Numerical solution for optimal control of the reactiondiffusion equations in cardiac electrophysiology
 Chamakuri Nagaiah,
 Karl Kunisch,
 Gernot Plank
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 nonlinear parabolic partial differential equation of reactiondiffusion 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 extracellular 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 (NCG) 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.
 Ashihara, T., Constantino, J., Trayanova, N.A. (2008) Tunnel propagation of postshock activations as a hypothesis for fibrillation induction and isoelectric window. Circ. Res. 102: pp. 737745 CrossRef
 Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O. (2008) A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in dune. Computing 82: pp. 121138 CrossRef
 Bourgault, Y., Coudiére, Y., Pierre, C. (2009) Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal. Real World Appl. 10: pp. 458482 CrossRef
 Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.F. (2006) Adaptivity in space and time for reactiondiffusion systems in electrocardiology. SIAM J. Numer. Anal. 28: pp. 942962
 Hairer, E., Wanner, G. (1991) Solving Ordinary Differential Equations II. Springer, Berlin
 Henriquez, C.S. (1993) Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21: pp. 177
 Hooks, D.A., Trew, M.L., Caldwell, B.J., Sands, G.B., LeGrice, I.J., Smaill, B.H. (2007) Laminar arrangement of ventricular myocytes influences electrical behavior of the heart. Circ. Res. 101: pp. e103e112 CrossRef
 Jafri, M.S., Rice, J.J., Winslow, R.L. (1998) Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys. J. 74: pp. 11491168 CrossRef
 Kroll, M.W., Swerdlow, C.D. (2007) Optimizing defibrillation waveforms for ICDs. J. Interv. Card Electrophysiol. 18: pp. 247263 CrossRef
 Lang, J. (2001) Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Springer, Berlin
 Leon, L.J., Roberge, F.A., Vinet, A. (1994) Simulation of twodimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics. Ann. Biomed. Eng. 22: pp. 592609 CrossRef
 Lions, J.L. (1969) Résolution de Quelques Problémes aux Limites Nonlinéaires. Dunod, Paris
 Nielsen, B.F., Ruud, T.S., Lines, G.T., Tveito, A. (2007) Optimal monodomain approximations of the bidomain equations. Appl. Math. Comput. 184: pp. 276290 CrossRef
 Nocedal, J., Wright, S.J. (2006) Numerical Optimization. Springer, New York
 Plonsey, R. (1988) Bioelectric sources arising in excitable fibers (ALZA lecture). Ann. Biomed. Eng. 16: pp. 519546 CrossRef
 Rogers, J.M., McCulloch, A.D. (1994) A collocationGalerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41: pp. 743757 CrossRef
 Sepulveda, N.G., Roth, B.J., Wikswo, J.P. (1989) Current injection into a twodimensional anisotropic bidomain. Biophys. J. 55: pp. 987999 CrossRef
 Sims, J.J., Miller, A.W., Ujhelyi, M.R. (1998) Disparate effects of biphasic and monophasic shocks on postshock refractory period dispersion. Am. J. Physiol. 274: pp. H1943H1949
 ten Tusscher, K.H., Panfilov, A.V. (2006) Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291: pp. H1088H1100 CrossRef
 Tung, L.: A bidomain model for describing ischemic myocardial DC potentials. Ph.D. Thesis, MIT, Cambridge, MA (1978)
 Vigmond, E.J., Weber dos Santos, R., Prassl, A.J., Deo, M., Plank, G. (2008) Solvers for the cardiac bidomain equations. Prog. Biophys. Mol. Biol. 96: pp. 318 CrossRef
 Weber dos Santos, R., Plank, G., Bauer, S., Vigmond, E.J. (2004) Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51: pp. 1960–1968
 Wikswo, J.P., Lin, S.F., Abbas, R.A. (1995) Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation. Biophys. J. 69: pp. 21952210 CrossRef
 Title
 Numerical solution for optimal control of the reactiondiffusion equations in cardiac electrophysiology
 Journal

Computational Optimization and Applications
Volume 49, Issue 1 , pp 149178
 Cover Date
 20110501
 DOI
 10.1007/s1058900992803
 Print ISSN
 09266003
 Online ISSN
 15732894
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Bidomain model
 Reactiondiffusion equations
 Ionic model
 Optimal control with PDE constraints
 Existence and uniqueness
 FEM
 Rosenbrock type methods
 NCG method
 Industry Sectors
 Authors

 Chamakuri Nagaiah ^{(1)}
 Karl Kunisch ^{(1)}
 Gernot Plank ^{(2)}
 Author Affiliations

 1. Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, Graz, 8010, Austria
 2. Institute of Biophysics, Medical University of Graz, Harrachgasse 21, Graz, 8010, Austria