Measurement and simulation of Joule heating during treatment of B-16 melanoma tumors in mice with nanosecond pulsed electric fields

Abstract

Experimental evidence shows that nanosecond pulsed electric fields (nsPEF) trigger apoptosis in skin tumors. We have postulated that the energy delivered by nsPEF is insufficient to impart significant heating to the treated tissue. Here we use both direct measurements and theoretical modeling of the Joule heating in order to validate this assumption.

For the temperature measurement, thermo-sensitive liquid crystals (TLC) were used to determine the surface temperature while a micro-thermocouple (made from 30 μm wires) was used for measuring the temperature inside the tissue. The calculation of the temperature distribution used an asymptotic approach with the repeated calculation of the electric field, Joule heating and heat transfer, and the subsequent readjustment of the electrical tissue conductivity. This yields a temperature distribution both in space and time.

It can be shown that for the measured increase in temperature an unexpectedly high electrical conductivity of the tissue would be required, which was indeed found by using voltage and current monitoring during the experiment. Using impedance measurements within $t_{after}=50\mathrm{\mu }\mathrm{s}$ after the pulse revealed a fast decline of the high conductivity state when the electric field ceases. The experimentally measured high conductance of a skin fold (mouse) between plate electrodes was about 5 times higher than those of the maximally expected conductance due to fully electroporated membrane structures $(G_{max}/G_{electroporated})\approx 5$. Fully electroporated membrane structure assumes that 100% of the membranes are conductive which is estimated from an impedance measurement at 10 MHz where membranes are capacitively shorted. Since the temperature rise in B-16 mouse melanoma tumors due to equally spaced ($\Delta t=2\mathrm{s}$) 300 ns-pulses with $E=40\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$ usually does not exceed $\Delta {\rm T}=3\mathrm{K}$ at all parts of the skin fold between the electrodes, a hyperthermic effect on the tissue can be excluded.

1. Introduction

The interaction of cells with pulsed electric fields can be divided into the immediate reaction to the electric field and long term effects that are mostly active reactions by the cell. Immediately, during the presence of the electric field, Joule heating [1] can kill cells by hyperthermia. Moreover, due to the shockwave, generated at the electrodes, membrane destruction is possible. If the integrity of membranes is compromised, cells may not recover and die. Long term effects include the accumulation of reactive oxygen species and apoptosis [2,3].

If a cell is subjected to an outer electric field of sufficient strength, electroporation at the plasma-membrane occurs which renders the membrane electrically conductive [4–7]. This high conductivity during electroporation in the presence of an electric field results in an elevated current flow through the cell resulting in a temperature rise due to Joule heating. Previous studies using long lasting high voltage pulses (up to 200 ms) at a field strength of $E=250\mathrm{V}/\mathrm{c}\mathrm{m}$ at the myocardium showed a considerable temperature rise reaching hyperthermia levels [8]. An even more dramatic temperature rise was reported for skin, especially the stratum corneum [9–13], where even for short pulse durations (1 ms) a significant temperature rise occurs, reaching the phase transition temperature of skin lipids (65 °C) [14,15].

In contrast to the well understood phenomenon of electroporation at applied field strengths below $E=1\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$ [16–18], the manipulation of cells with a much higher field strength ($E>10\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$) but with extremely short duration ($t_{pulse}<300\mathrm{n}\mathrm{s}$) yields effects that are not comparable with ‘classical’ electroporation [19]. One important difference is the extremely high electrical conductivity of cells and tissue during the presence of this very high electric field [20]. In conclusion, Joule heating cannot be excluded. However, both experiment and simulation show a negligible temperature rise making secondary effects like hyperthermia unlikely.

2. Material and methods

The temperature at the surface and inside murine skin tumors during the high voltage application was measured in two ways: 1) At the skin surface using temperature-sensitive liquid crystals; and 2) Within the bulk tumor with miniature thermocouples. Since the conductance of the tissue is crucial for Joule heating, it was investigated under high voltage conditions.

