Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 61

Shortly after the electroporation pulse ends, the pores start to reseal and
their number decreases exponentially (Neu and Krassowska 1999) with a time
constant, τ , of a few seconds. Therefore when considering the postpulse mass
transfer, we need to add a correction term that makes the area an exponen-
tially decaying function of time t where t = 0 at the end of the electroporation
pulse:

A p = πR p · N p e −t/τ (2.6)

Equation (2.6) is the final result of this stage and gives the total area of
the cell envelope for which the membrane is open and where ions and other
small molecules may pass quite freely. As we have mentioned earlier, the case
for large molecules is somewhat more delicate, and there only the number of
cells above a certain radius should be considered. If we focus our attention on
small molecules we can assume that the role of electroporation is completed
at this stage and we may start to consider the mass transfer mechanisms that
form the second component of the small-scale part of the model.
The membrane constitutes the principal barrier obstructing molecules from
entering or leaving the cell. However, the fact that some pores exist and that
the membrane no longer surrounds the entire perimeter of the cell does not
mean that molecules are completely free to travel. The details of molecular
motion through the pores are beyond the scope of this chapter. We will only
briefly state that interactions between the pore and the various molecules
that pass through it affect the nature of mass transfer through the pore and
may depend on the size and shape of the pore, the charge or polarity of the
molecules, and more. Molecules may sometimes pass through a relatively small
pore in a single-file fashion, which makes the process very different from that
of free-flowing molecules. Mechanisms of molecular uptake into electroporated
cells may include convection, diffusion, and even electrophoresis for charged
molecules when an electric field is applied. Convection is an interesting mecha-
nism to consider since cell swelling has been noticed following electroporation
due to water rushing into the electroporated cells (Abidor et al. 1994; Ivorra
and Rubinsky 2007). Electrophoresis has been suggested as the reason for the
increased uptake of charged molecules when low-voltage pulses are applied
following electroporation (Klenchin et al. 1991). Several studies have shown
that applying low-voltage pulses after the electroporation pulses increases the
amount of molecules such as DNA, which are introduced into the target cells.
Electrophoresis may be involved in this process (Satkauskas et al. 2002) in
a manner that resembles gel electrophoresis where DNA segments are moved
using an electric field. However, other studies (Liu et al. 2006) have questioned
whether this mechanism is indeed the reason for the experimental results. We
will not go into more details here but rather focus on the third and mechanism
mentioned earlier—diffusion. In the following paragraphs, we assume that the
driving force of molecular uptake of molecules into cells is the diffusion process
in which ions, for example, travel along their concentration gradient where the
membrane barrier does not exist.