The Forward and Inverse Problems: What Are They, Why Are They Important, and Where Do We Stand?

The inverse problem is a general term for the process of inferring the distribution of an inaccessible variable from an accessible one. An example is the tomographic reconstruction of the opacity of tissue within the body from measured projections.1 The forward problem is the computation of the measured variable from the source. The forward problem of electrocardiology is the calculation of electrical potentials generated by known sources at sites distant from those sources, typically through a volume conductor. The classic example is the computation of potentials at the body surface from a known set of sources, for example, an activation wavefront, in the heart. The companion and much more challenging problem is the ECG inverse problem. The task in the inverse problem is to use surface ECG recordings to compute the sources within the myocardium. The forward and inverse problems of ECG have been addressed with varying degrees of success for many decades.2 The two problems are very closely related to each other mathematically and biophysically, but because of the potential practical uses of the inverse formulation, it has been more aggressively addressed. The inverse problem is solved informally thousands of times per day, whenever a cardiologist makes an inference about the cardiovascular status of a patient by reading an ECG. However, the formal, technical solution is slow, approximate, and so far not useful in day-to-day practice. It is interesting to observe that clinical inverse solutions focus primarily on temporal sequences, often give special weight to cardiac repolarization, and use ECGs as a fast and inexpensive tool for gross evaluation. Conversely, formal inverse solutions have focused on spatial relationships, for example, the sequence of cardiac excitation, and have involved slow but elaborate analysis of detailed relationships among different regions of the heart. The latter has not been simply a matter of choice, because it is the spatial relationships that form the mechanistic basis of the origin of cardiac electrical signals. To date, formal solutions have contributed greatly to the understanding of the biophysical mechanisms by which electrical events within the heart lead to those observed on the body surface; however, this understanding so far has not led to improvements in most day-to-day measurements or evaluations of ECGs. The crux of the problem solution is the computation of a transfer function, usually in the form of a twodimensional matrix. The transfer function relates the torso potentials to the sources in the myocardium and can be established when both sets are known. This relationship is stated mathematically in Equation 1 in the second article by Ramanathan and Rudy in this issue of the Journal.3 The function contains information about the transformation of the source potentials by the geometric and electrical characteristics of the volume conductor that result in body surface electrograms. The inverse problem is solved, then, by calculating the inverse of the transfer function. The inversion of the matrix is “illposed,” meaning that there is too little information in the surface potentials to identify cardiac sources unambiguously and that small amounts of noise have a dramatic in uence on the results. To avoid these problems, it is common to introduce constraints to the system, essentially using assumptions about the sources or other information to make the mathematics tractable. An early example is the determination of the characteristics of an equivalent electrical dipole from the recorded vectorcardiogram.4 The mathematically precise de nition of the components of the equivalent dipole was given by Geselowitz,5 ,6 along with equations identifying the parts of the body surface signals that could not be accounted for by the dipole source. A variety of mathematical,7 statistical,8 and biophysical9 -1 1 constraints have been introduced and tested throughout the years. Since the introduction of minimally invasive procedures for the treatment of cardiac rhythm disorders, there has been a resurgence of interest in the inverse problem. With improved capabilities to eliminate selectively bits of myocardium that support reentrant or automatic tachyarrhythmias comes an increasing need for a method to identify anatomically the offending region with a minimum of invasion. The inverse problem is one approach. A robust and accurate solution for estimating myocardial potentials from surface recordings would be an important extension in the diagnosis and treatment of arrhythmias. There have been several limitations that have hinThis work was supported in part by Research Grants HL-33637 and HL-50537 from The National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland.