IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 11, NOVEMBER 20113279
Equivalent Pulse Parameters for Electroporation
Gorazd Pucihar, Member, IEEE, Jasna Krmelj, Matej Reberˇ sek, Tina Batista Napotnik, and Damijan Miklavˇ ciˇ c*
Abstract—Electroporation-based applications require the use of
specific pulse parameters for a successful outcome. When recom-
can be obtained by using equivalent pulse parameters. We deter-
pulses resulting in the same fraction of electroporated cells. Pulse
duration was varied from 150 ns to 100 ms, and the number of
pulses from 1 to 128. Fura 2-AM was used to determine electro-
poration of cells to Ca2+. With longer pulses or higher number
of pulses, lower amplitudes are needed for the same fraction of
electroporated cells. The expression derived from the model of
electroporation could describe the measured data on the whole in-
terval of pulse durations. In a narrower range (0.1–100 ms), less
complex, logarithmic or power functions could be used instead.
The relation between amplitude and number of pulses could best
be described with a power function or an exponential function.
We show that relatively simple two-parameter power or logarith-
mic functions are useful when equivalent pulse parameters for
electroporation are sought. Such mathematical relations between
pulse parameters can be important in planning of electroporation-
based treatments, such as electrochemotherapy and nonthermal
Index Terms—CHO cells, electropermeabilization, Fura 2-AM,
poorly membrane permeant, is used in various biotechnolog-
ical and biomedical applications, such as the introduction of
molecules into cells , , cell fusion , , tissue abla-
tion –, and sterilization of water and liquid food –.
In experimental settings, electroporation is normally performed
by placing a biological sample (e.g., cell suspension or a small
part of a tissue) between the electrodes and delivering a single
ating the electric field between them. The efficiency of electro-
poration can be interpreted differently in different applications
take of molecules, efficient electroporation is associated with a
high number of cells loaded with exogenous molecules that also
LECTROPORATION, as a method for increasing cell
membrane permeability to molecules that are otherwise
2011. Date of publication September 6, 2011; date of current version October
19, 2011.This work was supported by the Slovenian Research Agency (ARRS).
Asterisk indicates corresponding author.
G. Pucihar, J. Krmelj, M. Reberˇ sek, and T. B. Napotnik are with the
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana SI-
1000, Slovenia (e-mail: firstname.lastname@example.org; email@example.com;
*D. Miklavˇ ciˇ c is with the Faculty of Electrical Engineering, Univer-
sity of Ljubljana, Ljubljana SI-1000, Slovenia (e-mail: damijan.miklavcic@
Digital Object Identifier 10.1109/TBME.2011.2167232
survive the treatment, while electroporation efficiency in tissue
ablation and sterilization is related to killing the largest amount
of target cells or microorganisms. However, efficient electro-
poration is obtained only after careful adjustment of the pulse
parameters, among which the pulse amplitude, pulse duration,
and number of pulses have the largest impact on the outcome of
Each specific application of electroporation requires some-
what different settings of pulse parameters. In addition, pulse
orientation and density of cells, and other experimental condi-
tions, meaning that they can differ substantially even within a
given application of electroporation. To date, there have been
a vast number of different pulse protocols reported for various
applications of electroporation. For example, for the introduc-
tion of small molecules, pulses with amplitudes in the range
of 1 kV/cm and durations extending from hundred μs to ms
are used –. Larger molecules can be introduced us-
ing three different combinations of pulse parameters: 1) with
pulse amplitudes up to few kV/cm, lasting from few μs to
hundred μs , ; 2) with low pulse amplitudes of few hun-
dred V/cm but durations ranging into tens of ms ; 3) with
a combination of short high-amplitude pulses and long low-
amplitude pulses (mostly for the uptake of DNA) –. For
sterilization in food and drink industry, the pulse amplitudes
should be larger than 15 kV/cm in order to electroporate the
membranes of microorganisms, which are smaller than eukary-
otic cells, while pulse durations range from μs to ms , .
durations of several tens of ns and amplitudes of tens of kV/cm
or more are used –.
Sometimes, the specific pulse parameters required for effi-
cient electroporation are difficult to obtain. This might be due to
experimental setup (e.g., large samples of cells, high conductiv-
For these reasons, the pulse parameters can be usually adjusted
mentioned paragraph, similar outcomes of the experiment can
also be obtained by using equivalent pulse parameters. For ex-
ample, instead of using a number of short, high-voltage pulses,
one can either use longer pulses with lower voltage or adjust the
number of pulses by keeping the amplitude or duration of the
pulses unchanged. However, finding a suitable combination of
pulse parameters proved to be a difficult task, since simple rela-
tions, such as keeping the same energy of the pulses, turned out
to be inefficient , . Therefore, more specific functional
relations between the pulse parameters should be identified in
order to avoid the excessive amount of experiments and time
needed to determine the suitable parameters.
