A theoretical argument for extended interpulse delays in therapeutic high-frequency irreversible electroporation treatments

Abstract

High-frequency irreversible electroporation (H-FIRE) is a tissue ablation modality employing bursts of electrical pulses in a positive phase–interphase delay (d₁)–negative phase–interpulse delay (d₂) pattern. Despite accumulating evidence suggesting the significance of these delays, their effects on therapeutic outcomes from clinically-relevant H-FIRE waveforms have not been studied extensively.

Objective:

We sought to determine whether modifications to the delays within H-FIRE bursts could yield a more desirable clinical outcome in terms of ablation volume versus extent of tissue excitation.

Methods:

We used a modified spatially extended nonlinear node (SENN) nerve fiber model to evaluate excitation thresholds for H-FIRE bursts with varying delays. We then calculated non-thermal tissue ablation, thermal damage, and excitation in a clinically relevant numerical model.

Results:

Excitation thresholds were maximized by shortening d₁, and extension of d₂ up to 1,000 μs increased excitation thresholds by at least 60% versus symmetric bursts. In the ablation model, long interpulse delays lowered the effective frequency of burst waveforms, modulating field redistribution and reducing heat production. Finally, we demonstrate mathematically that variable delays allow for increased voltages and larger ablations with similar extents of excitation as symmetric waveforms.

Conclusion:

Interphase and interpulse delays play a significant role in outcomes resulting from H-FIRE treatment.

Significance:

Waveforms with short interphase delays (d₁) and extended interpulse delays (d₂) may improve therapeutic efficacy of H-FIRE as it emerges as a clinical tissue ablation modality.

I. Introduction

Electroporation is a biological phenomenon in which cells exhibit increased membrane permeability upon exposure to high amplitude electric fields. Increased permeabilization is presumably due to creation of defects in the cell membrane that increase transport of ionic species and macromolecules [1]. Depending upon characteristics of the applied field – such as amplitude and duration – these defects range in size from one to hundreds of nanometers, and can be tuned to target intracellular structures [2]–[4]. Electroporation can be implemented reversibly, whereby affected cells regain membrane integrity and recover following stimulus removal, or irreversibly, where cells die following treatment. Reversible electroporation has been used for decades to increase cellular uptake of poorly permeant or impermeant molecules, and to augment expression of specific genes by increasing transfection efficiency [1], [5]. The latter technique, known as irreversible electroporation (IRE), has recently emerged as a tissue ablation modality [6].

Unlike conventional ablative therapies in which temperature is manipulated to nonspecifically denature proteins, IRE directly affects cellular membranes without significant local heating, leaving the underlying tissue architecture intact. Due to this unique mechanism, IRE can be performed in cases where thermal ablation is contra-indicated by the presence of vasculature, ducts, or neural structures [7]–[9]. IRE also exhibits submillimeter precision and a sharp demarcation between untreated and ablated tissue, both of which reduce the likelihood of overtreatment or off-target damage [10], [11]. Despite these promising capabilities [12], [13], management of patients receiving IRE can be difficult. The long (70–100 μs) pulses easily stimulate cardiac myocytes, pain receptors, and skeletal muscle fibers, resulting in muscle contractions and potential arrhythmias [14], [15]. To avoid such complications, a neuromuscular blockade is administered prior to treatment and pulses are delivered in synchronization with the absolute refractory period of the heart [16]–[18]. Thus, despite its benefits over thermal ablation, this anesthetic protocol can prevent IRE from being recommended in some cases (e.g. “awake surgeries” or outpatient procedures) [19].

To further extend clinical capabilities of IRE, an alternative pulse delivery scheme termed high-frequency IRE (H-FIRE) has recently been introduced [20], [21]. H-FIRE replaces the long monopolar pulses with bursts of short (1–10 μs) bipolar pulses following a positive phase–interphase delay (d₁)–negative phase–interpulse delay (d₂) pattern (Fig. 1) [20], [22]. By changing the method of pulse delivery in this way, nerve excitation and muscle contractions are substantially reduced [22]–[24]. This is due to the local repolarization that occurs with each negative phase prior to activation of the minimum number of voltage-gated sodium channels (VGSCs) required for excitation [25], [26]. Additionally, the short bipolar pulses constitute a shift towards higher frequencies, which translates to more predictable ablation geometries and reduced likelihood of electrical arcing [27], [28]. A challenge that arises with H-FIRE waveforms is that lethal electric field thresholds (EFTs) are typically 1.5–3× higher than with IRE and are heavily dependent on the width of constitutive pulses [24]–[32]. For this reason, higher voltages must be applied across the electrodes to achieve similar ablation volumes, which has consequences on the amount of heat produced.

Fig. 1.

Representative idealized voltage waveform of an A) IRE pulse, B) Symmetric H-FIRE burst, and C) Variable delay H-FIRE burst. Each IRE pulse is characterized by an amplitude (1) and pulse width (T_p; 2), and is repeated at a specific interval (3). Symmetric HFIRE bursts (B) are comprised of short bipolar pulses - each with a positive pulse of amplitude (1) and width (2), an interphase delay (d₁; 4), and an identical negative pulse (i.e. ⑥=②+④+②+⑤). Each bipolar pulse is separated by an interpulse delay (d₂; 5) equal to (4), and is repeated N times to achieve a desired “energized time.” Variable delay H-FIRE bursts (C) are similar to (B) with the unique difference that (4) ≠ (5).

