Atte

Proceedings of the ASME 2020 Fluids Engineering Division Summer Meeting
FEDSM2020
July 13-15, 2020, Virtual, Online

ee
nd

O
dea

R

y
nl

FEDSM2020-20074y
op

C

DEFORMATION OF ERYTHROCYTE IN COUETTE TYPE OF PULSATILE SHEAR FIELD
Shigehiro Hashimoto1
Kogakuin University,
Tokyo, 1638677, Japan

ABSTRACTThe experimental methodology has been designed forevaluation of the cyclic deformation of an erythrocyte in thepulsatile shear field in vitro. To observe the deformation of thesuspended erythrocytes in the Couette type of the shear flow, arheoscope system has been manufactured. The system consists ofa pair of counter-rotating parallel disks and an inverted phasecontrast microscope. The human erythrocytes were suspended in
the dextran aqueous solution of high viscosity. The rotating
speed varies sinusoidally to make the pulsatile shear field. The
deformation of each erythrocyte was measured at the video
image of the rheoscope. The experimental results show that thesystem is available to measure the following behavior of anerythrocyte. The ellipsoidal shape of each erythrocyte varies
cyclically to follow the pulsatile cyclic shear field. The delay of
deformation phase of each erythrocyte in the cycle is related to
the frequency of the cyclic variation of the shear field. The delay
is also related to deformability of the erythrocyte.
Keywords: Biomedical Engineering, Shear Flow, Pulsatile
Flow, Erythrocyte and Deformation

Δv
ΔV
Δx
α
γ
η
ρ
Δω
τ

1. INTRODUCTION
An erythrocyte has large flexibility. An erythrocyte passes
through micro-circulation, of which the internal diameter issmaller than the diameter of the erythrocyte [1]. After circulation
through the blood vessels for days, the erythrocyte is damaged
and trapped in the micro-circulation systems in vivo.
Deformation of the erythrocyte has been investigated withvarious methodologies: the mechanism of deformation [2], theshape recovery process [3], fatigue [4], the correlation to thevelocity [5], and disease [6]. Deformation of the erythrocyte
observed in vivo and in vitro using various devices: a microchannel [7], a filter [8], a micro slit [9,10], and a rheoscope[11,12]. An erythrocyte deforms in the shear flow. While anerythrocyte is exposed to the shear flow, it shows tank tread
motion at the membrane [11], and ejects contents (hemolysis [12,13]) through the crevasse of its own membrane, before
fragmentation.
The repetitive stresses are applied to the erythrocyte during
circulation in vivo: in the pulsatile flow, and through themicrocirculation. The erythrocyte shows the viscoelastic
property, when it recovers to the original shape [3,4].
In the present study, the cyclic deformation of anerythrocyte has been measured microscopically in the pulsatile
shear field to detect the deformability of an erythrocyte in vitro.

NOMENCLATURE
alength of major axisblength of minor axisddistance between two parallel wallsffrequency of oscillations of diskP
deformation ratiordistance between rotational axis andobservation pointRradius of disk
Re
Reynolds numberS
delay ratio
TperiodΔT
delay
Vcircumferential velocity of disk
1

velocity difference between layers
circumferential velocity difference between
disks
distance between two layers
Womersley number
shear rate
viscosity of fluid
density of fluid
angular velocity difference between disks
shear stressContact author: shashimoto@cc.kogakuin.ac.jp

1

Copyright © 2020 ASME

2. MATERIALS AND METHODS
2.1 Rheoscope System
A rheoscope system has been manufactured to observe thedeformation of the suspended erythrocytes in the shear flow (Fig.
1). The system consists of a pair of parallel disks and an inverted
phase-contrast microscope (IX71, Olympus Co., Ltd., Tokyo).
The erythrocytes are sheared in the Couette flow between a pairof counter-rotating parallel disks, which are made of transparent
silica glass. The velocity of the middle plane between the counterrotating disks is zero in the shear field of Couette type flow. The
velocity of the erythrocyte floating near the middle plane in thefluid is low. The erythrocyte with the low velocity stays within
the observation area of the objective lens of the microscope fora long time.
The radius (R) and thickness of the disk is 40 mm and 5mm, respectively. The distance between the rotational axis andthe observation point (r) is 27 mm. Both disks are supported attheir rim by the ball bearing of 100 mm diameter. The edge ofthe lower disc has a wall to prevent fluid spills. The distance
between two disks (d) is adjusted between 0.08 mm and 0.12
mm. The distance is confirmed by the volume of the suspension
filled between the disks (between 0.3 cm 3 and 0.6 cm3), and by
the calibrated focus positions of the walls with the microscope.

