Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 60

size of the pores, usually by solving several differential equations. Solving the
equations is often a dicult task because many of the variables depend on each
other. For instance, the local transmembrane potential changes dramatically
as pores are created owing to the change in membrane permeability to ions.
As an example we shall take a single-cell model (Krassowska and Filev 2007)
and show how it may be used to determine the cell membrane permeability.
The number of pores, according to this model, is computed as a function of
V m using the pore density N :
dN
dt

= αe ( V m /V ep ) 2 1

N
N 0 e q ( V m /V ep ) 2

−

(2.3)

where α is the pore creation rate coecient (10 9 m − 2 sec − 1 ), V ep is the char-
acteristic voltage of electroporation (0.258 V), N 0 is the equilibrium pore
density for the membrane area at V m = 0 (1.5 10 9 m − 2 ), and q is an electropo-

ration constant ( q =2 . 46). For a known transmembrane voltage such as that
of equation (2.2) we can compute the pore density as a function of time.
The size of pores also plays an important role in determining the membrane
permeability for two reasons. First, larger pores will allow large molecules to
cross the membrane and effectively render the cell permeable to molecules
that would not be able to enter the cell if only smaller pores existed, regard-
less of the number of small pores. Second, the area of a pore depends on its
radius, and, thus, larger pores contribute more effective area through which
molecules can travel than smaller sized pores. In the current example, we
assume that all of the pores have the same size, in agreement with the model
we are using, where soon after the end of the pulse the pores shrink to the
minimal pore radius R p , and begin to reseal. This model does not adequately
describe reversible electroporation for transfection or other processes where
large molecules are introduced into cells over extended periods of time, but
for the sake of clarity we continue with this example and only note that
more elaborate models for long-term electroporation may be used in a similar
manner.
The amount of molecules that enter the cell under these assumptions
depends on the number of pores, which decrease rapidly after the electropo-
ration pulses is over since the pores start to reseal. During the pulse the pores
are already open and some of them are even larger than the minimal pore
radius we consider, but since the pulse duration is very short compared to the
phase of stable pores after the pulse, we will only take into account the mass
transfer that occurs after the pulse. We calculate the total area of the pores,
A p , from the number of pores per cell and the area of each pore:
A p = πR p ·

·

N p (2.4)

where N p is the number of pores at the end of the electroporation pulse and
is computed by integrating the pore density over the entire cell membrane
surface:

N p = NdS (2.5)