Figure 2 indicates the accuracy with which high-density potential maps can be reconstructed from clinical ground truth electrograms recorded during the relatively uniform spread of LA activation in coronary sinus pacing. Key features of activation were reconstructed from intra-atrial potentials sampled with a virtual 64-electrode noncontact catheter that occupied ∼67% of atrial cavity volume (Figure 2A). However, neither high resolution features of instantaneous potential maps (Figure 2B) nor high frequency components of regional electrograms (Figure 2F) were captured faithfully. Despite this, accuracy measures were high and surprisingly stable across a wide range of catheter dimensions with median CC = 0.92, nRMSE = 0.054 and ΔT = 1 ms for catheter:atrial volume ratios >0.3 with a 64-electrode basket catheter (Figures 2D–F). None of these measures improved with full or near contact between electrodes and endocardial surface, and nRMSE was highest between splines where the spacing of adjacent electrodes was greatest (Figure 2H), indicating that spatial distribution of electrodes on the surface that bounds the catheter is the problem here. Sampling density was increased by moving the catheter and synchronizing the electrograms acquired. The instantaneous potential map in Figure 2C was reconstructed from 192 electrograms “recorded” in 3 sequential steps by rotating the virtual 64-electrode catheter (relative volume ratio 0.67) around its axis in increments of 15°. This markedly improved the match between high-density ground truth and inverse maps (compare Figures 2A,C). The nRMSE between electrodes was substantially reduced (compare Figures 2H,I) and high frequency components of complex local electrograms were reconstructed accurately (Figure 2F). Median CC increased to ∼0.97 and median nRMSE halved across a wide range of catheter dimensions (Figure 2G).

In Figure 3, we present the effects of catheter dimension and electrode distribution on inverse solutions obtained with the MFS in simulated macroscopic reentrant activation of the LA that replicates features of atrial flutter. Catheter designs considered are a 64-electrode catheter with 8 uniformly spaced electrodes along 8 splines and a 130-electrode catheter that has 8 uniformly distributed electrodes on 16 splines with 2 additional polar electrodes. Once again, the centroids of catheters and LA chamber were aligned.

In the upper panel of Figure 3, an instantaneous ground-truth potential map (Figure 3A) is compared with corresponding inverse maps constructed from potentials sampled with a 64-electrode catheter (Figure 3B) and a 130-electrode catheter (Figure 3C). Endocardial potentials were poorly reconstructed in some regions of the LA with an 8 spline 64-electrode basket catheter but recovered more faithfully with a 16 spline 130-electrode catheter where electrode distribution is more uniform, spatially. As might be expected, errors with the 64-electrode catheter were greatest between splines near the equator where inter-electrode spacing was largest.

The correspondence between ground-truth and reconstructed surface electrograms was quantified for these two catheters over 3 consecutive activation cycles for a range of catheter dimensions, and results are presented in the lower panel of Figure 3. The accuracy with which unanchored reentrant rhythm could be reconstructed was consistently less than for more stable paced rhythms (compare Figure 3 with Figure 2) and it was affected more markedly by relative catheter dimensions. For each of the 3 metrics considered, performance was better at all catheter dimensions with 130-electrode catheters than with the 64-electrode catheters. For example, with 130-electrode catheters, CC approached a median of 0.97 [IQR 0.07] as catheter dimensions were increased, compared with corresponding values around 0.9 [IQR 0.19] with 64-electrode catheters (Figure 3D). Consistent with these results, nRMSE was reduced with increased catheter dimension reaching a median of 0.042 [IQR 0.055] for the 130-electrode catheter compared with 0.083 [IQR 0.099] for 64-electrode catheters (Figure 3E). Finally, ΔT was reduced to a median of 1 ms with a 130-electrode catheter compared with 2 ms for 64-electrode catheters (Figure 3F). All three metrics were relatively stable for catheter volumes >0.6 relative to LA volume.

The effects of noise on the accuracy of inverse endocardial potentials reconstructed with the MFS are summarized in Figure 4. Intra-atrial electrograms were “sampled” with 130-channel catheters during simulated macro-reentry with superimposed Gaussian noise at RMS voltages of 18, 56 and 178 µV. In general, addition of noise had little effect for catheter: LA volume ratios >0.5. However, inverse solutions were progressively degraded by noise at catheter volumes less than this (Figures 4B–D). Comparison of the representative electrograms in Figure 4A provides further insight into this finding. While SNR in reconstructed electrograms scales inversely with added noise, it is much worse for the smaller of the two catheters (6.54, 4.23 and 1.91 for RMS noise voltages of 18, 56 and 178 μV, respectively, compared with 69.56, 22.85 and 6.83 for the larger catheter). It is also noteworthy that while our inverse solutions do not recover higher frequency components in the ground truth electrograms when the catheter is small this is not systematically altered by noise.

