Introduction

Electroporation (EP) primarily involves the increased permeability of both the outer and the intracellular membranes of biological cells upon exposure to external electric pulses of sufficient duration and intensity1,2,3,4,5,6,7,8,9,10. Studies of electroporation in cells have a long history, and several reports of numerical simulations can be found in the literature11,12,13,14. Other more microscopic treatments that better include the inherent physics of the method have also been discussed15,16,17. The method is an effective non-viral approach for transfection that is independent of cell surface receptors. Other secondary EP induced influences pertain to transport of ions, macromolecules, and genes or drugs for chemotherapeutics18,19,20. Other downstream electric-field induced responses include increases in reactive oxygen species21,22,23,24, capacitive cellular entry of calcium that can disrupt intracellular calcium homeostasis and trigger a cascade of other events25,26,27,28, changes in the mitochondrial transmembrane potential (TMP), opening of the mitochondrial permeability transition pore (mPTP)29,30, and adverse effects on adenosine triphosphate (ATP) dynamics. A large influx of Ca2+ after EP can also deplete intracellular ATP and/or inhibit ATP production in mitochondria31. Pulsed electric fields have also been used for cell fusion32, microorganism inactivation33,34, and triggering of immunogenic processes35,36.

Reversible and irreversible EP are the two common modalities. Reversible EP is used to deliver impermeant molecules into cells37,38, and avoid permanent cell damage. Irreversible EP is an efficient ablation modality for treating tumors, especially those unsuitable for surgery or thermal ablation because of their anatomic location39,40. High field electric pulses are also increasingly applied in cardiac ablation to treat arrhythmias41. High-frequency irreversible electroporation (H-FIRE) can effectively treat tumors while mitigating pain and muscle contractions42,43,44,45.

The present study seeks to quantitatively assess the degree of EP at three specific membranes commonly encountered in the cellular structure driven by nanosecond electric pulsing: the cell plasma membrane, the endoplasmic reticulum (ER) membrane, and the mitochondrial membrane. All three membranes mentioned above control or influence calcium flux, and upon EP could lead to the introduction of supraphysiological concentrations of calcium in cells. Numerical simulations on such aspects have been reported in the literature26,46,47,48. For example, membrane poration would provide a way to induce Ca overload, which has been shown to be a novel cancer treatment modality49. Hence, the ER and the mitochondrion have been included as relevant organelles of interest in this study. Though the nucleus was represented by a sphere inside the ER surface, the process of electroporation in the nucleus was not included. This represents a modest first step and other organelles such as poration at the nucleus, golgi apparatus, etc. could be included in subsequent analyses.

Though calcium ions diffuse freely through pores in the outer mitochondrial membrane under normal homeostasis, their flow across the inner mitochondrial membrane (IMM), requires ion channels or transporters. However, it is quite likely that an external electric field can porate the IMM50, as we recently predicted. Calcium entry could then follow, leading to ATP depletion, which can trigger a cell death pathway. In fact, calcium EP has been proposed as an inexpensive and efficient cancer treatment modality49. Furthermore, many characteristics of tumor cells (e.g., evasion of apoptosis, limitless replication, insensitivity to anti-growth signals, and metastases), are also regulated or altered by calcium51,52.

Despite the conjecture and some promising reports53,54,55, realistic analysis and predictive modeling of EP, including three-dimensional cellular structural geometries and appropriate details of the inherent microstructure, has not been carried out to the best of our knowledge. Most simulations of electric field driven bioeffects, barring a few exceptions56,57,58,59, have relied on spherical cells for simplicity. In biology, spherical shapes are uncommon with protoplasts, murine myeloma cells60, and Streptococcus61 being rare examples of near-symmetric structures. A recent study, however, modeled the buildup of TMPs at the mitochondrion with its two membranes, including random geometric features of the cristae, within a Jurkat50. Here we attempt to go beyond the approximations and simplifications that have been used in the past. The goal is to develop a realistic, physics-based comprehensive simulation for electroporation taking account of a cell with some of its organelles, their realistic shapes, microscopic features such as the mitochondrial cristae, with eventual inclusion of multi-cellular treatments. The present contribution is a modest step in this direction.

