This study aims to present a numerical model to predict the electric field distribution in the presence of conductive or dielectric nanoparticles close to the cell membrane during electrochemotherapy treatment.
To solve the electromagnetic problem in transient conditions, finite element analyses are performed. Specifically, the field distribution during both the rise and the constant period of the pulse is investigated. To analyze the impact of the pulse dynamics, different rise times are considered.
The effect of a trapezoidal voltage pulse on the electric field distribution in a region of interest close to the nanoparticle is studied, considering nanoparticles made of conductive or dielectric materials.
The effect of different nanoparticle materials is studied, as well as the effect of the pulse rise time.
1. Introduction
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.
2. The finite element model
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).
The panel a schematic depicts a cell within cell medium between electrodes, with the lower electrode at 0 volts and insulating boundary conditions along the sides. Panel b plots voltage against time. The pulse rises from 0 volts to 100 volts during t r, remains at 100 volts through t p, then falls to 0 volts during t r. The upper duration spans 100 microseconds, the central point marks 50 microseconds, and the midpoint of the rising segment marks t r over 2. Panel c contains three finite element mesh views. Panel c 1 depicts a circular cell region spanning approximately minus 25 to 25 micrometres. Panel c 2 enlarges a nanoparticle measuring 500 nanometres in length and 100 nanometres in width across the cell membrane, with cell exterior and cell interior identified. Panel c 3 further enlarges the membrane mesh between approximately minus 0.30 and minus 0.26 micrometres horizontally and 9.98 to 10.02 micrometres vertically. Panel d depicts a circular cell with nanoparticle positions P 0 and P 45 separated by 45 degrees. Three R O I views depict horizontal, vertical, and diagonal nanoparticle orientations relative to the membrane, with L c marked in each view.(a) Geometry of the simulated model and (b) time evolution of the applied voltage pulse. (c) Mesh of (c1) the geometry with details at the (c2) boundary between the nanoparticle and the cell, and (c3) in the membrane region. (d) Position of nanoparticle close to the cell membrane. Zoom of geometry for a (left) horizontal, (center) vertical and at 45° (right) nanoparticle. Sample line Lc and ROI (red rectangle)
Source: Authors’ own work
The panel a schematic depicts a cell within cell medium between electrodes, with the lower electrode at 0 volts and insulating boundary conditions along the sides. Panel b plots voltage against time. The pulse rises from 0 volts to 100 volts during t r, remains at 100 volts through t p, then falls to 0 volts during t r. The upper duration spans 100 microseconds, the central point marks 50 microseconds, and the midpoint of the rising segment marks t r over 2. Panel c contains three finite element mesh views. Panel c 1 depicts a circular cell region spanning approximately minus 25 to 25 micrometres. Panel c 2 enlarges a nanoparticle measuring 500 nanometres in length and 100 nanometres in width across the cell membrane, with cell exterior and cell interior identified. Panel c 3 further enlarges the membrane mesh between approximately minus 0.30 and minus 0.26 micrometres horizontally and 9.98 to 10.02 micrometres vertically. Panel d depicts a circular cell with nanoparticle positions P 0 and P 45 separated by 45 degrees. Three R O I views depict horizontal, vertical, and diagonal nanoparticle orientations relative to the membrane, with L c marked in each view.(a) Geometry of the simulated model and (b) time evolution of the applied voltage pulse. (c) Mesh of (c1) the geometry with details at the (c2) boundary between the nanoparticle and the cell, and (c3) in the membrane region. (d) Position of nanoparticle close to the cell membrane. Zoom of geometry for a (left) horizontal, (center) vertical and at 45° (right) nanoparticle. Sample line Lc and ROI (red rectangle)
Source: Authors’ own work
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.
Electric properties of the materials; [NU] = not unit
| Model domain | Electrical conductivity σ [S m−1] | Relative permittivity εr [NU] | References |
|---|---|---|---|
| Cytosol | 0.13 | 60 | (Zudans et al., 2007) |
| Membrane | 5.3 10–6 | 12.8 | (Goldberg et al., 2018; Lamberti et al., 2015; Pavlin et al., 2005) |
| Culture medium | 1.3 | 80 | (Ivorra et al., 2010; Pavlin et al., 2005; Pucihar et al., 2001) |
| Au NP | 4.1 107 | 10 | (Gauthier, 1995; Zu et al., 2014). |
| Silica NP | 10–13 | 3 | (Ferry and Rode, 2025; “Properties”, 2026) |
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 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 and displacement current density .
