Name
Title | ||
|---|---|---|
Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology |
Abstract | ||
|---|---|---|
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 non-linear 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. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1007/s10589-009-9280-3 | Computational Optimization and Applications |
Keywords | Field | DocType |
Bidomain model,Reaction-diffusion equations,Ionic model,Optimal control with PDE constraints,Existence and uniqueness,FEM,Rosenbrock type methods,NCG method | Conjugate gradient method,Parabolic partial differential equation,Bidomain model,Mathematical optimization,Nonlinear system,Optimal control,Ordinary differential equation,Mathematical analysis,Delay differential equation,Elliptic partial differential equation,Mathematics | Journal |
Volume | Issue | ISSN |
49 | 1 | 0926-6003 |
Citations | PageRank | References |
10 | 0.62 | 4 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Chamakuri Nagaiah | 1 | 16 | 3.50 |
| Karl Kunisch | 2 | 1370 | 145.58 |
| G Plank | 3 | 201 | 33.05 |