, Bludszuweit predicted that the shear-stress fluctuation would exceed 100 Pa along the streamline inside commercially available centrifugal blood pumps (25,26). The randomly fluctuating shear flow generated inside the cardiovascular devices is expected to induce a large transmembrane stress and to cause rupture of the RBC membrane. How RBCs deform and withstand rupture in a fluctuating shear-flow field is of great interest for biomedical researchers to minimize mechanical trauma to RBCs in cardiovascular devices. The deformation capability of RBCs subjected to a cyclically reversing shear flow with high peak shear stress that simulates a randomly varying condition has not yet been studied.

This study was thus designed to provide both experimental and theoretical insight into the deformation and recovery process of normal human RBCs subjected to a cyclically reversing shear flow that simulated the randomly varying high shear field of various cardiovascular devices. To attain this goal, we built a cyclically reversing shear-flow generator (CRSFG) that could generate one-dimensional, cyclically reversing shear flow with its peak shear stress exceeding 100 Pa (27,28). The time course of deformation and recovery behavior of normal fresh human RBCs was studied to possibly uncover dynamic mechanical responses to a cyclically reversing shear How.

MATERIALS AND METHODS

Theory of oscillatory Couette flow between two parallel glass plates

First of all, the oscillatory Couette flow condition was a key in designing an experimental system. We designed a system that could exert an equal level of cyclically fluctuating shear stress to all the RBCs inside the flow lield. The experimental system is described in the following section. In this section, the theory of a reversing Couette How is reviewed (29).

In this study, we designed the experimental system to satisfy the conditions of Eq. 9 and to produce an oscillatory Couette How. The clearance width h of 30 µm between the parallel plates, kinematic viscosity of the fluid ? of 1.82 × 10^sup -4^ m^sup 2^/s and fluid density ? of 1100 kg/m^sup 3^ at the temperature of 24.5°C satisfied the relation shown in Eq. 9.

Preparation of human RBC-dextran PBS

Phosphate-huffered solution (PBS) was first prepared by dissolving a PBS powder (Wako Pure Chemical Industries, Osaka, Japan) in the distilled water. The dextran powder, with a molecular weight of 60,000-90,000 (dextran, low fraction, Acros Organics, Fairlawn, NJ) was mixed into the PBS with the weight percentage of 31. and then kept for few days in the refrigerator for better mixing. The viscosity of the dextran PBS was 0.20 Pa s at 24.5°C as measured using a rotating viscometer (DV-II Pro. Brooktield Engineering Laboratories. Middleboro. MA).

A 10-µl amount of fresh human whole blood was collected into an amicoagulated glass tube (Vitrex 1771, Modulohm A/S, Herlev, Denmark) by needle puncture from the fingertip of a healthy male volunteer and mixed with 1.0 ml of dextran PBS. The hematocrit of the RBC-dextran PBS was 0.45 vol %. and its viscosity was the same as that of the dextran PBS.

Cyclically reversing shear flow generator and microscope data acquisition system

Fig. 1 showed the schematic diagram of the experimental set-up that consisted of a CRSFG and a microscope data acquisition system. The arm of the slider-crank mechanism that was linked to the eccentric cam-motor system was attached to the upper movable plate of the parallel glass plate assembly (PGPA) between which the RBC suspension was inserted. Fig. 2 showed the assembled CRSFG system.

Two pieces of stainless steel spacer each having a thickness of 30 µm, a width of 12.7 mm, and a length of 150 mm (FGSM 0.03, Misumi. Tokyo, Japan) were fixed near the center of the stationary glass plate of the PGPA keeping the distance between them at 19 mm and right above the objective lens of the microscope. The stationary bottom plate (110 × 200 × 1 mm thickness) of the PGPA was fixed to the microscope stage (IX71, Olympus, Tokyo, Japan).

RBC suspension, 8.5-µl, was then placed in the space between the two spacers on the glass plate, followed by placement of the upper moving glass plate (76 × 52 × 1.3 mm thick, S9213, Matsunami Glass, Osaka, Japan) apparatus over the blood sample, and finally connection of the slider block. The eccentric cam-motor system made the 2.5-mm stroke oscillation of the slider block using the 164-mm-long link system between the cam and the slider block. The ball bearings with tight tit. one on the slider block and the other on the earn, allowed smooth conversion of the motor rotation into sinusoidal movement of the slider block. The upper glass plate attached to the slider block slid on the surface of the stainless steel spacers, exerting a high shear stress on the RBCs kept in the space. The PGPA thus secured the space of 30 µm where RBC suspension could he exposed to a cyclically reversing shear stress.