0018-9294/$26.00 © 2011 IEEE
3280 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 11, NOVEMBER 2011
of fluorescence (F345/F385) for cells in control (nonporated cells). (C) Cells
1 min after electroporation with a 250V/cm, 10 ms pulse. Brighter cells were
electroporated. Arrow denotes the field direction. Bar represents 20μm.
Monitoring electroporation. (A) CHO cells—bright field. (B) Ratio
The role of pulse parameters in the efficiency of electro-
poration was already systematically addressed in several stud-
ies , , , –. In some of these studies, the au-
thorsalsotriedtodetermine themathematical relations between
pulse parameters that lead to the same efficiency of electropo-
ration –, –. We describe them in more detail
in Section II. In general, different functional dependences be-
tween pulse parameters were reported and they were mostly
given for the relation between the pulse amplitude and the pulse
duration. Besides, the relations were obtained from relatively
narrow ranges of parameter values, and the parameters were
taken from different intervals. In our present study, we mea-
sured the relation between amplitude and duration, and between
amplitude and number of pulses that result in the same fraction
of electroporated cells. Pulse duration was varied in the range
from 150ns to 100ms and the number of pulses from 1 to 128
pulses (for 100 μs pulse duration). Mathematical relations from
the literature were then fitted to the measured data in order to
investigate whether these relations could be used to determine
equivalent pulse parameters on a wide range of their values.
The relations between pulse parameters obtained in this manner
are needed in specific electroporation-based applications, such
as treatment planning in electrochemotherapy, where equiva-
lent pulse parameters can be used if predicted parameters are
unavailable with the particular pulse generator.
II. MATERIALS AND METHODS
Chinese hamster ovary cells (CHO-K1) were plated in Lab-
Tek II chambers (Nalge Nunc Int., NY) or on cover glasses at
with 8% fetal calf serum, 0.15mg/mL L-glutamine (all three
from Sigma-Aldrich, Germany), 200 units/mL benzylpenicillin
(penicillin G), and 16mg/mL gentamicin and incubated in 5%
CO2at 37◦C. The experiments were performed 12–18 h after
plating, when most cells firmly attached to the surface of the
chamber or the cover glass and most of them did not yet divide
[see Fig. 1(A)].
B. Detection of Electroporation
To determine which cells were electroporated, a fluorescent
calcium indicator Fura 2-AM (Molecular probes, The Nether-
lands) was used. Fura 2-AM enters the cell through an intact
membrane, and is transformed in the cytosol into Fura 2, a
membrane-impermeant ratiometric dye. Electroporation results
in the entry of Ca2+ions into the cells, where their binding to
Fura 2 causes the change in the fluorescence of the dye. With
moderate pulse parameters, the cell membrane recovers after
electroporation (see Section II-D), and the cell stores the excess
Ca2+into its intracellular reservoirs or excludes it from the cy-
toplasm. The fluorescence thus returns gradually to the initial
value, allowing for another repetition of the experiment on the
Prior to experiments, the culture medium was replaced by a
staining solution, which was a mixture of fresh medium and
2μM of Fura 2-AM. After 25min of incubation at room tem-
perature, the staining mixture was washed with a fresh culture
medium to remove the excess dye. The culture medium con-
tains approximately 1mM of Ca2+, meaning that Ca2+ions
were present in the extracellular medium but were nearly absent
from the cytosol, as Ca2+do not readily cross the nonporated
Cells were observed under a fluorescence microscope (×40
objective, AxioVert 200, Zeiss, Germany) equipped with a
charge-coupled device camera and a monochromator (both Vis-
itron, Germany). The changes in intracellular concentration of
calcium, resulting from electroporation, were determined ratio-
metrically using MetaFluor 7.1 software (Molecular Devices,
GB), with the excitation wavelengths set at 345 and 385nm,
and the emission measured at 540nm for both excitation wave-
lengths. The ratio images were obtained by dividing the fluores-
cence image of cells excited at 345nm with the image excited at
of cells electroporated for Ca2+ions (% electroporated) were
field of view (% electroporated = 100(nF/nC)). The concentra-
tion of intracellular Ca2+was determined qualitatively by mea-
suring the ratio values for each electroporated cell for a period
of 1min after electroporation. These values were determined by
encircling the cells with regions of interest in MetaFluor and
integrating the ratio values within these regions. The maximum
ratios measured in each electroporated cell were averaged and
then presented on a graph.
