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Journal of Electrostatics 68 (2010) 261e274

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Modelling single cell electroporation with bipolar pulse parametersand dynamic pore radii

Sadhana Talele a,*, Paul Gaynor b, Michael J. Cree a, Jethro van Ekeran a

aDepartment of Electronic Engineering, University of Waikato, Private Bag 3105, Hamilton, New ZealandbDepartment of Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

a r t i c l e i n f o

Article history:Received 16 December 2008Received in revised form25 November 2009Accepted 15 February 2010Available online 19 March 2010

Keywords:ElectroporationAC pulsesPore radiiNumerical model

* Corresponding author.E-mail address: sadhana@waikato.ac.nz (S. Talele)

0304-3886/$ e see front matter � 2010 Elsevier B.V.doi:10.1016/j.elstat.2010.02.001

a b s t r a c t

We develop a model of single spherical cell electroporation and simulate spatial and temporal aspects ofthe transmembrane potential and pore radii as an effect of any form of applied electric field. The extent ofelectroporation in response to sinusoidal electric pulses of two different frequencies in a range of extra-cellular conductivity for two different cell radii are compared. Results show that pore radii tend to be morenormalized for AC fields. The relative difference in fractional pore area is reduced by the use of a 1 MHzsinusoidal applied electric field over a 100 kHz field.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Exposure of biological cells to electricfields can lead to a variety ofbiophysical and biochemical responses [1e7]. Electroporation (EP e

often also referred to as electropermeabilization) is a phenomenon inwhich applied electric field pulses create transient nanometre-scalepores in a cell membrane, when the transmembrane potentialexceeds a semi-critical value creating a state of high permeability[8,9]. These pores limit further growth of the transmembranepotential. The pores are long lived, sometimes surviving in themembrane for up to severalminutes, and providing pathways for themovementof large ions andhighmolecularweightmolecules suchasdrugs and DNA into the cell [8,9]. These properties have resulted inelectroporation as a common tool in biotechnology and are used inmanyapplicationsof biotechnology, biochemistry,molecular biology,medicine and other biological research [7e18].

Some of the applications are as follows.Electrochemotherapy (ECT): in cancer chemotherapy, some of the

drugs do not show their anti-tumour effects because of insufficienttransport through the cell membrane [7]. A combined use ofchemotherapeutic drugs and application of electric pulse is knownas electrochemotherapy and is useful for local tumour control.Especially, bleomycin has been reported to have shown a 700-foldincreased cytotoxicity when used in ECT [19,20]. This helps to

.

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achieve a substantial anti-tumour effect with a small amount of thedrug, which limits its side effects [21]. Bleomycin and cisplatin haveproven to be much more effective in electrochemotherapy than instandard chemotherapy when applied to tumor cell lines in vitro, aswell as in vivo on tumors in mice [22e24]. Clinical trials have beencarried out with encouraging results [11,13,25e28].

Electrogenetransfection (EGT): application of electroporation fortransfer of DNA into cells to effect some form of gene therapy, oftenreferred to as electrogenetransfection, is currently being applied insome clinical trials. It is presently considered to have large potentialas a non-viral method to deliver genetic material into cells, theprocess aimed at correcting genetic diseases [29].

Electrofusion (EF): under appropriate physical conditions, deliveryof electric pulses can lead to membrane fusion in close-contactadjacent cells. EF results in the encapsulation of both original cells’intracellularmaterial within a single enclosedmembrane and can beused to produce genetic hybrids or hybridomas [30]. Hybridomas arehybrid cells produced by the fusion of an antibody secreting stimu-lated B-lymphocytes, with a tumour cell that grows well in culture.The hybridoma is then able to continue to grow in culture, and a largeamount of specific desired antibodies can be recovered after pro-cessing. Electrofusion has proved to be a successful approach in theproduction of vaccines [31,32], antibodies [12], and reconstructedembryos in mammalian cloning [33].

Transdermal drug delivery (TDD): application of high-voltage pul-ses to the skin allows a large increase in induced ionic andmoleculartransport across the skin barrier [34]. This has been applied for

Table 1Geometric, electrical and electroporation parameters used in simulation for evolu-tion of pore radii.

