Mathematical Perspectives in Electroporation
Mathematical Perspectives in Electroporation
Electroporation-based technologies constitute a particularly interesting research area at the intersection of applied mathematics and bioelectrical engineering, with important applications in drug and gene delivery, electrochemotherapy, irreversible electroporation (IRE), and pulsed-field ablation (PFA). Their development requires both a detailed understanding of the electrical response of biological tissues and mathematical/numerical tools capable of describing the complex multiscale mechanisms underlying membrane permeabilization.
From the electrical-engineering perspective, electroporation is governed by the interaction between electric-field distribution, electrode geometry and placement, pulse amplitude, duration and repetition, and tissue conductivity and anisotropy. Electrical engineering researchers have developed experimental and computational frameworks for characterizing transmembrane voltage, conductivity changes and electric-field distributions, and for translating these quantities into treatment planning. In particular, numerical models mostly based on standard finite element methods have demonstrated that tissue conductivity evolves during electroporation, creating a feedback between the applied electric field and the resulting treatment zone. This coupling is essential for predicting lesion size and optimizing electrode configurations and pulse protocols.
These engineering challenges naturally lead to fundamental mathematical questions. At the cellular scale, models describe nonlinear membrane permeabilization and pore dynamics; at the tissue scale, cells form a heterogeneous microstructure whose electrical properties change dynamically during treatment. A key question concerns the rigorous upscaling from cell-scale models to tissue-scale models. Nonlinear homogenization and multiscale analysis therefore provide a framework for rigorously deriving effective tissue-scale models without explicitly resolving every cell. Recent works in this direction have been carried out and are currently ongoing. These developments demonstrate that electroporation raises genuine mathematical problems involving nonlinear PDEs, homogenization, anisotropy and evolving effective properties.
The interaction between the two disciplines is particularly valuable for technology development. Mathematical models can provide effective constitutive laws and predictive simulations, while electrical engineering supplies electrode systems, pulse generators, tissue characterization and experimental validation. Together, they enable the study and optimization of electrode placement, pulse waveforms, treatment margins and energy deposition. In cardiac PFA, for example, incorporating tissue anisotropy and cellular microstructure into computational models can improve predictions of lesion morphology and help explain how pulse parameters translate into selective tissue ablation.
Several promising research directions emerge from this interaction: nonlinear and dynamic homogenization, multiscale PDEs, and the use of hybrid deep-learning/physics-driven numerical methods to solve the nonlinear multiscale models accurately and rapidly. Ultimately, these approaches could enable patient-specific treatment planning in which mathematical models predict the required electric-field distribution and electrical engineers translate it into an optimized electrode configuration and pulse protocol.
Electroporation therefore offers a unique opportunity to combine mathematical theory with electrical-engineering innovation. The mathematical challenge is to understand and reduce the nonlinear multiscale physics; the engineering challenge is to implement and control the resulting electric fields in real biological systems. Their combination can lead to more predictive, efficient and controllable technologies for cancer treatment, drug delivery and cardiac ablation.
— Contributed by Clair Poignard
References
- Neal, Garcia, Robertson & Davalos. (2012). Experimental characterization and numerical modeling of tissue electrical conductivity during pulsed electric fields for irreversible electro-poration treatment planning.IEEE Transactions on Biomedical Engineering, 59, 1076–1085.
- Voyer, Silve, Mir, Scorretti, & Poignard. (2018). Dynamical modeling of tissue electroporation.Bioelectrochemistry, 119, 98–110.
- Kotnik, Rems, Tarek & Miklavčič — Membrane Electroporation and Electropermeabilization: Mechanisms and Models, Annual Review of Biophysics (2019).
- Garcia, Davalos & Miklavčič — numerical investigation of electrical and thermal effects in electroporation therapies (2014).
- Bihoreau Duchemin, Collin, Leguèbe & Poignard — Mathematical derivation of a homogenized bidomain model for pulsed field ablation (2026), Nonlinear Analysis, 92.
- Aycock & Davalos. (2019). Irreversible Electroporation: Background, Theory, and Review of Recent Developments in Clinical Oncology. Bioelectricity, 1, 214–234.
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