## 3 Results  
### Establishing the Comparison Framework  
To establish the comparison framework under the FDM the surface Laplacian estimates for the two TCRE configurations from Figure 1 are derived first. The derivation starts with calculating the three potentials on the TCRE surfaces: the central disc, middle ring, and outer ring.  
Figure 3: Schematic of the placement of the wireless sensor node at position CMV1 (comparable to V1) for electrocardiogram (ECG) recording.  
The potential on the central disc \(v_{CD}\) with a radius equal to \(r\) (Figure 1) is equal to the average of the potential \(v_{0}\) at the center of the disc and the potential \(v_{r}\) on the concentric circle with radius \(r\). The central disc potential is the same for both CIRD and LIIRD TCRE configurations as shown in Figure 1.  
\[v_{CD}=\frac{v_{0}+v_{r}}{2}\cong v_{0}+\frac{1}{8}r^{2}\Delta v_{0}+\frac{1}{ 96}r^{4}T_{4}+\frac{1}{2880}r^{6}T_{6} \tag{6}\]  
CIRD and LIIRD middle ring potentials are different. In Figure 1 the CIRD configuration places the middle ring between concentric circles with radii of \(4r\) and \(5r\) so the average of \(v_{4r}\) and \(v_{5r}\) is calculated.  
\[v_{MR}^{CIRD}=\frac{v_{4r}+v_{5r}}{2}\cong v_{0}+\frac{41}{8}r^{2}\Delta v_{0} +\frac{881}{96}r^{4}T_{4}+\frac{19721}{2880}r^{6}T_{6} \tag{7}\]  
However, for the LIIRD configuration, the average of potentials \(v_{3r}\) and \(v_{4r}\) is calculated instead.  
\[v_{MR}^{LIIRD}=\frac{v_{3r}+v_{4r}}{2}\cong v_{0}+\frac{25}{8}r^{2}\Delta v_{0 }+\frac{337}{96}r^{4}T_{4}+\frac{965}{576}r^{6}T_{6} \tag{8}\]  
Next, for both LIIRD and CIRD configurations the outer ring potential \(v_{OR}\) is calculated as the average of the potentials on concentric circles with radii of \(8r\) and \(9r\).  
\[v_{OR}=\frac{v_{8r}+v_{9r}}{2}\cong v_{0}+\frac{145}{8}r^{2}\Delta v_{0}+\frac {10657}{96}r^{4}T_{4}+\frac{158717}{576}r^{6}T_{6} \tag{9}\]  
Finally, \(v_{0}\) is canceled out. This is accomplished by taking the bipolar difference between \(v_{CD}\) and \(v_{MR}\). Since there are two different equations for the middle ring potential, the CIRD configuration is considered first.  
\[v_{MR}^{CIRD}-v_{CD}\cong 5r^{2}\Delta v_{0}+\frac{55}{6}r^{4}T_{4}+\frac{493}{ 72}r^{6}T_{6} \tag{10}\]  
Followed by the LIIRD configuration.  
\[v_{MR}^{LIIRD}-v_{CD}\cong 3r^{2}\Delta v_{0}+\frac{7}{2}r^{4}T_{4}+\frac{67}{ 40}r^{6}T_{6} \tag{11}\]  
Bipolar difference between the outer ring potential \(v_{OR}\) and the central disc potential \(v_{CD}\) which is the same for both configurations is taken as well.  
\[v_{OR}-v_{CD}\cong 18r^{2}\Delta v_{0}+111r^{4}T_{4}+\frac{5511}{20}r^{6}T_{6} \tag{12}\]  
These three bipolar differences are combined linearly to cancel out the 4 th order truncation term \(T_{4}\) and provide the surface Laplacian estimates for CIRD and LIRD configurations.  
\[\Delta v_{0}^{CIRD}\cong\frac{1}{r^{2}}\Big{[}\frac{37}{130}\Big{(}v_{MR}^{CIRD }-v_{CD}\Big{)}-\frac{11}{468}(v_{OR}-v_{CD})\Big{]}+\frac{163}{36}r^{4}T_{6} \tag{13}\]  
\[\Delta v_{0}^{LIIRD}\cong\frac{1}{r^{2}}\Big{[}\frac{37}{90}\Big{(}v_{MR}^{ LIIRD}-v_{CD}\Big{)}-\frac{7}{540}(v_{OR}-v_{CD})\Big{]}+\frac{173}{60}r^{4}T_{6} \tag{14}\]  
Considering the ratio of the respective 6 th order truncation term (\(T_{6}\)) coefficients for CIRD and LIIRD configurations the truncation error corresponding to the CIRD configuration is estimated to be 1.57 times greater than that of the LIIRD configuration.  
\[\frac{163}{36}\div\frac{173}{60}=\frac{815}{519}=1.57 \tag{15}\]  
Such ratios of truncation term coefficients for the lowest remaining truncation term order allow comparing any two TCRE configurations of the same size and with the same number of rings directly in terms of truncation error since according to [31] for Taylor series \"higher-order terms usually contribute negligibly to the final sum and can be justifiably discarded.\"