Symbol Value Description

a 15.0 (mm) Cell radiusCm 10�2 (Fm�2) Specific membrane capacitance [41]h 5.0 (nm) Membrane thickness [9,37,64]gm 1.9 (Sm�2) Specific membrane conductance [9]Vrest �80 (mV) Membrane rest potential [9]sin 0.3 (Sm�1) Intracellular conductivity [37]sex 1.2 (Sm�1) Extracellular conductivity [64]r* 0.51 (nm) Minimum radius of hydrophilic pores [40]rm 0.76 (nm) Minimum energy radius at Vm¼ 0 [40]T 295 (K) Absolute room temperature [9]n 0.15 Relative entrance length of pores [9]b 2.46 Pore creation constant [9]Vep 258 (mV) Characteristic voltage of electroporation [9]N0 1.5� 109 (m�2) Initial pore density [9]w0 2.65 Energy barrier within pore [9]j 1� 109 (m�2 s�1) Creation rate coefficient [9]wst 1.4� 10�19 (J) Steric repulsion energy [40]wed 1.8� 10�11 (J m�1) Edge energy [40]x0 1� 10�6 (J m�2) Tension of the bilayer without pores [41]x0 2� 10�2 (J m�2) Tension of hydrocarbonewater interface [9]Fmax 0.70� 10�9 (N V�2) Max electric force for Vm¼ 1 V [9]rh 0.97� 10�9 (m) Constant [40]rt 0.31� 10�9 (m) Constant [40]D 5� 10�14 (m�2 s�1) Diffusion coefficient for pore radius [41]

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274262

transdermal delivery of drugs, such as metoprolol [35], and alsoworks for larger molecules, for example, DNA oligonucleotides [35].

Electroinsertion (EI): another application of electroporation isinsertion of molecules into the cell membrane. As the electric fieldinduced membrane pores reseal, they entrap some of the trans-ported molecules. Experiments on electroinsertion suggest thepossibility of using the process to study certain physiologicalproperties of these cells and understanding aspects of the lipid-protein interactions of the cell plasma membrane [36].

Electroporation is an important technique for transport ofmacromolecules such as genes, antibodies, and drugs, into a hostcell. For effective transport, the pores should have sufficiently largeradii, remain open long enough, and should not cause membranerupture. In applications such as electrofusion and in vivo electro-poration, it is often necessary that cells of different radii be elec-troporated. Until recently, the development of theoretical models ofelectroporation lagged behind experimental research. In order tooptimize the efficiency of electroporation, it is important toconsider as many biological and physical aspects as possible and itis a necessity that a variety of electric field pulse parameters betried. Thus, a comprehensive model which can predict electro-permeabilization as a result of any form of applied electric fieldpulse and other important electroporation parameters is necessary.

A number of models have been reported in literature. Kotnik et al.[37] analyse the time evolution of induced transmembrane potentialwhen a cell is exposed to different time varying electric field shapes[37]. Their model uses passive complex dielectric material basedLaplace Transform equations to describe the transmembranepotential of the model cell. While this model is capable of producingresults for arbitrary waveforms, it does not include the dynamiceffects of membrane electropermeablization on transmembranepotential and pore density. Another theoretical model has beendeveloped for single cell electroporationwhen exposed to a unipolarsquare shape electric field pulse [9]. Their model considers electro-permeabilization as a dynamic process and describes the evolution oftransmembrane potential and pore density, assuming a set of definedelectroporation parameters. While this model includes the non-linear effects ofmembrane breakdown, it is not able to accommodatearbitrary pulse waveforms.

Another recent and substantially novel model based on nodalcircuit analysis techniques includes both basic non-linear effects ofelectroporation and can provide results for arbitrary pulse wave-forms in both single cell and tissue-like systems [38]. This modelwas primarily created to investigate the general response of cells toelectric fields, so it does not specifically concentrate on the finertemporal or spatial components of electroporation.

Recent reported studies [9,39] model EP to calculate trans-membrane potential and the pore density. These models considerthe non-linear behaviour of EP, which is due to the dynamics ofmembrane pore formation and its effect on the transmembranepotential in turn. However, these models assume a single non-varying pore radius around the cell membrane. Other models[40,41] include spatial and temporal aspects of pore radius evolu-tion. The electric fields used in these models were limited to uni-polar DC pulses. However, oscillating pulse protocols have beenreported to be used in electroporation based applications due to itsadvantage over DC pulse protocol [42] especially for electro-porating cells with nonhomogeneous sizes. There is a need todevelop a model which can consider AC fields.

The study presented here develops amodel of single spherical cellelectroporation that simulates spatial and temporal aspects of poreradius as aneffect of anygiven formof applied electricfield (includingunipolar or bipolar), and other important electroporation systemparameters. In particular, we first model a single spherical cell of15 mm radius, under the application of a 2 ms DC unipolar pulse and

two-cycles of a 1 MHz sinusoidal AC (bipolar) pulse. The trans-membrane potential and pore radius at various polar angular posi-tionsabout the cellmembraneandas a functionof timearepresented.

We then consider two cell radii (15 mm and 7.5 mm) and calculatefractional pore area (FPA) to compare the extent of elctropermabili-zation at varying extracellular conductivities and the effect of higherfrequency sinusoidal AC pulses.

The simulation results are used to compare the extent of elec-troporation in response to sinusoidal AC electric field pulses of twodifferent frequencies in a range of extracellular conductivity for thetwo cell radii. The results show that the pore radii tend to be morenormalized (less widely spread) when an AC field is used whencompared to a DC field. It is also observed that a 1 MHz bipolarsinusoidal applied electric field pulse reduces the relative differ-ence in fractional pore area between the two cell sizes compared toa 100 kHz pulse. However, a significantly higher amplitude isrequired to create the same level of average fractional pore area.

