Both the RE and the CC comparisons of the first set of simulations demonstrate the effects of adding inhomogeneities to the model. Note that in all cases, the average error did not get within 10% of the error between homogeneous and fully inhomogeneous models, suggesting that no single inhomogeneity can serve as surrogate for the complexity of the full model. Furthermore, only at anisotropy ratios at or above 10:1 did skeletal muscle play a significant role compared to the lungs and the subcutaneous fat. A further finding of note was that while fat located near the torso surface had a strong effect, fat with the same conductivity located on the epicardial surface exerted relatively minor influence. We note, however, that the fat pads used for this study covered only a portion of the epicardium and were quite small (from approximately 20 x 15 mm to 45 x 30 mm) compared to the sizes of the fat pad(s) seen in some patients. One would expect that as the size of a fat pad increased, it would play a more significant role in the overall rankings. Curious also was the finding that the inclusion of each lung individually resulted in slightly larger deviation from the homogeneous model than the combination of both lungs. Hence, a simple additive relationship among the inhomogeneities would seem unlikely. This finding is further supported by the result that adding some individual inhomogeneities resulted in larger deviations from the homogeneous torso than the full model, at least for a few instants over the course of the QRS. For example, in Figure 6 we see that the right lung had greater RE values for several time instants than the fully inhomogeneous case and in Figure 7 the left lung had the largest RE at one time instant.
While the results of selective removal of inhomogeneities reinforced some findings observed when individual inhomogeneities were added, they also revealed important differences. Common to both sets of results was that the effects of an inhomogeneity depended on the field distribution and hence changed sharply from instant to instant during the QRS. Likewise, the inhomogeneities that were located near the bottom of the average error plots--and thus lowest in order of influence--included the bone and individual fatpads in both sets of studies. Among the contrasts between addition and removal of inhomogeneities were the stronger influences of the removal of the 7:1 anisotropic muscle compared to the effects of adding it. Figures 9 and 16 reveal that the individual removal resulted in a relative error of about 14% while their addition to a homogeneous model changed the RE by less than 10%. The difference between removal and addition of inhomogeneities is further demonstrated by the result that in no case did the removal of a single inhomogeneity result in error appreciably greater than the homogeneous torso (although subcutaneous fat had slightly greater error in two time instants). The time course of the effect of an inhomogeneity also differed between addition and removal of single inhomogeneities. For example, addition of the great blood vessels played only a small role throughout the QRS, as shown in Figure 6; removal of the same inhomgeneity, on the other hand, resulted in fluctuating relative errors ranging from 2% to almost 20% over the course of the beat, as shown in Figure 15.
The shape of the histograms in Figures 9 and 16 also reveal differences between addition and removal of single inhomogeneities from the model. While the distribution in Figure 9 is relatively smooth for the addition of each individual inhomogeneity, the distribution in Figure 16 can be divided into three bands with clear differences in relative error between each band. From this, one can see that removal of subcutaneous fat, skeletal muscle (at anisotropy 7:1), and the lungs had the largest impact, with all the fatpads, fatpad 1 and the blood vessels at less than half the relative error and the other fatpads and bone having almost no influence on the accuracy of the forward solutions. A similar banding pattern is also seen in Figure 19.
The plots of the BSPMs from specific instants in time serve to provide some sense of the spatial differences that arose when the model conductivity was varied. In all the cases shown in Figures 13 and 20, the effects of addition or removal of even the single most important inhomogeneity did not change the coarse features of the maps; the maxima and minima remained in approximately the same locations and the overal topography of the maps was stable. The nature of the changes observed was in the shape of the features and in their amplitudes, with peaks and valleys typically becoming sharper as more complexity was added to the model. The findings have important consequences for the interpretation of our quantitative results and their practical application to forward and inverse problems. In many cases, the level of accuracy offered by even a homogeneous model may be adequate to reveal general features of the torso potentials from known epicardial signals. It is perhaps even more important to compare the level of error induced by a lack of fidelity in conductivity, as we have done, to that arising from lack of geometric resolution. A study that incorporates both these sources of error in a systematic manner with realistic torso geometry does not, to our knowledge, exist.
In the context of relevant results from the literature, our results support some aspects of previous studies [30, 11, 31, 12], with significant enhancement due to our ability to model anisotropy directly. A fundamental difference of this study from all others to our knowledge is that we computed the forward solution, i.e., the torso potentials from known epicardial potentials, not the inverse solution. Early studies by Messinger-Rapport and Rudy using an eccentric spheres model [30] suggested that the influence of idealized ``lungs'' and isotropic ``muscle'' layers was exclusively a change in amplitude of computed surface potentials with no shift in the topography of the distribution. While the simplicity of the geometry in this study permitted analytic solutions to the forward and inverse problem, it suffered from unrealistic symmetry and complete containment of the heart by the single, concentric lung layer. Stanley et al. conducted studies based on digitized models of an animal in which epicardial and torso surface potentials had been measured simultaneously [11] and found that skeletal muscle had the largest role in the accuracy of computed inverse solutions. Their numerical method, however, did not permit direct inclusion of anisotropy so they used a method suggested by McFee and Rush in which the isotropic muscle layer is artificially expanded radially by an amount related to the degree of anisotropy [32]. Their model also lacked subcutaneous fat, but did suggest that inhomogeneities close to the outer surface played the largest role in model accuracy, in general agreement with our results.
Among studies based on human data, reports by Walker and Kilpatrick [31] and Purcell et al. [33] noted the dominant influence of the lungs in inverse solution accuracy, as well as the relatively minor contribution of bone, again in qualitative agreement with our results. On the other hand, studies by Van Oosterom and Huiskamp on an inverse model based on the epicardial and endocardial activation sequence [12] suggest that removal of the lungs added perhaps a few percent to the relative error of their inverse solution. Much more critical were the blood filled chambers of the ventricles, which are unique to their model formulation and hence play no role in our simulations.
While our results suggest that accurate descriptions of tissue conductivity improves the quality of the simulations, this is not the only component of a complete forward solution. The level of discretization in the geometric model clearly plays a role in simulations and we have used 168,000 nodes and nearly one million elements for the results presented here. To evaluate, at least coarsely, the effects of node spacing, we created a second homogeneous model based on the same geometry as the original model but with only 62,000 nodes and then repeated the simulation of BSPMs. The results differed only slightly from those using the more finely meshed model (mean CC=.9991 and mean RE=4.8%). Although only a complete study of the effects of geometry on model results would confirm these results generally, the larger differences found due to conductivity changes suggests that our results are reasonably independent of model geometry.
None of the studies of inhomogeneities known to us have included the level of detail in the geometrical model or the range of accurately modeled tissues we have employed. Hence, while general concurrance of a few of our findings with previous investigations is encouraging, we feel our findings contain unique contributions to the modeling of bioelectric fields in realistic conditions.