We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the multiple traces formulation introduced in Henríquez et al. [Numer. Math. 136 (2016) 101–145]. Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations. Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achieved via a semi-implicit time-stepping scheme shown to be stable and second order convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin–Huxley dynamical model.
Mots-clés : Multiple traces formulation, semi-implicit time stepping, biological cells, exponential convergence
@article{M2AN_2018__52_2_659_0,
author = {Henr{\'\i}quez, Fernando and Jerez-Hanckes, Carlos},
title = {Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {659--703},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018019},
mrnumber = {3834439},
zbl = {1404.65148},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018019/}
}
TY - JOUR AU - Henríquez, Fernando AU - Jerez-Hanckes, Carlos TI - Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 659 EP - 703 VL - 52 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018019/ DO - 10.1051/m2an/2018019 LA - en ID - M2AN_2018__52_2_659_0 ER -
%0 Journal Article %A Henríquez, Fernando %A Jerez-Hanckes, Carlos %T Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 659-703 %V 52 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018019/ %R 10.1051/m2an/2018019 %G en %F M2AN_2018__52_2_659_0
Henríquez, Fernando; Jerez-Hanckes, Carlos. Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 659-703. doi : 10.1051/m2an/2018019. http://www.numdam.org/articles/10.1051/m2an/2018019/
[1] , , and , A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case. Differ. Integral Equ. 26 (2013) 885–912. | MR | Zbl
[2] , and , Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues. Nonlinear Differ. Equ. Appl. 23 (2016) 48. | DOI | MR | Zbl
[3] , , , and , Spectroscopic conductivity imaging of a cell culture. Asymptotic Anal. 100 (2016) 87–109. | DOI | MR | Zbl
[4] , and , Towards monitoring critical microscopic parameters for electropermeabilization. Quart. Appl. Math. 75 (2017) 1–17. | DOI | MR | Zbl
[5] , , , , , , , , and , LAPACK Users’ Guide. Vol. 9. SIAM (1999). | DOI | Zbl
[6] , and , On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders. J. Comput. Phys. 227 (2008) 1754–1771. | DOI | MR | Zbl
[7] and , Theoretical Numerical Analysis. Vol. 39. Springer (2005). | DOI | MR | Zbl
[8] , Boundary decomposition for Helmoltz and Maxwell equations I. Disjoint subscatterers. Asymptotic Anal. 38 (2004) 1–10. | MR | Zbl
[9] and , New currents in electrical stimulation of excitable tissues. Ann. Rev. Biomed. Eng. 2 (2000) 377–397. | DOI
[10] and , Peripheral nerve stimulation. Tech. Reg. Anesth. Pain Manag. 10 (2006) 79–88. | DOI
[11] and , Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982) 67–86. | DOI | MR | Zbl
[12] , , and , Spectral Methods. Scientific Computation. Springer-Verlag, Berlin (2006). | DOI | MR | Zbl
[13] and , A surface formulation or characteristic modes of material bodies. IEEE Trans. Antennas. Propag. 25 (1977) 789–795. | DOI
[14] , and , A surface-based electrophysiology model relying on asymptotic analysis and motivated by cardiac atria modeling. Math. Models Meth. Appl. Sci. 23 (2013) 2749–2776. | DOI | MR | Zbl
[15] , , , , and , Intracellular protein delivery and gene transfection by electroporation using a microneedle electrode array. Small 10 (2012) 1081–1091. | DOI
[16] , and , Multitrace boundary integral equations, in Direct and Inverse Problems in Wave Propagation and Applications. Vol. 14 of Radon Series on Computational and Applied Mathematics. De Gruyter, Berlin (2013) 51–100. | MR | Zbl
[17] , , and , Novel Multi-Trace Boundary Integral Equations for Transmission Boundary Value Problems, in Unified Transform for Boundary Value Problems: Applications and Advances, edited by and . SIAM (2015) 227–258. | MR | Zbl
[18] , Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19 (1988) 613–626. | DOI | MR | Zbl
[19] and , A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367–413. | DOI | MR | Zbl
[20] , and , An asymptotic two-layer monodomain model of cardiac electrophysiology in the atria: derivation and convergence. SIAM J. Appl. Math. 77 (2017) 409–429. | DOI | MR | Zbl
[21] , , and , Computational electrophysiology. Vol. 2 of Springer Series, A First Course in On Silico Medicine, Tokyo, Japan (2010). | MR | Zbl
[22] , , and , New modality for electrochemotherapy of surface tumors. Biotechnol. Biotechnol. Equip. 26 (2012) 3402–3406. | DOI
[23] and , Simulation of acoustic scattering by multiple obstacles in three dimensions. ANZIAM J. 50 (2008) 31–45. | DOI | MR
[24] and , A high-order algorithm for multiple electromagnetic scattering in three dimensions. Numer. Algor. 50 (2009) 469–510. | DOI | MR | Zbl
