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© 2015 The Korean Society of Rheology and Springer 163 Korea-Australia Rheology Journal, 27(2), 163-171 (May 2015) DOI: 10.1007/s13367-015-0016-x www.springer.com/13367 pISSN 1226-119X eISSN 2093-7660 Numerical investigation on the structural characteristics of multiple RBCs in a stenotic microcapillary under plasma-alcohol solution Aleksey Ni 1 , Taqi Ahmad Cheema 2 and Cheol Woo Park 1, * 1 School of Mechanical Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea 2 Department of Mechanical Engineering, GIK Institute of Engineering Science and Technology, Topi, Swabi Khabar Pakhtoon Khwa 23460, Pakistan (Received January 27, 2015; final revision received April 14, 2015; accepted April 15, 2015) Alcohol significantly affects blood rheology and influences the mechanical behavior of red blood cells (RBCs). Previous studies indicate that plasma-RBC and multiple RBC interaction are important indicators for atherosclerosis progression. Therefore, multiphysics interactions under highly viscous conditions of alcohol consumption and stenosis structure must be investigated. A 2D microcapillary model, wherein mul- tiple RBCs float through a stenotic structure, was established to investigate the effect of alcohol on cell motion and deformability under various flow conditions. Results show that the deformed cells inside the stenosis increase flow resistance and increase plasma velocity gradient in the pre and post stenotic regions. Moreover, the effect of the initial RBC position is important describing the RBC deformation pattern. The structural properties of RBCs may be significantly affected when the stenosis path is filled by cells that increase flow resistance. Keywords: red blood cell, alcohol, stenotic structure, microcapillary, plasma, deformability 1. Introduction Blood is a non-homogeneous fluid that mainly consists of blood cells, plasma, and nutrients. Blood circulation in microvessels delivers oxygen and nutrients to living tis- sues and removes metabolic wastes. Red blood cells (RBCs) have biconcave discoid shapes of 8 µm diameter and form 40 to 45% of the blood volume (Stoltz et al., 1999). When the diameter of RBCs are comparable with the vessel dimensions of capillaries, the co-interaction of RBCs and the interaction between RBCs and plasma sig- nificantly affects the overall properties of blood (Fung, 1993). For instance, the deformed shape of blood cells or the changes in plasma viscosity is a symptom of various diseases, such as acute myocardial infarction, malaria, and sickle cell anemia (Popel and Johnson, 2005; Gokturk et al., 2009; Cranston et al., 1984). The RBC membrane can undergo large deformations and the cell shape is squeezed to a diameter of 3µm while flowing through the vessel. Deformed RBCs may increase blood viscosity and flow resistance, thus resulting in myocardial infarction and apo- plexy (Mi et al., 2006). The circulation phenomenon can be aggravated under the presence of a pathological con- dition such as stenosis inside the microvessel, such as ste- nosis (Pozrikidis, 2003; Bagchi, 2007). In-vitro experiments have shown that alcohol changes the morphology of RBCs by changing plasma viscosity when mixed with blood plasma thus affecting the mechan- ical behavior of RBCs (Deng et al., 2005; Homaidan et al., 1984). In microcirculation, the effects of alcohol on RBCs increase in severity because of comparable dimen- sions within the vessel. Numerous researchers have investigated RBC deformability and blood flow velocity under the influence of alcohol (Hillbom et al., 1983; Shiraishi et al., 1994). However, most of these studies rely on the cation-osmotic hemolysis method to determine the relationship between plasma flow velocity and eryth- rocyte deformability (Gdovinova, 2001; 2002). Erythro- cyte deformability is plotted in terms of the percentage of hemolysis as a function of various concentrations of sodium chloride (NaCl) present in the incubated media (Gdovinova, 2006). Therefore, the influence of alcohol on local RBC deformation in microvessels has not been investigated. Recent developments in the field of computer simulation have enabled researchers to investigate local RBC defor- mation by using fluid structure interaction (FSI) methods. Researchers have numerically investigated RBC behavior without considering the effect of alcohol by using the FSI method (Secomb et al., 2007; Tsubota et al., 2006; Guo and Xiwen, 2011; Cho et al., 2011). These studies used the moving mesh-based arbitrary Lagrangian-Eulerian (ALE) method. However, moving mesh-based ALE dis- torts the mesh thus affecting solution accuracy and stabil- ity. One-way to avoid this situation is to place deformable objects into a frame, whereas the object deformation is calculated inside the frame. This method works well for a straight channel; however, some convergence problems may occur for complex geometries (e.g., a stenotic struc- *Corresponding author; E-mail: [email protected];
Transcript

