Abstract
Electroporation is a phenomenon used in the treatment of tumors by electrochemotherapy, non-thermal ablation with irreversible electroporation, and gene therapy. When treating patients, either predefined or variable electrode geometry is used. Optimal pulse parameters are predetermined for predefined electrode geometry, while they must be calculated for each specific case for variable electrode geometry. The position and number of electrodes are also determined for each patient. It is currently assumed that above a certain experimentally determined value of electric field, all cells are permeabilized/destroyed and under it they are unaffected. In this paper, mathematical models of survival in which the probability of cell death is continuously distributed from 0 to 100 % are proposed and evaluated. Experiments were performed on cell suspensions using electrical parameters similar to standard electrochemotherapy and irreversible electroporation parameters. The proportion of surviving cells was determined using clonogenic assay for assessing the ability of a cell to grow into a colony. Various mathematical models (first-order kinetics, Hülsheger, Peleg-Fermi, Weibull, logistic, adapted Gompertz, Geeraerd) were fitted to experimental data using a non-linear least-squares method. The fit was evaluated by calculating goodness of fit and by observing the trend of values of models’ parameters. The most appropriate models of cell survival as a function of treatment time were the adapted Gompertz and the Geeraerd models and, as a function of the electric field, the logistic, adapted Gompertz and Peleg-Fermi models. The next steps to be performed are validation of the most appropriate models on tissues and determination of the models’ predictive power.
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Acknowledgments
This study was supported by the Slovenian Research Agency (ARRS) and conducted within the scope of the Electroporation in Biology and Medicine European Associated Laboratory (LEA-EBAM). Experimental work was performed in the infrastructure center ‘Cellular Electrical Engineering’ IP-0510. The authors would like to thank Dr. Bor Kos for his helpful comments about fitting of the models and numerical modeling and Lea Vukanović for her help with the experiments in the laboratory.
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Appendix: Numerical Model of Temperature Distribution
Appendix: Numerical Model of Temperature Distribution
We calculated temperature distribution to determine that 90, 100 µs pulses of 4 kV/cm and 1 Hz repetition frequency do not induce significant Joule heating. A numerical model of a drop of cell suspension between parallel plate electrodes was made in Comsol Multiphysics (v4.4, Comsol, Sweden) using Electric Currents, Heat Transfer, and Multiphysics modules in time-dependent analysis. Electrodes were modeled as two blocks of 20 × 10 × 1 mm and a drop of cell suspension was modeled as a block of 7 × 7 × 2 mm. Two boundaries, one on each electrode, were modeled as terminals, with +400 V and −400 V in the first 90 s of the simulation, while the other boundaries were electrically insulated. Similar as in (Garcia et al. 2011a, b), only one pulse was applied for 90 s, but we multiplied the Joule heating by the duty cycle (duration/period) to adjust the amount of delivered energy. We ran the simulation for an additional 5 s after the pulse application to validate the model with our temperature measurements, since measurements of temperature were made within 5 s after the pulse application.
The change of conductivity due to cell electroporation was disregarded in the model, since our cell suspension was dilute. The values of parameters used in the simulation are shown in Table 8. The properties of the cell suspension (except for electrical conductivity, which is characteristic of our electroporation buffer) are the same as for water.
The model was validated with current and temperature measurements at 90, 100 µs pulses, 4 kV/cm, 1 Hz repetition frequency. The predicted current (3.3 A) was in the same range as the measured current (from 2.9 to 3.5 A). Temperature measured within 5 s after the pulse application was 37.0 °C, while the predicted temperature 5 s after the pulse application was 37.6 °C. The model thus adequately described our experiments.
The temperature distribution on the surface of the cell suspension and on the electrodes, and a slice through the drop of cell suspension after 90 pulses, is shown in Fig. 7. Since the temperature of the cell suspension does not exceed 42 °C, under our experimental conditions cell death can indeed be ascribed to electroporation.
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Dermol, J., Miklavčič, D. Mathematical Models Describing Chinese Hamster Ovary Cell Death Due to Electroporation In Vitro. J Membrane Biol 248, 865–881 (2015). https://doi.org/10.1007/s00232-015-9825-6
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DOI: https://doi.org/10.1007/s00232-015-9825-6