Biomedical Engineering Reference
In-Depth Information
for the electric field. The number of pulses, their duration, and the intervals
between pulses are some of the dominant factors. The model may use theo-
retical or experimental data to determine the results of the electroporation
protocol and use them in the following stages of the model. We shall proceed
with this example using the parameters of a single-cell electroporation model
(Krassowska and Filev 2007) and a detailed example of bleomycin mass trans-
fer with electroporation (Granot and Rubinsky 2008).
In this model, there is a single pulse, lasting 1 msec. The threshold for
electroporation is an electrical field 130 V/cm, and values of 1,000 V/cm and
above are assumed to cause irreversible electroporation. In this specific case,
we may notice that irreversible electroporation, which leads to cell death,
is not necessarily an unwanted effect, as the purpose of the procedure is to
introduce a lethal drug into cells with the intent of killing them. In fact, using
irreversible electroporation without any drugs for tissue ablation has been sug-
gested and successfully tested in several studies (Davalos et al. 2005; Miller
et al. 2005; Edd et al. 2006; Al-Sakere et al. 2007; Rubinsky 2007; Rubin-
sky et al. 2007). Still, care must be taken when choosing the electroporation
parameters to bring to a minimum the thermal effects caused by Joule heating
(Davalos et al. 2003).
During the pulse pores are created, and by solving equations (2.3) and
(2.5) we can obtain the number of pores in each cell as a function of the elec-
tric field in every region of the tissue. Shortly after the electroporation pulse
ends, the pores start to reseal and their number decreases exponentially with
a time constant of approximately 1.5 sec, as shown in equation (2.6). The dif-
fusion coecient D depends on the specific type of molecule we are interested
in. For bleomycin, D is assumed to be 10 4 mm 2 /sec, which is also a reason-
able value for many other molecules that have similar traits (Neumann et al.
1998).
Before the electroporation procedure begins, the drug is injected into
the tissue and allowed to diffuse and reach a uniform concentration. We
recall that very few molecules will be able to enter the cells at this stage.
The initial concentration of the drug is taken to be c ( t =0)=5
µ
M=
5 · 10 12 mol
mm 3 .
The next step is solving equation (2.10) to obtain the reaction rate and
determine how rapidly the drug is depleted from the extracellular medium
and absorbed by the cells. With this solution we turn to equation (2.11),
which gives the drug concentration at every point in the tissue and shows how
it evolves over time. To solve this equation we return to the finite elements
method and obtain a numeric solution. This analysis also allows us to calculate
the amount of drug that enters each cell. If we take bleomycin, for example,
it is highly toxic and a few hundred molecules that enter the cell are sucient
to cause cell death. The change in local concentration is mostly due to the
drug entering the cells with only a small effect by the diffusion of molecules
from other parts of the tissue. Thus, for each location we calculate the average
number of molecules that enter a cell in that small region, based on the change
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