The objective lens with 40× magnification (NA = 0.6, LCPlanFl 40×, Olympus) was used for imaging the RBCs. A high-speed video camera (MEMREMCAM fx-K3. Nac Image Technology, Tokyo. Japan) having a capability of 5000 frames/s with an exposure time of 1/20,000 s was attached to the microscope to capture the two-dimensional images of the RBCs. An eddy-current gap sensor (PU-07. Applied Electronics. Kanagawa, Japan) lixecl to the microscope stage detected the oscillating motion of the aluminum plate target attached to the slider block to provide continuous information on the glass plate movement.

EXPERIMENTAL PROCEDURE

The cyclically reversing shear flow al the frequencies of 1, 2, 3, and 5 Hz (motor speed of 60, 120, 180, and 300 rpm) was generated using the CRSFG to exert the peak shear stress of ~53, 108, 161. and 274 Pa, respectively, to the RBC suspension. All the experiments were performed at a temperature of 24.5°C.

For image analysis, RBCs staying at around the midpoint between the two parallel glass plates were selected. The microscope focusing procedure was as follows: initially the microscope was focused on the bottom glass plate, followed by focusing on the top glass plate through manipulation of the dial gauge. When the two end points were determined manually, the dial gauge was then adjusted to the midpoint between the two end-points. In this way. RBCs staying at around the midpoint between the two parallel plates were selected for dynamic deformation analysis.

Manual trigger signal (see Fig. 1) started saving RBC images captured by a high-speed video camera in a personal computer, simultaneously digitizing and storing eddy-current gap sensor signal on a digital oscilloscope with the sampling frequency of 19 kHz for a glass plate moving at 1 Hz, hut a frequency of 20 kHz for those moving at 2, 3, and 5 Hz, respectively.

Data analysis

RBC image analysis 1: L/W ratio

The DI was originally derived by Taylor to describe the elongation patterns of the two-phase emulsion (30). Later, the DI was adopted for analysis of RBC deformation. The curves for DI versus shear stress, however, tend to flatten in the high-shear-stress region, losing resolution in the region >50 Pa. On the other hand, the L/W ratio is better suited to show the intersample difference in the high-shear-stress region. Since the objective of this study was to investigate the deformation response of RBCs to a cyclically reversing shear flow with an extremely high peak shear stress exceeding 100 Pa. we decided to employ the L/W as an index to quantify RBC deformation.

RBC image analysis 2: RBC velocity (V^sub RBC^)

At first, the center of the RBC image was determined from the intersecting point of the major and minor axes of the ellipsoidal image. The displacement of the RBC central point over three video frames was then measured to calculate the V^sub RBC^, knowing that the elapsed time over three frames was 0.4 ms. The time course relation among the LfW, V^sub RBC^, and glass plate velocity (V^sub Plate^) was then examined in each condition.

Location of the focused RBCs

The location of the RBCs focused between the two parallel glass plates was estimated assuming that the How field between the two parallel glass plates was an oscillatory quasiCouette flow. Based on the theory of oscillatory Couette flow (Materials and Methods). V^sub RBC^ varies in a sinusoidal manner. The ratio of V^sub RBC^ to V^sub Plate^ can thus provide the approximate depth of the focused RBCs within the 30-µm clearance of the PGPA.

Derivation of V^sub plate^

The temporal gap sensor signal was converted to yield the position data of the glass plate using the predetermined calibration data. Since the gap sensor signal contained white noise and since the derivation of V^sub plate^ involved computation of the first derivative of the position signal with respect to time, signal smoothing was employed to minimize noise enhancement. The moving average was repeated tour times with different data samples each time. For the glass plate moving at 1 and 2 Hz, the first moving average involved 801 data points (400 data points before and after the current data point), the second 601 data points, the third 401 data points, and the fourth 201 data points. For the frequency of 3 Hz, 501. 401. 301. and 201data points were used, and finally, for the frequency of 5 H^sub z^, 301, 101,51, and 11 data points were used. The V ^sub plate^ was then obtained using the following difference equation (Eq. 12), which was derived as the fourthorder approximation of the convection equation in the area of fluid dynamics (31).