The numerical analyses of EP within the ER due to nanosecond electric pulses (NEPs) has been underreported in the literature, though De Angelis et al.62 did report on EP of the endoplasmic reticulum. The utility of ultrashort pulses in penetrating the outer plasma membranes of biological cells and causing EP at organelles was first shown by Schoenbach et al.9 and later expanded to other bioeffects63,64,65. It is thus conceivable that NEPs may be able to cause EP in both the ER and the mitochondrion. Furthermore, since the spatial distance between the ER and the mitochondrion is relatively small, mutual coupling and screening of the internal electric fields are likely to be important and needs to be probed.

In this contribution, we undertake the modest task of predicting the temporal evolution of TMP in response to an externally applied electric pulse. A three-dimensional (3D) structure is utilized for more realistic predictions. Time-dependent calculations using COMSOL were carried out to analyze the spatiotemporal development of the electric fields and the resulting EP at the outer plasma membrane and membranes surrounding the ER, the IMM and OMM. Our predicted results for EP at the ER are also compared to the report by De Angelis62, and exhibit good agreement. The details are discussed in the Results section. We account for changes in the membrane conductivity and the dynamic evolution of pore radii for self-consistency. Details of the simulation method used for the analyses are given next. Thereafter, the results are obtained, and an assessment of the predicted behaviors are presented. Finally, the conclusions, implications, and future research directions are discussed.

Methods

Modeling EP in irregularly shaped cells and/or organelles presents challenges since it requires accounting for the complex geometries and the disparate length scales and curvatures of their features. Any scheme for such a numerical study must be easily amenable to changes in shape, electrode positions, and cell densities, with acceptable accuracy; while allowing user-specified additions in cell number and degree of heterogeneity. Rather than develop analytical electric field models, we used the COMSOL Multiphysics suite to efficiently address these challenges. The generalized field solvers and dynamic meshing in COMSOL facilitate changes in the size or shape of cells or multi-electrode positions without placing heavy computational and analytical burden for each modification. The capability to use any arbitrary geometry would also allow for including data gathered from bio-medical images in future analyses.

As in our previous study50, the self-consistent calculations of EP involved several repetitive steps. First, the spatial distribution of voltage values at a set of discrete nodes within the entire simulation volume was obtained from the field solvers50. This then yielded a spatiotemporal electric field distribution (and hence, the TMPs) that would drive EP. The simulation model was thus based on a distributed circuit representation of a cell, its membrane, the various organelles and surrounding aqueous regions64,66,67,68,69,70. Membrane poration was incorporated by using the Smoluchowski model for time- and angular-dependent changes in membrane conductivity. The membrane conductivity was then updated using the local pore density distributions with details given elsewhere50,71,72.

It may be mentioned that using complete distributions n(ρ,θ,φ,t)73,74,75, which describe pores in four-dimensional space, is computationally intensive. Intricate geometries involving curved surfaces, or double membranes separated by a few nanometers, require fine spatial resolution, which increases the number of grid points, making the full pore-radius dependent calculations through the Smoluchowski equation very time consuming. Consequently, we have applied an asymptotic model of electroporation50,76,77 with a variable membrane tension78,79 to reduce computational expense. The time domain simulations used a 0.1 ns step, with COMSOL’s “Weak Form Boundary PDE” feature to solve the relevant partial differential equations50.

As already discussed, since high levels of cell calcium concentrations can lead to apoptosis80 and given that the ER is a calcium storage site81, it became germane to incorporate an appropriate COMSOL-based model of the ER. For completeness, the presence of mitochondrion as an additional sub-cellular structure was also included into the current analyses, thus enabling an assessment of proximity effects and electric field screening. While the mitochondrial structure with details of the cristae have been implemented and discussed previously50, the geometric construction for the ER requires some discussion. Confocal microscopy has obtained realistic three-dimensional (3D) geometrical representations of ER morphology, with reports of a threshold around 88 kV/cm for significant electroporation in the folds of the ER62. Furthermore, since calcium is mainly stored in the smooth ER82, we primarily focused on accurately modeling the smooth ER.