© 2015 The Korean Society of Rheology and Springer 163

Korea-Australia Rheology Journal, 27(2), 163-171 (May 2015)DOI: 10.1007/s13367-015-0016-x

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Numerical investigation on the structural characteristics of multiple RBCs

in a stenotic microcapillary under plasma-alcohol solution

Aleksey Ni1, Taqi Ahmad Cheema

2 and Cheol Woo Park

1,*1School of Mechanical Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea

2Department of Mechanical Engineering, GIK Institute of Engineering Science and Technology, Topi, Swabi Khabar Pakhtoon Khwa 23460, Pakistan

(Received January 27, 2015; final revision received April 14, 2015; accepted April 15, 2015)

Alcohol significantly affects blood rheology and influences the mechanical behavior of red blood cells(RBCs). Previous studies indicate that plasma-RBC and multiple RBC interaction are important indicatorsfor atherosclerosis progression. Therefore, multiphysics interactions under highly viscous conditions ofalcohol consumption and stenosis structure must be investigated. A 2D microcapillary model, wherein mul-tiple RBCs float through a stenotic structure, was established to investigate the effect of alcohol on cellmotion and deformability under various flow conditions. Results show that the deformed cells inside thestenosis increase flow resistance and increase plasma velocity gradient in the pre and post stenotic regions.Moreover, the effect of the initial RBC position is important describing the RBC deformation pattern. Thestructural properties of RBCs may be significantly affected when the stenosis path is filled by cells thatincrease flow resistance.

Keywords: red blood cell, alcohol, stenotic structure, microcapillary, plasma, deformability

1. Introduction

Blood is a non-homogeneous fluid that mainly consists

of blood cells, plasma, and nutrients. Blood circulation in

microvessels delivers oxygen and nutrients to living tis-

sues and removes metabolic wastes. Red blood cells

(RBCs) have biconcave discoid shapes of 8 µm diameter

and form 40 to 45% of the blood volume (Stoltz et al.,

1999). When the diameter of RBCs are comparable with

the vessel dimensions of capillaries, the co-interaction of

RBCs and the interaction between RBCs and plasma sig-

nificantly affects the overall properties of blood (Fung,

1993). For instance, the deformed shape of blood cells or

the changes in plasma viscosity is a symptom of various

diseases, such as acute myocardial infarction, malaria, and

sickle cell anemia (Popel and Johnson, 2005; Gokturk et

al., 2009; Cranston et al., 1984). The RBC membrane can

undergo large deformations and the cell shape is squeezed

to a diameter of 3µm while flowing through the vessel.

Deformed RBCs may increase blood viscosity and flow

resistance, thus resulting in myocardial infarction and apo-

plexy (Mi et al., 2006). The circulation phenomenon can

be aggravated under the presence of a pathological con-

dition such as stenosis inside the microvessel, such as ste-

nosis (Pozrikidis, 2003; Bagchi, 2007).

In-vitro experiments have shown that alcohol changes

the morphology of RBCs by changing plasma viscosity

when mixed with blood plasma thus affecting the mechan-

ical behavior of RBCs (Deng et al., 2005; Homaidan et

al., 1984). In microcirculation, the effects of alcohol on

RBCs increase in severity because of comparable dimen-

sions within the vessel. Numerous researchers have

investigated RBC deformability and blood flow velocity

under the influence of alcohol (Hillbom et al., 1983;

Shiraishi et al., 1994). However, most of these studies

rely on the cation-osmotic hemolysis method to determine

the relationship between plasma flow velocity and eryth-

rocyte deformability (Gdovinova, 2001; 2002). Erythro-

cyte deformability is plotted in terms of the percentage of

hemolysis as a function of various concentrations of

sodium chloride (NaCl) present in the incubated media

(Gdovinova, 2006). Therefore, the influence of alcohol on

local RBC deformation in microvessels has not been

investigated.