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 εr = 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).
3. Results
3.1 Field patterns in time
It is interesting to show the field pattern associated with Jc and Jd at selected time instants. Figure 1(d) shows the magnitude of the conduction, |Jc|, and displacement, |Jd|, 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.
The panel a columns correspond to 1, 5, 10, 11, and 50 microseconds during a voltage pulse. The upper row maps the magnitude of J c in amperes per square metre around a circular cell. J c remains low at 1 microsecond, increases near the cell at 5 microseconds, reaches its strongest concentration around the cell at 10 and 11 microseconds, and remains concentrated at 50 microseconds. The middle row maps the magnitude of J d in amperes per square metre. J d concentrates near the cell at 1 and 5 microseconds, decreases at 10 microseconds, and approaches the lowest mapped values at 11 and 50 microseconds. The lower row plots time from 0 to 100 microseconds. The pulse rises from 0 to its plateau by 10 microseconds, remains constant until 90 microseconds, and returns to 0 by 100 microseconds. Markers identify 1 microsecond near the baseline, 5 microseconds during the rise, 10 and 11 microseconds near the start of the plateau, and 50 microseconds at the plateau midpoint. Panel b repeats the five times for an enlarged membrane region centred near 10 micrometres. The J c maps increase from 1 to 10 microseconds, then decrease at 11 and 50 microseconds. The J d maps contain a narrow membrane band at 1 and 5 microseconds, a weaker band at 10 microseconds, and minimal values at 11 and 50 microseconds.Evaluation of the conduction current (upper line), |Jc|, and displacement current density (middle line), |Jd|, at 1, 5, 10, 11 and 50 µs in (a) the entire domain and (b) on the membrane (the rectangular strip is the membrane)
Source: Authors’ own work
The panel a columns correspond to 1, 5, 10, 11, and 50 microseconds during a voltage pulse. The upper row maps the magnitude of J c in amperes per square metre around a circular cell. J c remains low at 1 microsecond, increases near the cell at 5 microseconds, reaches its strongest concentration around the cell at 10 and 11 microseconds, and remains concentrated at 50 microseconds. The middle row maps the magnitude of J d in amperes per square metre. J d concentrates near the cell at 1 and 5 microseconds, decreases at 10 microseconds, and approaches the lowest mapped values at 11 and 50 microseconds. The lower row plots time from 0 to 100 microseconds. The pulse rises from 0 to its plateau by 10 microseconds, remains constant until 90 microseconds, and returns to 0 by 100 microseconds. Markers identify 1 microsecond near the baseline, 5 microseconds during the rise, 10 and 11 microseconds near the start of the plateau, and 50 microseconds at the plateau midpoint. Panel b repeats the five times for an enlarged membrane region centred near 10 micrometres. The J c maps increase from 1 to 10 microseconds, then decrease at 11 and 50 microseconds. The J d maps contain a narrow membrane band at 1 and 5 microseconds, a weaker band at 10 microseconds, and minimal values at 11 and 50 microseconds.Evaluation of the conduction current (upper line), |Jc|, and displacement current density (middle line), |Jd|, at 1, 5, 10, 11 and 50 µs in (a) the entire domain and (b) on the membrane (the rectangular strip is the membrane)
Source: Authors’ own work
3.2 Influence of the nanoparticle
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 Lc 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, tr/2 (being tr 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, tp/2 (i.e. 55 μs), for which a direct current phase is established. The electric field strength is evaluated on the Lc line for the time instant tr/2 and tp/2.
3.3 Horizontal-oriented nanoparticle
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.