Laboratory prototype of a Cliniporator device (IGEA, Italy),
a prototype of a microsecond square wave pulse generator with
a fast switch , and a prototype of a nanosecond pulse gen-
erator  were used to generate rectangular electric pulses
needed to electroporate the cells. Different generators had to be
used because all parameters could not be generated by a single
device. Either a single pulse with duration of 150 ns, 1 μs, 3 μs,
10 μs, 30 μs, 100 μs, 1 ms, 10 ms, 50 ms, or 100 ms or a train of
duration of 100 μs was delivered to cells. The pulse amplitude
and the current flowing through the cells were monitored with
an oscilloscope. The duration or the number of pulses in each
experiment was chosen randomly from the given set of parame-
ters. Attention was paid not to apply too many pulses to cells in
PUCIHAR et al.: EQUIVALENT PULSE PARAMETERS FOR ELECTROPORATION 3281
one experiment, especially when a train of pulses was delivered
(e.g., 64 and 128 pulses were never applied to the same cells).
For a given pulse duration or number of pulses, the pulse ampli-
tude was increased stepwise until ∼70% of cells (an arbitrarily
then transformed into equivalent voltage-to-electrode-distance
ratios (or electric field intensities E70) and plotted on a graph.
Between two successive increments of pulse amplitude we
waited for at least 5min for cell recovery (verified in a separate
experiment, see later), except if cells already became electropo-
rated. In this case, we first waited until the fluorescence of cells
returned to the initial value, and then, after additional 5min,
delivered the pulses with higher amplitude. During the experi-
stage to facilitate cell recovery.
Pulses longer than 1 μs were delivered to a pair of paral-
lel Pt/Ir wire electrodes with 0.8mm diameter and 4mm dis-
tance between them, which were positioned at the bottom of the
Lab-Tek chamber. The field distribution between the electrodes
was homogenous in the central region between the electrodes,
where the calculated field was equal to the applied voltage-to-
electrode-distance ratio . Different electrodes, with smaller
interelectrode distance, had to be used for pulses of 150 ns du-
ration, because of the high intensity of the field, required to
electroporate the cells with such pulses. The electrodes were
made of two adjacent 30μm thick flat gold layers mounted on
a microscope glass slide, and separated by 100μm . The
cover glass with cells was placed on top of the electrodes with
cells facing down.
D. Cell Recovery After Electroporation
To test if cells recovered during the 5 min delay between
two successive pulses, we performed an additional experiment.
Cells were prepared and incubated with the dye as described
medium but in a calcium-depleted modification of minimum
essential medium (MEM) (SMEM, Gibco) supplemented with
5μM of ethylene glycol tetraacetic acid (EGTA) to remove the
leading to 70% of electroporated cells was delivered, and 5min
later the medium was replaced with fresh SMEM supplemented
with 1mM CaCl2.
The ratio of fluorescence from cells did not change signifi-
cantly after electroporation in Ca2+depleted medium and re-
mained at the same level even after the addition of Ca2+, 5min
after electroporation. This shows that the fraction of electro-
porated cells (% electroporated) were not influenced by the
possible intracellular release of Ca2+(Ca2+induced Ca2+re-
lease) and that 5min interval was sufficient for cell membrane
recovery to Ca2+ions. The same pulse applied again resulted
in an increase of the ratio of fluorescence, confirming that the
previous pulse indeed electroporated the cells. Cell recovery af-
ter exposure to all investigated pulse parameters was verified in
the same manner.
E. Fitting the Relations Between Pulse Parameters to the
Experimentally determined pulse amplitudes leading to the
same fraction of electroporated cells at different pulse durations
(Systat, IL). To these data, various expressions were fitted using
SigmaPlot 8.0 and Matlab 7.5 (Mathworks, MA). These ex-
pressions were taken from the literature and are described later,
together with the parameter range from which they were de-
termined. Expressions (1)–(4) were obtained empirically, while
those given by (5) and (6) have a theoretical basis in models
Rols and Teissi´ e  investigated the threshold value of the
electric field E needed for electroporation of cells with pulses
lasting from tP= 2 to 100 μs and obtained a hyperbolic relation
between E and tP:
E = a +
Vernhes with coauthors  investigated the effects of electric
fields on the inactivation of amoebae. They determined the elec-
tric field E required to kill at least 95% of amoebae. For pulse
durations tPin the range from 50 μs to 100 ms, they obtained a
logarithmic relation between E and tP: E = a − b log (tP). In a
slightly modified form, this expression can also be written as
E = a − b log
In (2), the term tP/t0presents the duration of the pulse tP (ms),
normalized to unit of pulse duration t0= 1 ms in order to obtain
the dimensionless argument of the logarithmic function.
