1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN : 2278-800X, www.ijerd.com
Volume 5, Issue 9 (January 2013), PP. 22-29
Study the Window Effect of Rectangular Electrical Pulse
in Membrane Potential of Dielectric Model of Osteoblast
Cell under Different Microelectrodes
Swarup Sarkar1, Rajat Mahapatra2, Soumen Das3, Akash Ku Bhoi4
1,4
Applied Electronics and Instrumentation Department, Sikkim Manipal Institute of Technology
Majitar, Rangpo, Sikkim-737136, India
2
Electronics and Communication Engineering Department, National Institute of Technology
Durgapur, West Bengal, India.
3
School of Medical Science & Technology, Indian Institute of Technology
Kharagpur, West Bengal, India
Abstract:- With refarances of bilayer dielectric model, for the spherical osteoblast cell subjected to
expose on time domain electric field such as rectangular pulse, an equivalent circuit model is
presented. This paper presents the window effect of different linear and non linear electric field on
osteoblast cell in different microelectrode. It is found that because of different charging time constants
& relaxation time, different durations and different geometry of microelectrode have selective effects
on inner and outer membranes. We find that when duration of applied signal is reduced from
microsecond to nanosecond, and to pecosecond, the generated inner and outer membrane potential are
different & target induced is changed from the outer membrane to the inner membrane gradually for a
specific type electrode geometry. In this paper we also explore the comparative analysis of window
effect on cancer cell. The window effect gives preliminary explanation for various bioelectric effects
such as electroporation, intracellular electromanipulation and nanopores, providing help to the
applications of various cancer treatments.
Keywords:- osteoblast cells, window effect, linear and non linear electric field, charging time
constants, relaxation time and cancer treatment.
I. INTRODUCTION
Osteoblast cells are the most abundant and fundamental elements in bone, and their concentration in
serum is closely linked to bone metabolism and serves as a biological marker for the clinical assessment
of bone disease. For clinical investigation the dielectric property of osteoblast cell plays an important role. Like
other biological cell the dielectric properties of osteoblast cells are very remarkable. They typically display
extremely high dielectric constants at low frequencies, falling off in more or less distinct steps as the excitation
frequency is increased. Their frequency dependence permits identification and investigation of a number of
completely different underlying mechanisms, and hence, dielectric studies of osteomaterials have long been
important in electrophysiology and biomedical application. In drug delivery system the potential of the
membrane plays an important role to control the radius and density of generated pores which allows the drugs
enter into the cell. This concept is also applicable for clinical application in bone cell specially in osteoblast cell.
Now a days the rectangular ,triangular & sawtooth pulsed electric field, because of their unique biomedical
effects, have opened a new exposer to tumor treatment and become a research focus in the area of bio-
electromagnetics. Many experimental researches show that rectangular pulse with different parameters can cause
different bioelectric effects. Weaver et al found that in response to microsecond PEF (typical parameters: 1
kV/cm, 100 μs) [1-2], many reversible aqueous channels, which are often called pores (radius ≈ 20-110 nm),
appear at the cell membrane (outer membrane), while there is no obvious effect on the intramembranous
organelles. This physical procedure is termed electroporation, which can make cell membrane more permeable
to drug molecule, has been successfully applied to tumor treatment. On the other hand, Schoenbach et al found
that the nanosecond PEF with amplitude above 10 kV/cm can induce a series of cellular responses [3-9], which
are different from electroporation effect. This ns PEF can lead to poration of organelle membranes (inner
membrane), such as nucleolus, mitochondria, etc, without losing the integrity of outer membrane, and may
induce apoptosis of tumor cell. This phenomenon is termed intracellular electromanipulation (IEM). The ns PEF
might be a new and promising therapy for tumor treatment, but how to accurately decide its parameters is a key
point for its clinical application. Most recently, molecular dynamics modeling results and B16 cell experiments
22

2. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
[10] show that in response to pico second pulses (800 ps, 150 kV/cm, 1 MV/cm), reversible nanopores (radius ≈
0.4 nm) appear at the outer membrane of B16 cells. From the above experiments, it could be concluded that PEF
with different duration and intensity can cause various biomedical effects on cell, which suggests a window
effect between. external PEF and biological cell. In this paper, based on an equivalent circuit model of spherical
osteoblast cell, we studied the window effect in between rectangular pulse and transmembrane potential of
osteoblast cell in different microelectrode through time domain analysis .We also compair the window effect in
osteoblast cell with cancer cell.