The panel a contour map ranges from 0 to 250 volts per centimetre. Electric field values reach their minimum near the centre of the curved boundary and increase with distance from this region. Panel b ranges from 0 to 500 volts per centimetre. Values below the curved boundary approach 500 volts per centimetre, while values above it increase gradually from the boundary towards the outer region. Panel c plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 810 volts per centimetre before dropping to about 80 volts per centimetre at 0.1 micrometres, then increasing to about 95 volts per centimetre in the cell exterior. At 55 microseconds, the cell interior remains near 115 volts per centimetre before dropping to about 10 volts per centimetre, then increasing to about 45 volts per centimetre. Panel d plots the same sampling range at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre before dropping to about 35 volts per centimetre, then increasing to about 55 volts per centimetre. At 55 microseconds, the cell interior remains near 110 volts per centimetre before dropping to about 10 volts per centimetre, then increasing to about 50 volts per centimetre.Electric field strength for the case without NP position P0 in Figure 1(d) at (a) 55 μs and (b) at tr/2 = 0.5 μs, considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
The panel a contour map ranges from 0 to 250 volts per centimetre. Electric field values reach their minimum near the centre of the curved boundary and increase with distance from this region. Panel b ranges from 0 to 500 volts per centimetre. Values below the curved boundary approach 500 volts per centimetre, while values above it increase gradually from the boundary towards the outer region. Panel c plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 810 volts per centimetre before dropping to about 80 volts per centimetre at 0.1 micrometres, then increasing to about 95 volts per centimetre in the cell exterior. At 55 microseconds, the cell interior remains near 115 volts per centimetre before dropping to about 10 volts per centimetre, then increasing to about 45 volts per centimetre. Panel d plots the same sampling range at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre before dropping to about 35 volts per centimetre, then increasing to about 55 volts per centimetre. At 55 microseconds, the cell interior remains near 110 volts per centimetre before dropping to about 10 volts per centimetre, then increasing to about 50 volts per centimetre.Electric field strength for the case without NP position P0 in Figure 1(d) at (a) 55 μs and (b) at tr/2 = 0.5 μs, considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
The panels a and b map electric field around an elliptical nanoparticle above the curved boundary, with scales from 0 to 250 volts per centimetre. Panel a contains the lowest field above the nanoparticle and higher values farther from it, with local increases near its ends. Panel b contains low values above and below the nanoparticle, higher values near its ends, and the highest values within the nanoparticle. Panels c and d map the corresponding field with scales from 0 to 500 volts per centimetre. Panel c contains low values around the nanoparticle and values near 500 volts per centimetre beyond the curved boundary. Panel d contains a low field above and below the nanoparticle, local increases near its ends, and values near 500 volts per centimetre within the nanoparticle and beyond the boundary. Panel e plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 840 volts per centimetre in the cell interior, about 80 volts per centimetre in the first cell exterior segment, near 0 through the nanoparticle core, and about 20 volts per centimetre in the final cell exterior segment. At 55 microseconds, the corresponding values measure about 115, 10, 0, and 110 to 130 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, values measure about 400, 35, 0, and 20 volts per centimetre. At 55 microseconds, values measure about 115, 15, 0, and 110 to 130 volts per centimetre. Panel g plots the regions at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 790 volts per centimetre in the cell interior, decreases from about 80 to 0 in the first exterior segment, decreases from about 1000 to 940 through the nanoparticle core, and increases from 0 to about 30 in the final exterior segment. At 55 microseconds, values measure about 110, 0, 300, and 0 to 20 volts per centimetre. Panel h plots the regions at 5 and 55 microseconds. At 5 microseconds, values measure about 390, decrease from about 45 to 0, decrease from about 540 to 500, and increase from 0 to about 20 volts per centimetre. At 55 microseconds, values measure about 110, decrease to 0, remain near 300, and increase from 0 to about 20 volts per centimetre.Color plot of the electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle horizontally-oriented in position P0 in Figure 1(d) at (a)–(c) 55 μs and (b)–(d) at tr/2, considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
The panels a and b map electric field around an elliptical nanoparticle above the curved boundary, with scales from 0 to 250 volts per centimetre. Panel a contains the lowest field above the nanoparticle and higher values farther from it, with local increases near its ends. Panel b contains low values above and below the nanoparticle, higher values near its ends, and the highest values within the nanoparticle. Panels c and d map the corresponding field with scales from 0 to 500 volts per centimetre. Panel c contains low values around the nanoparticle and values near 500 volts per centimetre beyond the curved boundary. Panel d contains a low field above and below the nanoparticle, local increases near its ends, and values near 500 volts per centimetre within the nanoparticle and beyond the boundary. Panel e plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 840 volts per centimetre in the cell interior, about 80 volts per centimetre in the first cell exterior segment, near 0 through the nanoparticle core, and about 20 volts per centimetre in the final cell exterior segment. At 55 microseconds, the corresponding values measure about 115, 10, 0, and 110 to 130 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, values measure about 400, 35, 0, and 20 volts per centimetre. At 55 microseconds, values measure about 115, 15, 0, and 110 to 130 volts per centimetre. Panel g plots the regions at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 790 volts per centimetre in the cell interior, decreases from about 80 to 0 in the first exterior segment, decreases from about 1000 to 940 through the nanoparticle core, and increases from 0 to about 30 in the final exterior segment. At 55 microseconds, values measure about 110, 0, 300, and 0 to 20 volts per centimetre. Panel h plots the regions at 5 and 55 microseconds. At 5 microseconds, values measure about 390, decrease from about 45 to 0, decrease from about 540 to 500, and increase from 0 to about 20 volts per centimetre. At 55 microseconds, values measure about 110, decrease to 0, remain near 300, and increase from 0 to about 20 volts per centimetre.Color plot of the electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle horizontally-oriented in position P0 in Figure 1(d) at (a)–(c) 55 μs and (b)–(d) at tr/2, considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
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 tp/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 tr/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 Lc for the horizontal-oriented gold NP in P0 [Figure 1(d)] considering a sampling time equal to tr/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 tr/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 tr/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 tr/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.
