Numerical study on the effect of capacitively coupled electrical stimulation on biological cells considering model uncertainties

Electrical stimulation of biological samples such as tissues and cell cultures attracts growing attention due to its capability of enhancing cell activity, proliferation, and differentiation. Eventually, a profound knowledge of the underlying mechanisms paves the way for innovative therapeutic devices. Capacitive coupling is one option of delivering electric fields to biological samples that has advantages regarding biocompatibility. However, its biological mechanism of interaction is not well understood. Experimental findings could be related to voltage-gated channels, which are triggered by changes of the transmembrane potential. Numerical simulations by the finite element method provide a possibility to estimate the transmembrane potential. Since a full resolution of the cell membrane within a macroscopic model would lead to prohibitively expensive models, we suggest the adaptation of an approximate finite element method. Starting from a basic 2.5D model, the chosen method is validated and applied to realistic experimental situations. To understand the influence of the dielectric properties on the modelling outcome, uncertainty quantification techniques are employed. A frequency-dependent influence of the uncertain dielectric properties of the cell membrane on the modelling outcome is revealed. This may have practical implications for future experimental studies. Our methodology can be easily adapted for computational studies relying on experimental data.

As no experimental data were available for the dielectric properties of the cell culture medium used in our study, we searched for experimental data for a similar system. The temperature and frequency dependence of the conductivity have been discussed elsewhere 1 . These sources indicate that no significant frequency dependence has been observed between 50 Hz and 1 kHz. A study including higher frequencies has been carried out for sodium chloride 2 . Sodium chloride is the main component of many cell culture media and can thus be used for comparison. Fig. S1 shows the dielectric properties of a sodium chloride solution with a conductivity that resembles the usually assumed values for cell culture media. The values are based on the experimental data presented in a previous work 2 .
The electroquasistatic (EQS) approach is justified when the electromagnetic wavelength is much longer than the characteristic length d of the system 3 . This can be expressed by the inequality where k is the wave number, which is inversely proportional to the wavelength, and reads For heterogeneous systems as studied here, the values for permittivity ε as well as conductivity σ should be the maximal occurring values 4 . The different limits of (1) are shown in Fig. S2. The dielectric properties given in Table 1 Figure S1. Evaluation of the dielectric properties of NaCl solution with (expectedly) similar properties to the actual cell culture medium for the relevant frequency range. The properties of the NaCl solution were computed using the Cole-Cole model and experimental results that were reported in a previous work 2 .  Figure S2. Evaluation of the characteristic size d and the frequency values, from which on the EQS formulation is no longer applicable. The wavenumber k is used as formulated previously 3 . The inequality |kd| 1, which needs to be fulfilled to permit the use of EQS, is visualised for different |kd|. Furthermore, the characteristic sizes of the insulating layer and the cell culture medium are shown to highlight the frequencies from which on EQS should not be used.

2/15
Details of the numerical model When the membrane was meshed explicitly, we discretized the membrane such that it is represented by at least four layers of triangular elements and even finer at the triple point, where the membrane has a rather sharp edge (Fig. S3). Figure S3. Discretization of the cell membrane (between 6 and 6.005 µm) at the cell's apex. The symmetry axis is shown in red.
A critical point in the evaluation was located close to the triple point (Fig. S4). The same figure shows an example of how the TMP was evaluated.
In Fig. S5, we show exemplary points at which the TMP was evaluated.

Validation of the numerical approach
A first numerical study on the effect of electrical stimulation on the membrane has been presented by Taghian et al. 5 using a 2D domain of 50 µm height and 100 µm width with an abstract cell model (Fig. S6b). The capacitive coupling in this model can be understood in terms of an equivalent circuit model. Two capacitors (1 µm thick insulation covering the electrodes) are connected in series with a parallel RC circuit (48 µm thick cell culture medium filling the space between the electrodes) 5 . We used the analytical formula for the impedance of a cylindrical, (lossy) dielectric to describe the total impedance of the circuit. Here, ω is the angular frequency, d i is the thickness of the cylinder, r i its radius and ε * i its complex permittivity. The complex permittivity, ε * i = ε i − jσ i /ω, contains the permittivity ε i and the conductivity σ i . The electric field in the cell culture medium, which has shown to be influential on the TMP 5 , can be estimated by taking the ratio of the cell culture medium impedance and the total impedance multiplied by the imposed voltage difference between the two electrodes. All dielectric properties of the benchmark model are summarised in Table 1 of the manuscript. The deviation between the analytical and the numerical solution is negligibly small (Fig. S7). However, the relative error grows with decreasing frequency below 100 Hz.
To validate the cell model, the approximate method is compared against the full-fidelity model at prominent points along the cell membrane (Figs. S4 and S6). The TMP in the electro-quasistatic formulation is a phasor. Thus, its absolute value and phase are computed to check the validity of the approximate method in comparison to the so-far employed full-fidelity method. Apart from this comparison, we generally report the absolute value of the TMP as this is the property of interest in therapeutic applications.  Relative difference [%] Figure S7. Relative difference between the analytically determined and the numerical FEM result of the electric field strength in the cell culture medium for the benchmark system considered in previous works 5,6 . Due to the geometry and the boundary conditions, the field is homogeneous in the cell culture medium. The numerical result corresponds to the average field in the medium.
Firstly, the TMP was computed for the same dielectric parameters as in previous works 5, 6 (see Table 1 in manuscript) using the full-fidelity as well as the approximate model. The absolute value of the TMP along the circular part of the membrane is shown in Fig. S8. Up to about 1 MHz it remains constant and does not change along the membrane. From then on it starts to change, depending on the point on the membrane. Figure S8 shows that at the membrane apex (denoted by the blue line, i.e. an angle of 0°), the TMP increases from about 1 MHz and peaks at about 10 MHz before it decreases. A special point on the membrane is the triple point, where membrane, medium and insulator meet (see also Fig. S4). On the circular part, the triple point is located at an angle of 90°. Close to this point, the TMP drops continuously from about 1 MHz on and does not peak. The approximate and the full-fidelity method do not deviate significantly. For frequencies up to 100 kHz, the TMP along the bottom line (Fig. S10) is roughly 1.9 times larger than that along the circular part and is constant except for the triple point.
To assess the accuracy of the approximate method in a straightforward manner, we compared the relative error of the result at different points along the membrane with the results of the full-fidelity model as the best possible approximation. Corresponding figures can be found in Figs. S12-S15). For all points except the triple points, the relative error of the TMP remains below 0.1%. On the bottom line, the relative error is actually only about 10 −5 % for most of the frequencies. Close to the triple point, the difference increases and reaches more than 1% on the circular part and more than 0.1% on the bottom line for high frequencies close to 100 MHz, respectively. In contrast, the phase values are more sensitive to the computational method. For small frequencies up to 1 kHz, the results deviate even more than 100%. Note that the phase in this frequency region is close to 0°, and the absolute difference is thus only a few degrees (less than 4°for both the circular part and the bottom line). For larger frequencies, the relative error drops again below 1%.

Parameter dependence: membrane conductivity
The TMP along the bottom line for different conductivities is shown in Fig S16. In addition, its phase is shown in Fig. S17.

Uncertainty Quantification
The results of the UQ analysis of the basic model at the cell bottom are shown in Fig. S18. When using the experimentally determined dielectric properties of chondrocytes, the analysis provides different results for the cell bottom (Fig. S19).