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Towards Solid Tumor Treatment by Nanosecond Pulsed Electric Fields (p. 289-306)
Local and drug-free solid tumor ablation by large nanosecond pulsed electric fields leads to supra-electroporation of all cellular membranes and has been observed to trigger nonthermal cell death by apoptosis. To establish pore-based effects as the underlying mechanism to inducing apoptosis, we use a multicellular system model (spatial scale 100 µm) that has irregularly shaped liver cells and a multiscale liver tissue model (spatial scale 200 mm). Pore histograms for the multicellular model demonstrate the presence of only nanometer-sized pores due to nanosecond electric field pulses. The number of pores in the plasma membrane is such that the average tissue conductance during nanosecond electric field pulses is even higher than for longer irreversible electroporation pulses. It is shown, however, that these nanometer-sized pores, although numerous, only significantly change the permeability of the cellular membranes to small ions, but not to larger molecules. Tumor ablation by nanosecond pulsed electric fields causes small to moderate temperature increases. Thus, the underlying mechanism(s) that trigger cell death by apoptosis must be non-thermal electrical interactions, presumably leading to different ionic and molecular transport than for much longer irreversible electroporation pulses.
Key words: Nanosecond electric field pulses; Multicellular and tissue model; Dynamic pore behavior; Pore histograms; Membrane permeability; and Thermal effects.
Background and Introduction
Tissue ablation techniques to treat all cells of unwanted tissue for surgical, cosmetic, or other reasons are potentially valuable as minimally invasive clinical tools (1). Tissue ablation may be achieved by the application of RF electric fields that lead to cell death by overheating the targeted tissue. Strong electric field pulses that cause electroporation (EP) of cellular membranes, a universal mechanism by which cellular membranes become permeable to drugs, molecules, and genetic material, may alternatively be used for tissue ablation. Drug-assisted methods such as electro-chemotherapy (ECT) (2) and electro-genetherapy (3) are examples of such EP-based tissue ablation methods. There is now evidence, however, that drugs actually may not be necessary, since the impact of certain strong electric field pulses that cause EP itself is sufficient to trigger cellular mechanisms that lead to cellular death.
In a previous paper (4) we considered some fundamental mechanistic aspects that are relevant to solid tumor treatment by irreversible electroporation (IRE) (5-10). In short, IRE involves the application of pulses with a duration of typically hundreds of microseconds and an electric field strength on the order of a few kilovolts per centimeter (kV/cm). These IRE pulses are sufficient to cause what is known as conventional EP, for which pores in the plasma membrane (PM) of the cells are sufficiently large to facilitate molecular uptake and release. While those pulses reseal in most EP-based applications, IRE pulses are designed such that pores in the PM do not reseal - hence the PM is said to be irreversibly electroporated. Therefore, the barrier function of the PM is lost, leading to cellular death in a treated tissue region. Specifically, solid tumor treatment by IRE is observed to cause necrosis in the targeted cells (9).
Using Blumleln circuit-based or MOSFET-based pulsed power technologies, solid tumor treatment by much shorter nanosecond pulsed electric fields (nsPEFs) with durations of ten to several hundred nanoseconds and field strengths of tens or even hundreds of kV/cm has been demonstrated (11-14). In particular, Nuccitelli et al., (12, 14) used electric field pulses with durations of about 300 ns and with electric field strengths of 20 kV/cm and 40 kV/cm to demonstrate self-destruction of melanomas. Garon et al., (13) used electric field pulses that were even one order of magnitude shorter, namely 20 ns or less and field strengths up to 60 kV/cm. Their results showed decreased cell viability of a variety of human cancer cells in vitro, induction of tumor regression in vivo, and successful treatment of a human subject with a basal cell carcinoma for which they found a ?complete pathologic response?. Cell death does not appear to be due to immediate PM destruction, as expected for IRE pulses. Instead, it appears to be the result of delayed effects, which may be caused by the efflux of Ca2+ from intracellular stores (e.g., endosplasmic reticulum) that eventually cause apoptosis (13).
The concept of using nsPEF pulses as a therapeutic tool to treat solid tumors was first demonstrated by Beebe et al., (11) who reported the induction of apoptosis in solid tumors (mouse fibrosarcoma) ex vivo and the reduction of fibrosarcoma tumor size in vivo by nsPEF pulses up to 300 kV/cm and durations from 10 ns to 300 ns. Signs of apoptotic cell death comprise cell shrinkage, activation of caspases, persistent externalization of phosphatidylserine (PS) at the PM, and fragmentation of DNA (11). Apoptosis induction by nsPEF has also been observed for mammalian cancer cells in vitro (15).
There have been several other interesting observations about the responses of cells and tissues to nsPEF pulses. First, EP markers such as propidium iodide (PI) and ethidium homodimer, whose uptake has been traditionally used to indicate the membrane integrity and permeability changes in the PM were reported to be taken up by cells only in very small amounts. However, we have noted that no measurement sensitivity was established and that uptake may be below the detection limit (16). Second, effects in the cell interior have been observed that have not generally been reported for conventional EP conditions. Calcium release from the endoplasmic reticulum, cytochrome-c release from mitochondria, phosphatidylserine translocation at the PM, and caspase activation are examples of such observed intracellular effects (11, 17-22), Those effects can be readily understood in terms of the supra-EP hypothesis (23-26) which leads to a different degree of EP as will be described in the present paper.
The often stated motivation for the recent focus on nanosecond pulses is the low energy density that is delivered per pulse despite the much larger field than for IRE pulses. This argument appears misleading, however, since it is certainly not the dissipative energy that causes cell death, as is the case for RF tumor ablation, but a non-linear biophysical mechanism such as EP. It is arguably more important that nsPEF pulses lead to strong electric fields both inside and outside the cells and thereby significantly perturb organelle membranes in addition to the PM (17, 25).
The consideration of both IRE and nsPEF pulses thus provides the tantalizing prospect of designing specific electric field pulse protocols and treatments that lead to different cell death mechanisms, for example necrosis with IRE pulses and apoptosis with nsPEF pulses. The ability to determine the cellular death mechanism by an appropriate choice of electric field parameters may seem like an unnecessary choice for a patient in urgent need of tumor treatment. Yet, the apoptotic cell death pathway may provide certain advantages. Specifically, if secondary necrosis can be avoided then it should be possible to bypass non-specific damage to nearby tissue due to e.g., inflammation and/or scarring. It might also be possible to avoid the tumor lysis syndrome resulting from massive tumor necrosis.
