Please cite this article in press as: A. Soba, et al., Integrated analysis of the potential, electric field, temperature, pH and tissue damage generated by different electrode arrays in a
tumor under electrochemical treatment, Mathematics and Computers in Simulation (2017), https://doi.org/10.1016/j.matcom.2017.11.006.

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Mathematics and Computers in Simulation ( ) –
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Original articles

Integrated analysis of the potential, electric field, temperature, pH
and tissue damage generated by different electrode arrays in a tumor

under electrochemical treatment

Alejandro Sobaa,∗, Cecilia Suárezb, Maraelys Morales Gonzálezc,
Luis Enrique Bergues Cabralesd, Ana Elisa Bergues Pupoe, Juan Bory Reyesf,

José Pablo Martínez Tassée

a CNEA, Argentina
b Instituto de Física del Plasma, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires - CONICET, Buenos Aires, Argentina

c Departamento de Farmacia, Facultad de Ciencias Naturales, Universidad de Oriente, Santiago de Cuba, Cuba
d Dirección de Ciencia e Innovación, Centro Nacional de Electromagnetismo Aplicado (CNEA), Universidad de Oriente, Santiago de Cuba, Cuba

e Departamento de Física, Facultad de Ciencias Naturales, Universidad de Oriente, Santiago de Cuba, Cuba
f ESIME-Zacatenco, Instituto Politécnico Nacional, Ciudad de México, DF 07738, Mexico

Received 15 March 2017; received in revised form 15 September 2017; accepted 16 November 2017
Available online xxxxx

Highlights

• Simulation tool of the partial differential equations of two dimensional models in cancer.
• Calculation and simulations of the potential, electric field intensity, temperature and pH in the tumor.
• Integrated analysis to select adequately an electrode array for tumors.

Abstract

The Electrochemical treatment can be used for local control of solid tumors in both preclinical and clinical studies. In this
paper, an integrated analysis of the spatial distributions of the electric potential, electric field, temperature and pH together with
the acidic and basic areas are computed, via Finite Element Methods, to improve the geometrical description of electrode arrays
for a better electrochemical treatment. These physical quantities are generated by different polarization modes and shapes of
electrode arrays. Additionally, the equations over a rectangular two-dimensional domain, which represents the tumor tissue, are
solved. The results demonstrate how the electric potential, electric field, temperature and pH distributions depend strongly on
the electrode array. Furthermore, significant pH changes and temperature increments are shown after 60 min of treatment. The
numerical results agree with the analytical ones reported in the literature. It is concluded that the numerical solution method permits
to make an integral analysis, prediction and rapid visualization of the most important electrochemical variables that take place in

∗ Correspondence to: Comision Nacional de Energia Atomica-CONICET, Av. Gral Paz 1499, Buenos Aires, Argentina.
E-mail address: soba@cnea.gov.ar (A. Soba).

https://doi.org/10.1016/j.matcom.2017.11.006
0378-4754/ c⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights
reserved.

http://www.elsevier.com/locate/matcom
https://doi.org/10.1016/j.matcom.2017.11.006
http://www.elsevier.com/locate/matcom
mailto:soba@cnea.gov.ar
https://doi.org/10.1016/j.matcom.2017.11.006


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2 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

tumor destruction, thus, providing the possibility of a more effective therapeutic planning before electrochemical treatment is
conducted.
c⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights

reserved.

Keywords: Electrode arrays; Electric field intensity; pH fronts; Tissue damage

1. Introduction

Electrochemical treatment (EChT) and electrochemotherapy (ECT) are simple, safe, effective, and minimally
traumatic for patients with cancer. Furthermore, these treatments induce few side effects in the organism and they
can be used when solid tumors cannot be resected after thoracotomy and are not responsive to chemotherapy or
radiotherapy, i.e., conventionally inoperable. EChT consists in the application of a low-level direct electric current to
the tumor through implanted electrodes [8,26]. ECT, on the other hand, is the electroporation, short electric pulses
of high-voltage delivery associated with chemotherapy used to increase drug uptake significantly into the cell by the
electro-permeabilization of its membrane [30,47].

These therapies are being intensively studied and applied at present, even at the clinical level. In order to analyze
and improve potential EChT applications in clinics, there are necessary deeper studies about basic mechanisms of
action as well as the mode of establishing dose–response relationships and confident outcome prediction. In addition,
different electrode placements are used and optimal electrode distribution has not been determined yet [41]. In this
sense, many authors address their efforts to propose two-dimensional (2D) and three-dimensional (3D) analytical and
numerical models in order to improve the geometrical description of electrode arrays [1,3,2,11,37,40,42,41,58].

2D and 3D models have mainly focused on the spatial distributions of the electric potential (Φ), electric field
intensity (E) and temperature (T ) generated by different electrode arrays [3,30,42,57]. Nevertheless, the adequate
selection of an electrode array is based on the analysis of Φ and E [3,42]; T [57]; or Φ, E and T coupled [30].
2D and 3D models have been indistinctly used in EChT and ECT and their results demonstrate that Φ, E and T
spatial distributions depend on tumor characteristics (i.e., size, shape and electrical properties) [25,27,35,51,56] and
therapy parameters electrode array geometry (number, shape, location and polarity of electrodes, voltage applied to
the electrodes (V o), number, duration and frequency of pulses) [14,26,29,30,44,56,58]. Tumor electric conductivity
increases with the electric field and temperature during EChT/ECT application [13,14,18,30,41,47].

The preclinical and clinical studies demonstrate that Φ, E and T are not only involved in the destruction of
solid tumor but also pH change around electrodes. This later has been accepted as the main EChT antitumor
mechanism [52]. Electrochemical process [32], protein electro-denaturation fronts [39] and pH fronts [48] during
EChT/ECT are reported.

The above-mentioned suggests that the individual analysis of Φ, E and T spatial distributions constitutes a
necessary but not sufficient condition for choosing a given electrode array. Both 2D and 3D analytical and numerical
models reported in the literature do not deeply discuss how the pH spatial distributions and the changes in time of the
acidic and basic areas depend on the electrode array shape.

