Figure 5: Spatial tissue response based on the asymptotic electroporation model to a 1,500 V/cm trapezoidal pulse with 100 μs duration. Electric potential, electric field magnitude, transmembrane voltage, and pore density, as calculated from the asymptotic electroporation model, near the electrodes (A)-(D) during and (E)-(F) after the pulse. On each plot, 21 (for φ) or 11 (for E, ψ_PM, and N_p) contour lines are spaced evenly between the extreme values of the associated colorscale bar. Spatial response along centerline: (A) Electric field magnitude, (B) transmembrane voltage, and (C) pore density along the centerline (y = 0 mm). Times shown (from black to light gray) are 1, 4.6, 21, and 99 μs during the pulse, as well as 0.5 and 5 μs after the pulse. The white contour in (B) changes with time and denotes a field strength of 700 V/cm that indicates a reported borderline region between tissue that experiences IRE and reversible electroporation conditions (15).

The spatial extent of electroporation changes little after ∼5 μs (Fig. 5) because the membranes in unelectroporated regions of tissue have essentially reached their maximal Δψ_PM and in those regions where Δψ_PM does not exceed ∼1 V, little electroporation will occur on the time-scale of the 100 μs pulse (27). We believe this is a general feature of electroporation: local fields tend towards uniformity during a pulse. At the electrode interface, Δψ_PM peaks at 1.26 V at 0.32 μs and N_p reaches a pore density of 1.3 × 10¹⁵ m^-2. At the center of the tissue region, Δψ_PM peaks at the smaller value of 1.18 V at 1.41 μs and N_p reaches a smaller value of 1.5 × 10¹⁴ m^-2. Transmembrane voltages of 1 V and above can be maintained only for a short time, as REB causes the transmembrane voltage to decrease even during the pulse (26, 27). Consequently, while the tissue conductivity increases throughout the region between the electrodes, it increases most near the electrodes and least in the central region of tissue. Because of this gradient in tissue conductivity, the electric field becomes more uniform between the electrodes by the end of the pulse. Following the pulse, φ, E, and Δψ_PM rapidly decrease (Fig. 5E-G) with a complex discharge pattern. N_p remains elevated (Fig. 5H), and decays with an assumed 3 ms time-constant. As such, the perturbation of the tissue is long-lived, lasting much longer than the duration of the applied pulse, and molecular uptake or release may persist long after the pulse.

Figure 6 presents the total current (per system depth) through the entire tissue. We compare the passive tissue model with the active (electroporation-based) tissue model. The current is approximately twice as large for the asymptotic electroporation model (solid line) than for the passive model (dashed line) because electroporation in the active model increases the tissue conductivity by that factor. This is an average factor, as there are gradients in tissue conductivity as discussed in Figure 5. This means that the Joule heating is about two times bigger than in the passive model. After the initial current spike due to displacement currents that also coincides with the maximum in pore creation rate (REB), the passive model has an essentially flat current plateau, whereas the asymptotic electroporation model shows a characteristic slope in the current that indicates the slower creation of additional pores. We speculate here that the SE electroporation model would tend to increase this slope even more as pore expansion during the pulse leads to further membrane conductivity increases (57).


Figure 6: The total current (per system depth) flowing through the tissue system is shown as a function of time for the pulse of Figure 5. The current is approximately twice as large for the asymptotic electroporation model (solid line) than for the passive model (dashed line) because electroporation in the active model, on average, shows an increased tissue conductivity by that factor. After the initial current spike due to displacement currents, the passive model has an essentially flat current plateau, whereas the asymptotic electroporation model shows a slight upward slope in the current that indicates the slow creation of additional pores.

From the spatially distributed electrical response we can proceed to estimate local SAR values. Figure 7 shows the spatially distributed SAR in the multicellular model for the 700 V/cm pulse at the end of the pulse. Two different situations are compared, (i) passive membranes, and (ii) membranes described by the SE electroporation model. SAR is highly heterogeneous and localized, with large values around the membranes and the interstitial space. Within the intracellular space, SAR contribution is much stronger when electroporation occurs, as the current is partially driven through the cells.



Figure 7: Distributed SAR in the multicellular model, shown at the end of the 100 μs pulse with 700 V/cm, is concentrated at the PM and within ?hot? stripes along the field direction (perpendicular currents are much less pronounced); membrane sites are significant for temperature increase. Differences between (A) passive and (B) SE electroporation models are significant, in particular the intracellular region contributes to (B) as intracellular Joule heating occurs. The SAR scale is chosen to best compare the intracellular contribution to Joule heating, the maximum local SAR is about two orders of magnitude higher.

With the local SAR at our disposal, we can estimate the temperature change in the multicellular model. Many bioheat transfer problems in the context of pulsed electric fields have been modeled 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 (13, 60). 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 that scale. 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 (61) (as well as the metabolic heat generation) and consider the diffusion dominated bioheat equation:


Here, ρ, c, κ are the density, specific heat, and thermal conductivity of tissue, respectively, T is local tissue temperature, and t is time.

