Sage Journals: Discover world-class research

Abstract

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.

Keywords

irreversible electroporation apoptosis applied electric energy mathematical analysis H&E staining TUNEL assay

Introduction

Irreversible electroporation (IRE) has been used to treat various types of cancer clinically.^1

–6 The IRE is conducted through electrodes to deliver electrical energy that consists of a series of short-duration and high-intensity electrical pulses. The high-intensive electric energy produces irreversible structural changes in the cell membrane, which causes cells to undergo apoptotic pathways and consequently necrotic cell death.^7,8 This nonthermal mechanism of IRE is a big advantage, as compared to thermal ablation such as radiofrequency ablation, because the major blood vessel and nerve architecture as well as the extracellular matrix are preserved during IRE.^9

–13 This enables IRE-induced lesions to be rapidly healed within 2 weeks after treatment.¹⁴ This also enables a surgeon to apply IRE to complicated pancreatic cancer.¹

Planning how to apply IRE is important before practicing. Simulations on the electric field distribution were used to correlate the experimental data with the aim to predict the ablation area with electric field strength to be applied.^9,10,15 The ablated area reportedly increases with the applied electric field strength, characterizing the threshold of IRE in accompanied conductivity increase in applied tissue. However, these simulations have limitations to reflect significant parameters including applied electric energy. To overcome the complex geometry of targeted tissue, Bergues Pupo et al, Dev et al, and Čorović et al have used an array of needle electrodes^16
–18 or point current source¹⁹ for the electric fields. The mathematical models are in good agreement with the region of ablated tissue measured from histology of liver tissue.²⁰ However, there has been no report of histological and mathematical matching based on the electric energy in real tissue analysis.

In this study, we introduced the formula to express analytical electric energy and compared it to experimental results. Most studies assumed that the radius of electrodes was negligibly small, as compared to the distance between the electrodes,¹⁷ however, the exact electric field can be calculated.²¹ We performed the IRE on the liver tissue, measured the ablated area based on histological analysis, and compared it with the mathematical calculation from the electric energy assuming that the tissue is homogeneous. We focused on the static relationship between electric energy and the physiological response of targeted tissue mathematically and analytically at 10 hours after treating of IRE when apoptosis in IRE-applied tissue is severe.²²

Materials and Methods

Analytic Modeling of the Ablation Area

We developed a model that can predict the area of tissue to be ablated or damaged from apoptosis by electroporation to provide guidance during the electroporation experiment. The model considers time dependence of the critical electric field of ablation based on the concept of applied electric energy. The criterion of the ablation is that the local electric field strength exceeds a critical value.^23

–26 Assuming the time-independent electrical conductivity of tissue,⁸ this criterion results in a time-independent ablation area. However, in an experiment, the ablated area increases as the time of electric shock increases, and this has been empirically modeled by a reduction in the critical electric field over time.²⁷ The reduction is empirical and needs further clarification. In our modeling, we assumed that the reduction in the critical electric field originates from the applied electric energy, which is closely related to the electric field but increases over time. The energy is a function of electric field and time. At a constant time, the applied electric energy is a representation of the electric field, and so the criterion of critical electric field and its energy are the same. Our model thus reproduces previous research^23

–26 for constant time. As the time increases, the electric energy also increases. The applied electric energy is assumed to irreversibly cause pores in cell membrane and apoptosis.²³ In other words, we use the criterion of the critical electric field that reduces over time, and the reduction behavior is determined by the applied electric energy. This behavior applies to a certain range of the time and the field strength near experimentally used values. This is because an infinitely long time and an infinitely small electric field do not cause pore formation. In addition, the electric field criterion is a step function and predicts whether the ablation occurs. However, the criterion is statistical in the real situation²⁷ and was beyond the scope of this study.

The formulation of electric energy criterion is as follows. The electric power p per unit volume is E ²/ρ, where E is the electric field, and ρ is the resistivity of the tissue. The applied electric energy density w is p·t, where t is the time duration of the electric field application. Then

w = E^{2} \cdot t / ρ .

If w exceeds a critical value w_c , then ablation occurs. To predict the ablation area, we need to know both t and E.

