Multi-tissue Analysis on the Impact of Electroporation on Electrical and Thermal Properties

Abstract

Objective:

Tissue electroporation is achieved by applying a series of electric pulses to destabilize cell membranes within the target tissue. The treatment volume is dictated by the electric field distribution, which depends on the pulse parameters and tissue type and can be readily predicted using numerical methods. These models require the relevant tissue properties to be known beforehand. This study aims to quantify electrical and thermal properties for three different tissue types relevant to current clinical electroporation.

Methods:

Pancreatic, brain, and liver tissue were harvested from pigs, then treated with IRE pulses in a parallel-plate configuration. Resulting current and temperature readings were used to calculate the conductivity and its temperature dependence for each tissue type. Finally, a computational model was constructed to examine the impact of differences between tissue types.

Results:

Baseline conductivity values (mean 0.11, 0.14, and 0.12 S/m) and temperature coefficients of conductivity (mean 2.0, 2.3, and 1.2 % per degree Celsius) were calculated for pancreas, brain, and liver, respectively. The accompanying computational models suggest field distribution and thermal damage volumes are dependent on tissue type.

Conclusion:

The three tissue types show similar electrical and thermal responses to IRE, though brain tissue exhibits the greatest differences. The results also show that tissue type plays a role in the expected ablation and thermal damage volumes.

Significance:

The conductivity and its changes due to heating are expected to have a marked impact on the ablation volume. Incorporating these tissue properties aids in the prediction and optimization of electroporation-based therapies.

I. Introduction

ELECTROPORATION is a technique that has been used to reversibly permeabilize the membrane of a cell using an externally applied electric field [1]. This permeabilization is useful for several applications, including the delivery to and extraction of molecules from cells and the inactivation of bacteria [2]. Therapeutically, electroporation has been harnessed to enhance or enable delivery of genes, proteins, chemotherapeutics, or other molecules into cells [3].

Irreversible electroporation (IRE) later was found to be an effective technique for tumor ablation in multiple types of cancer [4], [5]. IRE has demonstrated particular efficacy in treating locally advanced and borderline pancreatic tumors [6], [7], attributable to its unique ability to induce non-thermal cell death, even in close proximity to vasculature and nerves. This poises IRE as a suitable standalone or adjuvant therapy to resection, with the goal of margin accentuation to make tumors more amenable to surgery. IRE has also shown promise in treating liver cancers, where it boasts benefits of avoiding thermal damage to large hepatic vessels [8]. Preclinical studies also demonstrate IRE as suitable for the treatment of intracranial disorders [9]; in addition to the development of a nonthermal lesion, high voltage PEFs induce voltage-dependent secondary volumes of blood-brain barrier (BBB) disruption [10]–[13].

Paramount to the success of these treatments is exposing tissue to electrical doses capable of inducing the desired electroporation effects. In the case of therapies utilizing the reversible regime, cells must experience a sufficient increase in the transmembrane potential in order to undergo permeabilization [14]. Similarly, the success of IRE is dependent on the cells undergoing an even higher increase in transmembrane potential to the point where they are unable to recover from the membrane disruption. Clinically, IRE treatments create this disruption through delivery of short, high-voltage pulsed electric fields (PEFs) using one or more needle electrodes inserted directly into the target tissue [15]. This needle configuration produces an electric field distribution (EFD) with relatively high field magnitudes immediately adjacent to the electrodes which decays centimeters away. Ensuring that the entire tumor is exposed to lethal IRE pulses may be aided by treatment planning models [16], [17]. Since treatment outcome is highly dependent on the applied EFD in the tissue, these numerical models must incorporate pulsing parameters such as the electrode spacing and shape, number of pulses, and applied electric potential to inform adequate field coverage. Additionally, the nature of electroporation causes the electrical conductivity of the tissue to be electric-field dependent. This non-linearity is due to the creation of low-resistance pathways during permeabilization [18]. Accounting for this non-linear conductivity is crucial for ensuring accurate predictions of the EFD [19], [20]. In a typical needle-electrode IRE treatment, the tissue close to the electrode(s) will exhibit a higher conductivity change as it is exposed to a higher electric field strength than distant tissue. Subsequently, this high local potential induces the highest current density, leading to increased Joule heating near the electrodes. These effects are important to include when incorporating ex vivo data collected at room temperature [21]. Thus, accurate prediction of the EFD requires the inclusion of α, the temperature coefficient of resistance which quantifies changes in conductivity due to change in temperature. Values for α exist in the literature, however, the conditions under which α is extracted (temperature, frequency, electrode setup, time after death, etc.) vary considerably and are performed on un-electroporated tissue. In this work, we examine the compounded effect that Joule heating has on the electrical conductivity when coupled with electroporation. We also demonstrate the non-linear behavior of dynamic conductivity changes over multiple pulses, suggesting a saturation of electroporation effects followed by linear increases due to Joule heating. As field-dependent conductivity is difficult to calculate from current output generated using a needle-electrode configuration [20], the use of parallel-plate electrodes simplifies the shape factor relating the current to the applied voltage [17]. Here, we utilize a parallel-plate configuration to characterize the dynamic conductivity of fresh porcine tissue during exposure to uniform PEFs. Determining this property from primary human tissue samples is preferable, but these are difficult to obtain, and minimizing variability due to variations in tissue acquisition time and tissue handling is challenging. Thus, animal models are often used as stand-ins for human tissue; cross-species variations are thought to have little impact in terms of electrical behavior [22]. Our objective is to evaluate and compare the dynamic electrical conductivity of brain, liver, and pancreatic porcine tissue using a parallel-plate configuration. This configuration enables the generation of conductivity functions that may be used to predict EFDs in other electrode configurations. Additionally, we present a method for quantifying the impact of Joule heating on the electrical conductivity during multi-pulse electroporation treatments, and report the resulting temperature coefficients for each tissue type. Furthermore, we illustrate the effects of electroporation and Joule heating on the resulting EFD by incorporating results from the parallel-plate experiments into a computational model of a typical two-needle electrode treatment. Additional computational models were constructed for validation of our dataset with existing preclinical data.

