Electric field distribution during electroporation process in presence of nanoparticles

Electroporation is used to improve cell membrane permeability using a voltage pulse to generate a local electric field to open holes and improve drug uptake (Gehl, 2003; Mir, 2001). It is well known that the electric field distribution depends on the electric properties of the material (Campana et al., 2018; Denzi et al., 2015; Kranjc and Miklavčič, 2016). Numerical models were used to investigate on a tissue scale (Corovic et al., 2013; Poignard et al., 2016) the effect of the presence of nanoparticles (NPs) in the system adopted for inducing the cell electroporation. To study the influence of NPs, the electromagnetic problem has to be studied at cell level, as in Guo et al. (2024) and Krassowska and Filev (2007). Lekner in Lekner (2014) theorized the influence of conductive NPs in electroporation application, evidencing that such NPs could enhance electroporation. The present study is conducted by means of numerical simulations to study the electric field distribution in the presence of NPs made of different materials. In fact, a different distribution of the electric field is expected if dielectric or conductive NPs are close to the cell membrane due to the difference in the electric properties of the materials.

In Chiaramello et al. (2021a), the effect of gold NPs has been studied. The problem was verified experimentally in Ghorbel et al. (2019), where an improvement of electroporation using conductive NPs is obtained. Nevertheless, some authors disagree with this thesis, as in Polajžer et al. (2025), where some formulations of Au NPs do not lead to an electroporation improvement. From the literature, no clear indications can be obtained concerning the role of the electric nature (conductive vs insulating) of the NP neighboring the cell in electroporation enhancement.

In this paper, the electric field distribution around the cell membrane in the presence of elongated NPs made of conductive or dielectric material is investigated. Moreover, the effects of the orientation of the NP with respect to a hypothesized spherical cell are studied. The gold NP with an elongated shape is the one studied by Lekner in Lekner (2014). To solve the electromagnetic problem and compute the electric field strength, among the different possible numerical approaches proposed in the literature (see, e.g. Sel et al., 2005), a dynamic current distribution problem is proposed in this paper, similar to the one suggested in Sieni et al. (2023). The analysis of the electric field distribution with and without NP gives information about the impact of the NP in proximity to the cell membrane, evaluated as a variation of the intensity of the electric field. The idea is to analyze the effect of a NP on the electric field distribution close to the cell membrane during the transient part of the voltage pulse. In particular, we have computed the electric field due to a trapezoidal voltage pulse with a predefined rise time applied to the system at two time instants, one belonging to the rise-time interval and a second one belonging to the part of the pulse where the voltage is constant. In this way, the transient effect during the pulse increment from 0 V to the plateau voltage is studied. Hence, the electromagnetic problem is solved during the transient, focusing on the rise-time interval. The electric field distribution in a region-of-interest (ROI) close to the cell membrane is evaluated and compared to the distribution obtained with a constant value of the voltage, i.e. solving a static conduction problem. Two different values of rise time of the trapezoidal voltage pulse are considered to investigate the electric field distribution in the ROI.

Figure 1(a) shows the two-dimensional geometry of a cell with cytosol and membrane, diameter of 20 µm and membrane thickness 7 nm. The cell is immersed in a 1 × 1 mm square domain representing the extracellular matrix made of cell culture medium. The two electrodes are supplied by a voltage pulse, as shown in Figure 1(b). This configuration of a cell between a couple of electrodes is a typical model proposed in the literature by different authors, for example, Chiaramello et al. (2021b), Guo et al. (2024), P. Lamberti et al. (2013) and Sieni et al. (2023).

