Second-order model of membrane electric field induced by alternating external electric fields
Abstract
With biological cells exposed to ac electric fields below 100 kHz, external field is amplified in the cell membrane by a factor of several thousands (low-frequency plateau), while above 100 kHz, this amplification gradually decreases with frequency. Below 10 MHz, this situation is well described by the established first-order theory which treats the cytoplasm and the external medium as pure conductors. At higher frequencies, capacitive properties of the cytoplasm and the external medium become increasingly important and thus must be accounted for. This leads to a broader, second-order model, which is treated in detail in this paper. Unlike the first-order model, this model shows that above 10 MHz, the membrane field amplification stops decreasing and levels off again in the range of tens (high-frequency plateau). Existence of the high-frequency plateau could have an important impact on present theories of high-frequency electric fields effects on cells and their membranes.
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As the frequency of the field increases, the high membrane field gain diminishes, allowing for higher field intensities to penetrate into the cell.
“as the frequency of the field increases, the high membrane field gain diminishes, allowing for higher field intensities to penetrate into the cell.”
From An Overview of Sub-Cellular Mechanisms Involved in the Action of TTFields (2016) · cites this paper as [30]
Observed wave patterns in electrically-stimulated AFs were very similar to the solitary waveforms recorded for electrically-stimulated non-linear transmission lines.
“The observed wave patterns in electrically-stimulated AFs [30] were very similar to the solitary waveforms recorded for electrically-stimulated non-linear transmission lines [110].”
From An Overview of Sub-Cellular Mechanisms Involved in the Action of TTFields (2016) · cites this paper as [30]
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“1– 100 MHz) caused by Maxwell-Wagner relaxation of cell membranes,(103) which means a peak of membrane dielectric loss and selective membrane excitation (heating) at this frequency(104) (reorientation of protein-bound water molecules, the motion of polar protein subgroups, the Maxwell-Wagner relaxation of the cell int…”
Transmembrane potential induced in a spherical cell model under low-frequency magnetic stimulation
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“In addition, the cellular membrane demonstrates a frequencydependent response to the field, because of its frequencydependent dielectric properties [22].”
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“The classical approach towards this model predicts a rapid field decrease in the MHz range due to the structural membrane dispersion [Gimsa, 2000; Kotnik and Miklavcic, 2000a].”
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Gao, L.-T., Feng, X., & Gao, H. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.02.034 · Cited by 31 · see citing works
“The material parameters used in our calculations are listed in Table 1, in which the values in the Cell column were experimentally measured from cells by Kotnik and Miklavcic [58] and Gowrishankar et al. [59], while those in the Vesicle column were measured from vesicles by Riske and Dimova [16].”
Yao, C., Mo, D., Li, C., Sun, C., & Mi, Y. (2007) · IEEE Transactions on Plasma Science · doi:10.1109/tps.2007.905110 · Cited by 31 · see citing works
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“For a cell membrane (index ‘m’) of 5–10 nm thickness, low dielectric values and extremely low electrical conductivities rm have been reported for frequencies <1MHz with increasing rm above 1MHz [22].”
BackgroundGowrishankar, T. R., Smith, K. C., & Weaver, J. C. (2012) · Proceedings of the IEEE · doi:10.1109/jproc.2012.2200289 · Cited by 28 · see citing works
“Accordingly, we first showed that a discretized approximation to a passive spherical membrane yields frequency-domain results [5] which were in good agreement with analysis by traditional analytical methods [9] (Fig.”
BackgroundMethodResultInfluentialElectric Fields Within Cells as a Function of Membrane Resistivity—A Model Study
Mossop, B. J., Barr, R. C., Zaharoff, D. A., & Yuan, F. (2004) · IEEE Transactions on NanoBioscience · doi:10.1109/tnb.2004.833703 · Cited by 28 · see citing works
“The variation in the literature is from approximately 100 to 10 000 cm [8], [9], [13], [15].”
A subnanosecond electric pulse exposure system for biological cells
Xiao, S. H., Semenov, I., Petrella, R. A., Pakhomov, A. G., & Schoenbach, K. H. (2016) · Medical & Biological Engineering & Computing · doi:10.1007/s11517-016-1516-7 · Cited by 27 · see citing works
“As the duration of the pulses is reduced into the subnanosecond range, capacitive properties of cytoplasm and external medium in addition to the conductive properties must be taken into account [16].”
BackgroundThe Cell in the Electric Field
Kotnik, T., Pucihar, G., & Damijan Miklavčič, A. (2011) · Springer eBooks · doi:10.1007/978-1-4419-8363-3_3 · Cited by 26 · see citing works
“To determine the time course of DY m induced by even higher field frequencies or even shorter pulses, the second-order extension of Schwan’s equation must be used, in which dielectric permittivities of the cytoplasm and the cell exterior also have to be taken into account [16–18].”
