ORIGINAL RESEARCH published: 16 May 2022 doi: 10.3389/fphys.2022.873049 Intracardiac Inverse Potential Mapping Using the Method of Fundamental Solutions Shu Meng 1, Nicholas Sunderland 1,2, Judit Chamorro-Servent 3, Laura R. Bear 4,5,6, Nigel A. Lever 1,7,8, Gregory B. Sands 1, Ian J. LeGrice 1,9, Anne M. Gillis 10, Jichao Zhao 1, David M. Budgett 1 and Bruce H. Smaill 1* 1 Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand, 2Bristol Heart Institute, University of Bristol, Bristol, United Kingdom, 3Department of Mathematics, Universitat Autònoma de Barcelona, Bellaterra, Spain, 4IHU Liryc, Electrophysiology and Heart Modeling Institute, Fondation Bordeaux Université, Bordeaux, France, 5Centre de Recherche Cardio-Thoracique de Bordeaux, Univ. Bordeaux, Bordeaux, France, 6Centre de Recherche Cardio-Thoracique de Bordeaux, INSERM, Bordeaux, France, 7Auckland City Hospital, Auckland, New Zealand, 8Department of Medicine, University of Auckland, Auckland, New Zealand, 9Department of Physiology, University of Auckland, Auckland, New Zealand, 10Libin Cardiovascular Research Institute, Calgary University, Calgary, AB, Canada Edited by: Introduction: Atrial fibrillation (AF) is the most prevalent cardiac dysrhythmia and Gernot Plank, percutaneous catheter ablation is widely used to treat it. Panoramic mapping with Medical University of Graz, Austria multi-electrode catheters can identify ablation targets in persistent AF, but is limited by Reviewed by: Vincent Jacquemet, poor contact and inadequate coverage. Université de Montréal, Canada Peter Michael Van Dam, Objective: To investigate the accuracy of inverse mapping of endocardial surface University Medical Center Utrecht, potentials from electrograms sampled with noncontact basket catheters. Netherlands Edward Joseph Vigmond, Methods: Our group has developed a computationally efficient inverse 3D mapping Université de Bordeaux, France technique using a meshless method that employs the Method of Fundamental Solutions *Correspondence: Bruce H. Smaill (MFS). An in-silico test bed was used to compare ground-truth surface potentials with b.smaill@auckland.ac.nz corresponding inverse maps reconstructed from noncontact potentials sampled with virtual catheters. Ground-truth surface potentials were derived from high-density clinical Specialty section: This article was submitted to contact mapping data and computer models. Cardiac Electrophysiology, a section of the journal Results: Solutions of the intracardiac potential inverse problem with the MFS are robust, Frontiers in Physiology fast and accurate. Endocardial surface potentials can be faithfully reconstructed from Received: 10 February 2022 noncontact recordings in real-time if the geometry of cardiac surface and the location of Accepted: 19 April 2022 electrodes relative to it are known. Larger catheters with appropriate electrode density are Published: 16 May 2022 needed to resolve complex reentrant atrial rhythms. Citation: Meng S, Sunderland N, Conclusion: Real-time panoramic potential mapping is feasible with noncontact Chamorro-Servent J, Bear LR, Lever NA, Sands GB, LeGrice IJ, intracardiac catheters using the MFS. Gillis AM, Zhao J, Budgett DM and Smaill BH (2022) Intracardiac Inverse Significance: Accurate endocardial potential maps can be reconstructed in AF with Potential Mapping Using the Method of appropriately designed noncontact multi-electrode catheters. Fundamental Solutions. Front. Physiol. 13:873049. Keywords: atrial arrhythmia, multi-electrode basket catheters, method of fundamental solutions, inverse mapping, doi: 10.3389/fphys.2022.873049 endocardial potentials Frontiers in Physiology | www.frontiersin.org 1 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping INTRODUCTION Accurate identification of regions in the heart which trigger ectopic activation and sustain reentrant arrhythmia is a critical step in effective interventional treatment of heart rhythm disturbance. Sequential contact mapping with catheters introduced percutaneously into one or more heart chambers is widely used for this purpose (Issa et al., 2019), but it can be time- consuming and works poorly in atrial fibrillation (AF) where rhythm is non-stationary. While multi-electrode basket catheters have been used for panoramic mapping in AF (Narayan et al., 2012; Pathik et al., 2018), it is difficult to position them so that the electrodes are uniformly distributed across the atrial surface and in contact with it (Oesterlein et al., 2016; Martinez- Mateu et al., 2018; Pathik et al., 2018). In principle, inverse methods can be used to reconstruct potentials on the endocardial surface of a cardiac chamber FIGURE 1 | Schematic representation of the intracardiac forward problem. Fictitious electrical sources (open circles) distributed around a virtual from electrograms recorded at electrodes that are not in boundary ΓV outside the surface ΓH that bounds a heart cavity ΩH generate contact with it if three-dimensional (3D) chamber geometry is current flux within the domain. This contributes to potentials recorded specified, the locations of electrodes are known and the forward with electrodes (closed circles) on a basket catheter. The electrodes lie on the problem, which describes the transfer relationship between open surface ΓC . measured and endocardial surface potentials, is specified accurately. However, the boundary mesh-based solution methods used previously to solve the intracardiac potential characteristic features of surface potential variation with an inverse problem have shortcomings that are discussed in detail electrode distribution appropriate to resolve it spatially (Meng elsewhere (Meng et al., 2017; Meng et al., 2022). With the finite et al., 2022). We argue that the sampling constraints identified in element method (FEM), the transfer matrix is sparse, inherently this study apply to noncontact intracardiac mapping in general. ill-conditioned and time-consuming to evaluate (Pullan et al., 2005). On the other hand, boundary elements (BEMs) are not robust for measurement points near the heart wall, particularly MATHEMATICAL BACKGROUND when surface topology is complex as is commonly the case with the atria. (Pullan et al., 2005). Noncontact