Numerical simulation of the pairwise interaction of deformable cells during migration in a microchannel

Leukocytes and other circulating cells deform and move relatively to the channel flow in the lateral and translational directions. Their migratory property is important in immune response, hemostasis, cancer progression, delivery of nutrients, and microfluidic technologies such as cell separation and enrichment, and flow cytometry. Using our three-dimensional computational algorithm for multiphase viscoelastic flow, we have investigated the effect of pairwise interaction on the lateral and translational migration of circulating cells in a microchannel. The numerical simulation data show that when two cells with the same size and small separation distance interact, repulsive interaction take place until they reach the same lateral equilibrium position. During this process, they undergo swapping or passing, depending on the initial separation distance between each other. The threshold value of this distance increases with cell deformation, indicating that the cells experiencing larger deformation are more likely to swap. When a series of closely spaced cells with the same size are considered, they generally undergo damped oscillation in both lateral and translational directions until they reach equilibrium positions where they become evenly distributed in the flow direction (self-assembly phenomenon). A series of cells with a large lateral separation distance could collide repeatedly with each other, eventually crossing the centerline and entering the other side of the channel. For a series of cells with different deformability, more deformable cells, upon impact with less deformable cells, move to an equilibrium position closer to the centerline. The results of our study show that the bulk deformation of circulating cells plays a key role in their migration in a microchannel.

Figure 1

Schematic of the simulated problem. Two interacting cells in a rectangular microchannel. The cells may have different deformability and different spacing in the $x$ direction (translational direction) and/or the $z$ direction (lateral direction). They are subject to shear flow created by a pressure difference between inlet and outlet. Periodic boundary conditions have been applied at the inlet and outlet. No-slip boundary conditions were applied to the channel walls in the $y$ and $z$ directions. Chanel height $H$ and width $W$ are 70 and 140 μm, respectively. Channel length $L$ is variable.

Figure 2

The swapping of two identical cells during migration in a microchannel. (a) Time evolution of the cell lateral positions (solid lines) and cell-to-cell translational distance ($x$ distance/$D$, blue dashed line) when the initial lateral separation distance ($z$ distance/$D$) is 0.286 (initial lateral position $z_0$/$H$ for cells 1 and 2 is 0.371 and 0.314, respectively). Black dotted lines are the $z$ position of single migrating cells (scenario A). (b) Flow field at $t$/$t^{*}$ = 27.4 with the center of cell 1 used as a reference point. (c) Snapshots of cell shapes at $t$/$t^{*}$ = 19.8, 27.4 and 35. Here, the $x$ coordinate of the cell 2 spheroid is calculated relative to that of the cell 1 spheroid. The channel length is 30$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 2.0 and 4.0, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 3

The passing of two identical cells during migration in a microchannel. (a) Time evolution of the lateral positions of the cells (black solid lines) and cell-to-cell $x$ distance (blue dashed line) when the initial $z$ distance/$D$ is 0.5 ($z_0$/$H$ for cells 1 and 2 is 0.414 and 0.314, respectively). (b) Snapshots of cell shapes at $t$/$t^{*}$ = 20, 22, 23, and 24. (c) Flow field at $t$/$t^{*}$ = 22 with the center of cell 1 used as a reference point. The channel length is 30$D$; the initial translational position $x_0$/$H$ for cells 1 and 2 is 2.0 and 4.0, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 4

The effect of flow rate (a), cell viscosity (b) and cell position (c) on the critical $z$ distance of two interacting cells. $D$/$H$ = 0.2. (a) $z$/$H$ for the lower cell is 0.32, γ = 10${}^4$. (b) $z$/$H$ for the lower cell is 0.32, Re = 10. (c) γ = 10${}^4$, Re = 10.

Figure 5

The interaction of two identical cells located in different halves of the microchannel. (a) Time evolution of the cell lateral positions (solid lines) and cell-to-cell $x$ distance (blue dashed line) when $z_0$/$H$ for cells 1 and 2 is 0.629 and 0.314, respectively. (b) Flow field at $t$/$t^{*}$ = 32.5 with the centroid of cell 2 used as a reference point. The channel length is 30$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 2.0 and 4.0, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 6

(a) Diagram of a long series of cells located in the same half of the channel. (b) Time evolution of the cell lateral positions (solid lines) and cell-to-cell $x$ distance (blue dashed line) when $z_0$/$H$ for cells 1 and 2 is 0.303. The channel length is 10$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 0.7 and 1.3, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 7

(a) Diagram of a long series of cells located in different halves of the channel. (b) Time evolution of the cell lateral positions (solid lines) and cell-to-cell $x$ distance (blue dashed line) when $z_0$/$H$ for cells 1 and 2 is 0.28 and 0.72, respectively. The channel length is 6$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 0.59 and 0.61, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 8

The migration of a series of identical periodically spaced cells (two lines of cells with a large initial $z$ distance between them). The initial lateral position $z_0$/$H$ for cells 1 and 2 is 0.186 and 0.386, respectively. The channel length is 6$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 0.9 and 0.3, respectively. Re = 10, γ = 10${}^4$, $D$/$H$ = 0.2.

Figure 9

Time evolution of the lateral positions of two cells with different deformability. The initial lateral position $z_0$/$H$ for cells 1 and 2 is 0.257 and 0.357, respectively. The channel length is 10$D$, and the initial translational position $x_0$/$H$ for cells 1 and 2 is 0.5 and 1.5, respectively. Re = 10, ${\gamma }_1$ = 5 × 10${}^4$, ${\gamma }_2$ = 5 × 10${}^3$, $D$/$H$ = 0.2.

Figure 10

Lateral equilibrium position of a series of identical periodically spaced cells as a function of $x$ distance for different cell diameters. Dash-dot lines show the equilibrium position values according to scenario A (no interaction).