Cell trapping in Y-junction microchannels: A numerical study of the bifurcation angle effect in inertial microfluidics

Abstract

The majority of microfluidic technologies for cell sorting and isolation involve bifurcating (e.g., Y- or T-shaped junction) microchannels to trap the cells of a specific type. However, the microfluidic trapping efficiency remains low, independently of whether the cells are separated by a passive or an active sorting method. Using a custom computational algorithm, we studied the migration of separated deformable cells in a Y-junction microchannel, with a bifurcation angle ranging from 30° to 180°. Single or two cells of initially spherical shape were considered under flow conditions corresponding to inertial microfluidics. Through the numerical simulation, we identified the effects of cell size, cytoplasmic viscoelasticity, cortical tension, flow rate, and bifurcation angle on the critical separation distance for cell trapping. The results of this study show that the trapping and isolation of blood cells, and circulating tumor cells in a Y-junction microchannel was most efficient and least dependent on the flow rate at the bifurcation angle of 120°. At this angle, the trapping efficiency for white blood cells and circulating tumor cells increased, respectively, by 46% and 43%, in comparison with the trapping efficiency at 60°. The efficiency to isolate invasive tumor cells from noninvasive ones increased by 32%. This numerical study provides important design criteria to optimize microfluidic technology for deformability-based cell sorting and isolation.

I. INTRODUCTION

The structure and remodeling dynamics of the cytoskeleton can be altered in diseased cells, leading to changes in cell deformability and motility.¹ For example, carcinoma cells become highly deformable and invasive after the “epithelial-to-mesenchymal transition” due to the replacement of cytoskeletal keratins with vimentin.^2–6 If the cells with altered mechanical phenotype (“mechanotype”) are present in blood, deformability-based microfluidics can isolate them from normal cells and other cellular components of whole blood^7–9 and then measure their mechanotype.^10,11 Together with low sample volume requirement, these features make this technology advantageous for diagnostics of cancer,⁹ blood diseases,¹² and systemic inflammation.¹³

In microfluidic sorting, cells with different viscoelasticity and size are separated across the flow because they drift to different lateral positions during perfusion in the microchannel.^14,15 This is the essence of passive cell sorting methods that rely on the hydrodynamic force and the overall geometry of the channel (i.e., hydrodynamic filtration,^6,16,17 deterministic flow,^18,19 inertial flow^20–26). The lateral separation of the cells can also be achieved by active sorting methods that use external forces (i.e., magnetic fields,^27–30 dielectrophoresis,^31,32 acoustic fields,^33–36 and optical forces^37,38). Active sorting has a higher sensitivity and specificity but lack throughput, relatively expensive, and has an increased risk of contamination.^39–41 Passive sorting is label-free, has higher throughput, and is easier to operate. However, this approach requires significant theoretical work to identify the design criteria for efficient isolation of cells.

Both active and passive methods utilize microchannels with Y- and other type of junctions to trap the separated cells in channel branches (daughter channels). The efficiency to isolate the cells of a specific type from a heterogeneous suspension of cells (e.g., blood), referred to as “trapping efficiency,” critically depends on how far the cells are separated in both longitudinal (along the flow) and lateral directions before they reach the junction. Low trapping efficiency (25% for passive and up to 55% for active methods) remains a big issue for microfluidic cell sorting.^{17,35,36,42,43} The trapping efficiency is often artificially increased by spiking the blood with cells of interest (e.g., cancer cells) along with decreasing the blood hematocrit.^{9,23,33,44–47} Such manipulations of the biological samples make microfluidic sorting less clinically reliable, especially for cancer diagnostics where the goal is to isolate rate circulating tumor cells (CTCs).⁹ This issue points out that current microfluidic devices cannot efficiently trap the cells into channel branches even with a large separation distance between the cells achieved.⁴⁷

In inertial microfluidics, where the cell suspension is subjected to a flow at a Reynolds number (Re) between 1 to 100,^20,48 separation between the cells is achieved by a hydrodynamic lift force.^17,49 The lateral equilibrium position is strongly influenced by cell size, but it is also a function of the cell’s mechanical properties, thus allowing the use of inertial microfluidics for deformability-based cell sorting.^14,24,47 Different channel geometry (straight, curved, spiral, alternating curves, pillars, ridges, and contraction/expansion) has been explored to improve cell sorting by this technology, but the outlet design remains largely unchanged.^{19,25,43,50–55}