Krassowska and coworkers  exposed the cells to pulses
with durations tP ranging from 50 μs to 16 ms and determined
the amplitude of the field E required to kill 50% of the cells.
They proposed a relation of the form E = atb
argument of the power function
In (3), the term tb
in (ms) and t0= 1 ms.
The same relation was obtained by Abram and coworkers
for inactivation of Lactobacillus plantarum  in the pulse
duration range from 0.85 to 5.1 μs.
Huiqi He with coworkers  performed single-cell mea-
surements of electroporation-mediated uptake of molecules of
For each investigated molecule, the relation between the thresh-
tPwas determined. These relations formed three-parameter ex-
P, which can be
E = a
Pwas replaced with (tP/t0)b, where tPis given
E = a + b exp(−ctP).
Saulis derived the expression for the fraction of electropo-
rated cells FP using equations for kinetics of pore formation,
3282 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 11, NOVEMBER 2011
originating from the theory of electroporation . The slightly
modified expression for FP yields
FP(E,tP) = 1 − exp(−kf(E)tP)
where the rate of pore formation kf(E) is given by
K1ER cos ϕ
1 − exp
and τ is the time constant of membrane charging 
(2λoλi)/(2λo+ λi) + (Rλm)/h.
In (5a)–(5c), ν is the frequency of lateral fluctuations of lipid
molecules, R is the cell radius, a is the area per lipid molecule,
ΔW0is the energy barrier for pore formation at zero membrane
voltage, kB is Boltzmann’s constant, T is the absolute temper-
ature, Cmis the capacitance of the membrane, r∗is the radius
of the pore, εm and εw are the relative permittivities of the
membrane and the water inside the pore, respectively, ΔΦ is
the resting membrane voltage, λo, λi, and λmare the conduc-
tivities of the extracellular medium, cytoplasm, and membrane,
respectively, and h is the membrane thickness. For a fixed value
of FP(in our case FP= 0.7), the relation between E and tPcan
be determined by numerically solving (5). The values of these
parameters together with their descriptions are given in Table I.
The parameters marked with “#” were changed during the fit-
ting of (5) to the measured data. For spherical cells, the fitted
parameters were ΔW0and r∗, as suggested in . For attached
cells, the fitted parameters were ΔW0, r∗, R, K1, and K2, with R,
K1, and K2reflecting the size of the attached cell, the shape of
the cell, and the influence of the shape on the membrane charg-
ing, respectively (for a spherical cell R = 6.5μm, K1= 1.5, and
Neumann  derived the relation between the electric field
E needed to electroporate 50% of green algae cells Chlamy-
domonas reinhardtii and the pulse duration tP from the interfa-
WP : WP = const · E2tP, he derived the following relation:
where tPeffis the effective pulse duration given by 
3 + exp
For pulses with tP >> τ, where τ is given by (5c), tPeffequals
tP. When experimental data were plotted in terms of E2versus
1/tPeff, it turned out that (6) was piecewise linear: in the range
from 100 to ∼500 μs and in the range from 500 μs to 16
DESCRIPTION OF THE PARAMETERS USED IN (5) AND THEIR VALUES
ms . Since the purpose of this study was to find continuous
expressions, which would describe the data in the whole range
of tP, (6) was not included in the fitting. Instead, the expression
was used in the analysis of the data in Section IV.
A. Relation Between the Pulse Amplitude and Pulse Duration
The pulse amplitudes leading to electroporation of roughly
longer pulses, lower amplitudes are needed to maintain roughly
the same fraction of electroporated cells [see Fig. 2(B)]. The
relation is strongly nonlinear since pulses shorter than 1 ms
require progressively higher amplitudes for the same effect. For
example, if 137V/cm was sufficient to electroporate cells with
a 100 ms pulse, the field had to be increased to 575V/cm to
electroporate cells with a 100 μs pulse, and up to 10 kV/cm
to obtain the same effect with a 150 ns pulse. To display the
nonlinear relation between the pulse amplitude and the pulse
duration, the same results are plotted also on a linear scale [see
which reflects the change in intracellular Ca2+concentration,
varies with pulse duration. While it remains at rather constant
value in the range from 30 μs to 10 ms, the ratio decreases for
pulses, which are out of this range [see Fig. 2(C)].