The circumferential velocity difference at the observation
point is calculated by Eq. (2).
ΔV = r Δω

In Eq. (2), r is the distance between the rotational axis andthe observation point, and Δω is the angular velocity difference
between the parallel disks.
With a DC (direct current) stepping motor, the angularvelocity of each disk (ω < 0.1 rad s–1) is controlled in one way.
The rotating directions of two disks are counter direction(clockwise and counterclockwise) each other. The absolutevalues of the angular velocities of two disks are adjusted to thesame value each other.
The angular velocity is periodically (period: between 0.67 s
and 7 s) varied with the sinusoidal mode in one way. The cyclic
fluctuation of the velocity of the disk makes the cyclic
fluctuation of the shear rate (γ) between disks. At the observation
point (r = 27 mm), the shear rate (γ) is lower than 6.8×10 s–1,
calculated with Δω (< 0.2 rad s–1) and with d (between 0.08 mm
and 0.12 mm) in the present experiment.
2.2 Erythrocyte Deformation
To apply the high shear stress on each erythrocyte at the low
shear rate, erythrocytes were sparsely suspended in a dextran(molecular weight between 2×105 and 3×105) aqueous solutionof high viscosity. The erythrocytes were sheared in the Couette
flow between two counter-rotating parallel disks at 298 K. The
human erythrocytes collected from the author were used in theexperiment.
The shear stress (τ) in the suspension is calculated by Eq.
(3).

Upper diskRr
Middle

(2)

d

τ=ηγ

(3)

Erythrocyte

In Eq. (3), η is the viscosity of the fluid, and γ is the shear
rate between the disks.
Variation is made on the shear stress (< 4 Pa) with η(between 0.02 Pa s and 0.06 Pa s) and with γ (lower than 6.8×10s–1) in the present experiment. The viscosity of the dextran
aqueous solution was measured with a cone and plate type ofviscometer.
The deformation of cells was observed with the microscopeand recorded with a video camera system. The long-focus
objective lens is used to observe erythrocytes suspended near the
middle plane. The stationary images were captured from thevideo movie every 0.033 s.
Each erythrocyte deforms from the biconcave shape to theellipsoid shape in the shear field. On the microscopic image, theoutline of each erythrocyte was traced, and the contour of eachcell was approximated to ellipsoid. On the ellipsoid, the lengthof the major axis (a), and the minor axis (b) were measured (Fig.
3). The ratio of axes is calculated as the deformation ratio (P) by
Eq. (4).

v
Lower disk

Microscope

FIGURE 1: RHEOSCOPE WITH COUNTER WITH COUNTERROTATING DISKS.

The shear rate γ is calculated by the following equation.
γ = ΔV / d(1)

In Eq. (1), d is distance between two parallel disks, and ΔV
is the circumferential velocity difference between the upper andthe lower disks. At the observation point with the same distancefrom the rotational axis, the shear rate γ is constant between two
disks regardless of the distance from the surface of the disk in
the Couette type of the flow.

P=1 － b/a2

(4)

Copyright © 2020 ASMEAt the circle, P = 0. As the ellipsoid becomes flat, P
approaches to unity. As the shear rate fluctuates periodically, thedeformation ratio of each erythrocyte fluctuates periodically. The
cyclic fluctuation of the deformation ratio of each erythrocyte
was polynomially approximated to the curve by at the timecourse tracings (Fig. 4).
The velocity of each erythrocyte was calculated by tracing
the same erythrocyte by the image every 0.033 s (Fig. 3). The
cyclic fluctuation of the velocity of each erythrocyte waspolynomially approximated to the curve by at the time course
tracings (Fig. 4). The period of the cyclic shear rate (T [s]) is
confirmed by the movement of each erythrocyte. The delay (ΔT[s]) (Fig. 2) of the deformation of each erythrocyte is calculatedby the phase difference of the approximated cyclic curves
between the velocity and the deformation ratio. The delay ratio(S) was calculated by Eq. (5).
S = ΔT / T

direction of the major axis (a) of each ellipsoid is parallel to theflow.
Figures 4a-4c exemplify the tracings of the deformation
ratio (cross) and the velocity (rhombus) of each erythrocyte.
Some erythrocytes deform to the deformation ratio of 0.65 under
the shear stress field lower than 4 Pa. Data are scattered in Fig.
4: the deformation ratio between 0.01 and 0.9, and the velocity
between － 40 μm/s and 100 μm/s. Data were able to beapproximated to the cyclic curve (red lines in Fig. 4). From theapproximate curve, the period (T) and the delay (ΔT) (Fig. 2)
were calculated at each erythrocyte. Each period was in the range
between 0.7 s and 7 s. Each delay ratio was in the range between
0.1 and 0.6.
(A)

