## 4 Discussion
Although a direct comparison with values observed in literature is not possible, as no other study has focused on the use of NPs when considering time-varying magnetic fields for cell permeabilization, a similar result was observed in [24], where the authors found that if using Au NPs during electroporation protocol, the electric field enhancement was much higher for isolated NPs than for highly aggregated NPs. All the considered configurations involved single NPs and grouped NPs equally spaced around and placed at very short distance, equal to 20 nm, from the cell membrane. To assess the influence of the distance from the cell membrane, two additional simulations considering the 24 spherical NPs placed at distances equal to 80 nm and 120 nm from the cell membrane were carried out. The results showed that, as expected, the E field values found along the cell membrane decreased for greater distances: maximum E values obtained for distances of 80 nm and 140 nm were equal to 30% and 11%, respectively, of the value obtained for a distance equal to 20 nm. Similarly, mean E values for distances of 80 nm and 140 nm were found to be equal to 37% and 21%, respectively, of the value obtained for a distance equal to 20 nm. Of course, a drastic decrease in the amplitude of the electric field in proximity of the cell membrane would result in a decreased probability of pore opening. These results, well expected, confirmed that the proximity of the Au NPs to the cell membrane could play a key role in the local enhancement of the electric field, and thus in the probability of enable pore opening. The second step in the experimental approach consisted of time-dependent simulation of the transmembrane potential and of the pore opening dynamic when a spherical cell was stimulated by the electric field induced by the time-varying magnetic field and enhanced by the presence of the Au NPs. As a first finding, when no NPs were considered, the electric field values induced by the time-varying magnetic field were not high enough to cause the transmembrane potential \(V_{m}\) to overcome the critical threshold for pore opening, thus not enabling the cell membrane permeabilization, even if the cell was located at a small distance from the coil. This finding, well expected, was coherent with results by our previous study [19] and by other modelling studies [28; 29], in which the authors found that with half-sinusoidal signals with frequencies lower than 500 MHz and amplitude equal to 5.5 T, the maximum \(V_{m}\) value was far below the critical value for pore opening. When considering the enhancement due to the presence of NPs, the results were completely different: both for cells placed a distance of \(d_{1}\) and \(d_{2}\) from the coil, the presence of NPs enabled the cell permeabilization. As expected, a great difference in the transmembrane potential \(V_{m}\) and in the pore density was observed if considering \(E^{99\text{th}}_{\text{AIIBox}}\), \(E^{99\text{th}}_{\text{Max}}\), and \(E^{99\text{th}}_{\text{MIN}}\). When the cell was placed in \(d_{1}\), considering both \(E^{99\text{th}}_{\text{AIIBox}}\) and \(E^{99\text{th}}_{\text{Max}}\), the cell permeabilization in all the considered NP configurations was obtained preferentially at cell poles, while a wider area of permeabilization was obtained mainly when considering \(E^{99\text{th}}_{\text{MAX}}\), i.e., when considering a very localized enhancement of the electric field, thus hypotesizing a very short distance between the cell membrane and Au NPs. This is coherent with findings of [20], in which the authors found that considering Au NPs during electroporation protocols shows strong increases in \(V_{m}\) with the presence of multiple NP clusters, demonstrating that enhanced pore opening could be possible even over sites far away from the poles, with proper distance between NPs cluster and cell membrane.