AXISYMMETRIC AND ASYMMETRIC BEHAVIORS

OF A RED BLOOD CELL IN CAPILLARIES

Ting Ye and Hua Li

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue

Singapore 639798, Republic of Singapore

Keywords: Red blood cell, Membrane force, Tank treading, Capillary flow, Computational biomechanics.

Abstract: The axisymmetric and asymmetric behaviours of a red blood cell (RBC) in capillaries are investigated

numerically by developing a two-fluid model, in which the membrane force is considered to describe the

RBC deformation. The quantitative validations with the experimental and theoretical results are provided,

and good agreements are found in the deformation index and deformed RBC shapes. The present results

show that the RBC experiences the axisymmetric motion if the membrane force is balanced between the

RBC cusps, otherwise the asymmetric motion occurs. The characteristic parachute shape of deformed RBC

is observed in the axisymmetric motion, while the tank-treading motion of RBC membrane is generated in

the asymmetric motion. As the capillary diameter increases, the decrease in RBC length is accompanied by

an increase in RBC width.

1 INTRODUCTION

Red blood cells (RBCs) play an essential role in

delivering oxygen to the body tissues via blood flow

through capillaries in all vertebrates and some

invertebrates. In human, a healthy mature RBC is

biconcave disk 8 µm in diameter and 2 µm in

thickness, which consists of cytoplasm enclosed by a

thin membrane (Evans and Fung, 1972). The RBC

membrane can experience stretching and bending

deformations subject to the blood flow, thereby it is

mainly responsible for the mechanical and

rheological behaviours of the RBC. It is well known

that RBCs are involved in many diseases, such as

the sickle-cell anemia resulted from the RBC

abnormality, and the capillary blockage due to the

RBC fragments. Therefore, it is of great importance

to study the behaviours of a RBC in capillaries for

revealing the mechanism of RBC deformation and

providing the insight into the RBC fighting against

relevant diseases.

Generally, a RBC in a capillary experiences an

axisymmetric motion, where the RBC is

axisymmetric with respect to the central axis of the

capillary due to the very small diameter (Secomb,

1987). In this motion, the RBC gradually deforms

from a biconcave shape into a parachute one. This

parachute shape is the characteristics of a RBC in a

capillary, which guarantes the RBC traversing

through various capillaries successfully, including

the smaller capillaries compared with the

undeformed RBC. The previously published works

(Secomb, 1987, Tsukada et al., 2001, Jeong et al.,

2006, Tomaiuolo et al., 2009) demonstrate that the

deformability of RBC in capillaries depends on the

RBC velocity and capillary diameter largely. With

increasing the RBC velocity or decreasing the

capillary diameter, the RBC width becomes

narrower accompanied by an increase of the length.

Thus, even though the capillary is very narrow, the

RBC may be squeezed through it successfully.

Instead of an individual RBC, Pozrikidis (2005)

numerically analyzed the axisymmetric motion of a

file of RBCs through capillaries using a boundary

integral method, and also examined the effects of the

cell spacing and capillary radius. However, if the

capillary diameter is quite large, the RBC shape

cannot be assumed to be axisymmetric any more

with respect to the central axis of the capillary. As a

result, the RBC will undergo an asymmetric motion.

Secomb and Skalak (1982) studied the asymmetric

motion of a RBC in a two-dimensional capillary

based on the lubrication method. They pointed out

that the tank-treading motion of the RBC membrane

usually accompies this motion. In other words, the

RBC membrane always rotates around the

Ye T. and Li H..

AXISYMMETRIC AND ASYMMETRIC BEHAVIORS OF A RED BLOOD CELL IN CAPILLARIES.