To determine if functional relations given with (1)–(5) could
describe the measured results, we fitted each of these equations
to our data [see Fig. 2(A)]. As the figure shows, most of the
equations could not be adequately fitted to the data. The only
curve that could at least qualitatively describe the data is the
one given by (5). The remaining four equations either could
not reproduce the increase in pulse amplitudes at shorter pulse
PUCIHAR et al.: EQUIVALENT PULSE PARAMETERS FOR ELECTROPORATION3283
for electroporation with a single pulse. (A) Field amplitudes leading to elec-
troporation of roughly 70% of cells at pulse durations ranging from 150 ns to
100 ms. Black circles are the measured data presented as means ± SD (N = 9).
The inset shows the same data on a linear scale of tP. The curves are the fitted
(1) (dashed gray): E70%= 520.9 V·cm−1+ 1.43 V·cm−1ms/tP, Equation (2)
(solid gray): E70%= 397.0 V·cm−1(tP/t0)−0.22. Equation (4) (dashed black):
E70%= 573.0 V·cm−1+ 12 760 V·cm−1exp(−2024 ms−1·tP). Equation (5)
(dotted black): r∗= 0.64nm, ΔW0 = 44.8 kBT. (B) Corresponding fraction
of cells electroporated to Ca2+. (C) Ratio of fluorescence (F345/F385) due to
intracellular change in Ca2+.
Relation between the field amplitude E70 and the pulse duration tP
found that (2), (3), and (5) could be fitted to the data reasonably
well (see Fig. 3), with the best fit obtained for a two-parameter
logarithmic function (2) and a two-parameter power function
We also estimated the maximum increase of temperature of
mined from the assumption that electrical energy is transformed
into heat completely
Here, U is the amplitude of the pulse, I is the current through
cell suspension, tP is the pulse duration, NP is the number of
V is the volume of the medium (V = 1mL), and cp is the
specific heat capacity of the medium (cp= 4186J·kg−1·K−1).
The maximum temperature increase of 1.5K (from the initial
293K) is generated by the longest, 100 ms pulse, and is well
below the temperature rise that could harm the cells.
tP in the range of pulse durations from 10 μs to 100 ms. The black circles
present the measured data (same as in Fig. 2), while the curves are the fits
of (2) (solid black), (3) (solid gray), and (5) (dotted black) to the data in the
given range. For E in (V·cm−1) and tP in (ms), the parameters yield Equa-
tion (2): E70%= 460.1V·cm−1− 161.6 V·cm−1·log (tP/t0); (3): E70%=
385.1 V·cm−1·(tP/t0)−0.19; (5): r∗= 0.65nm, ΔW0= 44.8 kBT.
Relation between the field amplitude E70 and the pulse duration
B. Relation Between the Pulse Amplitude and the Number
The pulse amplitudes leading to electroporation of approxi-
mately 70% of cells after the exposure to a train of 1, 2, 4, 8, 16,
32, 64, or 128 pulses, with each pulse lasting 100 μs, are shown
amplitude needed to obtain the same fraction of electroporated
cells decreases. Trains with less than 16 pulses require increas-
ingly higher pulse amplitudes to maintain the same fraction of
electroporated cells. At the same time, with higher number of
pulses, the ratio of the fluorescence of cells (F345/F385) de-
creases [see Fig. 4(C)]. The results show that the decrease of
pulse amplitude can be compensated by increasing the number
Functional relations given by (1)–(4) were again fitted to the
measured results. Despite the fact that these equations were
primarily used to describe the relation between the pulse ampli-
tude and the pulse duration, they can also be used to represent
the relation between the amplitude and the number of pulses.
Namely, except for (5) and (6), these equations were obtained
empirically and do not reflect any physical process related to
electroporation. The curves obtained after fitting (1)–(4) to the
data are shown in Fig. 4(A). In this case, the relation between
amplitude and number of pulses could be best described by a
two-parameter power function (3) or a three-parameter expo-
nential function (4).
Again, we calculated the change in the temperature ΔT of
the medium using (7). In this case, the maximum increase in
the temperature was 0.7K and was generated by a train of 128
cells and is in practice probably even lower, due to the relatively
low pulse repetition frequency of 1Hz, which allows for the
cooling of the medium between successive pulses.