2. Model of a single cell

Consider a spherical cell of radius a with intracellular conduc-tivity sin and specific membrane conductance gm, immersed inamediumof conductivity sex.When unacted upon the cellmaintainsthe membrane rest potential Vrest across its membrane. This systemis exposed to an externally applied time varying electric field E(t).Azimuthal symmetry about the axis of the applied electric field isassumed. The physical parameters used in the model and their cor-responding mathematical symbols are summarised in Table 1.

2.1. Transmembrane potential

It is assumed that the intracellular and extracellular regions arecharge free. The homogeneous externally applied electric field ofstrengthE(t) isusedasaboundarycondition, thepotential beingfixedat

fð3a; qÞ ¼ �3aEðtÞcos q; (1)

on a sphere of radius 3a surrounding the cell, where q is the polarangle. Since there are no free charges inside the region, thepotential obeys Laplace’s equation

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 263

V2f ¼ 0; (2)

except at the cell membrane at a radial distance of r¼ a, where thepotential is discontinuous because of the abrupt change inconductivity and the approximated infinitely thin membrane. Thusinternal and external potentials can be defined as,

Fðr; qÞ ¼�Finðr; qÞ 0 � r < a;Fexðr; qÞ a < r < 3a

: (3)

The current across the membrane is used to relate the internal andexternal potentials [9], that is,

�br,ðsinVFinða; qÞÞ ¼ �br$ðsexVFexða; qÞÞ ¼ CmvVm

vtþ Jm; (4)

wherebr is theunit outward radial vector,Cm is the specificmembranecapacitance of the cell, Jm is the current density at the cell membranedue to existing pores andVm is the transmembrane potential given by

VmðqÞ ¼ Finða; qÞ � Fexða; qÞ (5)

The current density Jm through the cell membrame consists ofthree terms, namely

Jm ¼ Jion þ Jsml þ Jlge; (6)

where Jion¼ gm(Vm� Vrest) is the ionic current density through theintact cell membrane [9], Jsml is the additional current density dueto the small (i.e. less than 1 nm radius) pores and Jlge is the addi-tional current density due to the larger pores (i.e. greater than 1 nmradius). The reasoning behind these two latter terms is explained inthe following.

It is assumed that on application of a sufficient external electricfield to the cell small pores are formed initially and that the currentdensity Jsml through these small pores is given by [9],

Jsml ¼ NismlðrÞ; (7)

where N is the pore density of the small pores formed and isml(r) isthe diffusion current through a single pore of radius r (true for poresless than 1 nm radius only). A previously derived expression for isml[9], based upon the NernstePlanck equation, is used for small pores,and is,

isml ¼pr2spsvmRT

Fh$

ðevm � 1ÞðG�evm � GþÞ; (8)

with

G� ¼ w0ew0�nvm � nvmw0 � nvm

: (9)

Here sps is the conductivity of the aqueous solution that fills thepore (approximated by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin sex

p), F is Faraday’s constant, R is the

universal gas constant, T is the absolute temperature, h isthe thickness of the cell membrane, n is the relative entrance lengthof the pore, w0 is the energy barrier inside the pore and vm is thenondimensional transmembrane potential [9] given by

vm ¼ Vm

�FRT

�: (10)

Eq. (8) for pore current isml accounts for the electrical interac-tions between the ions and the pore wall [9].

Initially formed small pores evolve in size if appropriate condi-tions exist. Now, assumeQ larger pores exist, theqth largerpore beingof radius rq with rq> 1 nm by definition. Each larger pore carriescurrent ilge through the membrane, thus the total current density Jlgethrough the Q larger pores is

Jlge ¼ 1 XQilge

�rq�; (11)

Aq¼1

where A is the corresponding cell surface area. For these largerpores, the current-voltage relationship assumes that the trans-membrane potential Vm occurs across the sum of pore resistance Rpand the series input resistance Rin [40,43], and is as follows:

ilgeðrÞ ¼ Vm

Rp þ Rin; (12)

where

Rp ¼ hpspsr2q

; (13)

and

Rin ¼ 12spsrq

: (14)

2.2. Formation of pores

Small pores exist with an initial pore density of N0 when there isno externally applied electric field. On application of the externalelectric field, small pores are assumed to be formed with theminimum energy radius rm¼ 0.76 nm and satisfy a pore density N(t) that evolves according to [9]

dNdt

¼ jeðVm=VepÞ2�1� N

NeqðVmÞ�; (15)

where j is the pore creation rate coefficient and Neq is the equi-librium pore density for a voltage Vm and is given by

NeqðVmÞ ¼ N0ebðVm=VepÞ2 : (16)

Here Vep is the characteristic voltage of electroporation and b isthe pore creation constant [44].