Recent developments in the field of computer simulation

have enabled researchers to investigate local RBC defor-

mation by using fluid structure interaction (FSI) methods.

Researchers have numerically investigated RBC behavior

without considering the effect of alcohol by using the FSI

method (Secomb et al., 2007; Tsubota et al., 2006; Guo

and Xiwen, 2011; Cho et al., 2011). These studies used

the moving mesh-based arbitrary Lagrangian-Eulerian

(ALE) method. However, moving mesh-based ALE dis-

torts the mesh thus affecting solution accuracy and stabil-

ity. One-way to avoid this situation is to place deformable

objects into a frame, whereas the object deformation is

calculated inside the frame. This method works well for a

straight channel; however, some convergence problems

may occur for complex geometries (e.g., a stenotic struc-*Corresponding author; E-mail: [email protected];

Aleksey Ni, Taqi Ahmad Cheema and Cheol Woo Park

164 Korea-Australia Rheology J., 27(2), 2015

ture in the channel). The only solution to restore the mesh

quality of a distorted mesh is to use the automatic re-

meshing method. Some, the numerical studies that have

investigated RBC deformation in microvessels do not con-

sider the use of the automatic re-meshing method. More-

over, most previous studies use single cells to describe the

structural properties of RBC while floating in plasma;

therefore, such studies are limited to the single cell

behavior. On the contrary, the actual morphology of RBCs

is influenced by mutual RBC interaction (Chen et al.,

2011). Therefore, multiple RBC motion and deformation

should be investigated, including the dominant viscous

effects caused by alcohol in microvessel (Vahidkhah and

Fatouraee, 2012).

This study aims to address the aforementioned concerns

by using a 2-D stenotic capillary model. The present work

is the extension of our previous research on the deform-

ability and behavior of a single RBC in the stenotic micro-

channel under the influence of plasma-alcohol solution

(Ni et al., 2013). The previous work purposed ALE mov-

ing mesh method to solve the fluid structure interaction of

RBC and alcohol. The present study used the same mov-

ing mesh methodology to model the motion of multiple

RBCs in the stenotic micro capillary with the effect of

highly viscous alcohol conditions. The problem of reduced

mesh quality because of the distorted mesh is addressed

by implementing an automatic re-meshing method. The

mechanical behavior of multiple cells with fluid dynamic

characteristics are computed and compared with the nor-

mal blood flow. The results present the importance of the

alcoholic effect on RBC behavior in a stenotic channel by

evaluating RBC deformation, RBC mutual interaction and

RBC-plasma interaction.

2. Model development and methods

The 2D biconcave RBC shape is considered in this study

because of its optimal combination of surface area to vol-

ume ratio (Guo and Xiwen, 2011). Moreover some of the

distinct deformation characteristics of RBCs can be eval-

uated easily by using this model (Zhodi and Kuypers,

2006). Any loss in the shape of the RBC may indicate a

problem in RBC function. From a computational point of

view, the use of a 2D simulation model is advantageous

because of its reduced computational time and complex-

ities. The contraction effect in stenotic blood vessel with

single or multiple RBCs can be completely analyzed by

3Dmodeling. However, 3D modeling will result in larger

distorted mesh that will reduce the mesh quality and affect

the accuracy of the results with large computational time.

In contrast, the 2D modeling with taking account the lim-

itations and certain assumptions is well suited for such

fluid-solid problems and can reproduce the investigated

phenomena more precisely (Guo and Xiwen, 2011). More-

over, it was also reported by many researchers that the

deformation and behavior of RBCs depend on shear rate

of the surrounding plasma and viscosity ratio between

external (plasma) and internal fluids (cytoplasm) (Mohan-

das et al., 1980; Linderkamp and Meiselman, 1982).

