## 4 Discussion
Although a larger electrode size corresponds to worse spatial resolution associated with a given recording configuration (bipolar, tripolar,...) [26; 27], the greater contrast obtained in the P-wave with the LIIRD TCRE configuration suggests better spatial resolution of this configuration compared to the bipolar one which is consistent with the finite element method modeling results from [23]. This improved ability to capture atrial activity facilitates its study, providing a tool of great clinical interest. LIIRD TCRE could provide more accurate cardiac activation patterns over bipolar concentric ECG and 12-lead ECG, and facilitate the diagnosis of atrial fibrillation in which a standard 12-lead ECG is insufficient to guide clinical management [32]. In fact, in a very recent study LIIRD TCRE provided the highest P-wave contrast (normalized amplitude) compared to bipolar, linearly decreasing inter-ring distances TCRE and quadripolar Laplacian estimates at CMV1 with significant differences from all the 12-lead standard ECG signals [33]. This ability is not compromised by obtaining the LIIRD TCRE Laplacian estimate using NDM or FDM, since they provide almost identical results in the equation coefficients and, therefore, in estimate signals obtained using Equations (16) and (17) respectively. On the other hand, having used a high-pass filter for the elimination of baseline drifts may have slightly reduced the relative amplitude of the low frequency cardiac components such as the P-wave and the T-wave. The use of more modern signal processing techniques [34], that respect to a greater extent the cardiac components, could lead to greater contrast in the study of these waves. The limitation of this study is that, for the validation of human ECG data, emulation of CIRD TCRE of the same size as the emulation of LIIRD TCRE using the real life CRE in Figure 2 was not feasible. The only other TCRE configuration of the same size that could be emulated was the linearly decreasing inter-ring distances one (using bipolar ECG signals \(U_{3}-U_{1}\) and \(U_{4}-U_{1}\)) from [23]. However, with the NDM and FDM based Laplacian estimates for LIIRD TCRE being virtually identical there is no reason to expect the NDM and FDM based Laplacian estimates for CIRD TCRE to be significantly different. Therefore, the CIRD over LIIRD truncation term coefficient ratios (based on NDM and FDM) corresponding to the ratios of truncation and, therefore, of Laplacian estimation errors will also be virtually identical in case when both the numerators and the denominators are. This human ECG data based result is consistent with the analytic result from Figure 6.  
In the future, this comparison framework will be developed into a comprehensive optimization problem including and directly comparing all the possible CRE designs of the same size and with the same number of rings. If this FDM based optimization problem cannot be solved analytically it will be solved numerically as was the NDM based one in [24] for a wider range of numbers of rings and percentile values. Solutions of this problem could inform the design of CREs by maximizing the accuracy of their Laplacian estimates and could be particularly useful for designs dissimilar to the overly simplified NDM. The second direction of future work is related to investigating the potential of using fractional Laplacian for electrophysiological measurement applications instead of currently used regular Laplacian [35].