Estimation of the time-varying shear stress (t)

The time-varying shear rate was obtained by dividing the V^sub plate^, by the clearance of 30 µm. followed by derivation of the time-varying shear stress (t) by multiplying the shear rate with the viscosity of the suspension 0.200 Pa s. The time course changes of the relation between L/W and t of RBCs were then analyzed.

RESULTS

Fig. 3. a-d. showed the sample images of RBCs for the cyclically reversing frequency of (a) 1 Hz, (b) 2 Hz, (c) 3 Hz, and (d) 5 Hz. For each frequency, the left panel (d) denoted RBC images with minimal L/W: the middle panel (ii) those at the zero shear stress (t^sub 0^) when the V^sub plate^ was zero: and the right panel (iii) those with the maximal L/W. The RBCs were moving in the horizontal direction.

In Fig. 4, a-d, panel t showed the plate movement and its velocity for one second, panel ii the plate velocity and RBC velocity for one cycle, and panel iii the instantaneous shear stress and L/W of RBC during deformation and recovery phases. In the figures, the movement of the glass plate toward the right, away from the gap sensor on the microscope stage of Fig. 2. was denoted as the positive direction. The zero time in each ligure denoted the moment when the glass plate was closest to the gap sensor, V^sub plate^ (t^sub 0^). Both V^sub RBC^ and V^sub plate^ varied like sine waves. The correlation coefficients for the linear regressions between the V^sub RBC^ and V^sub plate^ signals was computed and shown in Fig. 5, a-d. Since the correlation coefficients between them were ~1.0, it could be said that the V^sub RBC^ followed the V^sub plate^ closely without any time lag, but with its amplitude being ~60% of the V^sub plate^ amplitude. The slope of the linear regression line a (V^sub RBC^ = a × V^sub plate^) between V^sub RBC^ and V^sub plate^ was 0.64, 0.57, 0.63, and 0.59 for 1, 2. 3, and 5 Hz, respectively, of plate frequency. The ratio of the V^sub RBC^ to the V^sub plate^ at the time of the maximal level of V^sub plate^ (V^sub plate^, MAX) was 0.64 ± 0.07, 0.54 ± 0.05, 0.62 ± 0.04 and 0.59 ± 0.03 for 1 Hz, 2 Hz, 3 Hz, and 5 Hz, respectively, showing good agreement with the slopes of the linear regression lines. The RBC locations within the glass plates were then speculated to be 19.2, 17.1, 18.9, and 17.7 µm from the stationary bottom plate (Table 1) derived from the slopes of the linear regression equations of Fig. 5, a-d.

The L/W changes shown in Fig. 4, a (iii)-d (iii) did not follow the patterns of V^sub plate^ change exactly, but they showed asymmetrical response patterns consisting of fast deformation and slow recovery phase. The deformation phase was comprised of three periods: 1). a pre-rapid-elongation period lasting ~0.1 normalized fractional time (absolute time was normalized to one-cycle time); 2), rapid and linear elongation periods between 0.1 and 0.14 and between 0.6 and 0.64 of the normalized fractional time: and 3), a slow elongation period. The L/W during the rapid elongation period increased fairly linearly with lime and shear stress (Fig. 6, a and h). During this period, the shear-stress level also increased linearly with time (Fig. 6 r). The slope of the linear regression lines with respect to time in Fig. 6 a increased from 34.9 to 89.4, 128.7, and 209.6/s, whereas those with respect to shear stress in Fig. 6 h decreased from 0.14 to 0.092, 0.066, and 0.039/Pa with the frequency.

Fig. 7 showed the mean L/W change rate during the rapid and linear elongation period (0.1 and 0.14 of the normalized cycle time) in comparison to the recovery period (0.46 and 0.58 of the normalized cycle time) as a function of the cyclically reversing frequency. The L/W change rates in both periods increased as a function of frequency, with the elongation rate more than twice the recovery rate.

An interesting finding was that dynamic L/W response lagged behind the shear-stress change during the shape recovery phase. The RBCs remained elongated even when the shear stress reached the zero level and continued to further decrease even after the shear stress started to increase in the other direction. Table 2 summarized the time lag. The mean lag time was 80 ins for 1 Hz, 40 ms for 2 Hz, 32 ms for 3 Hz, and 16 ms for 5 Hz, whereas the percent lag time normalized to one-cycle time was 8.0% for 1 Hz, 2 Hz, and 5 Hz, and 9.7% for 3 Hz.