2.1. Mice

Hairless SKH-1 mice were obtained from Charles River (Boston MA) and housed in the animal facility of the Eastern Virginia Medical School (EVMS). Four tumors (B-16, murine melanoma) were induced by injecting 10⁶ cells in four locations under the skin on the back. The tumors were allowed to grow for several days until a size of several millimeters in diameter was reached. For the experiment, the mice were anesthetized by isofluorane inhalation according to a protocol approved by the EVMS IACUC.

Since the entire experiment for temperature measurement within the tumor using a thermocouple lasted only a few minutes, no temperature control was used. For measuring the surface temperature, the mice were placed into a thermostated chamber.

2.2. High voltage generator

An LC-ladder pulse-forming network in the Blumlein configuration [21] was triggered using a pressure-driven metal piston. The charging voltage was set to 1 < U/kV < 8. Pulse trains of 100 pulses with a regular spacing of 1 pulse every other second (0.5 Hz) were applied. Both, the voltage and current were measured using a digital oscilloscope.

2.3. Bulk temperature measurement using a miniaturized thermocouple

For achieving minimal size and fast response, we used a thermocouple rather than a fiber-optic transducer [22]. The thermocouple was made from copper and constantan wire (Metrohm, Riverview, FL) with a diameter of 30 μm (Fig. 1B). Establishing the junction was tricky. We charged a 10 μF-capacitor to 5 V and discharged it through the wires held shorted with a tweezer. With some luck, the brief but high current flow welded the both wires together.

Fig. 1.

Thermocouple (right) placed in tumor with electrodes contacting the skin fold with the tumor. The syringe needle necessary for inserting the thermocouple is removed and rests at the measurement device (not shown).

Since the thermocouples are inside the tumor during high voltage application (up to 2.5 kV), everything had to be electrically isolated. A small, battery-powered device had a first conditioning amplifier and a subsequent VCO (voltage controlled oscillator). The transfer function of the device was adjusted so that the frequency range between 2 kHz and 12 kHz represented a temperature range between 25 °C and 40 °C. A LED at the output of the VCO was connected to fiber optics. A second fiber was placed at the high voltage switch catching the flashover during switching for synchronization between pulse protocol and measurement. A two-channel light receiver converted the light pulses into an electric signal which was recorded using the soundcard of a computer placed a few meters away from the high voltage in order to prevent damage from the strong voltage transients. The entire setup was controlled with a microcontroller with all the peripheral components isolated optically via fiber optics. Light sensitive TRIAC switches were used for switching the pulser by opening a valve which pressurized a metallic piston. This in turn closed the switch.

Disk-shaped plane-parallel plate electrodes with a diameter of 5 mm arranged in a custom made clamp similar to a clothespin were used in order to contact a skin fold containing the tumor. The distance between the electrodes was determined by measuring the space between the edges of the plastic clothespin using a caliper. The thickness of the skin fold was with some variations d = 0.5 mm. Having for instance $U=2\mathrm{k}\mathrm{V}$ applied, resulted in $E=40\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$ when simply dividing voltage by the distance. The special distribution of electric field and temperature within the skin fold between the electrodes was estimated by simulation (Section 2.5).

For the placement of the thermocouple, we penetrated the tumor with a syringe needle. The thermocouple was inserted into the tumor through the needle and the needle was gently removed. This could be simply observed against a Teflon screen, back-illuminated using a white LED (Fig. 1A). This was especially useful because of the dark tumor and the unpigmented skin of the hairless mouse, allowing a precise placement of the electrodes.

2.4. Measurement of the surface temperature

Since a high temporal and spatial resolution was envisioned, we did not use an infrared measurement but registered the color change of Licristal, a temperature-sensitive liquid crystal (TLC) (Hallcrest, Glenview, IL). The crystals were arranged as slurry with bead-like structures 10 μm in diameter. Since the measurement requires a high sensitivity, we used a center temperature of 32 °C and a temperature span of ±5 K. This means, the dye is colorless at 29.5 °C. It turns blue – green – orange – red between 29.5 °C and 34.5 °C. Above 34.5 °C it is colorless again. The dye responses as fast as 1 ms. The response time in the experiment, taking the heat transfer from heated bulk to the surface into account is unimportant since the heat production is similar during an experiment for all parts. Local and temporal hot spots (e.g. at stratum corneum) however, may occur. In order to ensure a high contrast during the measurement, we painted the skin surface black (permanent marker) and added a thin layer of the TLC [8,23]. For calibration we did the same using a piece of paper and observed the color change in a temperature-controlled chamber (Fig. 2).

