Ž .Bioelectrochemistry and Bioenergetics 47 1998 11–18

Development of a dosage method for electrochemical treatment oftumours: a simplified mathematical model

Eva Nilsson a,), Jaak Berendson a, Eduardo Fontes b

a ( )Department of Chemical Engineering and Technology, Applied Electrochemistry, Royal Institute of Technology KTH , S-100 44 Stockholm, Swedenb Eka Chemicals R&D, P.O. Box 13000, S-850 13 SundsÕall, Sweden

Received 2 April 1998; revised 17 June 1998; accepted 19 August 1998

Abstract

Ž .If the principal destruction mechanism behind electrochemical treatment ECT of tumours is related to the destructive reactionproducts formed in the electrochemical processes at the electrodes, a mathematical model of these processes should be a powerful tool indeveloping a reliable dosage method. In this present study, a simplified mathematical model of the electrochemical processes, occurringduring ECT, is presented. The model is based on electrode kinetics and transport equations of ionic species in dilute solutions. Theanalysis focuses on tissue surrounding a spherical platinum anode, which is treated as an aqueous solution of sodium chloride containing abicarbonate buffer system. The considered electrochemical reactions are chlorine and oxygen evolution, while the considered homoge-neous chemical reactions are the water protolysis and buffer reactions. The validity of the model is investigated by comparing simulatedpH profiles with pH profiles and lesion sizes, reported from in vivo experiments. This paper indicates that by putting a proper set of inputparameters into quite a simple mathematical model, it is possible to predict the size of a lesion produced through ECT. The model gives avery good qualitative and a fairly good quantitative description of the pH profile, obtained in tissue surrounding the anode after ECTtreatment. q 1998 Elsevier Science S.A. All rights reserved.

Ž .Keywords: Electrochemical treatment ECT ; Electrotherapy; Mathematical modelling; Tumour; Direct current

1. Introduction

Ž .The development of electrochemical treatment ECT oftumours as a clinically acceptable therapy, has been some-what hindered by uncertainties regarding the destructionmechanism of the tumours. This brings about the lack ofan effective dosage method. A number of different hy-potheses on the destruction mechanism behind ECT have

w xbeen proposed in the literature 1–6 . Still, there is nodoubt that the electrochemical processes at the electrodesform locally destructive reaction products, which are trans-ported into the tissue surrounding the electrodes.

When platinum is used as electrode material in biologi-cal tissue, the main reactions at the anode are oxygen andchlorine evolution:

2H O°O q4Hqq4ey 1Ž .2 2

2Cly°Cl q2ey 2Ž .2

) Corresponding author. Tel.: q46-8-790-65-06; fax: q46-8-10-80-87;e-mail: eva@ket.kth.se

The main reaction at the cathode is hydrogen evolution:

2H Oq2ey°H q2OHy 3Ž .2 2

Consequently, the dominating reaction products that arelocally destructive at the anode, are hydrogen ions andvarious oxygen and chlorine containing species. Hydroxylions and molecular hydrogen are the destructive reactionproducts at the cathode.

If the principal destruction mechanism behind ECT isrelated to the destructive reaction products formed in theelectrochemical processes at the electrodes, a mathematicalmodel of these processes should be a powerful tool in thedevelopment of a reliable dosage method. Such a modelshould be able to predict concentration profiles of sub-stances dissolved in the tissue, and the potential profilewithin the tissue itself.

w xBerendson et al. 7,8 have presented several estima-tions of the spreading of chlorine and hydrogen ions, intissue surrounding a platinum electrode, during ECT treat-ment. These estimations, as well as experimental observa-

w xtions 6,9 , indicate that the spreading of hydrogen ionsdetermines the extent and size of the destruction zone

0302-4598r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved.Ž .PII: S0302-4598 98 00157-3

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–1812

around the anode. However, these calculations must beconsidered as primary estimations, since the influence ofpotential field on electrode kinetics and spreading of ionicspecies was neglected.