In Fig. 8, the two-dimensional T-L/W relationships of the data in Fig. 4 are shown. The shear stress t was calculated by multiplying the lime-varying shear rate by the lluid viscosity. The positive value of t denoted that the flow was toward the right in Fig. 2. The t-L/W patterns rotated in the counterclockwise direction in the right-half plane (the glass plate moving toward the right in Fig. 2). but in the clockwise direction in the left-half plane, showing the characteristic points for V^sub plate,zero^ (A), L/W^sub MIN^ (B and D, respectively), and t^sub MAX^ (C and E. respectively). The L/W^sub MAX^, and L/W^sub MIN^ occurred twice in each cycle, one for either direction of the plate movement. Table 3 showed the numerical data for characteristic points on the t-L/W curves. From Fig. 8. it was clear that the amplitude of L/W (L/W^sub AMP^ = L/W^sub MAX^ - L/W^sub MIN^) increased with the cyclically reversing frequency. The value of L/W at the zero stress point t^sub 0^ showed a tendency to increase from 2.77 ± 0.24 to 3.20 ± 0.30. 3.48 ± 0.33. and 4.44 ± 0.45 with the frequency.

Fig. 9 a shows the L/W^sub MAX^ L/W^sub MIN^, L/W at t^sub 0^ (L/W^sub 0^), and L/W^sub AMP^ versus cyclically reversing frequency. The value of L/W^sub MAX^ increased from 4.06 ± 0.45 to 4.60 ± 0.33, 4.87 ± 0.46. and 6.34 ± 1.04. whereas L/W^sub MIN^ increased from 1.70 ±0.14 to 1.81 ± 0.16. 2.16 ± 0.31. and 2.40 ± 0.24. and L/W^sub 0^ from 2.77 ± 0.14 to 3.20 ± 0.16. 3.48 ± (1.31. and 4.37 ± 0.24, respectively. Also shown in Fig. 9 b are DI^sub MAX^, DI^sub MIN^. DI^sub 0^, and DI^sub AMP^ (DI^sub MAX^ - DI^sub MIN^) for the purpose of comparison with L/W. The DI value approached 1.0 and was not a sensitive indicator for shape change. In particular. DI^sub AMP^ decreased as the frequency increased.

The L/W^sub AMP^ of RBCs normalized to the amplitude of t (t^sub AMP^) was displayed in Fig. 10 a. As the reversing frequency increased, the frequency response expressed by the ^-/WAMP/TAMP diminished. Likewise. DI^sub AMP^ normalized to the shear-stress amplitude also diminished with frequency (Fig. 10 h).

DISCUSSION

In this study, for the first time, the dynamic response of RBCs to a cyclically reversing shear flow with the peak shear stress level >100 Pa has been quantified using a specially built cyclically reversing shear flow generator. The validity of the results, however, depends on assumptions and questions related to the CRSFG system and data analysis. The first half of the discussion thus addresses the validation of 1). the stability of the PGPA. which is supposed to generate a quasi-Couette flow in the space ot 30 µm; 2), the cell orientation in the shearflow field; and 3), L/W as a parameter to express the deformation of RBCs. In the latter half of the discussion, the dynamic deformation and recovery response of RBCs to a cyclically reversing shear How will he analyzed, and a simple model comprised of an elastic linear element for the deformation phase and a parallel combination of a spring and dashpot for the nonlinear recovery phase is proposed. The discussion also touches upon the shape recovery phenomenon in comparison to previous works, followed by future perspectives.

Validation and verification of the experimental conditions

Stability of the PGPA system and oscillatory quasi-Couette flow generation

In analyzing the dynamic response of RBCs to a cyclically reversing shear flow, we made an assumption that the flow field of RBC suspension between the two plates was steady anil oscillatory quasi-Couette. The following three findings were obtained to verify the stability of the PGPA and the assumption of an oscillatory quasi-Couette flow. 1). The time course V^sub RBC^ measured at the midpoint in the PGPA showed the cyclic sinusoidal fluctuation with its amplitude ~0.60 of the V^sub plate^, (0.64 ± 0.07. 0.54 ± 0.05. 0.62 ± 0.04. and 0.59 ± 0.03 for 1 Hz, 2 Hx. 3 Hz, and 5 Hz, respectively) (Fig. 4 a ii-d ii, and Table 1), 2), The L/W of RBCs. both maximal and minimal levels, increased almost linearly as the reversing frequency increased (Fig. 9 a), indicating the stability of the PGPA over the frequency range between 1 and 5 Hz. 3), In addition, the V^sub RBC^ followed the V^sub plate^ without any time lag (Fig. 4, a ii-d ii, and linear regression analysis in Fig. 5, a-d). These three findings supported the stability of the PGPA and the theory that the CRSFG would successfully generate an oscillatory quasi-Couette flow in the 30-µm space of the PGPA.