The aim of this paper is to present a novel and integrated numerical model that describes how Φ, E, T and pH
spatial distributions, and acidic and basic areas change with the electrode array geometry during EChT and ECT
applications. For this, electrode arrays with circular, elliptic, parabolic and hyperbolic shapes are used, as reported
in [42]. Furthermore, Φ and E spatial distributions are compared with those previously reported in [3,42].

2. Model description

2.1. Model assumptions

It is assumed that:

1. 6 electrodes (three positive electrodes-named anodes and three negative electrodes-named cathodes) are placed
on electrode array with circular, elliptical, parabolic or hyperbolic shape (Fig. 1).



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 3

Fig. 1. Electrode configurations. (a) Conic sections: circle (e = 0), ellipse (e < 1), parabola (e = 1) and hyperbola (e > 1). (b) Electrode
arrays tested in the model. Circle (I), ellipse (II, e = 0.85), parabola (III, e = 1) and hyperbola (IV, e = 2). F, D, a, m, rn and θn are defined
in Section 2.4. d the smallest distance between two consecutive electrodes with alternate polarities. Positive and negative electrodes are specified
by the + and − signs, respectively.

2. The physical quantity Φ can be considered 2D away from the tips of the electrodes (long, uniform thickness and
fully inserted into the superficial tumor). Therefore, it is possible to examine Φ distribution on a plane [3,2,42].
Besides, E spatial distributions in 3D models with needle electrode arrays are similar to the values obtained in
2D [1].

3. A 2D conductive, linear, homogeneous, isotropic medium (tumor) is modeled. This assumption is a sim-
plification commonly accepted in this kind of models, as in other 2D [11,42] and 3D [30] models. The
tumor is considered as a single medium by two main reasons: first, the major alterations produced by
EChT and ECT occur inside it [25,32,51]. Furthermore, tissue damage occurs mainly around and between
electrodes, as in tumors [9,12,40,51] and potato [20,47]. Second, histopathological findings [25] and theoretical
simulations [2,26] reveal that the surrounding healthy tissue is minimally affected during and after EChT/ECT.
This is not significant for the treatment because E induced in this tissue is low [12,26,42] and the electric current
density in healthy tissue is smaller than 10 mA/m2 [26] due to its high electrical impedance.

4. Tissue relative electrical permittivity and the magnetic induction effect are neglected as non-dominant
effects.

5. The anode and cathode are considered H+ and OH− sources, respectively.

2.2. Electrode polarization modes

For simulations, three electrode polarization modes are used. Mode 0 consists in two positive upper electrodes (2
and 3), two negative bottom electrodes (5 and 6) and two neutral central electrodes (1 and 4) (see electrode numeration
in Fig. 1). Mode 1 is alternated polarization between consecutive electrodes (Fig. 1b). Mode 2 consists in three
consecutive electrodes with the same polarity and the other three with the opposite one. For parabolic/hyperbolic
arrays, mode 1 alternates polarization between electrodes. On the contrary, mode 2 keeps one polarization in one
branch of the parabola/hyperbola, and the opposite in the other branch.



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4 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

2.3. Electric potential, electric field, temperature and pH spatial distributions

Mode 1 is used for simulations of Φ (in V), E (in V/cm), T (in ◦C) and pH spatial distributions generated by these
four electrode array shapes for 1 h of treatment. The equations to calculate Φ (solution of Poisson nonlinear equation
for the entire region outside the electrodes), Eq. (1); E (gradient of scalar electric potential), Eq. (2); T (stationary
bioheat equation), Eq. (3) and pH (diffusion equation, see Eq. (4)) are given by

∇ • (σ (E, T )∇Φ) = 0 (1)
→

E = −∇Φ (2)

∇ • (k∇T ) − wbcbρb(T − Ta) + q ′′′
+ σ (E, T )|∇Φ|

2
= 0 (3)

∇ • (Di∇Ci ) =
∂Ci

∂t
(4)

with

σ (E, T ) = σ0 [1 + σE (E) + σT (T )] (5)

σE (E) =

⎧⎪⎪⎨⎪⎪⎩
3.5 E > Eirrev

1.0 + 2.5
E − Erev

Eirrev − Erev
Eirrev ≥ E ≥ Erev

1.0 Erev > E

(6)

and

σT (T ) = aT [T − Ta] (7)

where σ (E, T ) is the mean electric conductivity of the tumor during EChT/ECT [18] whereas σ 0 is its mean basal
electrical conductivity. σE (E) and σT (T ) are the average electrical conductivities due to E and T induced in this
tissue by EChT/ECT, respectively [47]. Erev (230 V/cm) is the reversible electric field intensity. Eirrev (350 V/cm)
is the irreversible electric field intensity [10]. aT (0.032 ◦C−1) is a temperature coefficient that depends on the
thermal characteristics of the material (a generic biological tissue in this case) [18]. Ta is the arterial temperature
(37 ◦C). k (0.564 W m−1 K−1), cb (3840 J kg−1 K−1), ρb (1039 kg m−3), wb (0.00715 s−1) and q ′′′ (10.437 W m−3)
represent the tissue thermal conductivity, specific heat capacity, mass density of the blood, blood perfusion rate and
the metabolic heat generation rate (all living tissue generates heat by the inner metabolic activity of the cells that
constitutes the tissue), respectively [18]. Di means the diffusion coefficient of the species i (H+ and OH−) whereas Ci

is the concentration of these species. t denotes the time. These two ionic species are selected because during electric
stimulation, the electrons react with water molecules at the cathodic side to produce hydroxyl ions, while at the anodic
side, protons are formed. Thus, gradients of H+ and OH− ions across the tissue are formed between the anodic and
cathodic interface.

