Impact of width and spacing of interdigitated electrode on impedance-based living cell monitoring studied by computer simulation

The conventional IDE sensor consists of two electrodes, working electrode (WE) and counter electrode (CE) where a large number of electrode fingers are arranged together to form a comb-like structure as shown in Fig. 1(a). A sinusoidal voltage is applied on the WE while the CE is grounded. Figure 1(b) shows the simulation model representing the unit domain of repetition in Fig. 1(a). The model illustrates the case where N = 4 and W = S = 15 μm. The parameters used for modeling are retained from our previous report,³²⁾ so that the transition from the previous work to the present work can be easily associated without any ambiguity. The unit domain has a fixed length of 120 μm, height of 240 μm, depth of 30 μm and only one cell is present in the simulation domain. The cell density on the actual IDE is therefore 250 cells mm^−12,32) and it remains unchanged for the given domain area. The model is simulated for the three cases when N = 2, 4 and 6. Then, the W and S can be calculated by 120 μm/2N when W = S. The periodic boundary condition is applied on the adjacent faces of the y–z plane of the simulation model, while the mirror boundary condition is imposed on the faces on the x–z plane. This in fact demonstrates the efficient simulation of the actual IDE repetition due to structural symmetry. The electrode material used will not have significant influence on the impedance, and hence ignored in this work.

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The living cell is modeled as a hemisphere representing an adherent cell as shown in Fig. 1(c). The cell diameter is 10 μm.³³⁾ The gap between the cell and the electrode surface is 100 nm.³⁴⁾ The effect due to nucleus is not examined here as it has less influence on the ECIS signal.

The differential form of Maxwell's equations are solved using the finite element method using the COMSOL Multiphysics® 5.3a with AC/DC module.³⁵⁾ The parameter values were chosen based on our previous work as given in Table I.³²⁾ The frequency was swept between ${10}^{2}\,{\rm{Hz}}\lt f\lt \,{10}^{9}\,{\rm{Hz}},$ and the user controlled meshing was implemented.

In actual experimental setup, the living cells are distributed over random positions on the IDE surface. However, this situation is difficult to simulate due to high computational burden. For this reason, we have given a simplified model to take into account of the geometrical periodicity in IDE. We know that the probability of the existence of cells is uniform in space. Thus, it is relevant to take an average over all the possible cell positions within the domain in Fig. 1(b). Here, it will be sufficient to take the cell position average from the center of W to the center of S due to structural symmetry.³²⁾ As impedance value is unchanged by the cell position in y direction, we only consider possible cell positions in x direction. Considering that each situation could occur in parallel in the actual IDE sensor surface, we simulate impedance values for each possible cell positions in x and take a total impedance as parallel connections of those, as shown in Fig. 2. Then, the total impedance can be expressed as the weighted linear combination of admittances as follows:

$$\begin{eqnarray}&&\,{Z}_{{\rm{total}}}{}^{-1}=\displaystyle \sum _{i=1}^{M}{w}_{i}\displaystyle \frac{1}{{Z}_{i}}/\displaystyle \sum _{i=1}^{M}{w}_{i},\left\{\begin{array}{l}{w}_{i}=1,\,{\rm{for}}\,i=1\,{\rm{and}}\,M\\ {w}_{i}=2,\,{\rm{for}}\,i=2,3,4,\,\ldots ,\,M-1\end{array}\right.,\end{eqnarray} \tag{ 1 }$$

where i represents possible cell positions in x between the center of W and center of S, ${Z}_{i}$ is the $i{\rm{t}}{\rm{h}}$ simulated impedance value, and ${w}_{i}$ is the $i{\rm{t}}{\rm{h}}$ weight originating from symmetry.³²⁾ The cells located at position 1 (center of W) and M (center of S) have the weight equal to 1. Likewise, the weight of the cells located from positions 2 to M − 1 are doubled. We set M = 9 as an optimum value to account for the precision and computation time. The simulation results in the next section will indicate such averaged impedance, ${Z}_{{\rm{total}}}.$

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Figures 3(a)–3(c) shows the sensitivity plot where W and S are varied for N = 2, 4 and 6 respectively. The sensitivity is plotted as a function of frequency for three different W/S ratios. We may notice that the sensitivity curves show similar characteristics for the different cases of N. The reason for this is given in detail in our previous report³²⁾ by considering the sensitivity behavior at different frequency regions. The increase in sensitivity is observed for smaller W/S ratio for different electrode geometries. Figure 3(d) shows the peak sensitivity obtained from the sensitivity plot for different W/S ratios corresponding to N = 6. Later, the peak sensitivity values are extracted in a similar manner for each case of W/S corresponding to different values of N given in Table II. Thereafter, the peak sensitivity is plotted for all cases of N as a function of W/S. Figure 4 shows the peak sensitivity plot for different W/S ratios. The horizontal axis shows the W/S ratio corresponding to different N values. The vertical axis shows the peak sensitivity obtained from the sensitivity plot for different N in Fig. 3.

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At first, we notice that the sensitivity increases with the increase in the number of electrode fingers N. In other words, the sensitivity increases for smaller electrode geometry. It is evident that the number of electrode edges increases with the increase in N. This implies that the current density or the electric field is concentrated along the electrode edges as shown in Fig. 5 as a result of which the sensitivity increases with the presence of cell on the electrode edges.²⁸⁾

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Second, the results in Fig. 4 shows that the peak sensitivity increases when W/S > 1 and W/S < 1. Figure 6 shows the current streamline and surface plots along with the schematic of adherent cell for the cases when W/S ≫ 1, W/S = 1 and W/S ≪ 1 respectively. When W/S > 1, W increases and S decreases due to which the cell is always present on the electrode. In this case, the denser electric field along S and the cell coverage on the electrode edges as shown in Fig. 6(a) result in the gentle increase in peak sensitivity when W/S > 1. However, the sensitivity increase in the range of W/S < 1 is significant as seen in Figs. 3(a)–3(c) and 4. This is due to the complete coverage of the cell on the electrode as well as the electrode edges as shown in Fig. 6(c) as a result of which the impedance due to current density is higher than W/S > 1. This results in increased sensitivity. For W/S = 1, the cell can be present only on one of the electrode edges irrespective of the cell position as seen in Fig. 6(b). This is true when the cell size is comparable or smaller than the electrode size. For this reason, the sensitivity due to W/S = 1 is smaller than W/S > 1 or W/S < 1.

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Next, we see that the slope of the peak sensitivity curve in Fig. 4 decreases with the increase in N. As the electrode geometry becomes smaller, the changes due to W/S becomes negligible. This is due to the fact that the effect from the existence of the cell on the electrode edges becomes marginal once the cell has complete occupancy on the two electrode edges as shown in Fig. 6(c). Thereafter, the changes in impedance with respect to the further decrease in the electrode width in the presence of cell has less impact on sensitivity. For the case when N is small, the W and S are large in comparison to the cell size. Then the changes due to W/S has a greater influence as the cell coverage increases with the decrease in W and S. This in turn results in increased slope for smaller N as seen in Fig. 4.

We may also observe that the difference due to W/S is small compared to that due to N, as the number of the electrode edges for the given N remains unaltered considering the fact that the electrode edge has significant impact on sensitivity in comparison to the electrode ratio.