intracardiac potential mapping seeks to reconstruct Meshless methods (MMs) employing the Method of the potential distribution on the inner surface of a heart Fundamental Solutions (MFS) (Fairweather and Karageorghis, chamber from a discrete set of potentials recorded at known 1998) have been successfully used to solve the body surface points inside the chamber with a multi-electrode catheter. To potential inverse problem (Wang and Rudy, 2006). This solve this inverse problem, it is first necessary to formulate the approach is computationally efficient and robust in a uniform, corresponding forward problem. Here, we extend an approach isotropic domain, an assumption that is realistic for the followed by Wang and Rudy (Wang and Rudy, 2006), in which intracardiac problem. In this case, the MFS provides an the MFS was applied to inverse body surface potential mapping. inherently simpler representation of the forward problem than The formulation of the intracardiac forward problem is shown boundary mesh-based methods. in Figure 1. Potentials ϕ(xc ) are recorded at M points xc on the While there have been numerous systematic analyses of the surface ΓC that bounds the electrodes. A set of N fictitious sources efficacy of inverse body surface potential mapping (Ramanathan is positioned at locations {ξ i }N i1 along a virtual 3D boundary ΓV and Rudy, 2001; Cluitmans et al., 2017; Bear et al., 2018), there that lies outside the endocardial surface of the cardiac chamber. have been very few equivalent studies of intracardiac potential The linear combination of the Laplace fundamental solution over mapping (Meng et al., 2022) and no attempt to use the MFS in the sources on {ξ i }N i1 allows us to have an expression of the this setting. In the research reported here, we have used the MFS potentials in the source free volume ΩH contained in the cardiac to address the accuracy with which time-varying potentials on the chamber. It is assumed that 1) there are no electrical sources or endocardial surface of the left atrium (LA) can be reconstructed sinks within the heart cavity, 2) conductivity throughout the from electrograms recorded inside the chamber with basket domain is uniform and isotropic, 3) the electrical properties of the catheters where the electrodes may or may not be in contact basket catheter can be neglected, and 4) bioelectric processes are with the atrial wall. We conclude that accurate real-time quasi-static. panoramic potential mapping and 3D phase mapping are At any instant, the potential ϕ(x) at any point x in ΩH due to feasible with noncontact intracardiac catheters using the MFS. fictitious sources located on the virtual external boundary Γ V is However, to faithfully recover complex potential fields, such as those seen in AF on the endocardial surfaces of the atria, it is N ϕ(x)  a0 + i1 ai G(ξ i , x) (1) necessary to use catheters that are sufficiently large to capture Frontiers in Physiology | www.frontiersin.org 2 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping FIGURE 2 | In silico analysis of the effects of mapping catheter dimension and electrode distribution on MFS inverse solutions during pacing from distal coronary sinus. Ground-truth LA surface potential distributions during atrial activation are constructed from pace-synchronized contact recordings acquired with a high density electrode array. Corresponding potential fields in the LA cavity are estimated throughout the atrial activation cycle and “sampled” at locations of virtual basket catheter electrodes. Inverse surface potential maps are then reconstructed from these data and compared with ground truth maps. (A) Ground-truth LA surface potential distribution at one instant during activation - see red line in (C). Corresponding inverse potential map (B) reconstructed from potentials sampled with virtual 64-electrode basket catheter with 0.67 volume fraction relative to LA. Electrodes are distributed uniformly along 8 splines as indicated in (A). (C) Ground truth and reconstructed electrograms at 1 in (A,B). In the middle panel, CC (D), nRMSE (E) and ΔT (F) are presented as functions of relative catheter volume for inverse solutions over one activation cycle. Median values and IQR are given for inverse maps constructed from “recordings” at 64 sites (blue) and 3 sequential “recordings” at 64 sites following stepwise rotation of the catheter through 15° around its axis (red). Both maps in the lower panel relate to the same time as in (A,B). In (G), the distribution of nRMSE on the LA surface is compared with spline location for a potential map constructed from 64 full or near-contact “recordings”, while (H) corresponds to (B). Here, the inverse map was constructed from 3 sequential “recordings” at 64 sites with a relative catheter volume fraction of 0.67. (I) Ground truth and inverse electrograms at 2) in (H). Abbreviations: MFS, Method of Fundamental Solutions; LA, left atrium; CC, correlation coefficient; nRMSE, normalized root-mean-squared error; ΔT, activation time difference; IQR, interquartile range. where a0 is a constant and a  (a1 , . . . , aN ) is the instantaneous linear system of equations when they are equated to the measured source current at a source {ξ i }N i1 . G is the fundamental solution of potentials on the M electrodes of the catheter. It should be noted the Laplace operator in 3D that while the forward problem has been set up here for the 1 continuous surfaces ΓC and ΓH, this is not a requirement of the MFS. G(ξ, x)  (2) Solution of this system yields the associated current source 4π|ξ − x| densities on the fictitious external boundary Γ V and and {ξ i }N i1 are the 3D locations of the fictious sources and |ξ − x| corresponding potentials on the endocardial surface ΓH are is the Euclidean distance between x and ξ. Note that ai = σli for then estimated by using Eq. 1 again for x ∈ Γ H . This system is i = 1, N where li is the source current at ξi and σ is conductivity. inherently under-determined because the number of electrodes Potentials at xc on the surface ΓC that bounds the electrodes M is generally less than N, the number of fictitious sources needed therefore can be estimated by using Eq. 1 for x ∈ Γ c . This results in a to map