Numerical simulation of cell squeezing through a micropore by the immersed boundary method

Abstract

The deformability of cells has been used as a biomarker to detect circulating tumor cells (CTCs) from patient blood sample using microfluidic devices with microscale pores. Successful separations of CTCs from a blood sample requires careful design of the micropore size and applied pressure. This paper presented a parametric study of cell squeezing through micropores with different size and pressure. Different membrane compressibility modulus was used to characterize the deformability of varying cancer cells. Nucleus effect was also considered. It shows that the cell translocation time though the micropore increases with cell membrane compressibility modulus and nucleus stiffness. Particularly, it increases exponentially as the micropore diameter or pressure decreases. The simulation results such as the cell squeezing shape and translocation time agree well with experimental observations. The simulation results suggest that special care should be taken in applying Laplace-Young equation (LYE) to microfluidic design due to the nonuniform stress distribution and membrane bending resistance.

Keywords: Cancer Cell Detection; Immersed Boundary Method; Lattice Boltzmann Method; Microfluidics Design.

Figure 1:

A 2D illustration for the velocity interpolation and force spreading process used in the immersed boundary method. Fluid nodes (black squares), solid nodes (red circles). (a) The solid velocity u(X,t) is interpolated from four nearby fluid nodes enclosed by the dash line. (b) The restoring force F(X,t) induced by the solid deformation is spread out to the four nearby fluid nodes enclosed by the dash line.

Figure 2:

(a) The snapshot for the capsule squeezing through a constricted slit. The channel size was L = 18mm,W = 7.5mm,D = 3 mm. Two isosceles trapezoids were placed at L = 9.2 mm to form a rectangular slit. The capsule was a sphere with diameter of 3.6 mm. The background color showed the fluid velocity field. (b) The deformation of the capsule as it moved through the slit. The streamwise length of the capsule L/L₀ was plotted as a function of the membrane tip position $y_t^{*}=y_t/l_b$, where $y_t^{*}=0$ indicated the center of the slit along streaming direction. The experimental data was obtained from Ref.[64, Fig.3]

Figure 3:

The geometry of the channel with narrow pore for cell squeezing test.

Figure 4:

The initial setup of the cancer cell squeezing through a narrow pore. The cell with a diameter of 15 μm was positioned at 10 μm away from the inlet. The pore size is 10 μm in diameter and 3 μm in length located at x= 20 μm. A 15 Pa pressure difference was applied at the inlet and right outlet. The background color shows the velocity distribution.

Figure 5:

Snapshots of cell squeezing through the micropore at different times. Velocity magnitude (background, in LB dimensionless units), streamlines (yellow), membrane tension are shown in the figure.

Figure 6:

Snapshots of cell squeezing through a narrow pore at different time. Cell profiles in the plane sliced through the cell center are shown in the figures. Cells with different deformability (area compressibility modulus K, in units of μN/m) are shown in different colors. Yellow: K1 = 4020, Blue: K2 = 420, Red: K3 = 60, Green: K4 = 24, Black: K5 = 20. The background color shows the velocity distribution for the K2 case. The fluid distribution for cells with different deformability shows similar pattern. They are not shown here.

Figure 7:

The flow volume rate passing through the micropore for cells with different deformabilities.

Figure 8:

Tension changes in the tail membrane after the cell exits the micropore.

Figure 9:

The translocation time for cells passing through a micropore with different cell membrane compression modulus.

Figure 10:

The volume rate at the middle section of the micropore under different pressure difference for cells with deformability of 20μN/m squeezing through a micropore with diameter of 10μm. (a) volume rate; (b) normalized volume rate.

Figure 11:

The translocation time under different pressure for cells with deformability of 20μN/m squeezing through a micropore with diameter of 10μm.

Figure 12:

The volume rate at the middle section of the micropore for different pore size for cells with deformability of 20μN/m squeezing through a micropore under pressure difference of 15pa. (a) volume rate; (b) normalized volume rate.

Figure 13:

The translocation time through different micropore size for cells with deformability of 20μN/m under the pressure of 15pa.

Figure 14:

(a) A snapshot of cancer cell with a nucleus squeezing through a 10μm pore. The nucleus diameter is 6 μm, shown in red. (b) The normalized volume rate for the cancer cell with a nucleus at different nucleus membrane shear modulus.

Figure 15:

The micropore is blocked by the cells. The head and tail membrane form two curved surfaces(curves shown in the figure) that can be approximated as spherical caps with different radius r and R. The middle part of the membrane could be approximated as a cylinder with length l and radius r.

Figure 16:

Nonuniform tension distribution over the cell membrane. Concentrated high tension was observed on the membrane within the pore. The tension in μN/m are shown in the color bar.