A number of studies have explored non-pharmacological methods of lowering lethal EFTs for H-FIRE waveforms by modifying the delivery strategy or constitutive pulse width of burst waveforms. Sano et al. showed that with a fixed width of 2 μs, single bipolar pulses repeated at 25–100 Hz may exhibit lower EFTs than traditional bursts [33]. Additionally, reducing the width of alternate polarity pulses lowered the EFT to roughly half the value of corresponding symmetric bursts [31], but the charge imbalance caused muscle contractions similar to those resulting from IRE [34].

Investigations have shown that cell permeability and survival are also closely linked to the delays within (interphase delay, d₁) and between (interpulse delay, d₂) bipolar pulses (Fig. 1C) [35], [36]. This is presumably due to “bipolar cancellation” (BPC) mediated by assisted discharge, whereby the subsequent pulse of opposite polarity forces membrane repolarization rather than passively allowing the transmembrane potential to decay according to its time constant (τ_m) [37]–[39]. By increasing interphase and interpulse delays, the time the membrane is exposed to a critical transmembrane potential (usually ~ 1 V) is increased [22]. Thus, one might expect that longer delays would improve electroporation-based outcomes. On the other hand, it is known that increased interphase delays reduce the ability of anodic pulses to abolish action potentials elicited by the cathodic phase [25], [40], [41]. Additionally, pores are created on the order of nanoseconds, but several microseconds are required for a sufficient number of VGSCs to open to cause nerve excitation, so while seemingly related, the mechanisms by which electroporation and stimulation are suppressed with bipolar pulses are not identical [26]. Further, recent reports suggest that a complex relationship exists between these delays and electroporation effects that cannot be fully explained by assisted discharge, and that extensions of the interpulse delay alone may enhance biophysical outcomes from treatment [36]. From a stimulation perspective, many studies have investigated methods to maximize efficiency of charge-balanced waveforms for functional stimulation purposes, offering insight into the effects of interphase gaps, multiple pulses in series, etc., but most of these investigations have been for much longer pulses than those we are interested in [42]–[44]. To our knowledge, modifications of the interphase and interpulse delays have not been investigated as a method to reduce excitation caused by clinical H-FIRE protocols, nor have the resulting biophysical effects been explicitly examined.

Here, we seek to determine the effects of integrating a variable delay approach into clinically relevant H-FIRE waveforms. As the transition to H-FIRE was in part motivated to mitigate tissue excitation, in this paper we further evaluate the relationship between interphase and interpulse delays within H-FIRE waveforms and resulting clinical outcomes, specifically nerve stimulation, electric field distribution, and resistive heating. After placing constraints on the range of delays that could realistically be implemented in clinical protocols – taking ECG synchronization and typically-employed repetition rates into account (see supplementary material) – we computationally evaluate the effect of modified burst waveforms on neural excitation. We then use the results from the nerve excitation study to inform the range of delays yielding desirable outcomes, and construct a numerical model to estimate ablation area, temperature rise, and thermal damage resulting from a typical treatment with a variety of modified waveforms. We demonstrate that extended interpulse delays increase excitation thresholds for bursts with constitutive pulse widths of 1 and 5 μs. We also show that the temperature increase due to Joule heating is lowered as the interpulse delay is extended from 100 ns to 1 ms, and that the field redistribution with longer interpulse delays may augment ablation size even in the absence of a reduced lethal EFT.

II. Methods

A. Modified SENN model

To assess the response of a myelinated neuron to a temporally arbitrary electric field, we adopted the SENN framework introduced by Reilly et al. [48], but replaced the Frankenhaeuser and Huxley current equations with Hodgkin-Huxley type formulations computed for mammalian neurons [45]. This model can provide the nerve fiber response to any transient electric field with known spatial distribution. Because electroporation-based treatments are conventionally performed with needle electrodes, we chose to model a scenario representing a nerve terminus within the vicinity of the electrodes and in parallel with a given electric field contour, as originally proposed by Mercadal and colleagues [47]. Thus, for each waveform under study, we calculated the electric field required to initiate an action potential in a short nerve segment with 6 nodes of Ranvier exposed to a uniform field. By assuming a nerve terminus is present at all points in a given domain, these thresholds can be extrapolated to estimate the amount of tissue exposed to fields capable of inducing action potentials irrespective of electrode geometry and stimulus amplitude [47]. It is worth noting that peripheral motor neurons are excited at lower stimulus magnitudes than skeletal myocytes, so it is not necessary to consider direct stimulation of muscle cells themselves [49].

The equivalent circuit employed in this model, originally proposed by McNeal [46], is given in Fig. 2. The relative transmembrane potential at the n^th node is related to capacitive, ionic, and axial currents according to:

$$C_m\frac{d{V}_n}{dt}+I_{i,n}=G_a\left(V_{n-1}-2V_n+V_{n+1}+V_{e,n-1}-2V_{e,n}+V_{e,n+1}\right)$$ (1)

where C_m is the membrane capacitance of the node, V_n is the nodal transmembrane potential relative to the resting potential (V_n = V_i,n – V_e,n – E_r), I_i,n is the sum of ionic currents across the membrane at the node, G_a is the axoplasmic conductance, and V_e,n is the extracellular voltage at the node. At the ends of the fiber, we assume no axial current [47], giving:

$$C_m\frac{d{V}_n}{dt}+I_{i,n}=G_a\left(-V_n+V_{n\pm 1}-V_{e,n}+V_{e,n\pm 1}\right)$$ (2)

Fig. 2.