potentials faithfully onto ΓH. Frontiers in Physiology | www.frontiersin.org 3 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping METHODS surface area was similar in both, with a contour adjacent to the boundary in the former assigned to the mitral valve orifice. An established computational approach (Ramanathan and Rudy, Extracellular potentials were approximated from the 2001) was used to quantify the accuracy of inverse potential transmembrane currents computed at each 3D point at a mapping and key steps are illustrated in Figure 2. First, ground- sampling rate of 1 kHz (Supplementary Video S1—Simulated truth potential distributions were specified on the endocardial ground truth data—in the Supplementary Material). surface of the left atrium (LA). The associated field throughout The open-source software environment SCIRun (http://www. the LA was determined by numerical solution of Laplace’s sci.utah.edu/cibc-software/scirun.html) was used for FEM equation and potentials were sampled at points corresponding solutions of the 3D forward problem. (Burton et al., 2011). to electrode locations on open basket catheters with different Intracardiac potential fields were computed from the ground- electrode distributions across a range of dimensions (Figure 2A). truth surface potential distributions at successive time instants by Endocardial surface potentials were then reconstructed from the solving Laplace’s equation throughout ΩH . The intracardiac field sampled potentials using the MFS (Figure 2B) and compared was sampled at points corresponding to electrodes on two basket with ground-truth potential distributions. This process was catheter configurations with 1) 64 channels with 8 equally spaced continued through a complete atrial activation cycle for electrodes along 8 splines at equal radial angles, and 2) 130 stationary rhythms or over several cycles of reentrant activity. channels with 8 equally spaced electrodes along 16 splines at Ground-truth electrograms were also compared with equal radial angles and electrodes at upper and lower poles. corresponding inverse estimates at points across the Basket dimensions were uniformly scaled to vary catheter: endocardial surface (Figure 2F) to assess the accuracy with atrial volume ratio. The centroids of catheters and the LA which local activation timing information is reconstructed. chamber were aligned to allow maximum catheter expansion Clinical ground truth data used in this study were acquired as and to ensure reproducibility between results. Noise was imposed follows. CT imaging was performed in one patient undergoing by adding Gaussian noise independently to the electrograms catheter ablation to treat atrial flutter. The patient gave written recorded at each electrode with power set at realistic levels. informed consent and the study protocol was approved by the Signal-to-noise ratio (SNR) is quantified as the ratio of root- Melbourne Health Research and Human Ethics Committee. LA mean-squared (RMS) voltages of reconstructed electrograms geometry was segmented with the Ensite VerismoTM tool and and noise. registered with respect to the mapping system (Ensite Precision, Inverse solutions with the MFS were run with purpose-written Abbott). A decapolar pacing catheter was positioned in the code. The fictitious boundary was formed by uniform scaling of the coronary sinus (CS). A 20-pole LassoTM catheter was atrial surface mesh and sources were associated with each node. introduced into the LA via trans-septal puncture and used to Inflation was quantified as the relative volume difference between ΓV collect 3,200 time-referenced, spatially-registered contact and ΓH. Solutions were stable across the inflation range 2–10% unipolar electrograms across the LA during pacing (300 ms (Supplementary Figure S1) and the value 6% was selected as interval) from the distal CS at a sampling rate of 1 kHz. The optimal in the results presented here. Inverse endocardial potential LA shell was refined to 5,000 vertices and potentials were distributions for intracardiac potentials “sampled” with virtual interpolated by Dirichlet energy minimization (Botsch et al., catheters were obtained using zero-order Tikhonov regularization 2010). (Tikhonov and Arsenin, 1977) employing the L-curve method to Ground truth data representing polymorphic reentrant atrial calculate the regularization parameter (Hansen, 2010). activation were simulated. Atrial surface geometry was Phase maps were constructed using the approach outlined by reconstructed in an anaesthetized sheep (crossbred female, Kuklik et al. (Kuklik et al., 2017) Sinusoidal recomposition was 53 Kg). All procedures were approved by the Animal Ethics applied to electrograms at each LA surface node and the Hilbert Committee of the University of Auckland and conform to the transformation was then used to estimate instantaneous phase at Guide for the Care and Use of Laboratory Animals (National these points. Institutes of Health publication no. 85-23). Gadolinium- Correspondence between ground-truth and reconstructed enhanced ECG-gated MRI images of the atria (1.0 × 1.0 mm2 potential maps were quantified by evaluating normalized RMS in-plane resolution approximately parallel to the atrio-ventricular error (nRMSE) and correlation coefficient (CC). valve plane and 1.6 mm between slices) were acquired with a 3T  Siemens Magnetom SkyraTM scanner. LA endocardial surface  N i i1 ∅GT − ∅R  i 2 geometry was segmented using Amira (Thermo Fisher Scientific) nRMSE  N 2 and CC and a 3D triangular surface mesh (1,529 nodes) was fitted to the i1 ∅iGT  LA with pulmonary veins and left atrial appendage truncated. N i i1 ∅GT − μGTN ∅iR − μR  Ground-truth potential distributions representing polymorphic    i1 (3) N N i 2 i 2 i1 ∅GT − μGT  i1 ∅R − μR  reentrant atrial activation were modeled on this geometry as follows. Meandering spiral wave reentry was simulated on an isotropic 2D monodomain with Fenton Karma activation kinetics where N is the number of surface points compared, ∅iGT and ∅iR (Fenton and Karma, 1998) using a standard cross-field S1-S2 are ground-truth and reconstructed potentials at surface point i, stimulus protocol (Pandit et al., 2005). Points on the 2D domain