II. Methods

A. Experimental Animals

The overall health of each animal was examined upon arrival. Both male and female pigs were used in this study, with weights between 45 and 272 kg. Pigs were sedated via intramuscular injection of a Xylazine/Telazol mix in the lateral neck, behind the base of the ear, using a 20–22 gauge and up to 1 inch long needle. After 15 minutes, or until fully sedated, the pigs were given an intracardiac injection of euthanasia solution directly into the heart using an 18 gauge, 4 inch long needle fitted to a syringe. Euthanasia was performed in the same room as sedation. The studies were approved by the Institutional Animal Care and Use Committee and were carried out in compliance with the NIH Guide for the Care and Use of Laboratory Animals.

B. Conductivity and Temperature Measurements

The pancreas, liver, and brain were excised and immediately placed into a room temperature (20 to 25 °C) modified phosphate-buffered saline (PBS) solution. The modified PBS solution was designed to mimic blood in regards to the electrical conductivity and osmolarity. The solution was composed of the following: 77.5 mM NaCl, 1.52 mM KCl, 14.5 mM Na₂HPO₄, 1.0 mM KH₂PO₄, and 35.4 g/L of sucrose. Experiments were performed within 60 minutes following tissue excision in order to reduce the impact of degradation on the measured tissue properties. Prior to measurement, tissue samples were removed from the PBS, dried using a paper tissue, further sectioned into smaller pieces and placed into a cylindrical cavity within an insulating polydimethylsiloxane (PDMS) mold until just full. The thickness of the mold varied between 0.5 and 0.7 cm, with a constant radius of 0.3 cm. Tissue was carefully placed to ensure the full volume of the mold was occupied to promote electrical contact with electrodes. Next, the mold and tissue were placed between two parallel-plate electrodes with 2 cm² surface area (BTX, Harvard Apparatus, Cambridge, MA).

A fiber optic temperature probe (LumaSense, Inc., Santa Clara, CA) was inserted through a hole in the side of the insulating PDMS mold to the center of the tissue in order to monitor temperature. Temperature values were recorded at either 2 or 4 Hz, with most of the measurements taken at 4 Hz. A 25 V, 100 μs PEF was applied in order to determine the initial DC conductivity of the tissue. This voltage was chosen so as not to induce electroporation, yet produce a high enough output current to be detectable by the current probe. Then, a total of 100 PEFs with 100 μs widths were delivered at 1 Hz. The voltage was applied such that an electric field at either 250, 500, 750, 1000, 1500, 2000, 2500, or 3000 V/cm was generated. Voltage and current were captured during each pulse using an oscilloscope (DPO2012, Tektronix Inc., Beaverton, Oregon), a 1000× high voltage probe (Enhancer 3000, BTX, Holliston, MA), and a 10× current probe (3972, Pearson Electronics, Palo Alto, CA). A schematic depiction of the experimental setup is shown in Fig. 1.

Fig. 1. Parallel-plate configuration enables evaluation of dynamic electrical conductivity and thermal properties.

A cylindrical porcine tissue sample is placed within an insulating mold to ensure a simple shape factor. Pulsed electric fields (PEFs) are applied using a commercial generator, and resulting waveforms are captured using a voltage and current probe. Additionally, temperature is monitored using a fiber optic probe (0.5 mm diameter).

