Figure 4: Size of the ?treated? region as a function of aspect ratio (d/D). (a) Volume (normalized by D³) of the revolved cross sectional area within a coaxial disk configuration where electric field strength (normalized by V/D) exceeds, from the top to the bottom curve, 10^-1.6, 10^-1.4, ?, 10^0.6. (b) Cross sectional area (normalized by D²) within the parallel needle electrodes configuration where the similarly normalized electric field strength exceeds, from the top to the bottom curve, 10^-1.8, 10^-1.6, ?, 10^0.2. Bold curves are for 0.1 and 1 in both graphs. The asterisk on each curve denotes the point where the region containing electric field strengths above a certain level (different for each curve) splits in two, or in other words where the particular level set (in normalized electric field strength) passes through the central point in the geometry. Circles in (b) along the curve E/(V/D) = 0.1 correspond with the three columns in Table II that report discrepancies between the 2D assumption (infinitely long parallel needle electrodes) and the fully 3D case.

For example, consider the disk electrode case from Figure 2c, where d/D = 0.4 and V/D = 400 V/cm and assume that the IRE threshold is 800 V/cm. The volume of tissue that will experience electric fields in excess of 800 V/cm can be estimated by locating approximately where the curve E/(V/D) = 2 or 10^0.3 exists. Since there is no curve where E/(V/D) is exactly 10^0.3, we must interpolate between the closest neighboring curves, which are the third (10^0.2) and second (10^0.4) curves from the bottom. Subsequently, locate the y-coordinate on the appropriate curve of constant E/(V/D) in Figure 4a, where the x-coordinate (d/D) is 0.4. From this we can see that the appropriate coordinates on Figure 4a would be (x = 0.4, y ∼ 0.3). Using this information we can approximate the volume of treatment (Vol/D³ ∼ 0.3) as 0.3 cm³. This matches our expectations from the extents of the 800 V/cm contour visible in Figure 2c, which is nearly equivalent to the tissue directly between the 10 mm diameter electrodes (0.314 cm³). It is also interesting to note that the inter-electrode gap (d) in this example is near-optimal for maximizing the size of the 800 V/cm region, given a 400 V pulse, according to Figure 4a.

The pair of needle electrodes in Figure 2e can be similarly analyzed. In Figure 2e, d/D = 10 and V/D = 10 kV/cm. Assuming the same target 800 V/cm electric field strength, locate the appropriate curve where E/(V/D) = 0.08 or 10^-1.1, which lies between the fourth and fifth curves from the top (10^-1.2 and 10^-1.0, respectively). Thus, the appropriate coordinates on Figure 4b would be (x = 10, y = 30). The treatment area is then 30 D², or 30 mm².

The asterisks on each curve denote the point where the region containing electric field strengths above a certain level (different for each curve) splits in two. Since this point lies to the right of the asterisk, there are actually two 15 mm² treated areas. This can be verified by seeing that the radius of the 800 V/cm contour around each electrode in Figure 2e is about 2.2 mm. To maximize the area with 800 V/cm electric field strength, it can be inferred by tracing the curve of constant E/(V/D) to the left, that the optimal d/D ratio would be about 7. This corresponds with an inter-electrode separation of 7 mm, at which point the treated area would be about 50% larger than area achieved with the 10 mm spacing shown in Figure 2e.

For this figure, areas and volumes with electric field strengths greater than a particular value were computed numerically by integrating E > cutoff across the entire tissue, which produces an area or volume since this Boolean expression is either 1 or 0 at all points. One minor comment is that it was necessary to multiply this integrand by 2πr to obtain treated volumes for the disk electrodes case, where (r) is the radial distance. It is important to remember when using needle electrodes that the treated area along various planes perpendicular to the electrodes will vary, with the largest regions of any particular electric field strength existing on the midplane (L/2); except for some edge effects at the distal and proximal ends of the electrode active surfaces, which may increase the size of some of the higher electric field strength regions locally. This manifests as a somewhat further extent of the treated region in the space peripheral to the electrodes than would be indicated from Figure 4b, an effect that becomes more prominent as L/d decreases. Note also that the treated area begins to separate along the electrode-electrode midline from the top and bottom towards the center as d/D is increased beyond the optimal values from Figure 4b. Refer to the 400 V/cm surface in the 3D pane of Figure 4b for an example.

