Analysis of an optimal control problem for the tridomain model in cardiac electrophysiology. (English) Zbl 1234.49004
Summary: In the present paper, an optimal control problem constrained by the tridomain equations in electrocardiology is investigated. The state equations consisting in a coupled reaction-diffusion system modeling the propagation of the intracellular and extracellular electrical potentials, and ionic currents, are extended to further consider the effect of an external bathing medium. The existence and uniqueness of solution for the tridomain problem and the related control problem is assessed, and the primal and dual problems are discretized using a finite volume method which is proved to converge to the corresponding weak solution. In order to illustrate the control of the electrophysiological dynamics, we present some preliminary numerical experiments using an efficient implementation of the proposed scheme.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 92C50 | Medical applications (general) |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
optimal control; finite volume approximation; convergence; cardiac electrophysiology; tridomain modelReferences:
| [1] | Andreianov, B.; Bendahmane, M.; Karlsen, K. H., Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperbolic Differ. Equ., 7, 1-67 (2010) · Zbl 1207.35020 |
| [2] | Andreianov, B.; Bendahmane, M.; Ruiz-Baier, R., Analysis of a finite volume method for a cross-diffusion model in population dynamics, M3AS Math. Models Methods Appl. Sci., 21, 307-344 (2011) · Zbl 1228.65178 |
| [3] | Aslanidi, O. V.; Benson, A. P.; Boyett, M. R.; Zhang, H., Mechanisms of defibrillation by standing waves in the bidomain ventricular tissue with voltage applied in an external bath, Phys. D, 238, 984-991 (2009) · Zbl 1165.92306 |
| [4] | Belik, M. E.; Usyk, T. P.; McCulloch, A. D., Computational methods for cardiac electrophysiology, (Ayache, N., Computational Models for the Human Body, Handbook of Numerical Analysis (2004), Elsevier: Elsevier North-Holland), 129-187 |
| [5] | Bendahmane, M.; Karlsen, K. H., Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1, 185-218 (2006) · Zbl 1179.35162 |
| [6] | Bendahmane, M.; Karlsen, K. H., Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl. Numer. Math., 59, 2266-2284 (2009) · Zbl 1165.92005 |
| [7] | Bendahmane, M.; Bürger, R.; Ruiz-Baier, R., A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology, Numer. Methods Partial Differential Equations, 26, 1377-1404 (2010) · Zbl 1206.92004 |
| [8] | Bendahmane, M.; Bürger, R.; Ruiz-Baier, R., A finite volume scheme for cardiac propagation in media with isotropic conductivities, Math. Comput. Simulation, 80, 1821-1840 (2010) · Zbl 1192.92002 |
| [9] | Bourgault, Y.; Coudière, Y.; Pierre, C., Existence and uniqueness of the solution for the bidomain model used in cardiac electro-physiology, Nonlinear Anal. Real World Appl., 10, 458-482 (2009) · Zbl 1154.35370 |
| [10] | Colli Franzone, P.; Savaré, G., Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, (Lorenzi, A.; Ruf, B., Evolution Equations, Semigroups and Functional Analysis (2002), Birkhäuser: Birkhäuser Basel), 49-78 · Zbl 1036.35087 |
| [11] | Coudière, Y.; Pierre, C.; Rousseau, O.; Turpault, R., 2D/3D discrete duality finite volume scheme (DDFV) applied to ECG simulation, (Eymard, R.; Herard, J. M., Finite Volumes for Complex Applications (V) (2008), Wiley), 313-320 · Zbl 1374.92004 |
| [12] | Dos Santos, R. W.; Dickstein, F., On the influence of a volume conductor on the orientation of currents in a thin cardiac tissue, (Lecture Notes in Comput. Sci., vol. 2674 (2003), New York), 111-121 |
| [13] | FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1, 445-465 (1961) |
| [14] | Haasdonk, B.; Ohlberger, M., Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN Math. Model. Numer. Anal., 42, 277-302 (2008) · Zbl 1388.76177 |