As the cellular and tissue response to nsPEF pulses is so distinctively different from the response to IRE pulses, the object of the present paper is to seek an underlying mechanistic understanding of the tissue response to nsPEF pulses. This is achieved by using exactly the same tissue models as used previously for IRE pulses (4), for which we determine the local electric fields and currents in the treated tissue and then study the biophysical response in terms of pore densities and pore sizes that lead to cell permeability changes, as well as the resulting temperature change in the tissue model. Throughout, we attempt to make objective comparisons between responses to nsPEF pulses and the much longer and weaker IRE pulses.
The basic methods and tissue models are the same as used previously (4) only the electric field pulse waveforms are different. In this way we can straightforwardly compare the mechanistic response at the cellular and tissue level to IRE pulses that lead to necrotic cell death with nsPEF pulses that lead to apoptotic cell death. To facilitate understanding we have restated our methods here.
As previously (4) we use the transport lattice (TL) method, which allows for a convenient description of electrical, chemical, and thermal behavior in a complex biological geometry that may contain tissue inhomogeneities and anisotropies. Basic features of the TL method have been presented elsewhere (23-25, 27-29). For the study of nanosecond pulses to treat solid tumors, we use two system models, a multicellular model of irregular cells and a tissue model, and consider their electrical and thermal responses to two representative nsPEF pulses. Each system model represents rabbit liver tissue but on a different spatial scale. Although experiments with nsPEF pulses on tumors have been performed for different cells and tissues, a liver tissue model is adopted here in order to compare the distinct features of tissue responses due to nsPEF pulses with those of IRE pulses.
As described previously (4) and mostly repeated here for ease of reading, the system geometry, shown in Figure 1A, is based on a drawing motivated by a tissue section image. It features a layer of 20 non-spherical liver cells with 14% interstitial fluid volume (30). The hepatocytes have an average cell size diameter of 21.7 μ (31). A corresponding TL (101 nodes × 101 nodes) was constructed as a large electric circuit comprising ˜104 interconnected local circuit models for passive charge transport and storage (resistors, capacitors) within electrolytes and active elements (pumps, electroporation) at the membrane (Figure 1B). The linked local membrane and electrolyte models are distributed spatially and are connected to their nearest neighbors on a Cartesian lattice (23, 27). The local membrane models are interconnected at the regularly spaced nodes, with submodels that represent the PM and two contacting regions of electrolyte (Figure 1B) (23, 27). There is no transport in the z-direction in this two-dimensional model. The lattice spacing l, as well as the depth of the system model, is 1 µm, and the system has a spatial scale L of 100 μ × 100 µm. Voltages applied along the top and bottom boundary of the system model provide the applied uniform electric field, Eapp.
Figure 1:(A) Geometry of the multicellular model of a region of liver tissue with 14% interstitial space and an average cell dimension of 21.7 µs. The extracellular electrolyte is shown in blue, the cells in red. Also given is the direction of the applied electric field Eapp. (B) TL circuits: Functional local models that represent electrolyte (Mel), membrane (Mm), and the PM-electrolyte interface (Me+PM+i) in the 101 × 101 TL; the lattice spacing l, as well as the depth of the system model, is 1 µm, leading to a spatial scale of 100 µm × 100 µm; More details described elsewhere (23, 24, 27). The equivalent EP circuit representing the asymptotic electroporation model (left gray box) (23) and its extension to the SE (41) is solved at every local membrane site. The local pore distributions are discretized, and pore drift and diffusion determine the associated non-Ohmic pore conductance Gm that is input for the time-dependent membrane current Im(t) in module Mm. (C) Two circular electrodes in a 200 mm × 100 mm tissue region (only a small subregion of the entire simulation region is shown). The electrodes have radii re = 0.25 mm and separation Le = 10 mm (edge-to-edge). (D) The tissue system mesh (only a small subregion of the mesh close to the needle electrodes is shown here). (E) The tissue model cell unit comprises extracellular (e), membrane (m), and intracellular (i) regions in series and a parallel shunt region. The spatial and electrical parameters of the regions are labeled. (F) The passive tissue conductivity of the tissue model, obtained from an AC-frequency sweep in SPICE, is compared with experimental measurements on rat liver by Raicu et al., (35).
Multiscale Tissue Model
Local solid tumor treatment by electric field pulses is controlled by an appropriate placement of electrodes. In order to control the spatial extent of the tissue region that is being treated by nsPEF pulses, it is essential to know the electric field redistribution that results from the dynamic behavior of EP. Figures 7-9 show the distributed electric response to nsPEF pulses in time and space in the tissue model for the 20 kV/cm pulse with 325 ns duration, 40 kV/cm pulse with 325 ns duration, and 60 kV/cm pulse with 20 ns duration, respectively. Needle electrodes are placed as shown in Figure 1. The distributed electric response is represented during and after the application of the pulse by the electric potential ø (A, F), the local averaged electric field E (B, G), the intracellular electric field Eint (C, H), supplemented by the local transmembrane voltages ΔψPM (D, I), and the local pore densities Np (E, J). The latter is calculated using the asymptotic EP model wherein pore expansion is neglected. The PM charging rate, and by extension the time of onset of significant EP and reversible electrical breakdown (REB), is determined by the local electric field magnitude E, which in Figures 7-9 is shown to be largest near the needle electrodes and drops off quickly with distance. As such, the membranes in the regions of tissue nearest the electrodes charge fastest and electroporate first, reaching pore densities of about 1016 m-2, which is about two orders of magnitude higher than for IRE pulses (4). Subsequently, a wave of elevated ΔψPM with values above 1 V and pore creation moves outward from the needle electrodes into both the central region of the tissue between the electrodes and around the needle electrodes. Transmembrane voltages of 1 V and above can be maintained only for a short time, as massive pore creation leads to REB, which in turn causes the PM transmembrane potential ΔψPM to decrease even during the pulse (26, 39). In contrast to IRE pulses, however, ΔψPM does not relax to values significantly below 1 V during the pulse because of the overall short time scale of the pulse. After the pulse all electroporated cell membranes in the tissue are depolarized for the lifetime of the pores. Hence the transmembrane potential ΔψPM is basically zero for all electroporated cell membranes. As such, the overall electric potential ø is essentially zero throughout the tissue, resulting in a lower number of equipotential lines on the scale presented in Figures 7-9 (F and I) 1000 ns after the pulse.