For the simulations, the potential applied to the electrodes (V o) is 12 V (+6 V to anodes and −6 V to cathodes) [27].
V o is applied during 1 h. The distance d between adjacent electrodes is 0.5 cm, as in preclinical studies [18,44]
in order to compare our results with those reported in [42]. σ0 = 0.4 S/m [36], and C0

H+ = 1 × 10−7 mol/dm3,
C0

OH− = 1 × 10−7 mol/dm3, DH+ = 6.25 × 10−5 cm/s and DOH− = 3.52 × 10−5 cm/s [49] are used. Superscript 0
indicates initial concentration of specie i . All electrical conductivities are given in S/m.

The Pennes bioheat equation is the most accepted to calculate heat transfer in tissues [30]. Despite, Eq. (3) is used
because V o is direct and temperature transitory variations that appear during the first moments of therapy application
are neglected in regard to the thermal effects induced in the tumor by V o application. Moreover, variations of T
exhibit mainly due to the non-linearity and introduced in σ (E, T ) coefficients, as reported by Lacković et al. [30].
When time elapses, σ (E, T ) coefficients change by the tumor electrical property modifications because E and T are
modified due to V o application [18,47].

Eq. (4) considers hydrogen ions (H+) involve in the acidic pH and in water electrolysis (2H2O ↔ O2 +4H+
+4e−)

around the anode. Besides, this equation takes into account hydroxide ions (OH−) involve in the alkaline pH
and in water electrolysis (2H2O + 2e−

↔ H2 + 2OH−) around cathode, during EChT [32,36]. The acidic pH
and basic pH are explained by the formations of hydrochloric acid (H+

+ Cl− → HCl) and sodium hydroxide



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 5

(N+

a +OH−
→ NaOH), respectively. Sodium and chlorine ions are formed from the decomposition of sodium chloride

(NaCl → N+

a + Cl−) [32]. Mobilities and concentrations of H+ and OH− species modify their diffusion coefficients.
For each electrode array shape in mode 1 and for 1 h of EChT/ECT, pH spatial distributions around anodes

and cathodes are shown separately to examine how pH distributes around positive electrodes or negative electrodes.
Additionally, an average pH in each point (x, y) generated by all electrodes, for each electrode array shape, is given
to know how each tumor area is affected or not under EChT action. The tumor area is affected if 0 ≤ average pH ≤ 6
or 7 ≤ average pH ≤ 14. Conversely, the tumor area is not affected if 6 < average pH < 7.

2.4. Finite element methods

Finite Element Methods (FEM) are used to obtain the numerical solution of Eqs. (1)–(4) on a 2D rectangular
domain 10d ×5d/10d ×5d (where d is the electrode distance). The size of this rectangular domain is 5 × 2.5 cm and
guarantees that tumor boundary (Σ ) is far enough and does not disturb the solution of the problem. In this study, the
treated area (tumor tissue plus margin security area) is defined by the area covered by threshold E , pH or T values.
With the electrode configurations that are used here, tumor size is in the order of the square centimeter, a very realistic
tumor size [25]. This rectangular domain size is sufficient to know 2D spatial distributions of Φ, E, T and pH around
and between electrodes, as well as in zones away from them. It should be noted that all electrode arrays are inserted
inside this 2D rectangular domain. The mesh of 60 000 nodes is uniform with a mean size of element d/10 (0.05 cm).
The electrodes are considered as points because their diameters are lower than a node of the mesh and this FEM code
does not take into account the electrode radius.

The weak formulation is more general and allows an optimal treatment of the boundary conditions involving
derivatives. This numerical code is based on this formulation and allows treating Dirichlet conditions or other natural
ones easily. The Galerkin formulation is built on this weak formulation and an adequate mesh. Furthermore, this
weak formulation is applied to Eqs. (1), (3) and (4) to calculate Φ, T and Ci , respectively. This is important to
simultaneously know how these three quantities and E are affected for each electrode array shape. This allows an
adequate selection of electrode array shape. A set of standard discrete equations of the forms K (φi−1)φi

= f (φi−1)
(for Φ and T ) and Kφn+1

= Mφ̇ (for Ci ) results from this weak formulation. Eqs. (1) and (3) are coupled by the
Joule heating term (or heating power per unit volume, represented by σ (E, T )|∇Φ|

2) that appears in Eq. (3) [13,30].
Together with appropriate boundary and initial conditions, Eqs. (1) and (3) represent a complete formulation of the
problem in the form of two coupled partial differential equations. These equations are simultaneously solved by
the Picard iterative method [43,54]. Stable solutions for T and E are reached for seven iterations. When σ (E, T )
increases with T (see Eqs. (5) and (7)), as in other works [30,36], additional coupling between Eqs. (1) and (3) exists
(see Eqs. (5)–(7)). Eq. (4) is solved by the Crank–Nicolson iterative method [16], reaching its stable solution for seven
iterations.

Eqs. (1)–(4) are coupled because V o appears at their prescribed electrode boundaries. Eqs. (1) and (2) are solved for
prescribed voltage boundary conditions on the electrodes and insulating boundary conditions on the outer edges of the
domain. Eq. (3) is solved under boundary conditions of null flux and parameters reported in previous studies [18,53].
On the other hand, Eq. (4) is solved under boundary conditions of null flux together with the parameters mentioned
above [49]. Acidic and basic pH values are calculated in each finite element on the basis of concentrations H+ (−log10
[H+]) and OH−(−log10 [OH−]), respectively.

Pupo et al. [42] use the unifying principle for conic sections to calculate the leading-order potential (Φ0(z)) and
the leading-electric field intensity (E0(z)) in an equidistant distribution of electrodes over a conic curve, given by

Φ0 (z) =

N∑
n=1

Cn ln
(

a
z − zn

)
, (8)

E0 (z) =

N∑
n=1

Cn

(
a

z − zn

)
, (9)

with

zn = rneiϕn (10)

rn =
me

1 ± e cos θn
, (11)



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6 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

where N represents the total number of electrodes placed on the array and z is the position of the point where the
calculations are made. a is the electrode radius. zn is the position of the nth electrode of the array. m is the distance
between the focus (F) and the directrix line (D), as shown in Fig. 1a. The straight line passing through F and
perpendicular to D is assumed to be the prime direction, from which the angles are measured. rn and θn are the
polar coordinates of the nth electrode (with the origin on the point F). The parameter e is the conic eccentricity that
distinguishes the type of conic section: e = 0 (the locus is a circle), e < 1 (the locus is an ellipse), e = 1 (the locus is
a parabola) and e > 1 (the locus is a hyperbola) [42].

