Abstract-Permeabilization, when observed on a tissue level, is a dynamic process resulting from changes in membrane permeability when exposing biological cells to external electric field (E).In this paper we present a sequential finite element model of E distribution in tissue which considers local changes in tissue conductivity due to permeabilization. These changes affect the pattern of the field distribution during the high voltage pulse application.The presented model consists of a sequence of static models (steps), which describe E distribution at discrete time intervals during tissue permeabilization and in this way present the dynamics of electropermeabilization. The tissue conductivity for each static model in a sequence is determined based on E distribution from the previous step by considering a sigmoid dependency between specific conductivity and E intensity. Such a dependency was determined by parameter estimation on a set of current measurements, obtained by in vivo experiments. Another set of measurements was used for model validation. All experiments were performed on rabbit liver tissue with inserted needle electrodes. Model validation was carried out in four different ways: 1) by comparing reversibly permeabilized tissue computed by the model and the reversibly permeabilized area of tissue as obtained in the experiments; 2) by comparing the area of irreversibly permeabilized tissue computed by the model and the area where tissue necrosis was observed in experiments; 3) through the comparison of total current at the end of pulse and computed current in the last step of sequential electropermeabilization model; 4) by comparing total current during the first pulse and current computed in consecutive steps of a modeling sequence.The presented permeabilization model presents the first approach of describing the course of permeabilization on tissue level. Despite some approximations (ohmic tissue behavior) the model can predict the permeabilized volume of tissue, when exposed to electrical treatment. Therefore, the most important contribution and novelty of the model is its potentiality to be used as a tool for determining parameters for effective tissue permeabilization.

Abstract-In electrochemotherapy (ECT) electropermeabilization, parameters (pulse amplitude, electrode setup) need to be customized in order to expose the whole tumor to electric field intensities above permeabilizing threshold to achieve effective ECT. In this paper, we present a model-based optimization approach toward determination of optimal electropermeabilization parameters for effective ECT. The optimization is carried out by minimizing the difference between the permeabilization threshold and electric field intensities computed by finite element model in selected points of tumor. We examined the feasibility of model-based optimization of electropermeabilization parameters on a model geometry generated from computer tomography images, representing brain tissue with tumor. Continuous parameter subject to optimization was pulse amplitude. The distance between electrode pairs was optimized as a discrete parameter. Optimization also considered the pulse generator constraints on voltage and current. During optimization the two constraints were reached preventing the exposure of the entire volume of the tumor to electric field intensities above permeabilizing threshold. However, despite the fact that with the particular needle array holder and pulse generator the entire volume of the tumor was not permeabilized, the maximal extent of permeabilization for the particular case (electrodes, tissue) was determined with the proposed approach.Model-based optimization approach could also be used for electro-gene transfer, where electric field intensities should be distributed between permeabilizing threshold and irreversible threshold-the latter causing tissue necrosis. This can be obtained by adding constraints on maximum electric field intensity in optimization procedure.

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Sequential model of liver tissue electropermeabilisation around two needle electrodes was designed by computing electric field (E) distribution by means of the finite element (FE) method. Sequential model consists of a sequence of static FE models which represent E distribution during tissue permeabilisation. In the model an S-shaped dependency between specific conductivity and E was assumed. Parameter estimation of S-shaped dependency was performed on a set of current measurements obtained by in vivo experiments. Another set of in vivo measurements was used for model validation. Model validation was carried out in three different ways by comparing experimental measurements and modelled results. The model validation showed good agreement between modelled and measured results. The model also provided means for better understanding processes that occur during permeabilisation. Based on the model, the permeabilised volume of tissue exposed to electrical treatment can be predicted. Therefore, the most important contribution of the model is its potential to be used as a tool for determining the electrode position and pulse amplitude needed for effective tissue permeabilisation.

Information about electric field distribution in tissue is very important for effective electropermeabilization. In heterogeneous tissues with complex geometry, finite-element (FE) models provide one of alternative sources of such information. In the present study, modeling of needle electrode geometry in the FE model was investigated in order to determine the most appropriate geometry by considering the need for frequent FE model computation present in electroporation models. The 8-faceted needle electrode geometry proposed--determined on a model with a single needle electrode pair by means of criteria function--consisted of the weighted sum of relative difference between measured and computed total current, the relative difference in CPU time spent on solving model, and the relative difference in cross section surface of electrodes. Such electrode geometry was further evaluated on physical models with needle arrays by comparison of computed total current and measured current. The agreement between modeled and measured current was good (within 9% of measurement), except in cases with very thin gel. For voltage above 50 V, a linear relationship between current and voltage was observed in measurements. But at lower voltages, a nonlinear behavior was detected resulting from side (electrochemical) effects at electrode-gel interface. This effect was incorporated in the model by introducing a 50-V shift which reduced the difference between the model and the measurement to less than 3%. As long as material properties and geometry are well described by FE model, current-based validation can be used for a rough model validation. That is a routine assay compared with imaging of electric field, which is otherwise employed for model validation. Additionally, current estimated by model, can be preset as maximum in electroporator in order to protect tissue against damage.