Computational and Mathematical Methods in Medicine

Volume 2018 (2018), Article ID 9425375, 12 pages

https://doi.org/10.1155/2018/9425375

## Numerical Simulations of the Motion and Deformation of Three RBCs during Poiseuille Flow through a Constricted Vessel Using IB-LBM

^{1}Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China^{2}College of Mechanical and Electrical Engineering, Huzhou Vocational and Technical College, Huzhou 313000, China^{3}School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan 114051, China

Correspondence should be addressed to Yikun Wei and Chuanyu Wu

Received 15 November 2017; Revised 19 January 2018; Accepted 23 January 2018; Published 21 February 2018

Academic Editor: Xiaole Chen

Copyright © 2018 Rongyang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The immersed boundary-lattice Boltzmann method (IB-LBM) was used to examine the motion and deformation of three elastic red blood cells (RBCs) during Poiseuille flow through constricted microchannels. The objective was to determine the effects of the degree of constriction and the Reynolds (Re) number of the flow on the physical characteristics of the RBCs. It was found that, with decreasing constriction ratio, the RBCs experienced greater forced deformation as they squeezed through the constriction area compared to at other parts of the microchannel. It was also observed that a longer time was required for the RBCs to squeeze through a narrower constriction. The RBCs subsequently regained a stable shape and gradually migrated toward the centerline of the flow beyond the constriction area. However, a sick RBC was observed to be incapable of passing through a constricted vessel with a constriction ratio ≤1/3 for Re numbers below 0.40.

#### 1. Introduction

Red blood cells (RBCs) play an important role in blood flow in the human body, particularly in the transportation of oxygen from the lungs to every cell of the body. An adult RBC has a biconcave shape of diameter 6 *μ*m and thickness 2 *μ*m [1–6]. The RBC membrane is highly deformable, which enables the passage of RBCs through a blood vessel with a diameter smaller than that of the RBCs [7, 8]. The flow of RBCs through a blood vessel represents a typical fluid-structure interaction (FSI) problem, involving a complex interplay of fluid dynamics, elastic body, and a moving boundary [9]. A variety of accurate and efficient numerical methods have been proposed for the solution of a FSI problem involving a complex geometry, such as the arbitrary Lagrangian–Eulerian method [10], immersed interface method [11], immersed finite element method [12], immersed boundary method [13], and immersed boundary-lattice Boltzmann method (IB-LBM) [14–18].

Previous studies on the IB-LBM emphasized its potential advantages for the solution of FSI problems, namely, its simplicity, parallelizability, intrinsic kinetic and explicit calculations, and essential relative simplicity for handing complex, moving, and deformable geometries [14–18]. In recent years, the numerical investigation of the motion and deformation of RBCs in capillaries and arteries has received considerable attention [15, 16]. Zhang et al. [4] presented a numerical IB-LBM algorithm for investigating the microscopic hemodynamic and hemorheological behaviors of discrete RBCs in shear flows. Dadvand et al. [9] used the IB-LBM to numerically investigate the motion and deformation of healthy and sick RBCs in viscous shear flows. Shi et al. [19] proposed a two-dimensional (2D) elastic spring model of the RBC membrane based on the immersed boundary method, which was first introduced by Peskin [20] for the investigation of blood flow through heart valves. Krüger et al. [21] used a hybrid LB-IB-finite element method to simulate the tumbling and tank-treading-like motion of dense suspended RBCs in an external shear flow. The transient motion and deformation of healthy RBCs and PF-RBCs at different stages were examined in a simple 2D microchannel, with the RBCs moving along the center line of the channel [22, 23]. Sui et al. [24–26] used a combination of the IBM, a multiblock lattice Boltzmann model, and membrane mechanics to investigate the transient behaviors of elastic capsules and the deformation and aggregation of RBCs in a shear flow. Ma et al. [27] proposed an IB-LBM that considered the ultrasonic effect for the simulation of RBC aggregation and deformation in an ultrasonic field. They found that the action of the ultrasound waves on the pure plasma could induce a recirculation flow. The IB-LBM has also been used to numerically investigate the effect of the RBC deformability on the dispersion of the cells at physiological flow rates with respect to the hematocrit [28]. Further, the IB-LBM has been applied to quantitative analyses of the motion and deformation of the RBC membrane in a Poiseuille flow and its compression during passage through a stenotic microvessel, with a focus on the cell-cell interaction strength [2, 3, 29]. The flow of multiple RBCs through a microvascular bifurcation has also been simulated by the 2D IB-LBM and an RBC spring model [30, 31]. Other methods have been used for the same purpose, such as by Stamou and Buick [32] and Wang et al. [33]. Alizadeh et al. [17, 18] also used a hybrid IB-LBM to investigate the dynamics of healthy and sick RBCs during flows through a constricted vessel. The foregoing shows that the IB-LBM is effective for investigating the dynamics of RBCs in flows through constricted vessels and in relevant biomedical applications.

The present study represents further work about certain previous studies [30, 31], namely, an examination of the motion and deformation of RBCs by numerical simulation using the IB-LBM. The primary objective was a qualitative analysis of the effects of the degree of constriction in the vessel and the Re number on the physical characteristics of flowing RBCs. The RBC dynamics were extensively analyzed with respect to the degree of constriction, Re number, elastic modulus, and bending modulus. The IB-LBM was specifically used to examine the physical characteristics of three elastic RBCs. Flows through a simple straight vessel and a vessel with an annular bump were considered. The rest of this paper is organized as follows. Section 2 briefly describes the employed governing equations and numerical method. The detailed numerical results are presented and discussed in Section 3. Finally, the conclusions drawn from the study and the scope for further study are presented in Section 4.