DOI: 10.5220/0003105700970102

In Proceedings of the International Conference on Biomedical Electronics and Devices (BIODEVICES-2011), pages 97-102

ISBN: 978-989-8425-37-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

cytoplasm in the asymmetric motion, like the tank

treading. In order to understand the tank-treading

motion, Sugihara-Seki and Skalak (1988)

investigated the asymmetric flow of a file and two

files RBCs using the finite element method, in which

each RBC is assumed as a rigid cylinder post and

not located at the centerline of capillary. Although a

certain progress has been achieved so far, the

detailed knowledge of the flow behaviors of a RBC

in capillaries is still of interest, especially for the

asymmetric motion of RBC.

In the present work, a two-fluid system is

developed to model the flow characteristics of a

RBC in capillaries, in which the mechanical

behavior of the RBC membrane is taken into

consideration. Subsequently, validations of this

system are carried out by comparing with the

experimental and theoretical results. Finally, the

axisymmetric behaviors of a RBC in capillaries are

simulated systematically on the basis of the two-

fluid system. Apart from that, the asymmetric

motion of the RBC in the capillary is also analyzed

in detailed, as well as the effect of the capillary

diameter.

2 MODELS AND METHODS

Figure 1: 2D schematic diagram of a RBC in a capillary

subject to the plasma flow with parabolic velocity profile.

The 2D schematic diagram of a RBC in a capillary is

provided in Figure 1, in which a plasma flow with a

parabolic velocity profile passes through the

capillary along the x direction. The 2D RBC with

zero velocity is put in the plasma flow, whose shape

is expressed as the parametric form (Evans and Fung,

1972),

0012

cos ( sin sin ),

sin ,

xx r c c c

yy r

θθ

=+ + +

⎧

⎨

⎩

(1)

where (x

, y

) is the RBC centre, r the maximum

radius, and θ the polar angular in the range [0, 2π].

The three coefficients c

, c

and c

are usually taken

as 0.1035, 1.0013 and −0.5614, respectively. Due to

the stress of the plasma flow, the RBC will move

and deform.

2.1 Governing Equations

A two-fluid system is developed here to describe the

fluid states of the RBC and plasma. The material

properties of fluids inside and outside the RBC are

different, such as density and viscosity. By treating

these two fluids as a single fluid with variable

material properties, the incompressible Navier-

Stokes equations are used over the whole domain to

describe the motion of the single fluid, written as

() [( )],

ρμ

∇⋅ =

∂

+∇⋅ =−∇ +∇⋅ ∇ +∇ +

∂

⎧

⎪

⎨

⎪

⎩

uu u u F

(2)

where u is the velocity vector, ρ and µ are the

density and viscosity of fluid, t is the time, p is the

pressure. The membrane force F reflects the

interaction between the RBC and plasma as the

result of the membrane deformation, given by

()

(,) ( ) ,

tds

−=

∫

fxxF

(3)

where x and x

are the spatial variable and

membrane position of Γ(t), s is the membrane length,

δ(x) is the 2D Dirac delta function, and f(s, t) is the

membrane force strength.

The membrane force strength can be derived by

the shell model (Pozrikdis, 2003), in which the

membrane is treated as a thin shell with finite

thickness, allowed to undergo bending and

stretching deformations, expressed by

(,) ,

ddm dm

ds ds ds

κκτ

⎛⎞

=+−

⎜⎟

⎝⎠

+t n

(4)

where τ and m are the in-plane tension and bending

moment, t and n are the unit tangent and normal

vectors, and κ is the curvature of the membrane.

Introducing the constitutive equations, the in-plane

tension and bending moment are expressed as

and

(),

κκ

= −

(5)

where E

and E

are the shear modulus and bending

stiffness, ε is the membrane strain, and κ

the resting

curvature of the membrane.

2.2 Numerical Methods

In numerical simulations, the computational domain

is discretized first by the staggered grid system.

Based on this grid system, the governing equations

are divided into three parts to be solved, namely

BIODEVICES 2011 - International Conference on Biomedical Electronics and Devices

calculating the membrane force, tracking the

membrane and solving the Navier-Stokes equations.