Cell orientation

A question was raised concerning the accuracy of the dimensions of RBC images in the video frame because of cell orientation. RBCs may he randomly oriented under a reversing shear flow and the two-dimensional projected images may vary depending on the inclination level (5.10-13) of the RBCs in the flow field. The inclination level would change under the cyclically reversing shear flow. According to the reports by Keller and Skalak in 1982 (32) and Tran-Son-Tay in 1984 (10), the inclination level decreased when the elongation level or shear rate of the RBCs increased. In our observation, since the membrane of the RBCs was always under some level of shear stress, the inclination problem was expected to be minimal. We observed some RBCs with their shape partially out of focus right after their shapes returned to the minimal size under the oscillating frequency of 2 Hz and 3 Hz, whereas no specific events related to the orientation problem were observed at 1 Hz. In this study, therefore, special attention was paid to eliminate from data analysis those RBC images that were partially out of focus.

L/W versus DI

As for L/W to describe the dynamic responses of RBCs to a cyclically reversing shear flow with extremely high shear stress >100 Pa, the L/W^sub MAX^ and L/W^sub MIN^ values increased linearly with the increase in shear stress secondary to an increase in the reversing frequency (Fig. 9 a). The DI, however, did not provide a sensitive indication of the shape change, because it approached the value of 1.0. In particular, the DI^sub AMP^ decreased with the frequency in contrast to the L/W^sub AMP^, which increased, although both amplitudes normalized to the input shear-stress amplitude diminished with the frequency (Fig. 9 b)). From this standpoint, the selection of L/W was the right decision fur analyzing the shape change of RBCs at higher shear stress. Although the L/W showed better resolution than did the DI in describing a two-dimensional shape change of RBCs, it was only a ratio parameter between the longer and shorter axes assuming the RBC as an ellipsoidal shape. When the shape of the RBCs deviated from an ellipsoid at low shear stress as well as at extremely high shear stress, the L/W would fail to accurately represent the physical shape change. A better and more precise method of quantifying the morphological change of the RBCsunder varying shear stress would bring a better understanding of the RBC deformation and recovery process in a three-dimensional flow field.

Now the discussion will focus on the analysis of the dynamic RBC deformation and recovery under a cyclically reversing shear flow.

Dynamic response of RBCs to an oscillatory quasi-Couette shear flow

As for the response of RBCs to the oscillatory quasi-Coucttc shear flow, the L/W did not show sinusoidal fluctuations. although the time-course V^sub RBC^ and V^sub plate^ varied like sine curves (Fig. 4. d-d). The L/W showed asymmetrical response patterns comprised of a rapid deformation and a slow recovery phase. An interesting finding was that a time lag was observed between t^sub 0^ and L/W^sub MIN^ (Fig. 4, a iii-d iii. and Tables 2 and 3). Additionally, the L/W^sub 0^ increased almost linearly with the frequency (Figs. 8 (zero-stress point A) and 9 a. and Table 3). The RBCs remained elongated and L/W continued to decrease even after the shear stress was reduced to zero and after the flow direction was reversed.

The two-dimensional display between L/W and t as shown in Fig. S provided a better viewing of the dynamic deformation and recovers responses of RBCs to a cyclically reversing shear flow. The butterfly-like profiles of Fig. S were thought to represent the normal dynamic deformation and recovery processes of RBCs under a cyclically reversing shear flow. As the reversing frequency increased, the loop became extended in the horizontal direction, increasing the instantaneous stress and simultaneously shitting upward to a higher L/W level. The peak shear-stress level increased together with the L/W^sub MAX^ and L/W^sub MIN^, and hence, the amplitude L/W^sub AMP^ became greater as the frequency increased.