Equivalent circuit model for myelinated nerve fiber. The membrane a each node of Ranvier can be described by a capacitance (C_m), resistance (R_m), and resting potential (E_r). Nodal transmembrane current is a function of the difference between local extracellular (V_e) and intracellular potential (V_i). Axial conduction is dependent upon internodal resistance (R_a). Teal line indicates direction of uniform field with respect to fiber.

In this equation, ± indicates that at node 1 and N, potentials at only the current and nearest neighboring node (n + 1 and n − 1 , respectively) are considered. To determine the extracellular potential at each node, we employ the following equation, which describes the potential along a line in parallel to an applied electric field:

$$V_{e,n}=V_{e,1}+E\cdot L\cdot n$$ (3)

where V_e,1 is a reference voltage at the terminal node nearest the cathode, E is the applied field, L is the internodal separation, and n is the node number. For convenience, V_e,1 was set to 0 as it has a negligible effect on the response [50].

Ionic currents were calculated according to the Hodgkin-Huxley framework using the gating parameters introduced by McIntyre et al.[45] for a mammalian myelinated nerve fiber at 36°C (see supplementary material for details). Values of other parameters are given in Table I. Voltage waveforms were created using a custom function in MATLAB vR2019a (MathWorks Inc., Natick, MA, US) with rise/fall times of 100 ns each. Waveforms were scaled to create time-varying extracellular potentials at each node according to (3). Equations (1) and (2) along with the nonlinear ionic current equations at each node were integrated implicitly and solved using a timestep of 2 ns. The applied field was increased stepwise in increments of 0.25% until an action potential was detected. In the ablation model, tissue exposed to fields equal to or above this threshold was considered to be excited [47].

Table I.

Parameters used in modified SENN model

Quantity	Unit	Value	Ref.
Axoplasmic resistivity, ρi	Ω · cm	70	[45]
Membrane capacity, cm	μF cm−2	2	[46]
Fiber diameter, D	μm	1	[47]
Nodal gap width, W	μm	1	[47]
Axon diameter, da	-	0.7 D	[46]
Nodal diameter, dn	-	0.33 D	[47]
Internodal separation, L	mm	1.15	[47]
Membrane capacitance	μF	cmπdnW	-
Membrane conductance	Ω−1	$\pi d_a^2/(4{\rho }_{i}L)$	-
Rest Potential, Er	mV	−88	-

B. Determination of lethal EFTs in vitro

Lethal thresholds were characterized experimentally for each waveform given in Table III using established methods [51]. Briefly, disk-shaped collagen type I hydrogel constructs were fabricated and seeded with hepatocellular carcinoma cell line Hep G2 (ATCC^® HB-8065^™) at a concentration of 10⁶ cells/mL. Hydrogels were covered in culture media and incubated for 24 hours. Two needle electrodes (4 mm spacing, Ø = 0.9 mm) were used to treat gels with the given H-FIRE waveform or IRE in a mobile incubator maintained at 37 °C. Voltage (600 V), repetition rate (1 burst/s), energized time (100 μs), and number of bursts (100) were maintained constant. After treatment, media was replenished and cells were incubated for 24 hours prior to live/dead staining with calcein AM and propidium iodide, respectively. Finally, cells were imaged with a confocal microscope and ablation areas were measured in ImageJ (NIH). A 3D numerical model of treatment was constructed (Comsol Multiphysics v5.5) and a function relating electric field strength to area of exposure was created. Ablation areas were used as inputs to the function to estimate lethal EFTs. For more details, see supplementary material.

TABLE III.

H-FIRE Burst Waveforms Assessed in 2D Ablation Model

Tp (μs)	d1 (μs)	d2 (μs)	fc (kHz)	σ0 (mS cm−1)	Tb (ms)
1	1	0.1	322.6	1.5	0.15
1	1	1	250	1.3	0.20
1	1	10	76.9	0.81	0.64
1	1	100	9.71	0.53	5.1
1	1	1,000	0.997	0.41	49
5	1	0.1	90.1	0.84	0.11
5	1	1	83.3	0.82	0.12
5	1	10	47.6	0.72	0.20
5	1	100	9.01	0.53	1.0
5	1	1,000	0.989	0.41	9.1
100	N/A	N/A	10	0.535	0.1

C. Realistic In Vivo Ablation Model

To evaluate non-thermal ablation, temperature rise, and thermal damage resulting from treatment with symmetric and variable delay H-FIRE waveforms, a 2-D finite element model representative of a two-needle in vivo ablation was developed.

The steady-state electric potential ϕ in an electrically inhomogeneous medium was computed by solving a modified Laplace equation that considers local electrical conductivity σ:

$$\nabla \cdot (\sigma \cdot \nabla \varphi )=0$$ (4)

The boundary conditions at the electrodes were Dirichlet in type and set to ϕ = V₀ where the energized (leftmost) electrode meets tissue and ϕ = 0 on the rightmost electrode boundaries. The exterior boundary was Neumann in type and electrically insulative, which confined all current to the domain and provided a worst-case scenario to temperature estimations. The tissue temperature rises during treatment according to the Joule heating term p, calculated as σ|∇ϕ|². This term was scaled according to the energized time per burst (τ) and burst repetition rate (f), and added to the traditional Pennes bioheat equation to calculate the temperature distribution:

$$\rho c_p\frac{\partial T}{\partial t}=\nabla (k\nabla T)+{\omega }_b{\rho }_b{c}_b\left(T_a-T\right)+p\tau f$$ (5)

where ρ is the tissue density, c_p is the tissue heat capacity, T is the temperature, k is the tissue thermal conductivity, ω_b is the blood perfusion rate, ρ_b is the blood density, c_b is the heat capacity of blood, and T_a is the arterial temperature. Initially, the domain was set to a temperature of 30 °C and all boundaries, including electrodes, were considered adiabatic. Tissue metabolism was neglected as it contributes minimally to bulk temperature rise [52].