Therefore, in the present study, an efficient and numer-

ically robust 2D RBC model was used to study mechan-

ical behavior of RBCs in an alcoholic plasma solution

inside a microcapillary with a stenotic structure. More-

over, the present study considered 2D computational

model instead of axisymmetric simulation due to the

involvement of nonsymmetrical initial positions of RBCs

and to analyze the effect of RBCs distortion in stenotic

region to fully replicate the actual blood vessel geomet-

rical effects.

Fig. 1 shows the schematic of the 2D model (diameter

= 10 µm) used in this study with 45% stenosis. RBCs

move from left to right in the plasma-alcohol solution via

a microcapillary vessel with a length of 100 µm. The ste-

notic structure is 5 μm high and 20 μm long. The RBCs

are expected to deform during its motion by external shear

flow. Two factors may influence the deformation intensity:

a) Ratio of internal and external fluids viscosity

b) Shear rate

RBCs are considered as capsules made of an elastic

membrane that encloses a cytoplasm and are suspended in

a Newtonian fluid (plasma). Biconcave-shaped RBCs with

8 µm average diameter and 1 µm thickness were intro-

duced from the inlet of the channel (Evans and Fung,

1972). Gdovinova reported that alcohol changes the prop-

erties of plasma and decreases shear rate (Gdovinova,

2001; 2006). Therefore, the effect of alcohol on the

motion and deformation of RBCs could be modeled with

an increased plasma viscosity and decreased shear rate,

(i.e. the RBCs move slowly and exhibit a slight deforma-

tion).

Previous studies reported that the primary deformation

mode of RBC is folding or bending about the longitudinal

axis of the capillary rather than compression or stretching

in the plane of the original disk due to the nature of RBC

membrane(Secomb and Hsu, 1996; Bagge et al., 1980).

Past researchers also stated that the RBCs behavior and

position in microchannel depend on the shear rate (Fed-

osov et al., 2014; Tsukada et al., 2001; Lee et al., 2004).

Different values of shear rate resulted in different shapes

of RBC like the tumbling, slipper, snaking and so called

parachute shape which is considered stable for cell motion

and it is orthogonal to the channel wall under different

value of shear rate (Secomb and Hsu, 1996; Bagge et al.,

1980; Fedosov et al., 2014). Based on the above men-

tioned research findings, the present study assumed the

initial position of RBCs to be perpendicular to stream

lines. Three different types of initial positions for RBC

were considered in the present study i.e., a) the center of

Numerical investigation on the structural characteristics of multiple RBCs in a stenotic microcapillary......

Korea-Australia Rheology J., 27(2), 2015 165

RBC at the symmetry axis, b) the center of RBC above the

symmetry axis and c) the center of RBC below the sym-

metry axis.

The model geometry can be classified into elastic

deformable structures (i.e., for RBC membrane modeling

and fluids for plasma and cytoplasm). Therefore, the prob-

lem under consideration is a fluid-structure interaction

problem. The fluid flow was described by the incompress-

ible Navier-Stokes equations in the spatial coordinate sys-

tem:

, (1)

(2)

where uf is fluid velocity, ρf is fluid density, p is pressure

and f is the body force per unit volume, which is assumed

to be zero in the fluid dynamic calculation of this study.

The variable τ can be written as a function of shear

strain rate in a Newtonian fluid:

(3)

where ε is the strain rate tensor and can be evaluated as

follows:

. (4)

A homogeneous, incompressible, and Newtonian

plasma with a viscosity of µ = 1.2×10−3 Pa·s and density

of ρ = 1,060 kg/m3 was considered in a rigid vessel. The

cytoplasm of plasma density has a viscosity of µ = 6×10−3

Pa·s (Ramanujan and Pozdrikidis, 1998). To include the

effect of alcohol, plasma viscosity was increased to µ =

5×10−3 Pa·s while density remain the same ρ = 1,060 kg/

m3 (assuming the presence of alcohol at 1 mL/kg). The

viscosity ratio (λ) for normal blood was 5, whereas that for

alcohol-mixed blood was 1.2. The Reynolds numbers for

normal blood and alcohol-mixed blood were 0.0088 and

0.00021, respectively. The RBC capillary numbers in nor-

mal and alcoholic plasma solutions were 0.52 and 2.17,

respectively. In this study, the RBC models consisted of

cytoplasm enclosed by a hyperelastic membrane. The

interaction of the membrane with the fluid could be

described by the following elastodynamic equation:

(5)

where σ is the Cauchy stress tensor, and Fi represents the

component of body force acting at the fluid solid interface.

The ALE method was used to handle the dynamics of

the deforming geometry and moving boundaries to study

the structural deformation of cell membranes by fluid

flow. A hyperelastic smoothing method was applied to

model the motion of solid materials following the

deformed mesh by using a neo-Hookean material model.

(6)

where G and K are the artificial shear and bulk modulus,

respectively; J is the ratio of the deformed elastic volume

to the un-deformed volume; is the first invariant of the

left Cauchy–Green deformation; λ denotes the Lamé con-

stant.

The energy stored in the solid body after the structure

experiences deformation could be evaluated by using thes-

train energy density function W:

(7)

where S and ε are the second Piola–Kirchhoff stress and

Green–Lagrange strain tensor components, respectively.

A direct two-way coupling formulation between RBC

and plasma was made wherein the RBC surface represents

the fluid-solid interface. The following boundary condi-

tions were assigned on this interface:

Displacement: , (8)

Traction: , (9)

No-slip: (10)

where df, ds and σf, σs are the displacement and stress ten-

sors of the fluid and solid domains, respectively; rep-

resents the mean fluid velocity.

The RBC membrane was considered hyperelastic and an

average RBC Young’s modulus of 30 kPa was chosen in

this study (Rotsch et al., 1999). The literature value of

Poison’s ratio used during the simulation was 0.49.

3. Numerical implementation

Commercial software COMSOL Multiphysics (4.3) was

used to simulate the 2D model of a microcapillary channel

with a stenotic structure. The 2D model was discretized by

using triangular elements. Fig. 1 shows the schematics of

the model used to simulate the motion and deformation of

multiple RBCs. A fully coupled time-dependent solver

with different time steps was implemented to simulate the

motion and deformation process of RBCs. The use of

commercial code allowed the possibility to solve the tran-

sient problem of cells moving through the microvessel.

Cell movement was restricted to small displacements

when the mesh quality became too low to obtain accurate

∂ρ∂t------ + ∇ ρf uf( )⋅ = 0

ρf

∂uf

∂t------- uf ∇uf⋅+⎝ ⎠⎛ ⎞ = −∇p + ∇ τ⋅ + f

τ = 2με

ε = 1

2--- ∇uf( ) ∇uf( )T+( )

ρs

∂2us

∂t2

---------- = ∇ σ⋅ + ρsFi

W = G

2---- I 3–( ) +

K

2---- J 1–( )2

I

S = ∂W

∂ε--------

df = ds

σf = σs

U = d·s

U

Fig. 1. Schematic outlines of the 2-D model used in the calcu-

lations.

Aleksey Ni, Taqi Ahmad Cheema and Cheol Woo Park

166 Korea-Australia Rheology J., 27(2), 2015

computational results. The mesh quality was a dimension-

less parameter between zero and one; the quality of the

elements is better when the value is closer to the one (Frey

and Borouchaki, 1999). Another problem with low mesh

quality is the appearance of inverted mesh elements and

convergence issues, particularly in iterative solvers. There-

fore, the automatic remeshing procedure with a mesh

quality limit of 0.2 was used to overcome such problem.

In this case, the solver stops the calculation when the qual-

ity of the mesh reached the limit, restores the initial qual-

ity of the mesh, and then continues. Fig. 2 shows the

schematic of the simulated model with moving triangular

meshes.

In this study, the generalized minimal residual (GMRES)

method was adopted to avoid the stability and conver-

gence problems that often appear while solving FSI prob-

lems by using sub-iteration techniques. However, GMRES

is a computationally expensive solver and requires large

memory requirements. To obtain the forces acted on the

cell, the entire surface was integrated and the deformation

plots were obtained by using various post-processing tools.