and μGT and μR are mean values for ground-truth and were sampled and mapped onto the 3D surface mesh so that reconstructed potentials, respectively, across the surface. Frontiers in Physiology | www.frontiersin.org 4 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping Activation times (ATs) for ground-truth (ATGT) and In the upper panel of Figure 3, an instantaneous ground-truth reconstructed electrograms (ATR) were estimated as maximum potential map (Figure 3A) is compared with corresponding negative rate of potential change and the activation time inverse maps constructed from potentials sampled with a 64- difference ΔT at each surface point was evaluated as electrode catheter (Figure 3B) and a 130-electrode catheter (Figure 3C). Endocardial potentials were poorly reconstructed ΔT  |ATGT − ATR | (4) in some regions of the LA with an 8 spline 64-electrode basket Programs were written in C or in the MATLAB programming catheter but recovered more faithfully with a 16 spline 130- language (The Mathworks, Natick, Massachusetts). 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 RESULTS where inter-electrode spacing was largest. The correspondence between ground-truth and reconstructed Inverse Potential Mapping in Stationary surface electrograms was quantified for these two catheters over 3 Rhythm consecutive activation cycles for a range of catheter dimensions, Figure 2 indicates the accuracy with which high-density potential and results are presented in the lower panel of Figure 3. The maps can be reconstructed from clinical ground truth electrograms accuracy with which unanchored reentrant rhythm could be recorded during the relatively uniform spread of LA activation in reconstructed was consistently less than for more stable paced coronary sinus pacing. Key features of activation were reconstructed rhythms (compare Figure 3 with Figure 2) and it was affected from intra-atrial potentials sampled with a virtual 64-electrode more markedly by relative catheter dimensions. For each of the 3 noncontact catheter that occupied ~67% of atrial cavity volume metrics considered, performance was better at all catheter (Figure 2A). However, neither high resolution features of dimensions with 130-electrode catheters than with the 64- instantaneous potential maps (Figure 2B) nor high frequency electrode catheters. For example, with 130-electrode catheters, components of regional electrograms (Figure 2F) were captured CC approached a median of 0.97 [IQR 0.07] as catheter faithfully. Despite this, accuracy measures were high and surprisingly dimensions were increased, compared with corresponding stable across a wide range of catheter dimensions with median CC = values around 0.9 [IQR 0.19] with 64-electrode catheters 0.92, nRMSE = 0.054 and ΔT = 1 ms for catheter:atrial volume (Figure 3D). Consistent with these results, nRMSE was ratios >0.3 with a 64-electrode basket catheter (Figures 2D–F). reduced with increased catheter dimension reaching a median None of these measures improved with full or near contact between of 0.042 [IQR 0.055] for the 130-electrode catheter compared electrodes and endocardial surface, and nRMSE was highest between with 0.083 [IQR 0.099] for 64-electrode catheters (Figure 3E). splines where the spacing of adjacent electrodes was greatest Finally, ΔT was reduced to a median of 1 ms with a 130-electrode (Figure 2H), indicating that spatial distribution of electrodes on catheter compared with 2 ms for 64-electrode catheters the surface that bounds the catheter is the problem here. Sampling (Figure 3F). All three metrics were relatively stable for density was increased by moving the catheter and synchronizing the catheter volumes >0.6 relative to LA volume. 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 Effects of Noise on Inverse Potential (relative volume ratio 0.67) around its axis in increments of 15°. This Mapping markedly improved the match between high-density ground truth The effects of noise on the accuracy of inverse endocardial and inverse maps (compare Figures 2A,C). The nRMSE between potentials reconstructed with the MFS are summarized in electrodes was substantially reduced (compare Figures 2H,I) and Figure 4. Intra-atrial electrograms were “sampled” with 130- high frequency components of complex local electrograms were channel catheters during simulated macro-reentry with reconstructed accurately (Figure 2F). Median CC increased to ~0.97 superimposed Gaussian noise at RMS voltages of 18, 56 and median nRMSE halved across a wide range of catheter and 178 µV. In general, addition of noise had little effect dimensions (Figure 2G). 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 Inverse Potential Mapping in Reentrant representative electrograms in Figure 4A provides further Rhythm insight into this finding. While SNR in reconstructed In Figure 3, we present the effects of catheter dimension and electrograms scales inversely with added noise, it is much electrode distribution on inverse solutions obtained with the MFS worse for the smaller of the two catheters (6.54, 4.23 and 1.91 in simulated macroscopic reentrant activation of the LA that for RMS noise voltages of 18, 56 and 178 μV, respectively, replicates features of atrial flutter. Catheter designs considered are compared with 69.56, 22.85 and 6.83 for the larger catheter). It a 64-electrode catheter with 8 uniformly spaced electrodes along 8 is also noteworthy that while our inverse solutions do not splines and a 130-electrode catheter that has 8 uniformly distributed recover higher frequency components in the ground truth electrodes on 16 splines with 2 additional polar electrodes. Once electrograms when the catheter is small this is not again, the centroids of catheters and LA chamber were aligned. systematically altered by noise. Frontiers in Physiology | www.frontiersin.org 5 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping FIGURE 3 | Effects of mapping catheter electrode distribution on inverse solutions with MFS during macro reentry. LA surface potentials throughout 3 activation cycles in simulated atrial flutter are reconstructed from electrograms sampled inside the LA cavity with 64- and 130-electrode basket catheters and compared