Close to the cell membrane, a single NP shaped like a rod of size 500 × 100 nm is positioned. The NP geometry includes an external layer 7 nm thick. The applied voltage is a symmetric trapezoidal pulse 100 µs long, magnitude 50 V and a rise time from 1 to 10 µs. The voltage pulse length is typical of the standard protocols of an electrochemotherapy treatment (Gehl et al., 2018). The rise time of the voltage pulse is in accordance with that of typical generators used in the laboratory (Bertacchini, 2017; IGEA, 2026). In this case, at pulse plateau, an electric field strength of 500 V/cm was applied. This electric field strength corresponds to the minimum electric field for some type of cell to start electroporation (Mir, 2006; Mir et al., 2006). This value has been chosen to verify the NPs effect in terms of local enhancement of the electric field. The electrical properties of the materials used in the models are summarized in Table 1. Both electric conductivity σ, and dielectric permittivity ε are assumed to be linear, homogeneous and isotropic properties.

The problem geometry in Figure 1 was solved as an electric field problem by means of finite element analysis (FEA) (Meunier, 2008). The mesh, whose detail is shown in Figure 1(c), has approximately 290,000 triangular elements, whereas the ROI has approximately 95,000 elements. In the cell membrane, a layered mesh was designed and represented in Figure 1(c). This mesh guided the mesh in the area close to the cell membrane and NP. The same strategy was used in the external layer of the NP.

Because of the material properties (i.e. permittivity and conductivity, Table 1) and the time scale of the applied pulse in the order of microseconds, an electro-quasi-static formulation is solved (Haus and Melcher, 1989). It is a diffusion problem, where both conduction and displacement currents are considered while the electromagnetic induction is neglected:

where V is the scalar electric potential subject to appropriate boundary conditions and initial conditions. Specifically, at the electrodes, the Dirichlet conditions are forced, while Neumann conditions hold on the external boundary. Initial conditions correspond to zero field in the whole domain.

A posteriori, the electric field $E=−∇V$ is uniquely derived thanks to the assumption of the irrotational field.

It can be noted that the two terms in equation (1) take into account the effect of drift and diffusion, respectively, in charge transportation. Therefore, they can be described as equivalent current densities, specifically conduction current density $Jc=σ∇V$ and displacement current density $Jd=ε∂∂t∇V$⁠.

Equation (1) is solved using the Comsol Multyphysics AC/DC module (Link to Innovation Starts with Multiphysics SimulationLink to the website of COMSOL, COMSOL AB, Stockholm, Sweden). In particular, the conduction current module with a time-transient solution was used. In COMSOL, when the electric current module is used in time-dependent studies, the current density also includes the displacement current dD/dt. Any magnetic material is used in this problem, but the model used takes into account both conduction and dielectric current components as well as the transient effect related to pulse rise time. The current density components associated with conduction and displacement were evaluated in the entire domain and in the membrane domain. The electric field strength is sampled on the lines Lc in Figure 3 inside the ROI marked with the red rectangle, approximately sized 1,500 × 1,500 nm².

Material properties used in simulations are summarized in Table 1 and are taken from the literature. In particular, it was considered that cytoplasm conductivity depends on the cell type and it ranges between 0.02 and 1 S/m (Labeed et al., 2006; Wang et al., 2017; Zhao et al., 2014) and relative permittivity ε_r = ε/ε₀, where ε₀ = 8.854 10⁻¹² F/m is the absolute permittivity of the free space between 80 and 150 (Goldberg et al., 2018; Guo et al., 2022; Ye et al., 2010). Zudans reported a cytoplasm conductivity of 0.13 S/m (Zudans et al., 2007). Instead, the membrane conductivity has typical values in the order of 10⁻⁶ – 10⁻⁷ S/m; the relative permittivity is close to 10 (Goldberg et al., 2018; Lamberti et al., 2015; Pavlin et al., 2005). Culture media depend on the specific type and can range from low conductivity to high conductivity, typically 0.2 S/m or between 1 and 1.5 S/m, with a typical relative permittivity of 80 (Ivorra et al., 2010; Pavlin et al., 2005; Pucihar et al., 2001). Finally, the electrical properties of NPs have been selected according to the data reported in Ferry and Rode (2025), Gauthier (1995), “AZoM”, (2026) and Zu et al. (2014).

