Analytical and Computational Validation Models

The traditional method of validating a simulation is to identify a problem that is simple enough that an analytical or closed form solution exists. Then one can compare a numerically computed solution with the analytically known true result. At a minimum, comparison of results from the analytical and numerical solutions can be useful for identifying fundamental problems with either the concept or the implementation of the more general numerical approach. Because of their simplicity, however, analytical solutions may provide data that establish only a useful set of validation criteria, but clearly not a sufficient set. Any tenable numerical approach must mimic the analytical result, but this is not enough to ensure accuracy of the numerical approach under other, more realistic conditions. Computational cross validation may also be possible in cases in which two separate numerical or computational approaches can solve the same physical problem. Here, too, the range of application of one of the computational approaches may be limited so that validation is incomplete. For example, one can implement a discrete source simulation to generate potentials on both heart and torso surfaces and use the results to cross validate a different inverse solution that computes heart potentials from torso potentials for any source. The surface-to-surface solution is then validated only for the dipole source and not for the broad range of applications for which it was conceived. We provide more examples of both these approaches in this section.

One significant advantage of most analytical or computational validation approaches in inverse problems is the relative ease with which one can control the essential factors such as geometric accuracy or level of detail and--sometimes to a more limited extent--the source configuration. In general, on the other hand, the geometry must remain simple in order for the analytic solution to exist. Even for simple shapes, it is sometimes possible to vary aspects of the geometry to at least begin to approach some aspects of realistic conditions and thus permit some general statements about the associated cardiac conditions. For example, one can use sets of nested spheres to represent external and internal boundaries in the volume conductor and then vary the conductivity assigned to each region. Rudy et al. have shown that one can also impose some eccentricity to the position of the spheres and thus suppress some of the complete symmetry that limits concentric sphere models.[5,6] The ease with which one can handle the second, more difficult requirement for validation-the availability of source values and remote measured potentials-is the main attraction of these methods; however, fidelity to biological conditions is often quite limited.

The best known analytically tractable models in electrocardiology are the concentric and eccentric spheres models proposed by Bayley and Berry,[7,8,9] developed extensively by Rudy et al.,[6,10,11] and still in use today.[3,12,13,14,15,16] Rudy et al. used these models for more than simply validating their numerical solution. By varying source types and locations and geometric model eccentricity, error, and conductivity, as well as regularization functionals used for the inverse solutions, they developed several more fundamental hypotheses about the relationships between each of these factors and inverse solution accuracy. Some of these ideas have subsequently been validated by physical models and human clinical studies. However, the basic limitation of analytical models based on simple geometries remains the uncertainty about the influence of real anatomical structures. This, in turn, restricts the application of conclusions drawn from such studies to realistic physiologic situations.

One validation approach that permits more realism in the geometric model is to begin with simulated or measured source data, and then place them in a realistic, discrete geometric model. These sources and a numerical forward solution then provide synthetic torso data to which one can (must) add noise, and then apply inverse solutions. Validation then consists of comparing the inverse solutions to the known sources. The simulated source data can, for example, be calculated from dipole models,[17,18] taken from the literature,[19] measured in open chest or torso tank experiments[20,21,22,23,24,25] (see next section), or calculated by an initial inverse solution from measured torso surface data.[26,27,28,29,30]

The justification for this rather suspiciously circular validation strategy comes from the fact that electrocardiographic inverse solutions are much less accurate than the associated forward solutions. The main challenge of the inverse problem, in fact, is to succeed in restoring some portion of the accuracy of the associated forward solution, despite the fact that the inverse problem, unlike the forward problem, is ill-posed. Thus it is reasonable to validate an inverse solution with data that best matches the forward solution. Based on validation studies with analytic solutions,[11] it is probably safe to assume that the errors that arise in constructing a geometric model derived from a patient or animal play a much larger role than those from the numerical methods used in the forward solution. Additional errors such as movement during mechanical systole simply compound the problem. By forward computing the torso potentials, one can reduce--or at least control--errors due to geometry because these errors are present in the forward solution. This strategy makes it possible to focus the validation studies on the errors that arise in the step of creating the inverse solution from the forward solution, which is the major challenge in electrocardiographic inverse problems.

One weakness of this approach is that it neglects the effects of any errors in the problem formulation, i.e., the equations used to describe the relationship between sources and remote potentials. This omission occurs because the same problem formulation is used in both the forward and inverse solutions. There is a hybrid approach that uses dipole sources to calculate both epicardial and torso surface potentials directly based on a realistic geometry, but uses an explicit epicardial to torso surface forward model, based on a different formulation, to generate separate surface-to-surface forward and inverse solutions. Because the two formulations are different, the dipole source forward model does not match exactly the surface-to-surface model. One can then compare the directly against the inversely computed epicardial potentials and perform cross validation.[18,31] This model mismatch can reduce at least some of the symmetry inherent in other computational approaches and potentially provide less biased validation conditions, but, of course, is limited to simple dipolar source models.

A more realistic hybrid approach is that described by Hren et al., in which the source was not a dipole, but a cellular automata model of cardiac propagation.[32,33,34] Cellular automata models represent cardiac tissue as a regular mesh of ``cells'', each representing a region of approximately $1~{\rm mm}^{3}$ and use simplified cell-to-cell coupling and state transition rules to determine the activation sequence. Although not as detailed as the monodomain and bidomain models that are based on descriptions of membrane kinetics, cellular automata models have a long history in simulations of normal and abnormal cardiac activation.[35,36,37,38,39,40,41,42,43,44,45] In order to apply cellular automata models to validation of electrocardiographic inverse solutions, Hren et al. developed a method with which they assigned electrical source strength to the activation wavefront and were thus able to compute epicardial and torso potentials from the cellular automata model for realistic geometric models of the human torso.[32,46] They used this technique to both validate inverse solutions[33] and to examine the spatial resolution of body surface mapping.[47,48]

An obvious, significant strength of the computational and analytical approaches is that they do not require the expense and extensive infrastructure of experimental studies, but can all be performed on a computer using methods that are well described in the literature. Moreover, as the examination of the results leads to new hypotheses to test, requiring new data, it is obviously easier to repeat a computational experiment after modifying the conditions or changing parameters than it is to have to modify and repeat experimental studies.