In this chapter we address the topic of validating the performance of electrocardiographic inverse solutions. A well-defined and carefully implemented validation scheme is critical to the evaluation of any numerical method. The main goals of validation are typically to assess the accuracy, speed, reliability, and any other special characteristics of the method, as well as explore its utility in practical applications. The nature of the electrocardiography problem makes validation especially challenging. It is difficult to obtain ground truth measurements for comparison under conditions that faithfully replicate, or even approximate, useful application scenarios. At the same time, the complex nature of the source, the ill-posedness of the inverse problem, and the resulting numerical sensitivities and instabilities all add to the difficulties. Moreover, the very problem of defining what is a physiologically or medically relevant goal is a challenge in itself. Thus, from the perspective of validating an inverse solution that is applied to a patient, there is ambiguity in the interpretation of the results and thus difficulty in defining an appropriate error metric.
We begin with an outline of the necessary elements of a validation approach, a framework that we will apply repeatedly throughout the chapter. We then return to the topic of interpretation of inverse solution results and the resulting ambiguities that may arise.
Inverse solutions are based on a representation of the relevant geometry and a description of the sources one wishes to reconstruct. Examining a validation strategy, therefore, begins with an evaluation of these two main required elements of an inverse solution: a model of the geometry and a model of the relevant cardiac sources. Geometry information comes in two forms: simplified models, based on continuous functions such as spheres or cylinders, or more often discrete models from measured locations of electrode positions and points describing the anatomy of the thorax, heart, and, where appropriate, torso inhomogeneities. These points may lie only on the surface of the relevant organs or may span their interior. Obtaining geometry for validation often requires medical imaging techniques, segmentation of the tissue boundaries, and geometric model construction, with each step contributing some (hopefully known) error to the final result. Validation studies often include varying specific parameters such as resolution and accuracy of the geometric measurements, inclusion of inhomogeneous regions, and any anisotropy in the conductivity assigned to each region. The more control that is available over these parameters, both in the inverse problem formulation and the validation model, the more complete the testing that is possible. Validation can then expand beyond simply testing the accuracy of a particular inverse solution to general questions of what resolution, precision, or complexity is required of a geometric model to achieve a desired accuracy.
The second requirement for validation is a description of both the bioelectric source, in whatever form is dictated by the inverse solution formulation, and the remote signals, typically the body surface potentials, or catheter potentials for inverse solutions based on intracavitary potentials. It is the requirement for an accurate source description that generally poses the most difficult technical problem, because it often requires measurement of epicardial or endocardial potentials or cardiac activation times. Such measurements are often infeasible to make in humans and present considerable challenges even in animals. For example, opening the chest to access the heart will disrupt the torso and thus impair simultaneous cardiac and body surface measurements as well as create non-physiologic conduction conditions in the torso interior. As we shall see, the form of source information can be quantitative, based either on synthetic data or actual measurements, but also qualitative, often based on a particular electrophysiological feature available by means of other, non-electrical forms of measurement, invasive procedure, or prior medical history. Examples of these three source types are, respectively, echocardiograms from ultrasound, electrograms from cardiac catheterization, or knowledge of prior myocardial infarction from medical history. Combinations are obviously possible, too, such as medical history information based on previous invasive procedures.
A general consideration in obtaining both geometry and source descriptions for validation is the need to attach to each value an estimate of the accuracy with which it was obtained. For instance, if geometric measurements are only accurate within an error of 5 mm, then it is unreasonable to expect an inverse solution based on that geometry to have a spatial error that is any lower than 5 mm. Similarly, if ground truth source measurements are only accurate to within a given noise figure, then this places a lower bound on the accuracy of any comparison with computed inverse solutions. More generally, the error assigned to each measurement determines by some means, not usually as direct as the examples above, the limit in accuracy that can be reasonably expected from any inverse solution based on those measurements. In the case where geometric and/or source data is synthesized rather than measured, it is necessary to add noise prior to inverse calculations to simulate some degree of realistic conditions. Moreover, a study of the way the solution accuracy changes with variable amounts of noise can become a valuable part of the validation procedure.
There is another aspect of validation for inverse solutions that leads to both ambiguity in the interpretation of results and additional approaches to the validation problem. One obvious gold standard of validation for electrocardiographic inverse solutions might be to reconstruct the source signals, for example, the epicardial potentials, over the entire heart, in a human subject, with near perfect fidelity. Practically, this is an impossible standard because such a detailed measurement of the source is not possible in humans. Even if it were technically and ethically possible, however, there would be another ambiguity with such a comparison. Common mathematical formulations of the error in inverse solutions may not have a clear relationship to the true criteria of interest, diagnostic accuracy and precision. Even qualitative accuracy measures, such as examination of isopotential map sequences or the excitation pattern a premature excitation, have a certain clinical ambiguity; sufficient accuracy for a general diagnosis, for instance, may not be sufficient accuracy for a remediative procedure such as cardiac ablation. Hence, to validate an inverse solution requires a clear idea of the anticipated application and a set of associated requirements. There is no single gold standard, which is fortunate given that the obvious standard is not available.
A related practical challenge associated with validation of inverse problems is the immense range of different types of tests possible for a given formulation. One strategy with which to achieve a workable subset is to focus on the effects of specific parameters, for example, features of the geometric model such as resolution and accuracy. One can also select a specific type of source according to a particular application or related set of applications. It may be adequate to detect and localize a particular feature of the source that does not require an entire heart beat as, for example, when attempting to locate ischemic regions of the heart[1,2] or sites of earliest epicardial activation for diagnosis of ectopic arrhythmias.[3] What is essential to note in any validation strategy is that results achieved in one configuration or for one instant in time do not necessarily extend to other conditions or times.[4]
We continue our discussion with a review of a number of validation approaches that have been previously employed for electrocardiographic inverse problems. We organize these approaches in terms of physical models, purely computational techniques, and clinical experiments. The goal of the discussion is to make the reader aware of the specific requirements of validating electrocardiographic inverse solutions, and to suggest a range of feasible approaches that overcome at least some of the obvious obstacles to human validation.
We apologize at the outset to the authors whose work we do not cite. The
literature on validation strategies for electrocardiography spans at least
60 years and we will inevitably miss some of it.