As demonstrated by Audet⁵⁶ and Roberts,⁵⁷ partitioning of rigid particles in a Y-junction channel significantly depends on the bifurcation angle (angle between diverging daughter channels). Despite its importance, the effect of the bifurcation angle on cell trapping has not been systematically explored. Levine⁵⁸ studied the flow distribution in a dog mesentery with a narrow range of bifurcation angles (30°–80°) under Stokes and low inertia (Re < 200) flow conditions. Jaggi⁴⁴ analyzed the separation of red blood cells (RBCs) from whole blood in a microchannel with an asymmetric 90° T-junction (one side channel stemming from the main channel) under low inertial flow conditions. Most inertial microfluidic devices with Y-junction outlets have utilized a bifurcation angle of 60° or 90°.^59–64 The Stokes flow regime was used in experimental analysis of cell separation in Y-junction channels with a bifurcation angle of 45°, 60°, 90°, or 120°.^{42,57,65–71}

Theoretical work on particle, droplet, or cell separation in a Y-junction channel has been focused on Stokes flow, which is a reasonable assumption for transit of blood cells through microvasculature.⁷² Simple analytical models have been suggested to predict the distribution of red blood cells, treated as rigid particles, in a capillary network with a fixed bifurcation angle (90°).^73,74 Adam⁷⁵ described the mathematical relationship between the bifurcation angle and radii of the bifurcating vessels. Computational studies were focused on identifying flow profiles in Y- or T-junction channels with different bifurcation angles (from 30° to 180°) and/or Re but without the presence of cells or other particulates.^76–79 More recent computational work has been on the migration of a single rigid or deformable particle in a straight channel,¹⁴ Y-junction channel with a fixed bifurcation angle of 90°,^80–83 or 180° T-junction channel.⁸⁴ Only a few studies were about hydrodynamic pairwise interactions of cells for different channel geometries.^15,84,85 A lot of attention was on modeling suspension flow through a straight channel,²⁴ a Y-junction with a fixed bifurcation angle of 45°, 60°, or 90°,^86–89 or a 90° or 180° T-junction channel,^90–93 without analysis of individual particles or cells. While the bifurcation angle has a small effect on red blood cell distribution or flow resistance in microvasculature,^77,94 where Re ≪ 1, it becomes an important factor in inertial microfluidics.⁷⁷

In our computational work, we modeled pairwise cell migration into daughter channels of a symmetric Y-junction microchannel at different bifurcation angles (from 30° to 180°) and flow conditions. The cells with different deformability and size are considered under the assumption that they are already separated in the parent channel by either a passive or active sorting method. The objective of our study is to identify the critical longitudinal and lateral cell-to-cell separation distances for trapping of circulating cells (red blood cells or RBCs, white blood cells or WBCs, and CTCs).

II. METHODS

A. Computational model

The passive migration of deformable, viscoelastic cells in a microchannel was simulated by a custom fully three-dimensional computational algorithm, known as VECAM.^95,96 In this algorithm, the cell surface and intracellular interfaces (e.g., between the nucleus and cytoplasm) are tracked by the volume-of-fluid (VOF) method. The velocity field in the computational domain was determined from the solution of the continuity and Navier-Stokes equations with the values of physical parameters averaged over each mesh element containing multiple phases,

$$\begin{array}{c}\hfill \nabla \cdot \mathbf{u}=0,\hfill \\ \hfill \rho \left(\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}\right)=\nabla \cdot \mathbf{T}-\nabla p+\nabla \cdot \left[{\mu }_c\left(\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T\right)\right]+\mathbf{f}.\hfill \end{array}$$ (1)

Since the problem was to simulate cell migration in channels wider than the cell size, with the cells located far from the channel walls, the core phase of the cell (nucleus) and the adhesive force were not considered in this study. Thus, we modeled the cell cytoplasm as a one-phase viscoelastic continuum which had a Newtonian fluidlike component (cytosol) characterized by shear viscosity μ_c and polymeric component (cytoskeleton) characterized by shear viscosity μ_cs. The cytoskeleton is a source of cell viscoelasticity and an important contributor to the interfacial (cortical) tension of the cell. The viscoelasticity of the cytoskeleton was described in VECAM by the Giesekus constitutive relation,

$$\begin{array}{ccc}\hfill & \hfill & \lambda \left(\frac{\partial \mathbf{T}}{\partial t}+\left(\mathbf{u}\cdot \nabla \right)\mathbf{T}-\left(\nabla \mathbf{u}\right)\mathbf{T}-\mathbf{T}{\left(\nabla \mathbf{u}\right)}^T\right)+\frac{\lambda \alpha }{{\mu }_{cs}}{\mathbf{T}}^2\hfill \\ \hfill & \hfill & ={\mu }_{cs}\left[\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^{\text{T}}\right],{\mu }_{cs}=\lambda G.\hfill \end{array}$$ (2)

