Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 64

Equation (2.11) is solved using the diffusion coecient D and is subject
to initial conditions as well as boundary conditions. The initial concentration
of molecules in the extracellular space is assumed to be uniform throughout
the tissue, neglecting the volume of cells in which there are no molecules. The
boundary conditions can often be modeled as no mass flux at the boundary
of the tissue.
Using the FEM we can solve the spatially dependent time evolution of c
to determine how the molecules diffuse around the tissue and also to obtain
an estimate of the amount of molecules that enter cells in different regions. In
general, more molecules will disappear into cells in areas where electroporation
has created more pores, and molecules will diffuse from higher concentration
regions where electroporation is less effective to regions of more intense elec-
troporation. In the following section we shall work out a detailed example
of such a process for delivering chemotherapeutical drugs into a cancerous
tumor.
2.6 Studies on Molecular Medicine with Drug Delivery
in Tissue by Electroporation
A common example of using electroporation for delivering drug molecules is a
method called electrochemotherapy (Mir et al. 1991; Mir and Orlowski 1999).
Drugs such as bleomycin are effective in destroying malignant cells, but are
not able to eciently penetrate the membrane under normal conditions. Using
reversible electroporation the tissue around a tumor becomes permeable due
to the pores that are created in the cell membrane and bleomycin can enter
the cytoplasm. Once inside the cell the drug can act and destroy it. To find out
whether the cell will be harmed, we shall estimate the number of bleomycin
molecules that are expected to enter the cell.
The configuration we have chosen for this worked-out example is a two-
dimensional analysis of a homogeneous tissue where electroporation is induced
using two needle electrodes. When two long needles are inserted into the tissue
to be used as electrodes, the electric field between and around them may be
analyzed in two dimensions, neglecting the boundary effects at the edges of
the needles. We therefore use a cross-section in the middle of the electrodes
as depicted in Figure 2.7.
We solve equation (2.1) to obtain the potential at every point in the
model and thus the electric field. The equation was solved using the finite
element method with the code written in Comsol Multiphysics (version 3.3a,
www.comsol.com). The boundary conditions were defined as a potential of
1,000 V on one electrode and ground on the other electrode. The outer edges
of the tissue, which are far from the electrodes and the region of interest, are
assumed to be electrically insulating. When using finite elements ones needs