The multiscale tissue model accounts for the electrical response at both the microscopic (e.g., PM EP) and macroscopic (e.g., needle geometry) scales and the interplay between the two (32). The scale of the tissue system is three orders of magnitude larger than the scale of the above multicellular model and therefore a discretization of the system could not realistically resolve individual cells and membranes. The multiscale model uses representative simple cell models distributed throughout the system model to calculate the local cell and membrane response, and the macroscopic electrical transport properties are determined by the distributed models.
Tissue Level: The impedance between two nodes that may comprise a number of cells in a small local tissue region is equal to the impedance of a cell scaled to have the same relative dimensions, assuming that total tissue volume comprises a uniform grid of such cells. Thus, we use the simple cell model shown in Figure 1E, which has a membrane enclosed region of intracellular electrolyte surrounded by extracellular electrolyte (32). This simple model can be straightforwardly translated into an equivalent circuit. The membrane and each region of electrolyte have an associated conductivity σ and permittivity ε (Table I). Additionally, each electrolyte region has a tortuosity n to account for the structural complexity of tissue not otherwise represented by the model. The relative sizes were chosen such that 14% of the total volume was extracellular. The tortuosities n are used as free parameters in fitting the frequency-dependent rat liver tissue conductivity to that measured experimentally by Raicu et al., (35). By this choice of electrical and geometrical parameters, the model reproduces the static conductivity value and approximately the same trend in the frequency dependence (Figure 1F). The equivalent circuit for the simple cell model (Figure 1E) is placed between each pair of adjacent nodes in the mesh with electrical components scaled to the local mesh geometry. The effective conductance of the membrane changes in accordance with the local degree of membrane EP as determined by the distributed cell models.
Cell Level: An equivalent circuit for a single cell is created for each node in the tissue level mesh to determine the cellular response to the local electric field. Each of these circuits is distinct from the primary, macroscopic tissue-level circuit network, but all of the circuits are solved simultaneously. The asymptotic model of EP is used as described below. The voltage across the cell unit is equal to the local electric field magnitude, as determined from the electric potential of the nodes in the mesh, multiplied by the cell unit length lu. The distributed cell models determine the transmembrane voltage and pore density throughout the tissue domain as functions of time. The pore density then determines the membrane conductance used in the macroscopic transport network. Thus, there is continual feedback between the macroscopic (tissue level) and microscopic (cellular level) models: the macroscopic behavior determines the local electric field in the microscopic model and the subsequent behavior at the microscopic scale (e.g., EP) then determines the local electrical properties at the macroscopic scale.
Electrolyte and Membrane Models
The passive electric components for the electrolyte are resistors and capacitors (27) (Figure 1B). The membrane circuits (Figure. 1B) include components for charge storage and conduction, resting potential, and an EP model (23) (see below). This provides a convenient means for including the dm = 5 nm thick membrane in a TL of much larger scale. Close to a pore the membrane dielectric is treated as pure lipid and assigned a dielectric constant, εl, of 2.1. This choice recognizes that local membrane properties are relevant to pore formation. In contrast, the PM capacitance involves a spatial average over membrane lipid and protein regions resulting in a relative permittivity, εm, of 5. The extracellular electrolyte has a conductivity, σe, of 1.2 S/m while the medium inside the cell has a conductivity, σi, of 0.4 S/m. These conductivity values are identical to the validated macroscopic tissue model, wherein the effective conductivities are the ratio of the electrolyte conductivity and the tissue tortuosity (see Figure 1F and Table I). Following Lauger (36) we use a simplified, single resting potential source model (Figure 1B) comprised of an active voltage source, Vip, and source series resistance, Rip (23). Here, the fixed quantities Vip and Rip, together with a negligible conductance of the equilibrium pores (26), determine the membrane resting potential, ψPM, rest, in the absence of applied electric fields.
The transient aqueous pore hypothesis of EP is based on continuum models of membrane pores, electrostatic energy differences, and thermal fluctuations, usually in the form of the Smoluchowski equation (SE) (26, 37-39)
The SE describes the evolution of the local PM pore distribution n in terms of the number of hydrophilic pores and pore radius rp given by the diffusion constant Dp, and is used in the multicellular model. The asymptotic model (40) used in the tissue model, is an approximation to the SE-based models that disregards pore size change, and can be used to describe cell and tissue responses to nanosecond timescale pulses (24, 25). For longer ECT and IRE pulses, the asymptotic model approximates the system electrical response, whereby more 0.8 nm pores are created to increase the membrane conductance in response to elevated transmembrane voltages (26). These pores readily transport Na+, Cl-, and K+ ions that dominate extracellular and cytosolic conductivity, but not significantly larger molecules.
The implementation of the asymptotic EP model in terms of an equivalent circuit is described in detail elsewhere (23). This equivalent circuit can readily be generalized to include dynamic pore expansion and contraction based on the SE-based EP model (41). The extended equivalent circuit is given in Figure 1B and represents drift and diffusion in pore radius space from the minimum pore radius (rmin = 0.8 nm) to a maximal pore radius (considered here as rmax = 3 nm). From the pore distribution, we find the local non-Ohmic conductivity σm (26) that, together with the local ΔψPM(t), determines the time-dependent membranecurrent Im (t) as input into the membrane circuit Mm. This approach may describe the PM response for all types of EP pulses, starting from nanosecond pulses that lead to supra-EP to microsecond or longer pulses that give rise to conventional EP. Pore lifetimes reported in the literature vary over many orders of time (from milliseconds to hours). Because there is presently no mechanistic understanding of this large range, we use an illustrative experimental value of τp = 3 ms (42). Note however, that this paper only considers electric conditions during the pulse for which the value of the pore lifetime is not directly relevant. All parameters of this standard model of EP are given in Ref. (26).
The nsPEF applied to the multicellular and tissue models are taken as idealized trapezoidal versions of the pulse used experimentally by Nuccitelli et al., (12, 14) (55 ns rise and fall times, 215 ns plateau duration, 20 kV/cm and 40 kV/cm field strength) and Garon et al., (13) (2.5 ns rise and fall times, 15 ns plateau duration, and 60 kV/cm field strength).
The system model circuits are solved for the electric potential ø by means of Kirchhoff's laws using Berkeley SPICE 3f5. SPICE generates solutions that are processed and displayed in Matlab as equipotentials and distributions of electroporated regions (23-25, 29).
