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Electric field computation and measurements in the electroporation of inhomogeneous samples

  • Alessia Bernardis , Marco Bullo , Luca Giovanni Campana , Paolo Di Barba , Fabrizio Dughiero , Michele Forzan , Maria Evelina Mognaschi EMAIL logo , Paolo Sgarbossa and Elisabetta Sieni
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

In clinical treatments of a class of tumors, e.g. skin tumors, the drug uptake of tumor tissue is helped by means of a pulsed electric field, which permeabilizes the cell membranes. This technique, which is called electroporation, exploits the conductivity of the tissues: however, the tumor tissue could be characterized by inhomogeneous areas, eventually causing a non-uniform distribution of current. In this paper, the authors propose a field model to predict the effect of tissue inhomogeneity, which can affect the current density distribution. In particular, finite-element simulations, considering non-linear conductivity against field relationship, are developed. Measurements on a set of samples subject to controlled inhomogeneity make it possible to assess the numerical model in view of identifying the equivalent resistance between pairs of electrodes.

Keywords: Clinical electroporation; electric field model; equivalent circuit; resistivity identification; potato samples
PACS: 87.50.C; 07.05.Tp

1 Introduction

Electrochemotherapy (ECT) is a local cancer therapy that uses voltage pulses. Voltage pulses generate an electric field that reversibly permeabilizes cell membranes. This way the chemotherapeutic drugs uptake is improved [1, 2, 3]. This therapy is currently applied to skin tumors, such as melanomas, or chest wall recurrence of breast cancer [4, 5].

The electric field is applied to the patient according to standard procedures, by means of needle electrode, formed e.g. by 7 or 8 needles, that covers a surface of about 1-2 cm2 [2, 3]. ECT standard protocols uses 8 voltage pulses per electrode pair (rectangular pulses, 100 µs long at 5 kHz) in order to generate an electric field. The local strength of the electric field in the tissue depends on tissue inhomogeneity. In fact, in the treatment area the electric field intensity is homogeneous if electrical properties of the tissue are homogeneous. Conversely, if the electrical properties of the tissue are inhomogeneous, the electric field intensity depends on the local electrical resistivity. This behavior is not unusual in tumor tissues where both myxoid and fibrotic components can be found in the tumor area [6, 7]. Moreover, in clinical practice the two needles of a pair can be inserted in tissues with different electrical resistivity.

This paper systematically investigates the problem of the discontinuity of the tissue electrical properties by means of numerical simulations and experiments on suitable phantoms. Phantoms are used to design different electrical properties, varying the electrical resistivity. Finally, the electrical resistance is evaluated both experimentally, starting from voltage and current measurements, and using Finite Element simulations.

2 Measurement procedure

The inhomogeneous phantom used in experiments is obtained using materials with different electrical resistivity. One of these materials is a slice of potato. This tuber is used in order to show the electroporation occurrence. In fact, it is well known that potato becomes dark just few hours after cell electroporation [8, 9, 10, 11, 12] if the tuber is preserved at room temperature. Using voltage and current data recorded by means of the pulse generator, the sample resistance is computed [6, 7]. The results obtained with the experimental tests are compared with numerical results obtained using a 3D model.

2.1 Simulation model

The numerical model used in simulations is shown in Figure 1, and is composed of an internal parallelepiped (1×3.5×5 cm) divided into two sub-volumes immersed in a second parallelepiped (1×5.5×7 cm). The electric field is generated between two cylindrical electrodes 9 mm long, with diameter 0.6 mm, separated by d =7.3 mm. The two internal sub-volumes are characterized by an electrical resistivity ρi(E) dependent on the tissue type. The two needle electrodes are positioned in the potato sample and in the gel, respectively. A voltage of 730 V is applied between the needle pairs by means of Cliniporator EPS01-3.x version (Igea Clinical Biophysics S.p.A., Carpi, Modena Italy). The generator load can be designed as a resistive network schematized in Figure 2.