The electric field of the tissue is calculated with the following assumptions. Usually, the electrodes’ length is larger than the distance between the electrodes, and the effect of tissue surface can be neglected. So, we assumed a 2-dimensional problem with 2 cylindrical electrodes and zero potential at infinity. As a first approximation, the applied electric energy is cumulative. Hence, we solved the problem for an electric field time independently and considered the effect of time later.

The simplified electroporation problem has the analytic solution of electric field. The electric potential Φ (x, y) is obtained by solving the Laplace equation,

\frac{\partial^{2} Φ (x, y)}{\partial x^{2}} + \frac{\partial^{2} Φ (x, y)}{\partial y^{2}} = 0,

according to the boundary condition described above. We assumed the homogeneity of tissue, so that the dielectric constant did not enter the Equation 2. D is the distance between the centers of the electrodes, and a the radius of the electrodes (Figure 1A). The voltages of the left and right electrodes are V ₀ and −V ₀, respectively. The electric field of the parallel electrodes, or the conducting cylinders, is calculated using the method of image charges.²⁸ Let $d = (D + \sqrt{D^{2} - 4 a^{2}}) / 2$ , d_i = a²/d, ${\vec{r}}_{1} = (x - (D / 2 - d_{i})) \hat{i} + y \hat{j}$ be the position vector from an image line charge to a point (x, y) and ${\vec{r}}_{2} = (x + (D / 2 - d_{i})) \hat{i} + y \hat{j}$ the position vector from another image line charge to a point (x, y), $r_{1} = | {\vec{r}}_{1} |$ and $r_{2} = | {\vec{r}}_{2} |$ . Then, the potential out of the electrodes is given by

Φ (x, y) = \frac{V_{0}}{\ln (a / d)} \ln \frac{r_{2}}{r_{1}},

and the electric field is given by

\vec{E} (x, y) = - \vec{\nabla} Φ (x, y) = \frac{V_{0}}{\ln (a / d)} (\frac{{\vec{r}}_{1}}{r_{1}^{2}} - \frac{{\vec{r}}_{2}}{r_{2}^{2}}) .

Figure 1.

The photo of the electrodes used in the experiment.

The solution of electric field, the applied voltage, and time duration of pulses gives applied electric energy density and ablation area. Let n be the number of the pulses, and Δt the time duration of the pulse. From Equation 1,

w (x, y) = {[E (x, y)]}^{2} n Δ t / ρ,

where, $E = | \vec{E} |$ . The curve of points where the electric energy density is the critical value w_c or w (x, y) = w_c is the ablated area’s boundary, the boundary is explicitly obtained as

y (x) = \pm \sqrt{- b^{2} - x^{2} + 2 b \sqrt{{(V_{0} \sqrt{n})}^{2} k + x^{2}}},

where, b = D/2 − d_i, and k = Δt/[ρw_cln(a/d)] . The area of ablation is obtained by numerically integrating the area in the boundary using Mathematica software (Wolfram Research, Champaign, Illinois). The ablated area is determined by the geometry of the system (b, a/d) and the applied voltage and the time $(V_{0} \sqrt{n Δ t})$ . The voltage and the number of pulses combine to the form of $V_{0} \sqrt{n}$ and determine the area. Different V ₀ and n may result in the same ablation area if $V_{0} \sqrt{n}$ are the same. The critical electric energy density w_c and the resistivity ρ are the properties of the tissue, and their product is obtained by fitting the experimental ablation area versus $V_{0} \sqrt{n}$ .

In short, if we assume that the ablation occurs for the applied electric energy density larger than the critical value, then the areas of ablation for given applied voltages, V ₀, and numbers of pulses, n, are the same for same $V_{0} \sqrt{n}$ . The boundary of ablated area is obtained analytically.

In Vivo Experimental Procedure

This study was approved by the Institutional Animal Care and Use Committee at the Seoul National University College of Medicine. Eight-week-old male Sprague Dawley rats weighing 250 to 350 g were used. All of the animals received humane care from properly trained professionals in compliance with the institutional guidelines of Seoul National University and current international laws and policies. The rats were maintained in a conventional animal colony and were adapted for a period of 2 weeks to provide relaxation before testing.