In Eqs. (8) and (9), Cn are constants to be determined from the boundary conditions, which may be determined
by solving non-homogeneous system of linear equations (see details in Aguilera et al. [3] and Pupo et al. [42]).
Consequently, Cn depends on potential on the j th electrode ( j = 1, . . . , N ). Besides, in Eq. (8), a constant term is
added if the number of electrodes is odd to satisfy conservation of the current [11]. rn can easily be shown to have a
general expression in polar coordinates similar in the three curves if the origin of coordinates is located in the conic
focus. It should be remarked that in the case of a hyperbola this equation represents only one of its branches (the
one whose focus is at the origin), rather than the entire curve. The plus sign corresponds to the left branch of the
hyperbola [42].

Aguilera et al. [3] proposed a complex variable (z = x + iy) and demonstrate that the total electric potential can be
written as a sum of multi-poles of all electrodes. The higher terms in multipole series can be neglected as the distance
between electrodes is larger than the electrode radius. So, Φ0(z) and E0(z) can be used. These authors also showed
that the multipole series for the electric potential and the electric field agree well.

To test the validity of this numerical code, Φ and E spatial distributions are compared with Φ0(z) and E0(z)
reported by Pupo et al. [42] (for four electrode array shapes and mode 1). Besides, Φ and E are compared with Φ0(z)
and E0(z) reported by Aguilera et al. [3]. This comparison is made for the electrode arrays with circular (e = 0) and
elliptical (e = 0.85) shapes, and mode 0, and along three different paths: (1) through electrodes 6 and 2; (2) through
electrodes 4 and 1; and (3) through electrodes 3 and 2 (Fig. 1). For Φ0(z) and E0(z) simulations, a = 0.215 mm is
considered [3].

2.5. Relative acidic and basic areas

The relative acidic area around anode (ratio of the tumor area with pH ≤ 6 relative to the total tumor area, Aacidic)
and the relative basic area around cathode (proportion of the tumor area with pH ≥ 8 relative to the total tumor area,
Abasic) are computed for estimating the tumor necrosis area induced by anodes and cathodes. Aacidic vs. t (Aacidic-t) and
Abasic vs. t (Abasic-t), during 1 h of EChT/ECT, are also shown to determine how the tumor necrosis area changes over
time during therapy.

2.6. Other simulations for the four electrode arrays in modes 1 and 2

The maximum T (Tmax) versus V o and maximum E (Emax) versus V o generated by different electrode array modes
are studied. Electrode array modes are circular mode 1 (A1), circular mode 2 (A2), elliptic mode 1 (A3), elliptic mode
2 (A4), parabolic mode 1 (A5), parabolic mode 2 (A6), hyperbolic mode 1 (A7) and hyperbolic mode 2 (A8) arrays.
These modes are compared by means of Tmax versus V o, Emax versus V o, Aacidic-t and Abasic-t. These simulations and
comparisons are made for 1 h of EChT/ECT and varying V o in the experimental range (2 to 12 V) [27,56].

2.7. Comparison criteria

The maximum difference (Dmax) and the Root Means Square Error (RMSE) are used for the comparison, given by

Dmax = max |Fi − G i | (12)

RMSE =

√ M∑
i=1

(Fi − G i )
2

M
. (13)

M (number of points), G i and Fi depend on the comparison performed. When the results of the FEM code are
compared with those reported by Pupo et al. [42], M = 33 000 points inside each electrode array and G i and Fi are



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 7

the i th calculated values of Φ/E and Φ0(z)/E0(z), respectively. For the case that the data obtained with the FEM code
and those reported by Aguilera et al. [3] are computed, M = 100 and G i and Fi are the i th calculated values of Φ/E
and Φ0(z)/E0(z) along each path, respectively. When Tmax vs. V o plot generated by the four electrode array shapes
for modes 1 and 2 are compared among themselves, M = 6, G i are the i-th calculated values of Tmax for an electrode
array and mode given, whereas Fi are those for another electrode array shape and mode specified. This same analysis
is also made for Emax vs. V o, Aacidic-t or Abasic-t plot generated by these four electrode arrays and two modes. In these
three cases, G i and Fi have the same interpretation but for Emax, Aacidic or Abasic.

Expressions to calculate the values of Φ, E,Φ0(z), E0(z), T, Ci , pH, Aacidic, Aacidic-t, Abasic, Abasic-t, Dmax and
RMSE, and the Picard and Crank–Nicolson iterative methods are implemented by an own code written in Fortran
90/95 language (Fortran 90/95 Programming Manual, Tanja van Mourik, 2005) and executed on a I7-class computer
with 8 threads and 8 Gb of Ram memory. Each calculation takes five to ten minutes. For the calculations, millimeter,
centimeter and decimeter unities are converted in meter. Moreover, the matrix of Φ, E, T and pH for each electrode
array are processed in a 256-core processor HPC with 256 GB RAM, to show their spatial distributions. The figures
are shown in GNU Octave 4.0 (free software, License 2015-05-29). As part of the GNU Project, GNU Octave is free
software under the terms of the GNU General Public License. The website is http://gnu.org/software/octave. Free
Software Foundation funds the GNU Project. Developer(s): John W. Eaton and many collaborators. This machine is
acquired by the Flemish Development Cooperation through VLIR-UOS (Flemish Interuniversity Council-University
Cooperation for Development of Belgium) in the context of the Institutional University Cooperation programme with
Oriente University, Santiago de Cuba, Cuba.