Figure 1 3D geometries: (a) three volumes with needles in potato and gel and (b) two volumes with both the needles in potato.
Figure 1

3D geometries: (a) three volumes with needles in potato and gel and (b) two volumes with both the needles in potato.

Figure 2 Electrical schema of the model.
Figure 2

Electrical schema of the model.

The electric field in the different volumes of the model is computed using Finite Element analysis solving a static conduction problem [13, 14, 15, 16, 17, 18]. Dirichlet conditions were imposed to each electrode’s surfaces [14, 16, 17, 18], i.e. constant electric potentials V1 = –V/2 and V2 = +V/2, respectivelly.

In order to obtain more accurate results [19, 20], the numerical model was solved considering potato resistivity dependent on electric field strength [10]:

ρ(E)=(σ0+σ1−σ021+tanhkvE−Eth)−1(1)

The typical range for conductivity is between 0.18 and 0.97 S/m, values measured on ‘electropored’ and ‘not electropored’ potatoes, respectively. Eth was evaluated experimentally.

The field problem was solved by means of a commercial software based on Finite Elements. A field model is of utmost importance when field quantities have to be evaluated and quantified precisely [21, 22].

2.2 Gel phantom preparation

The gel phantoms were produced according to a slightly modified procedure based on that proposed in [20] for the preparation of low-cost tissue-mimicking materials (spinal cord, SC and dura, D). Gelatin, water, agar, corn flour, potato flour, glycerin, sodium azide (NaN3) and sodium chloride (NaCl) were used as received. The list and amount of starting ingredients for the production of the phantoms is reported in Table 1 [23, 24].

Table 1

List of ingredients for the preparation of phantom materials.

D2DF1SC1
Glycerin2.00 g2.00 g0.90 g
Corn flour25.0 g-30.0 g
Potato flour-25.0g-
Gelatin--0.60 g
Agar1.00 g1.00 g-
Sodium azide0.80 g0.40 g0.30 g
Sodium chloride0.20g0.20 g-
Water(i) 20 ml + (ii) 50 ml

The procedure for the preparation of the tissue-mimic material follows four steps [23]:

  1. In a 50 ml beaker, the glycerin is added to 20 ml of deionized water at room temperature, then the corn or potato flour is slowly added while stirring vigorously;

  2. In an 800 ml beaker, the gelatin or agar is heated gradually to 90° C in a water bath with 50 ml of deionized water and NaCl. Heating is maintained until gelatin or agar is dissolved into a thick mixture;

  3. The viscous mixture from step (i) is added to the one obtained from step (ii). Heating and stirring are applied until the whole mixture turns semi-solid;

  4. After heating, the mixture is cooled to 50° C and NaN3 is added with mechanical stirring. The semisolid material is cast into boxes with suitable sizes for the experiments.

2.3 Material resistivity identification

The electrical resistivity, ρ, of the gel phantoms and potato tubers was evaluated experimentally using a plastic box with rectangular section (11 x× 9 mm, thickness 11.3 mm) and an electrode with two plate at L = 7.5 mm with rectangular geometry (width 10 mm) as shown in Figure 3. The voltage was applied to the electrode plates by means of a voltage pulse generator (manufactured by Igea, Carpi, Italy) and the corresponding current is measured. From voltage and current data the resistance was computed as in [7] and the resistivity was evaluated using Ohm’s law and the model for resistance, R, of a parallelepiped sample with length L (7.5 mm) and section A (10 × 11.3 mm):

ρ=RAL(2)
Figure 3 Experimental set-up for resistivity.
Figure 3

Experimental set-up for resistivity.

In order to evaluate gel resistivity the applied voltage pulses had an amplitude of 100 V (E = 130 V/cm) and 760 V (E = 1000 V/cm)).

