Vibration of eukariotic Cells in Suspension induced by a low-frequency
electric Field: A mathematical Model
A. ZEHE, A. RAMÍREZ
Facultad de Cs. de la Electrónica
Benemérita Universidad Autónoma de Puebla,
Apdo. Post. # 1505, 72000 Puebla, Pue.
MEXICO
Abstract:- When exposed to an external electric a.c. field, the physical mechanics of cell membranes is an essential
aspect of cell deformation and shape oscillations. Dielectrodeformation is related to the induction of a dipole moment
due to electric charges on the opposite boundaries of the cell. The dielectric response of induced dipoles are
resonance phenomena, which are expected to occur at intrinsic cell frequencies due to forced cell vibrations. It has
been shown, that higher harmonics of oscillation frequencies go along with the possible appearance of double- or
even triple-peak formation in low-frequency impedance spectra. Literature data of experimental low-frequency
impedance spectroscopy confirm this finding.
Key Words:- forced cell vibrations, bioimpedance spectra, human blood
1. Introduction
Dielectric relaxation spectroscopy in frequency or time
domain has gained considerable attraction for studying
dielectric properties of biological cell- and particle
suspensions [1]. Biological systems are complex. Blood,
for instance, is an aqueous solution of many
substructures, different in composition, geometrical
arrangement and size. The interaction with an external
electric field involves multiple relaxation processes,
including interfacial polarization around the cells.
The interface polarization in the case of biological cells
is related to the complex dielectric permittivity of the
cell structural parts, which has led to the physical
presentation of a cell by shelled models [1,2]. Multilayer
ellipsoidal cell models exposed to electric and magnetic
fields up to 100 MHz have been studied [3]. Given the
spherical shape of lymphocytes with a thin cell
membrane and a spherical nucleus, which by itself is
covered by a thin nuclear envelope, a double-shell
model has been assumed and characterized by both time
domain dielectric spectroscopy and computer modeling
[2]. It is known, that the conductivity and the
permittivity of nucleoplasm is about twice that of
cytoplasm. This fact together with the volume
differences of both can play an important role in the
interpretation of bioimpedance spectra. While
lymphocytes possess a spherical double-shell structure,
human erythrocytes are single-shelled and of discoid
shape. Nevertheless are dielectric responses in an
external electrical field frequently treated within the
framework of a spherical model. The highly oblate
spheroid with semi axes a=b>>c is transformed under
the action of an electrical field in a three-axial ellipsoid
with semi axes a>b>>c (a is parallel to the external
field). A shelled cell model with one or several
membranes embracing electrolytes with mobile dipoles
or charge carriers implies also the possibility for a
mechanically vibrating oscillator. Given the elevated
oscillator mass, this could happen only at far lower than
microwave frequencies.
In the present study frequency-domain dielectric
spectroscopy in the low frequency range of 100 Hz…1
MHz was considered in order to test the viability of an
oscillatory excitation of cell structures by a low intensity
external electric field.
z
2. Dielectric Model of Forced Cell
Resonance
2.8 
m
y
Forced vibrations of an oscillator result, when an
external oscillatory force of frequency  is applied to a
particle subject to an electric field. The double-shell
lymphocyte bears similarities in the physical concept of
motion, where the external force acts on both, the inner
(nucleoplasm) sphere and the outer (cytoplasm) sphere.
Given the different geometrical and dielectric
characteristics, the response of each might be different
(Fig.1a). In the case of erythrocytes only one shell exists
and the oscillator model could be simpler, although the
more complex shape makes it more difficult in the
experimental approach (Fig.1b). Let us suppose, that a
shelled spherical cell containing a certain dipole charge
density is exposed to an harmonically variable electric
field. It is easy to imagine, that the elastic properties of
the nuclear envelope and the cell membrane provide for
a restoring force after a deformation due to the external
field action on the mobile ions inside the sphere plasmas
has occurred.
nuclear
envelope
nucleoplasm
= 60
= 0.5 S/m
m2
= 120
= 1S/m
m1
b
a
x
7.6 
m
Figure 1(b): Discoid shape of an erythrocyte of diameter
2R=7.6m and thickness at the rim 2L=2.8 m. a, b
represent semi axes (a=b=R) in a spheroidal approach
with small semi axis c in direction of z.
half axis of the field-deformed ellipsoid of radius R (i.e.
q=1 corresponds to a spheroid) characterizes both, the
elongation (contraction) deformation and the forces,
which determine the deformation [4].
The electric ponderomotive force is caused by the
applied time-dependent field E0(t), and contains the
dielectric properties of the cell:
membrane
Fpond ( , q )  V  F ( ) Eloc ( , q) 
2
cytoplasm
7 nm
40 nm
5.6m
f L (q)
. (1)
q
7m
Figure 1(a): Double-shelled lymphocyte cell as proposed
by Ermolina et al. [2];  is the dielectric constant,  specific electric conductivity, m1 - mass of the inner
nucleus (m1=91011g), m2- mass of the outer shell region
(m2=61011g), being mc= m1+m2 the total cell mass.
The periodic deformation of cells in an electric field
E0() with frequency  is determined by the distribution
of the electrical forces applied and the mechanical forces
generated in the deformed membranes and in the
adjacent layers of the cell. The mechanical forces
include elastic and viscous shear stresses in the
membranes. The axis relation q=R/L with L the longer
Here fL(q) is the depolarizing factor in direction of the
main axis L parallel to the external field vector E0. The
local field Eloc(, q) is the value of the applied field
inside the ellipsoid:
Eloc ( , q)  ( , q)  Eo ;
( , q) 
 m* ( )
 m* ( )  [ *p ( )   m* ( )]  f (q)
  (q )   m* ( )
(2)
The factor (q) has been calculated previously for the
general case of a dielectric object with arbitrary shape
[5]. The frequency dependence of the ponderomotive
electric field force is given by F()
F ( ) 
 p 2 2 p