3. Results

Figs. 2–5 show Φ (Figs. 2a–5a), E (Figs. 2b–5b), T (Figs. 2c–5c), basic pH around cathodes (Figs. 2d–5d),
acidic pH around anodes (Figs. 2e–5e) and average pH (Figs. 2f–5f) spatial distributions generated by circular,
elliptical (e = 0.85), parabolic and hyperbolic electrodes arrays, respectively. All these distributions are highly non-
homogeneous in the tumor and deeply depend on electrode configuration. These figures evidence that higher modular
values for Φ, E (45 ≤ E ≤ 50 V/cm) and T (7 ≤ T ≤ 52 ◦C) are observed at and around the electrodes and these
decrease among electrodes. pH reaches its smallest and greatest values at and around anode (pH ≤ 3) and cathode
(pH ≥ 12), respectively. Interestingly, for all electrode arrays, Aacidic (pH ≤ 6) and Abasic (pH ≥ 8) extend through the
tumor area, being more noticeable for Aacidic. For the tumor regions away from the electrodes (toward to Σ ), Φ and E
tend to zero, T decreases up to its physiological value (approximately 37 ◦C), and pH values tend to neutrality (nearly
7).

Φ and E spatial distributions obtained with the use of FEM code agree with those reported by Aguilera et al. [3],
Pupo et al. [42], for all 0 ≤ e ≤ 2 and spatial position along paths tested. Fig. 6 shows this good agreement for
e = 0.85 and three different paths. For the comparison of Φ and Φ0(z), Dmax = 0.0751 V and RMSE = 0.0513
V (paths through electrodes 6 and 2, Fig. 6a); Dmax = 0.0920 V and RMSE = 0.0096 V (paths through electrodes
4 and 1, Fig. 6c); and Dmax = 0.0004 V and RMSE = 0.0003 V (paths through electrodes 3 and 2, Fig. 6e). For
the comparison of E and E0(z), Dmax = 0.0060 V/cm and RMSE = 0.0049 V/cm ((paths through electrodes 6
and 2, Fig. 6b); Dmax = 0.0005 V/cm and RMSE = 0.0003 V/cm (paths through electrodes 4 and 1, Fig. 6d); and
Dmax = 0.0002 V/cm and RMSE = 0.0001 V/cm (paths through electrodes 3 and 2, Fig. 6f). All these comparisons
reveal small values of Dmax and RMSE. In addition, it can be verified that small values of Dmax and RMSE are obtained
for any paths between two electrodes when e = 0.85 and electrode arrays with 0 ≤ e ≤ 0.99 (results not shown).

For four electrode arrays and modes 1 and 2, Tmax and Emax values around electrodes non-linearly and linearly
increase as V o increases, respectively (results not shown). Below 4 V, these four electrode arrays generate the same
Tmax values. For V o ≥ 6 V, elliptical mode 2 and parabolic mode 1 arrays induce the lowest Tmax values while the
circular mode 1 and the hyperbolic mode 2 arrays induce the highest ones. The elliptical array produces the smallest
Emax values for all V o values, and parabolic and hyperbolic arrays mode 2 generate the highest ones. It can be seen
in Tables 1 and 2 that Tmax, Emax, Aacidic and Abasic depend on V o, electrode array shape and polarization mode for 12
V and 1 h of EChT/ECT. Table 2 shows that parabolic mode 2 and circular mode 2 electrode arrays induce the highest
and lowest relative areas, respectively.

Aacidic-t (Fig. 7a) and Abasic-t (Fig. 7b) increase with the time (during 1 h of treatment), for all electrode arrays and
modes 1 and 2, being more noticeable for Aacidic obtained with parabolic and hyperbolic arrays in both modes 1 and
2. Circular and elliptical arrays for mode 2 induced the lowest values of Aacidic and Abasic. These results confirm that

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Please cite this article in press as: A. Soba, et al., Integrated analysis of the potential, electric field, temperature, pH and tissue damage generated by different electrode arrays in a
tumor under electrochemical treatment, Mathematics and Computers in Simulation (2017), https://doi.org/10.1016/j.matcom.2017.11.006.

8 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

Fig. 2. Spatial distributions of (a) electric potential; (b) electric field intensity; (c) temperature; (d) basic pH around cathodes; (e) acidic pH around
anodes and (f) average pH generated by 12 V during 1 h of EChT (ECT) for a circular array of six electrodes in the mode 1.

Aacidic-t proved to be wider than Abasic-t and agree with those shown in Figs. 2d–5d. Furthermore, Fig. 7 reveals that
Aacidic-t turns out to be faster than Abasic-t. The comparison of these four electrode arrays for modes 1 and 2 shows small
values of Dmax and RMSE. Although the results are not shown, Aacidic, Abasic, Aacidic-t and Abasic-t increase non-linearly
as V o increases.

4. Discussion

The main merit of this study is the development of an integrated analysis of the coupling of V o with Φ; E; T ;
mobility and concentration of H+ and OH− ions; pH; σ (E, T ); Aacidic; Abasic; Aacidic-t and Abasic-t parameters; and
their effects on cellular death, unprecedented in the literature. From experimental point of view, this integrated analysis
requires an excessive handling of animals, expensive resources and a considerable amount of time. The results of this
paper are valid for a single stimulus of EChT/ECT, four shapes of electrode arrays (circular, elliptical, parabolic
and hyperbolic) and three electrode polarization modes. These agree with those reported in [3,42] and COMSOL-
Multiphysics. In addition, these theoretical results confirm that 2D spatial distributions of Φ, E, T and pH depend on
electrode array shape, as verified theoretically [42,41] and experimentally in potato [20,47] and tumors [12,40].



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tumor under electrochemical treatment, Mathematics and Computers in Simulation (2017), https://doi.org/10.1016/j.matcom.2017.11.006.

A. Soba et al. / Mathematics and Computers in Simulation ( ) – 9

Fig. 3. Spatial distributions of (a) electric potential; (b) electric field intensity; (c) temperature; (d) basic pH around cathodes; (e) acidic pH around
anodes and (f) average pH generated by 12 V during 1 h of EChT (ECT) for an elliptical array of six electrodes in the mode 1.

