85
Pulsed Electric Fields in Biological Cells and Membranes
Γ(r) as Γ(r) = Γ 0 [1–r 2 /r ∞
2 ], with r ∞ being a constant parameter. The varying tension is
perhaps more physical as it accounts for changes in the mechanical properties of cells
due to structural rearrangements and poration. The curves of Figure 2.6 reveal a distinct
shift toward larger pore radii for the fixed tension Γ 0 . For the two-pulse cases, an “idle
time” of 0.6 ns was used to separate the second voltage pulse from the first. The most
probable pore radius with the dynamic tension is predicted to be less than 0.55 nm, and
thus essentially small nanopores are predicted. This result indirectly agrees with several experimental observations of negligible propidium iodide (PI) dye uptake following nanosecond pulsing (Chen et al. 2004; Pakhomov et al. 2007; Kennedy et al. 2008).
A possible explanation is that the pore population formed by the ultrashort pulsing has
diameters smaller than the dye dimensions, and hence disallow dye uptake by the cells.
The predictions of Figure 2.6 also support the hypothesis that the pore diameters upon
ultrashort pulsing are small, so that the membrane is mainly permeable only to small
ions (or even nanoparticles), but not large molecules.
2.3.2 Continuum Approaches Applied to More
Complicated Cell Geometries
Most analyses and calculations of the TMP resulting from external pulsing have
assumed spherical cells. In biology, examples of such spherical shapes are vesicles,
protoplasts, murine myeloma cells (Gimsa and Wachner 1999), and some bacteria such
as Streptococcus (Batzing 2002). However, most other cells are nonspherical in nature. As
it turns out though, many cells deviating from the spherical shape do exhibit rotational
symmetry and can be represented either as prolate or oblate spheroids fairly accurately.
For example, mammalian red blood cells are close to an oblate spheroidal shape, while
retina photoreceptor cells (Radu et al. 2005), many bacteria such as Escherichia coli and
Pseudomonas (Gimsa and Wachner 1999), and yeasts (Asencor et al. 1993) roughly have
a prolate spheroidal geometry. This makes a compelling case, at least from the practical
standpoint, to examine bioelectric pulsing effects in such spheroidal cell shapes for more
realistic analyses and predictions. There have only been a few reports in the literature
on analyses for irregularly shaped cells (e.g., Pucihar et al. 2006) using finite-element
approaches to solve for the TMP, though electroporation was not explicitly considered.
Their method allows for arbitrarily shaped cellular geometries. A few other studies for
spheroidal cells (Kotnik and Miklavcic 2000a; Maswiwat et al. 2007) did not consider
electroporation in a self-consistent manner.
An evaluation of the TMP and possible membrane electroporation in spheroidal
cells arising from an ultrashort, high-intensity pulse has been reported (Hu and Joshi
2009). That study coupled the Laplace equation with the Smoluchowski theory of pore
formation and used double-shell models. Though the treatment of pores using the
Smoluchowski theory remains the same as for spherical cells, calculations for the potential are slightly different.
The schematic shown in Figure 2.7 represents a double-shelled prolate spheroidal
cell suspended in a medium. Because the spheroids have rotational symmetry in the
x-y plane, the arbitrarily oriented, externally applied electric field E 0 ( )
t can be assumed