Fig. 2.

Photograph of a mouse inside the temperature-controlled chamber with the electrode attached. The inset is the view though the agarose onto the skin surface covered with temperature sensitive liquid crystal (Licristal^™).

Although we made an illumination system with a ring of different LEDs (white, green, red), we finally used a bundle of glass fibers arranged around the objective and illuminated by a cold light source. This resulted in much better readability of the colors.

The electrode was designed for the highest possible homogeneity of the electric field at the site of the skin fold but still allowing observation in order to monitor the color change of the Licristal. It should be clear that the field inside an inhomogeneous object cannot be homogeneous. A 5 mm hole was drilled through both legs of a custom made clamp. At one side a polished stainless steel disk was inserted as counter electrode and at the other side a ring made from 0.5 mm silver wire was placed inside the hole. The entire hole was filled with a degassed agarose gel (1.5%) made with 140 mM PBS (phosphate buffered saline) and sealed with a thin cover slip. The agarose was placed under a vacuum during cooling which ensured good transparency for microscopic observation. For the better homogeneity of the electric field at the skin surface but still close enough for using a high magnification at the microscope, we placed the electrode ring 2.5 mm apart from the lower edge facing the skin which resulted by the application of $U=8\mathrm{k}\mathrm{V}$ (across skin fold and agarose) and $d_{skin}=0.5\mathrm{m}\mathrm{m}$ thickness of the skin fold in a field strength of about $E=26\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$. The actual field strength at the site of the tumor is even less due to the internal resistance of the pulser and the resistive behavior of the skin.

Since only a small temperature range of the Licristal could be used for the measurement, the entire setup (mouse, illumination, microscope objective) was placed in a temperature-controlled chamber at about 32 °C (Fig. 2). The wiring for the high voltage ensured safety for the low voltage illumination and the microscope camera. The high voltage pulse, current and voltage, were monitored using a digital oscilloscope. Moreover, the voltage together with the horizontal synchronization signal (HSYNC) of the microscope camera was traced with the soundcard of the computer for synchronization between video frames and high voltage pulse. All experiments were recorded (Fig. 2, inset) and subsequently processed in order to calculate the temperature during each frame. Regions with the damage of the Licristal layer were excluded from further processing.

2.5. Simulation

We used a theoretical simulation of this experiment to compare with our experimental results similar to methods in other studies [8,9,13,22,24]. Even though two quite differently designed experiments show similar results, we wanted to confirm that the basic laws of thermodynamics are not violated with our interpretation.

In order to save computing time, we used some simplifications. The geometrical model is shown in Fig. 3A. It is a cylindrical piece of the skin fold contacted at both sides by disk-shaped electrodes having a diameter of 5 mm (gray surfaces above and below the simulation volume). Assuming negligible field differences and heat transfer along a surface with constant radius (rotational symmetry), we limited the simulation to two dimensions which is sketched as the rectangular surface.

Fig. 3.

Geometrical model used for simulation of the heating inside a tumor during nsPEF application. The distribution of different materials within the simulation plane is shown below.

The heat capacity and heat conduction coefficients were taken from the literature [15], internet (e.g. http://www.itis.ethz.ch/itis-for-health/tissue-properties/database) and product specificatioins (Degussa Röhm Plexiglas Produktbeschreibung, Kenn-Nr. 211–1 and Merkblatt 821 Edelstahl Rostfrei). For the electrical conductivity we used values from our own investigation under different conditions (impedance without offset, small signal impedance with high voltage offset (dynamic impedance), voltage/current behavior during nsPEF-application) (Table 1).

Table 1.

Values used for simulation (σ_epore – conductivity for classically electroporated tissue, σ_{high cond.} – high conductivity during nsPEF, α – heat coefficient for conductivity at 25 °C, k – heat conductivity, c – heat capacity, ρ – density).

