Computational and Mathematical Methods in Medicine

Volume 2016, Article ID 2564584, 8 pages

http://dx.doi.org/10.1155/2016/2564584

## A Numerical Simulation of Cell Separation by Simplified Asymmetric Pinched Flow Fractionation

^{1}School of Life Science, Beijing Institute of Technology, Beijing 100081, China^{2}School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia^{3}Key Laboratory of Convergence Medical Engineering System and Healthcare Technology, The Ministry of Industry and Information Technology, Beijing Institute of Technology, Beijing 100081, China

Received 15 April 2016; Accepted 11 July 2016

Academic Editor: Yi Sui

Copyright © 2016 Jing-Tao Ma 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

As a typical microfluidic cell sorting technique, the size-dependent cell sorting has attracted much interest in recent years. In this paper, a size-dependent cell sorting scheme is presented based on a controllable asymmetric pinched flow by employing an immersed boundary-lattice Boltzmann method (IB-LBM). The geometry of channels consists of 2 upstream branches, 1 transitional channel, and 4 downstream branches (D-branches). Simulations are conducted by varying inlet flow ratio, the cell size, and the ratio of flux of outlet 4 to the total flux. It is found that, after being randomly released in one upstream branch, the cells are aligned in a line close to one sidewall of the transitional channel due to the hydrodynamic forces of the asymmetric pinched flow. Cells with different sizes can be fed into different downstream D-branches just by regulating the flux of one D-branch. A principle governing D-branch choice of a cell is obtained, with which a series of numerical cases are performed to sort the cell mixture involving two, three, or four classes of diameters. Results show that, for each case, an adaptive regulating flux can be determined to sort the cell mixture effectively.

#### 1. Introduction

Sorting various categories of particles from the mixture to achieve pure sample is of great importance in biological and medical engineering. With the rapid development of micro total analysis systems, small sample volume, high throughput sample processing, high efficiency, and precise particle fractionation are several representative requirements to guide the design of sorting scheme [1]. And correspondingly, a host of particle sorting techniques have been developed in these years: for example, the fluorescence-activated cell sorting [2–4], magnetic-activated cell sorting [5–7], dielectrophoresis sorting [8, 9], and size-dependent sorting [10–12]. The last one has received a remarkable attention attributing to its promising advantages of low cost, high efficiency, and being label-free. There are four typical size-dependent sorting methods that are generally reported, the deterministic lateral displacement [10, 13], the pinched flow fractionation (PFF) [14–16], the cross-flow filtering [17], and the inertial focusing sorting [18]. PFF is relatively simple because there is no extra and specific microstructure needed in the channel, and it has been used to sort polymer beads [14], microparticles [19], and emulsion droplets [20] and for blood cells [21] in recent years. In these above researches, an asymmetric pinched flow fractionation scheme (AsPFF) proposed experimentally first by Takagi et al. [19] is reported to perform a continuous separation and collection for 1.5~5 *μ*m particles; it bettered the traditional PFF remarkably, while there are still some aspects that could be improved, for example, to perform a hydrodynamic analysis and further develop an active and controllable cell or particle sorter.

In the present study, a numerical AsPFF cell sorter model is established with an immersed boundary-lattice Boltzmann method (IB-LBM), where the channel structure, the flow, the multiple sizes of cells, and their interactions are considered. Based on the model, cells with a prescribed size can be manipulated to enter a desired D-branch simply by regulating the flux of one D-branch (or the pressure of one outlet). The numerical results demonstrate that the numerical cell sorter is effective to perform an active and controllable cell sorting, which suggests an improved scheme of AsPFF and is valuable for guiding the experimental design of cell sorter on microfluidic chips.

