Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models

Aziz Belmiloudi; Aziz Belmiloudi

doi:10.3934/math.2019.3.928

AIMS Mathematics

2019, Volume 4, Issue 3: 928-983. doi: 10.3934/math.2019.3.928

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Research article Topical Sections

Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models

Aziz Belmiloudi ^,

Institut de Recherche Mathématique de Rennes (IRMAR), Université Européenne de Bretagne, 20 avenue Buttes de Coësmes, CS 70839, 35708 Rennes Cédex 7, France

Received: 18 March 2019 Accepted: 10 July 2019 Published: 26 July 2019
MSC : 49J35, 49K35, 35Q92, 35K57, 35B30, 92C50

Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Fréchet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.
- minimax control,
- multiple time-varying delays,
- electrotherapy,
- reaction-diffusion system,
- bidomain type model,
- parabolic-elliptic coupling,
- ionic model,
- cardiac electrophysiology,
- fluctuation,
- adjoint model,
- sensitive model,
- necessary conditions of optimality
Citation: Aziz Belmiloudi. Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models[J]. AIMS Mathematics, 2019, 4(3): 928-983. doi: 10.3934/math.2019.3.928

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Abstract

Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Fréchet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.

References

[1]	R. A. Adams, Sobolev spaces, Academic Press, New-York, 1975.
[2]	B. Ainseba, M. Bendahmane, R. Ruiz-Baier, Analysis of an optimal control problem for the tridomain model in cardiac electrophysiology, J. Math. Anal. Appl., 388 (2012), 231-247. doi: 10.1016/j.jmaa.2011.11.069
[3]	T. Ashihara, J. Constantino, N. A. Trayanova, Tunnel propagation of postshock activations as a hypothesis for fibrillation induction and isoelectric window, Circ. Res., 102 (2008), 737-745. doi: 10.1161/CIRCRESAHA.107.168112
[4]	O. V. Aslanidi, A. P. Benson, M. R. Boyett, et al. Mechanisms of defibrillation by standing waves in the bidomain ventricular tissue with voltage applied in an external bath, Physica D: Nonlinear Phenomena, 238 (2009), 984-991. doi: 10.1016/j.physd.2009.02.003
[5]	G. W. Beeler, H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol., 268 (1977), 177-210. doi: 10.1113/jphysiol.1977.sp011853
[6]	A. Belmiloudi, S. Corre, Mathematical modeling and analysis of dynamic effects of multiple time-varying delays on electrophysiological wave propagation in the heart, Nonlinear Analysis Series B: Real World Applications, 47 (2019), 18-44. doi: 10.1016/j.nonrwa.2018.09.025
[7]	A. Belmiloudi, Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies, Int. J. Biomath., 10 (2017), 1750056.
[8]	A. Belmiloudi, Dynamical behavior of nonlinear impulsive abstract partial differential equations on networks with multiple time-varying delays and mixed boundary conditions involving time-varying delays, J. Dyn. Control Syst., 21 (2015), 95-146. doi: 10.1007/s10883-014-9230-y
[9]	A. Belmiloudi, Robust control problem of uncertain bidomain models in cardiac electrophysiology, Journal of Coupled Systems and Multiscale Dynamics, 1 (2013), 332-350. doi: 10.1166/jcsmd.2013.1023
[10]	A. Belmiloudi, Stabilization, optimal and robust control. Theory and applications in biological and physical sciences, Springer-Verlag, London, 2008.
[11]	A. Belmiloudi, Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors, J. Math. Anal. Appl., 327 (2007), 620-642. doi: 10.1016/j.jmaa.2006.04.037
[12]	A. Belmiloudi, Minimax control problem of periodic competing parabolic systems with logistic growth terms, Int. J. Control, 79 (2006), 150-161. doi: 10.1080/00207170500483484
[13]	A. Belmiloudi, Nonlinear optimal control problems of degenerate parabolic equations with logistic time varying delays of convolution type, Nonlinear Anal-Theor, 63 (2005), 1126-1152. doi: 10.1016/j.na.2005.05.033
[14]	A. Belmiloudi, Robust control problems associated with time-varying delay nonlinear parabolic equations, IMA J. Math. Control I., 20 (2003), 305-334. doi: 10.1093/imamci/20.3.305
[15]	M. Bendahmane, K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218. doi: 10.3934/nhm.2006.1.185
[16]	G. A. Bocharov, F. A. Rihan, Numerical modelling in biosciences using delay differential equations, J. Comput. Appl. Math., 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4
[17]	M. Boulakia, S. Cazeau, M. A. Fernández, et al. Mathematical Modeling of Electrocardiograms: A Numerical Study, Ann. Biomed. Eng., 38 (2009), 1071-1097.