[25] and , An efficient algorithm for computing acoustic wave interactions in large -obstacle three dimensional configurations. BIT Numer. Math. 55 (2015) 117–139. | DOI | MR | Zbl
[26] and , A fully discrete H1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations. Numer. Algorithms 43 (2006) 355–383. | DOI | MR | Zbl
[27] and , Numerical analysis of spectral methods: theory and applications. Soc. Ind. Appl. Math. (1983). | Zbl
[28] , and , A Voronoi Interface approach to cell aggregate electropermeabilization. J. Comput. Phys. 332 (2017) 143–159. | DOI | MR | Zbl
[29] , and , Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation. Numer. Math. 136 (2016) 101–145. | DOI | MR | Zbl
[30] and , Multiple traces boundary integral formulation for Helmholtz transmission problems. Adv. Comput. Math. 37 (2012) 39–91. | DOI | MR | Zbl
[31] and , A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500–544. | DOI
[32] and , Modeling extracellular electrical neural stimulation: from basic understanding to MEA-based applications. J. Physiol. Paris 106 (2012) 146–158. | DOI
[33] , and , Current approaches to model extracellular electrical neural microstimulation. Front. Comput. Neurosci. 8 (2014) 13. | DOI
[34] , , and , “Classical” electropermeabilization modeling at the cell scale. J. Math. Biol. 68 (2014) 235–265. | DOI | MR | Zbl
[35] and , Mathematical Physiology I: Cellular Physiology. Springer-Verlag, New York (1998). | DOI | MR | Zbl
[36] , Linear Integral Equations. Vol. 82 Springer (1989). | DOI | MR | Zbl
[37] and , On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation.Numer. Math. 66 (1993) 199–214. | DOI | MR | Zbl
[38] , and , A second-order Cartesian method for the simulation of electropermeabilization cell models. J. Comput. Phys. 292 (2015) 114–140. | DOI | MR | Zbl
[39] , From Maxwell’s equations to the cable equation and beyond. Prog. Biophys. Mol. Biol. 85 (2004) 71–116. | DOI
[40] , Analytic Semigroups and Optimal Regularity in Parabolic Problems. Springer Science & Business Media (2012). | MR | Zbl
[41] , Making sense of complex phenomena in biology. Novartis Found. Symp. 247 (2002) 53–59; discussion 60–65, 84–90, 244–252. | DOI
[42] , Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press (2006). | DOI | MR | Zbl
[43] and , Global existence and uniqueness of a three-dimensional model of cellular electrophysiology. Discrete Contin. Dyn. Syst. 29 (2011) 1573–1636. | DOI | MR | Zbl
[44] , Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
[45] , , , , , , , , and High-efficiency gene transfer into skeletal muscle mediated by electric pulses. Proc. Nat. Acad. Sci. USA 96 (1999) 4262–4267. | DOI
[46] , and , Dependence of induced transmembrane potential on cell density, arrangement and cell position inside a cell system. IEEE Trans. Biomed. Eng. 40 (2002) 605–612. | DOI
[47] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Vol. 44. Springer Science & Business Media (2012). | MR
[48] , , , and , Continuous peripheral nerve blocks with stimulating catheters. Reg. Anesth. Pain Med. 28 (2003) 83–88. | DOI
[49] and , Integral equation solution of three-dimensional scattering problems, in Computer Techniques for Electromagnetics, Chap. 4, edited by . Pergamon, New York (1973) 159–263. | DOI
[50] and Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Vol. 3. Walter de Gruyter (1996). | DOI | MR | Zbl
[51] and , Periodic Integral and Pseudodifferential Equations With Numerical Approximation. Springer Science & Business Media (2013). | MR | Zbl
[52] and Boundary Element Methods. Springer-Verlag, Berlin, Heidelberg (2011). | DOI | MR | Zbl
[53] , , , and , Electrochemotherapy with bleomycin in the treatment of hypernephroma metastasis: case repeat and literature review. Tumori 86 (2000) 163–165. | DOI
[54] , and , Finite element analysis of cardiac defibrillation current distributions. IEEE Trans. Biomed. Eng. 37 (1997) 354–365. | DOI
[55] , Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer-Verlag, New York (2008). | DOI | MR | Zbl
[56] , The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23 (1986) 1–10. | DOI | MR | Zbl
[57] , , and , Electropermeabilization of cell membranes. Adv. Drug Deliv. Rev. 35 (1999) 3–19. | DOI
[58] Galerkin finite element methods for parabolic problems, in Springer Series in Computational Mathematics (2006). | MR | Zbl
[59] , , and , Modeling defibrillation of the heart: approaches and insights. IEEE Rev. Biomed. Eng. 4 (2012) 89–102. | DOI
[60] , , , , , , , and , Electro-quasistatic simulations in bio-systems engineering and medical engineering. Adv. Radio Sci. 3 (2005) 39–49. | DOI
[61] , Matrix formulation of electromagnetic scattering. Proc. of IEEE 53 (1965) 805–812. | DOI | Zbl
[62] and , Scattering from arbitrarily-shaped lossy dielectric bodies of revolution. Radio Sci. 12 (1977) 709–718. | DOI
Cité par Sources :