3284 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 11, NOVEMBER 2011
NP at pulse duration tP = 100 μs and pulse frequency of 1Hz. (A) Field
amplitudes leading to electroporation of roughly 70% of cells at NP ranging
from 1 to 128. Black circles are the measured data presented as means ± SD
(N = 12). The curves are the fitted expressions describing the relations between
E (V·cm−1) and NP. Equation (1) (dashed gray): E70%= 304.4 V·cm−1
+ 325.1 V·cm−1/NP. Equation (2) (solid black): E70% = 516.4V·cm−1
− 131 V·cm−1·log (NP). Equation (3) (solid gray): E70%
V·cm−1·NP−0.18. Equation (4) (dashed black): E70%= 288.5 V·cm−1+
347.4 V·cm−1·exp (−0.19NP). (B) Corresponding fraction of cells electropo-
Relation between the field amplitude E70 and the number of pulses
In this paper, we evaluated different reported functional rela-
tions between amplitude and pulse duration, and amplitude and
number of pulses that result in the same fraction of electropo-
rated cells and also investigated the level of cell electroporation.
Such relations could help researchers finding equivalent pulse
parameters for electroporation.
From our results, it follows that the change in the value of
one parameter can be compensated by carefully adjusting the
value of another parameter. For example, the decrease in the
pulse amplitude can be compensated by increasing the pulse
caseswhereexperimental settingsorpulsegenerator limitations
confine the pulse parameters within a given range of values. In
such cases, adjusted, i.e., equivalent pulse parameters could be
used to maintain the efficiency of electroporation.
Usually, the efficiency of electroporation is expressed either
as the fraction of electroporated cells, cell viability, or the up-
take of molecules into electroporated cells. In this paper, the
efficiency of electroporation was characterized by the fraction
of electroporated cells. Alternatively, we could choose the up-
take of molecules criterion, but in this case, fine-tuning of the
amplitude, required to maintain the constant uptake, would be
a more difficult task. Besides, Fura 2-AM might not be the
most suitable dye for measuring the uptake. Namely, the dye re-
sponds to changes in intracellular Ca2+, which can occur either
due to the inflow of extracellular Ca2+through the electropo-
rated membrane or due to the release of Ca2+from intracellular
reservoirs. The latter is difficult to estimate, especially because
the Ca2+release can be triggered by elevated cytosolic Ca2+
after electroporation and/or by nanosecond electric pulses .
Several studies have investigated the influence of pulse pa-
rameters on the efficiency of electroporation , , ,
the same efficiency can be obtained with different combinations
findings and further show that the same conclusions are valid on
a wider interval of parameter values, i.e., pulse durations from
150 ns to 100 ms, and the pulse number from 1 to 128.
A wide variety of mathematical expressions, describing the
relation between pulse parameters, can be found in the litera-
ture –, –, . These expressions range from
expressions, (1)–(6). The relatively large collection of different
mathematical expressions might be due to the fact that the re-
lations between parameters were determined within different
intervals of parameters and that the approaches used to derive
these expressions were different. For example, (1)–(4) were all
obtained empirically, (5) was derived from kinetics of pore for-
not describe the measured data reliably on the whole interval of
pulse durations. The largest disagreement between calculated
curves and measured data was observed for pulse durations
shorter than 10 μs, where pulse amplitudes sharply increased.
than the pulse duration, see (5c). Since (1)–(4) were determined
empirically, they are not physically related to the processes of
membrane charging and electroporation. Besides, (2)–(4) were
obtained for pulses longer than few tens of μs, where charg-
ing time of the membrane is insignificant with respect to pulse
durations considered, meaning that the extrapolation of these
expressions to shorter pulses essentially led to errors. However,
for longer pulses, (2), (3), and (5) could be fitted to the data with
better accuracy (see Fig. 3).
Equation (5) was the only equation that could, at least quali-
tatively, describe the measured data in the whole range of pulse
durations. The observed difference between theoretical predic-
tions of (5) and experiments at short pulse durations could be
partly explained by the fact that (5) was originally derived for
a spherical cell. Cells in our study were, for the purposes of re-
producible experiments, attached to the bottom of the chamber,
meaning that they were flat and had different shapes and sizes
(see Fig. 1). Under the same experimental conditions, attached
PUCIHAR et al.: EQUIVALENT PULSE PARAMETERS FOR ELECTROPORATION 3285
measured data for the case of a spherical cell (dotted curve) and an attached
cell (solid curve). Black circles are the measured data presented as means ± SD
(N = 9). Spherical cell: r∗= 0.64nm, ΔW0= 44.8 kBT. Attached cell: r∗=
0.32nm, ΔW0= 46.6 kBT, R = 15.9μm, K1= 1.24, K2= 2.56.