Specific Absorption Rate and Temperature Change
The specific absorption rate (SAR), defined as Joule heating, σ|∇ø|2, divided by the tissue (electrolyte) mass density r, is traditionally used in the electrical characterization of tissue to time-varying electromagnetic fields (e.g., cell phones and MRI exposures). At a given time point, SAR is calculated as follows. In a local volume of electrolyte, represented by model Mel (Figure 1A), a SAR contribution of V(t)2σel/(2rl2) is added at the two nodes (in the SAR array) between which the model is connected. Here, σel is either the intracellular or extracellular electrolyte conductivity, l is the lattice node spacing, and V = Δø is the voltage difference between the two nodes at time t. In the case of a membrane model (Mm; Figure 1A), the SAR contribution is (V1(t)2σe/(2rl2)) +ΔψPM(t)2σm (t)/(rdm2) + (V2 (t)2σi /(2rl2)) where V1(t) and V2(t) are the voltages across the two resistors that represent the extracellular and intracellular electrolytes contacting the membrane, and dm<< l.
The temperature change ΔT due to the application of electric field pulses may then be determined as follows. Bioheat transfer problems in the context of pulsed electric fields have been modeled typically by using the Pennes equation, which accounts for the ability of tissue to remove heat by both passive conduction (diffusion) and perfusion of tissue by blood (6, 43). Perfusion can be defined in a tissue region if a sufficient number of capillaries are present such that an average flow description is reasonable, and therefore a spatial scale of more than ˜ 100 µm must be considered. The multicellular model is below this scale for blood perfusion. In addition, on the short pulse time scale we consider here, thermal diffusion would not be fast enough on a ˜100 µm length scale to reach nearby capillaries. Indeed, a simple estimate for the diffusional spreading length on a 100 µs time scale is ~3 µm. For these conditions, we therefore can neglect perfusion (44) (as well as the metabolic heat generation) such that a diffusion-dominated bioheat equation
may be considered. Here, r, c, and k are the density, specific heat, and thermal conductivity of tissue, respectively, T is local tissue temperature, and t is time. As in Ref. (4), the thermal properties were assumed to be uniform (c = 3600 J/kg K; k = 0.512 W/m K; r = 1060 kg/m3) (6). The boundaries of the model were treated as thermal sinks. This ~100 µm separation of thermal sinks is about 2.5 times the spacing of capillaries. The above SAR is input into Eq. (2) to calculate the temperature at any corresponding time point. When the time interval between successive time steps (obtained from the electrical model) was larger than the minimum value, the SAR was interpolated to finer time intervals (0.1 l2cr/k) to avoid numerical instabilities.
To study the evolution of EP patterns in the multicellular model both in time and space, we first consider a single nsPEF. Figure 2 shows on the left the distributed electrical response to a trapezoidal 20 kV/cm pulse with 55 ns rise and fall times, 215 ns plateau duration (12). Specifically, the spatial distribution of equipotential field lines (black lines) are given by local values for the electric potential f inside and outside of the cells, of the multicellular model. The electric field is perpendicular to the equipotential field lines. White dots correspond to local PM sites with at least one local pore (corresponding to a pore density of 1012 m-2). The corresponding pore histograms, on the right, show the total number of pores in all cells of the multicellular model (bin width of 0.05 nm) at different time points during the pulse.
Figure 2 (left):Electrical response of the multicellular model to a trapezoidal 20 kV/cm pulse (55 ns rise and fall times, 215 ns pulse plateau) and correspondent pore histograms showing the total number of pores in the multicellular model (bin width 0.05 nm) at different time points. (A) Membrane charging phase during pulse rise time at t = 34 ns (when the applied field strength is about 11.5 kV/cm): Displacement currents at the PM dominate and lead to intracellular electrical fields essentially equal to the extracellular electric fields. Pore creation due to EP has started at this time at some PM sites and is shown by white dots that correspond to pore sites with more than 1 pore (equivalent to a pore density of 1012 pores per m2). (B) Still during the pulse rise time at t = 45 ns (when the applied field strength is about 15 kV/cm): EP has occurred on essentially all PM sites, regardless of the polar region of the membrane site and even for membrane sites not facing the polar side of the cell. (C) At t = 67 ns during the pulse and afterwards, basically no new pores are created at the PM, but the elevated PM transmembrane voltage of 0.8 - 1 V causes the present pores to start to grow in size. (D) Distributed electric response at t = 270 ns at the end of the pulse plateau does not change qualitatively from (C). The pore histogram shows still further pore expansion, but the short overall duration of the pulse restricts the pores to an averaged size of 1.05 nm.
Figure 3 (right):Electrical response of multicellular model to a trapezoidal 40 kV/cm pulse (55 ns rise and fall times, 215 ns pulse plateau) and correspondent pore histograms showing the total number of pores in the multicellular model (bin width 0.05 nm) at different time points. (A) Membrane charging phase during pulse rise time at t = 34 ns (when the applied field strength is about 23 kV/cm): Due to the larger Eapp compared to Figure 2, pore creation due to EP has occurred at essentially all PM sites regardless of the polar region of the membrane site and even for membrane sites not facing the polar side of the cell and is shown by white dots that correspond to pore sites with more than 1 pore (equivalent to a pore density of 1012 pores per m2). (B) Still during the pulse rise time at t = 45 ns (when the applied field strength is about 30 kV/cm): Elevated PM transmembrane voltages lead to pore expansion. (C) At t = 67 ns during the pulse and afterwards, basically no new pores are created at the PM, but the elevated PM transmembrane voltage of 0.8 - 1 V cause the present pores to further grow in size. (D) Distributed electric response at t = 270 ns at the end of the pulse plateau does not change qualitatively from (C). The pore histogram shows still further pore expansion, but the short overall duration of the pulse restricts the pores to an averaged size of 1.15 nm.
Figure 4:Electrical response of multicellular model to a trapezoidal 60 kV/cm pulse (2.5 ns rise and fall times, 17 ns pulse plateau) and corresponding pore histograms showing the total number of pores in the multicellular model (bin width 0.05 nm) at different time points. (A) At t = 2.8 ns already during the pulse plateau: Displacement currents at the PM dominate and lead to intracellular electrical fields essentially equal to extracellular electric fields. Pore creation due to EP has started at this time at some membrane sites, predominately in the non-polar regions of the cells. (B) At t = 3.6 ns: EP has occurred on essentially all PM sites, regardless of the polar region of the membrane site and even for membrane sites not facing the polar side of the cell. (C) At t = 11.2 ns during the pulse plateau, basically no new pores are created at the PM, but the elevated PM transmembrane voltage of 0.8 - 1 V causes the present pores to start to grow in size. (D) Distributed electric response at t = 17.5 ns at the end of the pulse plateau does not change qualitatively from (C). The pore histogram shows some further pore expansion, but the even shorter duration of the pulse compared to Figures 2 and 3 restricts the pores to an averaged size of 0.85 nm.
