## 4 Discussion  
This study focused on using a computational approach for the assessment of the electric field enhancement by using Au NPs in time-varying electromagnetic field cell membrane permeabilization, estimating the influence of the presence of Au NPs on transmembrane potential and on the pore opening dynamics. The simulated conditions were based on the in vitro experimental set-up described by [18]. To account for variability and uncertainty about geometries and relative placement and aggregations of the Au NPs, three different configurations were considered: spherical Au NPs equally spaced around the cell; cubic Au NPs, for accounting for the possible edge effect, equally spaced around the cell; and spherical Au NPs grouped in clusters. The experimental approach consisted of two steps: first, the electric field induced by the time-varying magnetic field and enhanced by different geometries of Au NPs was evaluated. As the enhancement of the electric field was greatly dependent on the position of the Au NPs with respect to the source, three values were considered: \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\), corresponding to the 99 th percentile of the electric field calculated on a cube containing the Au NPs, and \(E^{99\text{th}}\text{MAX}\) and \(E^{99\text{th}}\text{MIN}\), corresponding to maximum and minimum values, respectively, among those obtained calculating the 99 th percentiles of the electric field in 100 nm spherical neighborhood around each Au NP. The results showed that when the cell and the Au NPs are placed very near to the coil, at distance \(d_{1}=0.05\) mm, and when they are slightly further, at \(d_{2}=0.2\) mm, a great enhancement of the electric field was obtained thanks to the presence of the NPs. When considering the \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\), the maximum increase due to the presence of the NPs was equal to about 21,000% for both \(d_{1}\) and \(d_{2}\) and was observed when simulating spherical NPs equally spaced around the cell. Comparing \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\) to \(E^{99\text{th}}\text{MAX}\) and \(E^{99\text{th}}\text{MIN}\), i.e., comparing the 99 th percentile of the electric field obtained in the cube containing the Au NPs and the values obtained very near to the NPs, showed high variability. The maximum difference between \(E^{99\text{th}}\text{MAX}\) and \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\) was observed for the cubic NPs, as the \(E^{99\text{th}}\text{MAX}\) was 0.817 and 0.819 times greater than \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\) for \(d_{1}\) and \(d_{2}\), respectively, confirming previous findings about the edge effect and meaning that if a cubic Au NP was placed very near to the cell membrane, that specific point of the membrane would experience a very high electric field, thus enhancing the probability of pores opening. On the other hand, the maximum difference between \(E^{99\text{th}}\text{MIN}\) and \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\) was observed for the spherical NPs equally spaced around the cell, as between \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\) and \(E^{99\text{th}}\text{MIN}\), a decrement was observed almost equal to 98% for both \(d_{1}\) and \(d_{2}\). While all these findings describe the enhancement of the electric field obtained when each Au NP is placed so far from the others that is not influenced by their presence, the influence of the relative positions of NPs placed at a short distance was simulated by considering 40 NPs grouped in eight homogeneous clusters. Interestingly, although the number of simulated NPs was higher, i.e., 40 NPs in the cluster configuration vs. 24 NPs in the equally spaced configuration, the \(E^{99\text{th}}\text{A}\text{I}\text{B}\text{o}\text{x}\), \(E^{99\text{th}}\text{MAX}\), and \(E^{99\text{th}}\text{MIN}\) values in the cluster configuration were lower than those observed for the equally spaced spherical NPs.