#### 2. Governing Equations and Numerical Method

##### 2.1. Governing Equations

Consider an RBC with curved boundary immersed in the 2D viscous fluid domain . The point on RBC boundary is characterized by the Lagrangian parameters , and the fluid domain is represented by the Eulerian coordinates . The equations governing the incompressible flow and elasticity of the RBCs in an external force field are as follows [17, 18]:

In the above equations, , and , respectively, denote the fluid density, fluid velocity, fluid pressure, and dynamic viscosity; and are, respectively, the membrane forces acting on the RBCs at the Eulerian point and Lagrangian point ; and is a nondimensional Dirac delta function.

##### 2.2. Immersed Boundary-Lattice Boltzmann Method (IB-LBM)

A popular kinetic model, namely, the discrete Boltzmann equation in the Bhatnagar–Gross–Krook (BGK) model with a single relaxation time under an external force, may be reproduced as follows [9, 18, 22, 27, 29]:where is the equilibrium distribution function, is the distribution function, is the single relaxation parameter, is the time interval, is the particle velocity, and is a weight coefficient that is determined by the selected lattice velocity model. In the present study, a 2D lattice with nine velocity components, referred to as D2Q9, was employed. The formation of the D2Q9 lattice is illustrated in Figure 2.

The discrete velocity vectors of the 2D square lattice of D2Q9 can be expressed aswhere is the lattice speed and is the lattice constant. are the weight coefficients with the following values:

The equilibrium distribution function was chosen from the nine-velocity set model for 2D problems, as follows:where is the speed of sound.

An immersed boundary treatment of a nonslip boundary condition was adopted, wherein the boundary force is spread to the lattice points and the fluid lattice velocity is interpolated to the boundary points [18]. Figure 3 illustrates a 2D part of the membrane and the surrounding fluid. The interaction between the blood and the RBCs can be considered based on the relationship between the Lagrangian and Eulerian points using the following interaction equations [8, 9]:where is the Eulerian force of the fluid flow, is the Lagrangian force of the immersed boundary, and represents the cross-sectional profile of the immersed boundary of a discrete RBC. can be smoothly approximated by a continuous kernel distribution , as proposed by Peskin [20]:

The position of the RBC is updated explicitly:

The macroscopic density is evaluated as , the velocity as , the pressure as , and the viscosity as .

Equation (4) can be decomposed into the two following distinct parts that can be executed in succession.

Collision is

Streaming is

Here, represents the distribution function after the collision, with its execution followed by streaming of the resulting distribution to neighboring nodes.

A Chapman–Enskog expansion can be used to obtain the equations of the density and momentum from (4). To derive the classical fluid equations ((1) and (2)), two macroscopic time scales ( and ) and a macroscopic length scale are required. An execution of the streaming operation on the left-hand side of each of the classical fluid equations ((1) and (2)) obtained by the Chapman–Enskog expansion can be used to determine the inertial terms.

##### 2.3. RBC Model

A natural undeformed human RBC has a biconcave disk shape. The coordinates of the RBC cross-sectional profile can be described by the following equation [15]:where , , and . A physical model of the cross-sectional profile of an RBC is shown in Figure 4.

##### 2.4. Boundary Conditions

Three different boundary conditions were implemented in this study. A periodic boundary condition was applied to both the vessel inlet and outlet [1, 4, 18, 23, 29]; a nonslip boundary condition was applied to the solid-wall boundary of the vessel [7]; and a half-way bounce-back boundary condition was applied to the straight vessel walls.

The boundary conditions of the bottom and top walls are, respectively, expressed by the following equations:

The no-slip boundary condition on the fluid-solid interface is satisfied by making the velocity of any point on the solid surface equal to that of the adjacent fluid particle [9, 17, 25, 26].

#### 3. Simulation Results and Discussions

A model of a microvessel with an annular bump was constructed as shown in Figure 1. Numerical calculations were performed over 200 × 32 lattice nodes covering a physical space of 100 *μ*m × 15 *μ*m. A uniform square mesh with a nondimensional unit of was employed. The membrane of an RBC of ≈6 *μ*m in diameter and ≈2 *μ*m in thickness was represented by 100 elastic elements. The elastic modulus and bending modulus were, respectively, set to 6.0 × 10^{-3 }Pa·s and 2.0 × 10^{−19} Pa·s for a healthy RBC and 6.0 × 10^{-2 }Pa·s and 2.0 × 10^{−18} Pa·s for a sick RBC, while the nondimensional unit conditions and were set to 0.1 and 0.001 for a healthy RBC, respectively. The nonslip boundary condition was applied to the solid-wall boundary of the channel, while the immersed RBC elastic boundary and the periodic boundary conditions were, respectively, applied to the inlet and outlet of the channel. The physical problem is governed by the nondimensional Re number defined by , where is the RGBs radius and is the flow shear rate. The Re number was 0.1. To examine the motion and deformation of the three considered RBCs during flows through constricted vessels, five cases involving different degrees of constriction values (=*d/D*) were investigated. The initial positions of RBC I (upper), RBC II (middle), and RBC III (lower) were , , and , respectively (see Figure 1).