with ground-truth surface potentials. The upper panel presents typical results for catheters that bound a volume fraction of 0.67 relative to LA volume. These include (A) the ground-truth surface potential distribution at one instant during reentrant activation and corresponding potential maps reconstructed using electrograms sampled with (B) a 64- electrode basket catheter, and (C) a 130-electrode basket catheter. In the lower panel CC (D), nRMSE (E) and ΔT (F) are presented as functions of relative catheter volume for the 64-electrode catheter (blue) and 130-electrode catheter (red). Median values and IQR are given. Abbreviations: MFS, Method of Fundamental Solutions; LA, left atrium; CC, correlation coefficient; nRMSE, normalized root-mean-squared error; ΔT, activation time difference; IQR, interquartile range. Inverse Phase Mapping in Macroscopic and minor axes 25 and 23 mm, respectively) was initially located Reentry close to the origin of a simulated macro-reentrant circuit in the LA While potential maps during macro reentrant activity were (Figure 6A). The inverse solution (Figure 6B) was good near the reconstructed more faithfully using a 130-electrode catheter than a catheter, but much poorer over the rest of the LA. This is 64-electrode catheter with the same dimension, corresponding phase demonstrated in Figure 6C where CC is rendered on the LA maps in Figure 5A appear to carry very similar information about the surface; median CC is >0.9 in the RoI, but falls off rapidly with history of activation across the LA surface. This similarity in phase distance from this region. In the lower panel we present CC distribution was preserved throughout an extended sequence of (Figure 6D), nRMSE (Figure 6E) and ΔT (Figure 6F) in the RoI simulated electrical activity (Supplementary Video S1), where (red) and across the full endocardial surface (blue) for inverse phase singularities identified recovered with 64- and 130-electrode solutions constructed as the catheter was moved progressively catheters are also collocated. The correspondence with ground truth for along a line from the origin of the LAA to the inter-atrial septum phase maps obtained with noncontact catheters was maintained for a (Figure 6A). These figures demonstrate that regional mapping wider range of relative catheter volumes than for the potential maps in performance was excellent when the catheter was in or adjacent to Figure 3 above. However, CC was increased and nRMSE reduced with the RoI, but poor when the catheter was most distant from it. Global a 130-electrode catheter compared to a 64-electrode catheter mapping performance was best when the catheter was located (Figure 5B). This indicates that phase maps with the latter capture centrally, but significantly poorer in this case in the RoI. Of key features of wave front propagation in macro reentrant arrhythmia, particular interest, RoI performance was optimal when the catheter but aspects of the fine structure of the phase distribution are lost. 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]. Region-Of-Interest Potential Mapping With region-of-interest (RoI) mapping, a small catheter is positioned close to a region of the endocardial surface to reconstruct local DISCUSSION electrical activity. In Figure 6, we consider the accuracy with which regional electrical activity can be recovered using non- Summary contact catheters. This analysis was completed without adding This analysis of noncontact intracardiac potential mapping Gaussian noise. An 8-spline 64-electrode basket catheter (major extends an in-silico boundary mesh-based study previously Frontiers in Physiology | www.frontiersin.org 6 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping FIGURE 4 | Effects of noise on inverse solutions with the MFS during macro reentry. LA surface potentials throughout activation cycle in simulated atrial flutter are reconstructed from electrograms sampled inside the LA cavity using 130-electrode basket catheters with added Gaussian noise at different mean power levels. All data are compared with ground-truth endocardial potentials. In the upper panel (A) electrograms reconstructed from records acquired from a very small and a large basket catheter with 18, 56 and 178 μV of added noise are presented with the corresponding ground-truth electrogram. Catheter volumes relative to LA volume are 0.014 and 0.91, respectively. In the lower panel, CC (B), nRMSE (C) and ΔT (D) are presented as functions of relative catheter volume for the three noise levels. Median values and IQR are given. Abbreviations: LA, left atrium; CC, correlation coefficient; nRMSE, normalized root-mean-squared error; ΔT, activation time difference. reported by our laboratory (Meng et al., 2017; Meng et al., 2022). that surface electrograms can be reconstructed with acceptable Here we have investigated the accuracy with which endocardial accuracy (CC > 0.9, nRMSE ≤0.06 and ΔT ≤ 2 ms) during potential maps can be reconstructed from noncontact reentrant rhythm with basket catheters that fill only half of the electrograms recorded with a multi-electrode basket catheter cavity. A supplementary point that needs to be made here, is that using meshless methods that use the MFS, the first time that while relative catheter volume is an accessible measure of this has been done as far as we are aware. We demonstrate that dimension, it scales with the third power of radius for a fast, accurate noncontact potential mapping and phase mapping spherical basket catheter. Therefore, catheter volume increases are possible using this approach. However, the spatial frequency by 112.5% when its diameter changes from 35 to 45 mm. While of the electrical activity captured is determined by the distribution there was no contact between electrodes and LA wall for the of electrodes and in order to recover complex non-stationary centrally located catheter in Figure 3A (Catheter:Atrial volume rhythms, such as AF, the mapping catheter must address a ratio = 0.67), there was increasing (though incomplete) contact representative subvolume of the cardiac chamber. between them as relative volume expanded to ~0.9. Effects of Catheter Dimension Effects of Noise We have reported that