2.3. Evolution of pore radii

The pores that are initially created with minimum energyradius rm change in size to support minimizing the energy of theentire lipid bilayer [45]. The lipid bilayer energy wm is comprisedof four contributions: the electrostatic interaction between thelipid heads, pore edge energy, membrane tension and the electricforce acting on the pore, each of which are dependent on theradius of the pore. We examine each contribution in turn in thefollowing.

The lipid head groups which line the interior of a pore in the cellmembrame, tend to repel each other due to steric and/or electro-static interactions [45e47] contributing

wst

�r*rq

�4(17)

to the lipid bilayer energy. Here wst is the steric repulsion energyand r* is the minimum radius of a hydrophilic pores [44].

The appearance of a circular pore in a membrane is balanced bythe presence of two competing energy terms: reduction in theenergy barrier proportional to removal of pore area prq2 and increasein energy barrier by a linear edge component proportional to poreedge of length 2prq [48e50], thus contributing the two terms

2pwedrq � pxeff�Ap

�r2q (18)

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274264

to the lipid bilayer energy. Here,wed is the pore edge energy, and xeffis the effective tension of the membrane given by

xeff�Ap

� ¼ 2x0 � 2x0 � x0�1� Ap

A

�2; (19)

where x0 is the energy per area of the hydrocarbonewater interface[44,46], x0 is the tension of a membrane without pores and A is thetotal area of the lipid bilayer. A varying value of the total area Apoccupied by the pores at any given time is given by

Ap ¼XQq¼1

pr2q ; (20)

and contributes to a dynamic effect of pore formation bothtemporally and spatially.

The final contribution to the bilayer energy is due to themembrane potential [44]. Assuming the inner surface of a pore istoroidal [44,51], the electric force F acting on the pore is given by

Fðr;VmÞ ¼ Fmax

ð1þ rhrþrt

�V2m: (21)

This equation is a heuristic approximation [44] of the numericalsolution that the authors have computed for the electrical forceacting on a pore derived from first principles; here, rh and rt areconstants taken from reference [44]. This equation is thought to beappropriate for largerpores as it predicts that Fapproaches a constantvalue Fmax as the pore radius increases, rather than increase linearly[50], ordecrease to zero [52e54] as radius increases. The contributionto the bilayer energy is, of course, the integration of the force in thedirection of displacement.

Putting the above together, the lipid bilayer energy wm due to Qpores in the cell membrane is therefore given by

wm ¼XQq¼1

24wst

�r*rq

�4þ2pwedrq � pxeff

�Ap

�r2q

þZrq0

Fðr;VmÞdr35: ð22Þ

The rate of change of pore radii is dependent on the lipid bilayerenergy according to [40],

drqdt

¼ � DkT

vwm

vrq; q ¼ 1;2;.;Q (23)

whereD is the diffusion coefficient of the pore radius, k is Boltzman’sconstant and T is the absolute temperature. The numerical values ofthe various parameters used in the model are given in Table 1.

2.4. Limitations of the model

The model described in this chapter is capable of simulatingtransmembrane potential and pore radii formation/evolution ofa spherical cell when exposed to any form of electric field. Themodeldoes not include media permittivity. Including other shapes of cellsand the media permittivity would require substantial changes to themodel. The effect of other physical and chemical properties of elec-tropermeabilization system is not simulated. Although the electricalparameters have the greatest effect on electroporation dynamics,other properties may need to be included at some stage in an ulti-mate model of electroporation. Inclusion of such additional param-eters may provide additional information.

3. Numerical implementation

The model described above is implemented in MATLAB. Thefollowing description of model geometry is illustrated in Fig. 1.Azimuthal symmetry about the axis of the applied electric field isassumed. Extracellular space of thickness 2a is assumed. The intraand extracellular space is discretized using spherical coordinates,such that ri¼ iDr (i¼ 1,., 51) and qj¼ jDq (j¼ 0,.,24), thus Dr¼ 3a/51 and Dq¼ p/24. Due to azimuthal symmetry, discretization in q

results in 25 circular rings on the surface of the cell. Each circular ringis modelled as a single variable. Initially, each circular ring of the cellmembrane is assigned a pore density N0, of ‘small pores’, each of thepores having identical radius rm, the minimum energy radius. Eachring also has a transmembrane voltage Vm calculated as given below.

The evolution of pore radii is calculated as follows. Nj is the poredensity for the jth ring on the cell membrane. In accordance withEqs. (22) and (23), the pore radius evolves. An array of radii vari-ables, one element per large pore, keeps track of these larger poreradii. The algorithm computes drq=dt for the qth pore at every timestep. If this quantity is positive, small pores expand in size.

If dr/dt> 0, then for the jth ringwith area Aj, an integer number oflarge pore variables, namely floor(NjAj) are created. The small porepopulation is decreased by the corresponding number of large poresthat are created. To ensure that each pore is of a distinct value, thelarge pores are created at random radii normally distributed aboutthe minimum energy radius rm, with an extremely small standarddeviation (order of 10�20 m).

An explicit Euler method is used for solving the first orderdifferential equations.