[18]	Y. Bourgault, Y. Coudiere, C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Analysis: Real World Applications, 10 (2009), 458-482. doi: 10.1016/j.nonrwa.2007.10.007
[19]	A. J. V. Brandao, E. Fernandez-Cara, P. M. D. Magalhaes, et al. Theoretical analysis and control results for the FitzHugh-Nagumo equation, Electron. J. Differ. Eq., 164 (2008), 1-20.
[20]	N. Buric, D. Todorovic, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222.
[21]	N. Buric, K. Todorovic, N. Vasovic, Dynamics of noisy FitzHugh-Nagumo neurons with delayed coupling, Chaos, Solitons and Fractals, 40 (2009), 2405-2413. doi: 10.1016/j.chaos.2007.10.036
[22]	S. A. Campbell, Time delays in neural systems, In: V.K. Jirsa, A.R. McIntosh, (Eds.), Handbookof Brain Connectivity, Understanding Complex Systems, Springer, 65-90, 2007.
[23]	S. R Campbell, D. Wang, Relaxation oscillators with time delay coupling, Physica D, 111 (1998), 151-178. doi: 10.1016/S0167-2789(97)80010-3
[24]	J. O. Campos, R. S. Oliveira, R. W. dos Santos, et al. Lattice Boltzmann method for parallel simulations of cardiac electrophysiology using GPUs, J. Comput. Appl. Math., 295 (2016), 70-82. doi: 10.1016/j.cam.2015.02.008
[25]	R. H. Clayton, O. Bernus, E. M. Cherry, et al. Models of cardiac tissue electrophysiology: Progress, challenges and open questions, Prog. Biophys. Mol. Bio., 104 (2010), 22-48.
[26]	P. Colli-Franzone, L. Guerri, B. Taccardi, Modeling ventricular excitation: Axial and orthotropic anisotropy effects on wavefronts and potentials, Math. Biosci., 188 (2004), 191-205. doi: 10.1016/j.mbs.2003.09.005
[27]	P. Colli Franzone, G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level. In: Evolution Equations, Semigroups and Functional Analysis (eds.A. Lorenzi, B. Ruf), Birkhauser, Basel, 49-78, 2002.
[28]	P. Colli Franzone, L. Guerri, M. Pennacchio, et al. Spread of excitation in 3-D models of the anisotropic cardiac tissue. II: Effects of fiber architecture and ventricular geometry, Math. Biosci., 147 (1998), 131-171. doi: 10.1016/S0025-5564(97)00093-X
[29]	P. Colli Franzone, L. Guerri, S. Tentoni, Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field, Math. Biosci., 101 (1990), 155-235. doi: 10.1016/0025-5564(90)90020-Y
[30]	S. Corre, A. Belmiloudi, Coupled lattice Boltzmann simulation method for bidomaintype models in cardiac electrophysiology with multiple time-delays, Mathematical Modelling of Natural Phenomena (Topical Issues : Mathematical Modelling in Cardiology), In press.
[31]	S. Corre, A. Belmiloudi, Coupled lattice Boltzmann method for numerical simulations of fully coupled Heart and Torso bidomain system in electrocardiology, Journal of Coupled System and Multiscale Dynamics, 4 (2016), 207-229. doi: 10.1166/jcsmd.2016.1109
[32]	S. Corre, A. Belmiloudi, Coupled Lattice Boltzmann Modeling of Bidomain Type Models in Cardiac Electrophysiology, Mathematical and Computational Approaches in Advancing Modern Science and Engineering (eds. J. Bélair, et al.), Springer, 209-221, 2016.
[33]	H. Dal, S. Goktepe, M. Kaliske, et al. A fully implicit finite element method for bidomain models of cardiac electromechanics, Comput. Method. Appl. M., 253 (2013), 323-336. doi: 10.1016/j.cma.2012.07.004
[34]	M. Dupraz, S. Filippi, A. Gizzi, et al. Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans, Math. Method. Appl. Sci., 38 (2015), 1046-1058. doi: 10.1002/mma.3127
[35]	I. Ekeland, R. Temam, Convex analysis and variational problems, North-Holland,Amsterdam, 1976.
[36]	F. Fenton, A. Karma, Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation, Chaos, 8 (1998), 20-47. doi: 10.1063/1.166311
[37]	R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-465. doi: 10.1016/S0006-3495(61)86902-6
[38]	L. Glass, Synchronization and rhythmic processes in physiology, Nature, 410 (2001), 277-284. doi: 10.1038/35065745
[39]	A. Griewank, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Method. Softw., 1 (1992), 35-54. doi: 10.1080/10556789208805505
[40]	C. Henriquez, Simulating the electrical behavior of cardiac tissue using the bidomain model, Critical Reviews in Biomedical Engineering, 21 (1993), 1-77.
[41]	D. A. Hooks, K. A. Tomlinson, S. G. Marsden, et al. Cardiac microstructure: Implications for electrical propagation and defibrillation in the heart, Circ. Res., 91 (2002), 331-338. doi: 10.1161/01.RES.0000031957.70034.89