In order to calculate the membrane force, the

membrane is discretized by a set of Lagrangian

particles. Any two neighbouring Lagrangian

particles are connected by a straight line as a

membrane element. Thus, the strain and curvature of

each membrane element are obtained easily,

followed by the membrane force strength in Eq. (4)

by the finite difference method. Figure 2 shows

comparisons of the membrane force strength at a

fixed state between the numerical and exact results,

in which good agreement is observed. After that, Eq.

(3) is approximated by the trapezoidal rule so that

the calculation of the membrane force is completed.

Figure 2: Comparisons of the membrane force strength

between the numerical and exact results. The top (a) and

bottom (

b) figures correspond to the non-dimensional x-

and

y-components of membrane force strength,

respectively.

The material properties inside and outside RBC

are different, such that the membrane needs tracking

to distinguish the RBC and plasma. Besides, the

membrane position should be provided for

calculating the membrane force by Eq. (3). For these

two purposes, the binary level set method (Lie et al.,

2006) is used to track the membrane, in which a

binary level set function is defined to distinguish the

plasma and RBC, and governed by a convection

function. By solving the convection function using

the WENO scheme (Shu, 1997), the membrane can

be tracked. However, the accuracy of tracking the

membrane is not good due to the numerical

dissipation. In order to overcome the disadvantage,

the Lagrangian particles scattered to calculate the

membrane force are used again to correct the binary

level set function.

Finally, a hybrid method coupled SIMPLER

(Patankar, 1981) and SIMPLEC (Van-Doormaal and

Raithby, 1984) is developed to solve the Navier-

Stokes equations, after the material properties of

fluid are updated. In this method, a relative accurate

pressure field is obtained first by the SIMPLER idea

as the initial iterative pressure, and then the velocity

field is computed according to the momentum

equations. Generally, this velocity field cannot

satisfy the continuity equation with the sufficient

accuracy. Hence, a velocity correction is provided

by the SIMPLEC idea as the second step, which is

derived by the continuity equation. These two steps

are repeated as an iterative process until both the

continuity and momentum equations converge. Thus,

the pressure and velocity fields are updated at the

current time step.

3 RESULTS AND DISCUSSION

3.1 Validation

In this section, the model is validated by comparing

the present numerical predictions with the

previously published experiment (Jeong et al., 2006,

Tomaiuolo et al., 2009) and theoretical (Secomb,

1987) results.

Figure 3: Comparison of the predicted deformation index

with the experimental result (Jeong

et al., 2006).

Jeong et al. (2006) investigated experimentally

the RBC deformation in the rat mesenteric

capillaries. In their work, the effects of the diameter

and length of capillary are examined, as well as the

RBC velocity on the deformation index, defined as

the ratio of the length to diameter of the deformed

RBC (Tsukada et al. 2001). Here, we focus on one

of them, the relationship between the deformation

index and the velocity of RBC in a capillary with the

diameter of 6.2 µm. The comparison between the

present numerical and experimental results is shown

in Figure 3, in which a good agreement is found in

the increasing trend of deformation index. The

AXISYMMETRIC AND ASYMMETRIC BEHAVIORS OF A RED BLOOD CELL IN CAPILLARIES

average deformation index of RBC in the capillary is

about 1.58 very closed to the value of 1.55 reported

by Jeong et al. (2006). However, the deformation

index is smaller than that of Jeong et al. (2006) at

the higher velocity, which can be explained that the

deformed RBC in the experiment is no longer

axisymmetric at the higher velocity.

Furthermore, a comparison of the RBC shape at

the steady state is illustrated in Figure 4, in which

the experimental (Tomaiuolo et al., 2009) and

theoretical (Secomb, 1987) shapes are obtained in

the capillaries with diameters of 6.6 and 6.0 µm,

respectively. As expected, a good agreement is

provided in the deformed shapes, although there

exist slight differences resulted from the different

capillary diameter.