	σepore/Sm−1	σhigh cond./Sm−1	α/K−1	k/W(mK)−1	c/J(kgK)−1	ρ/kg/m−3
Stainless steel	1.5 × 106	1.5 × 106	0	15	500	8000
Plexiglass	10−13	10−13	0	0.19	1470	1190
Tumor	0.17	0.85	0.026	0.49	3421	1109
Viable dermis	0.11	0.52	0.026	0.37	3391	1088
Stratum corneum	0.009	0.04	0.026	0.21	1918	905
Connective tissue	0.1	0.5	0.026	0.39	2372	1027

The model in Fig. 3B is drawn to scale (voxel size 65.5 × 67.5 × 25.5 μm³) with the exception of the thickness of the epidermis (thin layer between electrodes and the skin fold) including the stratum corneum. The stratum corneum is the outer, dead layer of the skin and has a thickness of about 15 μm which is less than the voxel size. Since the stratum corneum has distinct behavior (low electrical conductivity) we accounted for the larger thickness by adjusting the conductivity in proportion to the thickness (25.5 μm/15 μm = 1.8). The simulation is based on a network of complex impedances [9,25]. Using Kirchoff’s laws, we solved the matrix for all sub-currents and voltages. The entire simulation package was written in MATLAB 5 (The MathWorks, Natick, MA). Here we assigned each voxel not only with electrical admittance (reciprocal of the impedance) but also with heat conduction and heat capacity as well as the temperature coefficient for the electrical conductivity. The field and temperature at the edges with the exception of the voxels at the center of the rotational symmetric model were modeled using the gradient along the last three voxels at the edge by a cubic approximation. Therefore, we did not underestimate the temperature rise due to ideal sink or overestimate it due to an open boundary. For the center pixels we assumed mirror behavior which results mathematically in an open boundary (no current or heat flow into the mirrored voxels). This ensures an optimum between computational demand and accuracy.

First, all voxels are initialized (temperature, admittance, temperature coefficient) for electrical conductivity (real part of admittance, heat capacity, heat conductivity). Next, zero matrixes for temperature difference and electric potential were created.

The simulation used an iterative procedure. After the calculation of the potential and current distribution, the temperature change within the grid time (10⁻¹² s) was calculated taking Joule heating and heat flow over the voxel boundaries into account. Due to the short time, a linear approximation due to nearly adiabatic heating was sufficient for negligible (<0.1%) error. The error was tested by successively decreasing the time grid (takes incredible calculation time). The ‘right value’ for heat flow was calculated using a simple geometry consisting only of two neighboring compartments per step. For the boundaries and the medium temperature rise expected here, a time grid of 10⁻¹² s was appropriate.

With the new temperature, we adjusted the electrical conductivity of each voxel and returned to the calculation of the potential distribution.

Some simplifications were made, partially to save computational time but also due to lack of information:

• The temperature coefficient for the electrical conductivity was partly derived from the literature but also from conductivity/temperature measurements for electrolytes and ‘standard’ tissue from pig (electrolyte, muscle, fat, connective tissue, skin, liver). For the tumor, we assumed the temperature coefficient of liver. Most temperature coefficients showed some variability but were between 0.019 K⁻¹ and 0.032 K⁻¹ with respect to 25 °C. No reliable results were found for skin and fat. Due to the uncertainty at this point, the temperature coefficient was in the final simulations set at 0.026 K⁻¹ for all parts of the skin fold containing electrolytes.

• We had no information about the electrical conductivity of tumor tissue (B16-tumor) in vivo during high voltage exposure. Although we had big tumors (Ø > 5 mm) for assessing this behavior, they disintegrated easily when probed. Thus, we used the value of liver tissue which is a good approximation.

• A shockwave, generated during the pulse was ignored in all simulations.

• Convection due to blood vessels was not considered for heat transport. This is appropriate, since in other experiments a blockage of local blood flow was observed in response to nsPEF.

• Due to the known increase of electrical tissue conductivity during the elevated electric field application (10%–15%), we used an average of the conductivity measured at the beginning and at the end of the pulse.