The animals were anesthetized with an intraperitoneal injection of 10 mg/kg Zoletil (Virbac, Carros, France) solution that was administrated throughout the procedure. A 15-mm midline abdominal incision was made under sterile conditions after shaving, and subsequently, the liver was entirely exposed. The electrodes fabricated with stainless steel wires with diameter of 100 μm, length of 5 mm, and center-to-center distance of average 2 mm (±0.16) were carefully inserted perpendicularly to the surface of liver to a depth of 5 mm (Figure 1).²² Then, 8, 32, and 64 rectangular direct current pulses of 100 μs width were sequentially applied at a frequency of 1 Hz with applied voltages of 150, 350, and 500 V/cm,²² which prevents heating around tissue near the electrodes because of higher dissipation of thermal energy as reported in a previous study,²² using a pulse generator (Nano-porator; SolcoBiomedical Ltd, Co, Korea). As a result of simulation for temperature distribution, these electric conditions did not cause thermal damage to liver tissues.²² A total of 36 rats were used. An IRE lesion was created per lobe of liver in rat, having a total of 3 lesions in a liver, to isolate the effect of electric pulses of neighboring lesions. The location of lesion was marked with tissue ink.²² Subsequently, the abdominal wall was sutured, and the animals were kept alive until 10 hours at which time the applied tissues undergo severe apoptosis.^7,22 All animals were then checked every couple of hours to ensure that they recovered from surgery and stayed healthy without complications. The regions from the liver tissues that had been treated as well as the untreated regions distant to the lesions were harvested after 10 hours of IRE, and the animals were then euthanized. The liver segments were fixed in 10% (vol/vol) buffered formalin, dehydrated, embedded in paraffin, and then sectioned with a microtome with an axial cut through the plane normal to the direction of the electrode length, without decreasing the strength of the electric fields that were applied, as in the previous study.²² Each sample was stained with hematoxylin and eosin (H&E) to assess the state of necrosis, and terminal deoxynucleotidyl transferase dUTP nick end labeling (TUNEL) assay kit (S7100; Millipore, Temecula, California) was used to assess apoptosis. Terminal deoxynucleotidyl transferase dUTP nick end labeling is a method for detecting DNA fragmentation that results from apoptotic signaling cascades by labeling the terminal end of nucleic acids.

The slides for the TUNEL assay were imaged with an image acquisition system (Olympus BX-51 with digital imaging system [Image-Pro plus 4.5]), and the area colored brown by TUNEL staining was then analyzed via Matlab to quantify the degree of apoptosis in the region of interest.

Results

Figure 2B shows the macroscopic images of H&E staining of liver tissue at 10 hours of IRE operation. In this case, the electrode diameter, the distance between the electrodes, the applied voltage, and the number of electric pulses were 0.1 mm, 1.0 mm, 1 kV, and 64 pulses, respectively. The H&E staining delineates the effects of IRE around the electrodes as expressed in a previous study.²² The affected area includes the tissues surrounding the electrodes and between the electrodes. The lesions at the site of electrodes (the dotted circle “E”) become significantly injured, which is microscopically eosinophilia, as compared to the normal parenchyma. The eosinophilia is ascribed to the chromosomal shrinkage of cells and demolished collagen structures. The boundary between demolished and normal tissues is a congested zone, which may be ascribed to the regeneration process. The capillaries near the electrodes are preserved within the destructive tissue zone.

Figure 2.

The geometry of the 2 electrodes and the macroscopic image of irreversible electroporation (IRE)-treated liver tissue with hematoxylin and eosin (H&E) staining at 10 hours post-IRE operation. A, “a” is the radius of an electrode, “D” is the distance between the centers of the electrodes, and “d” is the distance between the image charges. B, In this experiment, “a” is 0.1 mm, “D” is average 2.0 mm (±0.16), the applied voltage is 500 V, and the number of electric pulses is 64. The dotted circles (“E”) denote the positions of electrodes, and the arrows indicate the blood capillary that was preserved within destructed tissue. C, The image of the right side electrode “B.” Tissue around electrode has a distinct boundary with normal tissue and dense aggregation of many chromosomes due to destructive extracellular matrix, unlike normal morphology of tissue.