#### 2. Models and Methods

##### 2.1. Mathematical Models

In the numerical model, the fluid motion is solved by LBM with D2Q9 lattice model. The discrete lattice Boltzmann equation of a single relaxation time model is [26–28]where is the distribution function for particles of velocity at position and time , is the time step, is the equilibrium distribution function, is the nondimensional relaxation time, and is the body force term. In the two-dimensional nine-speed (D2Q9) model [29], are given as follows:where is the lattice spacing. In (1), and are calculated by [26, 30]where are the weights defined by , for to , and for to , is the velocity of the fluid, is the speed of sound defined by , and is the body force acting on the fluid. The relaxation time related to the kinematic viscosity of the fluid is in terms of

Once the particle density distribution is known, the macroscopical quantities, including the fluid density, velocity, and pressure, are then computed from

Although the lattice Boltzmann method is original from a microscopic description of the fluid behavior, the macroscopic continuity (6) and momentum equations (7) can be recovered from it through the Chapman-Enskog multiscale analysis [31]. Then the LBM maybe can be viewed as a way of solving the macroscopic Navier-Stokes equations:

For the IB-LBM frame, the fluid motion is first solved by LBM; then the position of immersed boundary can be updated within one-time step of through [32]where is the position of the cell membrane at time . is the membrane velocity and is the fluid velocity. is the lattice side length; is the nearby area of the membrane defined by the Delta function [33–35]:where

In (9) and (10), denotes the total dimension of the model. The fluid-structure-interaction is enforced by the following equation [27, 32, 33, 36]:where is Lagrangian force acting on the ambient fluid by the cell membrane. In the present study, the cell model is proposed aswhere is the tensile force, is the bending force, is the normal force on the membrane which controls the cell incompressibility, and is the membrane-wall extrusion acting on the cell. The four force components are [33, 37–39]where , , , and are the constant coefficients for the corresponding force components. In (15), is the evolving cell area, is the reference cell area, and is unit normal vector pointing to fluid. In (16), is the position of the vessel wall, and is the cut-off distance of the effective scope in the membrane-wall interaction.

##### 2.2. Physical Model and Simulation Setup

The geometry model of for cell sorting is illustrated in Figure 1, which consists of 2 upstream branches (U-branches), 1 transitional channel, and 4 downstream branches (D-branches). The U-branches and D-branches branches are labeled with the numbers, as well as the corresponding inlets and outlets. The two U-branches are perpendicular and symmetrical about the center line of the transitional channel. The transitional channel connects the U-branches and a circular buffer area which assembles the entrances of the four D-branches. The D-branches 1 and 4 are straight, while 2 and 3 are folded for the convenience to conduct the boundary condition of outlets; 1 and 4 are also symmetrical about the center line of the transitional channel, as well as 2 and 3. The entire length and width of device are 458 *μ*m and 400 *μ*m, respectively. The width of inlet 1 and inlet 2 is 70.71 *μ*m. The width of pinched segment is 30 *μ*m. The width of outlet 1, outlet 4, and unfolded part of outlets 2 and 3 is 26 *μ*m. The width of folded part of outlets 2 and 3 is 23 *μ*m. is defined as [19], where is the flux of a D-branch, is the pressure difference between the buffer center and the outlet, and is the flow resistance produced by the microchannel. In order to allocate the flow averagely for all the D-branches under the same pressure boundary conditions, s in all D-branches should be equal. A way to make be equal is described as two steps. First, set the pressure of all outlet to be the same. Second, change the length of the folded part of D-branches 2 and 3 until the stable flows of all outlets are equal. When sorting different size of cells, set the pressure of outlets 1, 2, and 3 to be the same, while the pressure of outlet 4 is regulatable, and the flows of D-branches can be reallocated by altering the outlet pressure. To quantify the the capacity of the reallocation of flow by regulating the flow of outlet 4, we define , where bigger means bigger flow through outlet 4 and smaller flow through 1, 2, and 3. In addition, since the flow resistance in each D-branch is the same, the flow is in proportion to ; that is, regulation of flow can be simply realized by regulating the pressure difference; this means that also can be defined as .