Numerical Investigation of a Novel Wiring Scheme Enabling Simple and Accurate Impedance Cytometry

Abstract

Microfluidic impedance cytometry is a label-free approach for high-throughput analysis of particles and cells. It is based on the characterization of the dielectric properties of single particles as they flow through a microchannel with integrated electrodes. However, the measured signal depends not only on the intrinsic particle properties, but also on the particle trajectory through the measuring region, thus challenging the resolution and accuracy of the technique. In this work we show via simulation that this issue can be overcome without resorting to particle focusing, by means of a straightforward modification of the wiring scheme for the most typical and widely used microfluidic impedance chip.

1. Introduction

Microfluidic impedance cytometry is a label-free technique for analysing single particles and cells at high throughput [1,2]. It is used in different biological assays, including particle sizing and counting, cell phenotyping, and disease diagnostics (e.g., [3,4,5,6]). In a typical impedance chip, suspended particles flow through a microchannel and two electrode pairs are used to measure the variation in channel impedance induced by the passage of a particle. The impedance change is exploited to characterize particle properties, in addition to particle counting. As an example, information on the size, cell membrane, and intracellular conductivity of biological cells are obtained, depending on the frequency of the stimulating AC voltage [7]. However, like in a Coulter volume measurement, the recorded signal depends not only on the intrinsic properties of the particle, but also on its trajectory through the channel [8]. This is due to the non-uniformity of the electric field in the sensing region, and produces blurring of measured particle properties [9,10,11], thus challenging the accuracy and the resolution of the technique.

Great effort has been devoted to overcoming or mitigating this issue. In particular, electrode designs that reduce the non-uniformity of the electric field (e.g., [12,13]), as well as particle focusing mechanisms (e.g., [14,15]) have been proposed. However, the former cannot completely remove the positional dependence, whereas the latter increase the complexity of the system. Recently, we proposed a method to correct the measured particle properties via a simple compensation procedure [16,17]. That method hinges on the use of a metric encoding particle trajectory, obtained by exploiting either five pairs of facing electrodes [16] or five coplanar electrodes [17]. However, multi-electrode approaches require a larger sensing region, thus increasing the number of particle coincidences for a given sample concentration [18].

The aim of this work is to demonstrate in silico that the standard and simple impedance chip comprising two electrode pairs can also provide a metric encoding particle trajectory which is effective in solving the positional dependence issue. This is achieved by means of a straightforward change in the wiring scheme proposed in Reference [19] with respect to the usual and widely used operation mode. No particle focusing systems are required, and the idea applies to both the facing and coplanar electrode configurations.

The method is presented and validated by exploiting a virtual laboratory based on the finite element method. In particular, an in silico particle sizing experiment is performed, showing excellent discrimination performance of both designs. In addition, it is shown that, contrary to the conventional one, the novel wiring scheme yields an accurate estimate of particle velocity. The relationship among estimated particle size, velocity, and cross-sectional position is also studied. The numerical investigation performed in this paper is instrumental to have a clear and deep understanding of the related experimental activity. The latter is reported in the companion paper [19], and closely agrees with the present simulation study.

2. Operating Principle

The geometric model of a standard microfluidic impedance chip with two pairs of facing electrodes is shown in Figure 1a, and its conventional wiring scheme is shown in Figure 2a. An AC voltage is applied to the top stimulating electrodes (E1 and E3), and the differential current flowing through the bottom measuring electrodes (E2 and E4) is collected. The use of a differential measurement scheme instead of an absolute one increases the signal-to-noise ratio and reduces the effect of electrode polarization [7]. The passage of a flowing particle is recorded as a pair of opposite pulses with the same amplitude. In fact, the recorded trace exhibits mirror symmetry with respect to the center of the sensing region [20,21], and the differential current is maximal (respectively, minimal) when the particle is approximately at the center of the E1–E2 (respectively, E3–E4) electrode pair (Figure 2b, curve 2). Pulse amplitude measured at low-frequency stimulating AC voltage is a measure of particle volume [1]. However, the electric field within the channel is non-uniform, and therefore the magnitude of the measured electrical current depends on particle trajectory height (i.e., y-coordinate of the center of the flowing particle). A further contribution to pulse amplitude comes from the cross current between the left and right pairs of electrodes (i.e., current flowing through a measuring electrode, E2 or E4, originating from the diagonally opposite stimulating one, respectively E3 or E1). As a consequence of the cross current, pulse amplitude is higher when the particle flows closer to the measuring electrodes than to the stimulating ones (Figure 2b, curves 1 and 3) . Therefore, the system is asymmetric top to bottom [10].