Blackstone et al.83,84 proposed an ER model consisting of a network of sheets and tubules in which tubules branch off from the surface of the rough ER and extend into a complex network where they intersect and continue branching out into the cell. Figure 1(a) of Goyal and Blackstone84 shows that the rough ER surrounds the nucleus in a series of stacked “sheets”, whereas the tubules come off the surface of the rough ER exterior. The tubules intersect at three-way junctions before branching out into the cell in a complex network. To simplify the geometry to a degree, we disentangled the geometric complexities by only modeling the first layer of smooth ER tubules coming off the surface of the rough ER membrane before the tubules intersect and branch out into the cell. This allowed us to simulate electroporation in the ER tubules without requiring substantial computational resources. The tubules were modeled as cylindrical objects branching off the outer rough ER membrane in a curved shape. Tubule location and orientation were chosen in a pseudo-random manner based on a built-in random function generator to accurately model the branching nature of the tubule structure. Figure 1(a) shows the geometry created to represent this. As shown, many cylindrical tubules branch off a spherical surface representing the outer membrane of the rough ER. Thus, the ER geometry of Fig. 1(a) is similar to that shown in Fig. 1 of Goyal and Blackstone84, without including the inner details or the nucleus sub-areas. Figure 1(b) shows the geometry used to represent the mitochondrion consisting of a double membrane in which the cristae are folded into the inner portion of the organelle.

Fig. 1
figure 1

(a) Smooth ER geometry featuring cylindrical tubules coming off the surface of the rough endoplasmic reticulum. (b) Mitochondrion geometry featuring a double membrane with cylindrical cristae structures extending into the mitochondrion. This figure is similar to that shown previously by our group50.

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Previous studies have reported the electrical (including permittivity and conductivity) and physical properties (membrane thickness and size) of the ER and mitochondrion62,84,85,86,87,88. Additionally, there are well established parameters for the electrical characteristics and physical size of typical Jurkat cells89,90,91,92,93 and for the electrical characteristics of typical copper electrodes94. Table 1 provides the values used to model this system.

Table 1 Parameters used in the simulations.
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We modeled the Jurkat cell as an ellipsoid with a major axis of 7 μm and length of 5 μm along the minor axes in the transverse directions50. Cells were placed in a “block” of suspension medium with dimensions of 20 μm × 20 μm × 20 μm. Finally, simulations were performed for each organelle individually and for both organelles together to probe individual and collective effects. In simulations for analysis of the individual ER or the mitochondrion, the organelles were placed at the cell center. For simulations of both organelles, the mitochondrion was offset from the center by 4 μm in the negative x-direction and the ER offset from the center by 2.5 μm in the positive x-direction.

Electroporation experiments have been carried out using needle electrodes95, which enhance local electric fields and thus aid poration. Needle electrodes can penetrate the skin, while also affecting a smaller area of tissue than patch electrodes, which makes them ideal for clinical treatments involving electroporation. Using multiple pairs of needle electrodes also permits sequential pulsing, which leads to a stronger and more uniform electroporation treatment across a larger tissue fraction. In this study, we modeled needle electrode tips by using conductive disks in the suspension medium, and thus our modeling was for in vitro conditions. Figure 2 shows two geometrical needle electrode models developed for the electroporation simulations. The top and bottom electrodes were positive and negative, with the zero node/potential taken to the center of the box. In this context, Neal et al.95 used a much larger electrode diameter than presented here (0.5 mm diameter versus 5 μm diameter). However, the shape is representative and could scale as required. The use of smaller electrodes here, leading to nonuniform electric fields drivers, is of interest in its own right, as local field enhancement can be a desirable feature for augmenting the bio-effects. In this context, the use of bio-compatible nanoparticles in the literature74,96,97 is precisely aimed at the creation of such inhomogeneities and locally enhanced electric fields.

