Membranes--Membranes are responsible for the
dielectric properties of tissues and cell suspensions at
RF's, as demonstrated by studies involving cell suspensions.
Yeast, blood, bacteria, pleuropneumonia-like organisms,
vesicles, and cellular organelles have been extensively
investigated by many investigators, including Fricke (1923),
Cole (1972), and Schwan (1957). This work has led to a
detailed understanding of the role of cell membranes in the
polarization processes of biological media in the RF range.
(The relatively simple geometrical shapes of cells in
suspensions facilitated this understanding.) The principal
mechanism for dielectric polarization at RF's and below is
the accumulation of charges at membranes from extra- and
intracellular fluids. For spherical particles, the following
expressions were derived (Schwan, 1957):
for the limit values of the simple dispersion that characterizes the frequency dependence. The time constant is
In these equations, Cm and G m are capacitance and
conductance per square centimeter of the cell membrane; R is
the cell radius; is the cellular volume fraction, and i =
1/ i and a = l/ a are the conductivities of the cell
interior and suspending medium. The equations apply for
small-volume fractions, , and assume that the radius of the
cell is very large compared with the membrane thickness. More
elaborate closed-form expressions have been developed for
cases when these assumptions are no longer valid (Schwan and
Morowitz, 1962; Schwan et al., 1970), and an exact
representation of the suspension dielectric properties as a
sum of two dispersions is available (Pauly and Schwan, 1959).
If, as is usually the case, the membrane conductance is
sufficiently low, Equations 4.2-4.5 reduce to the simple
forms to the right of the arrows.
A physical insight into Equations 4.2-4.5 is gained by considering the equivalent circuit shown in Figure 4.4, which displays the same frequency response defined in these equations. The membrane capacitance per unit area, Cm, appears in series with the access impedance, i + a/2, while the term a (1-1.5 ) provides for the conductance of the shunting extracellular fluid. Hence, the time constant, , which determines the frequency where the impedances 1/CmR and (i + a/2) are equal is given as Equation 4.5. Using typical values of i, a ~ 0.01 mho/cm, Cm = 1 F/cm2, R = 10 m, and = 0.5, with Equations 4.2-4.5 we see that the dispersion must occur at RF's and that its magnitude, s - , is exceptionally high.
From experimental dispersion curves and hence values of the four quantities s, ,
Equivalent circuit for the -dispersion of a cell suspension and corresponding plot in the complex dielectric constant plane (Schwan and Foster, 1980).
(s - ), and , the three quantities Cm, i, and a, can be determined with an additional equation available to check for internal consistency. Values for extracellular and intracellular resistivities thus obtained agree well with independent measurements. Dispersions disappear as expected after destroying the cell membranes, and their characteristic frequencies are readily shifted to higher or lower frequencies as intracellular or extracellular ionic strengths are experimentally changed. This gives confidence in the model, whose validity is now generally accepted.
This work led to the important conclusion that the
capacitance of all biological membranes, including cellular
membranes and those of subcellular organelles such as
mitochondria, is of the order of 1 F/cm2. This value
is apparently independent of frequency in the total RF range;
at low audio frequencies, capacitance values increase with
decreasing frequencies due to additional relaxation
mechanisms in or near the membranes. These mechanisms will
not be discussed here and have been summarized elsewhere
(Schwan, 1957; Schwan, 1965a).
From the membrane capacitance, we can estimate values for
the transmembrane potentials induced by microwave fields. At
frequencies well above the characteristic frequency (a few
MHz), the membrane-capacitance impedance becomes very small
by comparison with the cell-access impedance (i + a/2) ,
and the membrane behaves electrically like a short circuit.
Since intracellular and extracellular conductivities are
comparable, the average current density through the tissue is
comparable to that in the membrane. For an in situ field of 1
V/cm (induced by an external microwave-field flux of about 10
mW/cm²), the current density, i, through the membrane is
about 10 mA/cm² since typical resistivities of tissues
are of the order of 100 -cm at microwave frequencies. Thus
the evoked membrane potential, V = i/jCm, is about 0.5 V
at 3 GHz and diminishes with increasing frequency. This value
is 1000 times lower than potentials recognized as being
biologically significant. Action potentials can be triggered
by potentials of about 10 mV across the membrane, but (dc)
transmembrane potentials somewhat below 1 mV have been
recognized as being important (Schmitt et al., 1976).
If f < < fc, the total potential difference applied
across the cell is developed across the membrane capacitance.
In this limit, the induced membrane potential, V, across a
spherical cell is V = 1.5 ER, where E represents the applied
external field. Thus the cell samples the external-field
strength over its dimensions and delivers this integrated
voltage to the membranes, which is a few millivolts at these
low frequencies for cells larger than 10 m and external
fields of about 1 V/cm. These transmembrane potentials can be
Go to Chapter 4.1.2
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Last modified: June 24, 1997
© October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 78235-5301