## 4 Discussion  
Steps similar to the ones in [25] can be used to estimate the Laplacian for any multipolar CRE configuration where the widths of concentric rings and the radius of the central disc are nonnegligible (FDM). In this study, the proof of concept from [25] has been developed into a framework for direct comparison of any two specific CRE configurations of the same size and with the same number of concentric rings. Ratios of the lowest remaining order truncation term coefficients were used as a measure to base the comparison on since in [23] and [24] these ratios have been shown (using finite element method modeling) to be a predictor of the corresponding Laplacian estimation error. Namely, differences of less than 5% between the modeling derived ratios of Relative and Maximum Errors of Laplacian estimation and respective analytic ratios of truncation term coefficients have been reported for combinations of LIIRD, CIRD, and linearly decreasing inter-ring distances TCREs and quadripolar CREs [23] as well as for their quadratically increasing inter-ring distances counterparts [24]. The proposed comparison framework is implemented for the LIIRD and CIRD TCRE configurations. Obtained results confirm that a relatively small change in the geometry of the TCRE (moving the middle ring from 3\(r\)-4\(r\) to 4\(r\)-5\(r\) for a TCRE with radius of 9\(r\) in Figure 1) increases the Laplacian estimation error by more than 50% (approximately 1.57 times greater for the CIRD TCRE compared to the LIIRD one). Moreover, this study directly compared the truncation term coefficient ratios obtained for the FDM TCRE configurations with the respective ratios obtained for the NDM in [23]. Once the scale was adjusted to make the TCRE configurations for the FDM (Figure 1) and the NDM (Figure 5) comparable there was no statistically significant difference between the truncation term coefficient ratios from Figure 6. This comparison provides insight into the effect of including additional parameters such as the radius of the central disc and the widths of concentric rings into the CRE model. The fact that for TCRE configurations assessed in the current study the ratios of truncation term coefficients corresponding to FDM and NDM respectively differed by less than 5% suggests consistency between the two models. This is to be expected since FDM and NDM are also consistent in terms of the highest truncation term order (2\(n\)) that can be cancelled out for a CRE with \(n\) rings as was shown in [22] and [25] respectively. However, an argument can be made that this may change for CRE designs that are drastically different from the NDM. For example, CRE design with variable widths of individual concentric rings that are significant compared to the total radius of the CRE. For such a CRE design only FDM is likely to produce meaningful results. Finally, an intuitive result that larger differences between CRE configurations may result in larger differences in respective Laplacian estimation errors was suggested. For example, difference between the CIRD and the LIIRD configurations equivalent to 1/9 of the radius of the TCRE in case of FDM (Figure 1) resulted in a smaller CIRD over LIIRD error ratio of 1.57 while a more substantial difference of 1/6 of the radius of the TCRE between the same two TCRE configurations in case of NDM (Figure 5) resulted in a larger CIRD over LIIRD error ratio of 2.25. QRS amplitude are commonly used to quantify this contrast. In traditional 12-lead ECG recordings, the highest normalized P-wave amplitudes are obtained in Lead I, Lead III, and V1 [30], with values between 0.09 and 0.14. In the case of CREs, the highest contrast is obtained in position equivalent to V1. Bipolar ECG signals picked up at that area have yielded normalized P-wave values of 0.18 [12] and 0.16 [20] with electrodes of external diameter equal to 15 mm and 42 mm respectively. In this study, normalized P-wave amplitude of 0.25 was obtained for LIIRD TCRE emulation with external diameter equal to 45.6 mm.