Dynamic model of tissue electroporation on the basis of biological dispersion and Joule heating

TL;DR

The paper addresses the limitation of static tissue electroporation models by introducing a dynamic framework that couples three physical effects: biological dispersion (via a multipole Debye representation implemented with ADEs), temperature-dependent conductivity (via Joule heating), and electroporation dynamics (a three-state pore model with $P_0$, $P_1$, $P_2$). Implemented in COMSOL Multiphysics, the model uses potato tuber tissue data to calibrate 12 parameters across the three states and demonstrates accurate prediction of tissue conductivity for $E$ from $10$ to $100$ kV/m, with only a small thermal rise (max $igl( riangle T igr) ightarrow 0.89^ ext{°C}$). The three-state dynamics explain the rapid initial rise in conductivity through $P_0$ and $P_1$, followed by slower accumulation via $P_2$, while the temperature rise modestly modulates the conductivity. Overall, the model enables time-domain simulations of electroporation current at the tissue scale and provides insights into membrane-level effects relevant for optimizing electrochemotherapy and related PEF applications.

Abstract

Electroporation is a complex, iterative, and nonlinear phenomenon that is often studied by numerical simulations. In recent years, tissue electroporation simulations have been performed using static models. However, the results of a static model simulation are restricted to a fixed protocol signature of the pulsed electric field. This paper describes a novel dynamic model of tissue electroporation that also includes tissue dispersion and temperature to allow time-domain simulations. We implemented the biological dispersion of potato tubers and thermal analysis in a commercial finite element method software. A cell electroporation model was adapted to account for the increase in tissue conductivity. The model yielded twelve parameters, divided into three dynamic states of electroporation. Thermal analysis describes the dependence of tissue conductivity on temperature. The model parameters were evaluated using experiments with vegetal tissue (Solanum tuberosum) under electrochemotherapy protocols. The proposed model can accurately predict the conductivity of tissue under electroporation from 10 kV/m to 100 kV/m. A negligible thermal effect was observed at 100 kV/m, with a 0.89 °C increase. We believe that the proposed model is suitable for describing the electroporation current on a tissue scale and also for providing a hint on the effects on the cell membrane.

Figures (5)

• Figure 1: The experimental sample and the rotated 2D axisymmetric geometry created in the simulator. (a) Experimental setup. (b) 3D representation. (c) Transversal view. (d) Frontal view. The geometry is centred at the origin.
• Figure 2: Shape of sigmoidal functions $\beta_0$ and $\beta_1$ for potato tissue with parameters of Table \ref{['tbl:model-parameters']}.
• Figure 3: Experimental and simulation results for the electroporation experiments following the ESOPE guidelines (eight pulses 100 long at 5). Only the first, second, and eighth pulses are shown; see Supplementary Information for the full set of pulses. Magnitude of the PEF protocol at (a)10, (b)20, (c)30, (d)40, (e)50, (f)60, (g)80, and (h)100. The overshoot of the in vitro current in the opposite direction in the PEF transitions is a common parasitic effect in the experimental setup. The circled numbers indicate the pulse number. Exp Avg is the experimental average, CI is the confidence interval (95%) and Std Dev is the standard deviation.
• Figure 4: Concentration of electroporation dynamic states $P_0$, $P_1$, and $P_2$ for 20 and 100 at the centre of the sample.
• Figure 5: (a) Temperature rise due to Joule heating at the centre of the sample. (b) Apparent conductivity with thermal and electroporation influences at the centre of the sample. For the apparent conductivity, only the first, second, and eighth pulses are shown; see Supplementary Information for the full set of pulses.