We study the behavior of the electromagnetic field in a biological cell modeled by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain the first-order conditions in a simple way, and numerical simulations validate the theoretical results. Asymptotic transmission conditions at any order are given in the last section of the paper. This paper extends to the time-harmonic Maxwell equations the previous works presented in [30, 33, 31, 6].
Marc Duruflé 1 ; Victor Péron 1 ; Claire Poignard 1
@article{CML_2011__3_2_325_0,
author = {Marc Durufl\'e and Victor P\'eron and Claire Poignard},
title = {Time-harmonic {Maxwell} equations in biological cells {\textemdash} the differential form formalism to treat the thin layer},
journal = {Confluentes Mathematici},
pages = {325--357},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {2},
year = {2011},
doi = {10.1142/S1793744211000345},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000345/}
}
TY - JOUR AU - Marc Duruflé AU - Victor Péron AU - Claire Poignard TI - Time-harmonic Maxwell equations in biological cells — the differential form formalism to treat the thin layer JO - Confluentes Mathematici PY - 2011 SP - 325 EP - 357 VL - 3 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000345/ DO - 10.1142/S1793744211000345 LA - en ID - CML_2011__3_2_325_0 ER -
%0 Journal Article %A Marc Duruflé %A Victor Péron %A Claire Poignard %T Time-harmonic Maxwell equations in biological cells — the differential form formalism to treat the thin layer %J Confluentes Mathematici %D 2011 %P 325-357 %V 3 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000345/ %R 10.1142/S1793744211000345 %G en %F CML_2011__3_2_325_0
Marc Duruflé; Victor Péron; Claire Poignard. Time-harmonic Maxwell equations in biological cells — the differential form formalism to treat the thin layer. Confluentes Mathematici, Tome 3 (2011) no. 2, pp. 325-357. doi : 10.1142/S1793744211000345. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000345/
[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959) 623–727.
[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Comm. Pure Appl. Math. 17 (1964) 35–92.
[3] C. Balanis, Advanced Engineering Electromagnetics (John Wiley and Sons, 1989).
[4] D. Boffi, M. Costabel, M. Dauge, L. Demkowicz and R. Hiptmair, Discrete compact- ness for the p-version of discrete differential forms, accepted for publication in SIAM J. Numer. Anal.
[5] G. Caloz, M. Dauge, E. Faou and V. Péron, On the influence of the geometry on skin effect in electromagnetism, Computer Methods Appl. Mech. Eng. 200 (2011) 1053– 1058.
[6] S. Ciuperca, M. Jai and C. Poignard, Approximate transmission conditions through a rough thin layer. The case of the periodic roughness, Eur. J. Appl. Math. 21 (2010) 51–75.
[7] M. Costabel, M. Dauge and S. Nicaise, Singularities of eddy currents problems, preprint of the Newton Institute, 2003.
[8] M. Costabel, M. Dauge and S. Nicaise, Corner singularities of Maxwell interface and eddy current problems, in Operator Theoretical Methods and Applications to Mathe- matical Physics, Oper. Theory Adv. Appl., Vol. 147 (Birkhäuser, 2004), pp. 241–256.
[9] M. Dauge, I. Djurdjeviˇc, E. Faou and A. Rössle, Eigenmode asymptotics in thin elastic plates, J. Math. Pures Appl. 78 (1999) 925–964.
[10] M. Dauge, E. Faou and V. Péron, Comportement asymptotique à haute conductivité de l’épaisseur de peau en électromagnétisme, C.R. Acad. Sci. Paris Sér. I Math. 348 (2010) 385–390.
[11] M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976), translated from the Portuguese.
[12] B. Doubrovine, S. Novikov and A. Fomenko, Géométrie Contemporaine. Méthodes et Applications. I (MIR, 1982), Géométrie des surfaces, des groupes de transformations et des champs. [Geometry of surfaces, groups of transformations and fields], translated from the Russian by Vladimir Kotliar.
[13] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry — Methods and Applications, Part I, Graduate Texts in Mathematics, Vol. 93 (Springer-Verlag, 1992), 2nd edn., The geometry of surfaces, transformation groups and fields, translated from the Russian by Robert G. Burns.
[14] B. Engquist and J. C. Nédélec, Effective boundary condition for acoustic and electro- magnetic scattering thin layer, Technical Report of CMAP, 278, 1993.
