Biomedical Engineering Reference
In-Depth Information
The electroporation process has left the membrane unsealed and an area
described by equation (2.6) is now a part of the cell boundary where an
aqueous pathway for molecules exists. With no obstructions to impede their
motion, molecules that are found in large concentrations in the extracellu-
lar medium will now diffuse into the cell. When no molecules are added or
removed from the system the diffusion equation describes the concentration
of our molecules, c , as a function of space and time:
∂c
∂t =
( D
c )
(2.7)
where D is the diffusion coe cient. Although we shall use a similar version of
this equation later in this chapter for the large-scale diffusion of molecules in
the tissue, for the small-scale part of the model we will simplify our analysis
a little further. We are interested in the number of molecules that enter a cell
by diffusing through the aqueous pores across the cell membrane. To obtain
that, we take the diffusion flux J, which is given in dimensions of amount of
molecules per unit area per unit time and multiply by the total electropo-
rated area A p . This results in the amount of molecules that enter the cell per
unit time. According to Fick's first law of diffusion for an external molecule
concentration c ex and an internal concentration c in , the flux is given by
J = −P · ( c in
c ex )
(2.8)
where P is the permeability of the molecules through the membrane pores.
The permeability is often determined experimentally and is a measure of how
well molecules may flow through the pore under certain conditions. For very
large pores and very small molecules it could be approximated by the ratio
of D , the diffusion coe cient, and δ the membrane thickness P = D / δ . This
corresponds to a more common version of Fick's first law for the flow in one
dimension where x is in the direction normal to the pore area:
D ∂c
∂x
J =
(2.9)
For the limit of δ going to zero, in the case of a very thin membrane, P c
in equation (2.8) approaches D times the partial derivative of c along x in
equation (2.9). Nevertheless, actual values of the permeability depend on a
mixture of parameters such as the size of the molecule and the interaction
between the molecule and the pore so this estimate may serve as an upper
limit but in realistic scenarios the permeability may be much lower.
In many instances of reversible electroporation the concentration of
molecules inside the cell, c in , is much lower compared to that of the exter-
nal concentration. The goal of the process is in fact to increase the internal
concentration. In certain cases c in may be taken to be zero if, for example,
molecules in the cytoplasm bind very rapidly to some cellular compartment
and are not free to diffuse inside the cell (Granot and Rubinsky 2008).
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