Electric field of a six-needle array electrode used in drug and DNA delivery in vivo: analytical versus numerical solution

Determining tissue conductivity in tissue ablation by nanosecond pulsed electric fields

Nanosecond pulsed electric field (nsPEF) causes the permeabilization of the cell membrane and has been used to non-thermally treat cancerous tissues. As increased permeabilization in membranes were reported to be accompanied by impedance changes, the ablation effect of nsPEF on tissues can be monitored via the changes in tissue conductivity. In this study, effects of nsPEF on biological tissues were evaluated by determining the conductivities of potato and 4 T1-luc breast tumor tissues ex vivo from a murine model subjected to multiple 100-ns, 1-10 kV pulses. Using a four-needle electrode system with a calibrated electrode constant of 1.1 ± 0.1 cm, the conductivities of tissues was determined from both the network-analyzer measurement, before and after treatment, and voltage-current measurement in real-time. The conductivity of the potato tissue was measured for a frequency range of 0.1-3 MHz, and it increased with increasing pulse number and voltage amplitude. The conductivity of the tumor tissue was also observed to increase with pulse number and pulse voltage over a similar frequency range. In addition, the linear correlation between the ablation area in a treated potato tissue and the conductivity change in the tissue suggests that conductivity analysis of biological tissue under treatment could be a fast and sensitive approach to evaluate the effectiveness of a nsPEF treatment.

Continuum analysis to assess field enhancements for tailoring electroporation driven by monopolar or bipolar pulsing based on nonuniformly distributed nanoparticles

Recent reports indicate that nanoparticle (NP) clusters near cell membranes could enhance local electric fields, leading to heightened electroporation. This aspect is quantitatively analyzed through numerical simulations whereby time dependent transmembrane potentials are first obtained on the basis of a distributed circuit mode, and the results then used to calculate pore distributions from continuum Smoluchowski theory. For completeness, both monopolar and bipolar nanosecond-range pulse responses are presented and discussed. Our results show strong increases in TMP with the presence of multiple NP clusters and demonstrate that enhanced poration could be possible even over sites far away from the poles at the short pulsing regime. Furthermore, our results demonstrate that nonuniform distributions would work to enable poration at regions far away from the poles. The NP clusters could thus act as distributed electrodes. Our results were roughly in line with recent experimental observations.

Optimal Design of Needle Array for Effective Drug Delivery

Recently, the multi-needle drug injection has been adopted to overcome the shortcomes of conventional single-needle injection, enhancing the efficiency of drug delivery. However, the effect of needle array on the efficacy of drug delivery has not been fully elucidated. In this study, the interactions of drug analogous solution injected from a pair of needles were analyzed to examine the design criteria of effective multi-needle devices for drug delivery. Temporal and spatial variations of relative contents of the solution in the tissues were compared according to the distance between two adjacent needles (DN). As the DN increases from 5 to 20 D, where D is the needle diameter, the solution from each needle encounters 3.5 times faster, and 4.22 times more solution was accumulated. At the same time, the effective spreading area was continuously increased from 54.2 to 177.8 mm² and RCS gradient decreases from 0.087 to 0.037, due to the overlapping effect of the spreading solution from neighboring needles. Finally, based on the experimental results, an optimal design criterion of needle array for effective drug delivery was proposed. The present results would be helpful in the design of multi-needle injection devices and eventually offer advantage to patients with effective drug delivery.

Anistropically varying conductivity in irreversible electroporation simulations

Background

One recent area of cancer research is irreversible electroporation (IRE). Irreversible electroporation is a minimally invasive procedure where needle electrodes are inserted into the body to ablate tumor cells with electricity. The aim of this paper is to propose a mathematical model that incorporates a tissue’s conductivity increasing more in the direction of the electrical field as this has been shown to occur in experiments.

Method

It was necessary to mathematically derive a valid form of the conductivity tensor such that it is dependent on the electrical field direction and can be easily implemented into numerical software. The derivation of a conductivity tensor that can take arbitrary functions for the conductivity in the directions tangent and normal to the electrical field is the main contribution of this paper. Numerical simulations were performed for isotropic-varying and anisotropic-varying conductivities to evaluate the importance of including the electrical field’s direction in the formulation for conductivity.

Results

By starting from previously published experimental results, this paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude.

The anisotropic formulation predicts roughly a five percent decrease in ablation size for the monopolar simulation and approximately a ten percent decrease in ablation size for the bipolar simulations. This is a positive result as previously reported results found the isotropic formulation to overpredict ablation size for both monopolar and bipolar simulations. Furthermore, it was also reported that the isotropic formulation overpredicts the ablation size more for the bipolar case than the monopolar case. Thus, our results are following the experimental trend by having a larger percentage change in volume for the bipolar case than the monopolar case.

Conclusions

The predicted volume of ablated cells decreased, and could be a possible explanation for the slight over-prediction seen by isotropic-varying formulations.