Table II examines this discrepancy between the treatment volume predicted for the case of 1 mm diameter parallel needle electrodes by the 2D model and that for the full 3D model, where the three columns correspond to the circles marked along the curve where E/(V/D) = 0.1 in Figure 4b. From these data, which were computed for the same geometries as in Table I, it is clear that the 2D model is more accurate as L is increased. Increasing d, however, has a more complex effect on this error; the 2D model at an inter-electrode gap of 7.5 mm is most accurate in this case but is least accurate for d = 10 mm (note that d = 7.5 mm is the point in Figure 4b where the 2D treatment region splits in two). This can be understood by considering that the erosion of the treated region directly between the electrodes, above and below the midplane (L/2), offsets much of the additional volume in the spaces peripheral to the electrodes, so that it is not the actual shape of the treated region that is most accurate for d = 7.5 mm, but only the total volume. The column where d = 5 mm is fairly accurate in both the total volume and the shape of the treated region, so that the 2D model is in fact most useful to determine treatment volume when L/d is large and where d/D is below the point at which the treatment region splits. Note also that the case where L and d are both 10 mm matches the 100 V/cm surface depicted in Figure 1b.



The Effect of Tissue Heterogeneity

In a treatment, if the impedance distribution in the targeted region is homogenous, the results in Figure 4 can be applied directly to estimate the size of the treated region as a function of electrode geometry and voltage applied. However, there can be factors that would make the targeted domain heterogeneous, such as the presence of large blood vessels, multiple tissue types, or tissues with anisotropic properties, such as muscle. Figure 5 is an example of how tissue heterogeneity can affect the electric field distribution. Under these circumstances, the user would need to use our guidelines to make their own model including this heterogeneous conductivity and tailor our guidelines to their specific procedure.

Figure 5 provides three essential examples of how local heterogeneities in tissue conductivity affect the electric fields that develop within the nearby tissue. The first graph (Fig. 5a) shows a 5-mm diameter sphere when one fifth background conductivity is present between the electrodes. As a result of the higher impedance path for current traveling through the inclusion along the electrode-electrode axis, the current density directly between the electrode surfaces to the left and right sides of the low conductivity sphere, has fallen substantially. This has caused the 800 V/cm surface surrounding the electrodes to become locally smaller. The electric potential in this space is, therefore, much closer to the nearby electrode voltage and as a result will cause other regions of tissue to experience a higher than normal electric field as the electric potential establishes continuity. This is evident based on the appearance of a separate 800 V/cm surface within the inclusion with an hourglass shape. At this location, the electric potential will traverse the extra voltage necessary to connect the opposite regions that were varying more gradually than normal. In this particular case, the higher electric field strength corresponds with a focusing conical flow of a relatively small amount of current into the surface of the inclusion from one side and leaving in a similar fashion from the opposite side, where the net current flow is in the direction of the electrode-electrode axis.

The second graph (Fig. 5b) reveals the effect when the inclusion is five times more conductive than the surrounding tissue. In this figure, the path impedance between the electrodes and through the inclusion is substantially less than normal, leading to elevated electric field strengths in the spaces between each electrode and the inclusion. This is evident from the extension of the 800 V/cm and 1600 V/cm surfaces to the inclusion surface. In these two regions, the effect is that the electric potential is further removed from the electrode voltages than normal. This will cause the electric potential to vary gradually over the high conductivity inclusion in order to monotonically connect the two regions at opposite ends. The result is that the electric field strength inside the inclusion (and extending above and below its center) will become smaller than normal, a fact that is illustrated by the appearance of a hole in the 400 V/cm region that is centered within the high conductivity inclusion. The mechanism for the difference in this case is a shunting of extra current from surrounding normal tissue into one hemisphere and out of the other, where net current of course travels along the electrode-electrode axis. However, relatively little voltage drop will occur within the inclusion since it has a five-fold elevated conductivity and the extra current is not sufficient to offset this fact. Finally, the reason the hole extends into the normal tissue is that the high conductivity inclusion is providing a local short-cut for charges traveling near the inclusion surface but within the normal tissue.

The third graph (Fig. 5c) shows what will happen when a thin shell of low-conductivity tissue surrounds a core of high-conductivity tissue, as is the case within regions bounded by an endothelium, such as blood vessels. The path impedance from one electrode to the other and through the center of the inclusion will be between that for the two previous cases, but is approximately equal to the case in Figure 1b, which can be seen by the nearly undisturbed shapes of the primary electric field surfaces. Though the electric potential and local current densities are relatively unchanged in the space between the electrodes and the inclusion boundaries, the core-shell structure has an interesting effect of focusing the electric field onto certain portions of the shell, while reducing it greatly within the core. If the two regions are quite different in terms of their conductivities, this can be understood by considering that the potential difference across the entire inclusion will be split almost entirely between the two primary locations where current either enters or exits the core through the shell. In Figure 5c, this can be observed as the opposite halos of 1600 V/cm electric fields that correspond with conical influx or efflux of current from one electrode to the other and through the inclusion. Notice also that the hole in the 400 V/cm surface is less pronounced than in Figure 5b, since the concentrated current visible as the 800 V/cm and 1600 V/cm extended surfaces in Figure 5b is spread and greatly diminished by the low conductivity shell, and the hole does not protrude into normal tissue since the shell prevents the local short-cut discussed for Figure 5b.