The novel wiring scheme investigated in this work is shown in Figure 2c: the AC voltage is applied to diagonally opposite stimulating electrodes (E1 and E4), and the differential current flowing through the remaining measuring electrodes (E2 and E3) is measured. In this way, a particle flowing in the lower (respectively, upper) half of the channel (Figure 2c, trajectory 1 (respectively, 3)) is located closer (respectively, farther) to the bottom measuring electrode E2 than to the top stimulating one E1 when detected by the left pair of electrodes. Then, it is located farther (respectively, closer) to the top measuring electrode E3 than to the bottom stimulating one E4 when detected by the right pair of electrodes. Accordingly, the left (respectively, right) pulse of the recorded signal is expected to have higher amplitude than the right (respectively, left) pulse (Figure 2d, curve 1 (respectively, curve 3)). Particles traveling through the middle of the channel produce pulses with equal amplitude (Figure 2d, curve 2). This suggests that (i) the average of pulse amplitudes is not affected by top–bottom asymmetry (i.e., it is the same for trajectories 1 and 3 in Figure 2c), thus yielding a better estimate of particle volume than either single pulse amplitude; and (ii) the difference of pulse amplitudes encodes particle trajectory height. In fact, the relative difference of pulse amplitudes (i.e., pulse amplitude difference divided by pulse amplitude average) is here chosen in order to obtain a metric independent of particle size.

An asymmetric bipolar Gaussian template (Figure 3) can be conveniently fitted to signal traces (Figure 2d) in order to extract the left- and right-pulse amplitudes. The template is obtained as the difference of two shifted Gaussian pulses with different amplitude and width, as follows:

$$s\left(t\right)=g_1(t-t_c+\delta /2)-g_2(t-t_c-\delta /2),$$

(1)

with:

$$g_1\left(t\right)=a_1{e}^{-t^2/\left(2{\sigma }_1^2\right)},g_2\left(t\right)=a_2{e}^{-t^2/\left(2{\sigma }_2^2\right)}.$$

(2)

This template depends on the following parameters: central time moment, $t_c$; transit time, $\delta$; pulse width controls, ${\sigma }_1$ and ${\sigma }_2$; pulse amplitude controls, $a_1$ and $a_2$.

The cube root of the mean value of the pulse amplitude controls, $(a_1+a_2)/2$, yields an estimate D of the particle diameter d:

$$D=G{[(a_1+a_2)/2]}^{1/3},$$

(3)

where G is a gain factor depending on chip geometry, buffer conductivity, and electrode double-layer impedance. However, the estimate D—referred to as “electrical diameter” in the following—suffers from a mild positional dependence issue, turning out to be greater for trajectories 1 and 3 than for trajectory 2 (Figure 2d).

The pulse amplitude relative difference $\Delta$ is introduced:

$$\Delta =\frac{a_2-a_1}{(a_1+a_2)/2}.$$

(4)

It is shown in Section 4 that $\Delta$ provides an electrical estimate Y of actual particle trajectory height y:

$$Y=\alpha \Delta H,$$

(5)

where H is channel height, $\alpha$ is a calibration coefficient depending on chip geometry, and the origin of the y-axis is chosen on the channel axis. The estimate Y is referred to as “electrical height” in the following. Moreover, the metric $\Delta$ is exploited to compensate for the spread in D induced by the positional dependence issue, thus obtaining a “corrected electrical diameter” $D-\mathrm{corr}$ (Equation (9)).

By comparison, in the conventional operation mode, a symmetric bipolar Gaussian template is used [20] (i.e., $a_1=a_2=a$ and ${\sigma }_1={\sigma }_2=\sigma$ are enforced in Equation (1)). The electrical diameter D is computed by means of Equation (3), but the pulse amplitude relative difference $\Delta$ vanishes, not conveying additional information.

Using either wiring scheme, the transit time $\delta$ yields an estimate of the particle velocity v [16,24], namely the “electrical velocity”:

$$V=L/\delta ,$$

(6)

where L is the peak-to-peak distance of signal traces (Figure 2b,d), approximated by the centre-to-centre spacing of the electrodes. Comparing the simulation traces in Figure 2d with those in Figure 2b, it appears that the peak-to-peak distance of traces provided by the new wiring scheme is less sensitive to particle trajectory than the one provided by the conventional wiring scheme. In fact, as shown in Section 4, the novel wiring scheme yields a more accurate estimate of particle velocity than the conventional wiring scheme.

The conventional and new wiring schemes discussed above can also be implemented in the microfluidic impedance chip with coplanar electrode configuration shown in Figure 1b. The field is generated by two pairs of lateral metal electrodes (E1–E2 and E3–E4), and is guided along access channels to the active area in the main channel. The equipotential surfaces at the apertures of the access channels behave as vertical liquid electrodes injecting the current into the main channel [22,23].