Despite the vast difference of Φ, E, T and pH spatial distributions generated by these electrode array shapes, their
respective Aacidic and Abasic are nearly the same, suggesting that they can be interchangeably used in EChT and ECT
in modes 1 and 2. In contrast, electrode arrays with circular and elliptical shapes and electrode hyperbolic array [42]
have been suggested to treat cancer. Nevertheless, we should be very careful with these statements because Aacidic and
Abasic are not reported after EChT treatment. A further study is required to describe Aacidic-t and Abasic-t during and
after exposure time of EChT. Significant destruction of potato [20] and tumor [8,9,12,25,40,44,52,51,56] is observed
after EChT application. Additionally, this potato destruction is also observed when ECT is conducted [40,47].

Acidic pH around anode, basic pH around cathode and unaffected tumor areas away from the electrodes agree with
the experimental results [32,51]. It has been experimentally demonstrated that tumor areas that have not been affected
by EChT/ECT cause the tumor to re-grow. These areas can be avoided by the use of multiple needle electrodes, as
reported in [1,26,29]. Besides, the fact that Φ and E values tend to zero, and T and pH reach their physiological values
for regions near Σ explains the minimal adverse effects observed in the surrounding healthy tissue [25,56]. This fact
justifies, in part, why a single medium is used, in addition to arguments given in the assumption 3.



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tumor under electrochemical treatment, Mathematics and Computers in Simulation (2017), https://doi.org/10.1016/j.matcom.2017.11.006.

10 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

Fig. 4. Spatial distributions of (a) electric potential; (b) electric field intensity; (c) temperature; (d) basic pH around cathodes; (e) acidic pH around
anodes and (f) average pH generated by 12 V during 1 h of EChT (ECT) for a parabolic array of six electrodes in the mode 1.

As a result of V o application, mobility and concentration of H+ and OH− ions may be due to water electrolytic
decomposition around electrodes [21,28,32] and/or water self-dissociation from dissociation–association processes
(2H2O → H3O+

+ OH−), where H3O+ is the hydronium ion [4–7,46]. The formation of H3O+ and OH− ions
has been suggested when EChT is applied [21]. Disruption of the water structures between and within cells by
necrosis due to EChT/ECT action may be involved in these results, in agreement with Davidson et al. [15]. This
fact is in correspondence with the inflammation and edema observed during EChT application and the first days
post-treatment [8,9,25].

The induction of H3O+/H+ ions and the hydrochloric acid around anodes may explain Aacidic. However, the
formation the OH− ions and sodium hydroxide around cathodes justify why Abasic, in agreement with [32,42,48,50].
Aacidic and Abasic may be influenced by the flux of interstitial water from anode to cathode and other charged ionic
species involved in the complex electrochemical processes induced in the tumor. This flux may cause dehydration at
the anode and hydration at the cathode, as reported in [12,32].

Electrode polarization may influence over Aacidic and Abasic by the change in the concentration of the electroactive
species at surface electrode, over-potential (by transfer and slowness of the charge transfer process through electrode



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 11

Fig. 5. Spatial distributions of (a) electric potential; (b) electric field intensity; (c) temperature; (d) basic pH around cathodes, (e) acidic pH around
anodes and (f) average pH generated by 12 V during 1 h of EChT (ECT) for a hyperbolic array of six electrodes in the mode 1.

interface), reaction over-potential (by the existence of different chemical reactions that happen in the tumor), oxide
metal and organic substance layers absorbed over the electrodes [24,28]. These results may be argued because the
increase of the concentration of the substance that oxidize in the anodic surface make that the electrode potential
of the anode increases and the decrease of the concentration of the substance that reduces at the cathodic surface
make the electrode potential of the cathode decreases. In addition, protons under influence of the electric field and the
concentration difference should move from anode to cathode.

Stilinger [46] report a significant declination of OH− (hydrated OH− (OH−(H2O)n)) and an increase of H+

(hydrated H+ (H+(H2O)n)) when these two types of ions act anomalously under a perturbation (i.e., E, T and
pH induced by V o application), probably by association–dissociation processes in water. Besides, it well known
that the size, concentration and mobility of H3O+/H+ ions are smaller, higher and faster than those of OH− ions,
respectively [50], explaining why Aacidic is higher than Abasic and Aacidic-t is faster than Abasic-t. These two results
confirm the inverse ratio between concentrations of H+ and OH− ions ([H+][OH−] = Kw = 10−14) and agree with
those reported by Olaiz et al. [37]. The fact that the acidic area prevails may impact in antitumor effect during and



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tumor under electrochemical treatment, Mathematics and Computers in Simulation (2017), https://doi.org/10.1016/j.matcom.2017.11.006.

12 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

Fig. 6. Solutions obtained with this numerical code (dotted lines) and those reported by Aguilera et al. [3] (solid lines) generated by an electrode
elliptical array (e = 0.85) in mode 0, for the electric potential (V , left) and electric field intensity (V/mm, right), as function of spatial position
along three different paths: (a–b) through electrodes 6 and 2; (c–d) through electrodes 4 and 1 and (e–f) through electrodes 3 and 2. Electrode
numeration is shown in Fig. 1. Voltage applied to the electrodes is 12 V for 1 h of treatment. The results of Aguilera et al. [3] contain 401 points
and those obtained with this numerical code contain 301 points for Φ and Φ0(z) (Fig. 6a,c,e), and 100 points for E and E0(z) (Fig. 6b,d,f).

after EChT application [52] and in both inflammatory and immune responses [23,31], thus corroborating that a correct
manipulation of the tumor pH may have important therapeutic implications, as McCarty and Whitaker [34] reported.