II. THE EQUIVALENT DIELECTRIC MODEL OF OSTEOBLAST CELL
A. Time domain analysis
To study the window effect of osteoblast cell we consider the bilayer structure of the cell [15] [16] qnd
verify the time domain analysis of outer & inner membrane potential describe by C .Yao et al.
Fig 1. Equivalent circuit model of cell subjected to PEF (Ө =0)
From the above Figure, the complex-domain relationship between cb the cell voltage V and the
external field E is
𝑽(𝑆) = 𝑓𝑅𝑐𝐸(𝑆) ----------( 1)
The term f [19] is a compensation factor considering the
effect of extracellular medium and cytoplasm, etc. Since the
cell parameters are always under the conditions of
𝛾 𝑜 , 𝛾 𝑐 >> 𝛾 m and C c , 𝑅 >> 𝑑 , the term f can be approximated by a constant, f≈ 1.5. Therefore, the
transmembrane potential across the cell membrane is
1.5𝑅𝑐
𝑉𝑚 𝑠 = . 𝐸(𝑆) ---------------(2)
𝜏𝑐𝑙𝑙 𝑆+1
Actually, the charging course of the outer membrane is affected by both the extracellular and
intracellular mediums.
According to the reference [14], the charging time constant 𝜏𝑐𝑒𝑙𝑙 of the outer membrane is given by Schwan
1 1 𝜀𝑖 𝜀𝑚
𝜏𝑐𝑒𝑙𝑙 = + 𝑅𝑐. ----------------(3)
2𝛾0 𝛾𝑐 𝑑𝑚
Where, 𝜀𝑖 𝑑notes the vacuum permittivity. Therefore, at any given θ, the ratio of transmembrane
potential across the
outer membrane to field intensity in complex-domain is
1.5𝑅𝑐 cos 𝜃
𝐻𝑚 𝑠 = .
𝜏𝑐𝑙𝑙 𝑆+1
On the other hand the transmembrane potential across the inner membrane is
1
𝑉𝑛 𝑆 = 𝑆𝐶𝑛 – (4)
1 + 1 𝑆𝐶𝑛
𝐺𝑛
Therefore, at any given θ, the ratio of transmembrane potential to field intensity in complex-domain is
1.5 𝜏𝑐𝑒𝑙𝑙 𝑅𝑛 cos 𝜃
𝐻𝑛 𝑆 =
𝜏𝑛𝑢𝑐 𝑆+1 + 𝜏𝑐𝑒𝑙𝑙 𝑆+1
The charging time constant 𝜏𝑛𝑢𝑐 of inner membrane is[14]
1 1 𝜀𝑖 𝜀𝑛𝑚
𝜏𝑛𝑢𝑐 = + 𝑅𝑛 ---------------- (5)
2𝛾c 𝛾𝑛𝑐 𝑑𝑛
23

3. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
According to the dielectric cell parameters, it find out that 𝜏𝑐𝑒𝑙𝑙 > 𝜏𝑛𝑢𝑐.
Outer membrane:
According to the transfer functions defined by equations (2) and (5), the time responses of inner and outer
membranes to
a given rectangular pulse electric field E(s) can be obtained
𝑉𝑛(𝑡) = 𝐿−1 𝐻𝑛 𝑆 . 𝐸 (𝑆) -------(6) &
𝑉𝑚(𝑡) = 𝐿−1 𝐻𝑚 𝑆 . 𝐸 (𝑆) -------(7)
Put the value E(S)= ------- in equation no (6) & ( 7) we get the outer membrane potential (𝑉𝑚 𝑡 ) 𝑖s
𝑡 𝑡−𝜏
𝑉𝑚 𝑡 = 1.5 𝑅𝑐 𝐸 𝑡 −𝑒 𝜏𝑐𝑒𝑙𝑙 − 1 𝑡 − 𝜏 + 𝑒 𝜏𝑐𝑒𝑙𝑙 . 1 𝑡 − 𝜏 cos 𝜃-------( 8 )
At the end of the pulse, the transmembrane potential of outer membrane is
𝜏
𝑉𝑚 = 1.5𝑅𝑐𝐸 1 − 𝑒 − 𝜏𝑐𝑒𝑙𝑙 cos 𝜃--------------------------------------(9)
From the equation no (9) it is come to know that if 𝜏 𝑖𝑠 very low with respect to 𝜏𝑐𝑒𝑙𝑙 than outer membrane is
poorly charged and when 𝜏 >= 4 𝜏𝑐𝑒𝑙𝑙 than the outer membrane is fully charged.