It is interesting to show the field pattern associated with J_c and J_d at selected time instants. Figure 1(d) shows the magnitude of the conduction, |J_c|, and displacement, |J_d|, components of the current density, excited by an electric pulse with a rise time of 10 µs. The patterns are shown at different time instants, considering only the cell without NPs, in an 80 × 80 µm square. In particular, the following time instants have been considered: 1, 5, 10, 11 and 50 µs. It can be noted that the conduction current density [upper line in Figure 1(d)] increases until the pulse plateau and reaches its maximum strength at the plateau. In contrast, the displacement current density [middle line in Figure 1(d)] is larger during the pulse rise time (at 1, 5 and 10 µs) than during the plateau (at 11 and 50 µs), when it becomes negligible. All in all, the maximum value of the conduction current is orders of magnitude larger than the maximum value of the displacement current.

The distribution of the current density, conduction and displacement components, as a function of time, is shown in the region close to the membrane (a 100 nm × 100 nm²) in Figure 2(b). In the membrane region, the displacement component of the current density [middle line Figure 2(b)] is relevant (∼2,000 A/m²) during pulse rise time and negligible in the pulse plateau.

In this section, the results related to different NPs with different positions along the cell membrane for the two different pulse rise times are reported.

The analyzed NPs are positioned close to the cell membrane facing the positive electrode and along the radial direction at 45° with respect to the normal direction to the positive electrode [Figure 1(d)]. The electric field strength is evaluated in the ROI shown as a red rectangle in Figure 1(d), and along the line L_c passing through the cell interior, cell exterior and the NP core. Both horizontal and vertical oriented NPs are considered.

Considering the trapezoidal voltage pulse in Figure 1(b), the electric field strength is investigated for different time instants, i.e. in the middle of rise time, t_r/2 (being t_r the rise time of the applied pulse) for which the derivative of the displacement vector is not null, and in the middle of the pulse plateau, t_p/2 (i.e. 55 μs), for which a direct current phase is established. The electric field strength is evaluated on the L_c line for the time instant t_r/2 and t_p/2.

The colormap of the electric field strength evaluated at 55 µs, i.e. in the plateau zone of the pulse, without the NP (Figure 3) is compared to the one with the two different NPs, made of gold and dielectric material, respectively. The field is evaluated considering the pulse with 1 µs of rise time and the NP in position P₀ in Figure 1(d). Figure 4 shows differences in the electric field distribution due to the NP positioned close to the cell membrane. These variations highlight the impact of NPs on local electric field distribution in electroporation conditions.

In the case of the gold NP, the most prominent feature is the electric field strength approaching zero within the NP core, a good conductive material, evidenced by the deep blue region observed in Figure 4(a). The NP acts as a shield due to the rapid redistribution of free charges. Whereas, for a dielectric NP, the electric field approaches the maximum value [Figure 4(b)]. Consequently, the electric field is forced to bypass the NP, leading to a high electric field depicted by the intense red and orange regions primarily around the NP edge, increasing the electric field at NP extremities with respect to the case without NP. Then, the dielectric NP modifies the electric field strength close to the membrane.

The dielectric NP core shows a significant enhancement of the electric field [red color, Figure 4(a)] because the NP electric permittivity and conductivity are lower than those of the surrounding medium. High electric field magnitudes are also visible immediately surrounding the NP, particularly at its upper surface. The electric field directly close to the cell membrane appears to have a different distribution compared to the gold NP and without NP cases. Then, both NPs locally modify the electric field in different ways.

Figure 4(e)–(h) shows the different behavior of the electric field strength in proximity to the NP: the electric field at t_p/2, when the transient effect of the derivative of the displacement field is null. As expected for conductive materials, the electric field inside the gold NP core is null [Figure 4(e) and (f)]. However, in the surrounding media close to the cell membrane, the electric field is enhanced only at the NP edge. Instead, considering a dielectric NP during pulse rise time, the electric field is enhanced in the proximity of the cell membrane [Figure 4(g) and (h)]. Moreover, it is to be noted that the red color in the color scale corresponds to a double electric field strength with respect to that obtained for the direct current conduction case, in Figure 4(a). Consequently, during the pulse rise time, the electric field in proximity to the membrane is higher than that in the direct conduction case. This suggests an enhancement of the electric field strength in the presence of the NP during pulse rise time, due to the derivative of the displacement field.

