The mathematical basis for imaging cardia electrical function

The essential difficulty of the source-imaging problem in electrocardiology is that the data (body surface potentials) depend continuously on the properties of complex uniquely determined equivalent sources (e.g., epicardial potentials) while such sources do not depend continuously on the data (precluding utility of direct inversion of the operator relating sources and data). This is in distinction to the standard digital tomographic imaging problems in radiology, which superficially resemble the electrocardiographic imaging problem in their requirement for solution of a linear integral equation. From the mathematical standpoint, the electrocardiographic imaging problem requires that an operator with a continuous inverse be constructed from the original operator. This may be accomplished either by slightly perturbing the original operator (the regularization approach), or by performing a calculation on the data (based on inherent source evolution singularities) which allows restriction of the admissible solution domain to a compact set (the topological approach). An understanding of the mathematical questions fundamental to this imaging problem is helpful in assessing its present status and in identifying promising directions for the development of a clinically useful technique.