noncontact mapping performance Median CC was decreased and median ΔT was increased with deteriorates progressively as catheter dimensions are reduced reduced catheter dimension (Figures 4B,D) when Gaussian noise relative to those of the cardiac chamber and that this was added but there was no corresponding effect on median degradation becomes more marked when activation is complex. nRMSE (Figure 4C). The reconstructed electrograms in Neither of these findings is surprising. With increasing distance Figure 4A provide explanation for these results. Because from the heart surface, intracardiac potentials associated with local electrograms recorded toward the centre of the LA cavity with activation are progressively attenuated and blurred, and a small central catheter are attenuated, the noise added to them information lost in this process cannot be recovered fully. markedly reduces SNR. This is reflected in the reconstructed Furthermore, as catheter dimension is reduced, information is surface electrograms presented in the left-hand panel of captured from a decreasing subset of the cavity volume which may Figure 4A, where SNR is low and is reduced progressively as not fully reflect local activity. More striking, perhaps, is the finding noise amplitude increases. The recorded electrograms are also Frontiers in Physiology | www.frontiersin.org 7 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping 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). Electrode Distribution and Recovery of Complex Activation Patterns 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, FIGURE 5 | Phase maps of inverse solutions with MFS during macro- high spatial frequencies cannot be recovered and low frequency reentry. LA surface potentials throughout activation cycle in simulated atrial flutter are reconstructed from electrograms sampled inside the LA cavity using artifacts (aliasing) may occur (Shannon, 1949). An example of 64- and 130-electrode basket catheters. The centrally located catheters this is provided in Figure 2 where apparent fractionation of the occupy 67% of LA volume. (A) Ground truth phase map at one instant during electrogram reconstructed between adjacent splines with a 64 activation compared with corresponding phase maps for surface potentials electrode basket catheter (see blue trace in Figure 2F) reconstructed from electrograms recorded with 64-electrode and 130- disappears with more dense sampling in that region (compare electrode basket catheters. (B) CC and nRMSE presented as functions of relative catheter volume for the 64-electrode catheter (blue) and 130-electrode red trace with ground truth electrogram in Figure 2F). While catheter (red). Median values and IQR are given. Abbreviations: MFS, Method compressed sensing approaches can collect and represent sparse of Fundamental Solutions; LA, left atrium; CC, correlation coefficient; nRMSE, signals with many fewer sampling points than indicated by the normalized root-mean-squared error; IQR, interquartile range. 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 smoothed in this case and the frequency content of the ground- for inverse reconstruction of higher frequencies from sparsely truth surface electrograms is not recovered by inverse mapping. sampled signals without introducing excessive noise. However, the overlap between ground-truth and reconstructed It should be noted that metrics such as CC and nRMSE used electrograms is affected less by noise than might have been here to quantify the correspondence between ground-truth and expected. With increased noise power, the deviation between reconstructed potentials do not take the time-history of the reconstructed and ground-truth electrograms can increase, but electrograms into account. This provides additional there is also greater instantaneous overlap between the two. With information about the spread of electrical activation across the a large catheter (right-hand panel in Figure 4A), the magnitudes heart surface as illustrated by the phase maps in Figure 5. The of recorded electrograms are substantially greater and there is phase map shown here for a 64-electrode catheter is very similar much less smoothing. As a result, surface electrograms are to that presented for a 130-electrode catheter. Both correspond recovered more faithfully with much less impact of added noise. closely to ground truth phase maps throughout the activation Inverse methods are prone to instability and error in the sequence (Supplementary Video S1). Because phase mapping presence of noise. The fact that this is not the case here further identifies local activation as an abrupt standardized transition reinforces the fact that the transfer matrix used is inherently well- from +π to −π and imposes relatively uniform spatio-temporal conditioned and the regularization procedures adopted are variation around this, confounding effects of local variation in appropriate. However, the Gaussian noise introduced here is a potential magnitude are removed. very narrow representation of the problems faced in practice with It is also important to acknowledge that sampling density is inverse potential mapping. Artifacts in the unipolar signals used affected by catheter dimensions. For instance, for a 64-electrode for this purpose include common-mode electrical noise, far-field 25-mm diameter spherical catheter, inter-electrode spacing along Frontiers in Physiology | www.frontiersin.org 8 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping FIGURE 6 | Comparison of region-of-interest inverse potential mapping with global mapping. LA surface potentials during 3 activation cycles of simulated macroscopic reentry were reconstructed from electrograms sampled inside the LA cavity with a small 64-electrode basket catheter. The catheter was initially positioned with some electrodes touching the wall at the junction with the LAA and then moved rightward until electrodes made contact with the inter-atrial septum. (A) Representative ground truth potentials on cavity surface also showing broken white line along which catheter is moved from origin of LAA (relative catheter position 0) to inter-atrial septum (relative catheter position 1). (B) Inverse solution