3. Results and discussion

The temperature measurement within the bulk tumor was done with different voltages applied. Although the pulse generator could deliver 8 kV, we only obtained up to 2.2 kV at the electrodes due to the internal resistance of the pulse forming network. In order to prevent flashover and skin damage, the skin was wetted using ECG-electrode paste. Although different voltages were tested, only the application of 2.2 kV-pulses (average, measured) was repeated several times with similar results. The device was calibrated using a temperature range between 25 °C and 40 °C. A test with an agarose disk of known electrical conductivity showed the stability of the measurement under the harsh condition of high voltage application.

Two slightly different results are shown in Fig. 4A and B where a temperature rise of up to 4 K was found for a 0.5 Hz pulse train with 2.2 kV (average) and 300 ns pulse length. Counter experiments using 150 V/1 ms-pulses yielded a temperature rise up to 8 K but did not plateau after 50 pulses (Fig. 5).

Fig. 4.

Temperature increase inside the tumor due to 2.2 kV/300 ns pulses with 2 s interpulse spacing. The maximum temperature was reached after about 40 pulses. At the end of the experiment after 100 pulses, 29.4 °C was reached for the upper trace and 32.2 °C for the lower one. The dots depict the time of pulse application.

Fig. 5.

Temperature increase at a skin fold due to the application of 50 long lasting pulses of 150 V/1 ms for comparison the nsPEF-application.

An elevated temperature was evident even more than a minute after the last pulse was given. The thermal relaxation time of the entire system (skin fold + electrode system) was around 65 s. With the exception of the first pulse, where under similar condition a higher temperature jump was found, a typical jump of 0.3 K–0.8 K for a single pulse was reproducible. The higher temperature increase for the first pulse was attributed to the drop in voltage with subsequent pulses because of the voltage divider between the pulse-forming network and the skin fold.

3.1. Surface temperature measurement

This approach has significant limitations such as the longer heat transfer from the bulk surroundings to the skin surface and additional heat production within the agarose gel. However, it is not only a good indicator of the overall temperature increase but also provides the spatial distribution of the temperature elevation.

Overall, we observed good homogeneity with no localized regions with elevated electrical conductivity. This is in line with findings on skin electroporation where the localized transport regions reached a high density but only small dimension when the pulse voltage was increased [26]. Moreover, the current flow during electroporation was almost homogeneous.

The temperature rise presented in Fig. 6 was derived from a video (30 fps) with averaging over the observed surface.

Fig. 6.

Temperature rise at the surface of a skin fold using temperature sensitive liquid crystal (Licristal). The 10 pulses (26 kV/cm, 300 ns) were applied every 2 s starting at 4 s.

3.2. Elevated conductance during the presence of high electric field

Using impedance measurements together with electrical characterization during a high voltage application, a large increase in tissue conductance during the pulse was found [27]. This conductance is up to 5 times higher than that of the conductance expected from the impedance measurement at 10 MHz (Fig. 7). The conductance at 10 MHz is relevant since membranes only negligibly influence the current flow above this frequency due to the very low capacitative resistance. This can act as an upper case estimation for completely electroporated (100% electropores) membranes. It should be noted that the very high conductance during the pulse does not result from uncorrected displacement currents.

Fig. 7.

Conductance, G, of a skin fold depending on the applied field strength measured with 300 ns-second pulses. The conductance at 10 MHz without field application was 4.3 mS (line). Other than in experiments with parallel temperature measurement, the distance between the electrodes was 3 mm owing to the different construction of the holder. Conductive gel was used to fill the gap between electrode and skin fold. The apparatus for parallel assessment of impedance and high voltage behavior is described in [27].

It is feasible to estimate the maximum conductance increase, $\mathrm{\Delta }{G}_{max}$, expected for a skin fold of a mouse between plate electrodes as well as the field strength for 50%—increase in conductance $\left(E_{0.5}\right)$. This approach assumes nonconductive structures (e.g. lipids and charged molecules fixed in membranes) to be transformed into conductive or at least more conductive ones. For instance, an insulating membrane becomes electroporated at moderate field strength and at higher field strengths it disintegrates into micelle structures. Increasing the field strength further can even cause the dissociation of micelles yielding single molecules of high mobility. The field dependent conductance, $G\left(E\right)$, is

$$G(E)=G_0+\frac{\Delta G_{max}{e}^{b\left(E^2-E_{0.5}^2\right)}}{1+e^{b\left(E^2-E_{0.5}^2\right)}}$$ (1)