Here, u = (u, v, w) is the velocity vector, ρ is the mass density, T is the extra stress tensor that represents the cytoskeletal contribution to the intracellular stress, p is the thermodynamic pressure, G is the shear elastic modulus of the cytoskeleton, and λ is the cytoplasmic relaxation time. The cytoplasmic viscosity μ_cp = μ_c + μ_cs and μ_c = μ_ext = 1 cP, where μ_ext is the shear viscosity of the extracellular fluid, which is assumed to be a Newtonian fluid. Since we do not investigate the effects of shear thinning on cell deformation, the Giesekus mobility factor α was assumed to be zero in the current study, thus reducing the Giesekus model to the Oldroyd-B fluid one.

The cortical tension force per unit volume f was calculated in each mesh element by the continuous surface force (CSF) method using the following formula:

$$\mathbf{f}=\sigma \kappa ∥\nabla c∥\mathbf{n},$$ (3)

where σ is the cortical tension; n = ∇c/||∇c|| is the outward unit normal; κ = κ(t, x) = −∇·n is the local mean curvature; c = c(t, x) is the concentration function that takes the value of 1 inside the cell, between 0 and 1 at the interface and 0 outside the cell; and x = (x, y, z) is the position vector in the Cartesian coordinate system. We assumed that all phases had the same mass density and neglected gravity.

B. Rheological model of a living cell

It is important to mention that the model of a viscoelastic cell with cortical tension has been previously used by us to simulate the passive migration and hydrodynamic interactions of blood cells in a straight channel^14,15 and by others to study micropipette aspiration or microcirculation of blood cells.^97–99 Single-phase viscoelastic cell models have been employed for both theoretical and experimental analysis of WBCs, CTCs, and RBCs.^{1,97,99–103} While the liquid capsule model (a Newtonian fluid with an infinitesimally thin elastic solid membrane) is preferred for modeling the rheological behavior of RBCs,^100,104 it is only valid for describing the cell in which there exists a clear separation between the cytoskeleton and the plasma membrane. Experimental evidence, however, points out that hemoglobin, which makes 95% of the RBC internal structure, aggregates and forms strong bonds with spectrin in the RBC cortical layer with aging of RBCs.^105–107 Therefore, in “old” RBCs, which are more present in blood than “young” RBCs, cell deformability is strongly dependent on bulk viscoelastic properties rather on surface elasticity, a feature of the capsule model. Hemoglobin aggregation and binding to the cell membrane also characterize RBCs in blood diseases such as sickle cell disease.^12,108 WBCs and CTCs are nucleated cells with a dense three-dimensional cytoskeleton tightly attached to their cortical layer. Viscoelasticity of these cells is always defined by its bulk, as we discussed previously.^1,96

C. Simulation setup

The microchannel geometry of our computational model with a single inlet and two outlets is shown in Figs. 1(a) and 1(b). Before placing the cells, fully developed flow with a constant pressure gradient (−dP/dx) was established by imposing the plane Poiseuille flow solution for the inlet velocity (Table I), zero pressure at the outlets, and no-slip conditions at the walls. The Reynolds number was calculated using the inlet velocity U, extracellular fluid viscosity μ_ext and density ρ_ext, and the hydraulic diameter of the inlet D_h,

$$\text{Re}=\frac{{\rho }_{ext}U{D}_h}{{\mu }_{ext}},D_h=\frac{4\times \text{Area}}{\text{Perimeter}}.$$ (4)

FIG. 1.

