Limitation of the Inverse Problem in Body Surface Potential Mapping

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Abstract
The inverse problem in electrocardiography is ill-conditioned, and small noise included in the measured potentials causes large errors in the solution. Since the inverse problem is mostly described as a linear problem, the entire problem has often been treated in terms of a transfer matrix. The degree of linear independence among the vectors in the transfer matrix, which is directly related to the stability of the solution, is well represented by the singular values of the transfer matrix. By means of the singular value decomposition of the transfer matrix, the stability of solution to the inverse problem has been discussed when the potential data contain noise or the transfer matrix includes some error. We have derived expressions of maximum possible error magnification and a root-mean-square error magnification and, in terms of these parameters, found that only 4 equivalent cardiac dipoles or only 15 independent epicardial potentials can be estimated from body surface potentials when they are measured with an accuracy as high as 99 percent.