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2015, 12(3): 537-554. doi: 10.3934/mbe.2015.12.537

Cell scale modeling of electropermeabilization by periodic pulses

1. 

Inria Bordeaux Sud-Ouest, 200, Rue de la Veille Tour, 33405 Talence, France

Received  November 2013 Revised  January 2015 Published  January 2015

In this paper, we focus on the behaviour of periodic solutions to a cell-scale electropermeabilization model previously proposed by Kavian et al. [6]. Since clinical permeabilization protocols mostly submit cancer cells to trains of periodic pulses, we investigate on parameters that modify significantly the resulting permeabilization. Theoretical results of existence and uniqueness of periodic solutions are presented, for two different models of membrane electric conductivity. Numerical simulations were performed to corroborate these results and illustrate the asymptotic convergence to periodic solutions, as well as the dependency on biological parameters such as the cell size and the extracellular conductivity.
Citation: Michael Leguèbe. Cell scale modeling of electropermeabilization by periodic pulses. Mathematical Biosciences & Engineering, 2015, 12 (3) : 537-554. doi: 10.3934/mbe.2015.12.537
References:
[1]

L. Calmels, B. Al-Sakere, J. P. Ruaud, A. Leroy-Willig and L. M. Mir, In vivo MRI follow-up of murine tumors treated by electrochemotherapy and other electroporation-based treatments, Technology in Cancer Research and Treatment, 11 (2012), 561-570. doi: 10.7785/tcrt.2012.500270.

[2]

M. Cisternino and L. Weynans, A parallel second order cartesian method for elliptic interface problems, Communications in Computational Physics, 12 (2012), 1562-1587.

[3]

K. DeBruin and W. Krassowska, Modelling electroporation in a single cell. I. Effects of field strength and rest potential, Biophysical Journal, 77 (1999), 1213-1224. doi: 10.1016/S0006-3495(99)76973-0.

[4]

A. Gothelf, L. M. Mir and J. Gehl, Electrochemotherapy: Results of cancer treatment using enhanced delivery of bleomycin by electroporation, Cancer Treatment Reviews, 29 (2003), 371-387. doi: 10.1016/S0305-7372(03)00073-2.

[5]

R. P. Joshi, Q. Hu and K. H. Schoenbach, Dynamical modeling of cellular response to short-duration, high-intensity electric fields, Dielectrics and Electrical Insulation, IEEE Transactions on, 10 (2003), 778-787. doi: 10.1109/TDEI.2003.1237327.

[6]

O. Kavian, M. Leguèbe, C. Poignard and L. Weynans, "Classical'' electropermeabilization modeling at the cell scale, Journal of Mathematical Biology, 68 (2014), 235-265. doi: 10.1007/s00285-012-0629-3.

[7]

M. Leguèbe, Modélisation de L'électroperméabilisation à L'échelle Cellulaire (French), PhD thesis, Université de Bordeaux, 2014.

[8]

M. Leguèbe, C. Poignard and L. Weynans, A Second-Order Cartesian Method for the Simulation of Electropermeabilization Cell Models, Technical Report RR-8302, Inria, 2013.

[9]

M. Leguèbe, A. Silve, L. M. Mir and C. Poignard, Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments, Journal of Theoretical Biology, 360 (2014), 83-94.

[10]

J. Li and H. Lin, The current-voltage relation for electropores with conductivity gradients, Biomicrofluidics, 4 (2010), 13206. doi: 10.1063/1.3324847.

[11]

L. M. Mir, Electrogenetransfer in clinical applications, Wiley-VCH Verlag GmbH & Co. KGaA, (2006), 219-226.

[12]

O. M. Nesin, O. N. Pakhomova, S. Xiao and A. G. Pakhomov, Manipulation of cell volume and membrane pore comparison following single cell permeabilization with 60- and 600-ns electric pulses, Biochimica et Biophysica Acta - Biomembranes, 1808 (2011), 792-801. doi: 10.1016/j.bbamem.2010.12.012.

[13]

J. Neu and W. Krassowska, Asymptotic model of electroporation, Physical Review E, 53 (1999), 3471-3482. doi: 10.1103/PhysRevE.59.3471.

[14]

V. Péron, Modélisation Mathématique de Phénomènes Électromagnétiques dans des Matériaux à Fort Contraste (French), PhD thesis, Université Rennes 1, 2009.

[15]

B. Poddevin, S. Orlowski, J. Belehradek and L. M. Mir, Very high cytotoxicity of bleomycin introduced into the cytosol of cells in culture, Biochemical Pharmacology, 42 (1991), S67-S75. doi: 10.1016/0006-2952(91)90394-K.

[16]

C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Mathematical Methods in the Applied Sciences, 32 (2009), 435-453. doi: 10.1002/mma.1045.

[17]

G. Pucihar, J. Krmelj, M. Reberšek, T. B. Napotnik and D. Miklavčič, Equivalent pulse parameters for electroporation, IEEE Trans Biomed Eng, 58 (2011), 3279-3288. doi: 10.1109/TBME.2011.2167232.