The time step needs to bemuch smaller than the reciprocal of thehighest frequency used for simulation. All the results presented hereuse a time step of 2 ns, which is small enough to provide an accuracythat does not noticeably improve with smaller time steps [55].

For the calculation of Vm, the potential field F(r, q, t) is modelledin and around the cell as follows to calculate Fin and Fex.

The finite differencemethod is used to solve Laplace’s equation ina sphere surrounding the cell to give the electric potential. Whendiscretized on r and q as described above, Laplace’s equationbecomes a large set of linear (algebraic) equations relating the valuesof F at the discrete points. The external electric field is used asa boundary condition at r¼ 3a. Current across the membrane is usedas a boundary condition at the cell membrane.

Laplace’s equation written in spherical coordinates with theassumption of azimuthal symmetry can be reduced to,

v2F

vr2þ 2

rvF

vrþ 1r2

v2F

vq2þ cotq

r2vF

vq¼ 0: (24)

For points on the boundary of this region, at r¼ 3a, F is madeequal to the value set by the boundary condition. For interior pointsnot on the membrane, the derivatives are approximated by finitedifferences of the cell, which gives the system of linear equations,

vF�ri; qj

�vr

zF�riþ1; qj

�� F�ri�1; qj

�2Dr

(25)

v2F�ri; qj

�vr2

zF�riþ1; qj

�� 2F�ri; qj

�þ F�ri�1; qj

�Dr2

(26)

vF�ri; qj

�vq

zF�ri; qjþ1

�� F�ri; qj�1

�2Dq

(27)

v2F�ri; qj

�vq2

zF�ri; qjþ1

�� 2F�ri; qj

�þ F�ri; qj�1

�Dq2

: (28)

Fig. 1. Spherical cell model geometry showing the radius and angle discretization used in the numerical model.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 265

Substituting the discrete derivatives above into Laplace’s equa-tion (Eq. (2)) and simplifying gives,

� 2hDr2 þ r2i Dq

2iF�ri; qj

�þ riDq2½ri � Dr�F�ri�1; qj

�þ riDq

2½ri þ Dr�F�riþ1; qj�þ Dr2

1� 1

2cot

�qj�Dq

F

�ri; qj�1

�þ Dr21þ 1

2cot

�qj�Dq

F�ri; qjþ1

� ¼ 0: ð29Þ

In order to include the boundary condition involving the currentat the cell membrane, the radial derivatives of Fin and Fex must beestimated. Assume a is the index such that ra¼ a (that is, the radialindex that gives the point on the cell membrane, see Fig. 1).Consider v=vrFinðr; qjÞwhich is approximated at r ¼ a� 1=2Dr andr ¼ a� 3=2Dr by

vFinvr

����ða�Dr2 ;qjÞ

zFin

�ra; qj

�� Fin�ra�1; qj

�Dr

(30)

and

vFinvr

����ða�3Dr2 ;qjÞz

Fin�ra�1; qj

�� Fin�ra�2; qj

�Dr

(31)

respectively. We can estimate v=vrFinða; qjÞ by linearly extrap-olating from these two points out to the membrane itself, sothat

vFin���� z

vFin��� þ 1

�vFin

��� � vFin��� �

vr ða;qjÞ vr

��ða�Dr

2 ;qjÞ 2 vr

��ða�Dr

2 ;qjÞ vr

��ða�3Dr

2 ;qjÞ

z3Fin

�ra; qj

�� 4Fin�ra�1; qj

�þ Fin�ra�2; qj

�: (32)

2Dr

Similarly v=vrFexða; qÞ is computed at the membrane similarlyto give,

vFex

vr

����ða;qjÞz�3Fex

�ra;qj

�þ4Fex�raþ1;qj

��Fex�raþ2;qj

�2Dr

: (35)

All these finite difference equations along with the boundaryconditions are expressed as a system of linear equations. They aresolved at each time step, using the LU-decompositionmethod to givethe solution of the Laplace’s equation. The value of Vm is calculated asVm¼Fin�Fex at the membrane at each qj.

Depending on the parameter values a high number of poresmaybe formed. The total simulation time depends on the number oflarge pores created by the applied electric pulse. Strong pulses maycreate many pores, thus requiring long simulation times. To speedup computation in the event of such large numbers of poresforming (i.e. >104), pores are ‘created in groups’ of identical poreradius. A single radius variable describes the radius of all pores ina group, thus reducing the number of radius variables required. Thesize of this group is selected as required, based on the total numberof pores and number of variables the computer can handle.

Fig. 2. Transmembrane potential evolution at angles 0e60� around the cell membranefor a DC applied electric field pulse of 94.7 kV/m magnitude.