MethodAshour, M., Valizadeh, N., & Rabczuk, T. (2021) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2021.113669 · Cited by 25 · see citing works
Advances in pulsed electric stimuli as a physical method for treating liquid foods
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Chaos control and synchronization of two neurons exposed to ELF external electric field
Wang, J., Zhang, T., & Che, Y. (2006) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2006.03.098 · Cited by 24 · see citing works
“In this paper, based on results of space–time characteristics of trans-membrane voltage [9,10], the variation of cell trans-membrane voltage exposed to extremely low frequency (ELF) electric field is analyzed.”
BackgroundCell membrane electropermeabilization with arbitrary pulse waveforms
Flisar, K., Puc, M., Kotnik, T., & Damijan Miklavčič, A. (2003) · IEEE Engineering in Medicine and Biology Magazine · doi:10.1109/memb.2003.1191453 · Cited by 23 · see citing works
Multi-parameter Hopf-bifurcation in Hodgkin–Huxley model exposed to ELF external electric field
Jiang, W., Yanqiu, C., Fei, X., & Li, L. (2005) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2005.02.042 · Cited by 23 · see citing works
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“The strategy to overcome transmembrane dependence on the radius of microscopic structures for pulses with a lower frequency spectral content was the application of a second-order model of induced transmembrane potential (TMP) exploiting nanosecond pulses at higher intensity MV/m (NsEFPs) [162,168].”
MethodWang, M., Kao, M.-F., & Jang, L. (2011) · Review of Scientific Instruments · doi:10.1063/1.3594550 · Cited by 22 · see citing works
Kim, D., Chauhan, M., Kim, M., Jeong, W. C., Kim, H. J., Serša, I., Kwon, O. I., & Woo, E. J. (2014) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2014.2361689 · Cited by 22 · see citing works
“However, at high frequencies, the membrane capacitance allows alternating currents to pass through and as the membrane effect disappears, the current flows everywhere according to local ionic conductivity [7], [16], [28]–[30].”
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Nanoheating without Artificial Nanoparticles
Vincze, G., Szigeti, G. P., & Andócs, G. (2015) · Biology and Medicine · doi:10.4172/0974-8369.1000249 · Cited by 21 · see citing works
“56 MHz (carrier frequency of mEHT) is more than 100 [21,62].”
Visualization of Dynamic Sub-microsecond Changes in Membrane Potential
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Martín, S. M. S., Sebastián, J. L., Sancho, M., & Álvarez, G. (2006) · Bioelectromagnetics · doi:10.1002/bem.20234 · Cited by 18 · see citing works
“…potential [Bifano et al., 1984; Gedde and Huestis, 1997; Glaser, 1998], and intensive research has been conducted to determine the transmembrane potentials and field strengths induced by external RF fields [Grosse and Schwan, 1992; Kotnik and Miklavcic, 2000a,b, and references therein].”
BackgroundKim, S., Lee, E., Woo, E. J., & Seo, J. K. (2012) · Inverse Problems · doi:10.1088/0266-5611/28/7/075004 · Cited by 17 · see citing works
“The approach presented in this paper can be applied to studies in electroporation or electropermeabilization where transmembrane potentials are of major concern [12, 18, 19, 9, 33].”
MethodA 3-D Hybrid Finite Element Model to Characterize the Electrical Behavior of Cutaneous Tissues
Hartinger, A. E., Guardo, R., Kokta, V., & Gagnon, H. (2009) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2009.2036371 · Cited by 17 · see citing works
Biological dosimetry of magnetic resonance imaging
Guisasola, M. C., Desco, M., Millán, O., Villanueva, F. J., & Garcia-Barreno, P. (2002) · Journal of Magnetic Resonance Imaging · doi:10.1002/jmri.10099 · Cited by 17 · see citing works
A Theoretical Analysis of the Effects of Tumor-Treating Electric Fields on Single Cells
Li, X., Yang, F., Gao, B., Yu, X., & Rubinsky, B. (2020) · Bioelectromagnetics · doi:10.1002/bem.22274 · Cited by 16 · see citing works
“In theoretical analysis of electrical effects on the cell, traditionally, most works have focused on the cell trans‐membrane voltage (TMV) calculation [Schwan, 1957; Kotnik and Miklavčič, 2000a].”