In a previous article, a basic mathematical model of theelectrochemical processes, occurring during ECT, was for-

w xmulated and visualised 10 . The model focused on thetissue surrounding a spherical platinum anode, and thetissue was treated as an aqueous solution of sodium chlo-ride. In this paper, the basic mathematical model is furtherextended to take into account the limiting effect that abuffer system has on the spreading of hydrogen ions. As afirst approximation, only the bicarbonate buffer system isimplemented into the model. The validity of the model isinvestigated by comparing simulated pH profiles with pHprofiles and lesion sizes, obtained after ECT treatment of

w xmammary tissue in rats 9 . To additionally evaluate themodel, simulated pH profiles are compared with results

w xfrom an ECT study made by Samuelsson et al. 2 , inwhich lesion sizes in rabbit lung and liver tissues wereinvestigated.

2. Problem definition and basic assumptions

The analysis focuses on the tissue surrounding a spheri-cal platinum anode when applying a constant direct cur-rent. The cathode is assumed to be situated at a distancefar enough from the anode, to ensure that cathodic reactionproducts do not affect the tissue domain being studied. Thespherical geometry of the model and the symbols used to

designate its geometrical dimensions are shown in Fig. 1.The electrolyte domain, denoted as V , is bounded by innerand outer spherical boundary surfaces, with the radii ra

and r , respectively. The inner boundary, EV , representsr a

the surface of the spherical anode, while the outer bound-ary, EV , represents a spherical surface at a distance larger

enough from the anode to assure a constant concentration.Inputs to the model are the current density applied to

the anode, and the sizes of the anode and electrolytedomain. The tissue is treated as an aqueous solution ofsodium chloride containing a bicarbonate buffer system. Abicarbonate buffer system in an aqueous solution is charac-terised by the following reaction scheme:

kB1,fq yH qHCO ° H CO 4Ž .3 2 3kB1,b

kB2,f

Žaq .H CO ° CO qH O 5Ž .2 3 2 2kB2,b

kB3,f

Žaq . Žg .CO ° CO 6Ž .2 2kB3,b

where k , k , k and k , k , k are the rateB1,f B2,f B3,f B1,b B2,b B3,b

constants in the forward and backward directions, respec-tively. The bicarbonate system in a living body works asan open system. This basically means that the concentra-tion of dissolved CO is maintained to be constant in the2

body fluids. Any excess CO is taken up by the erythro-2

cytes in the blood. It is then transported through thesystemic capillaries to be finally released to the alveolargas, during the transit of blood through the pulmonarycapillaries.

Fig. 1. Spherical geometry of the model and symbols used to designate its dimensions. r is the space coordinate, r is the anode radius and r is the radiusa r

of the outer surface boundary.

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–18 13

If the tissue is subjected to an external acidic load, asoccurs in ECT treatment, excess Hq will react with HCOy,3

Ž . Ž .thus, shifting reactions 4 – 6 to the right. It has beenreported that the blood acid base status remains normal

w xduring ECT treatments 9,11 . This indicates that the ex-cess produced CO is not large enough to affect the2

overall CO concentration in the blood. However, the local2

concentration of CO is, during short periods, most likely2

larger than normal.Ž .In this model, it is assumed that the CO aq concentra-2

tion is kept constant throughout the solution domain. ThisŽ .assumption makes it possible to disregard reaction 6 in

the derivation of the model equations. The fact that the rateconstants k and k are very large—k being ofB1,f B1,b B1,f

the order of 5P1010 My1 sy1 and k of the order ofB1,b7 y1 w x1P10 s 12 —motivates the assumption that reaction

Ž .4 is close to equilibrium. In view of these assumptions,Ž . Ž .the buffer reaction schemes 4 – 6 can be reduced to the

over-all reaction:

kB,fq yŽaq .H qHCO ° CO qH O 7Ž .3 2 2

kB,b

with the following reaction rate, y :B

y sk c qc y yk c c 8Ž .B B ,f H HCO B ,b CO Žaq . H O3 2 2

were c denotes the concentration. The rate constants aregiven by k sk K and k sk where K isB,f B2,f B1 B,b B2,b B1

Ž .the equilibrium constant of reaction 4 .The bicarbonate concentration in the model is set to 27

mM. This choice of buffer concentration is justified by thefact that bicarbonate concentration in plasma and intersti-tial fluid is 25.8 and 27.1 meq dmy3 H O, respectively2w x13 . The total ionic concentration of the aqueous solutionis set to 0.16 M, and pH is set to 7.4. The consideredelectrochemical reactions are chlorine and oxygen evolu-tion, while the considered homogeneous chemical reac-tions are the water protolysis reaction:

HqqOHy°H O 9Ž .2

Ž .and the buffer reaction 7 . Furthermore, a number ofassumptions, apart from those mentioned above, are madein order to facilitate the solution of the problem. These

Ž .assumptions are: 1 Two mechanisms; diffusion and mi-gration, are assumed to contribute to the transport of the

Ž .ionic species. Convection is neglected. 2 The solutionnear the electrode is assumed to be saturated with respectto oxygen and chlorine. Small gas bubbles, with an inter-nal pressure of 1 atm containing either oxygen or chlorine,are assumed to be in equilibrium with the platinum elec-trode surface. The gas bubbles do not influence the con-

Ž .ductivity of the electrolyte. 3 The electrode kinetics arebased on reaction mechanisms proposed in the literature.The following mechanism is used to evaluate the electrode

w xkinetics of the oxygen evolution reaction 14 :

4 x MqH O°MOH qHqqey 10aŽ .Ž .Ž .2 ads

2 x 2 MOH °MOqMqH O 10bŽ .Ž .Ž .ads 2

2 MO°2 MqO 10cŽ .2

where M denotes the platinum electrode surface. It isŽ .assumed that reaction 10a is the rate determining step,

Ž .and that reactions 10b,c are close to equilibrium. Themechanism for the chlorine evolution reaction is repre-

w xsented by 15 :

2 x MqCly°MCl qey 11aŽ .Ž .Ž .ads

2 MCl °Cl q2 M 11bŽ .Žads. 2

Ž .Here, the discharge reaction 11a is assumed to be the rateŽ .determining step and reaction 11b is assumed to be close

Ž .to equilibrium. 4 Electro-osmotic effects are neglected.In reality, electro-osmosis can be disturbing as it causesloss of water around the anode. However, the effect can becounteracted by regular infusion of physiological sodiumchloride solution into this zone.

3. Model equations

The model is based on the transport equations of ionicspecies in dilute solutions and the equations of electrodekinetics. A detailed derivation of the basic transport equa-tions and electrode kinetics, applied in this model, is

w xpresented in an earlier article 10 . The model includes sixunknown parameters; namely, five concentrationsŽ . Ž .q q y y yc ,c ,c ,c ,c and the potential field F .Na H Cl OH HCO 3

Consequently, six model equations must be derived in theelectrolyte domain and boundaries.

3.1. Domain equations

Differential material balances are formulated for fourspecies:

E c zi i2sD= c q u =P c=F qR ,Ž .i i i i iE t zi

isHq,Cly,OHy,HCOy 12Ž .3

where

R qsk c qk c c yk c qc yH w ,b H O B ,b CO Žaq . H O w ,f H OH2 2 2

yk c qc y 13Ž .B ,f H HCO 3

R ys0 14Ž .Cl

R ysk c yk c qc y 15Ž .OH w ,b H O w ,f H OH2

R y sk c c yk c qc y 16Ž .HCO B ,b CO Žaq . H O B ,f H HCO3 2 2 3

Here, D denotes the diffusion coefficient, t the time, u theionic mobility, z the number of charges carried by an ion

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–1814

and R represents the production of a species through ahomogenous chemical reaction. k , k and k , kw,f B,f w,b B,b

denote the respective forward and backward rate constantsof water protolysis and bicarbonate buffer reactions, re-spectively. Conservation of electric charge and the condi-tion of electroneutrality form the two remaining domainequations:

5 52=P z u c=F q z D= c s0 17Ž .Ý Ýi i i i i iž /

is1 is1

5

z c s0. 18Ž .Ý i iis1

3.2. Boundary and initial conditions

Charge transfer and mass transport at the anode surfaceproceed at the same rate for all species involved in theelectrode reactions. For the other species, mass transport isequal to zero at the anode surface. Thus:

z n ii i j jyD = c Pn y u c =FPn sŽ . Ž .i i i iz n Fi j

isHq,Cly js I,II 19Ž .zi

yD = c Pn y u c =FPn s0Ž . Ž .i i i izi

isNaq,OHy,HCOy 20Ž .3

where n is the number of electrons, n is the outward unitnormal vector, n is the stoichiometric coefficient and F isFaraday’s constant. i denotes the current density con-j

tributed by reaction j. I represents oxygen evolution whileII denotes the chlorine evolution reaction. The currentdensity, contributed by oxygen evolution, is given by:

F FqEŽ .eq ,Ii s i exp yI 0,I ½ 5½ 2 RT

1F FqEŽ .eq ,I4

qy P C exp 21Ž .Ž .O H ½ 52 52 RT

while for chlorine evolution the analogous expression is:

F FqEŽ .eq ,IIyPi s i C exp yII 0 ,II Cl ½ 5½ 2 RT

1F FqEŽ .eq ,II2y P exp 22Ž .Ž .Cl ½ 52 52 RT

Here, Cscrcb and Psprpb, where p is the pressureand b denotes equilibrium. E is the potential differenceeq

between the solid and liquid phases, at equilibrium condi-

tions and relative to a reference, and is found throughNernst’s equation. i denotes the exchange current density,0

T the absolute temperature, and R the universal gasŽ . Ž .constant. Eqs. 19 and 20 and the electroneutrality con-

Ž .dition 18 provide the anode boundary conditions.At the outer boundary, concentration and potential gra-

dients are given by:

= c Pns0 23Ž .i

5iPnŽ .q z u c =FPn s0 24Ž . Ž .Ý i i iF is1

where i is the total current density.Initially, there are no concentration gradients through-

out the electrolyte, thus:

= c s0 at ts0 25Ž .i

and c sc0. The initial potential profile is solved from thei iŽ .domain equations and boundary conditions, using Eq. 25 .

4. Numerical solution and input data

The operator = is written in spherical coordinates, andby assuming rotational symmetry, the model is reduced toa one-dimensional problem with the space coordinate r.The space derivatives are approximated by the finite differ-ence method and the time derivative is approximated bythe trapezoidal rule. The obtained non-linear equation sys-

w xtem is solved by Newton’s method 16 , using sparsematrix routines included in MATLABw.

The very steep slopes of the potential and concentra-tions fields, close to the anode, give rise to numericalproblems. In order to avoid these, a new dimensionlessspace coordinate, x, is introduced:

r rr rxs ; 1FxF 26Ž .

r ra

The transformation leads to shorter step lengths close tothe anode and longer step lengths further out in the elec-trolyte domain. By this procedure, the finite differenceapproximation is able to describe the large gradients at theanode surface.

The time step length, D t, is chosen in order to producea stable numerical solution. The equations are discretisedonto a one-dimensional mesh, uniform in the space coordi-nate x. The mesh is chosen so that no grid points appearon the boundaries. Two imaginary grid points are intro-duced at one-half step lengths, D xr2, outside the bound-aries.

All the input parameters are given at 378C, since themodel results are to be compared with experimental resultsobtained in vivo. The initial concentrations of the ionicspecies, treated in this model, are shown in Table 1, andcorrespond to an aqueous solution with a total ionic con-

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–18 15

Table 1Initial concentrations used in the model

q q y y ySpecies Na H Cl OH HCO3

0 y7.4 y6.2c rM 0.16 1P10 0.133 1P10 0.027

centration of 0.16 M with pH 7.4. The initial hydroxyl ionw xconcentration is calculated using pK s13.6 17 . In orderw

to fulfil the electroneutrality condition, the exact initialsodium ion concentration is calculated from c0

qs0.16yNaŽ 0 0 .y qc yc .OH H

Thermodynamic and kinetic parameters used in themodel are listed in Table 2. The forward rate constant ofwater re-association is taken from the literature, while thebackward rate constant is obtained from k sw,b

k K rc . The forward rate constant of the bufferw,f w H O2

reaction is found in the literature and is valid in diluteaqueous solutions. The product k c is calculatedB,b CO Žaq .2

Ž .from the following equilibrium expression of reaction 7 ,at initial conditions:

k c0qc 0

yB ,f H HCO 3k c s 27Ž .B ,b CO Žaq .2 cH O2

Values for the diffusion coefficients at 378C were calcu-lated using the following relationship:

D m D m25 25 37 37s 28Ž .