[42]	D. A. Israel, D. J. Edell, R. G. Mark, Time delays in propagation of cardiac action potential, Am. J. Physiol., 258 (1990), H1906-17.
[43]	J. Jia, H. Liu, C. Xu, et al. Dynamic effects of time delay on a coupled FitzHugh-Nagumo neural system, Alexandria Engineering Journal, 54 (2015), 241-250. doi: 10.1016/j.aej.2015.03.006
[44]	J. Keener, J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 2009.
[45]	K. Kunisch, M. Wagner, Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1077-1106.
[46]	G. Lines, M. Buist, P. Grottum, et al. Mathematical models and numerical methods for the forward problem in cardiac electrophysiology, Computing and Visualization in Science, 5 (2003), 215-239.
[47]	J. L. Lions, Equations Differentielles Operationnelles, Springer, NewYork, 1961.
[48]	J. L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Tome 1&2, Dunod, Paris, 1968.
[49]	C. H. Luo, Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction, Circ. Res., 68 (1991), 1501-1526. doi: 10.1161/01.RES.68.6.1501
[50]	C. H. Luo, Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes, Circ. Res., 74 (1994), 1071-1096. doi: 10.1161/01.RES.74.6.1071
[51]	J. Roger, A. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE T. Biomed. Eng., 41 (1994), 743-757. doi: 10.1109/10.310090
[52]	C. C. Mitchell, D. G. Schaeffer, A two-current model for the dynamics of cardiac membrane, B. Math. Biol., 65 (2003), 767-793. doi: 10.1016/S0092-8240(03)00041-7
[53]	C. Nagaiah, K. Kunisch, G. Plank, Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology, Comput. Optim. Appl., 49 (2011), 149-178. doi: 10.1007/s10589-009-9280-3
[54]	A. Panfilov, R. Aliev, A simple two-variable model of cardiac excitation, Chaos Solitons and Fractals, 7 (1996), 293-301. doi: 10.1016/0960-0779(95)00089-5
[55]	M. Penet, H. Guéguen, A. Belmiloudi, Artificial blood glucose control using a DDE modelling approach, IFAC Proceedings Volumes, 47 (2014), 2076-2081. doi: 10.3182/20140824-6-ZA-1003.01233
[56]	A. J. Pullan, M. L. Buist, L. K. Cheng, Mathematically modelling the electrical activity of the heart-from cell to body surface and back again, World Scientific, Singapore, 2005.
[57]	A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (2nd ed.), Springer-Verlag, Berlin, 2007.
[58]	B. Roth, Meandering of spiral waves in anisotropic cardiac tissue, Physica D: Nonlinear Phenomena, 150 (2001), 127-136. doi: 10.1016/S0167-2789(01)00145-2
[59]	S. Rolewicz, Funktionalanalysis und Steuerungstheorie, Springer, Berlin, New York, 1976.
[60]	R. F. Sandra, M. A. Savi, An analysis of heart rhythm dynamics using a three-coupled oscillator model, Chaos, Solitons and Fractals, 41 (2009), 2553-2565. doi: 10.1016/j.chaos.2008.09.040
[61]	E. Scholl, G. Hiller, P. Hovel, et al. Time-delayed feedback in neurosystems, Philos. T. R. Soc. A, 367 (2009), 1079-1096. doi: 10.1098/rsta.2008.0258
[62]	T. Seidman, H. Z. Zhou, Existence and uniqueness of optimal controls for a quasilinear parabolic equation, SIAM J. Control Optim., 20 (1982), 747-762. doi: 10.1137/0320054
[63]	O. Sharomi, R. Spiteri, Convergence order vs. parallelism in the numerical simulation of the bidomain equations, Journal of Physics: Conference Series, 385 (2012), 012009.
[64]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer Science&Business Media, 2010.
[65]	J. Sundnes, G. Lines, X. Cai, et al. Computing the electrical activity in the heart, Springer, Berlin, 2006.
[66]	R. Temam, Navier-Stokes equations, North-Holland, Amsterdam, 1984.
[67]	L. Tung, A bi-domain model for describibg ischemic myocardial d-c potentials [Thesis/Dissertation], Massachussets Institute of Technology, 1978.
[68]	K. H. Ten Tusscher, A. V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model, Am. J. Physiol-Heart C., 291 (2006), H1088-H1100.
[69]	M. Veneroni, Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Anal-Real, 10 (2009), 849-868. doi: 10.1016/j.nonrwa.2007.11.008
[70]	E. J. Vigmond, M. Hughes, G. Plank, Computational tools for modeling electrical activity in cardiac tissue, J. Electrocardiol., 36 (2003), 69-74. doi: 10.1016/j.jelectrocard.2003.09.017
[71]	E. J. Vigmond, R. W. dos Santos, A. J. Prassl, et al. Solvers for the cardiac bidomain equations, Prog. Biophys. Mol. Bio., 96 (2008), 3-18. doi: 10.1016/j.pbiomolbio.2007.07.012
[72]	D. Wu, S. Zhu, Stochastic resonance in FitzHugh-Nagumo system with time-delayed feedback, Phys. Lett. A, 372 (2008), 5299-5304. doi: 10.1016/j.physleta.2008.06.015

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