C. Data Analysis

The reported electrical conductivity was quantified as the average conductivity at the final 5 μs of each 100 μs pulse; this value was used so as to avoid the transient capacitive effects present at the beginning of the pulse. A MATLAB (vR2019a; MathWorks Inc., Natick, MA, US) script was written to extract the conductivity across all 100 pulses for each individual sample. Temperature versus time data were imported, and a 20-point (corresponding with 5 s) moving average was used to smooth the data. Since parallel-plate electrodes were used to apply IRE treatments across a cylindrical tissue sample, the conductivity was calculated based on the current level at the end of each IRE pulse according to (1).

$$\sigma =\frac{l}{R\cdot A}=\frac{I\cdot l}{V\cdot A}$$ (1)

Here, σ is the electrical conductivity, R the tissue resistance, A the cross-sectional area, l the sample thickness, V the applied voltage, and I the induced current. Initial DC conductivity values were calculated based on the 25 V pulse delivered prior to high-voltage IRE pulses (Table I).

TABLE I.

DC Conductivity at 37 °C and Temperature Coefficient

Property	Tissue	Mean	SD	Min	Max	Median
${\sigma }_{DC}\left[\frac{S}{m}\right]$	Pancreas	0.113	0.041	0.058	0.214	0.101
	Brain	0.142	0.032	0.091	0.221	0.136
	Liver	0.122	0.025	0.077	0.172	0.120
$\alpha \left[\frac{\%}{{}^{°}C}\right]$	Pancreas	2.0	0.71	1.2	3.3	1.9
	Brain	2.3	0.43	1.6	2.7	2.5
	Liver	1.2	0.30	0.51	1.6	1.3

The increase in conductivity due to temperature rise is typically defined using the temperature coefficient of resistance α. This parameter defines the percent change in tissue conductivity solely due to increases in temperature. However, in the present case, electroporation effects are expected to introduce compounded changes in conductivity. Thus, we quantify these changes in the same way as α is typically calculated, but here we use the symbol β to indicate that this quantity is not equivalent to α. Because the collected conductivity and temperature data is discrete in time, β must be calculated over a range of pulses rather than as an instantaneous quantity. The temperature coefficient β over a range of pulses was calculated in MATLAB according to (2),

$$\beta =100\cdot \frac{{\sigma }_2-{\sigma }_1}{{\sigma }_1\cdot \left(T_2-T_1\right)}$$ (2)

where σ₁ and T₁ are the conductivity and temperature at the starting point of the range, and σ₂ and T₂ are the conductivity and temperature at the ending point of the range.

D. Numerical Analysis

All experimental data were collected in a parallel-plate configuration in order to isolate specific electric field magnitudes and determine electrical and thermal tissue responses. In order to translate these findings to a clinically relevant case, a 3D numerical model was constructed to determine the tissue-specific differences in the EFD and Joule heating effects. This model incorporated a needle electrode array which is typical of clinical IRE treatments [23]–[25]. Thus, we are using the experimental parallel-plate data as an input to a 3D model with a needle configuration. The finite element model was constructed in COMSOL Multiphysics v5.5 (COMSOL Inc., Stockholm, Sweden); here each tissue was modeled as a 12×10×8 cm ellipsoid. Two monopolar electrodes were constructed in COMSOL with the following dimensions: 1 mm diameter, 1 cm electrode exposure, 1.5 cm electrode spacing, and a 5 cm insulating handle. The tissue domain was assigned the electrical and thermal properties of either brain, liver, or pancreatic porcine tissue as determined from the above described parallel-plate electrode characterization experiments. Values for thermal conductivity (k), specific heat (c_p), and density (ρ) were collected from the literature for each specific tissue type (Table II), while the α values determined from the above analysis (Table I) were used. Finally, the electrode properties were set to those of steel (AISI 4340) within the COMSOL Material Library.

TABLE II.

Thermal Properties used in COMSOL Simulations

Parameter	Pancreas	Brain	Liver	Source
$k_{tissue}\left[\frac{W}{m\cdot K}\right]$	0.51	0.51	0.52	[26]
$c_{p,tissue}\left[\frac{J}{kg\cdot K}\right]$	3164	3630	3540	[26]
${\rho }_{tissue}\left[\frac{kg}{m^3}\right]$	1087	1046	1079	[26]
${\omega }_b\left[\frac{1}{s}\right]$	0.0139	0.0097	0.0155	[26]
$c_{p,blood}\left[\frac{J}{kg\cdot K}\right]$	3840			[27]
${\rho }_{blood}\left[\frac{kg}{m^3}\right]$	1060			[28]

The simulated IRE protocol consisted of an applied electric potential of 2500 V between the two electrodes, a pulse duration of 100 μs per pulse, and 100 pulses delivered at 1 Hz. A mesh refinement was performed until the computed error was less than 2%; this error was quantified as the error in the electric field magnitude at the midpoint between the two electrodes from sequential mesh refinements. The final mesh contained a total of 545,402 elements with 1,463,828 degrees of freedom, as determined from an Extra Fine mesh setting in COMSOL.

