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Electromagnetic Fields in Biological Systems
evolution. The problem of coupling harmonic electric fields to biological cells and the
transmembrane voltage induction was first studied in the early 80s (Schwan 1983). Later
contributions (Foster and Schwan 1995 and Foster 2000) extended this study to radiofrequency (RF) field exposures of spherical cells. Essentially, the potential induced by
an external field across the membrane of a spherical cell in suspension, ∆Φ, could be
calculated using the following simple analytical expression (Schwan 1983):
∆Φ = 1.5 E r cos(θ)/[1 + r {G m + iωC m }(ρ i + ρ a /2)]
(2.3)
where r is the radius of the cell, E the external electric field value, ω the angular frequency of the external excitation, θ the angle at any location on the membrane with
respect to the applied field direction, C m the membrane capacitance per unit area, G m the
membrane conductance per area, and ρ i and ρ a the resistivities of cell interior and exterior electrolyte, respectively. Results for the TMP arise in a straightforward manner if
one divides the cell into concentric shells to represent the interior cytoplasm, the membrane sheath, the outer extracellular region, etc. The Laplace equation is then solved
for each region to obtain the potential distribution and transmembrane voltages. The
potential in various regions has the following general form:
Φ(r,θ) = −A i r cos(θ) + B i r 3 cos(θ)/r 2
(2.4)
For the outermost region, A i = E and for the inner most region, B i = 0.
Other model analyses for sinusoidal electric excitations have been reported, including
the studies by the University of Ljubljana group (Kotnik and Miklavcic 2000b). Some
of their more recent investigations (Kotnik and Miklavcic 2006) probed the conditions
under which induced voltages at organelle membranes could exceed those established at
the cell plasma membrane. It was shown that upon a suitable choice of the cell electrical parameters, potentials could be induced across subcellular membranes that would
exceed those across the outer cell membrane. Frequency-domain analyses yielded
insights into the dependence of the voltage induction on the electric and geometric
parameters. Particularly, they showed that if the organelle interior was electrically more
conductive than the cytosol or if the organelle membrane had a lower dielectric permittivity than the cell membrane, then transmembrane voltages across the organelle
membranes could exceed those across the outer cell membrane. Time-domain analysis
was then used to determine the temporal evolution of TMPs by pulses with rise times
and durations in the nanosecond range. For example, the temporal response of the
membrane voltages to an applied fast-rising (1 ns) trapezoidal electric field (Kotnik and
Miklavcic 2006) showed that for the first 117 ns the voltage across a 3-μm diameter
spherical organelle would exceed that across the plasma membrane for a 10-μm cell.
Although insightful, the primary drawback of analytical approaches is their inability
to address the dynamic changes in cell parameters in a self-consistent manner. For example, application of a transmembrane voltage over time leads to increases in the membrane
conductance due to localized electroporation. Consequently, the G m and C m parameters
of Equation 2.3 change, and one can no longer obtain analytical, closed-form solutions.
Under such conditions (as are routinely encountered under intense field, short-duration