Modeling Electric Distribution With Gis

[Rosetta Ockman]

Electroporationbased therapies and treatments (e.g. electrochemotherapy, gene electrotransfer for gene therapy and DNA vaccination, tissue ablation with irreversible electroporation and transdermal drug delivery) require a precise prediction of the therapy or treatment outcome by a personalized treatment planning procedure. Numerical modeling of local electric field distribution within electroporated tissues has become an important tool in treatment planning procedure in both clinical and experimental settings. Recent studies have reported that the uncertainties in electrical properties (i.e. electric conductivity of the treated tissues and the rate of increase in electric conductivity due to electroporation) predefined in numerical models have large effect on electroporation based therapy and treatment effectiveness. The aim of our study was to investigate whether the increase in electric conductivity of tissues needs to be taken into account when modeling tissue response to the electroporation pulses and how it affects the local electric distribution within electroporated tissues.

We built 3D numerical models for single tissue (one type of tissue, e.g. liver) and composite tissue (several types of tissues, e.g. subcutaneous tumor). Our computer simulations were performed by using three different modeling approaches that are based on finite element method: inverse analysis, nonlinear parametric and sequential analysis. We compared linear (i.e. tissue conductivity is constant) model and non-linear (i.e. tissue conductivity is electric field dependent) model. By calculating goodness of fit measure we compared the results of our numerical simulations to the results of in vivo measurements.

The results of our study show that the nonlinear models (i.e. tissue conductivity is electric field dependent: σ(E)) fit experimental data better than linear models (i.e. tissue conductivity is constant). This was found for both single tissue and composite tissue. Our results of electric field distribution modeling in linear model of composite tissue (i.e. in the subcutaneous tumor model that do not take into account the relationship σ(E)) showed that a very high electric field (above irreversible threshold value) was concentrated only in the stratum corneum while the target tumor tissue was not successfully treated. Furthermore, the calculated volume of the target tumor tissue exposed to the electric field above reversible threshold in the subcutaneous model was zero assuming constant conductivities of each tissue.

Our results also show that the inverse analysis allows for identification of both baseline tissue conductivity (i.e. conductivity of non-electroporated tissue) and tissue conductivity vs. electric field (σ(E)) of electroporated tissue.

Our results of modeling of electric field distribution in tissues during electroporation show that the changes in electrical conductivity due to electroporation need to be taken into account when an electroporation based treatment is planned or investigated. We concluded that the model of electric field distribution that takes into account the increase in electric conductivity due to electroporation yields more precise prediction of successfully electroporated target tissue volume. The findings of our study can significantly contribute to the current development of individualized patient-specific electroporation based treatment planning.

The aim of our study was to investigate whether the increase in electric conductivity of tissues needs to be taken into account when modeling tissue response to the electroporation pulses and how it affects the local electric field distribution within electroporated tissues. We also attempted to identify the nature of functional dependency of tissue conductivity on electric field. We therefore compared linear (i.e. tissue conductivity is constant) and non-linear (i.e. tissue conductivity is electric field dependent) model to the experimental data in vivo[56]. The study was performed for both single tissue (one type of tissue, e.g. liver) and composite tissue (several types of tissues, e.g. subcutaneous tumor). Our computer simulations were performed by using three different modeling approaches: 1. inverse analysis (IA) [60, 61], 2. nonlinear parametric analysis (NPA) [62] and 3. sequential analysis (SA) [58]. The algorithm for IA modelling approach has been previously developed by [60, 61] to model different physical phenomena in wide range of research and industry including biomechanics, pharmaceutical industry and space [63]. In this paper we introduced and successfully applied the IA approach for the first time in the field of modelling of biophysical processes that occur during tissue electroporation.

The results of all three modeling approaches are then compared to the results of our previous experimental measurements in vivo[56]. The results of our present study show that the nonlinear models fit experimental data better than linear models. We also demonstrate that tissue conductivity as a function of local electric field need to be taken into account when an electroporation based therapy or treatment is planned.

