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Pulsed Electric Fields in Biological Cells and Membranes
finite-element models in order to gain useful insights into the role played by various
factors on the temperature. Others have explored the thermal effects of various geometric, biological, and electroporation pulse parameters (Becker and Kuznetsov 2006)
including the blood vessel presence and size, plate electrode configuration, and pulse
duration and frequency. Using a three-dimensional transient finite volume model of
in vivo parallel plate electroporation of liver tissue in the presence of a blood vessel, the
target region tissue was found to have lower thermal damage. This was attributed to
heat conduction away from the region by the blood vessel. Increase in the plate spacing
increased the thermal damage as the volume to surface area ratio increased. The results
of dividing a long pulse into shorter pulses of equal intensity and frequency showed that
increasing the time over which energy can be introduced into the system resulted in
lower maximum target region temperatures and significantly decreased thermal dose
values. Similarly, it was found that decreasing pulse frequency resulted in lower thermal
damage and lower maximum temperatures.
Some important parameters from a practical standpoint are the exposure duration,
SAR, AD, pulse shape and width, and the repetition rate. Although the specific effects
and bioresponses are complicated to predict and quantify, the thermal effects can be
obtained from the pulse duration and SAR. Mathematically, the SAR (in W/Kg) can
be obtained from the expression σ E 2 /ρ, where E is the local electric field created by
the external excitation, σ the conductivity of the biomaterial, and ρ the material density; while the AD is the product of the SAR, the pulse duration, and number of pulses.
Experimental data on cell subject to the ultrashort, high-intensity pulses have shown
temperature increases of less than 3°C when subjected to a single ∼330 kV/cm pulse of
10-ns duration. However, trains of 20–400 pulses were shown to deliver sufficient energy
to cause temperature increases in excess of 6°C, as shown in Figure 2.12.
Calculations on slightly longer 60-ns electric pulses having a 60 kV/cm intensity show
peak temperature increases of about 3°C. Applying the distributed circuit model discussed in Section 2.3.1 to a spherical cell of radius 10 μm consisting of cytoplasm (ε = 60ε 0 ,
σ = 0.13 S/m), outer medium (ε = 80ε 0 , σ = 0.6 S/m), and a 5-nm-thick plasma membrane
(ε = 8ε 0 , σ = 5.3 × 10 −6 S/m) yielded the curves as shown in Figure 2.13. Changes in local
conductivity due to membrane poration were included and though not shown here, large
increases by orders of magnitude were predicted over time. For example, the highest conductivity was predicted to be about 0.007 S/m. The power density values are shown in
Figure 2.13 as a function of time for five different angular locations. The initial power
dissipation is negligible because the membrane acts as a virtual open circuit with the
displacement current being the dominant mechanism. Later (∼40 ns and beyond), the
conduction current develops as pores begin to form and increase both in numbers and
radial size. Due to the nonlinear increases in conductance and pore densities over time,
the power density attains a peak and then begins to reduce. The decrease can be associated with a collapse of the local TMP as the pores effectively produce an electrical “short.”
Based on the power density, changes in temperature as a function of time and angular
location were obtained. Values for the density ρ M and membrane specific heat c M were
chosen to be 10 3 kg/m 3 and 2 × 10 3 J/kg/Kelvin, respectively, in keeping with reports in
the literature (Croce et al. 2010). Figure 2.14 shows the temperature profile. Interestingly,
the largest temperature increase, which occurs at the poles (i.e., θ = 0°), is predicted to