Optimal design of deterministic lateral displacement device for viscosity-contrast-based cell sorting

We solve a design optimization problem for a deterministic lateral displacement (DLD) device to sort same-size biological cells by their deformability, in particular to sort red blood cells by their viscosity contrast between the fluid in the interior and the exterior of the cells. A DLD device optimized for efficient cell sorting enables rapid medical diagnoses of several diseases such as malaria since infected cells are stiffer than their healthy counterparts. The device consists of pillar arrays in which pillar rows are tilted and hence are not orthogonal to the columns. This arrangement leads cells to have different final vertical displacements depending on their deformability and therefore it vertically separates the cells. The pillar cross section, the tilt angle of the pillar rows, and the center-to-center distances between pillars are free design parameters. For a given pair of viscosity-contrast values of the cells, we seek optimal DLD designs by fixing the tilt angle and the center-to-center distances. So the only design parameter is the pillar cross section. We propose an objective function such that a design minimizing it provides efficient cell sorting. The objective function is evaluated by simulating the cell flows through a device using our two-dimensional model [Kabacaoğlu et al., J. Comput. Phys. 357, 43 (2018)]. We solve the optimization problem using a stochastic optimization algorithm. Since the algorithm converges in $O\left(1000\right)$ iterations and our high-fidelity DLD model is expensive to evaluate the objective function, we propose a low-fidelity DLD model that enables fast solution of the problem. Finally, we present several scenarios where solving the optimization problem finds designs that can separate cells with similar viscosity-contrast values. These designs have cross sections that have features similar to a triangle.

Figure 1

Top views of two DLD designs for deformability-based cell sorting. Flow is from left to right. Vertically aligned pillars (in gray) form columns and the pillar rows are tilted with respect to the flow axis. The red cell is stiffer than the blue cell. (a) The conventional design cannot sort the cells because both cells zigzag. (b) Solving a design optimization problem leads to a design with the same tilt angle of the pillar rows as in (a) but a triangular pillar cross section and narrower gaps that can sort the cells since the soft cell can move along the tilted pillar rows (i.e., in the displacement mode) and the stiff cell can zigzag.

Figure 2

(a) Domain of a cell flow in a DLD device and (b) top view of the lattice of pillars with arbitrary cross sections (in gray) in a DLD device. (a) The interior and boundary of the $i\text{th}$ pillar are denoted by ${\mathrm{\Omega }}_i$ and ${\mathrm{\Gamma }}_i$ (${\mathrm{\Omega }}_0$ and ${\mathrm{\Gamma }}_0$ are the ones of the exterior wall) and ${\omega }_k$ and ${\gamma }_k$ stand for the interior and boundary of the $k\text{th}$ cell, respectively. (b) Here $G_x$ and $G_y$ denote the gap sizes and $\theta$ is the tilt angle of the pillar rows. Fluid flows in the $x$ direction (horizontal) and each pillar column is shifted in the $y$ direction (vertical) by $\mathrm{\Delta }\lambda$ with respect to the previous column. We set the gap sizes such that they are the spacings between the circumferential rectangles. In addition, $h_{\mathrm{p}}$ and $w_{\mathrm{p}}$ are the height and the width of the rectangle, respectively. The center-to-center distances in the $x$ and $y$ directions between the rectangles are ${\lambda }_x=G_x+w_{\mathrm{p}}$ and ${\lambda }_y=G_y+h_{\mathrm{p}}$, respectively.

Figure 3

(a) High-fidelity model results in large number of unknowns, which renders its use for optimization impractical. (b) We develop a low-fidelity model, which is constructed as follows. Sidewalls in the high-fidelity model pass through where the vertical gap size between the pillars is minimum (i.e., $G_y$) on the imaginary columns on the left and on the right. We assign a parabolic velocity ${\mathit{U}}_h$ on the imaginary gaps on ${\mathrm{\Gamma }}_h$ (blue arrows in the left figure) and zero velocity on ${\gamma }_h$. Then we solve (8) to obtain the density ${\mathbf{\zeta }}_h$ on $\mathrm{\Gamma }={\mathrm{\Gamma }}_h\cup {\gamma }_h$. Finally, we compute the velocity ${\mathit{U}}_l$ at $\mathbf{x}\in {\mathrm{\Gamma }}_l$ due to the density ${\mathbf{\zeta }}_h$ using (9). We impose ${\mathit{U}}_l$ as a Dirichlet boundary condition when simulating cell flows using the low-fidelity model as in the right figure. The $y_f$ on the right figure is the vertical distance between the displacing cell's center and the top of the pillar underneath it at the end of the simulation; $\alpha$ is the inclination angle of the cell.

