Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 55

invasive device and the need to shape the effective field in such a way as to
treat certain parts of the tissue without affecting other neighboring regions.
Therefore, we need to expand the experimental data to interpolate the con-
ductivity changes for other electric field values. We assume, for the purpose
of this model, that below 450 V/cm there are no significant changes in con-
ductivity because the transmembrane potential of the cells would be below
the electroporation threshold. We further assume that above 1,500 V/cm the
conductivity changes reach a saturated value and will not increase any fur-
ther. This is a reasonable assumption because tests have shown (Weaver and
Chizmadzhev 1996) that above a certain threshold the effect of the membrane
on conductivity is negligible and the conductivity value is similar to that of a
configuration with no membrane at all.
A further assumption is that the conductivity increases linearly with elec-
tric fields between 450 and 1,500 V/cm. An increase in the conductivity has
been observed in several studies (Pavlin et al. 2005; Pavlin and Miklavcic 2008)
although not enough data has been studied to validate the linearity assump-
tion. However, this simple model seems close enough to available experimental
results to make it a good basis for a first model. Figure 2.6 depicts the results
of combining these assumptions with the empirical results of electroporation
in rat liver tissue. Every line represents the percent of change in conduc-
tivity relative to the initial value for a different number of pulses. The first
pulse represents the changes after a single pulse and the last line the changes
after the entire set of eight pulses. For nonuniform electric fields, different
parts of the tissue experience electroporation in a different manner, and the
changes in conductivity at each region would depend on the local electric
field.
This model can be used to calculate the effects of electroporation on tissue
conductivity for various configurations. To obtain the electric field for each
point in the model we need to solve the Laplace equation:

∇ ( σ ∇ u ) = 0 (2.1)

where σ is the electrical conductivity and u is the electric potential. An ana-
lytic solution is possible only for a limited number of cases so usually numerical
methods are applied. One of the most useful tools to solve the Laplace equa-
tion is the finite element method (FEM) in which the domain is divided into
very small homogenous elements for which the equation can be easily solved
(Vauhkonen et al. 2001). The boundary conditions are obtained by assuming
that the boundaries of the tissue are electrically insulated and that the elec-
trodes have a known and constant potential as determined by the voltage we
apply. The initial conductivity of the tissue at every point σ is assumed to be
known and usually taken to be homogeneous.
The solution of the Laplace equation is u for every point, and by taking the
gradient of u we obtain the electric field. Using the functions of Figure 2.6 we
have the conductivity after the first pulse of electroporation. This stage in the
process may be refined by using an iterative procedure (Pavselj et al. 2005)