Although the levels of the L/W^sub MAX^, L/W^sub Min^ L/W^sub 0^ and L/W^sub AMPX^ all increased with the frequency as shown in Fig. 9 A, the L/W^sub AMP^ normalized to t^sub AMP^ decreased with the frequency, reaching a plateau at ~5 Hz (Fig. K) </). Figs. 9 and 10 together disclosed that the RBCs continued to change their shape by increasing L/W in response to a cyclically reversing shear flow, hut the normalized change of their L/W^sub AMP^ with respect to the shear-stress amplitude diminished upon reaching the plateau. The frequency response characteristics of RBC shape change to the reversing shear How diminished with the frequency.

The deformation phase consisted of three periods: 1), a pre-rapid-elongation period lasting 0.1 normalized time unit; 2). rapid and linear elongation periods: and 3), a slow elongation period. During the rapid and linear elongation periods, L/W increased fairly linearly to both lime and shear stress changes (Fig. 6, a and b). It should be noted that during this period, the shear stress increase rate was remarkably high with 242. 968, 1961, and 5376 Pa/s (Fig. 6 r). Regardless, the L/W increase was rather linear, exhibiting an elastic property. The slope of the linear regression lines (d(L/W)/dt) for the L/W-time relationships increased from 34.9 to 89.4, 128.7, and 209.6/s, whereas d(L/W)/dt for the L/W-strcss relationships decreased from 0.145 to 0.092, 0.049, and 0.040/Pa with the increase in frequency from I to 5 Hz. This result implies that the faster the shear rate, the faster is the RBC response in terms of L/W change, whereas the sensitivity of L/W change with respect to shear stress change diminishes if the initial L/W and shear levels are larger (Fig. 10 a).

Fig. 11 summarizes the characteristic points on the t-L/W curves. Also shown are the speculated curves tor the L/W versus steady uniform shear stress. Since the L/W increased linearly to the shear stress level during the rapid elongation period in our experiment (Fig. 6 b), but since the maximal L/W under the uniform shear cannot be attained under a cyclically reversing shear flow, particularly at higher reversing frequency, the L/W level in the steady uniform flow would be higher than the L/W^sub MAX^ of the reversing shear flow at higher frequency. The L/W^sub MAX^ of the reversing flow, however, should converge to that of the uniform shear flow at lower frequency.

An interesting finding was the lag time observed between the zero shear time and the L/W^sub MIN^ time. The RBCs remained elongated even if the external shear stress level returned to zero, and the L/W level continued to decrease for a little while even though the flow direction was reversed and the shear-stress level started to increase. At the time of zero shear stress, the energy stored in the RBC membranes to recover their original shape probably overrode the external tensional stress. The L/W. therefore, continued to decrease even though the external shear stress started to increase. As summarized in Table 2. ~8.0% of the time after the zero-shear-stress time for each frequency except 3 Hz (9.7%). the dynamic equilibrium between the intracellular recovery force and extracellular shear stress occurred to reverse the L/W change process. Further increase in the extracellular shear stress thus exceeded the cell's recovery force, and hence, RBCs started to elongate.

In the following section, a possible mechanical model to explain the dynamic response patterns of RBCs to a cyclically varying external shear flow is presented.

A simple mechanical model to explain dynamic RBC responses

Based on the experimental findings, a simple mechanical model as shown in Fig. 12 was proposed. The model consists of two phases, the deformation and recovery phases. The deformation phase consists of three periods: a pre-rapid-elongation period (Fig. 12 a, S1), a rapid elongation period (Fig. 12 a, S2), and a slow elongation period (Fig. 12 a, S3). An elastic linear element (Fig. 12 b) represents the rapid elongation period, and a parallel combination of a spring and dashpot (Fig. 12 c) represent the nonlinear recovery phase. A question was raised, however, concerning the validity of the model to describe the entire process of the deformation and recovery phase. To verify the validity of the model, the rapid elongation period was expressed as a constant-modulus model:

dt/dx = K,

where x is the displacement and K is the clastic modulus as obtained from the inverse of the slope in Fig. 6 b. Solving the above equation for t, we obtain

t = Kx + C,

where C is the intercept of the shear-stress axis. Solving for displacement x, we obtain

x = (t - C)/K

By substituting t values from the experiment and initial shear-stress values for the linear portion, L/W curves were generated and are shown in Fig. 13. Although the model showed good agreement with the experimental data for the rapid elongation period, it deviated afterward from the experimental value as the shear-stress level increased. The deviation would possibly represent the alteration of the membrane elastic modulus depending on the membrane strain and shear-rate level, as reported by other workers (10,12).