(5)

The delay ratio (S) is zero, when the cyclic deformation
occurs at the timing synchronous to the periodically fluctuating
velocity. The delay ratio (S) approaches to unity, when the delayof the deformation extends. When the cyclic deformation occurs
at the counter phase of the cycle of the shear rate, the delay ratio
becomes 0.5.

vbaIntensity
1.53

v’

Velocity
1

T

0.52
0
-0.51

(B)

ΔT
Deformation ratio

-1
-1.50
0

3

6

9
Time [s]FIGURE 2: DEFORMATION RATIO (DOTTED LINE) AND
VELOCITY (SOLID LINE): PERIOD (T), DELAY (ΔT)

3. RESULTS
Figure 3 exemplifies the image of erythrocytes in the shear
field of the rheoscope system manufactured in the present study.
The media moves from left to right in the front layer (v), while
the media moves from right to left in the rear layer (v’). The
erythrocytes near the middle layer are able to be observed for along time within the focused area of the microscope, while theerythrocytes are sheared in the Couette flow. Each erythrocyte
deforms from a biconcave to an ellipsoid in the shear stress field.
The ellipsoid is elongated, as the shear stress increases. The

0.05 mm
FIGURE 3: DEFORMED ERYTHROCYTES IN SHEAR FIELD IN
SUCCESSIVE IMAGES: (A) AND (B): VELOCITY OF CELL (v)3

Copyright © 2020 ASME

Deformation Ratio

1

80

0.6
40
0.4
0

0.2
0
5.00

0

6

8

A micro channel can simulate the microcirculation system[7]. The effects of shear flow on cells were observed in theprevious studies. The micro-slit is also useful for observation of
deformation of a cell [14].
The cone and plate type instruments are used to makeuniform Couette type flow [13]. There are several reasons why
the parallel disks type is chosen for the rheoscope in the present
study. It is easier to make transparent disk with a flat surface.
The distance between two disks is uniform, so that it is easier tomaintain the distance between disks constant. Because thehardness of the disks supports the preciseness of the gap distance
between two disks, the glass is selected for the material of disks
in the present study.
The shear rate between two parallel disks increases inproportion to the distance from the rotational axis. The shear ratevaries with less than 5 percent in the observation area of 1 mm
square, of which distance from the rotational axis is 27 mm. The

Velocity [μm/s]

Deformation Ratio

4

PERIOD (T): DOTTED LINE, APPROXIMATE STRAIGHT LINE

0.7
9.00
Time [s]2

FIGURE 5: RELATIONSHIP BETWEEN DELAY RATIO (S) AND

0.2
0
6.00

13.00

Period (T) [s]

0.8

0.8

9.00 11.00
Time[s]

0.0

0.9

0.4

0.6
7.00

0.5

AND VELOCITY (RHOMBUS): APPROXIMATE CURVE(DEFORMATION RATIO (DOTTED LINE) AND VELOCITY(SOLID LINE)): PERIOD 2.7 s, DELAY RATIO 0.09

0.6

0.7

0.2

Delay ratio (S)FIGURE 4a: TRACINGS OF DEFORMATION RATIO (CROSS)

1

0.8
0.4

1.0

-40
11.00

7.00
9.00
Time [s]0.6

AND VELOCITY (RHOMBUS): APPROXIMATE CURVE(DEFORMATION RATIO (DOTTED LINE) AND VELOCITY(SOLID LINE)): PERIOD 7 s, DELAY RATIO 0.34

120

0.8

0.9

FIGURE 4c: TRACINGS OF DEFORMATION RATIO (CROSS)

Velocity [μm/s]

Deformation Ratio

1

0.8

0
5.00

4. DISCUSSION
The mechanism of erythrocytes deformability was a target
for previous studies [2]. Deformability of erythrocytes can be anindex for diagnostics [6]. Both the shear rate and the shear stressgovern destruction of erythrocytes in the shear flow [13].

