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

WILLIAM M. SMITH, PH.D., and ROGER C. BARR, PH.D.*

From the Departments of Biomedical Engineering and Medicine, The University of Alabama at Birmingham, Birmingham,Alabama; and the *Departments of Biomedical Engineering and Pediatrics, Duke University, Durham, North Carolina

Editorial Comment

The inverse problem is a general term for the processof inferring the distribution of an inaccessible variablefrom an accessible one. An example is the tomographicreconstruction of the opacity of tissue within the bodyfrom measured projections.1 The forward problem is thecomputation of the measured variable from the source.The forward problem of electrocardiology is the calcu-lation of electrical potentials generated by knownsources at sites distant from those sources, typicallythrough a volume conductor. The classic example is thecomputation of potentials at the body surface from aknown set of sources, for example, an activation wave-front, in the heart. The companion and much more chal-lenging problem is the ECG inverse problem. The task inthe inverse problem is to use surface ECG recordings tocompute the sources within the myocardium. The for-ward and inverse problems of ECG have been addressedwith varying degrees of success for many decades.2 Thetwo problems are very closely related to each othermathematically and biophysically, but because of thepotential practical uses of the inverse formulation, it hasbeen more aggressively addressed.

The inverse problem is solved informally thousandsof times per day, whenever a cardiologist makes aninference about the cardiovascular status of a patient byreading an ECG. However, the formal, technical solutionis slow, approximate, and so far not useful in day-to-daypractice. It is interesting to observe that clinical inversesolutions focus primarily on temporal sequences, oftengive special weight to cardiac repolarization, and useECGs as a fast and inexpensive tool for gross evaluation.Conversely, formal inverse solutions have focused onspatial relationships, for example, the sequence of car-diac excitation, and have involved slow but elaborateanalysis of detailed relationships among different regionsof the heart. The latter has not been simply a matter ofchoice, because it is the spatial relationships that form

the mechanistic basis of the origin of cardiac electricalsignals. To date, formal solutions have contributedgreatly to the understanding of the biophysical mecha-nisms by which electrical events within the heart lead tothose observed on the body surface; however, this un-derstanding so far has not led to improvements in mostday-to-day measurements or evaluations of ECGs.

The crux of the problem solution is the computationof a transfer function, usually in the form of a two-dimensional matrix. The transfer function relates thetorso potentials to the sources in the myocardium and canbe established when both sets are known. This relation-ship is stated mathematically in Equation 1 in the secondarticle by Ramanathan and Rudy in this issue of theJournal.3 The function contains information about thetransformation of the source potentials by the geometricand electrical characteristics of the volume conductorthat result in body surface electrograms. The inverseproblem is solved, then, by calculating the inverse of thetransfer function. The inversion of the matrix is “ill-posed,” meaning that there is too little information in thesurface potentials to identify cardiac sources unambigu-ously and that small amounts of noise have a dramaticin� uence on the results. To avoid these problems, it iscommon to introduce constraints to the system, essen-tially using assumptions about the sources or other in-formation to make the mathematics tractable. An earlyexample is the determination of the characteristics of anequivalent electrical dipole from the recorded vectorcar-diogram.4 The mathematically precise de� nition of thecomponents of the equivalent dipole was given by Ge-selowitz,5 ,6 along with equations identifying the parts ofthe body surface signals that could not be accounted forby the dipole source. A variety of mathematical,7 statis-tical,8 and biophysical9 -1 1 constraints have been intro-duced and tested throughout the years.

Since the introduction of minimally invasive proce-dures for the treatment of cardiac rhythm disorders, therehas been a resurgence of interest in the inverse problem.With improved capabilities to eliminate selectively bitsof myocardium that support reentrant or automatic tachy-arrhythmias comes an increasing need for a method toidentify anatomically the offending region with a mini-mum of invasion. The inverse problem is one approach.A robust and accurate solution for estimating myocardialpotentials from surface recordings would be an importantextension in the diagnosis and treatment of arrhythmias.

There have been several limitations that have hin-

This work was supported in part by Research Grants HL-33637 andHL-50537 from The National Heart, Lung, and Blood Institute, Na-tional Institutes of Health, Bethesda, Maryland.

J Cardiovasc Electrophysiol, Vol. 12, pp. 253-255, February 2001

Address for correspondence: William M. Smith, Ph.D., The Universityof Alabama at Birmingham, B140C Volker Hall, 1670 UniversityBoulevard, Birmingham, AL 35294-0019.Fax: 205-975-4720; E-mail:wms@crml.uab.edu

253

Reprinted with permission fromJOURNAL OF CARDIOVASCULAR ELECTROPHYSIOLOGY, Volume 12, No. 2, February 2001

Copyright ©2001 by Futura Publishing Company, Inc., Armonk, NY 10504-0418

dered the routine use of the inverse problem in practicalresearch and clinical applications. First is the complexityof the sensors and instrumentation required. The appli-cation of a large number of electrodes on the bodysurface in an accurately characterized geometric patternand the high signal-to-noise ratio required in the ECGsare technically demanding. However, with increasingsophistication in body surface mapping, these challengesare being addressed effectively.12

A more fundamental problem is establishing adequatevalidation of the solutions in order to build con� dence inthe results. It is clearly quite dif� cult to make simulta-neous measurements on endocardium, epicardium, orintramyocardium and on the body surface. In experimen-tal animals, the need to open the chest in order to applyan adequate number and distribution of electrodes un-avoidably disrupts the volume conductor and rendersmeasurements from the body surface suspect. This prob-lem is in addition to the changes resulting from simplyintroducing electrodes, often with metallic components,into the heart. Furthermore, even when measurementsthroughout the thickness of cardiac muscle are at-tempted, the number of measurement locations (hun-dreds) remains several orders of magnitude lower thanthe number of cells of the heart (billions), so that a greatmany details are known only approximately, if at all.

