Effect of Lateral Position on RBC Deformation Using Immersed Boundary Lattice-Boltzmann Method

Mehdi Navidbakhsh, Marzie Rezazadeh Dept. Mechanical Engineering

Iran University of Science and Technology Tehran, Iran

mnavid@iust.ac.ir, mrezazadeh@iust.ac.ir

shahrokh Rahmani,Hanie Monshizade Dept. Mechanical Engineering

Iran University of Science and Technology Tehran, Iran

rahmani_shahrokh@mecheng.iust.ac.ir,hmonshizade@mecheng.iust.ac.ir

Abstract— In this article the effect of lateral position on a circular red blood cell (RBC) deformation is investigated by two dimensional numerical simulation. Two different lateral positions (centerline and off-center) are considered. Duo to the difficulties in common numerical methods for simulating this process the lattice-Boltzmann method has been applied. The diameters of microchannel and RBC are 20 µm and 4 µm respectively. Results show that the centerline and off-center RBC are changed into parachute and elliptic shape respectively. Furthermore the results indicate that difference between off-center RBC and its initial shape is much more than centerline RBC.

Keywords- Lattice Boltzmann Method; RBC deformation; Lateral position numerical methods

I. INTRODUCTION

numerical methods can be used For simulating the processes that their modeling are associated with problems using continuous methods and due to difficulties associated with their experimental study[1].In recent years, the Lattice Boltzmann method (LBM) has developed into a powerful and promising numerical method for simulating blood flow [2]. This method resulted from the statistical physics in which the fluid is modeled as a pseudo-particles collection and such particles propagate over a discrete domain unlike conventional methods which consider the fluid in a continuous manner. During the last two decades, the LBM has been used to various kinds of complex problems including compressible flows, multiphase and multi-component fluids, particulate suspensions, reaction-diffusion system, wave motion and Poisson equations [3,4]. This method can be used for simulating blood flow in complex situations, platelets and artificial blood which continuous methods are encountered with difficulties for simulating them. In 1980, Frisch et al. [5] used a new technique for solving the two-dimensional Nervier-Stokes equation based on LBM. Migliorini et al. [6] applied a Lattice Boltzmann approach to determine the forces exerted on rolling leukocytes by red blood cells. Ladd [7] was the first to use the Lattice Boltzmann method to examine a solid particle in a fluid.

fluid-particle interaction problems,no-slip boundary condition is used and particles surface is determined with boundary points which consists of a collection midpoints connecting both

fixed-point. Due to change in computational boundaries during particle motion, some fluctuations in particle’s power and speed occur. Thus Lattice Boltzmann method and immersed boundary composition is used for simulating fluid-particle interaction.

One of the important issues that can influence flow properties of blood suspensions is red blood cell and its deformation through microchannel. Thus the goal of the present study is to develop a two-dimensional simulation for the motion and deformation of single RBC in both center and off-center positions in a microchannel

II. METHOD

A. Equations LBM has less difficulty in comparison with common

numerical methods because of having simplicity of the algorithm and complete stability for doing massively parallel computations. In LBM a fluid is modeled as pseudo-particles which can move in a lattice domain at discrete time steps.LBM consists three categories of particles; static particles which do not move at LB grids, and particles which move along the coordinate and diameter directions. In this paper 2D Lattice-Boltzmann model D2Q9 has been utilized for simulation (Figure 1).

Figure 1. D2Q9 Lattice-model

The discrete lattice Boltzmann equation (LBE) has the

form of

978-1-4673-3130-2/12/$31.00 ©2012 IEEE

Proceedings of The 19th Iranian conference on Biomedical Engineering (ICBME 2012), Tehran, Iran, 21-22 December 2012

272

1( , ) ( , ) ( ) ( , ) ( , )eq

i i i if x c t t t f x t f x t f x t

(1)

Where , ( , )if x t , ( , )eqif x t are dimensionless relaxation,

density distribution function, and equilibrium distribution function, respectively.Viscosity can be obtained from Nervier-Stokes equation as below”

21

( )2 sc t (2)

The density and the momentum u are defined by:

8

1i

if

(3)

8

1i i

iu e f

(4)

As mentioned in the introduction for modeling the

interaction effect between incompressible viscous fluid and moving boundaries, a body force term should be added into the Lattice Boltzmann equation to reach immersed-boundary Lattice Boltzmann method as below [8]

( , ) ( , )

1[ ( , ) ( , )]

i i i

eqi i i

f x e t t t f x t

f x t f x t tF

(5)

( , )eq

i if E u (6)

'1

2i ii

u e f f t (7)

'2 4

( . )1(1 ) .