These phenomena were more apparent as a form of apoptosis in TUNEL assay. Figure 3 exhibits the apoptotic appearance at 10 hours after applying IRE, that is, brownish by TUNEL staining. As expected, these apoptotic areas tend to be more widened with applied electric energy. Low electric energy induces apoptosis only at the area surrounding the electrodes, regardless of anode or cathode. Apoptosis at the area between the electrodes seems to start with the applied electric energy of w_c = 5.9 × 10⁵ J/m³, which should be an electrically critical point for entire destruction by electroporation. Figure 4 graphs the relationship between the apoptotic area and the applied electric energy. The apoptotic area increased almost linearly with rising applied voltages and applied pulsing numbers, that is, increasing with applied electric energy.

Figure 3.

Typical macroscopic images of irreversible electroporation (IRE)-treated liver tissue with terminal deoxynucleotidyl transferase dUTP nick end labeling (TUNEL) assay. The normal tissue shows no staining, whereas the IRE-treated tissues show the pathological apoptosis near the electrodes. With increasing applied electric energy, the apoptotic ablation area is gradually widened. The critical applied electric energy is 5.9 × 10⁵ J/m³ for all the cases. All images were magnified to ×4.

Figure 4.

A graph for the ablation area. The tendency of apoptotic area is increasing with applied electric energy at 10 hours after irreversible electroporation (IRE).

Figure 5 shows the fitting of electrically injured area, as compared to mathematically calculated apoptotic areas. The calculation for their fitting was based on the electric energy, that is, applied voltages and pulsing numbers (total duration time). In Equation 6, the ablation area was calculated by the curve of points with the critical electric energy. In the curve equation, ρw_c is an unknown value that was found by fitting the experimental ablation area versus $V_{0} \sqrt{n}$ in Figure 5. The predicted ablation area was calculated by the criterion of electric energy (Equation 6) with given geometry, $V_{0} \sqrt{n}$ and ρw_c . With the ρw_c found by the fitting, the areas of ablation contour for V ₀ and n used at experimental situation in Figure 2 were drawn in Figure 6. As expected, the ablation areas increased with applied electric energy, confirming that the critical point of entire apoptotic ablation was above electric energy density of 5.9 × 10⁵ J/m³, where the conductivity, 1/ρ, was 1.24 mS/cm from the result of Haemmerich et al.²⁹ The confirmation was reflected in Figure 7, which is the superposition of the appearances of apoptosis with the simulated electric energy density. Spots that the energy density is over 2.4 × 10⁸ J/m³ are observed to be entirely destructive.

Figure 5.

The fitting of the ablation area obtained in the experiment to that produced by the theoretical model. The model explains the trend in the ablation against a measure of applied electric energy $E_{av} \sqrt{n}$ , where E _av is the electric field strength and n is the number of pulses. The effect of time was considered as n.

Figure 6.

The contours of the electric energy density and the shape of ablated tissue according to the fitted theoretical model. The contours are similar to those of the ablation area obtained in the experiment, indicating that the theoretical model explains the trends in the ablated area. The arrow indicates the boundary of the ablated tissue, and the numbers in the last figure are the strength of the electric field normalized to the critical electric energy density for an ablation of 64 pulses. In other tissues, where the arrows are not indicated, the boundary can be found by comparing colors with those in the last subfigure as arrows. ps indicates pulses.

Figure 7.

The superposition of terminal deoxynucleotidyl transferase dUTP nick end labeling (TUNEL) assay with isoelectric energy density curves. The superposition clearly shows the consistency of the applied electric energy with the simulation, especially exhibiting that the entire destruction between electrodes starts at 5.9 × 10⁵ J/m³. The electric energy density above 5.9 × 10⁵ J/m³ causes the tissues to undergo apoptosis.

Discussion

As in previous reports,^7,22 the H&E staining showed a well-demarcated margin between the untreated and the treated regions with widespread pale eosinophilic cytoplasm, congestion in sinusoids, hemorrhage, pyknosis, karyorrhexis, and rupture of connective tissues in Figure 2. Some of the blood capillaries in the treated section were preserved without hemorrhage, which would contribute to a quick recovery of the tissues treated through microcirculation. Even though the sites of the electrodes are not exactly recognized, the loosened surrounding tissues indicate the location of the electrodes. The difference may be due to the presence of coagulative necrosis by thermal injury.³⁰ Recently, several studies on tissues near the electrode have shown such observations: reporting vaporization, carbonization, coagulation, enzyme denaturation, and so on, as a result of the application of electric energy.³⁰ However, electroporation did not always exhibit a thermal effect but a much weaker effect on the adjacent tissues in many other cases.