Figure 5: Effect of a heterogeneous tissue conductivity on the electric field distribution. Each graph depicts the surfaces of constant electric field strength (for 100, 200, 400, 800, and 1600 V/cm) that would result from the presence of a 5 mm diameter spherical inclusion, located halfway between the electrodes. The inclusion is composed of tissue with one fifth or five times the background electrical conductivity in graphs (a) and (b), respectively, while (c) the third graph is similar to (a) but with a nested 4 mm diameter sphere of five times normal conductivity. Color map, electrode geometry, pulse amplitude, and background tissue properties are identical to the similar 3D model of needle electrodes in Figure 1b.

The lessons from these three cases also apply in a general sense to any local perturbation in conductivity within a section of tissue where a macroscopic electric field has been imposed. Analogous to the electrode voltages are the bounding electric potentials, set up across the local control volume that includes the perturbation, and induced by the more global variation of impedance and ultimately the electrode voltages. In fact, the line of reasoning from Figure 5c can lead to an understanding of how local electric field strength translates into transmembrane potential. From the discussion of Figure 5c, one other important phenomenon becomes apparent. The high electric fields that will develop across the shell of such core-shell objects will cause electroporation locally long before it occurs in the surrounding tissues. This will provide an easier path for current to reach the high-conductivity core, leading to a much increased flow of current across the newly electroporated shell regions, and increasing somewhat the total amount of Joule heating. Depending on the details of the case, this could involve a pattern of microscopically focused large increases in temperature, visible as small paths of thermal damage across strategic points in the endothelium of large blood vessels or other lumen-filled cavities, a phenomenon that has been observed in recent animal studies of IRE (2) and may be the cause of the vascular lock observed even with presumably reversible pulses (32, 33, 34). Though not primarily responsible for the effectiveness of IRE, this phenomenon will enhance its therapeutic effect by causing tissues that survive the pulse itself to die later of local vascular occlusion (2).

Even with the large perturbations in conductivity from the three examples of Figure 5, neither the total power consumption (Fig. 5a-c drew 2.78 kW, 2.92 kW, and 2.81 kW as compared with 2.83 kW for Figure 1b, respectively, for σ = 0.2 S/m) nor the shapes of the lower electric field surfaces (100 V/cm and 200 V/cm) has been affected significantly. This is primarily due to the fact that the inclusions are far from the electrode surfaces, where the electric potential changes most gradually and where perturbations will have only a local effect on electric fields. This is not the case when poor electrode-tissue contact is present. In fact, this can become a controlling influence on the strength of the electric fields that will develop within the tissue, and its effects can be understood most simply by assuming that the effective voltage across the electrodes will diminish to an amount approximately equal to (K/σ)/(2z_e/A_e+K/σ) times the applied voltage difference, where z_e and A_e are the electrode impedance (in Ωm²) and area per electrode. This simplification makes the inherent assumption that current density is uniform across the electrode surfaces, but a more accurate prediction is possible through the incorporation into the numerical model of the complete electrode model (31), which connects the electrode voltage and impedance to the electric potential in the adjacent tissue through the normal current density across the electrode-tissue boundary as:



The effect of electrode impedance will diminish for larger electrode surfaces and when K is larger, but it will always be a source of some error and also of surface heating.

If there is concern that the impedance distribution of the tissue is not homogenous, the impedance distribution within the domain needs to be established to properly model and predict the area ablated using IRE. The user must make their own model as described in the methods section with an accurate knowledge of the distribution of σ as well as the electrode configuration. An example of a method to obtain the anatomy of the targeted region is to use an imaging technique such as MRI or CT. Once the tissues in the targeted region are identified, their properties can be taken from literature (35) if the user cannot measure the properties directly.

