Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

Abstract

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework
of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of
the Cauchy problem for the Laplace equation in R
3. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy
problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of
the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm
based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The
Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace
equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory.
In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.