Histological and Mathematical Analysis of the Irreversibly Electroporated Liver Tissue

Irreversible electroporation has clinically been used to treat various types of cancer. A plan on how to apply irreversible electroporation before practicing is very important to increase the ablation area and reduce the side effects. Several electrical models have been developed to predict the ablation area with applied electric energy. In this experiment, the static relationship between applied electric energy and ablated area was mathematically and experimentally investigated at 10 hours after applying irreversible electroporation. We performed the irreversible electroporation on the liver tissue of Sprague Dawley rats (male, 8 weeks, weighing 250-350 g). The ablated area was measured based on histological analysis and compared with the mathematical calculation from the electric energy, assuming that the tissue is homogeneous. The ablated area increased with the increase in applied electric energy. The numerically calculated contour lines of electric energy density overlapped well with the apoptotic area induced by the irreversible electroporation. The overlapped area clearly showed that the destructive threshold of apoptosis between electrodes is electric energy density level of 5.9 × 10⁵ J/m³. The results of the present study suggested that the clinical results of the irreversible electroporation on a liver tissue could be predicted through mathematical calculation.

Educational application for visualization and analysis of electric field strength in multiple electrode electroporation

BACKGROUND

Electrochemotherapy is a local treatment that utilizes electric pulses in order to achieve local increase in cytotoxicity of some anticancer drugs. The success of this treatment is highly dependent on parameters such as tissue electrical properties, applied voltages and spatial relations in placement of electrodes that are used to establish a cell-permeabilizing electric field in target tissue. Non-thermal irreversible electroporation techniques for ablation of tissue depend similarly on these parameters. In the treatment planning stage, if oversimplified approximations for evaluation of electric field are used, such as U/d (voltage-to-distance ratio), sufficient field strength may not be reached within the entire target (tumor) area, potentially resulting in treatment failure.

RESULTS

In order to provide an aid in education of medical personnel performing electrochemotherapy and non-thermal irreversible electroporation for tissue ablation, assist in visualizing the electric field in needle electrode electroporation and the effects of changes in electrode placement, an application has been developed both as a desktop- and a web-based solution. It enables users to position up to twelve electrodes in a plane of adjustable dimensions representing a two-dimensional slice of tissue. By means of manipulation of electrode placement, i.e. repositioning, and the changes in electrical parameters, the users interact with the system and observe the resulting electrical field strength established by the inserted electrodes in real time. The field strength is calculated and visualized online and instantaneously reflects the desired changes, dramatically improving the user friendliness and educational value, especially compared to approaches utilizing general-purpose numerical modeling software, such as finite element modeling packages.

CONCLUSION

In this paper we outline the need and offer a solution in medical education in the field of electroporation-based treatments, e.g. primarily electrochemotherapy and non-thermal irreversible tissue ablation. We present the background, the means of implementation and the fully functional application, which is the first of its kind. While the initial feedback from students that have evaluated this application as part of an e-learning course is positive, a formal study is planned to thoroughly evaluate the current version and identify possible future improvements and modifications.

Patient-specific treatment planning of electrochemotherapy: procedure design and possible pitfalls

Electrochemotherapy uses electroporation for enhancing chemotherapy. Electrochemotherapy can be performed using standard operating procedures with predefined electrode geometries, or using patient-specific treatment planning to predict electroporation. The latter relies on realistic computer models to provide optimal results (i.e. electric field distribution as well as electrodes' position and number) and is suitable for treatment of deep-seated tumors. Since treatment planning for deep-seated tumors has been used in radiotherapy, we expose parallelisms with radiotherapy in order to establish the procedure for electrochemotherapy of deep-seated tumors. We partitioned electrochemotherapy in the following phases: the mathematical model of electroporation, treatment planning, set-up verification, treatment delivery and monitoring, and response assessment. We developed a conceptual treatment planning software that incorporates mathematical models of electroporation. Preprocessing and segmentation of the patient's medical images are performed, and a 3D model is constructed which allows placement of electrodes and implementation of the mathematical model of electroporation. We demonstrated the feasibility of electrochemotherapy of deep-seated tumors treatment planning within a clinical study where treatment planning contributed to the effective electrochemotherapy treatment of deep-seated colorectal metastases in the liver. The described procedure can provide medical practitioners with information on using electrochemotherapy in the clinical setting. The main aims of this paper are: 1) to present the procedure for treating deep-seated tumors by electrochemotherapy based on patient-specific treatment planning, and 2) to identify gaps in knowledge and possible pitfalls of such procedure.

Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy

BACKGROUND

Electrotherapy is a relatively well established and efficient method of tumor treatment. In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola).

METHODS

Analytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions.

RESULTS

Both analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor.

CONCLUSION

The mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections.