Geometry of the computational model. (a) 3D view of the Y-junction channel used in numerical simulations. L, L₁, L₂ are the total length of the system (entrance channel and its branches) and the lengths of the entrance and branched channels, respectively. W₁ and W₂ are the widths of the entrance and branched channels. H is the height of the channels. The angle of the Y-junction (bifurcation) is denoted as θ. (b) Schematic picture of the channel midplane showing initial placement of cells. D is the diameter for the cell that is initially spherical. X is the initial separation distance between the cells in the flow direction. Z is the initial offset distance of the cell’s center from the centerline. (c) Illustration of three different scenarios for cell migration through channel branches: (1) subcritical, when both cells are in the same branch; (2) critical, where one of the cells is stuck at the junction; and (3) supercritical, when the cells travel into separate branches.

TABLE I.

Channel dimensions and flow properties used for simulations.

	Re = 1	Re = 10
Centerline velocity, U (cm/s)	1.43	14.29
Extracellular fluid viscosity, μext (Pa s)	0.001
Extracellular fluid density, ρext (kg/m3)	1000
Total channel length, L (μm)	350
Entrance channel length, L1 (μm)	140
Entrance channel width, W1 (μm)	70
Daughter channel width, W2 (μm)	35
Channel height, H (μm)	70

The single cell migration or the migration of two cells separated longitudinally by distance X and laterally by distance Z* = 2Z − R₁ − R₂ [Fig. 1(b)] was then simulated. The lateral separation distance is negative when the centroid-to-centroid distance 2Z is less than R₁ + R₂. One cell was positioned above the centerline and the other one symmetrically below the centerline, with distance Z between the cell centroid and the flow centerline (offset distance) [Fig. 1(b)]. Focusing the cells at the centerline and then positioning above and below the centerline are used in active sorting methods.¹⁰⁹ Cells being stacked vertically, horizontally spaced, and angled with respect to one another are typical configurations in magnetophoretic cell separation.³⁰ We assumed that the offset distance was unrelated to the hydrodynamic lateral position to account for the effect of the external forces in active sorting.¹¹⁰ The centroid of the cells was initially located at least 30 µm away from the inlet to prevent any perturbations that might arise from the inlet flow conditions. The transient perturbations of fully developed flow caused by cell placement in the entrance (parent) channel may lead to an unexpected drift of the leading cell toward the centerline. This transient drift was short (∼10 time steps at Re = 10 and 30 steps at Re = 1) and did not exceed 0.06% of the offset distance.

In the single cell simulation, the cytoskeletal viscosity was fixed at 10 Pa s, but deformability of the cell varied due to the differences in the relaxation time and Reynolds number, as shown in Table II. According to Eq. (4), the apparent cytoplasmic viscosity changed with time as

$${\mu }_{cp}^{+}=\left({\mu }_c+{\mu }_{cs}\right)\left[\frac{{\mu }_c}{{\mu }_c+{\mu }_{cs}}+\left(1-\frac{{\mu }_c}{{\mu }_c+{\mu }_{cs}}\right)\left(1-e^{-t/\lambda }\right)\right].$$ (5)

Since the time to reach the junction did not exceed 10 ms for all the flow conditions used in this study, Eq. (5) predicts that the apparent viscosity of the “stiff” cell with G = μ_cs/λ = 1000 Pa and λ = 0.01 s was 6.32 Pa s at this instant. For the “soft” cell with G = 50 Pa and λ = 0.2 s, the apparent viscosity was 0.49 Pa s, which was at least 13 times less than the apparent viscosity of the “stiff” cell.

TABLE II.

Material properties of idealized cell used for simulations.

	Soft	Stiff
Cytoskeletal shear elasticity, G (Pa)	50	1000
Cytoplasmic relaxation time, λ (s)	0.2	0.01
Cytoskeletal viscosity, μcs (Pa s)	10	10
Cortical tension, σ (pN/μm)	30
Density, ρ (kg/m3)	1000

Figure 1(c) shows three scenarios (subcritical, critical, or supercritical) for pairwise cell migration. In the single cell simulation, the cell was either stranded at the junction in the subcritical case or moved to an upper or a lower daughter channel at above the critical value of offset distance Z. The junction tip was sharp in most simulations, but we also investigated the effect of the curved tip junction on cell trapping, as done in other computational studies.^81,82,87,111 The curved tip was a semicircle. Its radius increased linearly with the bifurcation angle from 2 μm up to 7 μm to replicate the geometries of the other studies.

