79
Pulsed Electric Fields in Biological Cells and Membranes
pulsing), an alternate approach for modeling is to represent the electrical characteristics
of cells by a distributed equivalent circuit, and then perform node analyses to obtain the
potentials in the time domain. Distributed circuit solvers such as SPICE are not, however,
suited to the present problem for the following reasons: (1) In such circuit solvers, values of
the circuit parameters (e.g., resistors, capacitors, etc.) cannot be made time dependent; (2)
the stochastic nature of pore formation cannot be included; (3) the geometric dependence
of R and C makes it inconvenient to run simulations with dynamic variations in shapes
and sizes as pores form; and (4) soft thresholds inherent in the pore formation process are
difficult to implement. For example, some degree of poration can start at voltages lower
than the 1-V threshold generally assumed but it would then take more time. Also, given
biological variability, orientational effects, etc., all pores need not have a rigid threshold
value. (5) Finally, it is difficult to use circuit models to simulate the gradual resealing of
pores. The resealing process always takes a long time (microseconds or longer), and the
long-lived, diffusion-driven currents cannot be modeled by circuit simulators.
A simple lumped circuit representation with inclusion of the dynamic changes in cell
parameters has been discussed in the literature (e.g., Vasilkoski et al. 2006). In this model
for spherical cells, the transmembrane voltage is governed by the following relations:
(C m + C e ) d(ΔΦ)/dt = C e dV app /dt + V app /R e − ΔΦ[G m (n) + 1/R e ], for t ≤ t pulse (2.5a)
(C m + C e ) d(ΔΦ)/dt = G m (n) d(ΔΦ)/dt, for t > t pulse
(2.5b)
where C m and C e denote the capacitance of the membrane and the electrolyte, respectively,
and R e is the resistance of the aqueous medium surrounding the cell. Equation 2.5b
allows a straightforward calculation of the transmembrane voltage ΔΦ if the cell parameters are constant. However, quite generally, the membrane conductance G m (n) can
change nonlinearly as a function of time due to the applied voltage. Consequently, the
pore conductance becomes a function of time and is spatially nonuniform as dictated
by the pore-size distribution n(r,θ,t) over the membrane surface. Under such conditions,
the biophysical mechanisms of electroporation and evaluations of the pore densities at
biological membranes need to be taken into account, and these are described through
the Smoluchowski equation (Abidor et al. 1979). Details of this aspect, including pore
growth and evolution of the radial distributions are discussed later in this section.
A purely numerical approach was discussed (Joshi, Hu, and Schoenbach 2004) for
calculating the time-dependent potentials and current flows throughout the cell. It was
based on a time domain nodal analysis involving a dynamic, distributed circuit representation of a cell and its membrane structures. The entire cell can be broken up into
segments, and each segment represented by a parallel combination of a resistor (R) and
capacitor (C). For computational efficiency, azimuthal symmetry of spherical cells can
be used to map the three-dimensional structure into the r and Φ coordinates of a spherical system. This method is different from the Legendre polynomial models often used
for the electrostatic cases (Joshi et al. 2001) because it allows for dynamic flows and also
lends itself to the inclusion of membrane poration effects. For such analyses, volume and
shape changes of cells can be ignored because the external applied pulse is too short for
cells to deform much during the time interval. Application of Kirchhoff’s current law at
each node then yields a set of N coupled linear equations in the N node voltages.