Finally, an Arrhenius-type thermal dose equation was used to calculate the extent of thermal damage Ω:

$$\Omega =\int \xi e^{-E_a/RT}dt$$ (6)

where ξ is the frequency factor, E_a is the activation energy, and R is the universal gas constant [53]. To give a more intuitive description to the thermal damage arising from treatment, a probabilistic approach was used to estimate the percentage of cells P_Ω that would die upon reaching different values of Ω according to [54], [55]:

$$P_{\Omega }=100\cdot \left(1-e^{-\Omega }\right)$$ (7)

Using this approach, values of Ω = 1 and Ω = 4.0036 correspond to 63% and 99% cell death due to thermal damage, respectively. To calculate the area where cells were killed due to heat, thermal damage was conservatively approximated as areas where tissue was exposed to Ω ≥ 0.53 (41%), as this has been noted as the onset of thermal damage [55].

Tissue electrical conductivity exhibits complex behavior in response to applied electric fields [56]. This behavior is tissue specific and depends on the amplitude and duration of the applied field, as well as the number of pulses. Previous reports have shown that electroporation-induced conductivity changes can be represented by a sigmoidal curve that: (a) begins at a baseline conductivity σ₀ determined by the characteristic frequency of the waveform [57]; (b) exhibits a transition range related to the reversible and irreversible ablation thresholds of the waveform [58]; and (c) saturates to a final conductivity σ_f similar to the tissue’s conductivity in the upper end of the β-dispersion frequency range (1 kHz – 100 MHz) [59], [60].

To consider these dynamic changes, we used the model introduced by Sel et al. [58], in which the dependence of σ on the local field is given by:

$$\sigma (\overrightarrow{E})={\sigma }_0+\frac{{\sigma }_f-{\sigma }_0}{1+D\cdot e^{-(|\overline{E}|-A)/B}}$$ (8)

where D is a fitting parameter, and A and B are related to the electroporation effects of the waveform. They are calculated as:

$$A=\frac{E_0+E_1}{2}B=\frac{E_1-E_0}{C}$$ (9)

where E₀ and E₁ are the electric field thresholds for reversible and irreversible electroporation, respectively, and C is a sigmoid fitting parameter. We used values of C and D of 8 and 10, respectively, which were previously determined for liver tissue [58]. Baseline electrical conductivity (σ₀) was determined by evaluating a liver impedance model [61] at the characteristic frequency of each waveform, and electroporated conductivity (σ_f) was set to 0.32 S/m by evaluating the model at 10 MHz (Table III).

Our experimental results showed no difference in lethal EFT as a function of d₂, so E₁ was set to 1,030 V/cm for all waveforms with 1 μs pulses, and 658 V/cm for waveforms made up of 5 μs pulses – the average across all values of d₂ for either constitutive pulse width. The reversible threshold (E₁) was characterized for the 5-1-5-1 protocol (453 V/cm) and assumed constant regardless of d₂ or T_p. Thresholds were also characterized for a conventional IRE protocol to provide a comparison. E₀ for the IRE protocol was set to 282 V/cm, which was computed by scaling our experimentally determined E₁ (429 V/cm) to the ratio of E₀/E₁ reported in [58].

We also considered the effect of temperature on local conductivity. Thus, the overall conductivity is given by:

$$\sigma (\overrightarrow{E},T)=\left[{\sigma }_0+\frac{{\sigma }_f-{\sigma }_0}{1+D\cdot e^{-(|\overline{E}|-A)/B}}\right]\left(1+\alpha \cdot \left(T-T_{\text{ref }}\right)\right)$$ (10)

where α is the temperature coefficient of electrical conductivity and T_ref is a reference temperature at which the conductivity values were measured (T_ref = T_a = 37 °C)

Comsol Multiphysics v5.5 (COMSOL Inc., Stockholm, Sweden) was used to construct the model and solve equations (4)-(11). Tissue was modeled as a circle with radius 20 cm to ensure boundary distortions would not play a role in the results. Two circles (∅ = 1 mm) with center-to-center separation of 1 cm were used to represent the electrodes. For each waveform, a typical treatment consisting of 100 bursts was simulated with τ = 100 μs and f = 1 Hz. Thermal damage was computed 10 minutes after treatment ended, as thermal damage continues to accumulate until tissue temperature falls below approximately 43 °C. The applied potential was set to 2,500 V in all simulations except those used to find the equivalent potential in Fig. 9, in which case the voltage applied was varied until the area of excitation for the given variable delay waveform matched the area of excitation for the symmetric case. Material properties of liver tissue employed in the model can be found in Table II. Burst parameters are summarized in Table III.

Fig. 9.

Longer interpulse delays allow for higher applied voltages. Maximum voltages that can be applied while maintaining the same area of excitation as with symmetric interpulse delays (d₁/d₂ = 1 μs) are shown for bursts with A) 1 μs and B) 5 μs constitutive pulse widths. Also shown are the ablation areas resulting from treatment with the maximum voltage given for each waveform.

Table II.

