On boundary stimulation and optimal boundary control of the bidomain equations

Engwer

Journal of Computational Physics

2014

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Optimal control for cardiac electrophysiology based on the bidomain equations in conjunction with the Fenton-Karma ionic model is considered. This generic ventricular model approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potentials. However, it is challenging due to the appearance of state-dependent discontinuities in the source terms. A computational framework for the numerical realization of optimal control problems is presented. Essential ingredients are a shape calculus based treatment of the sensitivities of the discontinuous source terms and a marching cubes algorithm to track iso-surface of excitation wave fronts. Numerical results exhibit successful defibrillation by applying an optimally controlled extracellular stimulus.

Section: Solution Proceduresmentioning

confidence: 99%

Boundary control of bidomain equations with state-dependent switching source functions in the ionic model

Engwer

Journal of Computational Physics

2014

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“…The optimal control approach for the termination of reentrant waves in cardiac electrophysiology based on the bidomain model was discussed. The Luo-Rudy phase 1 membrane model was considered which represents excitability and refractoriness in a bio-physically more detailed as compared to the Fitz-Hugh-Nagumo model used in our previous studies [9]. Here the parabolic equation and the ODEs were solved on as a coupled system.…”

Section: Resultsmentioning

confidence: 99%

“…In previous work [7,8,6] the controller action representing the current delivered by electrodes was modeled as distributed force. Recently, we modeled the injected current via Neumann boundary conditions in the bidomain equations [9] using the simplified FitzHugh-Nagumo ionic model [23].…”

Section: Introductionmentioning

confidence: 99%

Application of optimal control to the cardiac defibrillation problem using a physiological model of cellular dynamics

Applied Numerical Mathematics

Plank

2015

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Optimal control techniques are investigated with the goal of terminating reentry waves in cardiac tissue models. In this computational study the Luo-Rudy phase I ventricular action potential model is adopted which accounts for more biophysical details of cellular dynamics as compared to previously used phenomenological models. The parabolic and ordinary differential equations are solved as a coupled system and an AMG preconditioner is used to solve the discretized elliptic equation The numerical results demonstrate that defibrillation is possible by delivering a single strong shock. The optimal control approach also leads to successful defibrillation and demands less total current. The present study motivates us to further investigate optimal control techniques on realistic geometries by incorporating the structural heterogeneity in the cardiac tissue.

“…For the numerical realization of this condition together with the PDEs one needs a special purpose numerical methods to solve the system uniquely. For this purpose, we adopted a stabilized saddle point formulation from the work of Bochev and Lehoucq [8], see [10,11] for the discussion and implementation details of this technique for the current problem. After the full discretization of the PDEs we obtain a system of linear algebraic equations and to solve the linear system we employed a Conjugate Gradient (CG) method with AMG preconditioner [7], which is developed using a greedy heuristic algorithm for the aggregation based on a strength of connection criterion.…”

Section: Computer Implementationmentioning

confidence: 99%

Primal-dual active set strategy for large scale optimization of cardiac defibrillation

Applied Mathematics and Computation

2017

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In this paper, the feasible study of the optimal control techniques for the cardiac defibrillation on the anatomical three spatial dimensional rabbit ventricle geometry in the presence of bilateral control constraints. The present work addresses the numerical treatment of multiscale and multidomain simulations of bidomain equations, the description of deriving optimality system, the applicability of primal-dual active set methods to treat the bilateral control constraints for solving such large scale optimization of cardiac defibrillation. The numerical results are demonstrated for the successful defibrillation study on 3D rabbit ventricle geometry by utilizing the less total currents, robustness of the optimization algorithm w.r.t to the variations in the model parameters, feasible study of the multiple smaller boundary control support and the numerical convergence of the optimization algorithm on the finer spatial grids. The parallel efficiency is demonstrated for the primal-dual active set optimization algorithm on such finer spatial grid.