BackgroundInfluentialWang, C., Ma, J., Jin, W., & Wu, Y. (2011) · Applied Mathematics and Computation · doi:10.1016/j.amc.2011.10.027 · Cited by 16 · see citing works
“[41] [42] [43] [44] [45] and I E ¼ C m dV E dt is used to measure the parameter current flowing through the capacitor due to the electric field stimulus.”
Hsieh, C.-H., Lu, C., Chen, W., Ma, B.-L., & Chao, C. (2017) · PLoS ONE · doi:10.1371/journal.pone.0188885 · Cited by 16 · see citing works
Integrated platform for culture, observation, and parallelized electroporation of spheroids
Bregigeon, P., Rivière, C., Franqueville, L., Vollaire, C., Marchalot, J., & Frénéa‐Robin, M. (2022) · Lab on a Chip · doi:10.1039/d2lc00074a · Cited by 15 · see citing works
Calculated plasma membrane voltage induced by applying electric pulses using capacitive coupling
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Zheng, W., Mi, Y., Ma, C., Zhang, M., Tang, S., & Wang, J. (2023) · Journal of Applied Physics · doi:10.1063/5.0139065 · Cited by 11 · see citing works
Dielectrophoretic Characterization of Dendritic Cell Deformability Upon Maturation
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Transmembrane Voltage Induced by Applied Electric Fields
Kotnik, T. (2016) · Springer eBooks · doi:10.1007/978-3-319-26779-1_8-1 · Cited by 10 · see citing works
Sowa, P. W., Mariyanats, A. O., Kiełbik, A., Rohlfing, A., Novickij, V., Kollotzek, F., Sigle, M., Marzi, J., Schenke‐Layland, K., Borst, O., et al. (2025) · EP Europace · doi:10.1093/europace/euaf217 · Cited by 9 · see citing works
“67,91 In cardiac tissue, intercellular electrical coupling restricts free ion movement and limits uniform charging, so electro-poration occurs predominantly near the cathode and at sites of abrupt conductivity changes—such as fibre junctions, fat inclusions, or scar tis-sue—where local potential…”
Modeling the excitation of nerve axons under transcutaneous stimulation
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Evaluation of Interactions of Electric Fields Due to Electrostatic Discharge With Human Tissue
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Determining a Pulse Coupling to Subcellular Components With a Frequency-Domain Transfer Function
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“The transfer function of electric fields between the extracellular region and the membrane has been found to be greater than 1 (amplification) [4].”
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Local Dosimetry at Cellular and Subcellular Level in HF and Millimeter-Wave Bands
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Pulsed Field Ablation for the Treatment of Atrial Fibrillation: A Review and a Look into its Future
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“This phenomenon amplifies the externally applied electric field, known as the Maxwell-Wagner effect, such that the critical magnitude needed across the cell membrane for pore formation is reached [ 47,48].”
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Mechanic stress generated by a time-varying electromagnetic field on bone surface
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“During time-varying field stimulation, the Bcomplex conductivity,^ defined as S = σ + jωε, was used to account for the dielectric permittivity of the material [28, 30, 48].”
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Li, X., Yang, F., He, W., & Rubinsky, B. (2019) · Biomedical Microdevices · doi:10.1007/s10544-019-0452-3 · Cited by 5 · see citing works
Analysis for relationship of transmembrane potential–pulsed electric field frequency
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Yi, G., Wang, J., Han, C., Deng, B., Wei, X., & Jin, Q. (2013) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2013.09.026 · Cited by 5 · see citing works
- Authors
- Kotnik, T., & Miklavcic, D.
- Affiliations
- Centre National de la Recherche Scientifique (FR)
- ORCID
- Kotnik, T.
- Venue
- IEEE Transactions on Biomedical Engineering
- Published
- 2000
- Volume
- 47
- Issue
- 8
- Pages
- 1074-1081
- ISSN
- 0018-9294 (ISSN-L)
- OpenAlex
- W2138651796
- Cited by
- 150
- References
- 36
- Citations by year
- 2026: 2 · 2025: 2 · 2024: 8 · 2023: 4 · 2022: 5 · 2021: 4 · 2020: 5 · 2019: 6 · 2018: 14 · 2017: 13 · 2016: 9 · 2015: 8 · 2014: 6 · 2013: 4 · 2012: 7
- FWCI
- 1.80
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Biomedical Engineering (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Biophysics › Electromagnetic Fields and Biological Effects
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Electrical and Electromagnetic Research
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Biotechnology › Microbial Inactivation Methods
- Keywords
- membrane electric field, AC electric field, second-order model, field enhancement, plasma membrane, electroporation, cell membrane permeability
- MeSH terms
- Biomedical Engineering, Cell Membrane, Electric Stimulation, Electromagnetic Fields, Membrane Potentials, Models, Biological
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