Figure 4: Comparison of the predicted RBC shape (a) with

the experimental (

b) (Tomaiuolo et al., 2009) and

theoretical results (

c) (Secomb, 1987).

3.2 Axisymmetric Motion of RBC

When the RBC shape is axisymmetric with respect

to the central axis of capillary, it will experience the

axisymmetric motion. Figure 5 shows the RBC

deformation behaviours in a capillary with the

diameter of 10 µm subjected to the maximum flow

velocity of 1.25 mm/s. At the initial state, a

biconcave RBC is located at the central axis of

capillary. In the presence of parabolic blood flow,

the RBC deforms gradually to a steady parachute

shape, convex in front (at leading surface) and

concave at the rear. Meanwhile, the RBC is

transported forward, and the RBC shape is always

axisymmetric.

Figure 5: RBC shapes in a capillary at different time,

where the number on the top of each snapshot indicates

the non-dimensional time

The axisymmetric deformation behaviour is

attributed to the distribution of membrane force

strength, as shown in Figure 6. With increasing time,

the membrane force strength increases at the sharp

cusps and the leading surface, such that the

deformation at these parts also becomes larger and

larger. At the leading surface the membrane force is

generated mainly by the stretching deformation,

which makes the RBC outward bulge at the leading

surface. However, it is attributed to the bending

deformation largely at the sharp cusps, squeezing the

sharp cusps narrower and narrower continuously. As

a result, the RBC deforms to a steady parachute

shape from the initial biconcave shape.

Figure 6: Distribution of membrane force strength at

different time, where the number on the bottom right of

each snapshot indicates the non-dimensional time.

Figure 7: Distribution of the non-dimensional flow speed

for a RBC in the capillary. The top (

a) and bottom (b)

figures correspond to the non-dimensional time of 2.0 and

4.0, respectively.

Figure 7 illustrates the contour plots of non-

dimensional flow speed in ten levels from 0.0 to

0.25. It is found that the flow speed inside the RBC

is larger than that in the carrier fluid, such that the

BIODEVICES 2011 - International Conference on Biomedical Electronics and Devices

100

RBC becomes rounder and rounder. Furthermore,

the flow speed inside the RBC decreases with

increasing time. Once the flow speeds inside and

outside the RBC maintain equally each other, the

RBC will achieve a steady state.

3.3 Asymmetric Motion of RBC

Apart from the axisymmetric motion, the RBC also

undergoes an asymmetric motion if the RBC shape

is not axisymmetric with respect to the central axis

of the capillary. In this section, the asymmetric

motion of the RBC is investigated by locating the

RBC near the central axis of the capillary.

Figure 8 depicts the motion and deformation of

the RBC, whose centre is placed above and below

the central axis of the capillary at the initial state.

The leading surface of RBC bulges gradually, while

the rear becomes more concave to maintain the area

conservation. In addition, the RBC tilts

anticlockwise when the RBC centre at the initial

state is above the central axis, as shown in Figure

8(a). However, the opposite tilt orientation is

observed if the RBC centre is below the central axis,

as illustrated in Figure 8(b). The tilt behaviours are

resulted from the parabolic distribution of the flow

velocity, which makes the upper and lower parts of

RBC undergo the unbalanced velocities. As the RBC

tilts gradually, the RBC membrane also rotates

around the RBC interior continuously, as found by

the motion of the representative node (hollow circle)

in Figure 8. This phenomenon is well known as the

tank-treading motion of the membrane.

Figure 8: The asymmetric motion of a RBC in the

capillary, where the top (

a) and bottom (b) rows are

obtained by adjusting the RBC centre slightly above and

below the central axis of capillary at the initial state, and

the hollow circles refer to the motion of a fixed

representative node.