The H&E staining has been usually utilized to evaluate the destructive area of IRE in liver tissue.^{2,20,22,30
–32} However, the evaluation was based on results from necrosis. In the present study, the ablation area was evaluated according to apoptosis using TUNEL assay. The TUNEL assay shows the breakage of DNAs, which is an important biochemical indicator of apoptosis. In addition, TUNEL staining provides the clear boundary between ablated area and nonaffected area by distinguishable color. As expected, the treated tissue exhibited fiercely apoptotic appearances at 10 hours after ablation, particularly around and/or between the electrodes in TUNEL assay.

Figure 4 shows the ablation area based on the apoptotic area, which analytically exhibited linearity in response to the increase in the applied voltage and the number of pulses. Figure 5 also supported the relationship of the total amount of electric energy with the ablation area. The electroporated ablation from the experiment was compared with our theoretical model, and the results showed consistency. The model’s electric energy threshold 5.9 × 10⁵ J/m³ is similar to those of previous studies: 2.6 × 10⁵ J/m³ for rabbit livers with 8 pulses of 100 µs duration and threshold electric field of 700 V/cm³³ and 1.2 × 10⁶ J/m³ for porcine livers with 99 pulses of 100 µs duration and threshold electric field of 423 V/cm.¹⁰

However, the histological analysis and mathematical simulation did entirely not produce identical results. The apoptotic areas for the 32 pulses at 700 and 1000 V/cm were quite similar, possibly as a result of the errors in the experimental procedures. Because the electrodes are flexible, the electrode positions at the histological section are slightly crooked even when the wire was guided by polydimethylsiloxane polymer.²² Thus, Figure 3 exhibits wider electrode sites for 8 pulses at 150 V/cm and narrower for 32 pulses at 150 V/cm. These could explain the errors in our analysis. In addition, liver tissue might be inhomogeneous since it involves cells, extracellular matrix, blood capillaries, and various kinds of blood cells. These properties can cause distortions in the homogeneous distribution of the electric field within the tissue, resulting in an insufficient delivery of electric energy needed to produce an irreversible transmembrane potential.³⁴ The injured spots for the electrodes in Figure 3 were asymmetrical, and noninfluenced areas were observed in the ablated tissue.³⁵ Also, the effect of conductivity change with time as electroporation progresses, or dynamic conductivity, should be explored.^{3,27,33,34,36} The change is caused by the opening of membrane pores. The conductivity change results in the change in the ablation area.

The strength of the applied electric field depends on the conductivity of the tissue between the electrodes as well as the cell, which affects the electrostatics of the system. Therefore, there were regions between the electrodes where IRE could not be achieved, even though a sufficient electric energy was applied to irreversibly cause damages to the cells between the electrodes. For these reasons, even though the same amount of electric energy was applied, the apoptotic rate had a slight variation. Figure 6 indicates the results calculated by assuming homogeneity of the tissue. Thus, the ablated areas were apparently symmetric for both electrodes, completely destroying the tissue starting at the electric energy of 5.9 × 10⁵ J/m³. As in Figure 7, this simulation was consistent with the experimental results, even though there were errors in the sites where the electrodes were introduced. Samples at which the electric energy of over 5.9 × 10⁵ J/m³ was applied (350 V/cm, 32 ps; 350 V/cm, 64 ps; 500 V/cm, 32 ps) appeared to be fully apoptotic up to the line of electric field of 500 V/cm for 64 ps and larger fields for smaller pulse numbers. Our model was simple and did not include properties of heterogeneity of applied tissue and thermal effects due to IRE, but the simulation based on electric energy was good. A similar tendency has been reported in other studies.³⁶ Thus, this simulation can help practitioners better design and predict the effects of electroporation as a clinical treatment.

