## 2 Materials and Methods  
### Simulated Experimental Set-Up  
Figure 1 shows the geometry of the source of the time-varying magnetic field, consisting of a circular coil with a radius equal to 3.07 mm, in which a time-varying electric current flows (Figure 1a). The biological cell and the Au NPs were assumed to be positioned in the hole at the center of the coil, surrounded by extracellular medium. The magnetic pulse was approximated as a sinusoid with frequency equal to 33 kHz and amplitude 5.5 T.  
The cell was modeled by a three-layered spherical model, composed of three homogenous and isotropic regions: the extracellular medium (0#), the membrane (1#), and the intracellular cytoplasm (2#). The external cell radius and membrane thickness were set equal to 10 \(\upmu\)m and 10 nm, respectively, while dielectric permittivity \(\varepsilon\) and conductivity \(\sigma\) values in the three cell regions are reported in Table 1.  
The Au NPs were modeled as solids with conductivity equal to 4.11 \(\times\) 107 S/m. Two geometries were simulated: spherical NPs with a radius equal to 40 nm, similarly to the approach described in [23], and cubic NPs with a side length equal to 40 nm, in order to take into account the irregular shape of real-life nanoparticles, which cause enhancement of the localized field, known as the edge effect [25].  
Three different conditions, simulating different relative positions of the Au NPs around the cell placed and shown in Figure 2, were considered: (1) 24 Au NPs were placed all around the cell, on the \(z=0\) plane, at a radial distance from the membrane equal to 20 nm and with an angular distance among them equal to 15\({}^{\circ}\) (Figure 2a); (2) the Au NPs were placed in the same positions of condition (1), but the Au NPs were simulated by a cubic geometry (Figure 2b); (3) 40 spherical Au NPs were placed near the membrane, grouped in 8 clusters, each composed of 5 NPs with the geometry shown in Figure 2c. The shorter distance between the membrane and the NPs of each cluster was set equal to 60 nm, while the clusters were spaced with and angular distance equal to 45\({}^{\circ}\). As a comparison, we analyzed a fourth condition in which the cell was exposed to time-varying electromagnetic fields but no Au NP was used. The three conditions \"with NPs\" and the one \"no NP\" condition were simulated when the cell was placed in two different positions, at distances \(d_{1}=0.05\) mm and \(d_{2}=0.2\) mm from the coil, for a total of eight conditions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\varepsilon\) **(F/m)** & \(\sigma\) **(S/m)** \\ \hline Extracellular medium & \(5.93\times 10^{-10}\) & 0.55 \\ Membrane & \(1.035\times 10^{-10}\) & \(1.26\times 10^{-7}\) \\ Cytoplasm & \(5.93\times 10^{-10}\) & 0.55 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dielectric properties of cell regions.  
Figure 1: Schematic view of the 3D geometry of the source of the time-varying magnetic field (**a**), of the 2D geometry (**b**), and of the spherical cell placed in an extra-cellular medium and the 2D electric field distribution (**c**).  
For all the eight considered conditions, the procedure involved two steps: first, the geometries shown in Figures 1a and 2 were simulated using the commercial software 5.6 COMSOL Multiphysics (AC/DC Module), the electric field induced by the time-varying magnetic field and enhanced by different geometries of Au NPs was evaluated, and the 99 th percentile of the module of the electric field was calculated (see Section 2.2). Second, using the electric field calculated in the first step, a time-dependent simulation of the pore formation dynamics was carried out, following the procedure described in Section 2.3.