Material properties used in ablation model

Quantity	Unit	Value	Ref.
Density, ρ	kg m−3	1,079	[62]
Heat capacity, cp	J kg−1 K−1	3,540	[62]
Thermal conductivity, k	W m−1 K−1	0.512	[63]
Temperature coefficient, α	% K−1	2	[63]
Blood perfusion, wb	s−1	9.27×10−3	[64]
Blood density, ρb	kg m−3	1,050	[62]
Blood heat capacity, cb	J kg−1 K−1	3,617	[62]
Frequency factor, ξ	s−1	2.08×1027	[36]
Activation energy, Ea	J mol−1	1.866×105	[36]

To ensure accuracy of the solution, the mesh was refined until less than a 1% difference was observed in the electric field distribution between refinements. The final mesh consisted of 226,499 vertices and 451,680 triangular elements. Simulations took place on a Dell OptiPlex 7071 with an 8-core Intel I9 processor and 32 GB of RAM.

III. Results

A. Modified SENN Model Behavior and Validation

Fig. 3 gives the strength/duration (S/D) curve for the modified SENN model employed here for single monopolar and bipolar pulses. The behavior of the model is in agreement with other published studies [47], [50], [65]. Namely, on this log-log plot, a linear relationship is observed between threshold amplitude and pulse width for both monopolar and bipolar pulses with durations up to roughly 500 μs. Beyond this point, thresholds decrease minimally and are nearly identical for both monopolar and bipolar pulses. Comparison of the overall behavior of the model to experimental studies was obtained through a least-squares fit to the standard equation for excitation of an ideal linear membrane:

$$E_t={\left[1-e^{-T_p/{\tau }_e}\right]}^{-1}$$ (11)

in which E_t is the threshold field for a monopolar pulse of duration T_p and τ_e is an empirical S/D time constant. According to (12), the SENN model used here exhibits a time constant τ_e of 626 μs, which is in agreement with reported experimental values [40], [66]. Additionally, the threshold for a conventional IRE pulse (100 μs monopolar) was 1.2 V/cm -nearly identical to that reported by Rubinksy and Golberg [67].

Fig. 3.

Strength/duration curve for myelinated nerve fiber in response to single monopolar and bipolar pulsed fields.

B. Effect of pulse width and delays on nerve fiber response

The neural response to varying interphase and interpulse delays is shown in Fig. 4. Although intuitive given the body of literature demonstrating amplified excitability with prolonged interphase gaps for single biphasic pulses, here it is explicitly demonstrated that short interphase delays result in the greatest stimulation thresholds. For symmetric 1 μs pulse bursts, stimulation thresholds decreased 24% and 41% when the interphase delay was increased to 1 or 10 μs from 0. For 5 μs pulse bursts, this effect was not as pronounced, but thresholds fell 9% and 32% across the same delay increases.

Fig. 4.

Longer interpulse delays (d₂), but not interphase delays (d₁), increase thresholds required to elicit action potentials in the SENN model of a myelinated nerve fiber. Threshold electric fields required for stimulation for A) 1 μs pulse bursts and B) 5 μs pulse bursts. note different ordinate scales. Thresholds calculated for single burst with energized time of 100 μs.

For bursts made up of 5 μs pulses, interpulse delays of 1 ms exhibited increased excitation thresholds approximately 60 % higher than with no interpulse delay, regardless of the interphase delay. Interestingly, for bursts with 1 μs constitutive pulse widths, holding the interphase delay to a minimum increases the relative gain in stimulation threshold that can be attained while lengthening the interphase delay. For example, a burst comprised of 1 μs pulses with d₁ of 1 μs and d₂ of 1,000 μs exhibits a threshold 2.44× that of a similar burst with d₂ maintained at 1 μs; extending d₁ to 10μs reduces the gain that can be achieved with this modification in d₂ to 1.95×. Finally, the dashed curve in Fig. 4 indicates that symmetric increases in both delays cause a sharp decline in excitation threshold with a minimum near 250 μs and a gradual recovery with further symmetric increases in d₁/d₂. The trends observed here suggest the optimal strategy for reducing nerve stimulation is to minimize d₁ while maximizing d₂. In light of this observation, subsequent simulations were performed with d₁ fixed 1μs to examine the effects of d₂ independently. We chose to fix d₁ to 1 μs for practical reasons; most generators used in clinical and pre-clinical studies employ fully-controlled solid-state switches (e.g. MOSFET, IGBT) which have a certain switching time, and when generating bipolar pulses, some time is required during polarity reversal to avoid shorting the DC power supply.

C. Effect of interpulse delay on physical response

To assess the effect of the interpulse delay on ablation, temperature rise, and nerve excitation in a realistic clinical setting, we first characterized thresholds required to ablate liver cells in a 3D collagen hydrogel, which informed subsequent construction of a numerical model representative of in vivo treatment. In this model, we applied H-FIRE waveforms with pulse widths of 1 or 5 μs and fixed interphase delay of 1μs, then varied the interpulse delay from 0.1 μs to 1,000 μs (Table III). See supplementary material for calculation of upper limit on d₂.

Fig. 5 illustrates the results of cell-laden hydrogel treatment with different variable delay H-FIRE waveforms. Due to generator limitations, the shortest interpulse delay used in experiments was 250 ns. A two-way ANOVA was performed to test the effect of pulse width and interpulse delay, and it was found that only pulse width contributed significantly (p < 0.0001) to different lethal EFTs. Due to this result, E₁ was set as the average of all protocols with a given constitutive pulse width, which was 1030 ± 182 V/cm for bursts with 1 μs pulses and 658 ± 116 V/cm for bursts made up of 5 μs pulses. See supplementary material for individual results.