After simplification of equation (6) the membrane potential of inner membrane as
And at the end of the pulse, the transmembrane potential of inner membrane is
𝑡 𝑡
1.5 𝜏𝑐𝑒𝑙𝑙 𝑅𝑛𝑢𝑐 𝐸 𝑡
𝑉𝑛 𝑡 = (𝑒 𝜏𝑐𝑒𝑙𝑙 − 𝑒 𝜏𝑛𝑢𝑐 ) cos 𝜃-------( 10 )
𝜏𝑐𝑒𝑙𝑙 −𝜏𝑛𝑢𝑐
From the equation no (10) it is come to know that if 𝜏 𝑖𝑠 very low with respect to 𝜏𝑐𝑒𝑙𝑙 & 𝜏𝑛𝑢𝑐 than
outer membrane is poorly charged and when 𝜏 >= 4 𝜏𝑛𝑢𝑐 than the the inner membrane is fully charged, but its
transmembrane potential of outer membrane is close to zero at the end of pulse.
The charging of the inner membrane is affected by the time constants (𝜏 𝑐𝑒𝑙𝑙, 𝜏 𝑛𝑢𝑐) of inner and outer
membranes. As 𝜏 𝑛𝑢𝑐 << 𝜏 𝑐𝑒𝑙𝑙, after the transmembrane potential across inner membrane reaching to its
maximum, it starts to rapidly decline even if the field still persists.these shifting of generating membrane
potential from outer to inner membrane shown in our study for osteoblast cell in rectangular electric pulse.
III. USED PARAMETER
A. Dielectric Value of Osteoblast Cell
parameter Cell parameters value
conductivity Extracellular medium 10 × 10−3
(S/m) Cell membrane 1.2 × 10−7
Cell cytoplasm 0.039s
Nuclear membrane 10 × 10−1
Nuclear cytoplasm 0.08s
Extracellular medium 80
relative Cell membrane 22
permittivity Cell cytoplasm 93
Nuclear membrane 22
Nuclear cytoplasm 93
Geometry Cell radius 12 µ𝑚
parameter Cell membrane thickness 0.006µm
(μm) Nuclear radius 6 µ𝑚
B. Simulation Tool
For fast and accurate simulation results, we chose the semiconductor, RF, and industry preferred NI
Circuit Simulator and lab-view, product version(2.5.1).Lab view is commonly used for complex analogue and
mixed-signal circuits .In an application, NI Circuit Simulator is used to study the circuit design .The analytical
and mathematical simulation is done by the latest version MATLAB 10.5 and Math AutoCAD. Similarly, the
electric field analysis of various electrode is done by COMsol multiphysics software.
C. Used Electrode
In our study we used different type of microelectrodes whose geometry and intra arrangement are
shown in fig:4 & specification and generated electric field intensity of the used microelectrodes are explore in
table I & II .
24

4. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
Table-I
(Electrode specification)
Length of electrode (l) 1mm
Width of electrode (w) 100 µm
Inter-electrode distance 100 µm
Thickness of electrode(d) 1µm
ε0 8.856 × 10−12 F
TABLE:II
(ELECTRIC FIELD INTENSITY CALCULATING FROM COMSOLE MULTI PHYSICS S/W)
Type of Rotation of Electric field
electrode electrode intensity(𝟏𝟎) 𝟒
Rectangular In line(c) ig:4 3.68
Shift (d) fig:4 3.75
Semicircular In line a)fig:4 4.22
Shift (b) fig:4 4.07
Saw tooth In line(e) ig:4 8.87
Shift (f) fig:4 3.61
Co-centric V/log(b/a)
Fig2. Configurations of electrode for FEA analysis.
IV. THE WINDOW EFFECT
A. SIMULATION OF THE WINDOW EFFECT IN BETWEEN OF RECTANGULAR PULSE AND
TRANSMEMBRANE POTENTIAL:
a. Rectangular Electrode
For Ta=1e-012 For Ta=1e-010
0.5 0.5
0 0
-0.5 -0.5
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
For Ta=1e-012 For Ta=1e-010
0.5 0.5
TMP in volts--------->
TMP in volts--------->
0 0
-0.5 -0.5
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
25

5. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
For Ta=1e-011 For Ta=1e-009
0.5 0.5
0 0
-0.5 -0.5
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
For Ta=1e-011 For Ta=1e-009
0.5 0.5
TMP in volts--------->
TMP in volts--------->
0 0
-0.5 -0.5
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
For Ta=1e-008
0.5
0
-0.5
-20 -15 -10 -5 0
10 10 10 10 10
For Ta=1e-008
0.5
TMP in volts--------->
0
-0.5
-20 -15 -10 -5 0
10 10 10 10 10
time in sec ------>
Fig: 3.Transmembrane potential of outer and inner membrane in different pulse width (Ta) & duration of
rectangular pulse in rectangular electrode.