D. Mesh refinement

All the data were produced on a computational mesh with cubic grid elements of 1.094 µm in size (64 elements per 70 µm, fine mesh). Additionally, the following sizes of the grid elements were tested in the single cell case at U = 14.29 cm/s (Re = 10): 2.188 µm (coarse mesh) and 0.729 µm (finest mesh). The finest mesh simulation ran seven times slower than the fine mesh but improved the accuracy by less than 8%, as measured by the percent difference in the single-cell Z_crit. The Z_crit difference was 37% between the coarse and finest meshes.

III. RESULTS AND DISCUSSION

A. Single-cell simulation

Figure 2(a) shows shape changes of a single soft cell traveling in a Y-junction channel. The critical offset distance Z_crit for cell entrance into one of the daughter channels was identified by using the vertical asymptote of the time vs initial distance curve [Fig. 2(b)] based on the fact that the time to enter a daughter channel became very large when the cell was stranded at the junction. Interestingly, this asymptote was shifted to the right with an increase in cell diameter. This likely occurred due to the increased interaction of a larger cell with its surroundings, leading to the deviation of the cell centroid from the initial flow streamline. This effect may also explain why Z_crit increased with the cell size [Fig. 2(c)]. For any cell size considered, Z_crit decreased with the bifurcation angle until reaching a local minimum at 120°.

FIG. 2.

Analysis of single cell migration through the Y-junction channel. (a) Snapshots of a cell traveling through a 90° bifurcating channel. (b) Time course for cell migration through a 45° junction as a function of the offset distance for different particle diameters. The dotted line refers to an asymptote of the time course curve for the 8 μm diameter cell used to determine the critical offset distance (Z_crit). The shaded region identifies the subcritical distance range for the 8 μm diameter cell. [(c) and (d)] Critical offset distance as a function of the bifurcation angle for (c) different cell diameters and (d) different cell elasticities and flow velocities. (e) Effects of cell elasticity, flow velocity, and cell diameter on the cell deformability in a Y-junction channel. (f) Change in critical offset distance between a sharp (red) and a curved junction (blue) for different bifurcation angles. The triangle marks the critical distance for the T-junction channel.

Interestingly, blood vessel branching theory⁷⁵ predicts that the optimal angle between mother and daughter vessels is close to 120°. According to this theory, the optimal angles between the first daughter and mother vessels (ϕ₁) and between the second daughter and mother vessels (ϕ₂) depend on the radii of the mother (r₀), first daughter (r₁), and second daughter vessels (r₂) as follows:

$$\text{cos}\left({\varphi }_1\right)=\frac{r_{\text{0}}^4+r_1^4-{\left(r_{\text{0}}^3-r_1^3\right)}^{4/3}}{2r_{\text{0}}^2{r}_1^2},\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi }_2\right)=\frac{r_{\text{0}}^4+r_2^4-{\left(r_{\text{0}}^3-r_2^3\right)}^{4/3}}{2r_{\text{0}}^2{r}_2^2}.$$ (6)

We can apply this cylindrical vessel analysis to our problem using the XZ-plane of our Y-junction channel, i.e., our r₁ = r₂ = W₂/2 = 17.5 μm and r₀ = W₁/2 = 35 μm. Then, Eq. (6) gives θ = ϕ₁ + ϕ₂ ≈ 126°.

The way the stiff cell responded to different flow conditions changed when the bifurcation angle transitioned through 120°. For example, Z_crit increased with Re at an angle less than 120°, but the opposite behavior was seen at an angle greater than 120° [Fig. 2(d), compare red and green lines]. Z_crit for the soft cell decreased monotonically with the angle at Re = 10 [Fig. 2(d), orange line]. However, at Re = 1, it had a local minimum at 120° [Fig. 2(d), blue]. When the stiff cell relaxation time increased to the value of the soft cell (0.2 s), the critical offset distance changed with the angle similar to that of the soft cell but was much smaller in magnitude (compare orange and gold lines for Re = 10 and blue and purple lines for Re = 1).

Figure 2(e) plots the cell deformation index (DI) vs the bifurcation angle at the instant when the cell just moved beyond the junction,

$$\text{DI}=b/a,$$ (7)

where a and b are the lengths of minor and major axes of the cell. DI changed with the angle only when the cell was soft and the flow velocity was sufficiently high (Re = 10). In this case, DI increased from its minimal value of 1.7 at 30° to the maximal value of 2.4 at 105° [orange line in Fig. 2(e)]. The “shape-dependent migration”¹¹² at which more deformable particles experience a larger deviation from streamlines may explain why the soft cell has a larger Z_crit than the stiff one [Fig. 2(d)]. The results shown in Figs. 2(d) and 2(e) indicate that more deformable cells are more sensitive to changes in flow rate and bifurcation angle. Thus, previous Stokes flow analysis of rigid particle migration in bifurcating vessels and channels⁷² may not be applicable to describe the behavior of deformable cells.

