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Derivation of nonlinear time-dependent macroscopic conductivity for an electropermeabilization model via homogenization

Tobias Gebäck, Ioanna Motschan-Armen, Irina Pettersson

TL;DR

This work develops a rigorous multiscale framework for reversible electropermeabilization by homogenizing Kavian et al.'s cell-scale model in a periodic tissue; the microscopic problem couples electrostatic potentials in intra- and extracellular spaces through a nonlinear, memory-bearing membrane transmission. Through a priori estimates and two-scale convergence on oscillating membranes, the authors derive a macroscopic problem with memory where the effective conductivity $\sigma_{\rm eff}$ depends nonlinearly on the electric field and evolves in time, capturing a characteristic drop due to membrane charging. A mixed analytical-numerical approach demonstrates that even with constant microscopic conductivities, tissue-scale conductivity exhibits sigmoid-like behavior in response to applied voltage and pulse duration, aligning with experimental observations. The framework provides a mathematically rigorous explanation for nonlinear, time-dependent conductivity dynamics in electroporated tissues and offers a computational pathway to evaluate $\sigma_{\rm eff}$ for realistic geometries. Overall, the paper advances the theoretical understanding of membrane-scale dynamics propagating to macroscopic tissue responses and supports applications in optimized electrical stimulation protocols.

Abstract

We study a phenomenological electropermeabilization model in a periodic medium representing biological tissue. Starting from a cell-level model describing the electric potential and the degree of porosity, we perform dimension analysis to identify a relevant scaling in terms of a small parameter $\ve$ - the ratio between the cell and the tissue size. The electric potential satisfies electrostatic equations in the extra- and intracellular domains, while its jump across the cell membrane evolves according to a nonlinear law coupled with an ordinary differential equation for the porosity degree. We prove the well-posedness of the microscopic problem, derive a priori estimates, obtain formal asymptotics, and rigorously justify the expansion combining two-scale convergence with monotonicity arguments. The resulting macroscopic model exhibits memory effects and a nonlinear, time-dependent effective current. It captures the nontrivial evolution of effective conductivity, including a characteristic drop reflecting the capacitive behavior of the lipid bilayer, in agreement with experimental data. Numerical computations of the effective conductivity confirm that, although microscopic conductivity is constant, tissue conductivity varies nonlinearly with electric field strength, showing a sigmoid trend. This suggests a rigorous mathematical explanation for experimentally observed conductivity dynamics.

Derivation of nonlinear time-dependent macroscopic conductivity for an electropermeabilization model via homogenization

TL;DR

This work develops a rigorous multiscale framework for reversible electropermeabilization by homogenizing Kavian et al.'s cell-scale model in a periodic tissue; the microscopic problem couples electrostatic potentials in intra- and extracellular spaces through a nonlinear, memory-bearing membrane transmission. Through a priori estimates and two-scale convergence on oscillating membranes, the authors derive a macroscopic problem with memory where the effective conductivity depends nonlinearly on the electric field and evolves in time, capturing a characteristic drop due to membrane charging. A mixed analytical-numerical approach demonstrates that even with constant microscopic conductivities, tissue-scale conductivity exhibits sigmoid-like behavior in response to applied voltage and pulse duration, aligning with experimental observations. The framework provides a mathematically rigorous explanation for nonlinear, time-dependent conductivity dynamics in electroporated tissues and offers a computational pathway to evaluate for realistic geometries. Overall, the paper advances the theoretical understanding of membrane-scale dynamics propagating to macroscopic tissue responses and supports applications in optimized electrical stimulation protocols.

Abstract

We study a phenomenological electropermeabilization model in a periodic medium representing biological tissue. Starting from a cell-level model describing the electric potential and the degree of porosity, we perform dimension analysis to identify a relevant scaling in terms of a small parameter - the ratio between the cell and the tissue size. The electric potential satisfies electrostatic equations in the extra- and intracellular domains, while its jump across the cell membrane evolves according to a nonlinear law coupled with an ordinary differential equation for the porosity degree. We prove the well-posedness of the microscopic problem, derive a priori estimates, obtain formal asymptotics, and rigorously justify the expansion combining two-scale convergence with monotonicity arguments. The resulting macroscopic model exhibits memory effects and a nonlinear, time-dependent effective current. It captures the nontrivial evolution of effective conductivity, including a characteristic drop reflecting the capacitive behavior of the lipid bilayer, in agreement with experimental data. Numerical computations of the effective conductivity confirm that, although microscopic conductivity is constant, tissue conductivity varies nonlinearly with electric field strength, showing a sigmoid trend. This suggests a rigorous mathematical explanation for experimentally observed conductivity dynamics.
Paper Structure (17 sections, 10 theorems, 175 equations, 9 figures, 3 tables, 2 algorithms)

This paper contains 17 sections, 10 theorems, 175 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Electropermeabilization model
  3. Dimension analysis
  4. Problem statement and homogenization result
  5. Well-posedness of the microscopic problem \ref{['eq:microscopic-prob']}
  6. Formal asymptotic expansions
  7. Homogenization
  8. A priori estimates
  9. Compactness
  10. Proof of the homogenization result
  11. Numerical simulations
  12. General solution algorithm
  13. Algorithm for the case of two parallel plates
  14. Computation of $A$
  15. Computation of $\mathcal{B}(t)$
  16. ...and 2 more sections

Key Result

Theorem 2.3

Under assumptions H1--H3, H5, $p_\varepsilon \to p_0$ strongly in $L^2(0,T; H^1(\Omega))$ and $(u_\varepsilon, [u_\varepsilon], X_\varepsilon)$ solving eq:microscopic-prob-truncated converges as $\varepsilon \to 0$ to the solution $(u_{-1}, [u_0], X_0)$ of eq:effective-problem-coupled-strong-intro i

Figures (9)

  • Figure 1: (a) $\Omega=\Omega_\varepsilon^c \cup \Gamma_\varepsilon \cup \Omega_\varepsilon^e$; (b) Periodicity cell $Y=Y^c \cup \Gamma \cup Y^e$.
  • Figure 2: Computational domain in the $(\tau, t)$-plane.
  • Figure 3: Simulations of $B$ given by equation \ref{['def:B']}.
  • Figure 4: Evolution of the effective conductivity $\sigma_{\text{eff}}$ for different pulse magnitudes.
  • Figure 5: Evolution of the effective conductivity $\sigma_{\text{eff}}$ for $|E| = 0 \, V/cm$.
  • ...and 4 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1: A priori estimates
  • ...and 14 more