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Pulsed Electric Fields in Biological Cells and Membranes
Depending on the processes involved and the inherent physics, the dielectric behavior
can be more complicated than the single time constant description. This could occur if,
for example, the kinetics are not of first order, or if there happen to be multiple relaxation processes. In such cases, the polarization needs to be expressed as a series involving various time constants as the following:
D(t) = D ∞ + D 1 [1 − e −t/τ1 ] + D 2 [1 − e −t/τ2 ] + …
(2.18a)
ε = ε ∞ + Δε 1 /[1 + jωτ 1 ] + Δε 2 /[1 + jωτ 2 ] + …
(2.18b)
where τ 1 , τ 2 , etc., are the various time constants associated with the various relaxation
processes. An empirical function in-line with Equation 2.17 that has been widely used
(Cole and Cole 1941) is as follows:
ε = ε ∞ + [ε(0) − ε ∞ ]/[1 + (jωτ) 1−α ] − jσ 0 /(ω ε 0 )
(2.18c)
The most important mechanisms that influence the dielectric properties are the following: (1) the interfacial polarization (the Maxwell–Wagner effects), (2) dipolar orientation, and (3) ionic diffusion (or counterion polarization). Of these, the first arises in the
context of mixtures and suspensions, and several reviews have already appeared on this
subject (e.g., Hanai 1968; Dukhin and Shilov 1974; Irimajiri, Hanai, and Inouye 1979).
The time constants are roughly the product of the resistance (R) and capacitance (C) of
the interfacial region, and the values are in the radio-frequency range. The orientational
contributions to dielectric relaxation have also been studied, and the applications to protein solutions have been reported (Debye 1929; Takashima 1969). The counterion polarization has time constants that roughly depend on L 2 /D, where L represents the diffusion
length and D the diffusion coefficient. Typical values lie in the audio-frequency regime.
These concepts can be applied to predict the TMP development across cell membranes
especially under ultrashort, pulsing conditions. For a spherical cell, the potentials developed in response to an ultrafast external electric field E 0 (t) are roughly solutions of the
Laplace equation. In the case of a simple, spherical single-shell model with three regions
consisting of the intracellular region of radius a, an extracellular region starting from
r > b, and membrane thickness b − a, the expressions are as follows:
Φ E (r,θ) = (–E 0 r + A/r 2 ) cos(θ), for r > b
(2.19a)
Φ M (r,θ) = (–B r + C/r 2 ) cos(θ), for b > r > a
(2.19b)
Φ I (r,θ) = (–K r) cos(θ), for a > r
(2.19c)
In Equation 2.19(a through c), A, B, C, and K are constants that need to be determined
by matching the potential and electric fields at the boundaries r = a and r = b, while Φ E ,
Φ M , and Φ I denote the voltages within the intracellular, membrane, and extracellular
regions. Algebraic manipulation finally leads to the following:
ΔΦ M = B(b − a) − C(1/b 2 − 1/a 2 )
(2.20a)