$G_0$ is the conductance at $E=0\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$ and $b$ is a factor depending on the effective polarization volume ${\mathrm{V}}_{\mathrm{p}}$. $\Delta \epsilon$ is the change of dielectric behavior (permittivity, energy conserving capability) due to the structural changes of the material between the electrodes.

$$b=\frac{0.5\Delta \epsilon {\epsilon }_0{V}_p}{k_{B}T}.$$ (2)

$k_B$ is the Boltzmann constant and $T$ is the temperature. Fitting the data in Fig. 7 using Eq. 1 yields $b\approx 0.013{\mathrm{c}\mathrm{m}}^2{\mathrm{k}\mathrm{V}}^{-2}$. The other parameters, $\Delta G_{max}\approx 19.4\mathrm{m}\mathrm{S}$ and $E_{0.5}\approx 17.1\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$ were derived from the experiment. ${\mathrm{G}}_0=2.4\mathrm{m}\mathrm{S}$ was obtained from the impedance measurement. Due to the complex nature of the skin fold between the electrodes, it is not reasonable to predict the electric field strength within a cell membrane. Moreover, since the pore concept for the interpretation of very high field effects ($E>10\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$) on cell membranes is not experimentally proven yet, we do not seek further interpretation for $b$.

3.3. Simulation

For the simulation we used the same settings as in the experiment assessing the bulk temperature of the tumor. In order to account for the higher electrical conductivity of the tissue due to permeabilizing the cell membranes, we used as conductivity the values measured at 10 MHz. This is a suitable approximation for the upper boundary since the reactance $X_C$ of the cell membranes is very low $\left(X_C=1/\left(2\pi fC\right)\right).C$ is the capacitance of the membrane and $f$ is the frequency. At high frequency, the membrane appears therefore capacitatively shorted, i.e. it corresponds to 100% pores within the membrane. With 2.2 kV applied to the electrodes, the field strength inside the tumor reached about 35 kV/cm (Fig. 8A) which is less than the expected 40 kV/cm. The reason for this discrepancy is mostly the lower resistance of the tumor compared to the surrounding tissue, especially the stratum corneum.

Fig. 8.

Simulated electric field distribution in a plane within the skin fold. Using boundaries similar to the experiment, around 35 kV/cm was reached within the tumor. The corresponding temperature rise during a 300 ns high voltage pulse is shown in the lower panel. The conductivity of the electroporated tissue was estimated from impedance measurement at 10 MHz where the membranes have a negligible resistance.

Assuming an adiabatic temperature rise during the 300 ns pulse duration, only 0.2 K temperature rise was found (Fig. 8B). This is quite surprising since the simulation had already applied an upper boundary approximation. Using the measured current flow together with the volume of the skin fold between both electrodes and an approximate heat capacity of this of water, we find a temperature rise of 0.4 K which is two times more than the simulated value. This suggests a much higher conductance of the tissue between the electrodes which is in line with findings presented in Fig. 7.

For selected tissue, the increased tissue conductance under high voltage condition was measured (Table 2).

Table 2.

Comparison of conductivity (magnitude), |σ|, of selected tissue (rat) at f = 10 MHz and during the presence of an outer electric field of E_applied = 30 kV cm⁻¹. The measured values are raw data without any correction for elevated temperature.

	|σ|10 MHz/Sm−1	|σ|30 kV cm – 1/Sm−1
Blood	0.6	1.4
Muscle (length orientation)	0.35	1.85
Muscle (perpendicular orientation)	0.3	1.9

Table 2 shows measured magnitude, $\left|\mathrm{\sigma }\right|$, for tissue samples from rat. We note, that the high conductivity does not result from an artificially high displacement current.

