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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.

MSC:
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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DUNE
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