On the thin-film-dominated passing pressure of cancer cell squeezing through a microfluidic CTC chip

Abstract

Detection of circulating tumor cells (CTCs) shows strong promise for early cancer diagnosis, and cell-deformation-based microfluidic CTC chips have been playing an important role. For the design and optimization of high-throughput CTC chips, the dynamic pressure drop in the microfluidic chip during the CTC passing process is a key parameter related to the device sensitivity and filtering performance and has to be given very serious consideration. Although insights have been provided by previous researches, there is still a lack of understanding of the fundamental physics and complex interplay between viscous tumor cell and the flow inside the microfluidic filtering channel. In this paper, the process of the viscous cell squeezing through a microchannel is modeled by solving the governing equations of microscopic multiphase flows, with the tumor cell modeled by a droplet model and the immiscible cell–blood interface tracked by the volume-of-fluid method. Detailed dynamics regarding the filtering process is discussed, including the cell deformation, flow characteristics, passing pressure characteristics as well as the relationship between the pressure drop across the device and the thin film formed in the filtration channel. Current simulation shows a good agreement with analytic results, and an analytical formula is proposed to predict the passing pressure in the microchannel. Our study provides insights into the fluid physics of a viscous cell passing through a constricted microchannel, and the proposed formula can be readily applied to the design and optimization of cell-deformation-based microchannels for CTC detection.

Appendix: derivation of pressure drop estimating formula

For calculation of the pressure drop in the filter channel, the balance of the fluids in the filter channel is expressed as

$$F_{i} - F_{o} = P*A = 2\pi r_{\text{ch}} L\tau_{\text{w}}$$

(15)

in which $F_{\text{i}}$ and $F_{\text{o}}$ are the forces imposed on the cross sections of filter channel inlet and outlet, respectively; $A$ is the effective area where the main pressure drop is imposed. According to the pressure distribution in the filter channel, the pressure is mainly imposed on the annular film which has the area $A = \pi (r_{\text{ch}}^{2} - r_{i}^{2} )$. Based on Eq. (15) and the hydrodynamic theory of the thin film Eq. (11), the theoretical pressure is expressed as

$$P_{\text{theoretical}} = \frac{{8\mu_{\text{f}} r_{\text{cham}}^{2} r_{\text{ch}}^{2} LU}}{{\left( {r_{\text{ch}}^{2} - r_{i}^{2} } \right) \cdot \left( {r_{\text{ch}}^{4} + \left( {\frac{1}{\lambda } - 1} \right)r_{i}^{4} } \right)}}$$

(16)

To consider the effects of deformation resistance and the interactions on the interface, a correction term $P^{{\prime }}$ needs to be added to the theoretical value

$$P = P_{\text{theoretical}} + P^{{\prime }}$$

(17)

$P^{{\prime }}$ is the correction term expressed as $P^{{\prime }} = K \cdot P_{\text{theoretical}}$, in which K is a coefficient determined by viscosity ratio and film thickness. Considering the contribution of both shear stress and the cell resistance to deformation, the annular film and viscosity are the major factors affecting the viscous pressure. We propose that the coefficient adheres to the simple form,

$$K = K(\varepsilon ,\lambda ) = k \cdot \varepsilon \cdot f(\lambda )$$

(18)

In which $k$ is a constant, $\varepsilon$ is the non-dimensional film thickness, and $f(\lambda )$ is a function of viscosity ratio. Considering the variable combined effects introduced by different viscosity ratio values, $f(\lambda )$ is assigned the form $f(\lambda ) = \ln (\lambda /\lambda_{0} )$, where $\lambda$ is the viscosity ratio and $\lambda_{0}$ corresponds to the case when viscous resistance balances with carrier–drop interactions, and pressure drop reaches the minimum level, with the $\lambda_{0}$ value ($\lambda_{0} = 36$) being obtained from simulation data. Based on Eq. (13), one can get the non-dimensional thin-film thickness expressed as

$$\varepsilon = e/r_{\text{ch}} = 0.4223 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}}$$

(19)

By applying least-square method to the simulation data, one can get the optimum value of $k = 6.32$. Thus, the formula for viscous pressure correction term is finally expressed as

$$P^{{\prime }} = 2.67 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \ln (\lambda /\lambda_{0} ) \cdot P_{\text{theoretical}}$$

(20)

Then, the pressure drop is expressed as

$$P = [1 + 2.67 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \ln (\lambda /\lambda_{0} )] \cdot P_{\text{theoretical}}$$

(21)

With Eqs. (16) and (21) can be rewritten as,

$$P = \frac{{[1 + 2.67 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \ln (\lambda /\lambda_{0} )] \cdot 8\mu_{\text{f}} r_{\text{cham}}^{2} r_{\text{ch}}^{2} LU}}{{\left( {r_{\text{ch}}^{2} - r_{i}^{2} } \right) \cdot \left( {r_{\text{ch}}^{4} + \left( {\frac{1}{\lambda } - 1} \right)r_{i}^{4} } \right)}}$$

(22)

Based on the definition of $r_{i} = r_{\text{ch}} - e$ and the film thickness $e = 0.4223 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \cdot r_{ch}$ [from Eq. (19)], Eq. (22) has the final form as,

$$P = \frac{{[1 + 2.67 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \ln (\lambda /\lambda_{0} )] \cdot 8\mu_{\text{f}} r_{\text{cham}}^{2} LU}}{{\left[ {1 - (1 - 0.4223 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} )^{2} } \right] \cdot \left[ {1 + \left( {\frac{1}{\lambda } - 1} \right) \cdot (1 - 0.4223 \cdot Ca^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} )^{4} } \right] \cdot r_{\text{ch}}^{4} }}$$

(23)