1Velocity [μm/s]

Figure 5 shows the relationship between the delay ratio andthe period of cyclic fluctuation of the shear stress on ten
erythrocytes collected from ten different cases. The dotted line
shows the approximate straight line on data. The delay ratio
scatters, which may show the property depends on individualityof each erythrocyte. Some erythrocytes deform within the shortdelay. Even in the cyclic flow with the long period (with the low
frequency) of seven seconds, the deformation of the erythrocyte
occurs with a certain ratio of the delay. In the cyclic flow withthe period shorter than one second (with the frequency higher
than 1 Hz), the deformation of the erythrocyte tends to bedelayed.

12.00

FIGURE 4b: TRACINGS OF DEFORMATION RATIO (CROSS)AND VELOCITY (RHOMBUS): APPROXIMATE CURVE(DEFORMATION RATIO (DOTTED LINE) AND VELOCITY(SOLID LINE)): PERIOD 4.9 s, DELAY RATIO 0.35

4

Copyright © 2020 ASME

shear rate in the observation area is approximately mean value inthe whole volume of the suspension between two disks.
In many cases, each erythrocyte keeps own position in thesame plane, so that the velocity of each erythrocyte shows thevelocity of the layer, where each erythrocyte is floating. In somecases, on the other hand, erythrocyte moves over the centerstationary plane, and moves to the counter direction (Fig. 4a).
In the shear field, the difference between velocities of twolayers (Δv) is calculated by Eq. (6).
Δv = γ Δxellipsoidal shape in the three-dimensional figure. The disk, on
the other hand, shows a circle or an ellipse in two-dimensional
projection according to the direction. The deformation ratio of0.65 shows that the erythrocyte is exposed to the shear stressaround 3 Pa in Couette flow in the present study. The erythrocyte
with high flexibility would make the cyclic deformation withvery short delay to follow the periodically fluctuated shear stress.
The periodical velocity fluctuation is used to trace the phase of
the cyclic shear stress fluctuation in Couette type of the flow inthe present study.

(6)
5. CONCLUSION
Using the rheoscope of a pair of counter rotating disks, thecyclic deformation of an erythrocyte has been measured
microscopically in the pulsatile shear field to detect the
deformability of an erythrocyte in vitro.

In Eq. (6), γ is shear rate, and Δx is the distance between twolayers: Δv is 68 μm s–1 at γ of 6.8×10 s–1 with Δx of 1 μm. Whenthe erythrocyte moves to the neighbor layer of the shear flow, thevelocity of the erythrocyte varies. The variation can cause
scattering of the velocity of the erythrocyte at the tracings (Fig.
4).
The dextran solution is applied to the suspension to inhibit
turbulence in the flow with increase of viscosity. Reynolds
number (Re) is a useful index for estimation of the turbulent flow.
Re = ρ d V / η

ACKNOWLEDGEMENTS
The author thanks to Mr. Ryo Muto for his assistance of theexperiment.
REFERENCES[1] Skalak, R. and Branemark, P.I. “Deformation of Red
Blood Cells in Capillaries.” Science Vol. 164 No. 3880
(1969): pp. 717-719.
[2] Brody, J.P., Han, Y., Austin, R.H. and Bitensky, M.
“Deformation and Flow of Red Blood Cells in a
Synthetic Lattice: Evidence for an Active Cytoskeleton.”
Biophysical Journal Vol. 68 (1995): pp. 2224-2232.
[3] Tajikawa, T., Imamura, Y., Ohno, T., Muranishi, F.,
Kubota, M. and Ohba, K. “Measurement and Analysis of
the Shape Recovery Process of Each Erythrocyte for
Estimation of Its Deformability Using the Microchannel
Technique: the Influence of the Softness of the Cell
Membrane and Viscosity of the Hemoglobin Solution
Inside the Cell.” Journal of Biorheology Vol. 27 No. 1-2(2013): pp. 1-8.
[4] Sakuma, S., Kuroda, K., Tsai, C.H.D., Fukui, W., Arai, F.
and Kaneko, M. “Red Blood Cell Fatigue Evaluation
Based on the Close-encountering Point between
Extensibility and Recoverability.” Lab on a Chip Vol. 14
No. 6 (2014): pp. 1135-1141.
[5] Tsai, C.H.D., Tanaka, J., Kaneko, M., Horade, M., Ito, H.,
Taniguchi, T., Ohtani, T. and Sakata, Y. “An On-Chip
RBC Deformability Checker Significantly Improves
Velocity-Deformation Correlation.” Micromachines Vol.
7 No. 10, 176 (2016): pp. 1-13.
[6] Baskurt, O.K., Gelmont, D. and Meiselman, H.J. “Red
Blood Cell Deformability in Sepsis.” American Journalof Respiratory and Critical Care Medicine Vol. 157 No.
2 (1998): pp. 421-427.
[7] Zheng, Y., Shojaei-Baghini, E., Azad, A., Wang, C. andSun, Y. “High-throughput Biophysical Measurement of
Human Red Blood Cells.” Lab on a Chip Vol. 12 No. 14(2012): pp. 2560-2567.