Modeling of electric field distribution in a single tissue during electroporation was carried out in liver tissue. We analyzed experimental data for plate and needle electrodes. Original experiments with plate electrodes were performed in rat liver, while the experiments with needle electrodes were performed in rabbit liver [56].

Modeling of electric field distribution in composite tissue (i.e. tissue composed of different types of tissues) was carried out in cutaneous tumor tissue. In vivo experiments were performed in mice with plate electrodes for two different types of cutaneous tumors: B16 and LPB sarcoma within the study by [56].

The high voltage pulses in all in vivo experiments were applied to the electrodes according to the electroporation protocol of eight 100 μs long pulses delivered to the tissues with 1 Hz repetition frequency. Within the experiments [56] the real time tissue electroporation control was performed by monitoring and measuring the electric current during the delivery of electric pulses. In order to evaluate the tissue electroporation efficiency during the pulse delivery the 51CrEDTA indicator was used for all tissues. The system for the real time tissue electroporation control and 51CrEDTA uptake measurements were described in our previous work [56]. The electric current measured during the in vivo experiments provides the information about the bulk tissue conductivity changes during the pulse delivery.

In order to compare and evaluate the results of our numerical models to the experimental data we calculated the goodness-of-fit measure R2 between numerically calculated and in vivo measured electric current I [A] at the end of the last delivered pulse (i.e. the 8-th pulse) to the tissue.

Electric field distribution in the models of tissue exposed to electroporation pulses U [V] can be determined by solving the equation (Eq. 1) for scalar electric potential. Namely, if we neglect the capacitive transient and time course of conductivity increase during the pulse, we may assume that the current density in tissue is divergence free and electric potential satisfies:

Measurement of electric current I [A] during delivery of eight rectangular electroporation pulses of 100 μs duration and 1 Hz repetition showed that after rapid capacitive transient (which is due to tissue capacitance) the current increased towards the end of the pulse [56, 52]. The rate of increase of electric current I [A] depends on the applied voltage U [V], which provides information about the changes in electric properties of the treated tissue (i.e. changes in electric conductivity of the tissue σ [S/m]). The rates of increase of I and σ thus allow us to identify the degree of tissue electroporation.

Numerical calculations of Eq.1 and tissue electroporation dynamics σ(E) in our study were performed with three different approaches that were based on finite element method: Inverse analysis (IA), nonlinear parametric analysis (NPA) and sequential analysis (SA). The software for IA was previously developed by Center for Computational Continuum Mechanics (Ljubljana, Slovenia) [61], while for the NPA and the SA we used Comsol Multiphysics software (version 3.5a, COMSOL, Sweden).

It is necessary to increase the voltage U from 0 V in small steps and use the solution from a previous step as the initial condition for the next step. This is taken care of automatically by the parametric solver. The parametric solver consists of a loop around the stationary solver. If the nonlinear solver does not converge it tries a smaller parameter step; if it does converge it determines the size of the next parameter step based on the speed of the convergence of the Newton iteration.

The SA in our study was based on the sequential permeabilization model proposed by [58], where changes in tissue conductivity were used as an indicator of tissue permeabilization. Namely, the dynamics of electroporation was modeled as a discrete process with the sequence of static finite element models. Each of the finite element models described the process in one discrete interval (each of the discrete intervals relates to a real, discrete, but undetermined time interval). In each static finite element model in sequence, the tissue conductivity was determined based on the electric field distribution from the previous model in the sequence according to the Eq. 12:

The numerical calculations obtained with NPA and SA by using COMSOL Multiphysics 3.5a software package were performed in the 3D Conductive Media DC application mode on a computer running Windows XP (64 bit) with a 3.00 GHz Intel Pentium D processor and 2 GB of RAM.

For the numerical calculations with IA we used AceGen (version AceFEM 3.101) system [61, 64, 65]. The calculations were performed on a computer running Windows XP (64 bit) with 2.67 GHz Intel Core i7 CPU and 6 GB of RAM.

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