Figure 8 gives the spatial distributed temperature increase ΔT in the multicellular model as calculated using Eq. [2] for the passive and SE electroporation model. The thermal properties were assumed to be uniform (c = 3600 J/kg K; κ = 0.512 W/m K; ρ = 1060 kg/m³) (13). The boundaries of the model were treated as thermal sinks. This ∼100 μm separation of thermal sinks is about 2.5× the spacing of capillaries. The SAR was input into Eq. [2] to calculate the temperature at the corresponding time point. When the time interval between successive time steps (obtained from the electrical model) was larger than the minimum value to guarantee numerical stability, the SAR was interpolated to finer time intervals (0.1 l²cρ/κ). Without thermal diffusion the maximum temperature rise would be 0.4 °C for 700 V/cm in the passive model compared to 0.8 °C for the SE electroporation model (not shown). However, with thermal diffusion, the temperature increase that is initially mostly confined to the interstitial regions spreads out throughout the multicellular system and decreases the local temperature spikes. In the SE electroporation model, even the intracellular regions showed an initial increase in temperature from baseline by local SAR before thermal diffusion leads to an intracellular temperature rise. We believe that this behavior, i.e., local temperature rise in the interstitial volume around the membranes and then thermal diffusion into the 86% intracellular space that contributed less to the Joule heating, is an effect not captured in any present bioheat equation, and leads to a reduced average temperature increase in tissue by electric field pulses. Comparing the electroporation with the passive thermal response we find that the temperature rise can be more than a factor of 2 larger in most regions and more than 2 orders of magnitude larger in some intracellular regions. But overall these temperature rises are small and are not expected to cause thermal damage. However, IRE protocols apply trains of pulses, not a single pulse, and the field strength is usually higher than 700 V/cm considered here. Hence larger temperature increases are expected.



Figure 8: Distributed temperature increase ΔT, shown at the end of the 100 μs pulse with 700 V/cm in the multicellular model. The temperature increase is initially concentrated within ?hot? stripes along the PM and the interstitial space due to local SAR; Membrane sites are significant for temperature increase. Differences between (A) passive model and (B) SE electroporation model: Intracellular SAR?s contribute in (B) and lead to temperature increase there by Joule heating. The ratio between the temperature increase ΔT of the SE electroporation model and the passive model is shown in (C). The largest differences occur within the cells. Same ratio in (D) on a limited scale, indicating a factor of 2 difference around membranes and the interstitial space. Note that none of these localized temperature rises are sufficient to cause thermal damage.

The local electropermeability, P_m(r_s,t), of the PM for molecule/ion (solute ?s?) of radius r_s depends on the time-dependent Δψ_PM and, in order to quantify molecular transport, can be computed for the membrane of each cell according to



In here, D_s is the solute?s diffusion coefficient. The permeabilization depends on the local pore distribution n(r_p,t) and contains all the relevant information on available pore sizes, and, thus, depends on field strength, field duration, number of pulses, but also resealing dynamics and pore lifetime. Each solute experiences a geometric hindrance inside the pore that is given by H(r_p,r_s), and, if electrically charged, is subject to the partition factor K(r_p, Δψ_PM) that is governed by the electrostatic interactions (Born energy) inside a pore (27). The overall cell permeability may be obtained from Eq. [3] by integrating over the entire cell. As an example, we will consider the membrane permeability P_m(r_s,t) to calcium, ATP, and propidium iodide in future work. One motivation is that uptake or release of certain biochemically important molecules may be relevant for their role in IRE-induced necrosis.

Discussion and Conclusion

We have presented multicellular and tissue models and their electric and thermal responses to representative IRE pulses. Such increasingly realistic models involve complex electrical interactions throughout the system on a sub-microsecond time scale and nanometer to micrometer length scale. The passive electrical properties of tissue are frequency-dependent (62), as higher frequencies allow for access to subcellular compartments. Tissue exposure by conventional electroporation pulses typically involves a broad band of frequencies (up to ∼1 MHz). However, most tissue models assume only a time- or frequency-independent passive tissue conductivity (13, 63), or spatial variation in tissue conductivity that is based on a series of static models and requires a mapping between electric field and conductivity, based on the very experimental results (42, 59) these models seek to predict. They, therefore, lack a mechanistic hypothesis of what is causing the dynamic conductivity changes, and do not provide predictive power, for example, with respect to tissue transport or if different pulse conditions are considered. Here, we have estimated the response of tissue to pulsed electric fields by taking into account dynamic displacement and dynamic conduction currents. Both of our models demonstrate strong redistribution of fields and currents that are electroporation-driven.