Fig. 2
figure 2

(a) Realistic cell geometry containing peripheral ER model along with the mitochondrion with one pair of electrodes. (b) Same geometry with three pairs of electrodes along three mutually orthogonal axes.

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Results

A previous COMSOL-based electroporation study of the ER with structure generated using 3D confocal microscopy62 showed that ~ 88 kV/cm was needed for EP in the folds of the ER. This is rather high compared to experimental work establishing that electroporation can occur at 0.1 kV/cm for larger human cells (e.g., muscle cells), or 1–2 kV/cm for microsecond pulse durations for smaller bacterial cells98,99,100. One might start by expecting the induced TMP (Vmem) to vary with the radius (R) of a spherical bioentity based on the Schwan Eq. 104: Vmem= (3/2) E R cos(θ) [1-exp(-t/τm)], where E is the local electric field, t the time, and τm the membrane charging time which roughly scales as: τm ~ R εm/d, where εm and d are the permittivity and thickness of the membrane102,103. Hence, at very small times Vmem~ (3/2) E cos(θ) {(t d)/εm}. However, such a simplified analysis could be inaccurate since it is based on the following assumptions: (i) Near zero membrane conductivities102, (ii) a spherical shape for the cell and all its organelles, and (iii) a concentric structure as has often been assumed104. In reality, analytic formulations are not possible, especially with non-spherical structures that are positioned off-center. Furthermore, the geometric intricacies and potential shielding by other organelles and cells can alter simple-minded relationships. Only finite-element based numerical analyses, as done here, can furnish a correct solution. Also, it may be mentioned that we were not aiming for a match with actual experiments, but rather to look for similar qualitative trends. Given that many aspects of the overall complicated cell structure were not included in the present model, one cannot hope to get a close comparison to mimic a real experiment. Nonetheless, our contribution pushes the boundaries for modeling, and we will aim for even more realistic simulations in the future.

Previous experiments27 have shown that the mitochondrial electroporative response depends on the rise-time of the applied pulse. This was hypothesized to arise from the higher displacement currents associated with a faster rising electric pulse. Other reports also indicate similar pulse-dependent bio-outcomes105. Our prior modeling established that nanosecond excitation could rapidly induce large TMPs in the mitochondrion50. Here for the simulations, we chose a 100 ns pulse with 20 ns rise and fall times and a 60 ns ON-time for the field stimulus.

We first discuss EP of a Jurkat cell containing a mitochondrion and the ER. For these simulations, one pair of electrodes was pulsed using a 100 ns pulse-width (as discussed above) for applied fields of 25 kV/cm, 75 kV/cm, and 125 kV/cm. TMP was then taken at a point on each surface normal to the applied electric field to ensure consistency in the calculations. Figure 3 shows the time-dependent TMP over a 150 ns duration at four locations: (i) the inner mitochondrial membrane (TMP-IMM), (ii) the outer mitochondrial membrane (TMP-OMM), (iii) the cell membrane (TMP-CM), and (iv) the endoplasmic reticulum membrane (TMP-ER).

Fig. 3
figure 3

Evolution of the transmembrane potential (TMP) in Volts over a 150 ns duration for points on the surface of the inner mitochondrial membrane (IMM), the outer mitochondrial membrane (OMM), cell membrane (CM), and the ER membrane (ER), with all points taken normal to the applied fields. (a) Points along the geometry that were chosen. Results for electric fields of (b) 25 kV/cm, (c) 75 kV/cm, and (d) 125 kV/cm.