Change in Effective Conductivity of Tissue Due to Electroporation

Researchers have shown that there is a change in tissue impedance during and as result of electroporation (13, 36, 37, 38). Even though such changes were not incorporated into our models, such changes would be important if the area is heterogeneous and especially for the needle configuration. These changes can be readily incorporated into the user?s numerical models (12, 37). Many electroporation devices monitor the current being applied, which can be used to measure the bulk conductivity of the tissue as described in Figure 3. Monitoring the impedance change during IRE is an active means for the physician to monitor the procedure. Furthermore, the region that has been irreversibly electroporated can be imaged using electrical impedance tomography (12, 37), to verify that the targeted region has been successfully treated.

Heat Dissipation After a Single Pulse

If it is necessary to take into consideration the thermal effects from a treatment, then other tissue properties such as the mass density, heat capacity and thermal conductivity are needed. If these properties cannot be directly measured, the properties of the tissue can be taken from the literature, for example from (35). It should be noted that the thermal and electrical conductivities of biological tissues are dependent on temperature and their dependence can be found in literature and incorporated into the models if necessary. Since IRE produces negligible heating, the change in conductivity is not significant. For example, in liver the thermal and electrical conductivities vary by about 0.25% and 1.5% per degree Celsius, respectively (35). Properties for metabolism and blood flow can be found in (39) even though it has been suggested that these factors have a negligible contribution to the overall temperature distribution as compared with Joule heating (22). Futhermore, blood flow can be neglected as revealed by the results presented in (2) that showed that perfusion stops during such a procedure, which may assist in inducing total necrosis of the tissue.

Figure 6 was generated to enable the duration, extent and degree of the transient hyperthermia following an electroporation pulse, applied between needle electrodes in any uniform tissue, to be predicted based on approximate knowledge of tissue electrical and thermal properties and the details of the pulse.

Figure 6: Dissipation of heat generated by a single pulse between two parallel needle electrodes. Graphs show the evolution of temperature with time at various points along the centerline between electrode centers. Temperature rise (ΔT) and time (t) are given in a dimensionless form to allow general use. Asterisks in above electrode depictions mark exact locations from where the temperature is reported in the several plots in each pane. In each case, the highest and lowest curves correspond to the electrode inner surface and midpoint between electrodes, respectively. Eight curves in (a) are for d/D = 20, six curves in (b) are for d/D = 10, and four curves in (c) are for d/D = 5. Exact positions of marked points can be found by starting with the point on the inner electrode surface (D/2 from electrode center) and marking subsequent points at intervals of aⁿ (D/2), where this factor is 1.252 for (a), 1.203 for (b), and 1.151 for (c). Top horizontal line designates the maximal temperature rise within the object just at the end of the pulse.

The application of a short electroporation pulse across needle electrodes will cause a sudden deposition of a highly non-uniform amount of heat (2). Immediately after the pulse (and to some extent during the pulse), this heat will begin to diminish in strength as it spreads throughout the tissue as described by the heat diffusion equation. The initial heat deposition can be seen as the dimensionless temperature rise at each marked point in the tissue at the far left of the three graphs in Figure 6. This heating is so non-uniform that points far from the electrode surface are not subjected to the more intense heating near to the electrodes until some time has passed, as can be observed from the bump in temperature that occurs at progressively later times as one looks at points further from the electrode surface. By the time the temperatures have stabilized at the far right side of the Figure 6 plots, an additional pulse can be applied with minimal cumulative heating of the tissues.

As an example, consider a 100 μs pulse of 1000 V as depicted in Figure 2e that is applied across a tissue with ρ = 1050 kg/m³, c_p = 3600 J/kgK, k = 0.5 W/mK, and σ = 0.2 S/m. The time factor on the x-axis of Figure 6 is then 0.00132 s^-1, and the temperature factor on the y-axis becomes 18.9 K^-1. We can then predict the maximum temperature rise, which occurs at the inner surface of the electrode just at the end of the pulse, by reading the height of the horizontal line in the appropriate graph (Fig. 6b), approximately 13. So, the temperature rise would be 13/(18.9 K^-1), or about 0.7 K. After 1 second (0.00132 on the x-axis), the temperature at this point (upper curve) will have fallen to 0.25 K, and the temperature at the middle of the tissue (lowest curve) peaks at 0.03 K, a full 30 seconds after the pulse (0.04/0.00132 s^-1). This pulse then is essentially non-thermal and could be applied at 1 Hz tens of times without causing thermal damage.

In the full 3D case, things would be somewhat different. It is clear that the amount of heat deposition will be higher than expected, which has already been demonstrated in the discussion of Figure 3b, and that this additional heating will be deposited primarily in the space peripheral to the electrodes (as can be seen from Fig. 1b) and locally at the electrode tip and junction with the shaft. But at the same time, this heat will be provided with a greater volume of cool tissue into which to spread. The result is that the most long-lived region of hyperthermia will be in the space between the centers of the electrodes and midway along their active lengths, where the 2D model is most accurate. To design pulse parameters to avoid thermal damage, the data in Figure 6 are, thus, sufficient for treatment planning in most cases.

