Continuum analysis to assess field enhancements for tailoring electroporation driven by monopolar or bipolar pulsing based on nonuniformly distributed nanoparticles

Recent reports indicate that nanoparticle (NP) clusters near cell membranes could enhance local electric fields, leading to heightened electroporation. This aspect is quantitatively analyzed through numerical simulations whereby time dependent transmembrane potentials are first obtained on the basis of a distributed circuit mode, and the results then used to calculate pore distributions from continuum Smoluchowski theory. For completeness, both monopolar and bipolar nanosecond-range pulse responses are presented and discussed. Our results show strong increases in TMP with the presence of multiple NP clusters and demonstrate that enhanced poration could be possible even over sites far away from the poles at the short pulsing regime. Furthermore, our results demonstrate that nonuniform distributions would work to enable poration at regions far away from the poles. The NP clusters could thus act as distributed electrodes. Our results were roughly in line with recent experimental observations.

Figure 1

Transmembrane potential (TMP) as a function of time for a spherical cell at eight different angular locations without any nanoparticles. Electroporation as taken into account for a pulse with maximum electric field amplitude of 60 kV/cm, rise and fall times of 1.5 ns, and an ON time of 70 ns.

Figure 2

Simulation results for the TMP as a function of time for one nanoparticle (NP) cluster located close to the cell membrane over the $9^{\circ }<\theta <{18}^{\circ }$ segment. The TMP values at seven different angular locations are shown. The $\theta ={18}^{\circ }$ location shows a drop in the TMP after about 50 ns, indicating a membrane poration.

Figure 3

Simulation results for the TMP as a function of time for one NP cluster located close to the cell membrane over the ${45}^{\circ }<\theta <{54}^{\circ }$ angular segment. The TMP values at the same seven angular locations as given in Fig. 2 are shown. The $\theta ={54}^{\circ }$ location shows a drop in the TMP after about 45 ns, indicating membrane poration over that region.

Figure 4

Evolution of the TMP over time with two NP clusters placed over the angular locations spanning $9^{\circ }<\theta <{18}^{\circ }$ and ${45}^{\circ }<\theta <{54}^{\circ }$. For consistency, the same seven angular locations over the spherical membrane as in the previous figures are chosen. The $\theta ={18}^{\circ }$ and $\theta ={54}^{\circ }$ sites are predicted to be porated.

Figure 5

A 70-ns snapshot of the pore density distributions versus pore radius with two NPs placed at 9–18 ° and 45–54 ° as discussed in Fig. 4.

Figure 6

Results for the dynamics with two NP clusters placed at different angular locations near the cell membrane. (a) Evolution of the TMP over time for NPs spanning $9^{\circ }<\theta <{18}^{\circ }$ and ${36}^{\circ }<\theta <{45}^{\circ }$, (b) TMP for NPs placed at the $9^{\circ }<\theta <{18}^{\circ }$ and ${54}^{\circ }<\theta <{63}^{\circ }$ angular positions, and (c) time-dependent TMP due to NPs spanning $9^{\circ }<\theta <{18}^{\circ }$ and ${72}^{\circ }<\theta <{81}^{\circ }$. (d) A 73-ns snapshot of the radial distribution $n$($r$) vs $r$ for the pore densities at the four angular locations of 18 °, 45 °, 63 °, and 81 °.

Figure 7

Results for the TMP over time at different angular locations with three NP clusters at angular locations spanning $9^{\circ }<\theta <{18}^{\circ }$, ${45}^{\circ }<\theta <{54}^{\circ }$, and finally ${72}^{\circ }<\theta <{81}^{\circ }$. The $\theta ={18}^{\circ }$ and $\theta ={54}^{\circ }$ as well as the $\theta ={81}^{\circ }$ locations are all predicted to be porated at about 56, ∼45, and ∼54 ns, respectively.

Figure 8

A 70-ns snapshot of the pore density distribution vs pore radius with three NPs placed at 9–18 °, 54–63 °, and 72–81 °.

Figure 9

Results for the TMP over time in response to a bipolar pulse at different angular locations with three NP clusters at angular locations spanning $9^{\circ }<\theta <{18}^{\circ }$, ${54}^{\circ }<\theta <{63}^{\circ }$, and ${72}^{\circ }<\theta <{81}^{\circ }$. The $\theta ={54}^{\circ }$, 81 °, and 18 ° locations are all predicted to be porated at the first cycle.

Figure 10

Snapshots of the pore density distribution versus pore radius with three NPs placed over the regions spanning 9–18 °, 54–63 °, and 72–81 °, in response to a bipolar external electric field at (a) $T=176\mathrm{ns}$ and (b) $T=412\mathrm{ns}$.