Table of Contents
Biological Tissues
Samples of liver, kidney and heart of rabbits of the Leporidae strain Oryctolagus cuniculus species, were used. All procedures performed with the animals were approved by the Research Ethics Committee on the Use of Animals  CEUA/UNIVILLE, (process 02/2019). The liver and kidney samples were obtained with 20 mm × 20 mm × 10 mm edges. The heart does not have a homogeneous structure and its interior is hollow due to the atria and ventricles. Thus, such an organ was fragmented in half, having dimensions of 10 mm x 10 mm and 5 mm thick.
Electroporation Protocols
The electroporation protocol used in chemotherapy treatments consists of the application of 8 pulses, with a duration of 100 μs and an interval of 1 s between the pulses.21 However, the objective of this work is to evaluate whether the electroporation model reliably represents the tissue conductivity variations under some different protocols as follow:

Protocol (A) 10 voltage pulses, time at high and low level 99.5 μs, rise and fall time 0.5 μs. Voltage values: 200, 500, and 800 V.

Protocol (B) 10 voltage pulses, time at high and low level 9.5 μs, rise and fall time 0.5 μs. Voltage values: 200, 500, and 800 V.

Protocol (C) Voltage ramp, with linear increase from 0 to 400 V within the following time intervals: 100, 200, 300, 400, 500, 600, 700, 800, and 2400 μs.
For the application of protocols A and B a system of parallel stainless steel needles separated by a distance of 5.4 mm was used. The needles have a diameter of 0.6 mm and a height of 15 mm. Thus, the maximum fields obtained are: 370, 926, and 1481 V/cm for voltages of 200, 500 and 800 V, respectively. In protocol C parallel needle electrodes with a spacing of 2.8 mm, needle diameter of 0.7 mm and a height of 10 mm were also used. Protocol A and B were applied to five tissue samples, while for protocol C three samples were used.
Electroporator and Thermal Measurements
Two electroporators were used, one used for protocols A and B with the capacity to supply 800 V and 10 A. Another for protocol C which consists of a linear amplifier with a capacity of 400 V and 10 A. Waveforms voltage and current were measured and digitized and sent to the computer for analysis. To read the temperature on the sample surface, a FLIR® thermal camera, model FLIR C2, was used. The camera was positioned 5 cm above the sample. Data acquisition was performed using the FLIR tools plus software.
Electrical Properties of Tissues
Dielectric dispersion in biological tissues occurs mainly due to the accumulation of hydrated ions on the cell membranes, but also due to dipole relaxation of water and polar macromolecules in tissue fluids.26
The Cole–Cole empirical model is widely used to describe the dielectric dispersion of biological tissues due to the use of a distribution of relaxation times to represent each dispersion band11:
$$ hat{varepsilon }left( omega right) = varepsilon_{infty } + mathop sum limits_{n} frac{{Delta varepsilon_{n} }}{{1 + left( {jomega tau_{n} } right)^{{1  alpha_{n} }} }} + frac{{sigma_{text{s}} }}{{jomega varepsilon_{text{o}} }}, $$
(1)
where ε_{∞} is the electrical permittivity at high frequencies, σ_{s} is the static conductivity, Δε_{n} is the magnitude of the electrical permittivity dispersion, ε_{o} is the vacuum permittivity, τ_{n} is the central relaxation time of the distribution, n is the index referring to the dispersion and α is a parameter to be adjusted empirically with a value less than the unit.
The conductivity and dielectric constant spectra of the intact tissue were obtained with the aid of the Agilent® impedance analyzer model 4294A. The samples of rabbit tissues were placed in a circular parallel plate system, with a thickness of 3 mm and a diameter of 15 mm. With the aid of a genetic algorithm from the Matlab® software, the parameters of the Cole–Cole model for the beta band dispersion were obtained. The genetic algorithm was configured with the following parameters: population size = 100, crossover = 0.85 and generations = 50. Such parameters of the Cole–Cole model are used in the computational simulation of tissue electroporation with ECM.
Dynamic Model of Electroporation
The dynamic model of electroporation was proposed by Ramos and Weinert27 and is based on the asymptotic model proposed by Debruim and Krassowska.7 According to the dynamical model, Eqs. (2) and (3) below show how to calculate the rate of variation of the tissue conductivity as a function of the electric field intensity:
$$ frac{{{text{d}}sigma_{text{p}} left( t right)}}{{{text{d}}t}} = frac{{sigma_{text{eq}}  sigma_{text{p}} left( t right)}}{{tau_{hbox{min} } + Delta tau e^{{  left( {Eleft( t right)/E_{2} } right)^{2} left( {sigma_{t} /left( {sigma_{t} + sigma_{text{p}} left( t right)} right)} right)^{2} }} }}, $$
(2)
$$ sigma_{text{eq}} = frac{{sigma_{text{o}} sigma_{t} }}{{sigma_{text{o}} + sigma_{t} e^{{  left( {Eleft( t right)/E_{1} } right)^{2} left( {sigma_{t} /left( {sigma_{t} + sigma_{text{p}} left( t right)} right)} right)^{2} }} }}, $$
(3)
where σ_{p} is the conductivity of electroporation, σ_{o} is the initial conductivity, E_{1} and E_{2} are electroporation thresholds and E(t) is the applied electric field. τ_{min} and Δτ are relaxation times. The σ_{t} parameter has important functions for the model. It limits the value of σ_{p} and permits to model the electric field division between cell membrane and surround environment resulting in the reduction of the transmembrane potential when the electroporation becomes intense.
Thermal Model
In the analysis of the biological electroporation, the evaluation of the electric field distribution has central importance to define the extension of the electroporated area. Besides, side effects must be considered in the planning of electroporation protocols, the most important being thermal heating.6 In our experiments, the calculation and measurement of the temperature distribution in the electroporated tissue served as an additional check for the quality of the electroporation model.
The mathematical model used for the transfer of heat in biological tissues is known as the heat transfer equation:
$$ nabla cdot left( {knabla T} right) + w_{text{b}} c_{text{b}} left( {T_{text{a}}  T} right) + lambda = rho c_{text{p}} frac{partial T}{partial t}, $$
(4)
where k is the thermal conductivity of the tissue, T is the temperature in Kelvin, w_{b} is the perfusion rate of the blood, c_{b} is the thermal capacity in the blood, T_{a} is the temperature of the artery, λ is the generation rate of metabolic heat, ρ is the density of the tissue and c_{p} is the specific heat of the tissue.23
The thermal model implemented in the simulators is based on Eq. (4) with some adaptations. The terms of blood perfusion and metabolic generation were neglected since the experiments were carried out in ex vivo samples. Besides, a thermal dissipation term due to the electric stimulus was included in the tissue volume represented by ρ_{diss}:
$$ nabla cdot left( {knabla T} right) + rho_{text{diss}} = rho c_{text{p}} frac{partial T}{partial t}. $$
(5)
Simulation
The simulations of biological electroporation were performed in two software: The ECM program developed in our research group and the commercial program COMSOL Multiphysics®.
The ECM was proposed by Ramos et al.,28 as a calculation method of the electric field and electric current distribution in a medium with low conductivity and high dielectric constant. It is based on lumped circuit elements, conductances and capacitances, that model the electrical conduction and polarization processes in the materials involved.26,27,^{–}28 To take into account the effects of dielectric dispersion in the ECM, the dispersion parameters of the β band were included.
In COMSOL Multiphysics® the following set of equations from the electromagnetic theory was solved in the time domain:
$$ nabla cdot vec{J} =  frac{partial rho }{partial t}, $$
(6)
$$ vec{J} = sigma vec{E} + frac{{partial vec{D}}}{partial t}, $$
(7)
$$ vec{E} =  nabla V. $$
(8)
The continuity equation is shown in (6). Equation (7) are the current density components with ( sigma vec{E} ) being the conducting current density, ( partial vec{D}/partial t ) the displacement current density. Conductivity σ is the sum of two components σ = σ_{s} + σ_{p}, where σ_{s} is the static conductivity of the tissue and σ_{p} the conductivity of electroporation. Equation (8) is the electrostatic relationship between the electric field and the electric potential.
In both simulators, the following boundary conditions are defined: electric potential defined at the border with a metallic electrode and null perpendicular electric field at the other borders of the analysis domain.
For the ECM simulation, the following rectangular discretization mesh was used: 100 divisions on the x and y axis, 10 divisions on the z axis, making a total of 100,000 elements. Thus, the edges of an element in the x and y direction are 9.9 × 10^{−5} m long and in the z direction 9 × 10^{−4} m. The COMSOL Multiphysics® software generated meshes with the following number of elements: Protocols (A) and (B)—48,969 elements; Protocol (C) 39,773 elements.