2.4 Data analysis

Voltage and current data recorded by ECT equipment were used in order to evaluate the equivalent resistance at the cylindrical needle ends as in [7]. For the sake of a comparison, considering the computation model in Figure 1, the resistance at the electrode was evaluated as the ratio between imposed voltage V and the integral of the normal component of current density J evaluated on the transversal section, S, of the current density flux

R=V/∫SJ¯⋅n¯dS(3)

Where J depends on the electric field and material conductivity, J = σE. This way the equivalent resistance at the needle ends is computed and depends on the electric field if the conductivity depends on electric field [10].

2.5 Evaluation of potato electroporation threshold

The electric field intensity used to obtain electroporation on the potato tuber was evaluated experimentally using the set-up in Figure 3. Potato cubes were positioned in the plastic support. For each sample, 8 voltage pulses at 5 kHz were applied. Different voltage amplitudes between 0 and 800 V (electric field between 0 to 1060 V/cm) were selected for each sample. Sample resistance and resistivity were computed from the voltage and current data. The potato color was observed 24 h after the pulse application. Samples were preserved at room temperature, by covering them with plastic film. The color of each potato sample was compared to its sample resistivity. Dark intensity was evaluated by means of ImageJ. For each image the median dark intensity was recorded. Experiments were repeated at least thrice.

2.6 Computational and experimental set-up

Using the models in Figure 1 three types of set-up of materials, potato and gels, were used in computational and experimental tests: (a) model in Figure 1(b) with volume Vp filled with potato, volume Vg with one gel from Table 2; (b) model in Figure 1(a) with volume V1 filled with potato, volume V2 with one gel from Table 2 and volume V3 empty; (c)model in Figure 1(a) with volume V1 filled with potato, volume V2 and V3 with one gel from Table 2.

Table 2

Resistivity and conductivity values for gels

(@130V/cm)(@1000V/cm)
Tissueρ[Ωm]σ[S/m]ρ[Ωm]σ[S/m]
D20.78 ± 0.091.30 ± 0.140.83 ± 0.031.20 ± 0.5
DF11.09 ± 0.020.92 ± 0.010.99 ± 0.061.01 ± 0.06
SC13.38 ± 0.060.3 ± 0.013.33 ± 0.20.3 ± 0.02

3 Results and discussion

Computational and experimental results are discussed below.

3.1 Gel electrical characteristics

Table 2 reports the gel resistivity and conductivity computed at two different electric field intensities using equation (3). The values evaluated from applying a uniform electric field at 130 V/cm and 1000 V/cm are comparable.

3.2 Potato characterization

Table 3 reports the resistivity for potato tuber evaluated by means of equation (3) varying the applied voltage. Each value is the average of at least 3 values. From these values of resistivity the conductivity values for electroporated (ρ1 = 1.05 Ωm, σ1 = 0.97 S/m) and not-electroporated (ρ0 = 6.6 Ωm, σ0 = 0.18 S/m) potato used in the equation (1) were evaluated. The average of dark intensity (DI) is also reported. In this case a scale between 0 and 255 was used, where a value close to 0 corresponds to a dark color.

Figure 4 Evaluation of electric field threshold for electroporation: (a) fit of experimental data and (b) potato color as a function of electric field.
Figure 4

Evaluation of electric field threshold for electroporation: (a) fit of experimental data and (b) potato color as a function of electric field.

Table 3

Potato resistivity, conductivity, and dark intensity (DI) at different voltage amplitude (electric field).

VE[V/cm]ρ[Ωm]σ[S/m]DI
00––115.7 ±5.7
1001306.60 ± 2.790.18 ± 0.0798.1 ±11.4
1502004.12 ± 0.890.25 ± 0.0587.1 ±12.2
2002652.29 ± 0.500.46 ± 0.1249.9 ± 5.9
4005301.52 ± 0.210.67 ± 0.0950.6 ± 8.8
5006601.47 ± 0.630.78 ± 0.2654.7 ± 7.4
6008001.14 ± 0.260.92 ± 0.2054.9 ± 5.3
7009301.12 ± 0.230.93 ± 0.1940.7 ± 5.0
80010601.05 ± 0.170.97 ± 0.1335.0 ± 5.0

