Multipole Approach to the Inverse Problem in Electrocardiology: Convergence of the Multipole Equivalent Generator on the Inhomogeneous Body Conductor

Abstract

The multipole approach to the inverse electrocardiological problem consists of estimating the multipole components of the cardiac electric generator, starting from the measured body surface potential. This paper presents a critical investigation of the basic premise for the applicability of the multipole approach, namely the convergence of the multipole equivalent generator for the heart on the surface of an inhomogeneous body conductor. As an extension to multipole theory, a criterion for the convergence is derived. Based on realistic models for the body conductor and the cardiac electric generator, we observe that the criterion is not strictly satisfied in realistic conditions. Numerical simulations with the same models point out that the multipole equivalent generator is indeed not convergent in the strict mathematical sense. On the other hand, we show that the multipole equivalent generator yields a rather close approximation of the electrocardiological potential for intermediate values of the order of the multipole generator. A discussion is given on how to explain the apparently ambiguous results for the estimation of cardiac multipole components.