A full description of electroporation leads to three major consequences with respect to IRE applications: First, a time-dependent pore population exhibits dynamic pore size changes during and after the pulse, thereby altering the transport of molecules of different size and charge into and out of each cell, which should be important to future understanding of why IRE leads to necrotic cell death. Second, the associated change in effective membrane electrical conductivity leads to intrinsic field redistributions that elevate the intracellular electric field. This itself may cause a biochemical effect inside the cell and lead to cell death by necrosis. And third, a tissue conductivity change leads to greater electric dissipation by Joule heating. Hence, a thermal threshold for the amount of tissue heating caused by IRE pulses cannot be obtained by passive models alone. Instead, an increase in electrical conductivity by PM electroporation must be included and is significant. Post-pulse conductivity measurements (14) do not show the full conductivity increase that is obtained during the pulses as pores will shrink and some may have decayed by that time.

Effective conductivity changes and local electric fields are not the only parameters relevant to understanding the tissue response to strong field pulses. In addition, and even more importantly, the local permeability P_m(r_s) with respect to a certain type of molecules ?s? is a relevant quantity that distinguishes different pulse parameter and waveforms. For example, nanosecond electric field pulses (64, 65) cause large changes in tissue conductivity and strong local electric fields, but the membrane permeability to molecules like propidium iodide or DNA may remain small. In contrast, conventional electroporation electric field pulses as discussed here for IRE will lead to much less dramatic conductivity changes, and smaller local electric fields, but the electropermeability P_m(r_s) for large molecules is significant. Previous tissue models that do not consider electroporation explicitly (13, 42, 59, 63) cannot assess molecular transport and thus biochemical tissue change.

Electropermeabilization is, thus, a major consequence of electroporation. Accumulation of pores often increases the PM permeability dramatically and is essential for cell uptake of external molecules, the main application of electroporation. A striking example is transfection of cells in vitro by electroporation-based DNA delivery, which was a seminal development (66), so much so that ?electroporation? is often incorrectly used to mean the complete process of successful delivery of DNA into the cytoplasm, transport to and into the nucleus, followed by expression. This multistep process is complex and has been optimized substantially in recent years, leading to electroporation-based non-viral gene transfer that appear superior to other transfection techniques (67, 68, 69).

Temperature estimates are crucial to limiting the IRE field parameters to a non-thermal intervention. The temperature increase in the tissue that results from IRE pulses can be found from local SAR values, and the bioheat equation (13, 60). For the time and length scales of the multicellular model considered here, a diffusion dominated bioheat Equation [2] may be employed. Most importantly, local SAR values as input from the SE electroporation model lead to higher local temperatures rises than with the passive model. These local hot stripes spread out throughout the whole system, and this process is limited by the thermal diffusion time scale.

Usually, ECT and IRE protocols employ trains of identical electric pulses. In terms of electroporation properties, it can be shown that a second pulse does not give rise to more pores, irrespective of pulse repetition frequency and pore lifetime. This is due to a memory effect of electroporation. In particular, the associated high-conductivity state, i.e., the specific number of pores and their size, is determined by the pulse amplitude and duration. A second pulse will interact with a higher conductivity membrane and add only as many pores to the pre-existing number of pores (that persist from the first pulse) such that the same total number of pores are created as with the first pulse, or even fewer if pore expansion is accounted for (Esser, unpublished).

The goal of tissue ablation by IRE is to cause complete necrotic cell death within a predefined and restricted local tissue volume. It is important to remember that ECT causes cell death predominantly by drug-induced apoptosis. This cell death mechanism may be more desirable ?in order not to produce large instantaneous necrosis, which would result in massive tumor necrosis and possible ulceration and wound appearance? (42). Advantages and disadvantages of the suggested methods must therefore be considered in a clinical setting and treatment.

Finally, conventional electroporation is cell-size dependent. In general, larger or more extended cells require smaller electric fields than smaller cells to cause a similar PM effect. In homogeneous tissue such as liver, it can be expected that all cells experience similar pore densities for a given field strength. However, cell size is heterogeneous in solid tumors, and here the smallest cell will dictate the field parameters that need to be employed. The cell size limitation is minimized (21, 22, 23, 24) for the local and drug-free tumor treatment of tissue by nanosecond electric fields pulses that were recently reported to cause self-destruction of skin melanomas (64) and to affect other human malignancies (65). Pulse trains at 0.5 Hz, each with field strengths of 40 kV/cm and 300 ns duration, reportedly caused tumor cell nuclei to rapidly shrink (electropyknosis) and tumor blood supply to stop (64). The non-thermal mechanism by which the melanoma cells die is unknown yet, but the sub-microsecond pulses are expected to cause nearly homogeneous supra-electroporation of all cells and their nuclear membranes (24).

Acknowledgments

Supported by NIH grant RO1-GM63857, and a Graduate Fellowship from the Whitaker Foundation to K.C.S. We thank K. G. Weaver for computer support.

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