Fig. 5.

Lethal thresholds are impacted by changes in pulse width but not interpulse delay. A) Representative confocal images of ablations produced by each combination of T_p and d₂ under study. Summarized ablation areas (B) and lethal thresholds (C) demonstrate differences due to T_p (p < 0.0001) but not d₂ (p > 0.01).

In the model, lengthening the interpulse delay (d₂) reduced the effective baseline conductivity due to the lower characteristic frequency of the burst. This effect is especially prevalent in bursts with constitutive pulse widths of 1 μs, but becomes less significant with pulse widths of 5 μs. As d₂ is extended to 100 μs or longer, effective baseline conductivity becomes identical regardless of the constitutive pulse width of the burst (Table III). Alternatively, the sigmoid transition range correlates with the relative difference between the reversible EFT and lethal EFT. At or above the lethal EFT, the conductivity of all burst waveforms with a given pulse width converges.

The electric field distribution during the last burst as a function of constitutive pulse width and interpulse delay is shown in Fig. 6A–D. As the electric field is a function of local conductivity, trends in the field distribution closely follow those of the conductivity as mentioned above. Of note is the increased vertical field exposure but slightly suppressed exposure moving horizontally outward from the electrodes with increased d₂. Panels (E) and (F) in Fig. 6 demonstrate the percent difference in amount of tissue exposed to varying electric fields compared to a symmetric burst with 1 μs intererphase and interpulse delays. These curves were developed by calculating the area of tissue exposed to fields equal to or greater than each field given on the x-axis, then calculating the relative difference between each variable delay curve and the symmetric curve. Areas of exposure to fields of 10 to about 1,500 V/cm are increased with longer interpulse delays. The exception to this trend occurs for a small range of fields near the midpoint of the transition zone in the dynamic conductivity curve for each waveform.

Fig. 6.

Extended interpulse delays increase the area of exposure to therapeutic electric field strengths. A-D) Electric field contours for bursts with constitutive pulse widths of (A, C) 1 μs and (B, D) 5 μs and interpulse delays of 0.1 μs (A, B) and 1,000 μs (C, D) are shown during the final burst. The white line delineates the area of ablation due to IRE. E-F) Percent difference in exposure of tissue to therapeutic electric fields for each variable delay waveform versus a symmetric burst with d₁/d₂ set to 1 μs and pulse widths of E) 1 μs and F) 5 μs. It is worth noting that E_rev and E_lethal are dependent upon applied pulse characteristics and were determined experimentally in this study.

The intersection points between the vertical lines and the curve for each waveform indicate the relative difference in area of reversible (E_rev) and irreversible (E_lethal) electroporation compared to a symmetric burst with both delays set to 1 μs. Thus, 1 μs pulse burst with d₂ = 1,000 μs give rise to a 17% and 4.9% increase in area of reversible and irreversible electroporation, respectively, while shortening d₂ to 0.1 μs reduces these areas by 2.7% and 0.6%, respectively. Bursts made up of 5 μs pulses exhibit similar trends, with ineases in area of 7.6% and 3.2%, respectively, with d₂ = 1,000 μs versus the symmetric case. By shortening d₂ to 0.1 μs with a 5 μs pulse width, the area of reversible electroporation is reduced by 1% while the irreversibly electroporated region increases 2.3%.

Fig. 7 illustrates the thermal response as a function of modifications in d₂. Panels A-D show the thermal damage distribution arising from treatment with different H-FIRE waveforms expressed as a percentage (P_Ω). For 1 μs pulse bursts, the area of thermal damage (Ω > 0.53) is reduced from 38.7 mm² to 34.6 mm² as d₂ is extended from 0.1 to 1,000 μs. With the same extension in d₂, thermal damage resulting from treatment with 5 μs pulse bursts is reduced from 40.2 mm² to 39.3 mm². Additionally, the maximum value of Ω is inversely related to d₂. Bursts with constitutive pulse width of 1 μs exhibit maximum values of 5.7 and 4.1 for d₂ = 0.1 and 1,000 μs. Applying the same changes in d₂, 5 μs burst waveforms exhibit maxima of 6.1 and 5.8, respectively. This can be attributed to higher temperatures adjacent to the electrodes with shorter delays due to the increased electrical conductivity.

Fig. 7.

Longer interpulse delays reduce thermal damage in therapeutic H-FIRE treatments. Thermal damage (%) contours for (A, C) 1 μs pulse bursts and (B, D) 5 μs pulse burst waveforms with interpulse delays of 0.1 and 1,000 μs, respectively. The white line indicates the onset of thermal damage.

D. Effect of interpulse delay on relative efficacy

Next, we introduce a metric to define the relative efficacy of a given waveform:

$${\text{R}}_{eff}=\frac{A_{IRE}}{A_{TD}}\frac{A_{IRE}}{A_{EXC}}=\frac{A_{IRE}^2}{A_{TD}\cdot A_{EXC}}$$ (12)

where R_eff is a dimensionless quantity representative of the efficacy of the waveform and A_IRE, A_TD, and A_EXC are the areas of irreversible electroporation, thermal damage, and excitation. This value can be used to compare the ability of each waveform to create ablations while limiting the extent of thermal damage and nerve excitation.