Fig. 3. Transmembrane potential evolution at angles 120e180� around the cellmembrane for a DC applied electric field pulse of 94.7 kV/m magnitude.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274266

3.1. Model validation

To confirm the accuracy and validate the model presented here,the simulations for the transmembrane potential Vm were initiallycompared with results obtained by calculating the transmembranepotential Vm using analytical solutions given by Holzapfel et al. [56].Solutions were obtained for applied bipolar sinusoidal pulse electricfields at amplitudes less than that required for onset of electro-poration, as the equations from [56] do not model the non-lineareffects of membrane electroporation. It was found that these resultsmatched very closely, to the extent that differences could only beobserved when plotting magnified waveform amplitude. Thisconfirms that the developed numerical model is a valid numericalcalculation technique for calculation of transmembrane potentialwhenbipolar sinusoidal pulses are applied.’ Themodel of Krassowskaand Filev [41] includes provision for non-linear electroporationdynamics including pore radius evolution to a unipolar electric fieldstep. The induced transmembrane potential on application of dcapplied field using the model presented in this paper is within 2% ofthat reported by [41], indicating that the pore dynamics includingnumber of pores and pore radii had been correctly calculated in themodel presented here.

Induced transmembrane potential and pore radii evolution toa bipolar electric field is not available in literature. The simulationresults of this paper present these.

4. Simulation results

Details of temporal and spatial evolution of transmembranepotential and pore radius for bipolar pulses have not been reported inthe literature to date. Thus, simulations are carried out for evolutionof transmembrane potential and pore radius for two applied electricfield pulses (DC andAC) for a single spherical cell. The time stepneedsto bemuch smaller than the reciprocal of the highest frequency usedfor simulation. All the results presented here use a time step of 2 ns,which is small enough to provide an accuracy that does not notice-ably improvewith smaller time steps [55]. Of particular interest is thecell charging time, pore creation time and pore radius evolution time[41]. Cell charging is the time fromfirst applicationof external electricfield to the formation of the first additional pore on the cellmembrane. Pore creation phase is time from the first pore to the timeof the last pore formed. The remaining time in the simulation is thepore radius evolution time.

4.1. Evolution of transmembrane potential, formation of pores, andpore radius evolution

Figs. 2e9 show the transmembrane potential and the poreradius evolution around the circumference of a 15 mm radius cell intwo individual cases: when a 2 ms, 94.7 kV/m DC electric field pulseis applied, and when a sinusoidal AC (1 MHz), 235 kV/m amplitudeelectric field pulse is applied for the same length of time. The peakamplitude (Ep) of the electric field in each case was set to providea terminal FPA at the end of 2 ms of approximately 0.015%(presented in [57] as being within good range of permeabilization).All the other parameters in the model are kept identical for bothsimulations.

Figs. 2 and 3 (DC case), and 6 and 7 (AC case) show the evolutionof the transmembrane potential at various positions around thecell. Figs. 4 and 5 (DC case) and Figs. 8 and 9 (AC case) show theevolution of pores at various positions around the cell. Each blackline represents evolution of the first few pores modelled individ-ually since there are only few additional pores formed at thebeginning. Each green (grey) line represents evolution of a group ofpores (groups of 20 pores in these simulations) asmany pores begin

to be created. The number in the right-hand corner of these figuresindicate the number of pores eventually formed at the particularpolar (q) position. At the angular positions not shown in thesefigures (q¼ 60e120�), there are no new pores created.

Fig. 4. Pore radius evolution at angles 0e60� around the cell membrane for a DC applied electric field pulse of 94.7 kV/m magnitude. Black lines represent individual pores, green(grey) lines represent groups of 20 pores.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 267

In the DC case, the time duration of 2 ms is chosen for the simu-lation so that the fine temporal features of the relationships betweenthe pore formation, pore radii evolution and transmembranepotential evolution can be seen. As observed in Figs. 4 and 5 the firstpore is formed at position q24 (at 0.42 ms) and the last pore for thisparticular pulse protocol is formed at position q6 (at 1.3 ms). Thisconstitutes thepore creationphase (0.42 mse1.3 ms). The initial period

when no pores are formed (0e0.42 ms) is the charging of the cellmembrane. Once transmembrane potential reaches a magnitude ofabout 1 V, pores start to form and expand/evolve. These time rangesfor charging and pore creation are specific to parameters chosen inthis simulation, and will vary with changes in the parameters.

In the case when an AC field is applied, as seen in Figs. 8 and 9,additional poresmay format the subsequent peakof transmembrane

Fig. 5. Pore radius evolution at angles 120e180� around the cell membrane for a DC applied electric field pulse of 94.7 kV/m magnitude. Black lines represent individual pores,green (grey) lines represent groups of 20 pores.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274268

potential. This happens if the transmembrane potential at theparticular position is beyond the threshold value. This increases thepore creation time as compared with DC pulses.