In summary, at t_r/2 with a 1 µs rise time, both NPs modify the distribution of the electric field around the NP. Gold NP enhances the electric field at the edge around its conductive body by redirecting the electric field lines, leading to a shielded interior. In contrast, the dielectric NP allows the field to permeate and concentrate within its volume, resulting in a different electric field distribution. An enhancement occurs both within and at the boundaries of the NP itself. The gold and dielectric NPs provide a different mechanism for influencing the electric field near the cell membrane during the initial phase of the pulse.

Figure 4(e) and (f), shows the electric field strength along the line L_c for the horizontal-oriented gold NP in P₀ [Figure 1(d)] considering a sampling time equal to t_r/2 (0.5 and 5 μs) for rise time 1 μs and 10 μs, respectively. The highest electric field occurs inside the cell during the pulse rise time. On the contrary, in the region outside the cell, the electric field strength at 55 μs is higher than the one at t_r/2. The electric field gap between the interior and exterior of the cell membrane, as expected, is comparable at 55 μs, in the middle of the pulse plateau for both the rise time [Figure 4(e) and (f), 1 μs and 10 μs rise-time]. Nevertheless, the electric field gap between the internal and external sides of the membrane is larger during the transient than at the plateau and larger for the lower rise time, 1 μs.

Figure 4(f) shows the case of gold NP for which a pulse with a rise time of 10 μs is applied. The electric field strength at t_r/2, i.e. 5 μs for a 10 μs pulse rise time, is higher than at 55 μs for the cell interior and between the NP and the cell membrane. Specifically, the electric field strength within the cell is around 120 V/cm at 55 μs, independently of the rise time, whereas during pulse rise time, the electric field strength is approximately 840 V/cm for 1 μs and 400 V/cm for 10 µs. In the region between the cell membrane and NP, the electric field strength is lower and under 100 V/cm in the case of a 1 μs pulse rise time and under 50 V/cm for a 10 μs rise time. This suggests that a faster rise time leads to a more intense electric field build-up across the cell and its vicinity, which is a critical aspect for electroporation.

Comparing the results considering the rise-time 1 µs or 10 µs [Figure 4(g) and (h), ] in the case of dielectric NPs horizontally oriented, evident differences in the electric field strength are observed at t_r/2 and 55 µs. In the case of dielectric NP, the electric field in the cell interior at 55 μs assumes the same value as in the case of gold NP. Therefore, the NP does not influence the electric field strength in the cell interior when the direct current phase is established. The difference with the gold NP is inside the NP core, which experiences the highest electric field strength and reaches 1,000 V/cm during the 1 µs pulse rise time, and in the region between the cell membrane and the NP, where the electric field decreases from the cell membrane moving toward the NP. In the region outside the NP toward the electrode, the electric field is influenced by the type of NP, and it is lower in the case of dielectric NP.

The comparison of the color plots representing the electric field strength at 55 µs, considering a voltage pulse with a 1 µs rise time for a gold and a dielectric NP in vertically-oriented position P₀, is shown in Figure 5.

Considering the vertically-oriented gold NP [Figure 5(a)], and the direct current case, i.e. at 55 μs, the electric field at the NP edge is lower with respect to the case without the NP, and has a different distribution. In particular, the gold NP reduces the size of the area with a low electric field value. Instead, considering the vertically-oriented dielectric NP, the electric field strength in the ROI is lower than that in the case without NP. Moreover, the electric field is less affected with respect to the horizontal NP case.

Considering the electric field strength around the vertically-oriented NP during the rise time of 1 μs pulse, the gold NP enhances the electric field between the NP and the cell membrane, while in the case of a dielectric NP, it is lower [Figure 5(c) and (d)].