During IRE treatment, two phenomena are known to affect the electric conductivity of tissue: 1) electroporation and 2) Joule heating. The electroporation effects are modeled by incorporating the dynamic conductivity curves quantified in this study to the tissue domain. Joule heating effects on electrical conductivity are modeled by incorporating the temperature coefficient of resistance α and coupling this with transient temperature increase. In the interest of comparing the EFD from compounded electroporation effects and Joule heating effects, four EFDs were computed as follows: 1) the static EFD (σ), consisting of no electroporation effects and no Joule heating effects; 2) the static EFD incorporating temperature effects on conductivity (σ(T)); 3) the dynamic EFD (σ(E)) accounting for tissue-specific electroporation effects in the absence of Joule heating effects on conductivity; and 4) the dynamic EFD (σ(E, T)) accounting for tissue-specific electroporation effects as well as Joule heating effects on conductivity. The electric potential distributions were computed by solving the governing equation (3) using a stationary solver:

$$\nabla \cdot (\sigma \nabla \varphi )=0$$ (3)

Here, ϕ represents the electric potential. A Dirichlet boundary condition was applied to the surface of one electrode with a potential of 2500 V while the other electrode surface was set to 0 V. A time-dependent study was carried out to account for heating incurred during multiple pulses. Two sources of heating were considered in the simulation: 1) Joule heating from the PEFs and 2) heat dissipation due to blood perfusion. Both sources were accounted for by incorporating a modified Pennes’ bioheat equation (4), in which a duty-cycle approach was used to average the thermal energy generated during a single 100 μs pulse over 1 s.

$$\rho c_p\frac{\partial T}{\partial t}=\nabla (k\nabla T)-{\omega }_b{c}_b{\rho }_b\left(T-T_b\right)+\frac{\sigma \cdot |\nabla \varphi {|}^2\cdot p}{\tau }$$ (4)

Here, ω_b represents the heat dissipation due to a distributed blood perfusion, p represents the pulse on-time (100 μs) and τ the pulse delivery period (1 s). Initial temperature as well as temperature of blood (T_b) were set to 37 °C. Diagonal elements of the conduction current tensor were defined by the experimentally determined α (Table I) with a maximum value of 1 S/m (5).

$$\sigma (E,T)=\sigma (E)\cdot \left\{1+\alpha \cdot \left(T-T_{blood}\right)\right\}$$ (5)

A thermal dose model, CEM₄₃, was used to establish the relative differences in thermal dosage when pulsing across each tissue [29]. CEM₄₃, equation (6), is used to represent the thermal dosage as the cumulative equivalent minutes referenced to 43°C.

$$CE{M}_{43}=\sum _{i=1}^N{R}_C^{\left(43-T_i\right)}{t}_i$$ (6)

Here, t is time in minutes and T the temperature. R_C is a constant taking on a value of 0.5 if T is above the breakpoint temperature of 43°C, or 0.25 if T is below 43°C. As a conservative approximation, a CEM₄₃ of 10 minutes is used to quantify thermal effects. It is important to note that the dose of tissue thermal damage depends on the tissue type [30]. Our assumption that thermal damage is incurred at CEM₄₃ 10 minutes in used to inform a relative comparison between the tissues investigated in this study.

III. Results

Tissue conductivity values were calculated for each electric field magnitude (n=5–6) for the 1^st and 100^th pulse in the IRE treatment. These values were adjusted to either 20 °C or 37 °C using the sample baseline temperature recording (range: 18.2 – 28.4 °C) and the mean temperature coefficient of conductivity (Fig. 2). For the 1^st IRE pulse, the conductivity values were fit to the sigmoid function shown in equation (7). Resulting fits and parameters for each tissue type are given in Fig. 4 and Table III, respectively. Due to variation in the actual voltage delivered by the pulse generator, the raw data (Fig. 2 a), c), e)) are plotted at the actual voltage applied. This voltage was used to calculate the actual electric field applied, and subsequently, the conductivity. For the purposes of plotting the average conductivity versus electric field magnitude and curve-fitting, the data were grouped according to the programmed electric field (Fig. 2 b), d), f)). In the first IRE pulse, brain tissue exhibited the smallest change in conductivity with increasing electric field magnitude (approximately 0.05 S/m at 37 °C), while pancreas and liver showed similar changes. The discrepancies between tissue conductivity are largest at high electric field values.

$$\sigma (E)={\sigma }_0+\frac{{\sigma }_f-{\sigma }_0}{1+e^{-A\cdot \left(E-E_{del}\right)}}$$ (7)