4.2. Frequency dependence of electropermeabilization ina heterogeneous population of cell radii/sizes

Owing to the variable nature of biological growth dynamics, allelectroporation applications treat cells of a range of sizes. Although

this range may only be in the order of a few percent, some applica-tionsmay have ranges exceeding 200 percent [58]. As such,whateverthe application, there may be a substantial benefit in being able tonormalize the degree of electropermeabilization (considered equiv-alent to the fractional pore area) with respect to cell radius/size.Earlier passive and dynamic modelling of bipolar electric fieldinduced transmembrane potential has indicated that as thefrequency of a constant amplitude applied electric field is increased,transmembrane potential reduces and becomes less dependent on

Fig. 6. Transmembrane potential evolution at angles 0e60� around the cell membrane for a two-cycle 1 MHz sinusoidal bipolar applied electric field pulse with 235 kV/m peakamplitude.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 269

cell radius [39,59]. In order to achieve a transmembrane potentialhigh enough for EP, the electric field amplitude must be increased athigher frequencies. To determine whether this effect translates todynamic electroporation modelling that includes pore radii vari-ability, two higher frequency electric field pulses of 100 kHz and

Fig. 7. Transmembrane potential evolution at angles 120e180� around the cellmembrane for a two-cycle 1 MHz sinusoidal bipolar applied electric field pulse with235 kV/m peak amplitude.

1 MHz were used for electropermeabilization simulation. Twosubstantially (yet realistically) different cell radii of 15 mmand 7.5 mmwere considered.

Figs.10 and 11 show the fractional pore area for 7.5 mmand 15 mmradius cells exposed to a two-cycle sine wave electric field pulse of100 kHz and 1 MHz respectively. The peak amplitude of the electricfield, Ep was set to provide an average terminal fractional pore area(for the two cell radii) of approximately 0.015% at a sex of 0.2 S/m. It isseen from Figs. 10 and 11 that the relative difference in FPA betweenthe two cell sizes at an extracellular conductivity of 0.2 S/m is 1.1 for100 kHz and 0.6 for 1 MHz. This relative difference increases as sexincreases to a limit of approximately 1. For a physiological mediumwith sex¼ 1.2 S/m (as found in in vivo EP applications), this relativedifference is 0.93 for the 1 MHz two-cycle sine wave electric fieldpulse and 1.0 for the 100 kHz electric field pulse when aiming ata mean fractional pore area of 0.015%.

5. Discussion

5.1. Evolution of transmembrane potential and pore radius

For a DC pulse protocol, it is seen that during the charging phase,transmembrane potential Vm has a larger magnitude at hyper-polarized pole (position q¼ 180�) as compared to the hypopolar-ized pole (position q¼ 0�) due to the negative membrane restpotential of �80 mV. As an effect of this, the poration on thehyperpolarized side occurs earlier than the hypopolarized side, inagreement with reported experimental results in the literature[60]. The model also predicts that the DC pulse creates more butsmaller pores on the hyperpolarized hemisphere (13,406 pores) asagainst fewer and larger pores on the hypopolarized hemisphere(8282 pores). Similar modelling predictions are reported by [41] inthe literature. This prediction agrees with the experimental resultsreported by [61], in which smaller molecules entered throughhyperpolarized end while larger molecules through the hypopo-larized end.

Fig. 8. Pore radius evolution at angles 0e60� around the cell membrane for a two-cycle 1 MHz sinusoidal bipolar applied electric field pulse with 235 kV/m peak amplitude. Blacklines represent individual pores, green (grey) lines represent groups of 20 pores.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274270

The number of pores at a particular position of the cell not onlydepends on the transmembrane potential, but also on the condi-tion of the remaining cell membrane. This is a plausible expla-nation as to why more pores are formed at q¼ 7.5� and q¼ 15� aswell as at q¼ 172.5� and q¼ 165� as compared to position q¼ 0�

and q¼ 180� respectively. The pores near to the non-polar regionstend to be larger in agreement with the modelling results pre-sented by [41].

5.2. Effect of field strength and frequency on pore dynamics

An important aimof this studywas to explore theeffects of bipolarpulse protocols and their parameters on electroporation. In generalthe peak transmembrane potential on the cell membrane increaseswith increasing unipolar field strength up to a maximum value ofaround 1.43 V. Transmembrane potential does not significantlyexceed this limit because the net amount of electroporation (which is

Fig. 9. Pore radius evolution at polar angles 120e180� around the cell membrane for a two-cycle 1 MHz sinusoidal bipolar applied electric field pulse with 235 kV/m peak amplitude.Black lines represent individual pores, green (grey) lines represent groups of 20 pores.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 271

measured by the final fractional pore area) depends sensitively ontransmembrane potential. The transmembrane potential increasesinitially by capacitive charging. Newpores in themembrane formandexpand/evolve due to electroporation and allow current to flow. Thisprevents further increase of the transmembrane potential.

Fractional pore area is also found to increase with increasingelectric field magnitude. This is because additional pores form untiltheir total area allows sufficient current to flow to stop the increase

of transmembrane potential. We also note that the radii of thelargest pores are smaller for stronger applied fields. The fractionalpore area increases in spite of this because, for strong fields, verymany smaller pores (radii in the range 1e5 nm) are formed.