2i i

i is s

e u e uF fc c

(8)

Where xct

and i are weight factors with values

4 / 9

1/ 9 for 1 4i (9) 1/ 36 for 5 8i

Neo-Hookean law for calculating RBC membrane stresses is written as follow:

2 2 2 21 2 1 2( )sW E h (10)

Where sE , h and 1 2, are elasticity module, thickness and

strains respectively.

'f is calculated using energy equation and given as

'

i

Wf

(11)

B. Boundary Condition If uo(x,t) is assumed to be a velocity distribution at the

boundary and fluid kept stationary close to the boundary, uo=0 is used to describes no-slip condition. For applying no-slip boundary condition in LBM, The bounce-back boundary condition is used. (Figure 2) (i.e. computational nodes are divided into solid an fluid nodes).

Figure 2. Bounce method for stationary solid boundary

Bounc-back boundary condition can be written as follow:

( , ) ( , )out ini if x t f x t x fluid (12)

C. Calibration

Physical parameters in SI system are defined according to Lattice Boltzmann units[8]. In this paper each Lattice Boltzmann unit is by micrometer.

26310 , 1000kgm

s m (13)

The other parameters are defined as follows

60.2 10s

mx l (14)

92.7 10t

st l (15)

188 10m

kgm l (16)

III. RESULT

A circular RBC deformation is simulated in a two-dimensional microchannel. The RBC diameter, channel height, RBC membrane shear modulus and Reynolds number are considered 2.08 and 20 µm, Es=6.3×10-6 N/m and Re=0.185 respectively. A parabolic velocity profile with Umax=11.25×10-

3m/s is considered for fluid flow. Two different lateral positions are considered 5 µm and 10 µm far centerline and off-center RBCs respectively. As it can be seen from figure 3(a), the shape of the RBC is changed into parachute shape in order to overcome to hydrodynamic resistance when it is

273

(a)

(b)

Figure 3. RBC deformation result of crossing the channel (a) center line RBC deformation; (b) off-center RBC deformation

located on the centerline of the microchannel. When the RBC is located off-center of microchannel, the shape is changed into elliptic shape and stretched along its diameter (figure 3 (b)).

The amount of deviation of RBC from its equivalent circle is investigated by using Mext which is described based on zero-order momentum. Variation of Mext for RBC in two positions is shown in figure (4).

For center line RBC, Mext is changed infinitesimally, at the beginning. As time goes on, the RBC deforms to a parachute shape and consequently the Mext increases, but for off-center RBC, Mext increases significantly much more than center line RBC at the beginning time. That is due to velocity difference in lower and upper parts of RBC. For off-center RBC, Mext reaches to its maximum value then decreases slightly as RBC

Figure 4. Mext changes versus time for center line and off-center RBC

migrates closer to center line and reduction of hydrodynamic resistance.

IV. CONCLUSION

In recent years, RBC deformation in microchannel has been studied through numerical methods by many authers. In this article the immersed-boundary Lattice Boltzmann method is used to the study of effect of lateral position on a circular RBC deformation in a microchannel. Results indicate that center line and off-center RBC are changed into parachute and elliptic shape. The LBM can be utilized to model the RBC deformation with different shapes in the future.

REFERENCES

[1] Y. Sui , Y.T. Chew , P. Roy , H.T. Low, “A hybrid method to study

flow-induced deformation of three-dimensional capsules”,J. Computational Physics 227,2008, pp.6351–6371,.

[2] S.Y. Chen, G.D. Doolen, “Lattice Boltzmann method for fluid flows”, Annu. Fluid Mech. 3 ,1998,pp.314–322.

[3] M. Yoshino, Y. Hotta, T. Hirozane, M. Endo, “ numericalmethod for incompressible non-Newtonian fluid flows based onthe lattice Boltzmann method”, J. Non-Newtonian Fluid Mech.147,2007,pp.69–78.

[4] Prosenjit Bagchi ,”Mesoscale Simulation of Blood Flo in Small Vessels”, Biophysical Journal Volume 92,2007,pp.1858–1877.

[5] Frisch U, Hasslacher B, Pomeau Y., Lattice–gas automata for the Navier–Stokes equations. Phys Rev Lett;56:1986,pp.1505.

[6] Migliorini et al., “Red blood cells augment leukocyte rolling in a virtual blood vessel”, Biophys. J. 83,2002,pp. 1834.

[7] A.J.C. Ladd, “Numerical simulations of particulate suspensions via a discredited Boltzmann equation”, J. Fluid Mech. 271, 1994,pp.285.

[8] Michael M. Dupin, Ian Halliday, Chris M. Care, and LyubaAlboul, Lance L. Munn,”Modeling the flow of dense suspensions of deformable particles in three dimensions”, Physical review E 75, 066707, (2007).

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