Mathematical Models of Mass Transfer in Tissue for Molecular Medicine Reversible Electroporation - Porous Media: Applications in Biological Systems and Biotechnology - page 57

conditions. This time the conductivity σ is the unknown and the potential u
is known only on the boundary, where the voltage measurement electrodes are
located. An additional boundary condition exists since we know the current
density on the boundary of the tissue under the current injecting electrodes.
Calculating the conductivity, given the currents when the potential on the
boundary is known, which is often referred to as the inverse problem, turns out
to be mathematically ill-posed (Lionheart 2004). This problem does not have
a unique solution for every input, which means that different internal con-
ductivity distributions result in the same boundary potentials. Furthermore,
solutions are usually not stable so very small changes in the input can result in
very large changes in the output. The principal way to overcome this problem
is to use some a priori knowledge in the form of a regularization matrix (Borsic
et al. 2002) that is introduced to the solution. A very common regularization
matrix relies on the a priori “knowledge” that the internal impedance is a
slowly varying spatial function, or in other words, that the impedance does
not change very much between adjacent parts of the tissue. This assumption
is very useful in solving the equations, but it obviously restricts the solution
in terms of spatial resolution. Small inhomogeneities are dicult to detect,
and it is essentially impossible to determine sharp borders between two tissue
types, even when their impedance is different.
FIGURE 2.7
Example of a reconstructed conductivity map after electroporation. Electro-
poration was performed using the two needle electrodes, which are the two
white discs surrounded by black circles. The conductivity is shown as different
shades where bright areas represent higher conductivity.