Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. (English) Zbl 1226.49024

Summary: The bidomain equations, a continuum approximation of cardiac tissue based on the idea of a functional syncytium, are widely accepted as one of the most complete descriptions of cardiac bioelectric activity at the tissue and organ level. Numerous studies employed bidomain simulations to investigate the formation of cardiac arrhythmias and their therapeutical treatment. They consist of a linear elliptic partial differential equation and a non-linear parabolic partial differential equation of reaction-diffusion type, where the reaction term is described by a set of ordinary differential equations. The monodomain equations, although not explicitly accounting for current flow in the extracellular domain and its feedback onto the electrical activity inside the tissue, are popular since they approximate, under many circumstances of practical interest, the bidomain equations quite well at a much lower computational expense, owing to the fact that the elliptic equation can be eliminated when assuming that conductivity tensors of intracellular and extracellular space are related to each other.
Optimal control problems suggest themselves quite naturally for this important class of modelling problems and the present paper is a first attempt in this direction. Specifically, we present an optimal control formulation for the monodomain equations with an extra-cellular current, \(I_e\), as the control variable. \(I_e\) must be determined such that electrical activity in the tissue is damped in an optimal manner.
The derivation of the optimality system is given and a method for its numerical realization is proposed. The solution of the optimization problem is based on a nonlinear conjugate gradient method. The main goals of this work are to demonstrate that optimal control techniques can be employed successfully to this class of highly nonlinear models and that the influence of \(I_e\) judiciously applied can in fact serve as a successful control for the dampening of propagating wavefronts.


49M25 Discrete approximations in optimal control
35Q93 PDEs in connection with control and optimization
92C40 Biochemistry, molecular biology
49J20 Existence theories for optimal control problems involving partial differential equations


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[1] Ashihara, T., Constantino, J., Trayanova, N.A.: Tunnel propagation of postshock activations as a hypothesis for fibrillation induction and isoelectric window. Circ. Res. 102(6), 737–745 (2008)
[2] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in dune. Computing 82(2), 121–138 (2008) · Zbl 1151.65088
[3] Bourgault, Y., Coudiére, Y., Pierre, C.: Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal. Real World Appl. 10(1), 458–482 (2009) · Zbl 1154.35370
[4] Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.F.: Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Numer. Anal. 28(3), 942–962 (2006) · Zbl 1114.65110
[5] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics. Springer, Berlin (1991) · Zbl 0729.65051
[6] Henriquez, C.S.: Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21(1), 1–77 (1993)
[7] Hooks, D.A., Trew, M.L., Caldwell, B.J., Sands, G.B., LeGrice, I.J., Smaill, B.H.: Laminar arrangement of ventricular myocytes influences electrical behavior of the heart. Circ. Res. 101(10), e103–e112 (2007)
[8] Jafri, M.S., Rice, J.J., Winslow, R.L.: Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys. J. 74(3), 1149–1168 (1998)
[9] Kroll, M.W., Swerdlow, C.D.: Optimizing defibrillation waveforms for ICDs. J. Interv. Card Electrophysiol. 18(3), 247–263 (2007)
[10] Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Lecture Notes in Computational Science and Engineering, vol. 16. Springer, Berlin (2001) · Zbl 0963.65102
[11] Leon, L.J., Roberge, F.A., Vinet, A.: Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics. Ann. Biomed. Eng. 22(6), 592–609 (1994)
[12] Lions, J.L.: Résolution de Quelques Problémes aux Limites Non-linéaires. Dunod, Paris (1969)
[13] Nielsen, B.F., Ruud, T.S., Lines, G.T., Tveito, A.: Optimal monodomain approximations of the bidomain equations. Appl. Math. Comput. 184(2), 276–290 (2007) · Zbl 1115.92005
[14] Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006) · Zbl 1104.65059
[15] Plonsey, R.: Bioelectric sources arising in excitable fibers (ALZA lecture). Ann. Biomed. Eng. 16(6), 519–546 (1988)
[16] Rogers, J.M., McCulloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41, 743–757 (1994)
[17] Sepulveda, N.G., Roth, B.J., Wikswo Jr., J.P.: Current injection into a two-dimensional anisotropic bidomain. Biophys. J. 55(5), 987–999 (1989)
[18] Sims, J.J., Miller, A.W., Ujhelyi, M.R.: Disparate effects of biphasic and monophasic shocks on postshock refractory period dispersion. Am. J. Physiol. 274(6), H1943–H1949 (1998)
[19] ten Tusscher, K.H., Panfilov, A.V.: Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291(3), H1088–H1100 (2006)
[20] Tung, L.: A bi-domain model for describing ischemic myocardial DC potentials. Ph.D. Thesis, MIT, Cambridge, MA (1978)
[21] Vigmond, E.J., Weber dos Santos, R., Prassl, A.J., Deo, M., Plank, G.: Solvers for the cardiac bidomain equations. Prog. Biophys. Mol. Biol. 96(1–3), 3–18 (2008)
[22] Weber dos Santos, R., Plank, G., Bauer, S., Vigmond, E.J.: Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51(11), 1960–1968 (2004)
[23] Wikswo Jr., J.P., Lin, S.F., Abbas, R.A.: Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation. Biophys. J. 69(6), 2195–2210 (1995)
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