Recently, a question was raised concerning the reliability of numerical simulation to predict the ablation of IRE.³⁷ Wimmer et al compared the numerical simulation and experiment of IRE and concluded that a numerical simulation underestimates the ablation area for some cases. First, it is possible that the applied electric energy criterion would improve their simulation. In their experiments, the voltage and the pulse application time were changed simultaneously, so it is difficult to predict whether the critical electric energy criterion, or the time-dependent critical electric field criterion, would fit better than the constant critical electric field criterion used in their numerical simulation. Second, it is likely that the conductivity change over electric field^{3,27,33,34,36} would improve it. To improve their simulation in the future, we compared an unsuccessful and a successful case. The unsuccessful case is the use of bipolar electrodes, which have the positive and negative voltage pole in 1 electrode (protocol no. 6 in³⁷). The successful case was the use of 2 parallel electrodes (protocol no. 4 in³⁷). The 2 cases have the same experiment conditions, except for the geometry of the electrodes. Since the difference is time independent, the conductivity change as a function of static electric field may improve their simulation.

The limitations of this study are that the thermal effects from electric fields were not reflected in the theoretical calculation when analyzing the apoptotic area, even though other studies have insisted that reversible electroporation, and even IRE, does not introduce thermal effects, even near the electrode. An additional study is required to accurately assess this effect in terms of the performance in simulations and actual experiments.

Here, we attempted to take into account the apoptotic area for liver tissue with applied electric energy densities. The experimental and theoretical results obtained could facilitate the application of electroporation as a clinical treatment.

Footnotes

Abbreviations

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Bagla

Papadouris

. Percutaneous irreversible electroporation of surgically unresectable pancreatic cancer: a case report. J Vasc Interv Radiol. 2012;23(1):142–145.

Kingham

Karkar

D’Angelica

. Ablation of perivascular hepatic malignant tumors with irreversible electroporation. J Am Coll Surg. 2012;215(3):379–387.

Neal Ii

Garcia

Kavnoudias

. In vivo irreversible electroporation kidney ablation: experimentally correlated numerical models. IEEE Trans Biomed Eng. 2014;62(2):561–569.

Usman

Moore

Talati

Watkins

Bilfinger

. Irreversible electroporation of lung neoplasm: a case series. Med Sci Monit. 2012;18(6):CS43–CS47.

Valerio

Stricker

Ahmed

. Initial assessment of safety and clinical feasibility of irreversible electroporation in the focal treatment of prostate cancer. Prostate Cancer Prostatic Dis. 2014;17(4):343–347.

Yarmush

Golberg

Serša

Kotnik

Miklavčič

. Electroporation-based technologies for medicine: principles, applications, and challenges. Annu Rev Biomed Eng. 2014;16:295–320.

Kim

Sung

Baik

. Changes of apoptosis in tumor tissues with time after irreversible electroporation. Biochem Biophys Res Commun. 2013;435(4):651–656.

Xiao

Yao

Liu

. Irreversible electroporation and apoptosis in human liver cancer cells induced by nanosecond electric pulses. Bioelectromagnetics. 2013;34(7):512–520.

Miklavcic

Semrov

Mekid

Mir

. A validated model of in vivo electric field distribution in tissues for electrochemotherapy and for DNA electrotransfer for gene therapy. Biochim Biophys Acta. 2000;1523(1):73–83.

10.

Sano

Neal

Garcia

Gerber

Robertson

Davalos

. Towards the creation of decellularized organ constructs using irreversible electroporation and active mechanical perfusion. Biomed Eng Online. 2010;9:83.

11.

Jiang

Davalos

Bischof

. A review of basic to clinical studies of irreversible electroporation therapy. IEEE Trans Biomed Eng. 2015;62(1):4–20.

12.

Scheffer

Nielsen

de Jong

. Irreversible electroporation for nonthermal tumor ablation in the clinical setting: a systematic review of safety and efficacy. J Vasc Interv Radiol. 2014;25(7):997–1011.

13.

Rossmeisl

Garcia

Pancotto

. Safety and feasibility of the NanoKnife System for irreversible electroporation ablative treatment of canine spontaneous intracranial gliomas. J Neurosurg. 2015;123(4):1008–1025.

14.

Garcia

Rossmeisl

Neal

. Intracranial nonthermal irreversible electroporation: in vivo analysis. J Membr Biol. 2010;236(1):1271–1236.

15.

Edd

Horowitz

Davalos

Mir

Rubinsky

. In vivo results of a new focal tissue ablation technique: irreversible electroporation. IEEE Trans Biomed Eng. 2006;53(7):1409–1415.

16.

Bergues Pupo

Reyes

Bergues Cabrales

. Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy. Biomed Eng Online. 2011;10:85.

17.