4. Results and Discussion

The motion and mechanical behavior of multiple RBCs

while moving through the microvessel with stenosis were

studied under various conditions of decreased shear rate

and increased plasma viscosity under the influence of

alcohol. The validation of the computational model was

conducted with a single RBC moving through a 6.2 µm

diameter capillary from the experiment of Jeong et al.

(2006) can be observed on Fig. 3. RBC deformation was

computed and the results were in agreement with the

experiment. The experimental value deviated from the

numerical value, particularly at high RBC velocities

because of the effect of retrograde flow in the non-uni-

form microcapillary.

4.1. Effect of initial RBC position, viscosity, and inletvelocity

The initial RBC position is important for the rheology of

plasma, and motion and the deformability of RBC. This

Fig. 2. Schematics of the moving mesh used for calculating the

motion of multiple cells through a microvessel. (a) Initial con-

dition of RBCs. (b) Condition of RBCs before entering the ste-

notic structure. (c) Condition of RBCs while entering the stenotic

structure. Fig. 3. (Color online) Comparison of numerically computed

RBC deformation with the in-vivo results of Jeong et al. (2006).

Fig. 4. (Color online) RBC motion and deformation at normal

plasma velocity and of viscosity (v=1×10−3 m/s, µ=1.2×10−3

Pa·s) with initialpositionsof RBCs centers. (a) At the symmetry

axis. (b) Above the symmetry axis. (c) Below the symmetry axis.

Numerical investigation on the structural characteristics of multiple RBCs in a stenotic microcapillary......

Korea-Australia Rheology J., 27(2), 2015 167

effect has been studied under various boundary condi-

tions. At the start of the simulation, multiple RBCs were

horizontally located at a distance of 11, 13.6, 16.2, and

18.8 µm from the inlet. Three different vertical positions

of RBCs were examined by placing the center of the RBC

at the channel symmetric axis and at a distance 0.6 µm

above and below the axis. The motion of the RBCs was

traced at four different times of 0, 0.015, 0.025, and 0.38

s. Fig. 4 shows the comparison between RBC motion and

deformation at a normal plasma viscosity of 1.2×10−3 Pa·s

with a velocity of 1×10−3 m/s at the inlet. The alcoholic

effect was studied by using plasma viscosity of 5×10−3

Pa·s while keeping other conditions constant (Fig. 5).

Moreover the effects of reduced velocity because of alco-

hol with the same viscosity was also investigated (Fig. 6).

RBCs start to deform as they approach the stenotic

structure because of the convergence of plasma stream-

lines in the narrow section of the channel. In all the cases

discussed in this study, RBCs tend to increase the plasma

velocity gradient not only in the stenotic region but also in

the pre and post stenotic regions by offering increased

flow resistance, when RBC reach the narrow section (t =

0.015 s). RBCs flow through the stenosis at a high plasma

shear rate with relative ease. However, RBCs require

increased time when the shear rate is low. In the stenosis,

the RBC shape was completely distorted; however, the

shape was restored to its original shape in the post stenotic

region. RBCs initially positioned with their centers at the

symmetric axis are symmetrically deformed at the center

irrespective of the fluid velocity and viscosity. This con-

dition can also be interpreted by a symmetric velocity gra-

dient increase in the plasma throughout the channel.

However, the deforming pattern of RBCs is different in

the other cases. The intensity of the deforming pattern

increases in the stenotic region with a non-symmetric

plasma velocity gradient. One main reason for this non-

parachute shape in the stenotic region is a ±45o turn before

their entrance into this region. Moreover, the deformation

of RBCs placed away from the channel symmetry axis is

small when the inlet velocity is reduced (Figs. 4-6). This

result shows the poor microcirculation that may become a

cause of deficiency of oxygen and nutrient in tissues and

organs. The deformed parachute shape of RBCs can be

observed in the non-symmetric cases when the plasma vis-

cosity is increased, thus indicating, that shear forces dom-

inate the stenotic region to increase membrane deformability.

