Biomedical Engineering Reference
In-Depth Information
r
u
E
FIGURE 2.8
A two-dimensional model of a single spherical cell in a uniform electric field.
E is the electrical field, r is the cell's radius, and θ is the angle between the
direction of the electric field and the specific location on the cell membrane.
assumption of a uniform electric field does not usually hold, as we have just
stated, but even with a nonuniform field on the large-scale tissue level, the
changes in the electric field at the small-scale cell level are generally negligible.
We can therefore assume that the electric field in the vicinity of the cell we are
about to examine is uniform, although the field strength will vary for different
regions of the tissue. The second assumption, treating the cell as a single cell
is more delicate. As Figure 2.4 illustrates, the cells in a solid tissue do not
resemble an ensemble of independent single cells, but have mutual effects on
each other in terms of geometrical shape, transmembrane potential, and more.
Nevertheless, we make this simplifying assumption for the sake of clarity in
the description of a general class of models. More accurate models can rely
on similar methods and treat various scenarios of packed cells in tissue using
models that are described in the literature (Esser et al. 2007).
As we have seen, the determining factor in cell membrane electropora-
tion is the transmembrane potential. To estimate this potential we examine a
two-dimensional model of a spherical cell, as shown in Figure 2.8. The trans-
membrane potential, V m , for such a cell with a radius r , found in a uniform
electric field E , will vary for different regions of the membrane. Specifically, it
will depend on the angle θ between the direction of the electric field and the
membrane region (Neumann et al. 1989).
V m =1 . 5 rE cos( θ )
(2.2)
This translates to large transmembrane potentials along the electric field axis,
where θ is close to 0 or 180 , and to a negligible transmembrane potential
near the equator of the cell, where θ is close to 90 or 270 . We can use one of
the several biophysical models (Neu and Krassowska 2003; Smith et al. 2004)
to estimate the size of pores at different regions of the cell and their time evo-
lution during the application of the electroporation pulse or pulses and after
the external field has been removed. Such models estimate the number and
Search WWH ::




Custom Search