| [15] | Hager, W. W.; Zhang, H., A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2, 35-58 (2006) · Zbl 1117.90048 |
| [16] | Hermeline, F., Approximation of 2D and 3D diffusion operators with discontinuous full-tensor coefficients on arbitrary meshes, Comput. Methods Appl. Mech. Engrg., 196, 2497-2526 (2007) · Zbl 1173.76362 |
| [17] | Hestenes, M. R.; Stiefel, E. L., Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49, 409-436 (1952) · Zbl 0048.09901 |
| [18] | Hinze, M.; Kunisch, K., Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim., 40, 925-946 (2001) · Zbl 1012.49026 |
| [19] | Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE Constraints, Math. Model. Theory Appl., vol. 23 (2009), Springer: Springer Netherlands · Zbl 1167.49001 |
| [20] | Keener, J.; Sneyd, J., Mathematical Physiology, vols. I and II (2009), Springer: Springer New York · Zbl 1273.92017 |
| [21] | Kelly, R.; Staines, A.; MacWalter, R.; Stonebridge, P.; Tunstall-Pedoe, H.; Struthers, A. D., The prevalence of treatable left ventricular systolic dysfunction in patients who present with noncardiac vascular episodes: A case-control study, J. Am. Coll. Cardiol., 39, 219-224 (2002) |
| [22] | K. Kunisch, M. Wagner, Optimal control of the bidomain system (I): The monodomain approximation with the Rogers-McCulloch model, Nonlinear Anal. Real World Appl., in press.; K. Kunisch, M. Wagner, Optimal control of the bidomain system (I): The monodomain approximation with the Rogers-McCulloch model, Nonlinear Anal. Real World Appl., in press. · Zbl 1256.49009 |
| [23] | K. Kunisch, M. Wagner, Optimal control of the bidomain system (II): Uniqueness and regularity theorems for weak solutions, SFB Report, Graz University, 2011.; K. Kunisch, M. Wagner, Optimal control of the bidomain system (II): Uniqueness and regularity theorems for weak solutions, SFB Report, Graz University, 2011. · Zbl 1280.49005 |
| [24] | Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0203.09001 |
| [25] | Nagaiah, C.; Kunisch, K.; Plank, G., Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology, Comput. Optim. Appl., 49, 149-178 (2011) · Zbl 1226.49024 |
| [26] | Nagaiah, C.; Kunisch, K., Adaptive and higher order numerical solution for optimal control of monodomain equations in cardiac electrophysiology, Appl. Numer. Math., 61, 53-65 (2011) · Zbl 1201.92026 |
| [27] | Nagumo, J. S.; Arimoto, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng., 50, 2061-2071 (1962) |
| [28] | F. Nobile, A. Quarteroni, R. Ruiz-Baier, An active strain electromechanical model for cardiac tissue, Int. J. Numer. Meth. Biomed. Engrg., doi:10.1002/cnm.1468; F. Nobile, A. Quarteroni, R. Ruiz-Baier, An active strain electromechanical model for cardiac tissue, Int. J. Numer. Meth. Biomed. Engrg., doi:10.1002/cnm.1468 · Zbl 1242.92016 |
| [29] | Sanfelici, S., Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology, Numer. Methods Partial Differential Equations, 18, 218-240 (2002) · Zbl 1002.65100 |
| [30] | Sundnes, J.; Lines, G. T.; Cai, X.; Nielsen, B. F.; Mardal, K.-A.; Tveito, A., Computing the Electrical Activity in the Heart (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1182.92020 |
| [31] | Tubino, F.; Solari, G., Double proper orthogonal decomposition for representing and simulating turbulence fields, J. Engrg. Mech., 131, 1302 (2005) |
| [32] | Veneroni, M., Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Anal. Real World Appl., 10, 849-868 (2009) · Zbl 1167.35403 |
| [33] | Volkwein, S., Nonlinear conjugate gradient methods for the optimal control of laser surface hardening, Optim. Methods Softw., 19, 179-199 (2004) · Zbl 1062.49034 |
| [34] | Zuazua, E., Controllability and observability of partial differential equations: Some results and open problems, (Handbook of Differential Equations: Evolutionary Equations, vol. III (2007), North-Holland: North-Holland Amsterdam), 527-621 · Zbl 1193.35234 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