When the junction tip was curved, Z_crit of the soft cell dropped. It continued to decrease with the bifurcation angle until reaching the sharp tip value at 180° [compare blue and red lines in Fig. 2(f)]. This result suggests that fabricated microchannels that have rough and sharp edges at their junctions may be inefficient in cell sorting because of high probability of cell capture at the junctions. Blood vessels have much smoother, curved bifurcation tips than microchannel junctions, which has an important physiological value in preventing excessive blood cell accumulation that can cause vessel blockage and tissue hypoxia.

B. Two-cell simulation: Identical cells

Figure 3(a) shows snapshots of two initially separated cells that moved into separate daughter channels. The leading cell (blue) disturbs the flow near the trailing cell (red), and if the separation distance between the cells is below the critical value, it forces the trailing cell to move in its wake into the same daughter channel. With an increase in the separation distance, the trailing cell may be stranded at the junction (critical case) or move into the daughter channel separate from that of the leading cell (supercritical case). As in the single-cell simulation, we used the time vs distance curves to identify the critical conditions for migration of two cells into separate daughter channels [compare red and orange lines in Fig. 3(b)]. These conditions include critical values of the longitudinal separation and offset distances, X_crit and Z_crit.

FIG. 3.

Migration of two identical cells through the Y-junction. (a) Snapshots of two cells at X = 5 μm and Z = 2.2 μm traveling into separate branched channels at 90°. (b) The time taken by cells to travel to branched channels for the three scenarios illustrated in Fig. 1(c). The region left of the dotted green line has both cells traveling to the same branched channel. The region right of the dashed blue line has each cell traveling to a separate branched channel. X_crit (Z_crit) is the critical value of X (Z) for cell separation. (c) Two-cell X_crit as a function of the bifurcation angle at Z = 1 μm and (d) two- and single-cell Z_crit as a function of the bifurcation angle at X = 5 and 10 μm for different diameters of the stiff cell. (e) Changes in Z_crit of two soft cells due to an increase in X from 5 to 10 μm and curving the junction.

The effects of cell size and bifurcation angle on the critical separation distances of two identical stiff cells at Re = 10 are displayed in Figs. 3(c) and 3(d). Both X_crit and Z_crit reached local minima at 120° and decreased with a decrease in cell diameter. As evident from Fig. 3(c), X_crit reached a lower limit of ∼2 μm at the offset distance Z = 1 μm. This means that the population of stiff cells cannot be split in a Y-junction microchannel if they are located close to the centerline and spaced longitudinally by less than 2 μm. When the cells were fully separated in the lateral direction [Z_crit > (R₁ + R₂)/2], the lower limit for X_crit decreased to zero across all bifurcation angles. If the longitudinal distance between the cells increased from 5 μm to 10 μm, Z_crit changed from 1.3 μm to 0.9 μm at 120° [red and black dotted lines in Fig. 3(d)]. Due to hydrodynamic interaction between the cells, the critical offset distance for pairwise migration was much less than that of the single cell (Z_crit = 0.18 μm, compare red and orange lines).

The soft cells were less sensitive to the longitudinal separation distance than the stiff ones [compare red and black lines in Figs. 3(d) and 3(e)]. Curving the junction tip slightly reduced the critical offset distance of these cells at a bifurcation angle less than 120° and had no effect on this distance at 120° or above [red and blue lines in Fig. 3(e)].

C. Two-cell simulation: Nonidentical cells

Figure 4 displays the critical offset distance for nonidentical cells. First, we kept the diameter of the leading cell, D₁, at 14 μm and varied the trailing cell diameter, D₂, from 14 to 8 μm. Both the cells had the same mechanical properties (they were stiff cells). For all bifurcation angles excluding 120°, there was a significant variation in Z_crit with the trailing cell diameter [Fig. 4(a)]. However, Z_crit reached its minimal value of 1.3 µm and did not change with D₂ at 120°. From these data, we can conclude that (1) if the cells are identical and not well separated, a 120° Y-junction microchannel will be able to split the cells into two equal populations and (2) if the cells have two different radii (R₁, R₂) and are laterally separated by a distance no less than $Z_{crit}^{*}=2Z_{crit}-R_1-R_2=2.6\mu \text{m}-R_1-R_2$, they will migrate into and thus be trapped in separate daughter channels of this microchannel. Note that $Z_{crit}^{*}$ is negative because R₁ + R₂ > 2Z_crit. This means that the cells do not need to be fully separated in the lateral direction to be sorted at a 120° junction: it is sufficient to have the lateral distance between the cell centroids greater than 2.6 μm and a small gap between the cells in the longitudinal direction.