Induction of Φ, E, T and pH in the tumor first produce microscopic damages (at cellular, molecular and electronic
levels) that are subsequently observed at macroscopic level (Acidic and Abasic). Besides, increase of Aacidic-t and Abasic-t
generated by all electrode arrays agree with the increased space-time necrosis area reported in tumors after EChT
treatment [8,9,25,52,51] and in potato during EChT [20] and ECT [47], suggesting the validity of these theoretical
results.

The temperature T induced in the tumor is due to Joule effect [30,47]. This Joule heating explains the skin erythema
in patients [25]. This erythema differs completely from the burn produced by radiotherapy and/or hyperthermia [30].
However, Cury et al. [12] conclude that EChT antitumor effect is by electrochemical processes and not by temperature.
Artemov et al. [4–6] suggest that concentrations of H3O+ and OH− do not depend on T . On the other hand, pH
decreases and the constant equilibrium for water solution (Kw) increases non linearly when T increases. As a result,
the dimensionless activity of the species H3O+, OH− and H2O depends non-linearly on T [7].

Non-linear rise of Tmax, linear increase of Emax as well as non-linear space-time enlargement of Aacidic and Abasic
as V o increases may be connected with non-linear increase between cell death around electrodes and V o [19,56],
together with the increase of tumor conductivity [18,26]. The increase of the tumor conductivity brings about



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 13

Table 1
Dmax and RMSE values resultant from comparison by pairs of all electrode arrays for: maximum temperature (Tmax) vs. applied voltage (V o),
maximum electric field intensity (Emax) vs. V o, relative acidic tumor area vs. time and relative basic tumor area vs. time.

Comparison between
electrode array modes

Tmax vs. V o Emax vs. V o Aacidic-t Abasic-t

Dmax (◦C) RMSE (◦C) Dmax (V/cm) RMSE (V/cm) Dmax RMSE Dmax RMSE

A1–A2 2.82 1.381 2.61 1.663 0.07 0.051 0.03 0.024
A1–A3 1.82 0.898 2.10 1.329 0.04 0.029 0.02 0.017
A1–A4 4.50 2.235 9.20 5.955 0.03 0.023 0.02 0.018
A1–A5 4.74 2.325 3.71 2.402 0.03 0.024 0.01 0.004
A1–A6 3.64 1.968 3.70 2.405 0.03 0.023 0.01 0.004
A1–A7 0.26 0.160 2.71 1.767 0.06 0.046 0.03 0.022
A1–A8 0.33 0.198 2.71 1.753 0.06 0.045 0.03 0.022
A2–A3 1.00 0.493 0.50 0.341 0.11 0.079 0.06 0.040
A2–A4 1.68 0.856 6.61 4.292 0.04 0.028 0.01 0.006
A2–A5 1.92 0.944 6.31 4.064 0.01 0.074 0.04 0.028
A2–A6 1.07 0.625 6.31 4.067 0.01 0.074 0.04 0.028
A2–A7 3.08 1.532 5.31 3.428 0.13 0.096 0.06 0.045
A2–A8 3.15 1.574 5.31 3.415 0.13 0.096 0.06 0.045
A3–A4 2.68 1.343 7.12 4.627 0.07 0.052 0.05 0.034
A3–A5 2.92 1.432 5.80 3.731 0.01 0.006 0.02 0.012
A3–A6 1.82 1.089 5.80 3.733 0.01 0.009 0.02 0.012
A3–A7 2.08 1.049 4.82 3.095 0.03 0.017 0.01 0.007
A3–A8 2.15 1.090 4.82 3.082 0.02 0.017 0.01 0.007
A4–A5 0.24 0.104 12.90 8.357 0.06 0.047 0.03 0.022
A4–A6 0.86 0.353 12.90 8.359 0.06 0.047 0.03 0.022
A4–A7 4.76 2.387 11.91 7.721 0.09 0.069 0.06 0.039
A4–A8 4.83 2.429 11.91 7.708 0.09 0.068 0.06 0.039
A5–A6 1.10 0.449 0.03 0.012 0.00a 0.002 0.00c 0.000e

A5–A7 5.00 2.476 1.00 0.636 0.03 0.023 0.03 0.018
A5–A8 5.07 2.517 1.00 0.650 0.03 0.022 0.03 0.018
A6–A7 3.90 2.122 1.00 0.639 0.03 0.023 0.03 0.018
A6–A8 3.97 2.164 1.00 0.652 0.03 0.022 0.03 0.018
A7–A8 0.10 0.051 0.05 0.029 0.00b 0.001 0.00d 0.001

Ai-A j (i ̸= j) represents the comparison by pairs of all electrode arrays used. A1, A2, A3, A4, A5, A6, A7 and A8 depict the circular mode
1, circular mode 2, elliptic mode 1, elliptic mode 2, parabolic mode 1, parabolic mode 2, hyperbolic mode 1 and hyperbolic mode 2 arrays,
respectively. Tmax and Emax are the maximum temperature and electric field generated by these electrode arrays. Abasic-t is the time dependence of
relative basic area and
a 0.008.
b 0.006.
c 0.0007.
d 0.002.
e 0.00001.

Table 2
Relative acidic (pH < 6) and basic (pH > 8) tumor areas for 12 V and
1 h of EChT generated by the different electrode arrays and application
modes tested.

Mode types Shapes of electrode arrays

Circular Elliptic Parabolic Hyperbolic

Mode 1 0.2301 0.2523 0.2651 0.2424
Mode 2 0.1921 0.2100 0.2657 0.2424

the increase of EChT effectiveness [25,26] and electrode polarization [35]. The chemical interaction between the
electrolyte molecules and the water ions happens to breakdown the electrophoretic effect, thus activating the water
ions for the DC conductivity. Besides, the increase of the electric current with voltage is explained from the increase
of this physical quantity [38].



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14 A. Soba et al. / Mathematics and Computers in Simulation ( ) –

Fig. 7. (a) Aacidic (acidic area) versus t (time) and (b) Abasic (basic area) versus t induced in the tumor by electrode arrays with circular, elliptical,
parabolic and hyperbolic shapes, for modes 1 and 2, and 12 V during 1 h of EChT (ECT).