Considering a vertically-oriented NP [Figure 5(e) and (f)] and a rise time of 1 or 10 µs, respectively, there are differences in the electric field strength (V/cm) evaluated along the line L_c at 0.5 or 5 µs, depending on the pulse rise time.

Figure 5(e) and (f), shows the electric field strength along the line L_c for a gold NP. The electric field inside the cell is the same found for the horizontally-oriented NP considering the same time instants, i.e. t_r/2 and 55 µs. At t_r/2, the electric field strength for the voltage pulse with 1 µs rise time increases moving from the cell to the NP. As in the case of horizontally-oriented NP, the electric field at the t_r/2 assumes a higher electric field value considering the shorter rise time, i.e. 1 µs, with respect to the longer rise time, i.e. 10 µs. In the NP core, there are no significant differences for the rise times considered. In contrast, with a 10 µs rise time, the electric field strength at t_r/2, in the region between the cell and NP, is superposed to the one evaluated at 55 µs, i.e. in the direct current conduction phase.

In the case of dielectric NP [Figure 5(g) and (h)], the electric field strength along the line L_c has a different behavior with respect to the case of gold NP. In the NP core, the electric field is not null, and in the region between the cell and NP, the electric field decreases close to the NP.

The electric field inside the cell is lower in the presence of a dielectric NP with respect to the gold NP, specifically, the difference is equal to 50 V/cm. This difference is more pronounced and reaches higher values with a faster voltage pulse rise time, suggesting that pulse dynamics are critical for the local field in the presence of gold or dielectric NPs.

Figure 6 shows the color plots representing the electric field strength at 55 µs [Figure 6(a) and (b)], and at t_r/2 [Figure 6(c) and (d)], considering a pulse with a 1 µs rise time for the NP in position P₄₅ [Figure 1(d)]. The behavior of the electric field is similar to the one of the horizontal NP with a lower electric field intensity.

Considering the gold and dielectric NP in position P₄₅ at 55 μs, the electric field strength sampled on the line L_c in the region between cell membrane and NP, the electric field strength is lower for the gold NP than for the dielectric NP [Figure 6(e) and (f)].

In the literature (Guo et al., 2022), some simulation results without NPs related to the electric field distribution close to the cell are reported and they are comparable with the obtained distribution. Moreover, the distribution of electric field, and an enhanced electric field, in the presence of a prolate NP is comparable to the one found by Lekner (Lekner, 2014).

Considering this analysis, it can be evidenced that the gold NP increases the electric field inside the cell during pulse rise time (Figure 4). These phenomena do not occur with dielectric NPs. Considering the different orientations of NP, it can be observed that the horizontally oriented NP improves the electric field strength in proximity of NP extremities with respect to the same NP vertically oriented during pulse plateau [Figure 4(e) and (f), for gold and Figure 4(g) and (h), for dielectric NP horizontal-oriented and Figure 6(e) and (f), for gold and Figure 6(g) and (h), for dielectric NP vertical-oriented]. Instead of then it occurs in the plateau part of the pulse, during pulse rise-time the electric field strength is enhanced in the area of NP extremities if compared to the no-NP case (Figure 3). Moreover, the electric field shows an evident difference at the extremity of the NP if gold [Figure 4(a)] or dielectric material is considered [Figure 4(c)].

The obtained results show how the electric field is modified in a region close to the cell membrane, a nonconductive region, in the presence of a gold and dielectric NP with different orientations. From the results, it appears that the NP orientation influences the electric field distribution in the region of analysis. In the simulated conditions, the gold or dielectric NPs used show opposite effects for the same applied voltage pulses. Moreover, considering the derivative of electric displacement in the computational model allows the proper calculation of the difference in the electric field distribution due to the presence of the NP during the pulse rise time. In the present work, it was evidenced that the pulse rise time can modify the transmembrane potential. Then, the proposed model is able to quantify the effect of displacement current during the pulse rise time in a more complete model since the pulse rise time is studied. In fact, the faster the pulse rise time, the higher the electric field gap is between the internal and external parts of the cell. In future work, different cells will be investigated, taking into account all their electrical properties.