Since we are unable to assess the high voltage behavior for each tissue separately, we adjusted the electrical conductivity of all tissues for further simulation by a factor of five. Due to the short pulse duration, nearly adiabatic heating occurs while the electric field is present resulting in simplified calculation:

$$\mathrm{\Delta }\mathrm{T}\left(t\right)=A\left(1-e^{-k_{on}t}\right)\approx k_{\mathit{heating}}t.$$ (3)

This approximation is true for sufficiently small $k_{\mathit{on}}$, where

$$1-e^{-k_{on}t}\approx 1-\left(1-k_{on}t\right)=k_{on}t.$$ (4)

Since both, $A$ and $k_{\mathit{on}}$ are unknown, we define $A{k}_{\mathit{on}}=k_{\mathit{heating}}=\Delta T/\Delta t$ as an empirical quantity. Practically, this means that nearly adiabatic heating occurs during the presence of the electric field and heat dissipation is negligible. While experimentally $k_{\mathit{heating}}$ between $1<\mathrm{K}/\mathrm{\mu }\mathrm{s}<2.7$ was found, simulation with values reasonable for the experimental condition yielded 4.3 K/μs (Fig. 9).

Fig. 9.

Simulation of the temperature rise during pulsed high voltage application at a skin fold shown in Fig. 3. The extremely high conductivity during the presence of the electric field was taken into account yielding higher temperature than shown in Fig. 8.

It should be noted that $k_{\text{heating}}$ depends on several factors, especially the material properties (heat capacity c, electrical conductivity $\sigma$, mass density $\rho$) and the electric field due to the high voltage application.

With the limiting case of $\mathrm{\Delta }\mathrm{T}<<\mathrm{T}$, the incremental rate of temperature increase is

$$\frac{\mathrm{\Delta }\mathrm{T}}{\mathrm{\Delta }t}=\frac{E^2\sigma }{c\rho }.$$ (5)

Using the experimental values for a tissue of mixed materials ($\sigma \approx 0.2\mathrm{S}/\mathrm{m}$ (electroporated tissue), $\rho \approx 1000\mathrm{k}\mathrm{g}/{\mathrm{m}}^3,c\approx 3\mathrm{k}\mathrm{J}/\left(\mathrm{k}\mathrm{g}\mathrm{K}\right))$ and a field strength of $E\approx 30\mathrm{k}\mathrm{V}/\mathrm{c}\mathrm{m}$, we obtain $k_{\mathit{heating}}=0.75\mathrm{K}/\mathrm{\mu }\mathrm{s}$ which is somehow less than experimentally found. Again, the reason lies in the extremely high conductivity during the presence of an electric field exceeding 10 kV/cm (Fig. 7.).

From the cooling, we can obtain the thermal relaxation of the system by fitting the equation

$$\mathrm{\Delta }\mathrm{T}\left(t^{\prime }\right)=\mathrm{\Delta }{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}{e}^{-k_{\mathit{off}}{t}^{\prime }}.$$ (6)

Surprisingly, simulation yielded $k_{\mathit{off}}=0.21{\mathrm{s}}^{-1}$ while the experimental values are between 0.015 s⁻¹ and 0.2 s⁻¹ with the tendency of faster cooling at the beginning which slows down within the first seconds.

The much faster relaxation in the simulation yielded a lower final temperature compared to the experiment. A reason for the experimentally found slower cooling may be the fat layer of the skin which was not contained in the model.

Even though the electrical conductivity value for the tissue involved is only an estimate, it helps us to interpret the experimental results where similar heating during the pulse application was found. A marked difference is the thermal relaxation time which was up to 65 s in the experiment but only 4.8 s in the simulation. This means that the thermal conductivity in our simulation was at some point overestimated. This resulted in a much faster plateauing of the temperature (about 5 pulses) which is in marked contrast to the experiment with 40–60 pulses. Moreover, the final temperature difference was experimentally higher. As found in former studies, the temperature rise within the stratum corneum is higher than that in the other tissues. However, the model used here did not allow us to resolve down to the level of the envelope of corneocytes. We tested the hypothesis of local temperature rise within single cell membranes but – at least in the simulation – we could not find a higher temperature which is also in agreement with the findings in [28].

4. Conclusion

Joule heating is biologically not significant for single pulses with nanosecond duration. This means, the cell killing effect of nsPEF does not involve hyperthermia, but is probably due to some other effects that the electric field has on intracellular organelles. We propose that the temperature rise beyond the expected range is due to a substantial reorganization of charged molecules within and in the vicinity of membrane structures resulting in the partial disintegration of these membranes into micelles and single molecules under the influence of the extremely high electric field strength.