We Plot the relationship curves between transmembrane potentials (at the end of the pulse) and
duration τ ( for rectangular electrode where E=3.68𝑋104 ),at different pulse width (Ta) of rectangular pulse for
outer and inner membrane of osteoblast cell , as shown in above Figure. It could be concluded that rectangular
pulse with different duration has obvious selective effect on the inner and outer membranes, i.e. window effect
,which are as follows.
(1) At Ta=10-9 s(nano second): As compared to time constants, the duration is so long that both the inner
and outer membranes can be fully charged. But the transmembrane potential across inner membrane is close to
zero at the end of the pulse. This field contains abundant signal components at low frequency, mainly acting on
the outer membrane.The field about 3.68𝑋104 V/cm can make the transmembrane potential across outer
membrane reach the critical value for poration. As a result, the outer membrane is porated, but the inner
membrane is not prorated.
(2) At 𝑇𝑎 = 10−10 𝑡𝑜10−11 (𝑠𝑢𝑏 𝑛𝑎𝑛𝑜 𝑠𝑒𝑐𝑜𝑛𝑑) i.e. τ = 4 τnuc<4 τcell: At the end of the pulse, the
transmembrane potential of the inner membrane is 0.25V, which is close to membrane potential of the outer
membrane, with almost similar effective action time. As external field with enough intensity may have similar
action on inner and outer membranes, so both of them might be expected to porate.
(3) At 𝑇𝑎 = 10−12 (𝑝𝑖𝑐𝑜 𝑠𝑒𝑐𝑜𝑛𝑑)i.e. 0.05 τnuc<τ<4 τnucFor the extremely fast charging speed, the inner
membrane fully charged and much greater transmembrane potential than the outer membrane. This field
contains abundant signal components at middle-high frequency, mainly acting on the inner membrane. The high
intensity may cause the inner
membrane porated, but the outer membrane not porated..
(4)At 𝑇𝑎 > 10−12 s ( above picosecond), i.e. τ<0.05 τnuc:The inner and outer membranes are extremely poorly
charged by the rectangular pulse which exhibit their filtering action on the extremely high frequency
components. But when responding to the strong electric field (several hundred kV/cm), the transmembrane
potentials can still reach the critical value, so the porations of the inner and outer membranes can also be
expected. Considering the extremely short action time, the pores should be different from those induced by
electroporation.
In summary, it is come to know from above figure that the very short pulse duration causes to low
transmembrane potential across outer membrane, while both the very short and very long pulse durations
decrease the transmembrane potential across inner membrane for osteoblast cell . This thoroughly exhibits the
band-pass and low-pass filter characteristics of inner and outer membranes, respectively which is also explore
through our frequency domain analysis.
26

6. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
b. Semicircular Electrode
For Ta=1e-010
For Ta=1e-012 1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1 -20 -15 -10 -5 0
10
-20
10
-15
10
-10
10
-5
10
0 10 10 10 10 10
For Ta=1e-012 For Ta=1e-010
0.2
TMP in volts--------->
0.2
TMP in volts--------->
0.1 0.1
0 0
-0.1 -0.1
-0.2 -0.2
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
For Ta=1e-009 For Ta=1e-008
1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
For Ta=1e-009 For Ta=1e-008
0.2 0.2
TMP in volts--------->
TMP in volts--------->
0.1 0.1
0 0
-0.1 -0.1
-0.2 -0.2
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
Fig: 4 transmembrane potential of outer and inner membrane in different pulse width & duration of rectangular
pulse in semicircular electrode.
We Plot the relationship curves between transmembrane potentials (at the end of the pulse) and
duration τ ( for semicircular electrode where E=4.22𝑋104 ),at different pulse width (Ta) of rectangular pulse
for outer and inner membrane of osteoblast cell, as shown in above Figure. It could be concluded that
rectangular pulse with different duration has obvious selective effect on the inner and outer membranes, i.e.
window effect. All the plots are as same as rectangular electrode only the value of membrane potential for both
outer & inner membrane are different but when Ta=10−10 a negative voltage is obtain in inner membrane due to
the presences of rest potential. Moreover, the negative transmembrane potential shown when Ta=10−10 is a
peculiar property of the inner membrane; in other words, the voltage across the inner membrane is
bipolar.Positive voltage produced due to the charging of the membrane. While the charges stored in the outer
membrane discharges at the negative edge of rectangular pulse, which is the forms of a discharge current. The
discharge current will charge the inner membrane again. As the discharge current is in the direction opposite to
the charged one, as a result, the transmembrane potential induced by discharge current is negative.