While potential maps during macro reentrant activity were reconstructed more faithfully using a 130-electrode catheter than a 64-electrode catheter with the same dimension, corresponding phase maps in Figure 5A appear to carry very similar information about the history of activation across the LA surface. This similarity in phase distribution was preserved throughout an extended sequence of simulated electrical activity (Supplementary Video S1), where phase singularities identified recovered with 64- and 130-electrode catheters are also collocated. The correspondence with ground truth for phase maps obtained with noncontact catheters was maintained for a wider range of relative catheter volumes than for the potential maps in Figure 3 above. However, CC was increased and nRMSE reduced with a 130-electrode catheter compared to a 64-electrode catheter (Figure 5B). This indicates that phase maps with the latter capture key features of wave front propagation in macro reentrant arrhythmia, but aspects of the fine structure of the phase distribution are lost.

With region-of-interest (RoI) mapping, a small catheter is positioned close to a region of the endocardial surface to reconstruct local electrical activity. In Figure 6, we consider the accuracy with which regional electrical activity can be recovered using non-contact catheters. This analysis was completed without adding Gaussian noise. An 8-spline 64-electrode basket catheter (major and minor axes 25 and 23 mm, respectively) was initially located close to the origin of a simulated macro-reentrant circuit in the LA (Figure 6A). The inverse solution (Figure 6B) was good near the catheter, but much poorer over the rest of the LA. This is demonstrated in Figure 6C where CC is rendered on the LA surface; median CC is >0.9 in the RoI, but falls off rapidly with distance from this region. In the lower panel we present CC (Figure 6D), nRMSE (Figure 6E) and ΔT (Figure 6F) in the RoI (red) and across the full endocardial surface (blue) for inverse solutions constructed as the catheter was moved progressively along a line from the origin of the LAA to the inter-atrial septum (Figure 6A). These figures demonstrate that regional mapping performance was excellent when the catheter was in or adjacent to the RoI, but poor when the catheter was most distant from it. Global mapping performance was best when the catheter was located centrally, but significantly poorer in this case in the RoI. Of particular interest, RoI performance was optimal when the catheter was ∼10 mm from its initial position with electrodes 9–20 mm from the LA wall; median CC was 0.96 [IQR 0.072], median nRMSE 0.09 [IQR 0.05] and median ΔT 0.89 ms [IQR 1.97 ms].

This analysis of noncontact intracardiac potential mapping extends an in-silico boundary mesh-based study previously reported by our laboratory (Meng et al., 2017; Meng et al., 2022). Here we have investigated the accuracy with which endocardial potential maps can be reconstructed from noncontact electrograms recorded with a multi-electrode basket catheter using meshless methods that use the MFS, the first time that this has been done as far as we are aware. We demonstrate that fast, accurate noncontact potential mapping and phase mapping are possible using this approach. However, the spatial frequency of the electrical activity captured is determined by the distribution of electrodes and in order to recover complex non-stationary rhythms, such as AF, the mapping catheter must address a representative subvolume of the cardiac chamber.

We have reported that noncontact mapping performance deteriorates progressively as catheter dimensions are reduced relative to those of the cardiac chamber and that this degradation becomes more marked when activation is complex. Neither of these findings is surprising. With increasing distance from the heart surface, intracardiac potentials associated with local activation are progressively attenuated and blurred, and information lost in this process cannot be recovered fully. Furthermore, as catheter dimension is reduced, information is captured from a decreasing subset of the cavity volume which may not fully reflect local activity. More striking, perhaps, is the finding that surface electrograms can be reconstructed with acceptable accuracy (CC > 0.9, nRMSE ≤0.06 and ΔT ≤ 2 ms) during reentrant rhythm with basket catheters that fill only half of the cavity. A supplementary point that needs to be made here, is that while relative catheter volume is an accessible measure of dimension, it scales with the third power of radius for a spherical basket catheter. Therefore, catheter volume increases by 112.5% when its diameter changes from 35 to 45 mm. While there was no contact between electrodes and LA wall for the centrally located catheter in Figure 3A (Catheter:Atrial volume ratio = 0.67), there was increasing (though incomplete) contact between them as relative volume expanded to ∼0.9.