In this paper, the occurrence of a stochastic resonance is not discarded; this happens for weak inputs, nonlinear
systems that involve endogenous noise sources and the existence of thresholds of certain parameters [17,22]. This
hypothesis may be argued because solid tumors are treated with small values of V o (weak input: 6 ≤ V o ≤ 12
V). They are nonlinear systems that involve endogenous noise sources by fluctuations and intrinsic instabilities [33].
Besides, thresholds of T (≥ 38 ◦C) [30,57], E (≥ 350 V/cm) [10], current density (≥ 1 000 mA/m2) [26] and pH (pH
≤ 4 around anode and pH ≥ 10 around cathode) [12,32] may be induced in the tumor, depending on EChT doses and
electrode parameters.

Furthermore, noise in the tumor may be determined by multiple collisions of charged chemical species (i.e., ions,
electrons, molecules) with neutral water molecules. These chemical species are accelerated by the electric field
induced in the tumor, mainly around electrodes. A possible mechanism to explain these collisions is from the
formation and disintegration of protonated water clusters, which takes place by adding or subtracting one water
molecule [55]. This may explain, in part, the hydration and dehydration of the water around anode and cathode,
respectively [32].

The above-discussed confirms the essential role of pH in the tumor necrosis, indicating that σ (E, T ),Φ and E are
also influenced by pH. As a result, σ (E, T ) in Eqs. (3) and (5) should be replaced by σ (E, T , pH). This means that
2D and 3D models (realistic or not) should include σ (E, T , pH), to design adequate electrodes arrays, which are used
in EChT, ECT and hyperthermia.

The underlying weakness of this numerical code is based on the use of a 2D model with lack of more realistic
conditions for tissue properties. Nevertheless, 2D models are used by several reasons: (1) they are feasible for the
integrated analysis of Φ, E, T and pH spatial distribution generated by these four shapes of electrode arrays. E spatial
distributions in 3D models with needle electrode arrays are similar to the values obtained in 2D [1,20,41]; and both
size and spatial distribution of tumor necrosis are similar around electrode for different very thin cuts of a tumor under
EChT [12,25,52,51] or ECT [40] action. In addition, Φ, E, T and pH spatial distributions are difficult to measure.

Although in the literature 3D anatomically realistic models solved with COMSOL-Multiphysics and similar
packages have been proposed [58], in this study, tissue realistic conditions are not considered because the electrical
and biological parameters of the tumors cannot be controlled by the performing physician. Besides, preclini-
cal [32,36,40,44] and clinical [25,27,56] studies suggest that V o strength, electrode array geometry and electrode
positioning are more important in EChT effectiveness than realistic tissue properties (electrical and anatomic). On the
other hand, the spatial pattern of tissue damage in 3D potato pieces (heterogeneous and anisotropic biological tissue)
adopts the same geometry of the electrode array (collinear, circular, elliptical, parabolic and hyperbolic shapes) [20].
This later is theoretically predicted in 2D and 3D homogeneous, isotropic and conductor media [15,46]. It is important



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A. Soba et al. / Mathematics and Computers in Simulation ( ) – 15

to point out that although a realistic model is not considered in this study, the mentioned in this paragraph does not
mean that tissue characteristics and parameters of electrode arrays do not affect cancer cure planning.

On the other hand, as a tumor is heterogeneous and isotropic, its bulk electrical conductivity is a tensor (3 × 3
symmetric matrix) [35]; however, it may be diagonalized (averaged conductivity between three principal axes) [45]
and therefore σ 0 may be used. The FEM code is valid when the surrounding healthy tissue (of mean basal electrical
conductivity, σ 1) is also considered. Therefore, it is important to take into consideration that at Σ , Φ must satisfy the
boundary conditions to ensure the continuity of the solution [26]. Moreover, the integrated analysis for 3D electrode
arrays on realistic or nonrealistic tissues may be extended.

All in all, the integrated analysis of Φ, E, T and pH proves the FEM code effectiveness and it can improve
understanding of electrode arrays, it can help the interpretation of experiments and it can allow the estimation of
magnitudes that are difficult to measure experimentally, all of which are essential to maximize the tumor destruction
with minimum damages to the organism.

In conclusion, the integrated analysis of the electric potential, electric field intensity, temperature, pH and
acidic/basic areas permits to know how they depend on electrode array shape and polarization modes. This is vitally
important to improve the geometrical description of electrode arrays for a better electrochemical treatment. This
analysis should be included for any 2D or 3D model.

Acknowledgments

We would like to give our special thanks to the Editor in Chief and reviewers of this article for their expert help
and invaluable feedback. In addition, the authors are grateful to Adrián Cabrera Ricardo, Eduardo Roca Oria, Rosa
Ivette Robles Matos and Yenia Infante Frómeta for their useful discussions on the subject. This work was partially
supported by the International European Cooperation in Science and Technology (COST Action TD1104) and the
Cuban Ministry of Higher Education (# 6.176). Juan Bory Reyes is partially supported by the EZIME-Zacatenco,
Instituto Politécnico Nacional in the framework of SIP programs. Dr. Alejandro Soba and Dr. Cecilia Suárez are
researchers at CONICET.

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http://refhub.elsevier.com/S0378-4754(17)30374-9/sb57
http://refhub.elsevier.com/S0378-4754(17)30374-9/sb58
http://refhub.elsevier.com/S0378-4754(17)30374-9/sb58
http://refhub.elsevier.com/S0378-4754(17)30374-9/sb58

	Integrated analysis of the potential, electric field, temperature, pH and tissue damage generated by different electrode arrays in a tumor under electrochemical treatment
	Introduction
	Model description
	Model assumptions
	Electrode polarization modes
	Electric potential, electric field, temperature and pH spatial distributions
	Finite element methods
	Relative acidic and basic areas
	Other simulations for the four electrode arrays in modes 1 and 2
	Comparison criteria

	Results
	Discussion
	Acknowledgments
	References