(7)

In Eq. (7), ρ is density of the fluid, d is distance between
two parallel disks, η is viscosity of the fluid, and V is thecircumferential velocity of the disk.
Re is smaller than 0.02 at ρ (103 kg m–3), η (> 0.02 Pa s), d(< 1.2×10–4 m), and V (< 2.7×10–3 m s–1) in the present
experiment. The number is smaller than unity, so that the
turbulent flow hardly occurs. The inertia effect is small aroundthe middle plane, where the velocity is close to zero.
In the present study, the shear rate is periodically varied tosimulate the pulsatile flow. Womersley number (α) is a usefulindex for estimation of the viscous effects on the pulsatile flow.
α2 = d 2 (2πf ρ / η)(8)

In Eq. (8), d is distance between two parallel disks, f is thefrequency of the oscillations of the disk, ρ is density of the fluid,and η is viscosity of the fluid.
Womersley number (α) is smaller than 0.01 (d < 1.2×10–4
m, f < 2 s–1, ρ = 103 kg m–3, and η > 0.02 Pa s) in the present
experiment. The number is smaller than unity, so that the inertial
effect is very small [15]. The single erythrocyte in the blood ishardly distinguished with the microscope, because the volume
ratio of erythrocytes in the blood is higher than 0.3. In the present
study, erythrocytes are dispersed in the dextran solution to makethem easy to be observed as each single cell. The density ofcontent in a cell increases with aging in vivo. In the previous
study, the younger cells were collected from 10 percent of thesupernatant section after centrifugation, where the older cells
were collected from 10 percent of the bottom section after
centrifugation [12].
In Fig. 3 in the present study, every cell in the twodimensional projection shows an ellipse, which relates to the5

Copyright © 2020 ASME

[8] Ariyoshi, K., Maruyama, T., Odashiro, K., Akashi, K.,
Fujino, T. and Uyesaka, N. “Impaired Erythrocyte
Filterability of Spontaneously Hypertensive Rats:
Investigation by Nickel Mesh Filtration Technique.”Circulation Journal Vol. 74 (2010): pp.129-136.
[9] Salehyara, S. and Zhu, Q. “Deformation and internal
stress in a red blood cell as it is driven through a slit by
an incoming flow.” Soft Matter Vol. 12 No. 13 (2016):
pp. 3156-3164.
[10] Hashimoto, S., Mizoi, A., Hino, H., Noda, K., Kitagawa,
K. and Yasuda, T., “Behavior of Cell Passing throughMicro Slit.” Proc. 18th World Multi-Conference on
Systemics Cybernetics and Informatics Vol. 2 (2014): pp.
126-131.
[11] Fischer, T.M., Stohr-Lissen, M. and Schmid-Schonbein,
H., “The Red Cell as a Fluid Droplet: Tank Tread-Like
Motion of the Human Erythrocyte Membrane in Shear
Flow.” Science Vol. 202 No. 4370 (1978): pp. 894-896.

[12] Hashimoto, S., “Detect of Sublethal Damage with
Cyclic Deformation of Erythrocyte in Shear Flow.”
Journal of Systemics Cybernetics and Informatics Vol. 12
No. 3 (2014): pp. 41-46.
[13] Hashimoto, S. “Erythrocyte Destruction under
Periodically Fluctuating Shear Rate; Comparative Studywith Constant Shear Rate.” Artificial Organs Vol. 13(1989): pp. 458-463.
[14] Takahashi, Y., Hashimoto, S., Hino, H. and Azuma, T.
“Design of Slit between Micro Cylindrical Pillars for
Cell Sorting.” Journal of Systemics Cybernetics andInformatics Vol. 14 No. 6 (2016): pp. 8-14.
[15] Caro, C.G., Pedley, T.J., Schroter, R.C. and Seed, W.A.
eds., The Mechanics of the Circulation, Oxford, Oxford
University Press (1978): pp. 57-60.

6Copyright © 2020 ASME