T T25 37

with the given diffusion coefficients being those that occurw xin water at 258C 18 . The viscosities were estimated to be

y2 y2 w xm s0.890 mNs m and m s0.692 mNs m 19 .25 37

The ionic mobilities are related to the diffusion coeffi-cients through the Nernst–Einstein equation:

z Fiu s D . 29Ž .i iRT

The standard electrode potentials used in the Nernstequation, E0 and E0 , are given relative to SHE. TheI II

partial pressures of oxygen and chlorine are assumed to be

Table 2Kinetic and thermodynamic input values to the model

Parameter Value Referencey1 y1 11Ž . w xk M s 1.5P10 20w,fy1 y1 5Ž . w xk M s 3.1P10 21B,f

Ž . w xc M 55.5 20H O22 y1 y5Ž . w xqD cm s 1.78P10 18Na

2 y1 y5Ž . w xqD cm s 12.5P10 18H2 y1 y5Ž . w xyD cm s 2.72P10 18Cl

2 y1 y5Ž . w xyD cm s 7.05P10 18OH2 y1 y5Ž . w xyD cm s 1.49P10 18HCO 3

y2 y10Ž . w xi A cm 1.0P10 220,Iy2 y3Ž . w xi A cm 1.0P10 150,II

0 Ž . w xE V 1.23 18I0 Ž . w xE V 1.36 18II

constant and equal to 1 atm. The exchange current densi-ties are obtained from experiments run in aqueous solu-tions.

5. Results and discussion

In this section, the model presented in this paper isw xcompared with a model described in an earlier article 10 .

These two models are hereafter referred to as the bufferand the sodium chloride model, respectively. In addition,the validity of the buffer model is investigated by compar-ing simulated pH profiles with results obtained experimen-tally in vivo.

5.1. Comparison between the buffer model and the sodiumchloride model

Simulations have been run in order to visualise thebasic difference between the buffer and sodium chloridemodels. In the sodium chloride model, tissue is approxi-mated by an aqueous solution of 0.16 M sodium chlorideat pH 7.4. The thermodynamic and kinetic parameters,used in the simulations, are those presented in this article,i.e., the parameters valid at 378C. The geometrical dimen-sions of the anode and the electrolyte domain are r s1a

mm and r s6 cm. The total anode current density is setr

to 100 mA cmy2 .Fig. 2 shows the simulated pH profiles in the electrolyte

domain after different times of electrolysis. The two mod-els predict about the same pH value adjacent to the anodesurface. The movement of the acidic zone, in the tissuesurrounding the anode, is considerably impeded by thebicarbonate buffer system. The pH profiles predicted bythe buffer model are much steeper than the profiles ob-tained from the sodium chloride model. The steep profilesare caused by the rapid reaction between bicarbonate and

Fig. 2. Simulated pH profiles obtained from the sodium chloride modeland the buffer model, after 10 and 30 min of electrolysis. The total anodecurrent density is 100 mA cmy2 .

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–1816

Fig. 3. Concentration of sodium, hydrogen, chloride and bicarbonate ions,as a function of r in the electrolyte, obtained from the buffer model, after10 min of electrolysis. The total anode current density is 100 mA cmy2 .

hydrogen ions, and the continuous diffusion and migrationof bicarbonate ions towards the anode. The behaviour of

w xthe buffer model is similar to a shrinking core model 23 .In this case, the shrinking part is an outer shell rather thana core.

Fig. 3 shows the concentration profiles of sodium,hydrogen, chloride and bicarbonate ions, obtained from thebuffer model. The time of electrolysis in this simulation is10 min. The concentration profiles are similar to the ones

w xobtained in the sodium chloride model 10 . The differencebetween the models shows up in the reaction zone, atrs6 mm in this figure. This difference is introduced bythe steep bicarbonate profile caused by the buffer reaction,which influences the profiles of the other ions through theelectroneutrality condition.