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The reduction in TMP of the cell membrane (CM) at ~ 0.0375 µs during the 25 kV/cm field in Fig. 3(b) indicates the onset of poration, which corresponds to the effective membrane shorting. The IMM exhibits a slight onset of EP earlier at ~ 0.03 µs with an imperceptible dip in the TMP. However, the growth and poration rates are small; hence, the TMP remains nearly flat until the pulse begins to turn off at 0.08 µs. In comparison, the OMM and the ER membranes do not porate at 25 kV/cm. For the 75 kV/cm case [Fig. 3(c)], the CM exhibits a stronger EP characteristic with a sharper fall-off in the TMP starting at ~ 0.022 µs, after an overshoot up to ~ 2.2 V. Furthermore, the TMP of the IMM increases sharply and begins to fall as EP begins. The IMM response is predicted to be faster than the CM characteristic. Additionally, the OMM exhibits signs of EP with a reduction in TMP at ~ 0.022 µs, similar in time to the dynamic onset at the CM. These results qualitatively agree with experimental observations of reductions in mitochondrial membrane potential in Jurkat cells30. However, an important distinction between possible indications of electroporation based on measured data and the simulation results reported here needs to be clarified. Here, the only quantity being tracked and predicted is the time-dependent variation in TMP values. A flattening of such a curve, or even a negative slope over certain time intervals, only points to possible poration at the local level. The pore formation, however, may be small in terms of its density, or the total area, and thus might not have tangible effects on measurable molecular transport. For example, the outflows of ions or the uptake of molecules such as propidium iodide (PI), which is a common marker for poration, might not be picked up in experiments. Furthermore, given the time needed for motion and flow in or out of cells, nothing would likely be discerned over one or even a few nanosecond pulses. Finally, cellular response could even occur after an initial strong pulse as has recently been shown by Asadipour et al.106, or might require a train of multiple pulses.

Moreover, experimental results30 could not definitively conclude whether the nanosecond electric pulses induced decrease in the mitochondrial membrane potential resulted from the direct electroporation of the inner mitochondrial membrane, the formation of mitochondrial membrane permeability transition pores, or from some other permeabilization mechanism. However, mitochondrial poration could be a likely cause. The ER, however, is predicted to undergo negligible poration despite the higher 75 kV/cm field. At 125 kV/cm, ER is predicted to reach a TMP greater than 1 V, which most likely corresponds to the cusp of EP initiation107,108. However, the TMP does not appear to decrease much (or perhaps the effect is masked by the start of the pulse decay). This result is on the same order of magnitude as a previous report62, indicating that the EP threshold for the ER is much higher. Our result could loosely be compared with a report by De Angelis et al.62. They reported TMP at the ER crossing the poration threshold of 1 V at an externally applied electric field of 130 kV/cm. Here, we predict similar results for electric fields of 125 kV/cm. It should be kept in mind, however, that the simulation reported by De Angelis et al.62 did not account for the complicated three-dimensional geometry that we employ in the present calculations. The present result, however, is indirectly suggestive of the observed calcium release process in cells subjected to external electric pulsing109,110,111. Since the ER itself is not likely to be easily porated, the observed increase in cellular calcium might arise from plasma membrane poration, possibly followed by CICR.

Next, the mitochondrion and ER were simulated separately inside a Jurkat cell at a 75 kV/cm field to study the EP response of each organelle. For these studies, each organelle was placed in the center of the cell and the needle electrode geometry [cf., Fig. 2(a)]. A stimulus in the z-dimension applied for the single pulse case, while the three-electrode pair geometry [cf., Fig. 2(b)] was chosen for applying a pulse in the x, y, and z dimensions for the three sequential pulsing case. Simulation results for the local electric field developed in the IMM [Fig. 4(a)], and on the OMM surface [Fig. 4(b)] are shown. The applied field was taken to be along the z-axis. The associated voltage profile at 50 ns (midway through the 100 ns applied pulse) is shown in Fig. 5(a), while Fig. 5(b) gives the electric field magnitudes with finer resolution in the y-z plane. Figure 4(a) shows that field enhancement occurs at the tips of the mitochondrial cristae normal to the applied field. The high fields occur only at the cristae tips for structures oriented along the z-direction, but much lower fields occur at cristae that are otherwise aligned. Furthermore, the electric field decreases slightly near the polar caps at the top and bottom surfaces of the mitochondrion in Fig. 4(b). As we show later, this occurs due to strong electroporation at these regions that leads to electrical shorting. The cross-sectional view of the voltage profile in Fig. 5(a) reveals large gradients at the cristae tip pointing along the z-axis. In Fig. 5(b), the field collapse in the IMM cristae is not predicted to be as large. This is due to the smaller size of the cristae, which does not translate into a large porated area.