Figure 4

Velocity magnitude and field in (a) HFDLD and (b) LFDLD without RBCs for a triangular pillar cross section. The gap sizes are $G_x=G_y=7.5\mu \text{m}$. The dimensions of the triangles are $h_{\mathrm{p}}=w_{\mathrm{p}}=17.5\mu \text{m}$.

Figure 5

Scenario 1: $({\nu }_1,{\nu }_2)=(5,10)$ and $\theta =0.17\mathrm{rad}$ (see also Table 1).

Figure 6

Scenario 2: $({\nu }_1,{\nu }_2)=(4,10)$ and $\theta =0.17\mathrm{rad}$ (see also Table 2).

Figure 7

Scenario 3: $({\nu }_1,{\nu }_2)=(5,50)$ and $\theta =0.17\mathrm{rad}$ (see also Table 3).

Figure 8

Scenario 4: $({\nu }_1,{\nu }_2)=(5,10)$ and $\theta =0.2\mathrm{rad}$ (see also Table 4).

Figure 9

(a) Phase diagram for the transport modes of the cells in the optimal designs for scenarios 1–3 as a function of viscosity contrast $\nu$. Displacing and zigzagging cells are demonstrated with blue triangles and red circles, respectively. (b) Phase diagram for sorting cells with the viscosity contrasts ${\nu }_1=5$ and ${\nu }_2=10$ using cylindrical pillars as a function of the gap sizes $G_x$ and $G_y$. We indicate pairs of gap sizes leading to separation with blue triangles and to no separation with red circles.

Figure 10

Cell trajectories in HFDLD for scenario 1: $({\nu }_1=5,{\nu }_2=10)$ and $\theta =0.17\mathrm{rad}$. The cell in blue is softer (${\nu }_1=5$) than the one in red (${\nu }_2=10$). The initial guess in (a) is in the no-separation mode, while the optimal design in (b) leads the soft cell to displace and the stiff cell to zigzag.

Figure 11

Cell trajectories in HFDLD for scenario 2: $({\nu }_1=4,{\nu }_2=10)$ and $\theta =0.17\mathrm{rad}$. The cell in blue is softer (${\nu }_1=4$) than the one in red (${\nu }_2=10$). The initial guess in (a) is in the no-separation mode, while the optimal design in (b) leads the soft cell to displace and the stiff cell to zigzag.

Figure 12

Cell trajectories in HFDLD with the optimal design for scenario 3: $({\nu }_1=5,{\nu }_2=50)$ and $\theta =0.17\mathrm{rad}$. The cell in blue is softer (${\nu }_1=5$) than the one in red (${\nu }_2=50$). The cell trajectories in the initial guess are in Fig. 1. The initial guess is in the no-separation mode, while the optimal design leads the soft cell to displace and the stiff cell to zigzag.

Figure 13

Cell trajectories in HFDLD for scenario 4: $({\nu }_1=5,{\nu }_2=10)$ and $\theta =0.2\mathrm{rad}$. The cell in blue is softer (${\nu }_1=5$) than the one in red (${\nu }_2=10$). The initial guess in (a) does not sort cells, while the optimal design in (b) leads the soft cell to displace and the stiff cell to zigzag.

Figure 14

Cell trajectories in HFDLD with the proposed triangular pillar cross section for (a) scenario 1, $({\nu }_1,{\nu }_2)=(5,10)$ and $\theta =0.17$ rad, and (b) scenario 2, $({\nu }_1,{\nu }_2)=(4,10)$ and $\theta =0.17$ rad. See Fig. 1 for scenario 3. The cell in blue is softer than the one in red. The proposed DLD design is able to sort cells for these scenarios.