The role of additional pulses in electropermeabilization protocols

Abstract

Electropermeabilization (EP) based protocols such as those applied in medicine, food processing or environmental management, are well established and widely used. The applied voltage, as well as tissue electric conductivity, are of utmost importance for assessing final electropermeabilized area and thus EP effectiveness. Experimental results from literature report that, under certain EP protocols, consecutive pulses increase tissue electric conductivity and even the permeabilization amount. Here we introduce a theoretical model that takes into account this effect in the application of an EP-based protocol, and its validation with experimental measurements. The theoretical model describes the electric field distribution by a nonlinear Laplace equation with a variable conductivity coefficient depending on the electric field, the temperature and the quantity of pulses, and the Penne's Bioheat equation for temperature variations. In the experiments, a vegetable tissue model (potato slice) is used for measuring electric currents and tissue electropermeabilized area in different EP protocols. Experimental measurements show that, during sequential pulses and keeping constant the applied voltage, the electric current density and the blackened (electropermeabilized) area increase. This behavior can only be attributed to a rise in the electric conductivity due to a higher number of pulses. Accordingly, we present a theoretical modeling of an EP protocol that predicts correctly the increment in the electric current density observed experimentally during the addition of pulses. The model also demonstrates that the electric current increase is due to a rise in the electric conductivity, in turn induced by temperature and pulse number, with no significant changes in the electric field distribution. The EP model introduced, based on a novel formulation of the electric conductivity, leads to a more realistic description of the EP phenomenon, hopefully providing more accurate predictions of treatment outcomes.

Figure 1. In vitro and in silico models.

a) Experimental setup. Arrangement of six electrodes inserted in a potato slice. b) Domain and mesh of the mathematical model generated by Comsol. The arrow indicates the point where all calculations presented in figure 2 were made (measurement area).

Figure 2. Temporal evolution of main variables.

a) Electric current density (A/cm ²), b) electric conductivity (mS/cm) and c) temperature (C) vs. pulse number for different square electric pulses (voltages of 500, 800, 1000, 1500 and 1700 V; 100 µs, 1 Hz). Circles: predicted values from the Fortran code. Dashed lines: experimental data from EP treatments in potato tissue. Standard errors were omitted for clarity. Solid lines: predictions from a model without the pulse-conductivity term for 1700 V.

Figure 3. Electric field distribution.

Predicted electric field distribution (V/cm) in the x-y plane generated by Comsol after an EP treatment with a) 2 and b) 8 square electric pulses of 1500 V, 100 µs and 1 Hz. Isolines correspond to 50, 100, 150, 200, 300, 400 and 600 V/cm. The arrow indicates the point where all calculations presented in figure 2 were made (measurement area).

Figure 4. Electropermeabilized area.

Dark (electropermeabilized) potato area (cm ²) vs. pulse number for different EP protocols (4, 8, 16 and 32 square pulses of 100 µs, 1 Hz with amplitudes of 500, 1000 and 1500 V). Bars: standard errors. Lines: logarithmic regressions. Images at the top of the figure correspond to potato tissue treated with 4, 8, 16 and 32 pulses of 100 µs, 1 Hz and 1500 V. Cathode position is indicated as white spots.