Fig. 4
figure 4

Simulated electric fields at a 50 ns instant due to a 100 ns, 75 kV/cm applied pulse along the z-axis. (a) Electric fields developed in the interior portion of the IMM, and (b) electric field on the exterior surface of the mitochondrion due to the same applied pulse.

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Fig. 5
figure 5

(a) Voltage after 50 ns in a cross sectional (Y-Z plane) view of the mitochondrion due to a 75 kV/cm, 100 ns pulse applied in the z-direction. (b) Electric field shown after 50 ns over the Y-Z cross section due to the same 75 kV/cm, 100 ns pulse.

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Applying multiple pulses can be expected to induce additional effects. For probing this situation, we considered three sequential 100 ns, 75 kV/cm pulses along the z-, x-, and y-directions of a mitochondrion, respectively. A 50 ns snapshot of the internal field on the IMM surface in Fig. 6(a) is identical to the snapshot in Fig. 4(a). Figure 6(b) and 6(c) show the field after 150 ns and 250 ns. Each successive applied pulse is seen to increase the electric fields. Here it may be mentioned that repetition rates in practice tend to be somewhat lower than used here. For example, typical application of bipolar pulses in the H-FIRE protocol could involve a 2 µs – 5 µs – 2 µs burst repeated until 100 µs of on-time, with about a 5 µs delay between pulses39,43,45. Another slightly higher repetition rate was used by Kiełbik et al.112, while Pakhomov et al.113, applied repetition rates of up to 4 MHz, which translate into a time scale of 0.25 µs. Here, our value at 0.1 µs is shorter, but not by a large amount.

Fig. 6
figure 6

Snapshots of the electric field distribution on the surface of the IMM for three sequential 100 ns, 75 kV/cm pulses applied along the z-, x-, and y-dimensions, respectively, at time instants of: (a) 50 ns, (b) 150 ns, and (c) 250 ns.

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Figure 7(a) and Fig. 7(b) show simulated pore densities (on a logarithmic scale) for a single pulse [along the z-axis as in Fig. 4(a)] and a sequence of three successive non-overlapping pulses applied along the z-, x- and y- directions, respectively, at the 300 ns time instant. As expected, a single pulse along the z-axis induces the highest poration at the polar caps with levels reaching above 1016 m− 2. The remaining regions are comparatively less porated with densities from 109 to 1012 m− 2. Figure 7(b) predicts that applying three successive pulses can cause both substantially higher pore formation and more uniform poration throughout the organelle compared to a single pulse. This ability to reach different regions and spatial positions through repetitive pulsing is critical for cancer treatments, which require affecting the entire tumor volume. Studies of repetitive pulsing, clever electrode geometries, and multi-pronged applicators114,115 have indicated that multi-pronged electrode configurations and judicious pulse sequencing would enhance the degree of cellular uptake.

Fig. 7
figure 7

(a) Logarithmic pore density formed in the interior of the mitochondrion after 300 ns due to a single 100 ns, 75 kV/cm applied pulse. (b) Logarithmic pore density for three successive 100 ns, 75 kV/cm pulses applied along the z-, x-, and y-axes, respectively, at 300 ns.

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A similar trend of more effective TMP build-up also occurred in the ER. We simulated an ER exposed to a single 100 ns, 75 kV/cm pulse along the z-direction, and then three non-overlapping successive 100 ns, 75 kV/cm pulses applied along the z-, x-, and y- axes (as assessed for the mitochondrion). Figure 8 shows the electric field on the surface of the ER after 50 ns due to the single pulse. The field is highest in a ring near the “poles” of the ER membrane (at the top and bottom of the ER) with respect to the z-axis, indicating poration at these locations. However, the complex cylindrical structure of the tubules makes it difficult for high fields to build up within these sub-structures, and hence, the electric fields in the tubules generally lower than at the membrane surface. This indicates the difficulty in reaching a critical TMP in the ER tubules.