3.4 Vertically-oriented nanoparticle
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.
The panels a and b map electric field from 0 to 260 volts per centimetre. Panel a contains a low field region around and above the nanoparticle, with a local minimum near the curved boundary and increasing values farther away. Panel b contains a broad low field region between the nanoparticle and boundary, with increasing values towards the outer region. Panels c and d map electric field from 0 to 180 volts per centimetre. Panel c contains reduced values around the nanoparticle sides and higher values farther away. Panel d contains higher values around the nanoparticle and a gradual decrease towards the boundary. Panel e plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 840 volts per centimetre, the cell exterior increases from about 90 to 580 volts per centimetre, and the nanoparticle core remains near 0 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase from about 20 to 420 volts per centimetre, and remain near 0 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre, the cell exterior increases from about 40 to 380 volts per centimetre, and the nanoparticle core remains near 0 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase from about 20 to 430 volts per centimetre, and remain near 0 volts per centimetre. Panel g plots the regions at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 800 volts per centimetre, the cell exterior decreases from about 80 to 0 volts per centimetre, and the nanoparticle core increases slightly from about 105 to 115 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, decrease from about 20 to 0 volts per centimetre, and increase from about 20 to 65 volts per centimetre. Panel h plots the regions at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre, the cell exterior decreases from about 40 to 0 volts per centimetre, and the nanoparticle core increases from about 50 to 65 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, decrease from about 20 to 0 volts per centimetre, and increase from about 20 to 55 volts per centimetre.Electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle vertically-oriented in position P0 at (a) and (b) 55 µs and at (c) and (d) tr/2 considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
The panels a and b map electric field from 0 to 260 volts per centimetre. Panel a contains a low field region around and above the nanoparticle, with a local minimum near the curved boundary and increasing values farther away. Panel b contains a broad low field region between the nanoparticle and boundary, with increasing values towards the outer region. Panels c and d map electric field from 0 to 180 volts per centimetre. Panel c contains reduced values around the nanoparticle sides and higher values farther away. Panel d contains higher values around the nanoparticle and a gradual decrease towards the boundary. Panel e plots electric field against a sampling line from 0 to 0.3 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 840 volts per centimetre, the cell exterior increases from about 90 to 580 volts per centimetre, and the nanoparticle core remains near 0 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase from about 20 to 420 volts per centimetre, and remain near 0 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre, the cell exterior increases from about 40 to 380 volts per centimetre, and the nanoparticle core remains near 0 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase from about 20 to 430 volts per centimetre, and remain near 0 volts per centimetre. Panel g plots the regions at 0.5 and 55 microseconds. At 0.5 microseconds, the cell interior remains near 800 volts per centimetre, the cell exterior decreases from about 80 to 0 volts per centimetre, and the nanoparticle core increases slightly from about 105 to 115 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, decrease from about 20 to 0 volts per centimetre, and increase from about 20 to 65 volts per centimetre. Panel h plots the regions at 5 and 55 microseconds. At 5 microseconds, the cell interior remains near 390 volts per centimetre, the cell exterior decreases from about 40 to 0 volts per centimetre, and the nanoparticle core increases from about 50 to 65 volts per centimetre. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, decrease from about 20 to 0 volts per centimetre, and increase from about 20 to 55 volts per centimetre.Electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle vertically-oriented in position P0 at (a) and (b) 55 µs and at (c) and (d) tr/2 considering a pulse with a rise time of 1 μs. Electric field along Lc for (e) and (f) gold and (g) and (h) dielectric nanoparticle for a pulse rise time of (e)–(g) 1 μs and (f)–(h) 10 μs
Source: Authors’ own work
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 Lc 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 Lc 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. tr/2 and 55 µs. At tr/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 tr/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 tr/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 Lc 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.