c. Sawtooth Electrode
For Ta=1e-011
2
For Ta=1e-010
2
1
1
0
0
-1
-20 -15 -10 -5 0
10 10 10 10 10 -1
-20 -15 -10 -5 0
10 10 10 10 10
For Ta=1e-011
1 For Ta=1e-010
1
TMP in volts--------->
TMP in volts--------->
0.5
0.5
0
0
-0.5 -0.5
-1 -1
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
For Ta=1e-013
2 For Ta=1e-012
2
1
1
0
0
-1 -1
-20 -15 -10 -5 0
10 10 10 10 10 10
-20
10
-15
10
-10
10
-5 0
10
For Ta=1e-013 For Ta=1e-012
1 1
TMP in volts--------->
TMP in volts--------->
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
27

7. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
For Ta=1e-009 For Ta=1e-008
2 2
1 1
0 0
-1 -1
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
For Ta=1e-009 For Ta=1e-008
1 1
TMP in volts--------->
TMP in volts--------->
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-20 -15 -10 -5 0 -20 -15 -10 -5 0
10 10 10 10 10 10 10 10 10 10
time in sec ------> time in sec ------>
Fig: 5.Transmembrane potential of outer and inner membrane in different pulse width & duration of rectangular
pulse in trangular electrode.
We Plot the relationship curves between transmembrane potentials (at the end of the pulse) and
duration τ ( for trangular electrode where E=8.87𝑋104 ),at different pulse width (Ta) of rectangular pulse for
outer and inner membrane of osteoblast cell, as shown in above figure. It could be concluded that rectangular
pulse with different duration has obvious selective effect on the inner and outer membranes, i.e. window effect.
All the plots are as same semicircular electrode only the value of membrane potential for both outer & inner
membrane are different.
V. DISCUSSIONS
Though the window effect is based on the spherical osteoblast cells in suspension, it could be expected
to extend to practical applications. The pulse duration and intensity jointly determine the selective effect on the
various parts of the cell, to induce different bioelectric and therapeutic effects. When the duration is reduced
from microsecond to nano- second, and to peco-second, the element induction is changed from the outer
membrane to inner membrane gradually. At the same time, the amplitude of field intensity should be increased
to reach the threshold potential across the membrane. Window effect may give preliminary explanation for
various bioelectric effects such as electroporation, and nanopores, and can be helpful in choosing suitable
duration intensity combination for inducing corresponding poration effects of the inner and outer membranes of
osteoblast cell.It is also observed that the window effect does not depends on the shape of electrode although
the peak value of membrane potential is changed with the geometry of microelectrode .Because when the
transmembrane potential is increased the threshold voltage is also increased in same ratio.The rectangular pulse
may also have selective effect on different cancer cells, for they have different cell parameters, i.e. different
time constants (τcell and τnuc). For the healthy and cancer cells of a same type, it is also possible to determine a
reasonable rectangular pulse effect , with its main action on cancer cells.
VI. COMPARITIVE STUDY
In our study we also compair the window effect of rectangular pulse in Cancer cell with osteoblast
cell and we find that in cancer cell when the pulse duration is bellow micro level no voltage will generate at
outer & inner membrane but in microlevel outer membrane is fully charged whereas inner membrane has no
response.This condition remain constant upto peco range .Above this the membrane voltage is shifted from
outer membrane to inner membrane.
VII. CONCLUSION
From the frequency domain analysis it is come to know that the inner and outer membranes act as a
band-pass and low-pass filter characteristics respectively, so they has different filter activity in different
frequency range with the influences of the external electric field .Depondeng on different field duration
decides different transmembrane potentials across inner and outer membranes, so the field has selective action
or window effect on them. In summary, the rectangular electric pulse has a window effect on the osteoblast
cells. When the duration is reduced from microsecond to nanosecond and to pecosecond, the pulse is induced
from the outer membrane to inner membrane gradually. At the same time, we also find that the induced
membrane potential is varied in different microelectrode to generate poors in the membrane.. Furthermore,
rectangular pulse may also have selective effect on different parts of a osteoblast or different type of cancer
cells. The study of window effect in between membrane and rectangular pulse should be useful for the drug
delivery system & experimental treatment for bone cancer.
28

8. Study the Window Effect of Rectangular Electrical Pulse in Membrane...
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