5.2. Comparison between the buffer model and experimen-tal results obtained in mammary tissue

The buffer model is evaluated by comparing simulatedpH profiles with results obtained after ECT treatment of

w xnormal mammary tissue in rats 9 . In these experiments,constant direct current was applied to the tissue by means

Ž .of two spherical Pt:Ir 9:1 electrodes, with a radius of 0.5mm. The spheres were joined to electrically insulated Pt:Irsticks of a radius of 0.25 mm. Three different currents and

Ž .coulomb dosages current multiplied by treatment timewere used in the experiments, see Table 3. The current waslinearly raised from zero to operating current in 2 min, andwas similarly decreased to zero at the end of the treatment.The diameter of the spherical dark brown lesion, producedaround the anode, was measured directly after treatment.Subsequently, pH measurements were performed with amicro-combination pH glass electrode encased in a stain-less steel needle. The tip of the needle was 1.3 mm indiameter and measurements were made in the centre of thelesion, and at every other millimeter from this point, until

Table 3Ž . w xExperimental lesion radii mean"1 standard deviation, nG7 9 , simu-

lated pH values, at the radius corresponding to the border of the meanlesion radii, and simulated lesion radii, at different treatment conditions

Treatment Experimental Simulated pH at Simulatedconditions lesion radiusr experimental lesion radiusr

mm lesion border mm

1 mA, 5 C 2.7"0.5 2.5 1.52.5 mA, 5 C 3.5"0.4 2.0 3.52.5 mA, 10 C 4.3"0.4 2.1 4.05 mA, 2.5 C 2.4"0.2 1.5 3.45 mA, 5 C 3.4"0.3 1.7 4.35 mA, 10 C 4.4"0.3 1.8 5.2

Simulated lesion radii are obtained from simulated pH profiles, using thecondition that pH-2 gives lesions.

well out into the region of healthy tissue. The last pHmeasurement, in the tissue surrounding the anode, was runabout 10–15 min after current shutdown.

Simulations were run using current densities and treat-ment times identical to those used in the experiments.Linear current ramps were used at the beginning and endof the simulations. The geometrical dimension of the an-ode and the electrolyte domain, used in the simulations,are r s0.5 mm and r s6 cm, respectively.a r

Fig. 4 shows the simulated pH profile when using ananode current density of 85 mA cmy2 and a total treat-ment time of 1120 s. These conditions correspond to atreatment using 2.5 mA and 5 C. Also presented in Fig. 4are the experimental profile and the profile obtained fromthe sodium chloride model. As expected, the acidic zonearound the anode is considerably overestimated by thesodium chloride model. The buffer model, on the otherhand, predicts an acidic zone of about the same radius as

w xFig. 4. Comparison between simulated and experimental pH profiles 9 .The total anode current density is 85 mA cmy2 and the total time ofelectrolysis is 1120 s. These treatment conditions correspond to a currentof 2.5 mA and a coulomb dosage of 5 C. The experimental values aregiven as the mean"1 standard deviation and ns7.

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–18 17

that found in experiment. However, the simulated profile ismore acidic than the experimental profile. One reason forthis discrepancy may be that the pH measurements werenot enacted immediately after current shutdown. Duringthe period between current shutdown and pH measure-ment, species dissolved in tissue fluids are transported bymeans of diffusion. Accordingly, the pH profile, in tissuesurrounding the anode, continues to change with time,following the current shutdown.

In order to investigate the influence of diffusion on thepH profile, a simulation in which =F , i and i were setI II

to zero, after electrolysis, was run. Fig. 5 shows thesimulated pH profile as a function of time, followingcurrent shutdown. Also shown in the figure is the experi-mental pH profile. The pH increases continuously in theacidified zone surrounding the anode. The experimentalpH profile agrees fairly well with that of simulated pH,obtained after 10–15 min of diffusion. This result isconsistent with the fact that the last pH measurement wasdetermined about 10–15 min after current shutdown. Theslight difference in shape between experimental and simu-lated profiles, might be caused by an overestimation of thebuffer reaction rate and mobility of the ionic species. Intissue, species are transported mainly in the extracellularfluid which only constitute a fraction of the total volumeof tissue. The transport properties in tissue could possiblybe treated analogous to porous media by introducing effec-tiÕe diffusivities and mobilities.

Simulations were run in order to investigate the correla-tion between simulated pH profiles and experimental le-sion radii. These results are presented in Table 3. Thesimulated pH, at a radius corresponding to the border ofthe lesions, correlates well to a value between 1.5 and 2.5.Consequently, the simulations indicate that dark brownacidic heamatin, which defines the size of the lesions inthese experiment, is formed at this pH. This result isconsistent with an experimental observation reported by

Fig. 5. Simulated pH profiles as a function of time following currentshutdown: 0, 5, 10, 15, and 20 min. The initial pH profile is similarlyobtained from the buffer model as in Fig. 4. Also shown in the figure, is

Ž . w xthe experimental pH profile mean"1 standard deviation, ns7 9 .