Fig. 8
figure 8

Electric field on the surface of the endoplasmic reticulum after 50 ns following the start of a single 100 ns, 75 kV/cm pulse applied in the \(\:z\)-direction.

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Previous studies observed that high applied fields were required to porate the complicated tubular structures of the ER62. Figure 9(a) shows a logarithmic pore density profile on the ER surface after 300 ns following the start of a single 100 ns, 75 kV/cm applied pulse; Fig. 9(b) assesses portion for three, non-overlapping successive 100 ns, 75 kV/cm pulses. As with the mitochondrion, greater pore formation was predicted across membranes for sequential pulses applied in the three mutually orthogonal directions than with a single pulse alone. Furthermore, less poration was predicted in the tubules than on the surface of the ER membrane, due to the complex structure and bends in the tubules.

Fig. 9
figure 9

Logarithmic pore density on the ER surface after 300 ns due to: (a) a single applied 100 ns, 75 kV/cm pulse, and (b) three successive 100 ns 75 kV/cm applied along the z-, x-, and y- directions.

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Progressing toward adequately modeling electroporation dynamics in a full cell requires placing organelles close to one another. This naturally raises the issue of mutual coupling, and the possibility of electric field modifications due to screening effects. To probe this aspect, a comprehensive calculation was carried out with both the mitochondrion and the ER placed together in a Jurkat cell and exposed to three sequential 100 ns, 75 kV/cm pulses applied along the z-, x-, and y- axes. Figure 10 shows the simulated pore density on a logarithmic scale. As expected, screening occurs between the organelles, which reduces the poration over portions of the cell that have the ER and mitochondrion lying near each other. The greatest poration, on the other hand, is predicted to occur on the membrane surfaces of each organelle along the common axes that are closest to the electrodes. Furthermore, very low pore formation is predicted in the smooth ER, in keeping with previous results.

In this context, it may be mentioned that electrical stimulation is usually used for targeted killing of tumor cells. Thus, instead of a single cell, one would typically have a collection of cells, or part of a diseased tissue or organ. In such a situation, it becomes imperative to ensure that the entire tumor mass is effectively treated by the electrical exposure. So then, use of multiple pulsing at a fixed location might not be optimal since the same localized spot would continually be getting treated. If the electrical stimulation was being used for electrochemotherapy, it would be more advantageous to porate and ensure drug uptake from different sides and regions of the affected tissue/organ. This was the reason and essence behind choosing the different orientations in our simulations. Obviously, for a more realistic and meaningful analysis, at least a multi-cell, multi-pronged analysis would become necessary. Use could also be made of nanoparticles that function as distributed electrodes for better and more efficient outcomes68,116.

Fig. 10
figure 10

Logarithmic pore density predicted to formed in the ER and mitochondrion together for three successive 100 ns, 75 kV/cm pulses.

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We note that placing both organelles inside the cell necessitated offsetting each from the cell center for positional accommodation. This put each organelle at a slightly different distance from the straight path joining the centers of the electrode pair oriented in the x-y plane. As a result, the ER remained unaffected along the y- and z- directions, with the electric fields from both pairs of electrodes in the x-z and y-z planes threading between the organelles. The mitochondrion was less affected by the location shift, most likely due to its smaller size. In any event, Fig. 10 shows that screening occurs with lower poration predicted over regions of the surfaces where the ER and mitochondrion face each other. Finally, the mitochondrion exhibits a more well-rounded overall pore density over the entire surface, whereas the ER would likely have a small EP patch only over a small region furthest removed from the mitochondrion and in closest proximity to an electrode.