Panel (A) in Figure 2 shows a time point during the pulse rise at t = 34 ns, which corresponds to a field strength of 11.5 kV/cm. Here, displacement currents at the PM dominate and lead to intracellular electrical fields that are essentially equal to extracellular electric fields. This can be judged by the almost equal distance of the equipotentials throughout the entire multicellular model, which is in strong contrast to IRE pulses where the cellular shape is still visible. Therefore the local electric field is not strongly perturbed by the presence of cell membranes that enclose a spatial cellular (or even organelle) region. In other words, the almost parallel equipotentials demonstrates an almost uniform electric field. In this sense, the cellular structure is almost electrically transparent. The creation of pores with a minimum pore size of 0.8 nm due to EP (26) has started at this time point at distinct membrane sites. Panel (B) shows the distributed electric response at a time point still during the pulse rise time at t = 45 ns, which corresponds to a present field strength of 15 kV/cm. EP has now occurred on essentially all PM sites, regardless of the cell size, but also regardless whether or not the local membrane site faces the electrodes (polar regions), demonstrating that supra-EP does not depend on the cell size and cell orientation towards the field direction (23, 24). At t = 67 ns during the pulse plateau the pore histogram in panel (C) shows that basically no new pores are created at the PM, but that elevated local PM transmembrane voltages PM in the range of 0.6 - 1.1 V (not shown) cause the small pores present to grow in size. The distributed electric response in panel (D) at the end of the pulse plateau at t = 270 ns does not change qualitatively from panel (C). The pores have further grown somewhat in size during the pulse, but the short overall duration of the pulse restricts the pores to small sizes, such that an averaged size of only <rp> = 1.05 nm is achieved. These findings about the pore size distribution provide further evidence that the asymptotic EP model which neglects pore expansion is a good approximation to describe the membrane response to nsPEF pulses (26).
As seen from panels (A)-(D) in Figure 2, the electric current is always being driven through both the interstitial and the intracellular space because of the strong contribution of displacement currents. A passive electric response without EP on this micrometer spatial scale can therefore hardly be distinguished from an active response with EP. But this is only the case for the distributed electric response, not for the local voltage values, and especially not for the transmembrane voltages ΔψPM (16). As such the applied electric field also perturbs the organelles not explicitly considered here. The strong and direct exposure of the cell interior to strong electric fields is a unique property of nsPEFs, and arguably responsible for the observed intracellular effects (11, 17-22). Turning this fact around, it is therefore important to determine the resulting intracellular electric field for any electric exposure to evaluate the possibility of cellular effects.
Figure 3 shows the distributed electrical response of the multicellular model to a trapezoidal pulse with the same duration as in Figure 2, but with a field strength of 40 kV/cm (12, 14). Due to twice the field strength, EP starts earlier in time such that at (A) t = 34 ns most PM sites have already been electroporated. The distributed electrical response in panels (B-D) is similar to that in Figure 2, but there are more pores that have grown in size from the minimum pore size of 0.8 nm. Although the mean pore size of rp> = 1.15 nm is only marginally larger for the 40 kV/cm pulse, there are about 100 pores with a size of 2 nm for all cells in the model (corresponding to a density of about 6 × 1010 m-2), which is significantly different from the pulse with 20 kV/cm.
Figure 4 shows, similar to Figures 2 and 3, the distributed electrical response of the multicellular model to a trapezoidal 60 kV/cm pulse with 2.5 ns rise and fall times and a 15 ns plateau duration (13) and correspondent pore histograms at different time points. Panel (A) shows the distributed electric response at t = 2.8 ns already during the pulse plateau. Displacement currents at the PM dominate here again and lead to intracellular electrical fields that are essentially equal to the extracellular electric fields and the applied field strength. The creation of minimum-sized pores due to EP has started at this time at distinct membrane sites. Notably, EP may not necessarily start at the polar side of a cell facing the electrodes as it is well-known for isolated cells (45), but rather in the interstitial space were current flow is strongest. Note that EP depends exponentially on the local transmembrane voltage squared (26). Panel (B) shows a time point at t = 3.6 ns where EP has occurred on almost all PM sites, with essentially the same spatial pore distribution as in Figure 2. At t = 11.2 ns during the pulse plateau, shown in panel (C), only a few more pores have been created at the PM, as can be best seen from the maximum value of the pore histogram, and the elevated values of the PM transmembrane voltage ΔψPM in the range of 0.6 - 1.1 V (not shown) cause the small pores present to start growing. The distributed electric response in panel (D) at the end of the pulse plateau at t = 17.5 ns does not change qualitatively from panel (C). The pore histogram shows that some pores have grown in size, but the even shorter overall duration of this pulse restricts the pore size to even smaller sizes, such that an averaged size of only <rp> = 0.95 nm is reached.
These above results in Figures 2-4 are in contrast to erroneous statements in the literature claiming that the PM is not affected by nsPEF pulses (11, 17-20). According to our supra-EP hypothesis, the PM must respond by the EP mechanism to avoid transmembrane voltages ΔψPM of several volts that cannot be sustained. All molecular dynamics (MD) simulations of EP and the experimental results of Frey et al., (46) are consistent with our models that show supra-EP. The PM is thus very strongly affected by the nsPEF pulses resulting in pore densities that are much higher than expected for longer conventional EP pulses. This supra-EP response, however, is a different response than conventionally known, since all pores remain of nanometer size only and leads to utterly different electropermeability values than for conventional EP, including IRE pulses.
The application of electric field pulses to tissue leads to the dissipation of energy into heat and thus a temperature increase,Δ. While this effect is wanted for RF tumor ablation and causes thermal damage leading to necrotic cell death, EP-based ablation methods attempt to only lead to a comparatively small ΔT. To evaluate ΔT we start from the spatially distributed electrical potential ø, as shown in Figures 2-4, and proceed first to estimate local SAR values (4) for (A) the 20 kV/cm pulse, (B) the 60 kV/cm pulse, and (C) the 40 kV/cm pulse. Figure 5 shows the spatially distributed SAR in the multicellular model for the nsPEF pulses at the end of the respective pulse. Note here the different SAR scale, for example the pulse with 20kV/cm (A) and with 60 kV/cm (B), which simply reflects the SAR ∝ |∇ø|2 dependence (see Methods section) and therefore leads to a factor of (|∇øB|2)/(|∇øA|2) = 9 difference, so that there is almost one order of magnitude less SAR for the 20 kV/cm pulse. Although the SAR values are unusually high (GW/kg), the impact is only for a duration of nanoseconds. As shown in Figure 5, local SAR values reach the largest values in the interstitial space that is perpendicular to the electrodes, where the electrical current is the strongest. The SAR values inside the cells are comparable, since the gradient of the electric potential ø reaches similar values there, as discussed above. Only the interstitial space that is more parallel to the electrodes shows smaller SAR values. Overall, the distributed SAR is much less heterogeneous and localized for nsPEF pulses than for the longer IRE pulses (4).