Another related challenge is the possibility that accu-rate solutions to the forward and inverse problems wouldrequire an unreasonably detailed speci� cation of theanatomy and physiology of each patient or subject. Notonly is the body habitus of each individual potentiallyimportant, but also the electrophysiologic characteristicsof the various layers of the volume conductor (skeletalmuscle, subcutaneous fat, lungs) might strongly in� u-ence the integrity of the computations. Thus, in additionto requiring detailed anatomic imaging with one of sev-eral widely available, but often costly, modalities, effec-tive solutions might require precise measurements ofconductivities in those intervening tissues.

These problems are partially addressed by two articlesby Ramanathan and Rudy3 ,1 3 in this issue of the Journal.The authors made very careful simultaneous measure-ments of epicardial and surface potentials using an iso-lated heart in a torso-shaped tank that has been estab-lished as a useful simulation of human anatomy.7 Thesemeasurements, along with the realistic thoracic geome-tries available from the National Library of Medicine’sVisual Human Project, allowed computation of transferfunctions for homogeneous and inhomogeneous torsosand a torso with a stylized lung geometry. In the � rstarticle,1 3 the known epicardial potentials and the varioustransfer functions were used to compute simulated torsopotentials and compared with those that had been di-rectly measured. In the second article,3 the computedtorso potentials then were used with the inverse formu-lation to compute epicardial potentials for the three tho-racic models. Simulated electrograms, activation se-quences, and potential maps were compared with theirexperimentally measured counterparts. In both articles,the in� uence of including thoracic inhomogeneities in

the simulation was evaluated. It is encouraging that, inboth cases, the use of the homogeneous, less realisticthoracic anatomy barely degraded the accuracy of thesimulations. A good approximation with the simpli� edtorso model was observed for different torso sizes, lungand skeletal muscle conductivities, and genders, evenwhen noise and geometric error were introduced into thecomputations. Of interest is the ability of the technique toresolve two simultaneously paced epicardial sites ap-proximately 6 cm apart, either with isochronal or isopo-tential maps, whereas only isochronal maps could distin-guish between two sites with 2.5 cm spacing. Theseresults lead to optimism that these techniques mightbecome practical without exhaustive anatomic and elec-trophysiologic knowledge determined on a subject-to-subject basis. Combined with earlier work in computingintramural activation from epicardial potentials,14 thetechnique could be very powerful for studying bioelec-tric phenomena in the heart in health and disease.

It is clear that the approach represented in these arti-cles will continue to be a valuable tool in clinical andbasic studies of cardiac excitation during normal condi-tions and arrhythmias. Inverse calculations will be ableto provide a tool to address mechanisms with minimalinvasiveness, and their application in carefully chosencircumstances will be invaluable. However, their role inthe practical diagnosis and therapy of rhythm disorders isless clear. Current therapeutic approaches require at leastinsertion of catheters, so that the advantage of a com-pletely noninvasive diagnostic capability is equivocal, atleast presently, and there are several competing technol-ogies from which to choose. Pace mapping1 5 has beenused for several years to identify areas responsible for themaintenance of arrhythmias. Rigid1 6 and in� atable17 -19

probe arrays have been used effectively to implement aform of the inverse problem. In this approach, endocar-dial potentials are computed from electrograms mea-sured by electrodes in the blood pool of the ventricles oratria. It is important to note that the mathematical andcomputer procedures for inverse problems remain thesame for all remote-sensing electrodes, whether they belocated on the body surface or within the heart, althoughthe latter arrangement offers advantages because they arecloser. Probes also have been designed to make directcontact with the endocardial wall after deployment in acardiac cavity,2 0 allowing direct measurement of poten-tials. An endocardial navigation and mapping system21

provides both anatomic and electrophysiologic charac-terization of the endocardium, identifying locations thatare candidates for ablation or other intervention.

The group from Case Western Reserve University thatauthored these two articles3 ,1 3 is one of a few laboratoriesthat are intensively pursuing improved methodologiesfor addressing the forward and inverse problems of elec-trocardiology. The work presented here is a further im-provement and validation of the mathematical and elec-trophysiologic techniques that are necessary in this area.These investigators and others in the area have bene� ttedby certain major advances in the overall climate of in-verse solutions that have occurred over the last decade.

254 Journal of Cardiovascular Electrophysiology Vol. 12, No. 2, February 2001

These include enormously more powerful computer sys-tems now widely available at low cost, the availability ofimaging systems that can locate the position of the heartwithin the thorax, and the development of good mathe-matical models of the electrical behavior of the individ-ual cell types (atrial, ventricular, and conduction system)that are present. Continued advances are taking a periodof years to come to fruition, but promise to lead to moreuniversal acceptance and use by the scienti� c commu-nity.

References

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Smith and Barr Editorial Comment 255