Calculated values of A_IRE, A_TD and A_EXC are shown in Fig. 8A. A minor increase in ablation area is observed with longer interpulse delays, while areas of excitation and thermal damage are reduced. Trends in R_eff versus d₂ are shown in Fig. 8B. For visualization, R_eff for bursts with either constitutive pulse width was normalized to the case in which d₂ = 1 μs. Here, it can be seen that R_eff increases exponentially with d₂ on a log scale. This increase is continuous for 1 μs pulse bursts, but is not obvious until d₂ reaches 100 μs for bursts with constitutive pulse widths of 5 μs. For 1 μs pulse bursts, increasing d₂ from 0.1 μs to 1,000 μs increased R_eff nearly 3-fold from 0.35 to 0.84. Similar modifications in d₂ for 5 μs pulse bursts increased R_eff from 0.23 to 0.36. For comparison, R_eff for the modeled conventional IRE protocol was 0.022, which was mainly dominated by the large area of nerve stimulation.

Fig. 8.

Longer interpulse delays increase therapeutic efficacy of H-FIRE bursts A) Areas of IRE, thermal damage, and excitation for 1 μs pulse bursts (left) and 5 μs pulse bursts (middle) as a function of d₂. Right panel gives areas for a conventional IRE protocol. B) Normalized R_eff as a function of d₂ for 1 μs pulse bursts (solid) and (D) 5 μs pulse bursts (dashed). Rightmost panel gives R_eff for a traditional IRE protocol. R_eff is normalized to value at d₂ = 1 μs (i.e. symmetric burst), and for IRE this value is normalized to T_p = 5 μs case.

Finally, a parametric analysis was performed for each variable delay waveform with extended d₂ to determine the maximum voltage that can be applied while maintaining the same excitation area as the case in which d₁/d₂ are symmetric and equal to 1 μs (Fig. 9). For symmetric waveforms comprised of 1 μs and 5 μs pulse bursts, the standard treatment with 100 bursts at 2,500 V produced ablations of 130 mm² and 236 mm², respectively, and stimulation areas of 1,298 mm² and 6,341 mm² For 1 μs burst waveforms, extending d₂ to 10 μs, 100 μs, or 1,000 μs gave maximum voltages that could be applied of 2,422 V, 3,450 V, and 4,878 V, respectively, while maintaining a stimulation area of 1,298 mm². At these voltages, ablation areas are 127 mm², 123 mm², and 418 mm², respectively. Waveforms with 5 μs constitutive pulses exhibit maximum voltages of 2,116 V, 2,777 V, and 3,724 V when d₂. is 10 μs, 100 μs, or 1,000 μs respectively. Applying bursts at these voltages gives ablation areas of 206 mm², 296 mm², and 472 mm², while maintaining a stimulation area of 6,341 mm².

IV. Discussion

In this study, we proposed that variable delays within H-FIRE bursts may be more clinically efficacious in terms of ablating tissue with reduced nerve excitation in comparison to current waveforms with symmetric delays. To evaluate the feasibility of such an approach, we analyzed the effects of variable delays on neural excitation for bursts with constitutive pulse width of 1 μs and 5 μs. From these findings, we fixed the interphase delay to 1 μs and performed a parametic sweep of the interpulse delay in a realistic two-needle treatment model to quantify the extent of ablation, excitation, and thermal damage.

The results suggest that the largest clinical effect will be reduced nerve excitation with extended interpulse delays. For waveforms with short constitutive pulse widths near 1 μs, interpulse delay has a substantial influence on the characteristic frequency of the burst, and thus the effective baseline conductivity and Joule heating produced from treatment. This difference in baseline conductivity as a function of d₂ is dependent upon the slope of the tissue’s impedance spectrum across the range of frequencies of interest. Thus, the effects of modifying the interpulse delay on electrical conductivity and Joule heating may vary between tissues.

As shown in Fig. 4, prolonging d₂ yields a marked increase in the electric field required to stimulate a nerve fiber compared to a burst with short, symmetric delays. However, it should be noted that with longer burst durations, it is possible that a single burst will initiate multiple action potentials in excitable cells exposed to threshold stimuli or greater. This is due to the fact that action potentials − and, in turn, refractory periods − in skeletal muscle and nerve are roughly 2 to 4 ms in duration. In the case that extremely long values of d₂ (250–1,000 μs) are not clinically efficacious, shorter delays that maintain a burst duration just below the absolute refractory period of muscle will likely still be desirable. For example, for a 5 μs pulse burst with 100 μs of energized time and d₁ set to 1 μs, d₂ can be extended to 146 μs assuming an absolute refractory period of 3 ms for muscle (see supplementary material for calculation). By maintaining a burst duration that can be delivered while the muscle is refractory, a single action potential would be generated in affected cells, mitigating synergistic effects such as frequency summation that might occur due to multiple action potentials firing in response to each burst. This approach has been taken for ECT protocols (8×100 μs pulses) and in an in vivo rat study, it was found that the number of contractions could be reduced from eight lighter contractions to a single stronger contraction by increasing the repetition rate to 100 Hz or more [68]. While further increases in frequency reduced the strength of the single contraction, this finding suggests that bursts with durations in the range of 70 ms will be feasible, but this relationship must be more rigorously explored to definitively place limits on d₂.