FIG. 4.

Effects of cell diameter (a), cytoskeletal shear elasticity (b), and cytoplasmic shear viscosity (c) on Z_crit at U = 14.29 cm/s and X = 5 μm for different bifurcation angles. The viscosity (μ_cp) was kept constant at 10 Pa s in (a) and (b). The shear elasticity (G) was kept constant at 1000 Pa in (c). Subscripts “1” and “2” refer to the leading and trailing cells, respectively. Subscript “1, 2” indicates identical values of cell properties.

Second, we considered the same-diameter cells that had different cytoskeletal shear elasticity but equal cytoplasmic viscosity [Fig. 4(b)] or different viscosity but equal shear elasticity [Fig. 4(c)]. When the leading cell was stiff, and the trailing cell was soft, Z_crit reached a local minimum at 120°, but its minimal value (1.5) was greater than that of two identical stiff cells [compare orange and blue lines in Fig. 4(b)]. With the soft cell leading and stiff cell trailing, the local minimum was not observed, and Z_crit was greater than that of identical soft cells at 90° or higher (compare red and green lines). The tenfold difference in the cytoplasmic viscosity between the leading and trailing cells had a minimal effect on Z_crit for an angle between 30° and 105° [Fig. 4(c)]. However, the critical offset distance was sensitive to the cytoplasmic viscosity at higher angles. Specifically, at an angle between 120° and 150°, the lowest and highest Z_crit were for identical cells with low (1 Pa s) and high (10 Pa s) viscosity, respectively (compare red and blue bars). With the leading cell of high viscosity and the trailing cell of low viscosity, Z_crit was slightly below the critical offset distance for identical high-viscosity cells (compare orange and blue bars). The critical offset distance further decreased when the leading and trailing cells were of low and high viscosity, respectively (green bars).

Figure 5 shows the effect of flow velocity (Reynolds number) on trapping of identical and nonidentical cells in a bifurcating microchannel. Two cases considered, U = 1.43 cm/s (Re = 1) and U = 14.29 cm/s (Re = 10), are, respectively, at the lower end and in the intermediate range of flow velocities used in inertial microfluidics. As seen in Fig. 5(a), it is difficult to sort highly deformable cells by this approach: the critical offset distance was highest for two identical soft cells and monotonically decreased with the bifurcation angle at Re = 10 [blue line in Fig. 5(a)]. If all the cells are highly deformable, then the Y-junction microchannel with an angle between 135° and 180° is best suited for their sorting. When Re = 1, a local minimum at 120° reappeared and there was not much of a difference in Z_crit between identical soft and stiff cells (compare orange and green lines). Thus, the trapping efficiency of inertial microfluidics becomes less sensitive to cell deformability at the lower end of its velocities.

FIG. 5.

Critical offset distance as a function of bifurcation angle (a), leading-to-trailing cell elasticity ratio (b), and leading-to-trailing cell cortical tension ratio (c) at different flow conditions (U = 1.43 cm/s and 14.29 cm/s). In (a), the leading and trailing cells are identical. In (b) and (c), the trailing cell elasticity and cortical tension are kept fixed.

A trend of increasing Z_crit with the flow velocity continued in a small angle junction microchannel when the identical cells were replaced with the cells of different shear elasticity [red and blue bars in Fig. 5(b)]. However, the effect of flow velocity was minimal at 120°: for a pair of cells with an elasticity ratio of 25 or 50 (the leading cell was stiffer), Z_crit was between 1.6 and 1.7 for Re = 1 and 10 (gold bars). Thus, with the use of 120° junction microchannels, the efficiency of inertial microfluidics for isolation of soft cells from stiff ones becomes independent on the flow velocity. The cortical tension has a similar effect on Z_crit as the shear elasticity [Fig. 5(c)]. The only difference is that Z_crit slightly decreased with the leading-to-trailing cell cortical tension ratio at 120° and Re = 10 (gold bars). This indicates that inertial microfluidics can sort the cells different in cortical tension better at higher velocity.

