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

Abstract

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.

Figure 1

Overview of electrical stimulation approaches for in vitro cell culture experiments. The Petri dish / insulator (grey) is shown together with the cell culture medium (blue), the cells (green) and the electrodes (black). (a) Direct contact stimulation, where the electrodes are in immediate contact with the cell culture medium. This may affect the sample, e.g. by altering the sample’s configuration as a result of chemical reactions at the electrode. Direct contact experiments are often performed using direct current (DC) signals or low-frequency waves of, for example, $20\text{Hz}$. (b) Capacitive coupling remedies this drawback by isolating the electrodes from the sample. The electrodes can, for example, be placed outside a Petri dish that contains the sample^,. Capacitive coupling requires higher frequencies (such as $60\text{kHz}$) to induce electric fields through the insulating material^–. (c) Semi-capacitive coupling refers to a set-up where one electrode is in contact and the other one is isolated from the sample.

Figure 2

2.5D model of a cell on a substrate exposed to capacitively coupled fields. The 3D equivalent of the 2.5D model (zoomed-in) is shown in (a). The cell (red) adheres to a substrate (yellow) which is a plastic insulator with a thickness of $1\mathrm{\mu }\text{m}$. The cell has a radius of $5\mathrm{\mu }\text{m}$ and its membrane a thickness of $5\text{nm}$. The 2D view of the 2.5D model is shown in (b). On the top and bottom boundaries of the domain, Dirichlet boundary conditions are applied to impose a net voltage difference. The boundary conditions mimic the electrodes, which are not explicitly modelled. Note that the electrodes are not in direct contact with the medium since they are covered by insulators. The other boundaries are electrically insulating. Material parameters for the cell cytoplasm and the culture medium are assigned as stated in Table 1. Different locations along the curved part of the cell membrane are denoted by the angle with the symmetry axis. Positions along the bottom part are denoted by the distance to the cell centre.

Figure 3

In 3D, cells with a semi-ellipsoidal shape were used to simulate the case of adherent cells. The model of two cells with a minimal distance of about $5\mathrm{\mu }\text{m}$ (zoomed-in) is shown in (a). The wireframe view of a single cell and the definition of lines along which the solution was evaluated are shown in (b). Points located on the small and large meridian were characterised by the angle between the reference vector (blue arrow) and the vector from the origin (red point) to the point on the membrane. The points along the bottom line were characterised by the distance to the centre point.

Figure 4

TMP along the curved part of the cell membrane for different membrane conductivities of $0\text{S}/\text{m}$, ${10}^{-7}\text{S}/\text{m}$, ${10}^{-5}\text{S}/\text{m}$ and ${10}^{-3}\text{S}/\text{m}$. The results were generated using the approximate method.

Figure 5

TMP along the curved part of the cell membrane for different membrane conductivities of $0\text{S}/\text{m}$, ${10}^{-7}\text{S}/\text{m}$, ${10}^{-5}\text{S}/\text{m}$ and ${10}^{-3}\text{S}/\text{m}$ when the cell was separated from the well bottom by a gap of 100 nm. The results were generated using the approximate method.

Figure 6

Left axis: Mean and $90\%$ prediction interval of the absolute value of the TMP at the cell apex for the basic model shown in Fig. 2. Right axis: First order Sobol indices for each uncertain parameter, i.e. the conductivity (dark green) and permittivity (orange) of the membrane, the conductivity (purple) and permittivity (brown) of the cytoplasm, and the permittivity of the coating (pink), respectively.

Figure 7

Left axis: Mean and $90\%$ prediction interval of the absolute value of the TMP at the cell apex for the basic model shown in Fig. 2. Right axis: First order Sobol indices for each uncertain parameter, i.e. the conductivity (dark green) and permittivity (orange) of the membrane, the conductivity (pink) of the cytoplasm, and the permittivity of the coating (brown), respectively.

Figure 8

(a) The electric field strength around the cells and the TMP on the cell surface are shown for the expectation values of the cell dielectric properties (Table 2). (b) Left axis: Mean and $90\%$ prediction interval of the absolute value of the TMP at the cell apex for the ellipsoid cell model shown in Fig. 3. Right axis: First order Sobol indices for each uncertain parameter, i.e. the conductivity (dark green) and permittivity (orange) of the membrane. The results were evaluated along the large meridian using the angle with the central normal vector as an indicator for the location [as indicated by the red arrow in (a)]. The results along the small meridian deviated only slightly and are not reported.

Figure 9

(a) The electric field strength around the cells and the TMP on the cell bottom are shown for the expectation values of the cell dielectric properties (Table 2). (b) Left axis: Mean and $90\%$ prediction interval of the absolute value of the TMP at the cell bottom for the ellipsoid cell model shown in Fig. 3. Right axis: First order Sobol indices for each uncertain parameter, i.e. the conductivity (dark green) and permittivity (orange) of the membrane. The results were evaluated along the bottom line of the cell using the distance to the centre as an indicator for the location [as indicated by the red arrow in (a)].