ΔT is comparable for the two nsPEF pulses with (A) 20 kV/cm and 325 ns duration and (B) 60 kV/cm and 20 ns duration with maximal ΔT values reaching 0.15°C, since the one order of magnitude difference in field strength |∇ø|2 is essentially compensated by the one order of magnitude difference in pulse duration Δt. However, the nsPEF pulse with (C) 40 kV/cm and 325 ns duration leads to fourfold larger values of ΔT with maximal values of up to 0.6°C. Therefore, ΔT due to a single nanosecond pulse may be comparable or even higher than for a single IRE pulse.
Figure 5:Distributed SAR in the multicellular model, shown at the end of the (A) 20 kV/cm pulse at 325 ns, the (B) 60 kV/cm pulse at 20 ns, and (C) the 40 kV/cm pulse at 325 ns. Note the different scales for (A), (B), and (C). Note further that the unusual large values SAR values are maintained for nanoseconds only. SAR is strong inside the cells and in the interstitial volume (stripes) along the direction of the applied electric field Eapp, but SAR is less strong along the interstitial volume that is perpendicular to the field direction (at polar regions of some cells). SAR at the PM is typically less than in the interstitial space and inside the cells.
Figure 6: Distributed temperature increase ΔT, shown at the end of a single (A) 20 kV/cm pulse at 325 ns, the end of a single (B) 60 kV/cm pulse at 20 ns, and the end of a single (C) 40 kV/cm pulse at 325 ns. ΔT is correlated with the SAR distribution (Figure 5) since the thermal relaxation on the nanosecond timescale is negligibly small. Thus ΔT after the pulse is initially concentrated within ?hot? stripes along the PM and the interstitial space due to local SAR. Note that none of these localized temperature rises due to a single pulse are sufficient to cause thermal damage.
Note again that the above analysis is for a single pulse only, whereas the reported therapeutic solid tumor treatment with nsPEF pulses is typically performed with multiple pulses. For example, Nuccitelli et al., (12) used about 100 pulses with a frequency of 0.5 Hz, and Garon et al., (13) used 65 to 1,000 pulses with a delivery rate of 20 MHz. Temperature estimates that are then provided based on the passive tissue conductivity are easily off by a factor larger than 2, since the overall tissue conductance is increased by this factor due to EP (4). Hence, the overall generated temperature change ΔT by a single nsPEF pulse is not negligibly small and from a non-thermal mechanistic point of view, it is indeed advisable in treatments to have rather slow delivery rates (< 1 Hz) to have thermal diffusion and perfusion set in and relax the generated temperature profile.
Multiscale Tissue Model
Local solid tumor treatment by electric field pulses is controlled by an appropriate placement of electrodes. In order to control the spatial extent of the tissue region that is being treated by nsPEF pulses, it is essential to know the electric field redistribution that results from the dynamic behavior of EP. Figures 7-9 show the distributed electric response to nsPEF pulses in time and space in the tissue model for the 20 kV/cm pulse with 325 ns duration, 40 kV/cm pulse with 325 ns duration, and 60 kV/cm pulse with 20 ns duration, respectively. Needle electrodes are placed as shown in Figure 1. The distributed electric response is represented during and after the application of the pulse by the electric potential ø (A, F), the local averaged electric field E (B, G), the intracellular electric field Eint (C, H), supplemented by the local transmembrane voltages ΔψPM (D, I), and the local pore densities Np (E, J). The latter is calculated using the asymptotic EP model wherein pore expansion is neglected. The PM charging rate, and by extension the time of onset of significant EP and reversible electrical breakdown (REB), is determined by the local electric field magnitude E, which in Figures 7-9 is shown to be largest near the needle electrodes and drops off quickly with distance. As such, the membranes in the regions of tissue nearest the electrodes charge fastest and electroporate first, reaching pore densities of about 1016 m-2, which is about two orders of magnitude higher than for IRE pulses (4).
Subsequently, a wave of elevated ΔψPM with values above 1 V and pore creation moves outward from the needle electrodes into both the central region of the tissue between the electrodes and around the needle electrodes. Transmembrane voltages of 1 V and above can be maintained only for a short time, as massive pore creation leads to REB, which in turn causes the PM transmembrane potential ΔψPM to decrease even during the pulse (26, 39). In contrast to IRE pulses, however, ΔψPM does not relax to values significantly below 1 V during the pulse because of the overall short time scale of the pulse. After the pulse all electroporated cell membranes in the tissue are depolarized for the lifetime of the pores. Hence the transmembrane potential ΔψPM is basically zero for all electroporated cell membranes. As such, the overall electric potential ø is essentially zero throughout the tissue, resulting in a lower number of equipotential lines on the scale presented in Figures 7-9 (F and I) 1000 ns after the pulse.
Figure 7:Spatial tissue response to the 20 kV/cm trapezoidal pulse with 325 ns duration and 55 ns rise and fall times showing the (A, F) electric potential ø, (B, G) averaged electric field magnitude E, (C, H) intracellular electric field Eint, (D, I) transmembrane voltage ΔψPM, and (E, J) pore density Np, as calculated from the asymptotic EP model, near the electrodes during (A)?(E) and after (F)?(J) the pulse, along the centerline (y = 0 mm). On each plot, 21 (for ø) or 11 (for E, Eint, ΔψPM, and Np) contour lines are spaced evenly between the extreme values of the associated colorscale bar. Times shown are 55, 100, 150, 200, and 270 ns during the pulse, as well as 0 and 1000 ns after the pulse. The white contour in (B) denotes an electric field strength E of 1.83 kV/cm, while the white contour in (C) denotes an intracellular electric field Eint of 0.84 kV/cm, and the white contour in (E) denotes a pore density Np of 1014 m-2. Note that the three white contours change their spatial position during the pulse but by construction all fall onto the same position at the end of the pulse.