Fig. 6. Simulated pH profiles as a function of coulomb dosage: 2.5, 5, 10,20, and 40 C. The total anode current density is 30 mA cmy2 .

w xIto and Hashimoto 24 . According to their study, acidicheamatin is formed at pH-2.

Ž .By using this condition pH-2 gives lesion in thesimulated pH profiles, the model’s ability to predict lesionradii is investigated. The simulated lesion radii are tabu-lated in Table 3. As can be seen from the table, a fairlygood agreement is obtained between simulated and experi-mental lesion radii. A slight discrepancy is found at low

Ž .and high current densities low and high currents . Onepossible explanation for the overestimation of the lesionradius at high current density is the effect of electro-osmo-sis. The electro-osmotic effect causes dehydration of tissuesurrounding the anode. This dehydration decreases theeffective transport properties of the ionic species. At low

Ž .current densities moderate pH close to the anode , produc-tion of Hq from the chlorine hydrolysis reaction:

Cl qH O°HClOqHqqCly 30Ž .2 2

becomes more prominent, and might have a significantinfluence on the pH profile.

Ž .Fig. 7. Simulated and experimental lesion radius individual data pointsw x2 as a function of coulomb dosage. The simulated lesion radii areobtained from the pH profiles shown in Fig. 6, using the condition thatpH-2 gives lesions.

( )E. Nilsson et al.rBioelectrochemistry and Bioenergetics 47 1998 11–1818

5.3. Comparison between the buffer model and experimen-tal results obtained in lung and liÕer tissues

To additionally evaluate the buffer model, simulated pHprofiles are compared with results from a study made by

w xSamuelsson et al. 2 , in which normal lung and livertissues in rabbits were treated with direct current. Theseexperiments were carried out with bullet-shaped platinumelectrodes of a 1.5 mm radii and 40 mm2 surface areas.The applied current varied between 6 and 12 mA duringeach of the treatments. Five different coulomb dosageswere used in the experiments: 2.5, 5, 10, 20, and 40 C.After each completed treatment, the width and length ofthe elliptical dark brown lesion, obtained around the an-ode, was measured.

The bullet-shaped anode is treated as a spherical anode,in the simulation, and the current density is set to 12r0.4s30 mA cmy2 . The geometrical dimension of the anodeand electrolyte domain, used in the simulations, are r s1.5a

mm and r s6 cm, respectively. Simulated pH profiles, inr

tissue surrounding the anode and at differing coulombdosages, are shown in Fig. 6. By assuming that a pH-2 iscausing formation of acidic heamatin, the model’s abilityto predict lesion sizes is evaluated. Predicted and measuredlesion radii, as a function of coulomb dosage, are pre-sented in Fig. 7. In this figure, a very good agreementbetween simulated and experimental lesion radii is ob-tained. Due to the fact that the experimental electrode isbullet-shaped and the model’s electrode is spherical, thecomparison between the experiments and simulation isslightly complicated. The simulated lesion radii lies be-tween the minor and major axis of the experimental ellipticlesions.

6. Conclusion

The results of this study indicate that by using a properset of input parameters, into quite a simple mathematicalmodel, it is possible to predict the size of a lesion pro-duced through ECT. Moreover, the model presented in thispaper increases the understanding of the transport andreaction processes of ionic species in ECT treatment. Themodel gives a very good qualitative and a fairly goodquantitative description of the pH profile, obtained intissue surrounding the anode, after ECT treatment.

Acknowledgements

This work was financially supported by The SwedishCancer Society. The working group for ‘Analysis of mech-anisms of electrochemical treatment of cancer’ includingRadiumhemmet, Huddinge University Hospital, and TheSwedish University of Agricultural Sciences, is acknowl-edged for their co-operation and valuable discussions.

Special thanks to Henrik von Euler for the unpublishedexperimental data. Philip Byrne is acknowledged for valu-able opinions on the manuscript.

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