3.5 45°-oriented nanoparticle
Figure 6 shows the color plots representing the electric field strength at 55 µs [Figure 6(a) and (b)], and at tr/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.
The panels a and b map electric field around the inclined nanoparticle from 0 to 300 volts per centimetre. Panel a contains low values within and beside the nanoparticle, with local peaks near its ends and values near 150 volts per centimetre farther away. Panel b contains values above 200 volts per centimetre within the nanoparticle, lower values near one end, and values near 150 volts per centimetre in the surrounding region. Panels c and d map electric field from 0 to 250 volts per centimetre. Panel c contains low values within and beside the nanoparticle, with peaks near both ends. Panel d contains values near 100 volts per centimetre within the nanoparticle and lower values near one end. Panel e plots electric field against a sampling line from 0 to about 0.23 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 790 volts per centimetre in the cell interior, about 60 volts per centimetre in the first cell exterior segment, near 0 through the nanoparticle core, and increases from about 55 to 190 volts per centimetre in the final cell exterior segment. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, remain near 0, remain slightly below 0 through the core, and increase from about 240 to 450 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, the field measures about 800 volts per centimetre in the cell interior, about 1050 volts per centimetre in the first exterior segment, decreases from about 1380 to 1100 volts per centimetre through the nanoparticle core, and decreases from about 800 to 750 volts per centimetre in the final exterior segment. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase to about 2400 volts per centimetre, decrease from about 2400 to 1720 volts per centimetre through the core, and decrease from about 1700 to 1600 volts per centimetre.Electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle in position P45 at (a) and (b) 55µs and at (c) and (d) tr/2 considering a pulse with a rise time of 1 μs
Source: Authors’ own work
The panels a and b map electric field around the inclined nanoparticle from 0 to 300 volts per centimetre. Panel a contains low values within and beside the nanoparticle, with local peaks near its ends and values near 150 volts per centimetre farther away. Panel b contains values above 200 volts per centimetre within the nanoparticle, lower values near one end, and values near 150 volts per centimetre in the surrounding region. Panels c and d map electric field from 0 to 250 volts per centimetre. Panel c contains low values within and beside the nanoparticle, with peaks near both ends. Panel d contains values near 100 volts per centimetre within the nanoparticle and lower values near one end. Panel e plots electric field against a sampling line from 0 to about 0.23 micrometres at 0.5 and 55 microseconds. At 0.5 microseconds, the field measures about 790 volts per centimetre in the cell interior, about 60 volts per centimetre in the first cell exterior segment, near 0 through the nanoparticle core, and increases from about 55 to 190 volts per centimetre in the final cell exterior segment. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, remain near 0, remain slightly below 0 through the core, and increase from about 240 to 450 volts per centimetre. Panel f plots the same regions at 5 and 55 microseconds. At 5 microseconds, the field measures about 800 volts per centimetre in the cell interior, about 1050 volts per centimetre in the first exterior segment, decreases from about 1380 to 1100 volts per centimetre through the nanoparticle core, and decreases from about 800 to 750 volts per centimetre in the final exterior segment. At 55 microseconds, the corresponding values measure about 110 volts per centimetre, increase to about 2400 volts per centimetre, decrease from about 2400 to 1720 volts per centimetre through the core, and decrease from about 1700 to 1600 volts per centimetre.Electric field strength for (a)–(c) gold and (b)–(d) dielectric nanoparticle in position P45 at (a) and (b) 55µs and at (c) and (d) tr/2 considering a pulse with a rise time of 1 μs
Source: Authors’ own work
Considering the gold and dielectric NP in position P45 at 55 μs, the electric field strength sampled on the line Lc 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)].
4. Conclusion
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.