 0 m   p
 1) 2 ( e ) 2  (
) 
 1 
(
4  m
m
m

1  ( ) 
2 1
e
e=0m/m.
(3)
Here p and m are the static relative permittivities of the
cell and the suspending medium, p and m are the static
conductivities, and 0 is the vacuum permittivity.
The elastic force of the cell membrane is given by
F
(q )  ½  S  (1  q )
2
(4)
with S the surface area of the object and  the shear
elasticity modulus.
The friction force Ffr (q, dq/dt) between the cell and the
surrounding medium is
F (q , d q /d t )  S   q
2
dq
dt
(5)
with s the surface viscosity coefficient of the
membrane.
Due to the small mass of the considered object and low
velocities of deformation dq/dt, it is safe to exclude the
inertia term in the equation of cell motion. All forces
acting on the cell will then fulfill the condition of
instantaneous equilibrium; i.e., its sum must be equal to
zero. Introducing 0=s/ as time of viscoelastic
relaxation of the membrane, and summing up the forces
given by ecs. (3,4,5), it follows
2

2 f ( q ) VE0 
2

1  F ( )  ( , q) 
q 
q
S 

.
dq
1  2 0
dt
(6)
q(t ) ~
cos(t   1 )
1   (E ) 
2
1


2
2
o

cos(2t   2 )
1  2 (E ) 
2
1
 ;
2
o

 ( Eo )   o /qo 1  Bx02 /q02 ; xo2  VEo2 /S
(7)
1 and 2 are phase angles of the harmonics, and related
to tan n=n(Eo).
3. Comparison to Experimental Spectra
If by any reason forced oscillations of cells give rise to a
dissipative process at a peak frequency  of the external
field, one should expect peak repetitions at 2, and
possibly at 3. Indeed would such a peak sequence in a
spectrum be indicative, that a forced oscillation of cells
has occurred.
In contrast to the relaxation type response of dielectric
objects with permanent dipoles, the dielectric response
of induced dipoles are resonant phenomena, where
energy resonance occurs at frequencies 0, which are
easily calculated from the expressions for ’() and
’’(), and result in 0  [2Ne2 /0m']⅓.
Here N is the number of induced dipoles per unit
volume, m is the particle (oscillator) mass and e the
elementary charge. (02-2)=(0+)(0-) was
substituted by 20 as a first-order approximation for
 0.
c
a
4
b
log(dielectric loss/a.u.)
This equation describes the time-dependent elongation
(contraction) of the ellipsoid, depending on geometric
(V, S), electrical (p, p) and mechanical (, s) cell
parameters, as well as on experimental properties of the
medium (m, m). As can be appreciated in ec. (1), the
cell deformation is caused not only by the electric field
amplitude E0, but by the dimensionless force parameters
3
2
1
1
2
(VEo2 / S ) . If we apply a time-dependent electric
field Eo(t)=E0[1+δcosΩt] of frequency Ω, modulation
depth δ, and consider the periodic changes of the axis
relation q(t) as forced oscillations of the ellipsoid, a
solution of ec. (1) is found, which contains such
oscillations not only at the frequency  but also at its
higher harmonics. Details of the calculations are given
in [4]. For small-signal modulation δ<<1 of the electric
field amplitude, one gets
0
2
3
4
5
6
log(frequenc y/Hz)
Figure 2: Spectra of dielectric loss D vs. frequency 
[1/s] of human blood, as published by Vázquez et al. [6].
The sample is described to be composed of (a) 0.1 ml
blood + 20 ml H2O; (b) 0.1 ml blood + 20 ml H2O + 0.5