Figure 9 illustrates the distribution of membrane

force strength at the non-dimensional time of 3.0 for

the RBC centre above and below the central axis of

the capillary. It is found that the membrane force

strength is significantly large at the sharp cusps. The

sharper the cusp is, the larger the membrane force

strength is. This is because that the bending moment

plays a dominant role in the membrane part with the

large curvature. For example, the curvature at the

lower cusp is larger than that at the upper cusp of

each deformed RBC in Figure 9. Hence, the

membrane force strength is also larger at the lower

cusp. The unbalanced membrane force strength at

the upper and lower cusps is one of the

characteristics of the RBC asymmetric motion,

which is different from the RBC axisymmetric

motion.

Figure 9: Distribution of the membrane force strength at

the non-dimensional time of 3.0 for the RBC centre above

(

a) and below (b) the central axis of the capillary.

3.4 Effect of Capillary Diameter

In general, capillary diameter is about 6-10 µm,

which has a significant effect on the RBC

deformation except the RBC velocity. In this section,

the effect of capillary diameter is examined by

simulating the motion and deformation of a RBC in

the capillaries with diameters of 6.2, 8.0, 10 and 12

µm, respectively. With consideration of the diameter

of biconcave RBC, an elliptical RBC with the same

area of the biconcave RBC is treated as the initial

shape in the capillary of 6.2 µm.

Figure 10: RBC shapes in the capillaries with the different

diameter at the different time. The numbers on the top of

figure indicate the non-dimensional time.

AXISYMMETRIC AND ASYMMETRIC BEHAVIORS OF A RED BLOOD CELL IN CAPILLARIES

101

Figure 10 shows the motion and deformation of a

RBC in the different capillaries. As the capillary

diameter decreases, the increase in RBC length is

accompanied by a decrease in RBC width.

Consequently, the deformation index increases with

decreasing the capillary diameter. In other words,

the RBC has to be elongated more when it passes

through a smaller capillary, thereby leading to the

increasing trend of the deformation index. This

increasing behaviour is attributed to the shear stress

of the fluid flow. At a given flow velocity, the shear

stress increases with decreasing the capillary

diameter, such that the RBC is stretched more

obviously at the smaller capillary. In addition, the

RBC in the large capillary moves further than that in

the small capillary, as indicated in Figure 10(b), (c)

and (d). This is because that the RBC is more

centralized on the central axis of the capillary when

the capillary diameter is larger. At the end, the RBC

shape deforms asymmetrically in a small capillary as

shown in Figure 10 (a), especially at the non-

dimensional time of 4.0. The reason for the

asymmetric behaviour is that the shear stress acting

on the RBC membrane is more unbalanced if the

RBC is closer to the capillary wall, which was also

reported by Jeong et al. (2006).

4 CONCLUSIONS

The present work concerns a numerical investigation

of the axisymmetric and symmetric motion of a

RBC in capillaries by developing a two-fluid system.

In order to describe the RBC deformation, the

membrane force is treated as a singular force

coupled into the two-fluid system. A quantitative

comparison with the experimental data is carried out

by examining the relationship between the

deformation index and the RBC velocity, yielding a

good agreement. Apart from that, the predicted RBC

shapes in the present work are compared with the

experimental and theoretical results published,

which also reasonably shows good agreements.

The axisymmetric behaviours of a RBC in the

capillary are simulated first, in which the

characteristic parachute shape is observed. By

analyzing the membrane force strength, it is found

that the RBC experiences an axisymmetric motion if

the membrane force is balanced between the upper

and lower cusp of the RBC. Then, the asymmetric

behaviours of a RBC are investigated by adjusting

the initial position of the RBC in the capillary, in

which the tank-treading motion of the RBC

membrane is reproduced and the membrane force

strength is not balanced any more. Finally, the effect

of the capillary diameter on the RBC deformation is

evaluated, where a decrease in RBC length

accompanied by an increase in RBC width is

observed with increasing the capillary diameter.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial

support from the Ministry of Education of Singapore

through Academic Research Fund (AcRF) Tier 1

under Project No. M52056029 (RGM 7/07).

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