Median CC was decreased and median ΔT was increased with reduced catheter dimension (Figures 4B,D) when Gaussian noise was added but there was no corresponding effect on median nRMSE (Figure 4C). The reconstructed electrograms in Figure 4A provide explanation for these results. Because electrograms recorded toward the centre of the LA cavity with a small central catheter are attenuated, the noise added to them markedly reduces SNR. This is reflected in the reconstructed surface electrograms presented in the left-hand panel of Figure 4A, where SNR is low and is reduced progressively as noise amplitude increases. The recorded electrograms are also smoothed in this case and the frequency content of the ground-truth surface electrograms is not recovered by inverse mapping. However, the overlap between ground-truth and reconstructed electrograms is affected less by noise than might have been expected. With increased noise power, the deviation between reconstructed and ground-truth electrograms can increase, but there is also greater instantaneous overlap between the two. With a large catheter (right-hand panel in Figure 4A), the magnitudes of recorded electrograms are substantially greater and there is much less smoothing. As a result, surface electrograms are recovered more faithfully with much less impact of added noise.

Inverse methods are prone to instability and error in the presence of noise. The fact that this is not the case here further reinforces the fact that the transfer matrix used is inherently well-conditioned and the regularization procedures adopted are appropriate. However, the Gaussian noise introduced here is a very narrow representation of the problems faced in practice with inverse potential mapping. Artifacts in the unipolar signals used for this purpose include common-mode electrical noise, far-field activity due to electrical activation of the ventricles which can mask local activity completely in AF and complexity in atrial electrograms that may be due to far-field atrial activity, inadequate spatial sampling or non-uniform electrical properties in the underlying substrate. That said there are many ways that more robust regional information can be extracted from these channels using signal processing methods that exploit temporal and spatial correlation among adjacent electrograms and wavelet-based methods which identify characteristic differences in the instantaneous frequency content of recorded electrograms (Zhao et al., 2013).

Our data show that ground truth potential maps based on simulated macro-reentrant activity were reconstructed more faithfully using a 130-electrode catheter than a 64-electrode catheter with the same dimension (Figure 3). Furthermore, when noncontact electrodes were within a few mm of the cavity surface, dimension had no effect on the efficacy of the inverse solution, which was wholly dependent on electrode distribution. This reflects the fact that the accuracy with which surface potentials can be reconstructed depends on whether the sampled potentials provide a faithful representation of the field addressed. If the electrode distribution is not sufficiently dense, high spatial frequencies cannot be recovered and low frequency artifacts (aliasing) may occur (Shannon, 1949). An example of this is provided in Figure 2 where apparent fractionation of the electrogram reconstructed between adjacent splines with a 64 electrode basket catheter (see blue trace in Figure 2F) disappears with more dense sampling in that region (compare red trace with ground truth electrogram in Figure 2F). While compressed sensing approaches can collect and represent sparse signals with many fewer sampling points than indicated by the Nyquist (Shannon, 1949) theorem, optimal sampling strategies are determined by regional spatial and temporal correlation (Long et al., 2011). Specialized regularization techniques are also needed for inverse reconstruction of higher frequencies from sparsely sampled signals without introducing excessive noise.

It should be noted that metrics such as CC and nRMSE used here to quantify the correspondence between ground-truth and reconstructed potentials do not take the time-history of the electrograms into account. This provides additional information about the spread of electrical activation across the heart surface as illustrated by the phase maps in Figure 5. The phase map shown here for a 64-electrode catheter is very similar to that presented for a 130-electrode catheter. Both correspond closely to ground truth phase maps throughout the activation sequence (Supplementary Video S1). Because phase mapping identifies local activation as an abrupt standardized transition from +π to −π and imposes relatively uniform spatio-temporal variation around this, confounding effects of local variation in potential magnitude are removed.

It is also important to acknowledge that sampling density is affected by catheter dimensions. For instance, for a 64-electrode 25-mm diameter spherical catheter, inter-electrode spacing along splines is ∼5 mm, while the curvilinear distance between splines at the equator is ∼10 mm. These measurements are doubled with a 50-mm diameter catheter. This explains the apparent reduction in median CC and the increase in median nRMSE and its interquartile range for 64-electrode catheters when relative catheter volume increases from 0.8–0.91 (Figures 3D,E, respectively). Here any improvement in accuracy associated with proximity to the wall is offset by reduced electrode density. In the clinical setting, attempts to achieve direct contact between electrodes and the heart surface can introduce additional error by deforming catheter splines and increasing the nonuniformity of sampling. (Oesterlein et al., 2016; Pathik et al., 2018). It follows that global mapping with multi-electrode basket catheters is more likely to produce reliable results when electrodes are not in contact with the heart wall than when attempts are made to achieve close contact.