Increases in conductivity due to the recorded temperature changes (shown in Fig. 3) were quantified according to (2) in sets of 20 pulses as described above. For each set of 20 pulses, a single β value was calculated for that range and plotted at the median temperature value for that set. Fig. 5 shows the calculated β values across all IRE treatments. The samples used to calculate β were those exposed to an electric field of either 1000, 1500, 2000, 2500, or 3000 V/cm. This is due to a lack of significant conductivity change and subsequent temperature rise in samples treated with lower than 1000 V/cm. For visualization of the effects of electric field magnitude on temperature changes, data were divided into two groups: high and low electric field magnitudes with a threshold of 2000 V/cm. The choice between the use of 2000 or 2500 V/cm as the dividing value for the dataset was chosen to represent the three relatively high electric field magnitudes (2000, 2500, 3000 V/cm) and two low electric field magnitudes (1000, 1500 V/cm). Additionally, 2000 V/cm marks an increase in temperature and conductivity. Liver tissue exhibited the highest overall β value across the temperature range achieved during treatment. Liver samples also displayed the largest difference in β between the two groups, while pancreas and brain exhibited similar behavior. It should be noted that these values include the change in conductivity due to electroporation in addition to the increase due to subsequent Joule heating. In terms of maximum temperature achieved, liver tissue reached nearly 40 °C while pancreas and brain did not rise above 34 °C.

Fig. 2. Dynamic conductivity behavior varies across the studied tissue types.

Raw electrical current values increase with increasing applied voltage for a) pancreas, c) brain, and c) liver. Conductivity at the end of the pulse was calculated for each point, then the percent difference from the initial conductivity was calculated. Then, each percent difference value was applied to the average initial conductivity for that tissue type. Finally, each value was adjusted to 37 °C. Shown are the resulting mean values and standard deviations for b) pancreas, d) brain, and f) liver. At each electric field magnitude, n=5–6 samples were treated for a total of N=90, 94, and 97 measurements for pancreas, brain, and liver respectively, including both the 1^st and 100^th pulse measurements.

Fig. 4. Sigmoidal functions were fit to the conductivity data sets for each tissue type based on the first IRE pulse, adjusted to 37 °C.

Curve fits show differences in dynamic conductivity dependent on tissue type. These functions may be incorporated in models to predict the EFD for a given electrode configuration. Fit parameters are shown in Table III.

TABLE III.

Sigmoid Fit Parameters

T [°C]	Tissue	σ0 [S/m]	σf [S/m]	A [cm/V]	Edel [V/cm]
37	Pancreas	0.118	0.268	2.50E-3	1738
	Brain	0.157	0.204	2.80E-3	1680
	Liver	0.132	0.237	2.80E-3	1620
20	Pancreas	0.093	0.197	2.50E-3	1738
	Brain	0.112	0.145	2.80E-3	1680
	Liver	0.113	0.195	2.80E-3	1620

Fig. 3. Temperature changes increase with larger electric field magnitudes.

Temperature at each pulse number recorded from the fiber optic temperature probe for a) pancreas, b) brain, and c) liver tissues. Mean change from the baseline temperature at the beginning of treatment is reported on the y-axes with standard deviations. Shown are n=3–6 samples for each electric field magnitude, with a total of N=39, 42, and 39 temperature curves for pancreas, brain, and liver respectively.

Fig. 5. Conductivity behavior with increasing temperature varies across tissue types.

Change in conductivity per °C, β, was determined over the course of 100 pulse IRE treatments for a) pancreas, b) brain, and c) liver porcine tissue. Each treatment was divided into sets of 20 pulses, where β was calculated over each range then plotted at the median temperature for that set of pulses. Data from treatments with applied electric field magnitudes greater than or equal to 2000 V/cm are plotted as solid circles, while those treated with less than 2000 V/cm are open circles.

In order to assess the impact of both electroporation effects and changing temperature on conductivity, the discrete-time derivative of conductivity was calculated at each pulse in the treatment, from 1 to 100. The derivative, σ′, sharply decreases until after approximately 20 pulses (Fig. 6). After this point, σ′ decays linearly for the duration of the treatment. Following the initial rapid increase, the conductivity saturates and takes on a more linear behavior (Fig. 6 a). This result is reflected in the behavior of the conductivity time derivative (Fig. 6 b) as well as the decrease and subsequent saturation of β values in the mid-range temperatures (Fig. 5). We hypothesize that these β values reflect the true α values for these tissues; that is, the percent change in conductivity for a single °C rise in temperature, independent of electroporation or other effects. In order to normalize the α values we report, we chose a temperature difference of 1 °C, then calculated the percent change in conductivity over that range. To ensure the value would not be influenced by electroporation effects, we chose the median pulse (50) to serve as the starting point for the calculation of α. We then find the corresponding conductivity and temperature values following an increase of 1 °C (Fig. 6), and calculate α as in (2).