reconstructed from potentials sampled at relative catheter position 0. (C) Correlation coefficient map for ground-truth and inverse solutions in this case. The RoI indicated is the smoothed boundary of the surface in (C) where CC ≥ 0.9. In the lower panel (D–F) present CC, nRMSE and ΔT, respectively, in the RoI (red) and across the complete LA surface (blue) for inverse solutions with the catheter at equi-spaced locations along the line in (A); catheter locations 0 and 1 are indicated in (A). Median values and interquartile range (IQR) are given in lower panel. Abbreviations: RoI, region of interest; LA, left atrium; LAA, left atrial appendage; CC, correlation coefficient; nRMSE, normalized. splines is ~5 mm, while the curvilinear distance between splines at maps based on the distance of electrodes from surface nodes the equator is ~10 mm. These measurements are doubled with a and the results of analyses such as those outlined here. 50-mm diameter catheter. This explains the apparent reduction in median CC and the increase in median nRMSE and its Potential Clinical Impact of These Findings interquartile range for 64-electrode catheters when relative The results of this study indicate that global electroanatomic catheter volume increases from 0.8–0.91 (Figures 3D,E, maps can be recovered faithfully in real-time from electrograms respectively). Here any improvement in accuracy associated recorded with noncontact multi-electrode basket catheters using with proximity to the wall is offset by reduced electrode meshless methods that use the MFS. Accurate specification of 3D density. In the clinical setting, attempts to achieve direct electrode locations with respect to cardiac anatomy is required for contact between electrodes and the heart surface can introduce inverse intracardiac mapping, but this is readily achieved with additional error by deforming catheter splines and increasing the current hybrid navigation technologies (Issa et al., 2019). Our nonuniformity of sampling. (Oesterlein et al., 2016; Pathik et al., findings indicate that, for optimal performance, catheters should 2018). It follows that global mapping with multi-electrode basket be located centrally within the cardiac chamber and address a catheters is more likely to produce reliable results when electrodes representative subvolume of the cavity (>50% in the data are not in contact with the heart wall than when attempts are presented here) with minimum contact between electrodes and made to achieve close contact. endocardial surface. The capacity to reconstruct spatially The results for RoI mapping are consistent with these complex activation patterns is limited by electrode observations. The relatively small 64-electrode catheter in distribution, but when heart rhythm is stable and repeated, Figure 6 recovered local electrical activity with a high level of more detailed maps can be reconstructed with sequential accuracy. Moreover, regional mapping produced best results alteration of electrode locations, for instance by catheter when electrodes were not in contact with the wall (median rotation/translation. Potentially, this could be more efficient CC, nRMSE and ΔT: 0.96, 0.09 and 0.89 ms, respectively). than sequential contact mapping with high density contact Global performance was much poorer, but it would be arrays because complete maps can be developed with relatively straightforward to quantify the uncertainty of reconstructed few iterations. For nonstationary heart rhythms such as AF, Frontiers in Physiology | www.frontiersin.org 9 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping however, the accuracy with which endocardial surface activation in both cases is limited by electrode distribution. Finally, while can be reconstructed is constrained by the spatial distribution of ground-truth data are based in part on clinical and simulated electrodes on the catheter for both contact and noncontact data, the accuracy of inverse intracardiac mapping has been mapping. Sparse sampling can lead to repeating artifact in confirmed computationally. While many of the assumptions reconstructed activation patterns that is incorrectly identified made in specifying the forward problem are entirely as rotors (Roney et al., 2017; Martinez- Mateu et al., 2018). reasonable, other are not. The electrical properties of the Williams et al. (Williams et al., 2018) reported that >1.0–1.5 blood in the cardiac chambers are isotropic and uniform but points/cm2 were needed on the endocardial surface to resolve they are certainly not the same as those in the myocardium spiral wave activity and this corresponds to an inter-electrode adjacent to the endocardial surface where the fictitious sources spacing of 2–3 mm - much denser than is the case for 64-electrode are located. More detailed experimental characterization of the basket catheters, particularly for equatorial electrodes on adjacent accuracy with which endocardial potentials can be reconstructed splines. As demonstrated here, more optimal electrode using inverse mapping is therefore needed to confirm the analyses distribution is achieved with catheters that have 16 rather than presented here. 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. CONCLUSION Inverse methods have been used for noncontact intracardiac electrical mapping in two commercial systems. (Schilling et al., This study demonstrates that atrial endocardial potentials can be 1999; Grace et al., 2019). The Ensite multi-electrode array reconstructed accurately from electrograms recorded with (Abbott) is used for noncontact potential mapping and noncontact multi-electrode basket catheters using a fast robust consists of 64 electrodes mounted on an inflatable balloon. inverse mapping approach that employs the MFS. This enables Consistent with the results reported here, validation studies efficient and potentially more precise capture of global and have shown that accuracy is inversely related to the distance region-of-interest potential maps than comparable contact between the array and the heart wall with poor recovery of mapping methods. Because data for all electrodes are used, it endocardial surface potentials when this distance is >25 mm is not necessary to maximize contact between catheter and the (Earley et al., 2006). More recently, instantaneous charge heart wall. This reduces the deformation of catheter splines which density distributions associated with atrial electrical activation occurs when direct contact is sought, thereby preserving a more have been constructed from electrograms recorded with uniform electrode distribution. However, we demonstrate that noncontact 48-electrode basket catheters (Acutus/Biotronik) conventional 8 spline catheters are suboptimal for instantaneous (Grace et al., 2019). This is based on a forward model that contact or noncontact mapping of complex rhythms, such as AF. 