[18]

J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04796-5.

[19]

S. Šatkauskas, F. M. André, M. F. Bureau, D. Scherman, D. Miklavčič and L. M. Mir, Electrophoretic component of electric pulses determines the efficacy of in vivo DNA electrotransfer, Human Gene Therapy, 16 (2005), 1194-1201.

[20]

A. Silve, A. Ivorra and L. M. Mir, Detection of permeabilisation obtained by micropulses and nanopulses by means of bioimpedance of biological tissues, in Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP), (2011), 3164-3167.

show all references

References:
[1]

L. Calmels, B. Al-Sakere, J. P. Ruaud, A. Leroy-Willig and L. M. Mir, In vivo MRI follow-up of murine tumors treated by electrochemotherapy and other electroporation-based treatments, Technology in Cancer Research and Treatment, 11 (2012), 561-570. doi: 10.7785/tcrt.2012.500270.

[2]

M. Cisternino and L. Weynans, A parallel second order cartesian method for elliptic interface problems, Communications in Computational Physics, 12 (2012), 1562-1587.

[3]

K. DeBruin and W. Krassowska, Modelling electroporation in a single cell. I. Effects of field strength and rest potential, Biophysical Journal, 77 (1999), 1213-1224. doi: 10.1016/S0006-3495(99)76973-0.

[4]

A. Gothelf, L. M. Mir and J. Gehl, Electrochemotherapy: Results of cancer treatment using enhanced delivery of bleomycin by electroporation, Cancer Treatment Reviews, 29 (2003), 371-387. doi: 10.1016/S0305-7372(03)00073-2.

[5]

R. P. Joshi, Q. Hu and K. H. Schoenbach, Dynamical modeling of cellular response to short-duration, high-intensity electric fields, Dielectrics and Electrical Insulation, IEEE Transactions on, 10 (2003), 778-787. doi: 10.1109/TDEI.2003.1237327.

[6]

O. Kavian, M. Leguèbe, C. Poignard and L. Weynans, "Classical'' electropermeabilization modeling at the cell scale, Journal of Mathematical Biology, 68 (2014), 235-265. doi: 10.1007/s00285-012-0629-3.

[7]

M. Leguèbe, Modélisation de L'électroperméabilisation à L'échelle Cellulaire (French), PhD thesis, Université de Bordeaux, 2014.

[8]

M. Leguèbe, C. Poignard and L. Weynans, A Second-Order Cartesian Method for the Simulation of Electropermeabilization Cell Models, Technical Report RR-8302, Inria, 2013.

[9]

M. Leguèbe, A. Silve, L. M. Mir and C. Poignard, Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments, Journal of Theoretical Biology, 360 (2014), 83-94.

[10]

J. Li and H. Lin, The current-voltage relation for electropores with conductivity gradients, Biomicrofluidics, 4 (2010), 13206. doi: 10.1063/1.3324847.

[11]

L. M. Mir, Electrogenetransfer in clinical applications, Wiley-VCH Verlag GmbH & Co. KGaA, (2006), 219-226.

[12]

O. M. Nesin, O. N. Pakhomova, S. Xiao and A. G. Pakhomov, Manipulation of cell volume and membrane pore comparison following single cell permeabilization with 60- and 600-ns electric pulses, Biochimica et Biophysica Acta - Biomembranes, 1808 (2011), 792-801. doi: 10.1016/j.bbamem.2010.12.012.

[13]

J. Neu and W. Krassowska, Asymptotic model of electroporation, Physical Review E, 53 (1999), 3471-3482. doi: 10.1103/PhysRevE.59.3471.

[14]

V. Péron, Modélisation Mathématique de Phénomènes Électromagnétiques dans des Matériaux à Fort Contraste (French), PhD thesis, Université Rennes 1, 2009.

[15]

B. Poddevin, S. Orlowski, J. Belehradek and L. M. Mir, Very high cytotoxicity of bleomycin introduced into the cytosol of cells in culture, Biochemical Pharmacology, 42 (1991), S67-S75. doi: 10.1016/0006-2952(91)90394-K.

[16]

C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Mathematical Methods in the Applied Sciences, 32 (2009), 435-453. doi: 10.1002/mma.1045.

[17]

G. Pucihar, J. Krmelj, M. Reberšek, T. B. Napotnik and D. Miklavčič, Equivalent pulse parameters for electroporation, IEEE Trans Biomed Eng, 58 (2011), 3279-3288. doi: 10.1109/TBME.2011.2167232.

[18]

J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04796-5.

[19]

S. Šatkauskas, F. M. André, M. F. Bureau, D. Scherman, D. Miklavčič and L. M. Mir, Electrophoretic component of electric pulses determines the efficacy of in vivo DNA electrotransfer, Human Gene Therapy, 16 (2005), 1194-1201.

[20]

A. Silve, A. Ivorra and L. M. Mir, Detection of permeabilisation obtained by micropulses and nanopulses by means of bioimpedance of biological tissues, in Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP), (2011), 3164-3167.

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