The results for RoI mapping are consistent with these observations. The relatively small 64-electrode catheter in Figure 6 recovered local electrical activity with a high level of accuracy. Moreover, regional mapping produced best results when electrodes were not in contact with the wall (median CC, nRMSE and ΔT: 0.96, 0.09 and 0.89 ms, respectively). Global performance was much poorer, but it would be straightforward to quantify the uncertainty of reconstructed maps based on the distance of electrodes from surface nodes and the results of analyses such as those outlined here.

The results of this study indicate that global electroanatomic maps can be recovered faithfully in real-time from electrograms recorded with noncontact multi-electrode basket catheters using meshless methods that use the MFS. Accurate specification of 3D electrode locations with respect to cardiac anatomy is required for inverse intracardiac mapping, but this is readily achieved with current hybrid navigation technologies (Issa et al., 2019). Our findings indicate that, for optimal performance, catheters should be located centrally within the cardiac chamber and address a representative subvolume of the cavity (>50% in the data presented here) with minimum contact between electrodes and endocardial surface. The capacity to reconstruct spatially complex activation patterns is limited by electrode distribution, but when heart rhythm is stable and repeated, more detailed maps can be reconstructed with sequential alteration of electrode locations, for instance by catheter rotation/translation. Potentially, this could be more efficient than sequential contact mapping with high density contact arrays because complete maps can be developed with relatively few iterations. For nonstationary heart rhythms such as AF, however, the accuracy with which endocardial surface activation can be reconstructed is constrained by the spatial distribution of electrodes on the catheter for both contact and noncontact mapping. Sparse sampling can lead to repeating artifact in reconstructed activation patterns that is incorrectly identified as rotors (Roney et al., 2017; Martinez- Mateu et al., 2018). Williams et al. (Williams et al., 2018) reported that >1.0–1.5 points/cm² were needed on the endocardial surface to resolve spiral wave activity and this corresponds to an inter-electrode spacing of 2–3 mm - much denser than is the case for 64-electrode basket catheters, particularly for equatorial electrodes on adjacent splines. As demonstrated here, more optimal electrode distribution is achieved with catheters that have 16 rather than 8 splines. While phase mapping may relax sampling requirements to some extent, it seems evident that improved catheter design is necessary for accurate panoramic mapping in AF.

Inverse methods have been used for noncontact intracardiac electrical mapping in two commercial systems. (Schilling et al., 1999; Grace et al., 2019). The Ensite multi-electrode array (Abbott) is used for noncontact potential mapping and consists of 64 electrodes mounted on an inflatable balloon. Consistent with the results reported here, validation studies have shown that accuracy is inversely related to the distance between the array and the heart wall with poor recovery of endocardial surface potentials when this distance is >25 mm (Earley et al., 2006). More recently, instantaneous charge density distributions associated with atrial electrical activation have been constructed from electrograms recorded with noncontact 48-electrode basket catheters (Acutus/Biotronik) (Grace et al., 2019). This is based on a forward model that relates intracardiac potential fields to secondary cellular sources associated with distributed membrane charge dipoles (Plonsey, 1982; Grace et al., 2019; Willems et al., 2019). Our analysis makes no assumptions about the cellular basis of electrical activation. Instead, we address how well endocardial surface potentials can be recovered from a limited number of electrograms recorded inside the heart cavity. We found that information is lost with noncontact mapping if the basket catheters used are too small and with both contact and noncontact mapping if the sampling density is not sufficient. These factors would be expected to impact the spatial resolution with which surface charge distributions can be recovered from noncontact electrograms also.

A limitation of this study is that error introduced by uncertainty of the 3D geometry of the heart surface relative to 3D electrode locations has not been explicitly considered. While this would be expected to amplify uncertainty associated with relative catheter size, electrode distribution and noise, we note that our formulation of the intracardiac inverse problem is surprisingly robust. A further limitation is that although our ground-truth data represent atrial rhythms of increasing complexity, they do not fully replicate the spatio-temporal disorder which characterizes AF. However, the analysis presented here demonstrates that the performance of contact mapping is matched by this inverse approach and that spatial resolution in both cases is limited by electrode distribution. Finally, while ground-truth data are based in part on clinical and simulated data, the accuracy of inverse intracardiac mapping has been confirmed computationally. While many of the assumptions made in specifying the forward problem are entirely reasonable, other are not. The electrical properties of the blood in the cardiac chambers are isotropic and uniform but they are certainly not the same as those in the myocardium adjacent to the endocardial surface where the fictitious sources are located. More detailed experimental characterization of the accuracy with which endocardial potentials can be reconstructed using inverse mapping is therefore needed to confirm the analyses presented here.