Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 63

The final stage we examine in the small-scale model is the process of mass
transfer, which happens in every cell in the tissue. This stage will serve as
the link between the cellular-scale part of the model and the tissue-scale part.
When developing the model of large-scale mass transport in the tissue, we
assume that the cells are infinitesimally small, and model them as a distributed
reaction rate. This means that at the cell level molecules that enter the cell
will never flow out and in a certain sense may be treated as if they have
been removed from the system. From the large-scale model point of view, it
is as if we have a mass sink that removes the molecules from the system,
but this sink is not a point sink, but rather a distributed sink through which
molecules leave the system in every region where electroporation occurred.
This essentially means that c in is zero at the beginning of the process and
remains zero throughout the electroporation procedure although molecules
are entering the cell.
The amount of molecules that disappear in each region depends on the
extent of electroporation in that specific location. A convenient way of describ-
ing this at the cellular scale is by modeling the cells as uniformly packed in
the tissue so that each spherical cell is contained in a cube. The edges of the
cube are equal to the diameter of the spherical cell, 2 r . It is a deviation from
our earlier assumption of a single cell for the electroporation analysis, but it
is suciently reliable for the purpose of this illustrative example. The reac-
tion rate that describes how much matter in terms of the molecules we are
interested in is removed from the system depends on the volume around the
cells where the molecules are initially found. A cube with edges as described
earlier has a volume of V 0 =(2 r ) 3 for which the reaction rate is the flow of
molecules per unit time divided by the surrounding volume:

R = JA p /V 0 (2.10)

To sum up the cellular-scale part of the model we have a single spheri-
cal cell surrounded by an extracellular medium with a high concentration of
some molecule whose internal concentration is zero, and which we would like
to introduce into the cell. By inducing an above-threshold transmembrane
potential pores are created in the cell's membrane and some of the molecules
diffuse into the cytoplasm. These molecules bind to internal structures in the
cell and are effectively removed from the system.
From the tissue-scale perspective we are dealing with two aspects. First the
electric field values in the tissue are determined by the large-scale configuration
of the electrodes and the tissue's properties. This analysis is the basis for the
local electric field in the surroundings of each cell in the cellular-scale model.
Once we have calculated the reaction rate as detailed earlier, we return to the
large-scale model for the tissue-scale diffusion equation. The concentration
throughout the extracellular space is calculated using the spatially dependent
variable R :

∂c
∂t −∇

( D ∇ c )= R (2.11)