Fig. 6. Electrical and thermal properties exhibit two distinct regions of behavior during the course of a 100-pulse treatment.

a) Conductivity increases with pulse number during a representative 2500 V/cm IRE treatment in porcine liver. For all three tissue types, the first discrete-time derivative of conductivity (b) decays exponentially, then decreases linearly. c) Alpha, the temperature coefficient, was determined by calculating the increase in conductivity for a 1 °C increase in temperature following the 50^th pulse. Data shown from treatments spanning electric field magnitudes from 1000–3000 V/cm.

A. COMSOL Simulation

To evaluate how differences in tissue properties affect EFDs clinically, the finite element method (FEM) was used to compare the three conductivity curves and calculated α values using COMSOL Multiphysics. Fig. 7 a) shows the resulting cross-sectional EFDs for the cases of static σ, σ(T), σ(E), and σ(E, T) for each tissue; isocontours were calculated at 200, 500, and 1000 V/cm. The magnitudes of these contours were selected to represent a range of electroporation-induced effects including: BBB disruption and reversible electroporation at ~200 V/cm [12], [31], nonthermal tissue ablation at ~500 V/cm [9], [32], [33], and a conservative threshold for ablation of 1000 V/cm [34]. In all cases, the electric field redistribution, either due to electroporation and/or thermal effects, resulted in larger volumes of tissue coverage in comparison to the static σ simulation. A volume integration was performed in COMSOL to calculate the total volume enclosed by the isocontours (Fig. 7 b).

Fig. 7. Incorporating dynamic conductivity and temperature increase results in larger electric field distributions than the static case.

a) COMSOL Multiphysics was used to compute electric field distributions for brain, liver, and pancreatic tissue assuming three different conductivity cases. Applied voltage was 2500 V for all cases, and isocontours were plotted at 200, 500, and 1000 V/cm. Top: the static electric field distribution was calculated assuming a constant conductivity value (σ₀) for each tissue type. Middle: incorporating the three electric field-dependent conductivity curves (σ(E)) resulted in slightly different distributions for the three tissue types. Bottom: the corresponding α values determined experimentally were used to account for the rise in conductivity (σ(E, T)) and subsequent redistribution of the electric field following 100 pulses. Thermal damage is reported as CEM₄₃ (red). b) Volumes enclosed by each isocontour were calculated and plotted.

The cases of static σ, σ(T), and σ(E) allow for direct comparison between the impact of electroporation and thermal effects on field redistribution. For pancreatic and liver tissues, electroporation effects resulted in larger changes in volume coverage as compared to thermal effects. In the case of brain, the thermal effects resulted in larger changes in tissue coverage at 200 V/cm when compared to electroporation effects. At 500 and 1000 V/cm, the brain isocontours for σ(E) were 0.47% and 5.6% larger than that of σ(T), respectively. Additionally, thermal damage was assessed by calculating the volume of tissue exposed to 10 cumulative minutes at 43 °C (CEM₄₃). These thermal dose volumes and their percentages of electroporation for each EF contour are shown in Table IV.

TABLE IV.

Predicted Volumes of Thermal Damage

Tissue	Case	CEM43, 10 min [cm3]	% of E200	% of E500	% of E1000
Pancreas	σ	0.021	0.20	0.61	1.96
	σ(T)	0.037	0.32	0.95	3.14
	σ(E)	0.12	0.73	2.07	6.06
	σ(E,T)	0.15	0.81	2.31	6.64
Brain	σ	0.17	1.62	4.91	15.89
	σ(T)	0.22	1.76	5.28	17.60
	σ(E)	0.23	1.86	5.49	17.42
	σ(E,T)	0.29	1.98	5.77	17.90
Liver	σ	0.044	0.42	1.27	4.11
	σ(T)	0.054	0.48	1.44	4.74
	σ(E)	0.12	0.81	2.33	7.10
	σ(E,T)	0.13	0.82	2.34	6.95

The model predicted thermal damage in close proximity to the electrodes. From the numerical case of σ(E, T), representing the highest predicted thermal dose, brain tissue exhibited the highest thermal damage volume, which is likely caused by the lower perfusion used for brain tissue. In this simulation, the thermal damage was restricted to a distance < 2 mm from the electrodes. At the 500 V/cm EF contour, brain tissue exhibited the largest percentage of ablation by thermal damage at 5.77% for the σ(E, T) case. Both pancreas and liver demonstrated thermal/electroporation percentages lower than 2% at the 500 V/cm contour.

IV. Discussion

The dynamic conductivity of tissue is an important metric to include in predictive models of electroporation. These data are useful especially for treatment models in both the reversible and irreversible electroporation regimes in order to predict the expected treatment zone ahead of time. In applications such as electrochemotherapy (ECT) and gene electrotransfer (GET), these conductivity data, along with a known electric field threshold, help to determine the region of reversibly electroporated cells. In the case of IRE, a region of target cells require exposure to a high enough electric field magnitude to undergo cell death. Due to this complexity of these treatments, visualization of a given electric field for a set of parameters is conducive to successful treatment.