3.3 Inhomogeneous models

Figure 5 shows an example of the conduction field map obtained by attributing constant resistivity values to gel (data in Table 2) and potato. In this case, non-linear conductivity was considered for the potato samples. The field asymmetry is evident. Figure 6 shows the corresponding potato pieces obtained in experimental validation. The differences in the amplitudes of the dark areas at the boundaries with the gel (right side of each piece) are evident. They correspond to differences in the positions of the 390 V/cm equi-field level (orange line in Figure 5) in the potato domain in the simulation results. In particular, in the DF1 case, the 390 V/cm line at the boundary between the potato and the gel has a diameter of about 0.75 cm, while in the SC1 case it has a diameter of about 0.35 cm. Considering the electric field level at 265 V/cm, i.e. the level at which electroporation starts, the level line diameters are close to 1 cm and 1.5 cm, respectively, similar to the experimental values in Table 4.

Figure 5 Conduction field map considering a non-linear conductivity for potato, σ1, and constant for gels. σ2= σ3 equal to (a) 0.3 S/m and (b) 1 S/m.
Figure 5

Conduction field map considering a non-linear conductivity for potato, σ1, and constant for gels. σ2= σ3 equal to (a) 0.3 S/m and (b) 1 S/m.

Figure 6 Potato 24 h after pulse application In the model with three materials, as shown In Figure 1(a), using gels (a) DF1 and (b) SC1.
Figure 6

Potato 24 h after pulse application In the model with three materials, as shown In Figure 1(a), using gels (a) DF1 and (b) SC1.

Table 4

Resistance simulated and measured at electrode extremities considering different set-up, DI and distance XA

Needle positionRs [Ω]Rm [Ω]XA [cm]DI
(a) Two needles in potato
Big potato140180.0 ± 28.01.33 ± 0.1650
Medium potato142165.8 ± 25.11.11 ± 0.2454
Small potato145178.6 ± 6.21.10 ± 0.1758
(a1) Two needles in potato as in Figure 1(b)
Potato + D2136172.0 ± 14.41.55 ± 0.0845
Potato + DF1137170.6 ± 19.11.47 ± 0.2856
Potato + SC1141180.5 ± 10.21.37 ± 0.1264
(b) One needle in potato, one in gel
Potato and D2106145.1 ± 2.91.80 ± 0.0644
Potato and DF1120145.8 ± 9.81.72 ± 0.2649
Potato and SC1261283.1 ± 18.31.13 ± 0.3466
(c) One needle in potato, one in gel + surrounding gel
Potato and D2 + D2104133.9 ± 18.51.55 ± 0.0949
Potato and DF1 + DF1121143.3 ± 14.61.74 ± 0.3252
Potato and SC1 + SC1264276.7 ± 16.71.06 ± 0.2870

Table 4 includes findings of the tests performed using different combinations of gels and potato subject to a voltage of 730 V. It reports the resistance evaluated by means of (3) from computational models, and the experimental resistance. Moreover, the amplitude of the dark area at the boundary with the gel (right side of the potato pieces) for two- or three-material models, or in the middle between the two needles, is reported.

4 Conclusions

Tissue inhomogeneity is a relevant characteristic of tumors. From this background, the proposed Finite Elements model takes into account tissue inhomogeneities. The experiments were performed on a potato and the simulations are carried out accordingly. A good match between experimental results and simulations was found. In future works the proposed model will be extended to cancer tissues.

Acknowledgement

Authors are grateful to Igea S.p.A. (Carpi(MO) Italy), for pulse generator and electrodes. Project granted by CPDA138001 (Padua University) and partially made possible thanks to COST TD1104 action (www.electroporation.net).

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Received: 2017-11-2
Accepted: 2017-11-12
Published Online: 2017-12-29

© 2017 A. Bernardis et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2017-0092
Keywords for this article
Clinical electroporation; electric field model; equivalent circuit; resistivity identification; potato samples
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