Discussion

To the best of our knowledge, simulation work at probing time-dependent electroporation via nanosecond pulses in a cancer cell with inclusion of organelles such as the ER and mitochondrion has not been reported in the literature. This aspect represents a new feature. Next, both the ER and mitochondrion are entities that have connections to calcium. The ER has an abundant supply of calcium that could load a cell via electroporation. On the other hand, increased calcium levels are critical to the mitochondria and can trigger cell death. Since the cellular calcium concentrations can be modulated by electroporative means, such modeling becomes technologically important via the calcium connection. Novel cancer modalities based on “calcium electroporation” have already been studied49. Based on such simulation work, optimizing parameters to efficiently kill tumor cells could be envisioned, and would have a very positive impact in biomedical engineering and bio-physics based cancer treatment.

Most simulations of electric field driven bioeffects, barring a few exceptions56,57,58,59, have used spherical cellular geometries or simple symmetrical structures for simplicity. However, realistically assessing cellular responses, especially including organelles with complex shapes such as the ER and mitochondria, make simple representations grossly inadequate. The present contribution takes a modest first step towards realistic numerical studies of a single cell containing multiple sub-cellular organelles with complex geometric structures, which can be extended and incorporate more organelles including details of electroporation at the nucleus, the golgi apparatus etc. We chose NEPs because they have been hypothesized to allow for electric field penetration into cells to affect organelles. A simulation technique based on a time-domain nodal analysis that incorporates membrane poration utilizing the Smoluchowski model with angular-dependent change in membrane conductivity was used. Details of the geometric structures of the ER and mitochondria were included.

The present calculations led to the following results/predictions: (i) The relative electric field thresholds for membrane poration at the cell membrane, the IMM and OMM, and the membranes of the smooth ER were all determined. Of these, the ER required the largest field for possible electroporation. (ii) Simulation results obtained here show that it is much easier to porate the cell membrane than the ER membrane, suggesting that the experimentally observed increases in intracellular calcium could likely result from the CICR process initiated by the breakdown of the cell membrane. Previous reports conjectured that ER membrane poration might directly cause the experimentally observed increases in cellular calcium66,110. However, our results support CICR. (iii) The present simulation results show that mitochondrial poration can occur at about the same (or slightly lower) applied fields than the plasma membrane. Of the two membranes, the IMM was predicted to porate more easily than the OMM, in line with previous experimental trends27. (iv) Multiple pulsing was probed to evaluate possible advantages for killing tumors in tissues (particularly in a larger volume). A single electrode (or prong) or individual pulse was shown to have a limited effect. We determined that applying three non-overlapping pulses in three mutually orthogonal directions induced stronger TMP build-up than a single pulse for both the ER and the mitochondrion. This supports the hypothesis that multi-pronged electrode applicators can maximize therapeutic benefits114. While the studies by Heller at al.114,115 employed needles of larger diameter, the inherent physics remains the same though some quantitative differences may possibly arise. In any event, here we have similarly shown the advantage of multi-pronged electrode applicators through simulations. (v) Screening and mutual coupling was studied through comprehensive simulations of both the smooth ER and the mitochondrion in a Jurkat cell. The electric field modifications led to highly nonuniform development of the TMP. The regions near and facing the external electrodes developed the largest TMP, while regions of individual organelles lying close to each other had lower TMPs. This demonstrated the spatial nonuniformities inherent in creating the electric fields and resulting variations in the degree of bioeffects. This also underscored the importance of applying multiple pulses, either simultaneously or in a non-overlapping manner, with electrodes distributed spatially to maximize the desired effects and improve treatment efficacy of electroporation. While more compete models may change the details of these simulations, the general trends would remain since all relevant biophysics has been included here.

As a final note, possible improvements and further use of our work in bioengineering could be carried out to include a number of additional features. The basic continuum analysis that hinges on an approach employing the Smoluchowski theory with self-consistent electric field calculations would continue to apply. However, extensions could include: (i) adding other organelles (e.g., including details of electroporation at the nucleus, the golgi apparatus, etc.), (ii) including ion flow, (iii) increasing the scope to probe tissues rather than a single cell, (iv) coupling local heating, as was done in one of our previous contributions66, (v) taking data made available from actual discretized images to refine the accuracy of geometric structures used in the calculations.