Figure 8: Spatial tissue response to the 40 kV/cm trapezoidal pulse with 325 ns duration and 55 ns rise and fall times showing the (A, F) electric potential oslash;, (B, G) averaged electric field magnitude E, (C, H) intracellular electric field Eint, (D, I) transmembrane voltage ΔψPM, and (E, J) pore density Np, as calculated from the asymptotic EP model, near the electrodes during (A)?(E) and after (F)?(J) the pulse, along the centerline (y = 0 mm). On each plot, 21 (for ø) or 11 (for E, Eint, ΔψPM, and Np) contour lines are spaced evenly between the extreme values of the associated colorscale bar. Times shown are 55, 100, 150, 200, and 270 ns during the pulse, as well as 0 and 1000 ns after the pulse. The white contour in (B) denotes an electric field strength E of 1.81 kV/cm, while the white contour in (C) denotes an intracellular electric field Eint of 0.81 kV/cm, and the white contour in (C) denotes an intracellular electric field Eint of 0.81 kV/cm, and the white contour in (E) denotes a pore density Np of 1014 m-2. Note that the three white contours change their spatial position during the pulse but by construction all fall onto the same position at the end of the pulse.
Figure 9: Spatial tissue response to the 60 kV/cm trapezoidal pulse with 15 ns duration and 2.5 ns rise and fall times showing the (A, F) electric potential ø, (B, G) averaged electric field magnitude E, (C, H) intracellular electric field Eint, (D, I) transmembrane voltage Δψ, and (E, J) pore density Np, as calculated from the asymptotic EP model, near the electrodes during (A)?(E) and after (F)?(J) the pulse, along the centerline (y = 0 mm). On each plot, 21 (for f) or 11 (for E, Eint, ΔψPM, and Np) contour lines are spaced evenly between the extreme values of the associated colorscale bar. Times shown are 2.5, 4, 7, 11, and 17.5 ns during the pulse, as well as 0 and 1000 ns after the pulse. The white contour in (B) denotes an electric field strength E of 1.6 kV/cm, while the white contour in (C) denotes an intracellular electric field Eint of 1.57 kV/cm, and the white contour in (E) denotes a pore density Np of 1014 m-2. Note that the three white contours change their spatial position during the pulse but by construction all fall onto the same position at the end of the pulse.
The intracellular electric field Eint shown in Figures 7-9 may be key to understanding the particular effects observed for nsPEF pulses. It may in particular provide insight about the amount of electric perturbation of organelle membranes and to the likelihood of organelle EP. The averaged Eint has been estimated in our tissue model from the macroscopic electric field E and the cell parameters according to
where E is the local macroscopic electric field magnitude, ΔψPM is the PM transmembrane voltage, and lcell is the cell size (see Figure 1). For values of E = 10 kV/cm, ΔψPM = 1 V present during the pulse as seen in Figures 7 and 8, and lcell = 22 µm we find Eint = 0.9 kV/cm, which on this time scale according to Ref. (24) is sufficient to electroporate larger organelles in cells, such as the endoplasmic reticulum. Note that for even larger applied field strengths (20 - 60 kV/cm) Eint becomes more and more equal to E.
Figures 7-9 show that the spatial extent of EP changes during the entire nsPEF pulse in strong contrast to IRE pulse conditions which reach a stationary profile of the pore density relatively early during the pulse plateau (4). Following the pulse, ø, E, Eint, and ΔψPM rapidly decrease with a complex discharge pattern. The pore density Np remains elevated (Figures 7-9 J), and decays with an assumed 3 µs time- constant. As such, the perturbation of the tissue is long-lived, lasting much longer than the duration of the applied pulse, and ionic and small molecular uptake or release may persist long after the pulse. It is also noteworthy that the electroporated region is larger for the pulses with a duration of 325 ns (Figures 7 and 8) than for the previously considered IRE pulses (4). As demonstrated here, the electroporated tissue is not only positioned between the first and second electrode, but spreads out radially from both needle electrodes. This is significant and has implications for target tissue to be treated, and to avoid tissue that should not be treated.
The white contours in Figures 7-9 highlight a spatial region with a fixed pore density with an associated fixed average electric field E and fixed intracellular electric field Eint. Note that the white contour lines by construction are chosen such that all correspond to the same spatial region at the end of the pulse plateau (t = 270 ns in Figures 7 and 8). However, they do not generally align at earlier time points, thus illustrating that instantaneous electric field magnitude is not a good predictor of the spatial extent of EP on short time scales.
For example, the white contour at 270 ns in panels (B, C, E) in Figure 7 indicate a spatial region with a pore density of 1014 m-2 that is related to an instantaneous electric field of E = 1.83 kV/cm and an intracellular electric field of Eint = 0.84 kV/cm. Note, however, that these white contours do not align at earlier times during the pulse. While the electric field E remains at this spatial region since it corresponds essentially to the applied field strength, which does not change during the pulse plateau, the intracellular electric field Eint has a tendency to move inwards as displacement currents decay during the pulse. The white contour showing a pore density of 1014 m-2, on the other hand, moves from a spatial region around the needles outwards and at the end of the pulse comprises a much larger tissue region. This means that a much larger tissue region is altered by nsPEF treatments.
The time delay between the presence of a particular electric field and the actual generation of pores which is related to the pore generation time, may be observed by comparing panels (B) and (E) at 55 ns and 270 ns. While the electric field of E = 1.83 kV/cm is present at the white contour shown in panel (B), it takes the whole duration of the pulse to actually generate pores there.
Figure 8 for the 325 ns pulse with 40 kV/cm shows the white contour lines correspond to (B) E = 1.81 kV/cm; (C) Eint = 0.81 kV/cm, and (E) Np = 1014m-2. The electroporated region is now larger than for Figure 7, but the overall electric behavior is similar to the weaker 20 kV/cm pulse.
Figure 9 shows that the overall pulse duration is so short that displacement currents have not significantly decayed and the generation of pores has been abruptly stopped at the end of the pulse. In other words, if the pulse were to continue for a longer duration, the spatial extend of EP would have been much larger. In Figure 9 the white contour lines correspond to (B) E = 1.6 kV/cm; (C) Eint = 1.57 kV/cm, and (E) Np = 1014m-2.