Comparison of the calculated curves obtained by fitting (5) to the
ical cells, meaning that the transmembrane voltage induced on
their membranes is generally lower, and the charging time of
the membrane can be higher. Together, both of these effects
could explain the disagreement between the curve calculated
for spherical cells and the data measured on attached cells. To
partially account for the effect of cell shape, we modified (5) by
also varying the parameters R, K1, and K2to reflect the change
in the size of the attached cell, the decrease of the voltage on the
membrane, and the increase in the time constant of membrane
charging, respectively. Equation (5) modified for an attached
cell was then again fitted to the measured data (see Fig. 5, solid
curve). Compared to the spherical cell, a better agreement with
the data was now obtained (cf., solid and dotted curve in Fig. 5).
The parameters R, K1, and K2in (5) have changed from R =
6.5μm, K1= 1.5, and K2= 1 (spherical cell) to R = 15.9μm,
K1 = 1.24, and K2 = 2.56 (attached cell), implying that the
voltage on larger but thinner attached cells is lower compared
to the voltage on smaller spherical cells, while the time constant
of membrane charging is higher. From our previously reported
calculations for attached cells, we estimate that these values are
reasonable . The fitted value for r∗(0.32nm) is within the
reported values obtained for lipid bilayers and erythrocytes (r∗
= 0.3–0.5nm, , , ), while the value of ΔW0(46.6
kBT) is slightly larger (ΔW0= 40–45 kBT , , ).
Despite the fact that (5) is based on the theory of electropo-
ration, and could also describe the dependence between pulse
amplitude and duration on the whole interval of pulse parame-
ters, the logarithmic and power functions, given by (2) and (3),
might be more practical in determining equivalent pulse param-
eters. Namely, these two equations contain only two parameters
to be fitted, and are also relatively easy to evaluate computa-
tionally, but they can be applied only for pulses lasting at least
and pulse duration given by (6), when plotted as E2versus 1/tP,
should be piecewise linear . For his relatively short interval
(6) to the measured data. The fitted parameter a in E2= a/tP is 1) a =
681.4 V2·cm−2ms; 2) a = 6446.6 V2·cm−2ms; 3) a = 0.69 kV2·cm−2ms.
Note that tP at short pulses was replaced with tPeffaccording to (6b).
Relation between the E2
70and 1/tP. The curves are the fits of
of pulse durations (tP = 100 μs–16 ms), he could discern two
domains of pulse durations in which the data could be described
with (6). When our results in Fig. 2(A) were transformed to
comply with (6a) and (6b), we were able to discern three such
domains, which are shown in Fig. 6. The first domain contains
pulse durations up to few μs, the second domain pulses in the
range of few ms, and the third domain pulses longer than 10
ms. Due to the wide interval of pulse durations, the results are
appear bent. Compared to the results of Neumann, we obtained
somewhat different domains of pulse durations, which can be
wider range of pulse durations (150 ns to 100 ms) and also due
to the fact that our cell population was heterogeneous. The
existence of the third domain, for the longest pulse durations
electric pulses, such as electrophoresis or electroendocytosis.
We should mention that theoretical expressions (1)–(6),
which were fitted to our experimental data, were initially de-
rived to describe different experimental assessments. They can
be divided into three groups: 1) electroporation of cells: (1), (5),
(6) , , ; 2) inactivation of amoebae/killing of cells:
(2), (3) , ; and 3) uptake of molecules: (4) . Among
the investigated expressions, our data were in agreement with
theoretical expression (5) from group 1 and expressions (2) and
(3) from group 2. This is an interesting observation, since our
investigated parameter, the fraction of electroporated cells, was
used only in the models from group 1.
A similar mathematical fitting of expressions was also per-
formed for the dependence of pulse amplitude on the number of
pulses. In this case, a theoretical expression based on the model
of electroporation is not explicitly stated. However, it might
be possible to derive such an expression by using the equa-
tions for distribution functions for cell forming and resealing
times, given in recent papers of Saulis  , and following
the directions in the same papers. The four expressions (1)–(4)
do not have any physical background that would relate them
3286 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 11, NOVEMBER 2011
to electroporation meaning that they can also be used to de-
scribe the relation between amplitude and number of pulses.
The two-parameter power function (3) and the three-parameter
exponential function (4) seem to best describe the dependence
between the amplitude and the number of pulses. One drawback
of using a larger number of pulses is that such protocol requires
more time to perform the experiment. For example, delivering
time to perform an experiment increases even further if an array
of electrodes is used, where pairs of electrodes are sequentially
activated. Increasing the pulse repetition frequency would solve
the problem of excessive duration, but might at the same time
result in increased heating of the sample . Our experiments
do not show these relations since they were performed with
100 μs pulse duration and at 1Hz pulse repetition frequency
only. It is, therefore, possible that different functional relations
would be obtained with other combinations of pulse durations
and pulse repetition frequencies.