This condition could be one of the main reasons for the

progression of various diseases depending on RBC

deformability (Késmárky et al., 2008).

In the two non-symmetric cases, the cells tend to rotate

in the post stenotic region and the low shear rate made the

cell motion unstable and lead to cell collision. Similar

observations of RBC deformation and rotation have been

reported in previous studies under situations with compa-

rable viscosities of plasma and cytoplasm (Korin et al.,

2007; Wang et al., 2008). The tendency of RBCs to rotate

Fig. 5. (Color online) RBC motion and deformation under alco-

hol-plasma viscosity (v=1×10−3

m/s, µ= 5×10−3

Pa·s) with initial

positions of RBCs centers. (a) At the symmetry axis. (b) Above

the symmetry axis. (c) Below the symmetry axis.

Fig. 6. (Color online) RBC motion and deformation under alco-

hol-plasma velocity (v=1×10−4

m/s, µ=1.2×10−3

Pa·s) with initial

positions of RBCs centers. (a) At the symmetry axis. (b) Above

the symmetry axis. (c) Below the symmetry axis.

Aleksey Ni, Taqi Ahmad Cheema and Cheol Woo Park

168 Korea-Australia Rheology J., 27(2), 2015

toward the channel walls is caused by the plasma velocity

distribution, which is similar to a pipe flow, and the geo-

metric imbalance of RBCs during deformation (Wang et

al., 2008).

4.2. Effects on RBC velocity in the stenotic structureFig. 7 shows the velocity magnitudes for the plasma and

the four RBCs while passing through the stenotic region

under various conditions. The RBCs were placed initially

with their centers at the symmetry axis, and the first RBC

was designated as Cell 1, i.e. the first cell to enter into ste-

nosis. In a similar manner, the last cell to enter stenosis is

Cell 4. The cells between Cell 1 and Cell 4, are designated

as Cell 2 and Cell 3, respectively. All RBCs have similar

velocity distribution in the stenotic structure with a slight

difference at the entrance and exit of the narrow section.

Each velocity profile can be divided into three main sec-

tions starting with an increasing, uniform, and decreasing

velocity.

The length of the increasing and decreasing velocity

regions varies from Cell 1 to Cell 4 and depends on the

flow resistance in the stenotic structure. For example, the

increasing velocity region for Cell 4 starts at the time

when the velocity of Cell 4 starts to decrease. Therefore,

a strong relative velocity relationship exists between the

cells while moving and deforming through stenosis for all

conditions employed in this study. A slight decrease in

plasma velocity was observed because of the flow resis-

tance offered by RBCs in the stenosis. No significant

effect of increased viscosity on RBCs velocity distribution

because of the addition of alcohol was observed (Fig. 7b).

However, a slow inlet plasma flow results in 10-fold

decrease in the maximum velocity magnitude of RBCs

while following the previous pattern of velocity distribu-

tion (Fig. 7c).

4.3. Effects of initial RBC position on the von-misesstress

To study the effect of the initial position of RBCs on

their structural characteristics, the von-Mises stress was

defined as a function of principle stresses σ1, σ2 and σ3 to

represent the maximum distortion energy:

. (11)

The computation of the von-Mises stresses of RBCs was

conducted in the stenotic regions for the normal plasma

and highly viscous alcoholic plasma solution (Fig. 6). The

placement of RBCs with their centers displaced from the

symmetric line has an insignificant effect on the stress dis-

tribution pattern (Figs. 8a, 8b). However, the magnitude of

the stress increased when the RBCs moved through the

alcoholic plasma solution (Figs. 8c, 8d). This result shows

that the cell offers increased resistance to deformation in

viscous flows, thus achieving the peak stress value in the

middle of stenosis.

At this stage, the pressure of the plasma is maintained at

the minimum value, to apply a deforming force on the

RBC membrane structure. An important feature that

affects the stress profile is the resistance offered by Cell 1

to the flow in the stenosis and the following cells. This

σVM = σ1 σ2–( )2 σ1 σ3–( )2 σ2 σ3–( )2+ +

2----------------------------------------------------------------------------

Fig. 7. (Color online) Velocity magnitudes of plasma and RBCs

inside the stenotic region. (a) Normal plasma solution. (b) Alco-

hol-plasma solution. (c) Normal plasma solution with reduced

inlet velocity.