Typically, two or more needle electrodes are used to apply the PEFs. In this case, the electric field generated in the tissue is not uniform. This gradient in the EFD complicates the calculation of conductivity for each electric field magnitude, because the measured electric current reflects the bulk conductivity of the tissue. Parallel-plate electrode configurations enable the exposure of each tissue sample to a single, homogeneous electric field magnitude, allowing us to determine its influence on conductivity. We chose to fix the electrode spacing and vary the applied voltage, allowing us to characterize the tissue response to a wide range of electric field strengths. When considering a typical needle configuration, both the voltage and spacing will have an impact on the resulting ablation; for example, 3000 V applied across 2 cm will yield different results than 1500 V applied across 1 cm, regardless of conserving the applied voltage-to-distance ratio of 1500 V/cm. However, parallel-plate experiments inform the construction of a spatial conductivity map that enables prediction of ablation volumes and expected current values in these configurations.

A drawback of using animal models is that they are a substitute for human tissue and may introduce some error in terms of the extracted properties. Additionally, the studies are conducted on excised tissue exposed to air, and at room temperature, all factors that could contribute to deviations from in vivo values. To better recapitulate in vivo tissue, we treated samples with IRE within an hour of resection. Additionally, the experimental setup was located in the same facility as the animals to ensure minimal delay between the tissue harvests and data collection. Previous reports indicate that tissue undergoes post-mortem changes after resection, mostly at low frequencies [22]. Specifically, Surowiec et al. report low frequency conductivity increases in kidney, spleen, and brain within 10 hours following treatment, while no changes in liver were observed [35]. Another potential source of error is a lack of pressure control between the two parallel plates. The PDMS molds used to contain the tissue have the ability to flex slightly, though the electrode holder arms are not tight enough to apply a large deal of pressure to the sample. Tissue types were rotated often during a single pig harvest, and some amount of pseudo-randomization was performed in which higher electric field magnitudes were applied early after death as opposed to later, and vice versa. However, we acknowledge that formal and complete randomization of the experiments would have accounted for variations in the properties due to time after the harvest as well as the size of the animals.

The DC conductivity values calculated in this work are in good agreement with existing reports [22], which supports the agreement of measured dynamic conductivity changes with prior work (Table V). Values for α do exist in literature, though these are not derived from tissue in an already electroporated state. Duck reports the temperature coefficients of conductivity to be 1.4, 3.2, and 1.5, for cow/pig pancreas, brain, and liver respectively, and provides a range of 1–3 in general for soft tissues [22].

TABLE V.

Comparison of Conductivity Increase to Previous Work

Tissue	Source	Animal	Methods	σ0 [S/m]	Increase Factor
Pancreas	This Work	Porcine	Ex Vivo	0.093	2.11
Brain	This Work	Porcine	Ex Vivo	0.112	1.29
	Garcia et al. 2010 [9]	Canine	In Vivo	0.12	2.5
Liver	This Work	Porcine	Ex Vivo	0.113	1.73
	Corovic et al. 2013 [36]	Porcine, Rat	In Vivo	0.091	2 or 3
	Sel et al. 2005 [20]	Rabbit	In Vivo	0.067	3.60
Kidney	Neal et al. 2012 [17]	Porcine	Ex Vivo	0.15	4.96

Regarding the calculation of β, we expected the value to be electric field-dependent due to the effects of electroporation present within the first several pulses. We suspected the higher magnitudes of electric field would induce a more rapid change in conductivity due to the greater extent of permeabilization in the cellular membranes, leading to a relatively higher value for beta. However, our results indicate that it is in fact the lower electric field magnitudes which induce a relatively higher change in conductivity for the same temperature rise. This may indicate that lower electric field magnitudes electroporate the cell membranes more efficiently in terms of generated heat. We note that our analysis of β is performed on raw temperature values rather than the change in temperature. The β values for brain tissue, for example, span a range of temperatures beginning about 5 °C higher than those of pancreas and liver. This indicates some degree of experimental variability in the starting temperature. However, we observe that liver tissue achieves the greatest range in temperatures while brain has the smallest. This result agrees with each tissue’s dynamic electrical behavior in that brain has the smallest conductivity change with increasing electric field magnitude. Higher conductivity values at high electric field magnitudes, as is the case with liver, would theoretically lead to more Joule heating due to higher currents generated, thus resulting in a higher temperature achieved over the course of treatment. One limitation of studying electrical properties ex vivo is that the tissue is typically tested at room temperature. Since the electroporation process is temperature dependent [42], this is expected to affect the threshold of cell death [43] and will thus effectively shift the sigmoidal inflection point to the right in comparison to the in vivo case. Our simulation results indicate that ablation volumes differ very little between pancreatic and liver tissue for a typical IRE treatment. This was an expected result due to the similarity between the calculated dynamic conductivity curves. We expected brain tissue to exhibit smaller dynamic changes in the EFD because of the relatively small increase in conductivity for larger electric field magnitudes. This difference is reflected in the slightly smaller volumes enclosed by the chosen isocontours. These results concur with numerical studies that indicate dynamic conductivity functions better predict the EFD when compared with the static case [9], [20], [36], indicating the importance of incorporating dynamic changes for these tissue types. To validate the model, we adjusted the configuration of the numerical model to match those of existing preclinical studies.