Another interesting finding originates from the functional re-
lations between pulse amplitude and pulse duration. Namely,
(2), (3), and (5) suggest that electroporation can be obtained at
any pulse amplitude provided that pulse duration is sufficiently
long. There have been many opposing reports in the literature
on the existence of the threshold pulse amplitude for electropo-
ration of cells. While experimental studies mostly reported that
electroporation could be obtained only with amplitudes above
a certain value –, theoretical studies, especially those
based on the theory of pore formation , , predicted that
electroporation is not a threshold phenomenon. Our results ob-
tained with small divalent Ca2+ions suggest that there might be
no threshold (see Fig. 2). On one hand, this might be explained
more related to processes such as electrophoresis or electrically
stimulated endocytosis, rather than to electroporation. On the
other hand, Ca2+ions are small compared to molecules used in
intense electroporation to cross the electroporated membrane,
which might be the reason for the observed threshold . This
was demonstrated by He and coworkers who investigated the
threshold values of the electric field for molecules of different
sizes and obtained higher thresholds for larger molecules .
In the last decade, electroporation with nanosecond electric
pulses has become increasingly widespread. Pulses in these ap-
plications can last less than 1 ns, while the number of pulses can
longer than 150 ns were considered. For even shorter pulses, the
readers should refer to a recent study from Schoenbach and
coauthors , where the authors derived the mathematical re-
lations between the pulse parameters and bioelectric effects of
Electroporation-based technologies and medical applications
and biomedicine , , –. Although the number of
ing the optimization of pulse protocols for specific application
are still open. Among them is determination of appropriate am-
plitude, duration and number of electric pulses that assure suc-
cessful application, or treatment with minimum possible side
effects. A review of the studies related to pulse parameters used
in different electroporation-based applications shows a number
of efficient combinations of pulse parameters. In our present
study, we demonstrated that the change in the value of a specific
pulse parameter can be compensated by carefully selected value
of the other parameter. In addition, we showed that the relation
between pulse parameters can be described by relatively simple
mathematical expressions, such as power, logarithmic, or ex-
ponential functions. On the basis of these functions, equivalent
pulse parameters that assure similar effectiveness of electropo-
ration can be selected. Such parameters can be extremely useful
in the process of electroporation-based treatment planning ,
where limitations of the electrical devices and position of the
electrodes have to be taken into account.
G. Pucihar would like to thank Dr. T. Kotnik for proofreading
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Gorazd Pucihar (M’10) was born in Ljubljana,
Slovenia, in 1976. He received the Ph.D. degree
in electrical engineering from the University of
Ljubljana, Ljubljana, Slovenia, and University Paul
Sabatier, Toulouse III, Toulouse, France.
His research work is focused on experimental in-
vestigation and numerical modeling of electropora-
tant Professor at the Faculty of Electrical Engineer-
ing, University of Ljubljana.
Jasna Krmelj was born in Ljubljana, Slovenia, in
1987. She is currently an undergraduate student of
biomedical engineering at the Faculty of Electri-
cal Engineering, University of Ljubljana, Ljubljana,
Matej Reberˇ sek was born in Ljubljana, Slovenia, in
neering from the University of Ljubljana, Ljubljana,
He is currently a Research Associate in the Labo-
ratory of Biocybernetics, Faculty of Electrical Engi-
electroporation devices and investigation of biologi-
cal responses to nanosecond electrical pulses.
and the Ph.D. degree in medical sciences from the
University of Ljubljana, Ljubljana, Slovenia.
She is currently a Research Associate in the Labo-
ratory of Biocybernetics, Faculty of Electrical Engi-
neering, University of Ljubljana. Her main research
interests include electroporation, especially in vitro
experimentation and investigation of biological re-
sponses to nanosecond electrical pulses.
Damijan Miklavˇ ciˇ c received the Ph.D. degree in
versity of Ljubljana, Ljubljana, Slovenia.
He is currently with University of Ljubljana as
a Full Professor and Head of the Laboratory of Bio-
cybernetics. Since 2007, he has also been the Head
of Department for Biomedical Engineering. In the
last few years, his research has been focused on
electroporation-based gene transfer and drug deliv-
ery, development of electronic hardware, and numer-
ical modeling of biological processes.