Using the conventional measuring scheme, the AC voltage is applied to stimulating electrodes E1 and E3, whereas using the new wiring scheme the AC voltage is applied to stimulating electrodes E1 and E4. In both cases, the differential current flowing through the remaining electrodes is measured. The electric field in the sensing region is nearly uniform along the channel height y. However, fringing field effects occur in the $xz$-plane and therefore the measured traces depend on particle lateral position x [23]. Using the new wiring scheme, the relative difference $\Delta$ between the amplitude of the opposite pulses of the signal trace is defined according to Equation (4), and turns out to encode the particle lateral position x. Accordingly, the “electrical position” X is given by:

$$X=\beta \Delta W,$$

(7)

where W is channel width, $\beta$ is a calibration coefficient depending on chip geometry, and the origin of the x-axis is chosen on the channel axis. The metric $\Delta$ can therefore be used to compensate for the spread in signal amplitude induced by the particle lateral position x.

3. Materials and Methods

The effectiveness of the proposed wiring scheme is shown by a numerical campaign, using a virtual laboratory that provides close-to-experimental synthetic data streams [25]. The following particle-sizing experiment is simulated: a mixture of insulating beads of 5, 6, and 7 $\mathsf{\mu }$m diameter (with size coefficients of variation CV = 2.5%, 1%, and 1%, respectively) is pumped through the virtual microfluidic chip assuming uniform distribution of particle centres across the channel cross-section. Both the facing electrode ($\mathcal{F}$) and coplanar electrode ($\mathcal{C}$) chips are considered, using both the conventional (${·}^{\mathrm{conv}}$) and the new (${·}^{\mathrm{new}}$) wiring scheme. The relevant synthetic data streams are denoted as follows: ${\mathcal{F}}_{\mathrm{mix}}^{\mathrm{conv}}$, ${\mathcal{F}}_{\mathrm{mix}}^{\mathrm{new}}$, ${\mathcal{C}}_{\mathrm{mix}}^{\mathrm{conv}}$, ${\mathcal{C}}_{\mathrm{mix}}^{\mathrm{new}}$.

Full details of the data streams’ generation are provided in Appendix A. Figure 4 shows one second of the syntectic data stream ${\mathcal{F}}_{\mathrm{mix}}^{\mathrm{new}}$, along with the zoom of three exemplary events.

The synthetic data streams were processed using an in-house software toolbox. First, event detection in the data stream was performed. Then, for each detected event, template fitting and feature extraction were carried out as follows. The asymmetric bipolar Gaussian template in Equation (1) was fitted to the differential signal traces recorded using the new wiring scheme. The electrical diameter D, electrical velocity V, pulse amplitude relative difference $\Delta$, electrical height Y (facing electrode chip) or electrical position X (coplanar electrode chip) were computed and compared with the actual particle properties (diameter d, velocity v, cross-section coordinate y or x). The histogram of the root mean squared error of the fit, normalized by the mean value of the pulse amplitude controls, $(a_1+a_2)/2$, is reported in Figure S1 of Supplementary Materials. On the other hand, the symmetric bipolar Gaussian template—obtained as a particular case of the asymmetric one—was fitted to the data streams recorded using the conventional wiring scheme. In that case, only the electrical quantities D and V are defined.

4. Results and Discussion

Figure 5a shows the histogram of the electrical diameter D obtained using the conventional wiring scheme. The three bead populations (5, 6, and 7 $\mathsf{\mu }$m diameter) partially overlap and are not clearly distinguishable. Three main peaks are present, but there is a significant spread and skewness. The three populations cannot be separated even by jointly using the electrical diameter D and the electrical velocity V (Figure 5b). Moreover, Figure 5c shows that the electrical velocity V is not an accurate estimate of the actual particle velocity v (correlation coefficient 0.90). In fact, the peak-to-peak distance L of signal traces used in Equation (6) depends on particle trajectory height (cf. curve 1 and 3 in Figure 2b), and thus the electrode pitch may be a poor approximation of L for off-centre particles. Therefore, a positional dependence of the electrical velocity V appears.

The electrical diameter D and the electrical velocity V cannot resolve the three bead populations, even using the novel wiring scheme (Figure 6a,b). However, the spread in estimated particle diameter D is reduced with respect to the conventional wiring scheme (cf. Figure 5a and Figure 6a). Moreover, the electrical velocity V turns out to be an excellent estimate of particle velocity v (Figure 6c, correlation coefficient 0.99). In fact, the peak-to-peak distance L of signal traces hardly depends on particle trajectory (cf. Figure 2d), and closely match the electrode pitch.

