Improving the Accuracy of Laplacian Estimation with Novel Variable Inter-Ring Distances Concentric Ring Electrodes

Abstract

Noninvasive concentric ring electrodes are a promising alternative to conventional disc electrodes. Currently, the superiority of tripolar concentric ring electrodes over disc electrodes, in particular, in accuracy of Laplacian estimation, has been demonstrated in a range of applications. In our recent work, we have shown that accuracy of Laplacian estimation can be improved with multipolar concentric ring electrodes using a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2. This paper takes the next step toward further improving the Laplacian estimate by proposing novel variable inter-ring distances concentric ring electrodes. Derived using a modified (4n + 1)-point method, linearly increasing and decreasing inter-ring distances tripolar (n = 2) and quadripolar (n = 3) electrode configurations are compared to their constant inter-ring distances counterparts. Finite element method modeling and analytic results are consistent and suggest that increasing inter-ring distances electrode configurations may decrease the truncation error resulting in more accurate Laplacian estimates compared to respective constant inter-ring distances configurations. For currently used tripolar electrode configuration, the truncation error may be decreased more than two-fold, while for the quadripolar configuration more than a six-fold decrease is expected.

Solving the general inter-ring distances optimization problem for concentric ring electrodes to improve Laplacian estimation

“…This paper continues our work toward improving the accuracy of Laplacian estimation via multipolar CREs derived using the (4 n  + 1)-point method proposed in [17] and modified for linearly variable inter-ring distances CREs in [18]. Prior to [18], inter-ring distances of a CRE were not considered to be a means of improving the accuracy of Laplacian estimation with, to the best of the author’s knowledge, all the previous CRE research having been based on assumption of constant inter-ring distances.…”

“…It was hypothesized that the ratios of constant inter-ring distances truncation term coefficients over the increasing inter-ring distances truncation term coefficients as well as the ratios of decreasing inter-ring distances truncation term coefficients over constant inter-ring distances truncation term coefficients calculated for TCRE and QCRE configurations will be comparable to the respective ratios of Relative and Maximum Errors of Laplacian estimation obtained using the FEM model. For constant inter-ring distances over increasing inter-ring distances, the truncation term coefficient ratios for the lowest nonzero truncation term for TCRE (sixth order) and QCRE (eighth order) configurations were calculated to be equal to 2.25 and 7.11 respectively which were comparable (difference of less than 5%) to the corresponding ratios of Relative and Maximum Errors obtained using the FEM model for TCRE (2.23 ± 0.02 and 2.22 ± 0.03 respectively) and QCRE (6.95 ± 0.14 and 6.91 ± 0.16) configurations [18]. For decreasing inter-ring distances over constant inter-ring distances, the coefficient truncation term coefficient ratios for the lowest nonzero truncation term for TCRE and QCRE configurations were calculated to be equal to 1.78 and 3.52 respectively which also were comparable (difference of less than 5%) to the corresponding ratios of Relative and Maximum Errors obtained using the FEM model for TCRE (1.75 ± 0.02 and 1.74 ± 0.03 respectively) and QCRE (3.41 ± 0.09 and 3.38 ± 0.11) configurations [18].…”

“…Prior to [18], inter-ring distances of a CRE were not considered to be a means of improving the accuracy of Laplacian estimation with, to the best of the author’s knowledge, all the previous CRE research having been based on assumption of constant inter-ring distances.…”

“…For TCRE with middle ring radius αr and outer ring radius r the optimal range of values for coefficient α was determined to be 0 <  α  ≤ 0.22 while for QCRE with the first middle ring radius αr , the second middle ring radius βr , and the outer ring radius r the optimal range of values for coefficients α and β was determined to be defined by inequalities 0 <  α  <  β  <1 and αβ  ≤ 0.21. Truncation term coefficient functions used to solve the general inter-ring distances optimization problem have been validated using ratios of truncation term coefficients for constant and linearly variable inter-ring distances TCRE and QCRE configurations from [18]. …”

Textile Concentric Ring Electrodes: Influence of Position and Electrode Size on Cardiac Activity Monitoring

“…Other configurations of electrode layouts and shapes are needed for better spatial resolution. To deal with this issue, modified electrode designs based on concentric ring electrodes (CREs) have been applied to several bioelectric signal recording and analysis, such as electrocardiographic (ECG) [11][12][13][14][15], electroenterographic (EEnG) [16], and electrohysterographic (EHG) signals [17]. The concentric ring design is a physical approximation to the Laplacian filter, which is the second spatial derivative of the measured potentials and essentially assigns more weight to changes in bioelectric dipoles under measurement points, to enable better differentiation between concurrent, closely spaced dipole sources [14].…”