[15] E. Faou, Elasticity on a thin shell: Formal series solution, Asympt. Anal. 31 (2002) 317–361.
[16] E. Faou, Multiscale expansions for linear clamped elliptic shells, Comm. Partial Dif- ferential Equations 29 (2004) 1799–1845.
[17] E. C. Fear and M. A. Stuchly, Modelling assemblies of biological cells exposed to electric fields, IEEE Trans. Biomed. Engrg. 45 (1998) 1259–1271.
[18] H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic Press, 1963).
[19] K. R. Foster and H. P. Schwan, Dielectric properties of tissues and biological materials: A critical review, CRC Biomed. Engrg. 17 (1989) 25–104.
[20] Y. V. Kurylev, M. Lassas and E. Somersalo, Reconstruction of a manifold from elec- tromagnetic boundary measurements, in Inverse Problems: Theory and Applications (Cortona/Pisa, 2002), Contemp. Math., Vol. 333 (Amer. Math. Soc., 2003), pp. 147– 161.
[21] Y. V. Kurylev, M. Lassas and E. Somersalo, Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems, J. Math. Pures Appl. 86 (2006) 237–270.
[22] Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal. 153 (2000) 91–151.
[23] J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, Travaux et Recherches Mathématiques, No. 17 (Dunod, 1968).
[24] D. Miklavˇciˇc, D. ˇSermov, H. Mekid and L. M. Mir, A validated model of in vivo electric field distribution in tissues for electrochemotherapy and for DNA electrotransfer for gene therapy, Biochim. Biophys. Acta 1523 (2000) 73–83.
[25] L. M. Mir, Electroporation of cells in tissues. Methods for detecting cell electroperme- abilisation in vivo, in Electroporation Based Technologies and Treatment. Proc. of the Int. Scientific Workshop and Postgraduate Course, 14–20 November 2005, Ljubljana, Slovenia, pp. 32–35.
[26] S. Mu˜noz, J. L. Sebasti´an, M. Sancho and J. M. Miranda, Transmembrane voltage induced on altered erythrocyte shapes exposed to RF fields, Bioelectromag. 25 (2004) 631–633 (electronic).
[27] L. Paquet, Problèmes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse Math. 4 (1982) 103–141.
[28] V. Péron, Modélisation mathématique de phénomènes électromagnétiques dans des matériaux à fort contraste, Ph.D. thesis, Université Rennes 1, 2009.
[29] C. Poignard, Rigorous asymptotics for the electric field in tm mode at mid-frequency in a bidimensional medium with a thin layer, preprint, 27 pp., September 2006.
[30] C. Poignard, Asymptotics for steady state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Methods Appl. Sci., DOI: 10.1002/mma.923.
[31] C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Methods Appl. Sci. 32 (2009) 435–453.
[32] C. Poignard, About the transmembrane voltage potential of a biological cell in time- harmonic regime, ESAIM: Proc., Vol. 26, 2009.
[33] C. Poignard et al., Approximate condition replacing thin layer, IEEE Trans. Mag. 44 (2008) 1154–1157.
[34] G. Pucihar, T. Kotnik, B. Valiˇc and D. Miklavˇciˇc, Numerical determination of trans- membrane voltage induced on irregularly shaped cells, Ann. Biomed. Engrg. 34 (2006) 642–652.
[35] G. Schwarz, Hodge Decomposition — A Method for Solving Boundary Value Problems (Springer, 1995).
[36] J. L. Sebasti´an, S. Mu˜noz, M. Sancho and J. M. Miranda, Analysis of the influence of the cell geometry and cell proximity effects on the electric field distribution from direct rf exposure, Phys. Med. Biol. 46 (2001) 213–225.
[37] D. ˇSel, D. Cukjati, D. Batiuskaite, T. Slivnik, L. M. Mir and D. Miklavˇciˇc, Sequential finite element model of tissue electropermeabilization, IEEE Trans. Bio. Engrg. 52 (2005) 816–827.
[38] T. Y. Tsong, Electroporation of cell membranes, Biophys. J. 60 (1991) 297–306.
[39] K. F. Warnick and D. V. Arnold, Electromagnetic Green functions using differential forms, J. Electromagnetic Waves Appl. 10 (1996) 427–438.
[40] K. F. Warnick, R. Selfridge and D. V. Arnold, Teaching electromagnetic field Green functions using differential forms, J. Electromagnetic Waves Appl. 10 (1996) 427–438.
Cité par Sources :