For the case of disk electrodes, the heating is much more uniform. Aside from the pair of rings with high electric field strengths (for example see the 1600 V/cm regions in Fig. 2c), the variation of electric potential in the space directly between the electrodes and from one electrode surface to the other is quite linear, so the Joule heating will be much less heterogeneous than was the case for needle electrodes. An unfortunate consequence of this is that the heat will take longer to diffuse from the treated tissue, so the best practice would be to limit the total input of heat over the course of several pulses and not to depend on heat diffusion or on blood flow to withdraw heat [the best possibility for removal of heat would be with the use of cooled electrodes (23)]. As a rule of thumb, the best practice would be to compute the power consumption from the cell constant in Figure 3a, then assume that all of this energy (power multiplied by pulse duration) is deposited within the cylinder of tissue directly between the electrodes, of diameter D, which will predict a bulk temperature rise according to the electrical and thermal properties of the tissue.

A convenient equation to estimate the increase in temperature for the plate configuration from the Joule heating is:



where Δt is the total duration of the pulses. This equation assumes no heat dissipation between the pulses, and no fringe effects at the electrode edge. Furthermore, this equation assumes that the biological properties are uniform and the contributions from blood flow, metabolic heat, and electrode heat dissipation are negligible.

Thermal Dose Assessment and Measuring Damage

One of the distinguishing features of irreversible electroporation is that it does not induce thermal damage (40, 41). To assess whether a particular set of voltage parameters will induce thermal effects in addition to irreversible electroporation, the thermal damage can be calculated. Thermal damage, Ω, is a time-dependent process described by an Arrhenius type equation:



where ξ is the frequency factor, E_a is the activation energy, and R is the universal gas constant (42, 43, 44). If the period of exposure is long, thermal damage can occur at temperatures as low as 42 °C. However, 50 °C is generally chosen as the target temperature (43).

For procedures involving time varying temperatures, thermal damage can be assessed by calculating the amount of time it would take to equivalently damage the tissue as if it was held at a constant temperature, typically 43 °C (45). The following expression is the duration necessary to hold the tissue at 43 °C to result in an equivalent thermal dose:



where T_t is the average temperature during Δt with R = 0.25 when T_t ≤ 43 °C and R = 0.5 when T_t > 43 °C (46, 47).

Conclusion

The goal of this work was to demonstrate the benefits of numerical models in designing protocols for irreversible electroporation surgery. The models provided can be used for guidance in the design of IRE protocols, and instructions with examples were also given on the creation of models for a custom treatment. Since IRE is a new technique to ablate undesirable tissue, the voltage parameters, specifically the electric field, to induce IRE in specific tissues is currently an active area of research. Our results demonstrate that the two most important things to consider when predicting the electric field distribution for an IRE treatment are the tissue conductivity distribution and the electrode configuration.

The figures present the influence of critical parameters, such as electrode size and shape of electrodes and tissue heterogeneity on the resulting electric field distribution. The figures are presented in a non-dimensional format, so the reader can directly plug their scenario specific pulse and geometry parameters into the plots to ensure the entire targeted area is above the electric field threshold to induce irreversible electroporation. If information is available with regards to the properties of the tissue being treated, information (such as thermal dissipation) can be approximated using the non-dimensional curves. We illustrate how the conductivity distribution, which may be due to the presence of large blood vessels, affects the electric field. Knowledge of the tissue properties and considering obstructions that might make the electric field distribution perturbed is important. Any perturbation can and must be incorporated into the models if known beforehand.

Although the IRE procedure is easy to perform, many surgical conditions are important to consider in the design and planning of IRE protocols. For example, the electric pulses can induce instantaneous muscle contraction. In this case, the surgeon may want to inject a muscle relaxant. While actually performing IRE, appropriate safety and preparation precautions must be taken. For example, needle and plate electrodes are typically used in conjunction with a conductive paste to ensure good electrical contact. Finally, the use of high voltage equipment is always a safety concern.

Studies have shown that the extent of electroporation can be imaged with MRI (27) and in real-time with electrical impedance tomography (12, 37). Irreversible electroporation has the advantages of being a tissue ablation technique that does not require adjuvant chemicals, it is easy, and has the capability of being monitored and controlled with diagnostic imaging. Furthermore, this study shows that the results from IRE surgery can be predicted beforehand through modeling to optimize the treatment.

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