Shape recovery phenomenon in comparison with the conventional reported studies

Previously reported studies of RBC shape recovery by Chien et al. (33), Hochmutli et al. (34). Sutera et al. (35). and Fischer (36) all concerned the response to a steplike removal of the constant tensional force applied to the cell membrane. On the other hand, the shape recovery shown in our study occurred under sinusoidal decrease of the extracellular shear level. The extracellular shear stress thus acted upon RBCs continuously by altering the tension level. This point is quite different from those of previous studies, where the applied force to RBCs was completely removed to observe the recovery process. As for the characteristics of the dynamic deformation of RBCs under cyclically reversing shear How, Chien's study, that ism RBCs subjected to a steplike aspiration pressure and removal by a micropipette, has shown similar response patterns to our finding (33).

Future perspectives

The membrane moduli µ^sub m^ and ?^sub m^ (where µ^sub m^ and ?^sub m^ are the shear moduli of elasticity and shear viscosity, respectively) are variable, depending on shear stress and strain rate as well (10,12). These parameters were derived based on the energy balance between the input power (W^sub p^) by extracellular shear condition, and the rate of energy dissipation in both the membrane (D^sub m^) and cytoplasm (D^sub c^). However, they were evaluated only under the steady shear condition (10,13). In the steady uniform flow condition, W^sub p^ would keep a constant value, whereas in our experiment, W^sub p^ would be variable under the varying shear flow. Consequently, the energy balance during the cycle would become variable and more complicated in comparison to the condition under the steady uniform shear flow. A more detailed theoretical consideration of RBC deformation under the reversing shear flow would require quantification of additional parameters, such as the tank-treacling frequency and the effect of the ratio of intra- to extracellular viscosity.

The dynamic deformation behavior of RBCs as demonstrated in this study can be expected to occur in cardiovascular systems such as the heart chamber and narrow vessels, and prosthetic devices where shear stress varies from time to time and acts upon RBCs from all directions. The deformation and recovery responses under a varying shear How may be different from cell to cell depending on their aging and pathophysiological states, including mechanical damages. The study of dynamic responses of various cells using our system will possibly clarify the pathophysiological status of blood cells related to various cardiovascular diseases and make preventive measures possible.

CONCLUSION

This study quantified the dynamic deformation and recovery responses of RBCs to a shear stress fluctuation with peak shear stress of 53, 108, 161, and 274 Pa using a cyclically reversing shear flow generator developed in our laboratory.

There were four important conclusions that could be drawn from this study. First, the V^sub RBC^ measured at around the midpoint of 30-µm parallel glass plate assembly followed the V^sub plate^ without any time delay with its amplitude ~60% of the V^sub plate^. On the other hand, L/W consisted of a rapid deformation phase, and a nonlinear shape recovery phase. The deformation phase consisted of three periods: 1), a pre-rapid-elongation period; 2), rapid and linear elongation periods; and 3), a slow nonlinear elongation period. There was a time lag between L/W^sub MIN^ and t^sub 0^, which was 8.0% of one cycle time, independent of the reversing frequency, for 1, 2, and 5 Hz; for 3 Hz, it was 9.7% of one cycle time.

Second, a simple mechanical model consisting of an elastic linear element during the rapid elongation period and a parallel combination of a spring and dashpot during the nonlinear recovery phase tit partially to describe dynamic deformation and recovery responses of RBCs to a cyclically reversing shear flow.

Third, the dynamic response behavior of RBCs under a cyclically reversing shear flow was different from the conventional shape change in which a steplike force is applied to and completely releases from the RBCs. The RBCs under a cyclically reversing shear flow are exposed to continuously varying shear flow to exhibit different dynamic responses.

Last, the current method was shown to be a simple approach for evaluating the RBC dynamic deformation and recovery process in response to rapidly varying shear flows that simulates the How patterns occurring in cardiovascular devices such as continuous-flow blood pumps.

We acknowledge the stimulating discussion of Associate Professor Hiroshi Mizunuma of Tokyo Metropolitan University, and Matsunami Glass Ind. of Tokyo, Japan, for their kind supply of material.

This study was partially supported by a grant-in-aid (14208103) to the principal investigator (S.T.) from the Japan Society for the Promotion of Science (JSPS). N.W. was supported by an Ishidu Shun Memorial Scholarship from April 2004 to March 2005 and by a research fellowship (17-3690) from the Japan Society for the Promotion of Science starting April 2005. N.W. is a Research Fellow (DC2) of the Japan Society for the Promotion of Science (JSPS).