Numerical investigation on the structural characteristics of multiple RBCs in a stenotic microcapillary......

Korea-Australia Rheology J., 27(2), 2015 169

flow resistance initially decreases the deformation resis-

tance of the cell structure depending on the cell number,

thus making the cell prone to rupture. Therefore, a stenotic

structure may initiate RBC rupture if a cell is blocked

inside the narrow section.

4.4. Effects on RBC transverse to the axial diame-ter ratio in the stenotic structure

The deformation characteristics of RBCs and their

mutual deforming interaction have been computed in

terms of the ratio of the RBCs transverse diameter to the

axial diameter. The RBCs were placed initially with their

centers at the symmetry axis. Fig. 9 shows the distribution

of this ratio with respect to time inside the stenosis region.

The graphs represent an initial drop in the deformation at

the entrance of the stenosis followed by a constant diam-

eter ratio in the middle of stenosis. The stenosis causes in

a significant pressure drop, thus leading to reduced mem-

brane deformation while moving fast.

On the contrary, the increase in plasma pressure results

in a sharp increase in RBC deformation at the end of the

stenosis region. The fast moving Cell 1 had the smallest

ratio of transverse to axial diameter for normal and highly

viscous plasma flows. For the case of reduced inlet plasma

velocity, the deformation pattern is reversed at the exit of

the stenotic region, which is a clear sign of the shear

effects for reduced plasma velocity. Other cells show sim-

ilar deformation patterns with relatively high magnitudes

at the start and exit of stenosis for normal and highly vis-

cous plasma flows, thus indicating the effects of flow

resistance caused by the number of cell ahead of one par-

ticular cell. However, the small inlet flow velocity at the

inlet has a dominant shear effect on RBC deformation

after the middle of stenosis.

5. Conclusions

We conducted a 2D numerical simulation to investigate

the motion and deformation of multiple RBCs moving in

a microcapillary filled with plasma alcohol solution. The

presented model can help explain the pathologies as a

result of alcohol consumption, which increases plasma

viscosity and reduces blood flow to a significant extent.

The effects of rheological properties on the mechanical

Fig. 8. (Color online) Von Mises stress distribution for multiple RBCs inside the stenosis region for normal plasma viscosity when

the cell centers are displaced toward the (a) top wall and (b) bottom wall. The alcohol-plasma solution viscosity caused by alcohol

when the cell centers are displaced toward the (c) top wall and (d) bottom wall.

Aleksey Ni, Taqi Ahmad Cheema and Cheol Woo Park

170 Korea-Australia Rheology J., 27(2), 2015

behavior and mutual interaction of RBCs were investi-

gated. Multiple RBC motion was traced through stenosis

on the basis of their initial positions in the microcapillary

by placing RBCs centers 0.6 µm above and below the

symmetry axis, in addition to the actual RBC position,

(i.e. the RBC center at the symmetry axis). An automatic

remeshing methodology was applied to handle the prob-

lem of mesh distortion problem during RBC deformation.

The results show significant differences in fluid and cell

velocities, thus indicating the slip effect of the RBC mem-

brane. The deformed cells inside the stenotic region

increased the flow resistance, thus causing the velocity

gradient to increase in the pre and post stenotic regions.

Furthermore, the results presented the effect of alcohol

through significant increases in shear forces, adding

another factor on the deformation of RBCs. Moreover, a

strong relative velocity was found to exist between the

cells during cell movement and deformation through ste-

nosis under all conditions employed in this study. The

structural analysis on the RBCs membrane reveals that a

stenotic structure may initiate RBC rupture if a cell is

blocked inside the narrow section. A future study on the

behavior of RBC and its interaction with deformable walls

will facilitate the further understanding of this phenome-

non.

Acknowledgements

This work was supported by the National Research

Foundation (NRF) of Korea grant and funded by the

Korean government (MEST) (No. 2012R1A2A2A01046099).

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