Activating function of needle electrodes in anisotropic tissue

We present an analytical solution for the electrical potential and activating function (AF) established by cylindrical needle electrodes in anisotropic tissue. We compare this activating function to (1) AF computed assuming line-source electrodes and (2) AF computed using a finite element program. The results show that when the fiber is two needle diameters away from the electrodes, the maximum of the AF for needle electrodes is 1.43-times larger than for line-source electrodes, which results in lower thresholds for stimulation and electroporation. Therefore, for fibers that are close to the stimulating electrodes, one would benefit from using the formula that accounts for the electrodes' geometry.

Field distribution and DNA transport in solid tumors during electric field-mediated gene delivery

Gene therapy has a great potential in cancer treatment. However, the efficacy of cancer gene therapy is currently limited by the lack of a safe and efficient means to deliver therapeutic genes into the nucleus of tumor cells. One method under investigation for improving local gene delivery is based on the use of pulsed electric field. Despite repeated demonstration of its effectiveness in vivo, the underlying mechanisms behind electric field-mediated gene delivery remain largely unknown. Without a thorough understanding of these mechanisms, it will be difficult to further advance the gene delivery. In this review, the electric field-mediated gene delivery in solid tumors will be examined by following individual transport processes that must occur in vivo for a successful gene transfer. The topics of examination include: (i) major barriers for gene delivery in the body, (ii) distribution of electric fields at both cell and tissue levels during the application of external fields, and (iii) electric field-induced transport of genes across each of the barriers. Through this approach, the review summarizes what is known about the mechanisms behind electric field-mediated gene delivery and what require further investigations in future studies.

Analytical and numerical quantification and comparison of the local electric field in the tissue for different electrode configurations

BACKGROUND

Electrochemotherapy and gene electrotransfer are novel promising treatments employing locally applied high electric pulses to introduce chemotherapeutic drugs into tumor cells or genes into target cells based on the cell membrane electroporation. The main focus of this paper was to calculate analytically and numerically local electric field distribution inside the treated tissue in two dimensional (2D) models for different plate and needle electrode configurations and to compare the local electric field distribution to parameter U/d, which is widely used in electrochemotherapy and gene electrotransfer studies. We demonstrate the importance of evaluating the local electric field distribution in electrochemotherapy and gene electrotransfer.

METHODS

We analytically and numerically analyze electric field distribution based on 2D models for electrodes and electrode configurations which are most widely used in electrochemotherapy and gene electrotransfer. Analytical calculations were performed by solving the Laplace equation and numerical calculations by means of finite element method in two dimensions.

RESULTS

We determine the minimal and maximal E inside the target tissue as well as the maximal E over the entire treated tissue for the given electrode configurations. By comparing the local electric field distribution calculated for different electrode configurations to the ratio U/d, we show that the parameter U/d can differ significantly from the actual calculated values of the local electric field inside the treated tissue. By calculating the needed voltage to obtain E > U/d inside the target tissue, we showed that better electric field distribution can be obtained by increasing the number and changing the arrangement of the electrodes.

CONCLUSION

Based on our analytical and numerical models of the local electric field distribution we show that the applied voltage, configuration of the electrodes and electrode position need to be chosen specifically for each individual case, and that numerical modeling can be used to optimize the appropriate electrode configuration and adequate voltage. Using numerical models we further calculate the needed voltage for a specific electrode configuration to achieve adequate E inside the target tissue while minimizing damages of the surrounding tissue. We present also analytical solutions, which provide a convenient, rapid, but approximate method for a pre-analysis of electric field distribution in treated tissue.

Electric field-mediated transport of plasmid DNA in tumor interstitium in vivo

Local pulsed electric field application is a method for improving non-viral gene delivery. Mechanisms of the improvement include electroporation and electrophoresis. To understand how electrophoresis affects pDNA delivery in vivo, we quantified the magnitude of electric field-induced interstitial transport of pDNA in 4T1 and B16.F10 tumors implanted in mouse dorsal skin-fold chambers. Four different electric pulse sequences were used in this study, each consisted of 10 identical pulses that were 100 or 400 V/cm in strength and 20 or 50 ms in duration. The interval between consecutive pulses was 1 s. The largest distance of transport was obtained with the 400 V/cm and 50 ms pulse, and was 0.23 and 0.22 microm/pulse in 4T1 and B16.F10 tumors, respectively. There were no significant differences in transport distances between 4T1 and B16.F10 tumors. Results from in vivo mapping and numerical simulations revealed an approximately uniform intratumoral electric field that was predominantly in the direction of the applied field. The data in the study suggested that interstitial transport of pDNA induced by a sequence of ten electric pulses was ineffective for macroscopic delivery of genes in tumors. However, the induced transport was more efficient than passive diffusion.

Feasibility of employing model-based optimization of pulse amplitude and electrode distance for effective tumor electropermeabilization

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.

Sequential finite element model of tissue electropermeabilization

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.