Rudy ECG Forward Problem Revisited 527 features usually used for map interpretation.” Moreover, the potential magnitude difference was greatly reduced by eliminating the reference offset and simple scaling of the maps. These results are consistent with the results of the Bear study, where amplitudes differed substantially (corrections for reference offset were not attempted) but “correlation coefficients between simulated and measured BSPMs were high (≈ 0.9) for most of the activation sequence.” This recent study used additional measures for comparing the simulated and recorded BSPMs, namely the distance between potential extrema and their relative orientation. There were differences in these measures; a careful inspection of the BSPMs in Ramsey et al shows similar differences between their simulated and measured maps as well (although they considered these differences in regions of low potential gradients to be of little significance). Inclusion of the torso inhomogeneities in Bear simulations reduced, but did not remove these differences. Given the low resolution and smoothing effect of body surface potentials and the loss of geometric relationships between cardiac electric sources in the BSPM, it is unclear whether these differences are meaningful to the interpretation of the BSPM in terms of underlying electrophysiological processes in the heart. The presence of noise in the recorded maps and inaccuracy of the closed epicardial and body surface geometries and potentials that are interpolated from a limited set of measurements could be the source of these differences. The ultimate objective of electrocardiography is noninvasive determination of electrophysiological events in the heart from body surface potential measurements. In a broad sense, this is the definition of the inverse problem of electrocardiography. Similar to the forward problem, the inverse problem can be defined in terms of various equivalent cardiac sources, including epicardial potentials. 18 The forward problem is mathematically well–posed, meaning that there is continuous dependence of the solution on the data. In contrast, the inverse problem is ill-posed in the sense that small errors in the data (measurement noise, geometry errors, and inaccurate conductivity values) can cause large unbounded errors in the solution. This necessitates the use of regularization techniques that impose physiologically based constraints or iterative schemes to stabilize the solution in the presence of these inaccuracies that are always present in the experimental and clinical environments.

Regularized and iterative inverse solutions have provided the theoretical basis for electrocardiographic imaging (ECGI; also called electrocardiographic mapping), which combines BSPM with noninvasive information about the heart–torso geometry to reconstruct noninvasively potentials, electrograms, activation sequences (isochrones), and repolarization patterns on the epicardial surfaces of the heart. 19 The results of Bear demonstrate a limited effect of the torso inhomogeneities, relative to a homogeneous torso, on the patterns of body surface potentials in forward problem simulations. Naturally, this raises the question whether it is important to include the torso inhomogeneities in ECGI applications. This requires accurate geometric determination and segmentation of patient-specific inhomogeneities and precise determination of their conductivities, accomplished noninvasively. The conductivities can vary substantially between patients and in