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 DATA AVAILABILITY STATEMENT analysis makes no assumptions about the cellular basis of electrical activation. Instead, we address how well endocardial The raw data supporting the conclusion of this article will be surface potentials can be recovered from a limited number of made available by the authors, without undue reservation. 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 ETHICS STATEMENT noncontact mapping if the sampling density is not sufficient. These factors would be expected to impact the spatial resolution The studies involving human participants were reviewed and with which surface charge distributions can be recovered from approved by the Melbourne Health Rseearch and Human Ethics noncontact electrograms also. Committee. The patients/participants provided their written informed consent to participate in this study. The animal Limitations study was reviewed and approved by the Animal Ethics A limitation of this study is that error introduced by uncertainty Committee of the University of Auckland. 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 AUTHOR CONTRIBUTIONS size, electrode distribution and noise, we note that our formulation of the intracardiac inverse problem is surprisingly SM and NS jointly completed all aspects of the in silico analysis robust. A further limitation is that although our ground-truth presented here and developed the computational pipeline used. data represent atrial rhythms of increasing complexity, they do JC-S is responsible for our use of the Method of Fundamental not fully replicate the spatio-temporal disorder which Solutions (MFS), completed 2D prototype studies using this characterizes AF. However, the analysis presented here approach and was influential in the direction taken in this demonstrates that the performance of contact mapping is work. LB wrote our initial 3D MFS code. SM, BS, IL, NL, AG, matched by this inverse approach and that spatial resolution GS, and JZ designed, performed and analyzed animal Frontiers in Physiology | www.frontiersin.org 10 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping experiments which provided data for this study. JZ contributed ACKNOWLEDGMENTS to the simulation of reentrant arrhythmia and provided phase mapping code. SM, NS, and BS generated the figures. SM, NS, We gratefully acknowledge the input of JM Kalman, Royal GS, and BS wrote the manuscript. BS and DB were responsible Melbourne Hospital, Parkville, Australia and University of for funding and direction of the project. All authors Melbourne, Parkville, Australia who provided high resolution contributed to manuscript revision, read, and approved the mapping data from one patient (identity protected). We also version submitted. thank Linley Nisbet for her expert assistance with animal studies carried out in Auckland. FUNDING SUPPLEMENTARY MATERIAL This study received financial support from the Health Research Council of New Zealand Programme Grant 13/77 and The Supplementary Material for this article can be found online at: New Zealand Ministry of Business and Innovation Project https://www.frontiersin.org/articles/10.3389/fphys.2022.873049/ Grant UOAX1717. full#supplementary-material Meng, S., Zhao, J., LeGrice, I., Lever, N., Sands, G., Bear, L., et al. (2017). Atrial REFERENCES Electro-Anatomic Mapping with a Novel Noncontact Approach. Comput. Cardiol. 44, 1–4. doi:10.22489/cinc.2017.105-291 Bear, L. R., LeGrice, I. J., Sands, G. B., Lever, N. A., Loiselle, D. S., Paterson, D. 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Targeting Nonpulmonary Vein Sources in Persistent Atrial Frontiers in Physiology | www.frontiersin.org 11 May 2022 | Volume 13 | Article 873049 Meng et al. Intracardiac Inverse Potential Mapping Fibrillation Identified by Noncontact Charge Density Mapping: The reviewer EJV declared a shared affiliation with one of the authors LRB at the UNCOVER AF Trial. Circ. Arrhythm. Electrophysiol. 12, e007233. time of review. doi:10.1161/CIRCEP.119.007233 Williams, S. E., Harrison, J. L., Chubb, H., Whitaker, J., Kiedrowicz, R., Rinaldi, C. Publisher’s Note: All claims expressed in this article are solely those of the authors and A., et al. (2018). Local Activation Time Sampling Density for Atrial Tachycardia do not necessarily represent those of their affiliated organizations, or those of the publisher, Contact Mapping: How Much Is Enough? Europace 20, e11–e20. doi:10.1093/ the editors and the reviewers. Any product that may be evaluated in this article, or claim europace/eux037 that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Zhao, J., Yao, Y., Huang, W., Shi, R., Zhang, S., LeGrice, I. J., et al. (2013). Novel Methods for Characterization of Paroxysmal Atrial Copyright © 2022 Meng, Sunderland, Chamorro-Servent, Bear, Lever, Sands, Fibrillation in Human Left Atria. Tobej 7, 29–40. doi:10.2174/ LeGrice, Gillis, Zhao, Budgett and Smaill. This is an open-access article distributed 1874120701307010029 under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) Conflict of Interest: The authors declare that the research was conducted in the and the copyright owner(s) are credited and that the original publication in this journal is absence of any commercial or financial relationships that could be construed as a cited, in accordance with accepted academic practice. No use, distribution or reproduction potential conflict of interest. is permitted which does not comply with these terms. Frontiers in Physiology | www.frontiersin.org 12 May 2022 | Volume 13 | Article 873049