Čorović

Pavlin

Miklavčič

. Analytical and numerical quantification and comparison of the local electric field in the tissue for different electrode configurations. Biomed Eng Online. 2007;6:37–50.

18.

Dev

Dhar

Krassowska

. Electric field of a six-needle array electrode used in drug and DNA delivery in vivo: analytical versus numerical solution. IEEE Trans Biomed Eng. 2003;50(11):1296–1300.

19.

Özdemir

. Current Cancer Treatment—Novel Beyond Conventional Approaches. 2011:585–617.

20.

Rubinsky

Onik

Mikus

. Irreversible electroporation: a new ablation modality—clinical implications. Technol Cancer Res Treat. 2007;6(1):37–48.

21.

Griffiths

. Introduction to Electrodynamics. Upper Saddle River, New Jersey: Prentice Hall; 1999.

22.

Choi

Kim

Chung

Kim

Park

. Preclinical analysis of irreversible electroporation on rat liver tissues using a microfabricated electroporator. Tissue Eng Part C Methods. 2010;16(6):1245–1253.

23.

Kotnik

Kramar

Pucihar

Miklavcic

Tarek

. Cell membrane electroporation—part 1: the phenomenon. IEEE Electr Insul Mag. 2012;28(5):14–23.

24.

Miklavcic

Corovic

Pucihar

Pavselj

. Importance of tumour coverage by sufficiently high local electric field for effective electrochemotherapy. EJC Suppl. 2006;4:45–51.

25.

Miklavčič

Beravs

Šemrov

Čemažar

Demšar

Serša

. The importance of electric field distribution for effective in vivo electroporation of tissues. Biophys J. 1998;74(5):2152–2158.

26.

Kranjc

Markelc

Bajd

. In situ monitoring of electric field distribution in mouse tumor during electroporation. Radiology. 2014;274(1):115–123.

27.

Garcia

Davalos

Miklavcic

. A numerical investigation of the electric and thermal cell kill distributions in electroporation-based therapies in tissue. PloS One. 2014;9(8):e103083.

28.

Cheng

. Field and Wave Electromagnetics. 2nd ed. New York: Addison-Wesley; 1989.

29.

Haemmerich

Staelin

Tsai

Tungjitkusolmun

Mahvi

Webster

. In vivo electrical conductivity of hepatic tumours. Physiol Meas. 2003;24(2):251–260.

30.

Dunki-Jacobs

Philips

Martin

. Evaluation of thermal injury to liver, pancreas and kidney during irreversible electroporation in an in vivo experimental model. Br J Surg. 2014;101(9):1113–1121.

31.

Marcan

Pavliha

Music

Fuckan

Magjarevic

Miklavcic

. Segmentation of hepatic vessels from MRI images for planning of electroporation-based treatments in the liver. Radiol Oncol. 2014;48(3):267–281.

32.

Burke

. Comparison of percutaneous ablation technologies in the treatment of malignant liver tumors. Semin Intervent Radiol. 2014;31(2):129–137.

33.

Sel

Cukjati

Batiuskaite

Slivnik

Mir

Miklavcic

. Sequential finite element model of tissue electropermeabilization. IEEE Trans Biomed Eng. 2005;52(5):816–827.

34.

Corovic

Lackovic

Sustaric

Sustar

Rodic

Miklavcic

. Modeling of electric field distribution in tissues during electroporation. Biomed Eng Online. 2013;12:16.

35.

Ben-David

Ahmed

Faroja

. Irreversible electroporation: treatment effect is susceptible to local environment and tissue properties. Radiology. 2013;269(3):738–747.

36.

Long

Bakos

Shires

. Histological and finite element analysis of cell death due to irreversible electroporation. Technol Cancer Res Treat. 2014;13(6):561–569.

37.

Wimmer

Srimathveeravalli

Gutta

. Planning irreversible electroporation in the porcine kidney: are numerical simulations reliable for predicting empiric ablation outcomes? Cardiovasc Intervent Radiol. 2015;38(1):182–190.

Histological and Mathematical Analysis of the Irreversibly Electroporated Liver Tissue

Abstract

Keywords

Introduction

Materials and Methods

Analytic Modeling of the Ablation Area

In Vivo Experimental Procedure

Results

Discussion

Footnotes

Abbreviations

Declaration of Conflicting Interests

Funding

References