Note that the reported borderline of a field strength of E = 700 V/cm between a region that experiences IRE and reversible EP is even further outside the white contours shown in Figures 7-9, but the pulse duration is too short to create any pores there. IRE treatments attempt to arrange electrodes and apply IRE pulses such that a particular tissue area that corresponds with the solid tumor is experiencing a specific field strength threshold, e.g., 700 V/cm (8). In particular, an iso-electric field line with 700 V/cm was suggested in Ref. (8) to indicate the border between a tissue region that is reportedly subject to reversible EP (E < 700V/cm) and IRE (E > 700V/cm). The tissue region defined by this particular IRE threshold is only an empirical region and is based solely on the particular IRE pulses applied to liver tissue (8). But if a different pulse with a different field strength and pulse duration were chosen, such as nsPEFs demonstrated here, or if tissue had cells not 22 µm on average in size, this approach would not work as it is not robust. If a clinician were to repeat the treatment of some tissue with a nsPEF pulse (12) it can be seen from Figures 7 and 8 there is clearly a region in the tissue which is exposed to the 700V/cm threshold, but this region is not necessarily electroporated or even undergo IRE because the time scale is much too short. This shows that the concept of ?critical potentials? in EP is incorrect. Instead, EP involves a ?critical event? (pore creation) that depends at least on time as well as on ΔψPM.
Figure 10: The total current (per system depth) flowing through the tissue system is shown as a function of time for the 20 kV/cm pulse (top), 40 kV/cm pulse (middle), and the 60 kV/cm pulse for a passive model, which is the tissue model without explicit EP, and for an active model, which is the tissue model with EP being taken account. Displacement currents dominate the total tissue current for the short time scale (high frequencies) of the pulse rise in (A) and (B) and the total duration of the pulse in (C). The longer the duration of the pulse and the more the displacement currents decay in time, the less agreement is obtained between the active and the passive model.
Figure 10 presents the total current It (per system depth) flowing through the tissue model as a function of time for the trapezoidal 20 kV/cm pulse with 325 ns duration (top), the trapezoidal 40 kV/cm pulse with 325 ns duration (middle), and the trapezoidal 60 kV/cm pulse with 20 ns duration (bottom). As previously, we compare the total current for an active EP-based tissue model with a passive tissue model without explicit EP (4).
Figures 10 A (top) and B (middle) show for the active tissue model an increase of tissue current during the pulse rise, followed by a peak, and then a gradual reduction of It during the trapezoidal pulse plateau followed by the decline of It during the pulse fall. The stepwise changes in the slope of the applied electric field pulse at t = 0 ns, 55 ns, 270 ns, and 325 ns cause abrupt changes in the total system current at these times as a result of stepwise changes in displacement currents. The passive model (no EP), in contrast, shows a strong decay of the tissue current even during the pulse plateau. Though this pulse is only about 300 ns, it is sufficiently long for some of the high frequency components to decay. As this happens, the membrane becomes increasingly significant in determining the total current flowing through the system. In the active model, the membrane impedance remains small because the many pores allow the flow of an ionic conduction current, and the total system current remains relatively steady. In the passive model, however, the impedance of the membrane (and therefore the tissue) grows as the high frequency components of the displacement currents decay and the fixed, tiny conductance of the membrane prevents a significant conduction current, and the total system current decreases.
In other words, while the large membrane conductance of the active model permits a large transcellular current in addition to the shunt current, the small membrane conductance of the passive model increasing excludes transcellular current and a larger fraction of the total current must flow through the shunt outside each cell. In terms of the frequency-dependent tissue conductivity shown in Figure 1F, the system basically starts of at the right side at f = 108 Hz at the beginning of the pulse and then during the pulse moves to the left to lower frequencies. The longer the duration of the pulse, the less agreement is observed between a passive and active tissue model prediction for the tissue current. Overall, the tissue current is always higher for the active tissue model, since EP increases the tissue conductivity by a large increase of the PM conductance. EP thus always leads to an increased Joule heating in the tissue in comparison to a passive model (4).
On the even shorter time scale of the pulse in Figure 10 C (bottom), the impedance of the membrane is much smaller than the electrolyte impedance. As such, the shift from displacement to conduction dominated transmembrane current (as pores form) in the active model has relatively little effect on the total system current flowing by the end of the pulse, and there is relatively little difference between the total system current flowing through the active and passive model systems.
The above predictions for the tissue current in the passive and active tissue models appear to be in somewhat better agreement for nanosecond pulses than for longer IRE pulses (4). This, however, is entirely due to the strong contribution of displacement currents in this temporal regime. Compared to the IRE pulse condition, nsPEF pulses show a more complex interplay between conduction and displacement currents to the total current (16). In particular, the PM impedance is initially largely determined by the PM dielectric properties because of the extremely low conductance of the PM when no EP has occurred yet and the high-frequency content of the pulse rising edge. The large increase in membrane conductance that results from the creation of minimum-sized pores causes a shift from a dielectric-dominated PM impedance to a conductive-dominated PM impedance. The high conducconductance state of the PM then leads to a continued penetration of the electric field and electric currents into the intracellular space, even when the high-frequency components decay (16), as demonstrated in Figure 10.
Therefore, the alleged similarity between the passive and active tissue currents shown in Figure 10 should not be misunderstood. Specifically, it is only superficially suggested that passive tissue models are more appropriate to study nanosecond EP pulses than it is for IRE pulses. On the contrary, a passive model is inadequate to describe the response at the PM, where EP occurs to self-limit the transmembrane voltage to values of about 1 V, and instead predicts tens of volts for the transmembrane voltages, far in excess of what a biological membrane can sustain (16).
Note in Figure 10 that the total tissue current being driven through the tissue is higher for the nsPEF pulses than for the previously considered IRE pulse. This indicates that nsPEF pulses generate an even higher tissue conductance change than IRE pulses, even though this fact is masked partially in Figure 10 by the strong contribution of displacement currents.
Discussion and Conclusion
The treatment of solid tumors by nsPEF pulses leads to supra-EP at the PM, a universal mechanism by which only small nanometer-sized pores at large densities perforate cell membranes, and a strong interplay of conduction and displacement currents inside the cells and the interstitial space of the tissue. As a consequence, the cell interior is affected by intracellular electric fields that are almost as strong as the applied field strength. What is often neglected in the discussion of EP is the resulting exposure of the interior of the cell, i.e., both the intracellular electrolyte and the intracellular organelles, to an electric field. The inevitability of creating an intracellular field should address the magnitude of the intracellular field, which may vary from unimportant to tremendously important. It is therefore understandable that nsPEF pulses cause pronounced intracellular effects, which presumably are responsible for triggering a different cell death mechanism (apoptosis) than longer IRE pulses (necrosis). Supra-EP at the PM leads to a conductance change of the membrane that exceeds that of longer IRE pulses. However, this does not transl
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