The density plots of electrical velocity V versus electrical diameter D reported in Figure 5b and Figure 6b exhibit peculiar shapes (one for each bead population). They depend on the combined effects of positional dependence of electrical diameter and velocity distribution inside the channel. Moreover, when using the conventional wiring scheme, a positional dependence of electrical velocity also comes into play. In order to gain insight into this feature, an in-depth analysis of the mapping of bead trajectories onto the $DV$-, $DY$-, and $YV$-planes is reported in Appendix B.

The proposed wiring scheme allows one to extract an additional metric—the pulse amplitude relative difference $\Delta$ (Equation (4)). As demonstrated in Figure 6d, the three bead populations are separated in the density plot of $\Delta$ against the electrical diameter D. In fact, three parabolic-shaped clusters are clearly identifiable that respectively correspond to the 5, 6, and 7 $\mathsf{\mu }$m diameter beads (respectively from left to right). Each cluster can be fitted to a quadratic function:

$$D=a[1+b{(\Delta -c)}^2],$$

(8)

where a is particle nominal diameter, and the constants b and c account for the variation in signal with particle height as determined from the metric $\Delta$. The fitting values of those parameters are listed in

Table 1. Facing electrode chip. Parameters of quadratic model equation $D=a[1+b{(\Delta -c)}^2]$ fitted to data plotted in Figure 6d.

d ($\mathsf{\mu }$m)	a ($\mathsf{\mu }$m)	b	c
5.0	5.01	0.90	−0.001
6.0	6.00	0.90	−0.001
7.0	7.00	0.85	−0.001
Mean	–	0.89	−0.001

. The constant c should vanish due to channel symmetry top to bottom, and its fitting values are in fact negligibly small. The constants b and c should be independent of particle sizes, which is clear from Table 1, where the differences are minor. The mean value for constants b and c was then used to calculate the parabolas shown in Figure 6d for the three different particle sizes, showing an excellent fit with the data.

Equation (8) was used to correct the raw data as follows:

$$D-\mathrm{corr}=\frac{D}{1+b{(\Delta -c)}^2},$$

(9)

where b and c are the mean values of the constants in Table 1. The corrected data is plotted in Figure 6g, showing an almost perfect Gaussian distribution. Fitting a Gaussian allows the CVs to be calculated as follows: 3.3%, 1.6%, and 1.4%, for the 5, 6, and 7 $\mathsf{\mu }$m diameter beads, respectively. This can be compared with the actual values of 2.5%, 1%, and 1%.

Comparing Figure 6b with Figure 6h, it appears that the simple compensation procedure in Equation (9) enabled by the proposed wiring scheme eliminates the height-dependent variation in electrical diameter D; i.e. all particles of a given size range have the same electrical diameter $D-\mathrm{corr}$, irrespective of trajectory through the channel. The density plot of $D-\mathrm{corr}$ against particle diameter d is reported in Figure 6i. The correlation coefficient turns out to be 0.98.

A further asset of the proposed wiring scheme is its ability to supply an electrical estimate Y of particle trajectory y-coordinate by proper scaling of the metric $\Delta$ (Equation (5), $H=28$ $\mathsf{\mu }$m, $\alpha =0.75$). This is suggested by the density plot of $\Delta$ against the electrical velocity V (Figure 6e), showing parabolic shapes typical of a laminar flow. The density plot of Y against the trajectory y-coordinate is reported in Figure 6f. The correlation coefficient turns out to be 0.98.

Analogous results are obtained using the coplanar electrode chip. They are shown in Figure S2, Figure S3, and Table S1 of Supplementary Materials.

5. Conclusions

The idea investigated via simulation in this work is as simple as it is effective: by a straightforward change in the electrical connections (the swapping of two wires), the standard microfluidic impedance chip achieves excellent performance without any particle focusing mechanisms. In particular, a pretty good estimate of particle velocity and a novel metric encoding particle trajectory are obtained from the signal traces. The new metric is exploited to obtain an accurate estimate of particle diameter, overcoming the positional dependence issue. In case of non-spherical particles (e.g., erythrocytes, budding yeasts, some blurring in estimated particle properties would remain, because particle orientation also comes into play. With respect to multi-electrode approaches [16,17], the use of a standard impedance chip equipped with just two electrode pairs reduces the volume of the sensing region, and therefore reduces the coincidence issue, enabling higher throughput. The use of a differential measurement—as in the conventional wiring scheme—attenuates the electrode polarization disturbance. Both facing and coplanar electrode designs are effective, and they could also be implemented on the same chip in order to achieve 3D particle localization. As a general design consideration, smaller channel geometries yield higher signal-to-noise ratio, but they are more prone to clogging issues. The best compromise must be identified according to the application at hand.