It is important to appreciate the mechanism by which prolonged values of d₂ suppress nerve fiber activation. Because these fibers are much longer than most cells in the tissue parenchyma, membrane charging occurs via a unique mechanism, relying on charge redistribution along the length of the fiber rather than the potential difference between the interior and exterior of the fiber at a given point [69]. As a result, while parenchymal cells exhibit charging constants on the order of 1 μs, membrane charging constants of nerve fibers are much longer, and are typically reported to be on the order of hundreds of microseconds [47], [70], [71]. Thus, extending d₂ to tens or hundreds of microseconds allows the nerve fiber membrane to discharge between subsequent bipolar pulses within the burst.

Excitation is also dependent on d₁, which must be shortened sufficiently for the negative phase of each bipolar pulse to adequately cancel VGSC activation by its corresponding positive phase [26]. Conversely, interphase delays on the order of the time constant of targeted cells increase time of exposure to a critical transmembrane potential, which has been shown to correlate with permeabilization [72]. This implies an optimal value of d₁ exists that mitigates VGSC activation while also allowing passive discharge of targeted cellular membranes. In this study, we fixed d₁ to 1 μs (~ τ_m) to demonstrate theoretically that these mechanisms can be exploited to apply higher potentials without increased stimulation, which will allow for larger areas of ablation and permeabilization. In practice, this strategy will require optimization, and to achieve desirable results without substantial temperature increases, thermal mitigation strategies will be imperative [73]–[75].

In our in vitro study, we found that lethal EFTs were only a function of the width of constitutive pulses. Bursts with 1 μs pulse widths exhibited a threshold near 1 kV/cm, approximately 57% higher than those with 5 μs widths, indicative of the bipolar cancellation present with very short pulses. Recent experimental data have found that interphase and interpulse delays mediate permeabilization and lethality of high-frequency pulse bursts, with longer delays typically magnifying biological effects. For instance, Valdez and colleagues found that bipolar nanosecond pulses with interphase delays greater than 10 ms resulted in permeabilization comparable to that of energy-matched monopolar pulses [39]. And Polajzer and colleagues demonstrated that assisted discharge can explain cell viability trends resulting from symmetric increases in interphase and interpulse delay, but specific cases of cell permeabilization deviated from theoretical projections [35]. Recently, increased cell death has also been achieved by independently lengthening the interpulse delay up to 1 ms for a single burst with 800 μs of energized [36]. Thus, while we did not find a significant impact of d₂ on lethal thresholds, it is possible that in other cell types or in bulk tissue, extended values of d₂ may exhibit lower thresholds than the symmetric bursts currently being used. Importantly, if this is realized clinically, R_eff will be further amplified as d₂ is prolonged.

Finally, it is important to acknowledge potential limitations of our approach. By assuming a constant perfusion rate, our simulations neglect to account for local disruptions in blood flow [76], and may underestimate temperature rise. We assumed electrical properties of tissue exposed to subelectroporative H-FIRE bursts could be defined by determining a discrete characteristic frequency at which the burst operates [57], [77], [78]. Next, we assumed this characteristic operating frequency f_c is defined as the inverse of the bipolar pulse period (Fig. 1). This assumption has been employed previously, but not for high-frequency bipolar waveforms with variable delays as introduced here [57]. Thus, it remains to be determined whether extensions in d₂ continuously generate reductions in f_c, or whether there is a limit to the influence of d₂ on f_c. Additionally, we calculated σ_f by assuming the β-dispersion of liver tissue plateaus at 10 MHz, which can be qualitatively concluded from the data reported in [61] and is consistent with theory [79], [80]. Importantly, so long as each waveform saturates to the same conductivity, lengthening d₂ will produce the benefits reported in this paper regardless of the value of σ_f.

It is critical to note that the nerve excitation results are independent of the assumptions underlying the conductivity curves constructed in this study, and in their own right provide valuable information toward optimizing delays within H-FIRE waveforms. However, it is also important to recognize that the nerve stimulation model has limitations. To facilitate translation to the 2D ablation models, we chose to characterize thresholds for excitation at the terminus of a 6-node fiber. This truncated fiber model could introduce inaccuracies in terms of the current distribution that would not be present in a longer fiber. Additionally, the theoretical time constant of our model is high (τ_e = 626 μs) compared to the original SENN model (τ_e = 120 μs) exposed to uniform-field excitation, which reflects the small diameter and short length of the fiber modeled here; however, as previously mentioned, this falls within the range of experimental values for mammalian nerve [40], [48]. Finally, it is important to point out that areas given for excitation are areas in which a nerve terminus could be excited if aligned with a relatively constant field at threshold or higher. While these areas build intuition and facilitate comparison, they are not precisely indicative of the areas of tissue that will be stimulated, but surrogates that represent the relative magnitude of excitation (and thus, contraction). Thus, it will be critical to corroborate our findings with experimental results in the future.

Conclusion

Irreversible electroporation is an emerging focal treatment modality for solid tumors and in cardiac ablation for treating atrial fibrillation [81], [82]. While promising results have been reported, adoption of IRE has been hindered by its complex anesthetic regimen and treatment protocols. H-FIRE has been introduced to overcome these limitations and drastically simplifies clinical procedures, but waveforms suffer from reduced ablation volumes compared to their IRE counterparts. For this reason, higher voltages are often desired during H-FIRE, but this increases the likelihood of thermal damage and muscle contractions. Here, we demonstrate that minor changes to the delays within H-FIRE burst waveforms may suppress neural excitation and Joule heating. It is also possible that these waveforms will increase ablation size by modulating local electrical conductivity. These modified waveforms be readily implemented without generator hardware modifications or systemic changes to existing treatment protocols.