Similar trends are seenwith bipolar (AC) pulse protocols at a fixedfrequency. However, to produce a given fractional pore area value,higher bipolar field strengths must be used compared with unipolar(DC) field strengths for the same overall pulse duration. More

0 0.05 0.1 0.15 0.20

0.005

0.01

0.015

0.02

0.025

0.03

Extracellular conductivity σex (S/m)

( aera erop lanoitcarFAPF

)%( )

cell radius of 15 μmcell radius of 7.5μm

Fig. 10. Fractional pore area versus extracellular conductivity for 7.5 mm, and 15 mm cellradius exposed to a two-cycle sinusoidal bipolar applied electric field pulse of 130 kV/mpeak magnitude at 100 kHz.

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274272

succinctly then, for a fixed electric field amplitude, fractional porearea decreases with increasing frequency, as does the total numberof pores. This is to be expected because electroporation can onlyoccur when the field strength exceeds a certain threshold. For

0 0.05 0.1 0.15 0.20

0.005

0.01

0.015

0.02

0.025

0.03cell radius of 15 μmcell radius of 7.5μm

( aera erop lanoitcarFAPF

)%( )

Extracellular conductivity σex (S/m)

Fig. 11. Fractional pore area versus extracellular conductivity for 7.5 mm, and 15 mm cellradius exposed to a two-cycle sinusoidal bipolar applied electric field pulse of 350 kV/m peak magnitude at 1 MHz.

bipolar (AC) fields at higher frequencies, the time intervals duringwhich electroporation can occur are clearly shorter.

For unipolar electric field pulses, the pores formed in electro-poration have widely distributed radii (around 1e8 nm), withsignificant contributions to the total fractional pore area from poreswith radii in different size-ranges. This is not the case for bipolarfields. For example, at 1 MHz, nearly all of the total fractional porearea is contributed by pores with radii 1e3.4 nm.

It is evident from Figs. 6 and 7 that, considering 0.015% fractionalpore area as the desirable fractional pore area, two-cycles of 1 MHzsinusoidal bipolar electric field reduces the relative difference infractional pore area between the two cell sizes compared to two-cycles of a 100 kHz bipolar electric field. However, a significantlyhigher amplitude is required to create the same level of averagefractional pore area. Lower values of extracellular conductivity arewarranted for better normalization of the degree of electroporation(fractional pore area), as higher values of extracellular conductivityincrease the relative difference in fractional pore area for cells ofdifferent radii/size. Since typical in vivo applications have an extra-cellular conductivity equal to the physiological conductivity in theregionof 1.2 S/m [37], it is unlikely that normalizationof thedegree ofelectroporation can be achieved in these applications through utili-zation of higher frequencyACpulses. Typically, at higher extracellularconductivity, the required magnitude of electric field also reduces.

6. Conclusion

The model is capable of simulating transmembrane potentialand pore radii formation/evolution when a cell is exposed to anyform of electric field. The results show that the pore radii tend to bemore normalizedwhen a bipolar sinusoidal AC field pulse is used ascompared to a DC field pulse of the same duration. As a result,larger pore radii formation is more likely when a DC unipolar fieldpulse is used. The results also show that better normalization of thedegree of electroporation, measured as fractional pore area FPA,especially at lower values of extracellular conductivity, can beachieved with use of higher frequency (1 MHz) bipolar electric fieldas compared to 100 kHz . Thus, choice of the electric field param-eters (magnitude and frequency) and the extracellular conductivityfor optimum electropermeabilization is a trade off betweenparameters like the desired pore radii values, and normalization offractional pore area between the required cell radii/size.

Themodelling results in this paper, especially the use of ac electricfield pulses, remain to be fully experimentally investigated. A recentpublication [62] demonstrates experimental results of electro-poration using ac electricfields and that efficient electroporationmaybe achieved using ac fields. This could be very well related to thesimulation result presented in Section 4.2, in which we have shownthat the relative difference in FPA between the two cell sizes (twosubstantially (yet realistically) different cell radii of 15 mm and7.5 mm) at an extracellular conductivity of 0.2 S/m is much lower for1 MHz electric field pulse, indicating that with proper choice ofexperimental parameters like extracellular conductivity andfrequency of electric field, many more cells can be electroporatedresulting in better efficiency of electroporation.

Analysis andmeasurements needhigh resolution both temporallyand spatially. Use of pulsed laser fluorescence microscopewith a submicrosecond imaging capability has been used by [63], although forDC electric fields. This method would be useful to observe trans-membrane potential formation with ac pulses as simulated in thispaper.

Pore size information can be inferred from selective moleculartransport across electroporated cell membrane as done by [61] fordc pulses. Through use of fluorescence measurement using flowcytometry [62], the level of molecular uptake by electroporated

S. Talele et al. / Journal of Electrostatics 68 (2010) 261e274 273

cells as well as % of cells electroporated in a heterogeneous mixtureof cell radii, can be measured. These measurements could berelated to the quantity FPA simulated in the paper presented here.

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