Overall, the models agree well with existing experimental data where current output is available (Table VI); otherwise, a relevant electric field threshold was chosen from the literature in order to compare ablation sizes and/or volumes. It should be noted that these numerical simulations were conducted assuming a homogeneous, isotropic tissue environment. Tissues may exhibit some degree of electrical anisotropy, which is likely to be most notable in brain tissue. During the preparation of the tissue during ex vivo experiments, efforts were made to isolate gray brain matter which has been shown to be macroscopically homogeneous [44]. Macroscopic structures such as blood vessels, bile ducts, cerebrospinal fluid, and other tissues were not included, though these would likely influence the electric field distribution [45] and should be accounted for in patient-specific treatment plans. In addition, blood perfusion was considered constant in the models, yet tumor tissue has been shown to undergo vasoconstriction during as little as eight ECT pulses [46], and effects of PEFs on blood perfusion have been demonstrated to linger following treatment [13], [47].

TABLE VI.

Comparison of Model Output to Experimental Data

Porcine Pancreas
Voltage [V]	Pulse Number	Frequency [Hz]	Pulse Width [μs]	Probe Exposure [cm]	Probe Spacing [cm]	Ablation Area (Day 0) [mm2]	Ablation Area (Day 1) [mm2]	Ablation Area (This Work) [mm2]	Electric Field Threshold Used [V/cm]
2000	70	1	70	1	1	164	204	253	500 [32]
2000	70	1	70	1	1	242	264	253	500 [32]
2250	70	1	100	1	1.5	294	382	358	500 [32]
2500	90	1	100	1	1.5	353	353	404	500 [32]
Canine Brain (This Work: Porcine Brain)
Voltage [V]	Energy [J]	Pulse Number	Pulse Width [μs]	Probe Exposure [cm]	Probe spacing [cm]	Average Current [A]	Ablation Volume [cm3]	Average Current (This Work) [A]	Ablation Volume (This Work) [cm3]	Electric Field Threshold Used [V/cm]
500	0.8	90	50	0.5	0.5	0.36 ± 0.2	0.258	0.882	0.195	502.5 [9]
1000	5.2	90	50	0.5	0.5	1.16 ± 0.2	–	1.98	0.494	502.5 [9]
1000	8.8	90	50	0.5	0.5	1.96 ± 0.2	0.599	1.98	0.494	502.5 [9]
2000	39.4	90	50	0.5	1.0	4.38 ± 0.2	–	3.71	1.78	502.5 [9]
Porcine Liver
Voltage [V]	Pulse Number	Frequency [Hz]	Pulse Width [μs]	Probe Exposure [cm]	Probe Spacing [cm]	Ablation Width [cm]	Ablation Depth [cm]	Ablation Width (This Work) [cm]	Ablation Depth (This Work) [cm]	Electric Field Threshold Used [V/cm]
2250	90	1	100	2	1.5	2.76 ± 0.35	1.53 ± 0.27	2.71	1.58	450 [39]
2650	90	1	100	2	1.5	3.0 ± 0.32	1.68 ± 0.25	2.92	1.87	450 [39]
3000	90	1	100	2	1.5	3.12 ± 0.35	1.99 ± 0.23	3.11	2.04	450 [39]
Voltage [V]	Pulse Number	Frequency [Hz]	Pulse Width [μs]	Probe Exposure [cm]	Probe Spacing [cm]	Average Current 1st 10 Pulses) [A]	Average Current (1st 10 Pulses, This Work) [A]
3000	100	0.83	100	2	2.5	13.6 ± 1.3	11.78

Source: Wimmer et al. [37]

Sources: Ellis et al. [38], Garcia et al. [9]

Source: Ben-David et al. [40]

Source: Applebaum et al. [41]

V. Conclusion

The EFD and accompanying heating are important considerations when applying PEFs in biological tissue. During therapies such as IRE, coverage of the tumor with a high enough electric field magnitude is crucial for successful treatment outcome [48]. Though the mechanism of PEF treatments is non-thermal in nature, enabling the conservation of major vasculature [49], overtreatment leading to deleterious Joule heating could potentially damage these structures [50]–[52]. Optimal deployment of PEFs in therapies such as IRE may be strategically planned by maximizing damage to tumor cells due to electroporation while minimizing thermal damage. Here, we present data for three different tissue types that may be incorporated into predictive models, which may be personalized to a given patient and treatment configuration.