ml ethanol, introduced into a parallel-plate capacitor
cell. (c) is a different donor.
In order to arrive at numerical values, we apply for N 
1024 m-3, m = 1.5  10-10 g, ' = 120, and  = 200 s-1,
which correspond to realistic experimental parameters of
a certain type of human blood cells. Here  is the
width of the single resonance line, and 0  1.7 kHz is
the experimentally determined [6] and by ec. (7)
reproduced resonance frequency.
A second peak could then be found at 3.4 kHz, which is
actually seen in Fig. 2 of the mentioned spectra.
Certainly, electrode polarization and thus the generation
of electrode artifacts deserves attention. The
measurements of conductive materials in the frequency
range 1 Hz to 10 MHz are not so straight-forward as in
higher frequency regions. Electrode polarization is a
manifestation of molecular charge organization, which
in the presence of water molecules and hydrated ions
occur at the sample-electrolyte interface. As discussed
previously [7], the effect increases with increasing
sample conductivity, and its consequences are more
pronounced on the capacitance of the ionic solutions and
biological samples. It has been found that in the case of
biological material the poorly conducting cells shield
part of the electrodes from the ionic current and reduce
polarization effects.
4. Discussion
The low-frequency dispersion occurring below a few
kHz is often assigned to counter-ion displacement about
molecular structural parts or cell membranes. Moreover
are biological tissues inhomogeneous and show
considerable variability in structure and composition,
and hence in the dielectric response. Although such
variations are natural and may be due to physiological
processes or other functional requirements, they make
the meaningful interpretation of the measuring results an
involved task. Additionally are relaxation processes of
the sample material often obscured by electrode
polarization. The implications of these effects deserve
particular consideration.
The appearance of pronounced peaks in low-frequency
impedance spectra of human blood suspensions seems to
support the forced oscillatory behavior of cells with the
induction of dipole moments corresponding to the
oscillation frequency of the external field. The dielectric
responses of induced dipoles are resonant phenomena.
The difference to other macromolecular structures arises
from the fact, that the cell electrolytes are constrained in
elastic membranes, which make them interact as a whole
with the harmonically oscillating electric field. The
elastic deformation of each single cell under the action
of the slowly varying external field from spherical to
oblate with harmonically forth and back changing the
direction of the internal polarization vector makes the
charges oscillate between two preferred localization
sites, or the dipoles to assume one of two discrete
orientations in space. Indeed would this picture represent
an array of polarizable macrostructures with mutual
interactions only by way of the counter-ions. The precise
effect of the latter is not yet clear.
Acknowledgements
Financial support of SEP-SESIC, México, under
contract #2003-21-001-023 is gratefully acknowledged.
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