Intracellular electroporation site distributions: modeling examples for nsPEF and IRE pulse waveforms

Abstract

We illustrate expected electroporation (EP) responses to two classes of large electric field pulses by employing systems models, one of a cell in vitro and the other of multiple cells in vivo. The first pulse class involves "nsPEF" (nanosecond pulsed electric fields). The durations are less than a microsecond, but the magnitudes are extremely large, often 10 kV/cm or more, and all of the pores remain small. The second class involves "IRE" (irreversible electroporation). Durations are many microseconds to several milliseconds, but with magnitudes smaller than 10 kV/cm, and a wide range of pore sizes evolves. A key feature of both pulse classes is non-thermal cell killing by multiple pulses without delivering external drugs or genes. For small pulses the models respond passively (no pore creation) providing negative controls. For larger pulses transient aqueous pore populations evolve. These greatly increase local membrane conductance temporarily, causing rapid redistribution of fields near and within cells. This complex electrical behavior is generally not revealed by experiments reporting biological end points resulting from cumulative ionic and molecular transport through cell membranes. The underlying, heterogeneous pore population distributions are also not obtained from typical experiments. Further, traditional EP applications involving molecular delivery are usually assumed to create pores solely in the outer, plasma membrane (PM). In contrast, our examples support the occurrence of intracellular EP by both nsPEF and IRE, but with different intracellular spatial distributions of EP sites.

Fig. 1

System model geometries. Left: Single cell with organelles; described elsewhere [9]. The long, inter-connected ”wiggly” black structures represent the endoplasmic reticulum (ER); the nucleus is the off-center yellow circular structure, and the five small elongated dark red structures are mitochondria. The nuclear envelope and mitochondrial membranes are double while the plasma and ER membrane are single, all with appropriate resting potential sources. Right: In vivo, multicellular with irregular shapes and sizes but no organelles; described elsewhere. [10], [11]. Papers with modeling details are all now accessible publicly [9]–[11].

Fig. 2

nsPEF example [4] with the digitized experimental pulse waveform and model responses. Upper left: Experimental pulse waveform (nominal 100 ns duration, because the small ragged tail is relatively unimportant to the highly non-linear EP response). Lower left: Transmembrane voltage, U_m(t) for the anodic (RED) and cathodic (BLACK) poles of the circular cell. The spike following rapid charging is due to a burst of conducting small pores. Upper row of four panels: Spatial distributions of EP sites (WHITE regions; >10¹³ pores/m²) and equipotentials (black lines/curves; equipotentials of each panel are scaled separately). The top of the system box is an idealized anode (zero electrochemical overvoltage) and the bottom is an idealized cathode, so that the applied field is uniform in the absence of the cell. Lower row of four top/bottom panel pairs: RED PM pore population histograms (bin width 0.05 nm) for the upper (anodic) side of the cell, with BLACK histograms for the lower (cathodic) side. Note the semi-logarithmic scale, showing most pores are small (in the rightmost panel, the peak at the minimum pore size r_p ≈ 0.8 nm, and falling off rapidly to one at r_p ≈ 1.6 nm for both RED and BLACK distributions).

Fig. 3

IRE example [7] with the pulse waveform and model responses. Upper left: IRE pulse trapezoidal waveform (100 μs nominal duration, 98 μs peak (plateau) duration, 2.5 kV/cm pulse with 1 μs rise and fall times). Lower left: Transmembrane voltage, U_m(t) for the anodic (RED) and cathodic (BLACK) poles of the circular cell. The spike following the rapid charging is due to a burst of conducting small pores that occurs during the rising phase of the pulse. Upper row of four panels: Spatial distributions of EP sites (WHITE regions; >10¹³ pores/m²) and equipotentials (black lines/curves). EP is seen in much of PM and some regions of ER. The nuclear and mitochondrial membranes are not electroporated. Lower row of four top/bottom panel pairs: RED PM pore population histograms for the upper (anodic) side of the cell, with BLACK histograms for the cathodic side. Here, the PM pores are larger (some as large as 3 nm), but are fewer in number compared to the nsPEF case of Fig. 2. A maximum radius (reflecting boundary) of 3 nm was imposed.

Fig. 4

The multicellular model response to the nsPEF pulse [4] should be compared to Fig. 2 top left. Upper row of four panels: Spatial distribution of EP sites (WHITE regions; >10¹³ pores/m²) and equipotential curves/lines (DARK BLUE). At the time designated t = 0 ns there is a small applied field (0.18 kV/cm) but neither the transmembrane voltages nor the time is sufficient to create pores in the irregularly shaped plasma membranes (PMs). At t = 15 ns pores have been created at many membrane sites, particularly those membranes that are parallel to the equipotentials (perpendicular to the applied field pulse). In contrast there are some, but relatively few, at parts of the PMs that are nearly parallel to the field. The equipotentials are essentially parallel, so that within the in vivo environment (relatively little extra-cellular water) supra-electroporation is closer to being maximized. At t = 100 ns many of the previously unporated PM regions now have pores, and the equipotentials remain nearly parallel (almost uniform field within the system). Finally, at t = 400 ns when the field pulse is greatly diminished but not zero, the porated sites remain unchanged, but the equipotentials are no longer parallel. This is consistent with decreased PM conductance (individual pore conductance is a function of U_m), due to partioning associated with the Born energy [10], [11]). Lower row of four top/bottom panel pairs: The plots show the histogram of all membrane pores in the systems model (here all PM pores). The membrane pores grow in size, but expand at most to about 1.8 nm radius, with the great majority having radii near 1 nm. Note that the left-most panel appropriately has zero pores.

Fig. 5

Multicellular (in vivo) model response to the IRE pulse [7]. Upper row of four panels: Spatial distributions of EP sites (WHITE regions; >10¹³ pores/m²) and equipotentials (DARK BLUE lines/curves). EP is mostly confined to membrane regions perpendicular to the applied field pulse (parallel to the equipotentials). Throughout the equipotentials are not parallel, due to the spatial variation in PM conductance and spatially heterogeneous EP sites. Note that the right-most panel has no equipotentials, consistent with the zero applied field at t = 110 μs. Lower row of four top/bottom panel pairs: Here, the PM pores are significantly larger (some 3 nm), but are fewer in number compared to the nsPEF case. Also note that the right-most panel shows a significantly contracted pore population. The pores have not disappeared 10 μs after the pulse ceases, but have shrunk in size to yield a thermalized (broadened) size distribution with a peak at 0.8 nm (cf. Fig. 4 at 15 ns).