D. Application to circulating cells

In the last simulation, the pairwise migration of RBCs, WBCs, noninvasive breast cancer cells (MCF-7 cell line), and invasive breast cancer cells (MDA-MB-231 line) in a Y-junction microchannel has been studied at different bifurcation angles and Re = 1. Table III shows the material properties used to describe these cells in the computational model. Without knowing the cytoplasmic viscosity and the relaxation time of breast cancer cells, their shear elasticity G was estimated from the measured values of the average modulus of elasticity (E)¹¹³ by using the following formula: G = E/3.

TABLE III.

Material properties of blood and circulating tumor cells used in the numerical simulation.

	RBC	WBC	Noninvasive cancer cell	Invasive cancer cell
	(Erythrocyte)	(Neutrophil)	(MCF-7)	(MDA-MB-231)	References
Diameter, D (μm)	8	8.5	18	18	11, 12, 114, and 115
Shear elasticity, G (Pa)	41	276.5	270.5	135.25	113, 115, and 116
Relaxation time, λ (s)	0.17	0.17	0.17	0.17	1 and 12
Cytoskeletal viscosity, μcs (Pa s)	7	47	46	23

For all pairs of cells considered, the critical offset distance was lowest at 120° (Fig. 6). This confirms that the most effective trapping of separated blood cells and CTCs can be achieved in a microchannel with a 120° junction. At this angle, WBCs and RBCs moved to different daughter channels if their center-to-center transverse distance 2Z was as less as 1.4 μm (red bar). This strong effect was due to a significant difference in cytoskeletal shear elasticity between WBCs and RBCs. For the MDA-MB-231/RBC, MCF-7/RBC, MDA-MB-231/WBC, and MCF-7/WBC pairs, this distance was slightly higher: 2.0–2.6 μm (blue, orange, green, and brown bars). These cell pairs differ in both shear elasticity and cell size (MCF-7/RBC, MDA-MB-231/RBC, MDA-MB-231/WBC) or in cell size only (MCF-7/WBC). The separation and trapping of invasive MDA-MB-231 cells from noninvasive MCF-7 cells required a much larger center-to-center distance: 4.2 μm (black bar). This cell pair has the same diameter and a moderate difference in shear elasticity.

FIG. 6.

Critical offset distance for isolation of RBCs, WBCs, and circulating breast cancer cells (noninvasive: MCF-7 and invasive: MDA-MB-231) in a Y-junction microchannel with angles from 60° to 150°, according to the computational model.

When compared to the standard 60° Y-junction microchannel, the trapping efficiency in the 120° microchannel has been improved by $\left(Z_{crit}^{60^{°}}-Z_{crit}^{120^{°}}\right)/Z_{crit}^{60^{°}}\times 100\%=46.2\%$ for the WBC/RBC pair, 42.9% for the MDA-MB-231/RBC pair, 54.5% for the MCF-7/RBC pair, 47.8% for the MDA-MB-231/WBC pair, 45.8% for the MCF-7/WBC pair, and 32.3% for the MDA-MB-231/MCF-7 pair.

IV. CONCLUSION

A simple and controllable way for deformability-based pairwise sorting and isolation of circulating cells is through the use of a Y-junction microchannel. The current devices that use this approach have a 45°, 60°, or 90° junction and require large separation distances to move the cells of different mechanotype into different daughter channels. This leads to low trapping efficiency. Our computational model predicted that the trapping efficiency could be significantly increased at the bifurcation angle of 120°. We demonstrated that this angle was optimal for isolating CTCs and blood cells such as RBCs and WBCs. Our numerical simulation also showed that Y-junction microfluidics was most effective for sorting of the cells that differed in cytoskeletal shear elasticity, cytoplasmic viscosity, cortical tension, and/or size. This work illustrates the need and ability of computational modeling in optimizing the design of microfluidic technology for cell sorting and isolation. Our future work will focus on modeling deformability-based cell sorting in inertial microfluidics with initial random separation distance between the cells, different geometries of the junction (asymmetric, more than two daughter channels) and its tip, and accounting for multiple distinct phases (nucleus, cytoplasm, etc.) in bulk rheology of the cell.