Improving the accuracy of Laplacian estimation with novel multipolar concentric ring electrodes

Abstract

Conventional electroencephalography with disc electrodes has major drawbacks including poor spatial resolution, selectivity and low signal-to-noise ratio that are critically limiting its use. Concentric ring electrodes, consisting of several elements including the central disc and a number of concentric rings, are a promising alternative with potential to improve all of the aforementioned aspects significantly. In our previous work, the tripolar concentric ring electrode was successfully used in a wide range of applications demonstrating its superiority to conventional disc electrode, in particular, in accuracy of Laplacian estimation. This paper takes the next step toward further improving the Laplacian estimation with novel multipolar concentric ring electrodes by completing and validating 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 that allows cancellation of all the truncation terms up to the order of 2n. An explicit formula based on inversion of a square Vandermonde matrix is derived to make computation of multipolar Laplacian more efficient. To confirm the analytic result of the accuracy of Laplacian estimate increasing with the increase of n and to assess the significance of this gain in accuracy for practical applications finite element method model analysis has been performed. Multipolar concentric ring electrode configurations with n ranging from 1 ring (bipolar electrode configuration) to 6 rings (septapolar electrode configuration) were directly compared and obtained results suggest the significance of the increase in Laplacian accuracy caused by increase of n.

Validating the Comparison Framework for the Finite Dimensions Model of Concentric Ring Electrodes Using Human Electrocardiogram Data

“…This paper continues a series of works concentrating on further improving the accuracy of Laplacian estimation via CREs by optimizing the number of concentric rings [22] and inter-ring distances (distances between the consecutive rings) [23,24]. Analytic results from [22][23][24] have been validated using finite element method modeling. Moreover, in [12] stencil printed TCRE prototypes resembling the linearly increasing inter-ring distances (LIIRD) design proposed in [23] were assessed on human electroencephalogram, ECG, and electromyogram data with obtained results suggesting enhanced spatial resolution and localization of signal sources.…”

“…Laplacian estimate coefficients are not unique for both NDM and FDM. For NDM they are determined as a null space vector of a truncation term coefficient matrix and therefore are not unique [22]. For FDM, as shown in Subsection 3.1 they are also a solution of a system of linear equations.…”

“…Specifically, 12-lead short-term ECG recordings with conventional disc electrodes are broadly used in clinics with benchtop equipment to diagnose a wide range of cardiac pathologies that can be dangerous and even deadly such as bundle branch block, AV blocks, sinus bradycardia or tachycardia and atrial/ventricular fibrillation [21]. This paper continues a series of works concentrating on further improving the accuracy of Laplacian estimation via CREs by optimizing the number of concentric rings [22] and inter-ring distances (distances between the consecutive rings) [23,24]. Analytic results from [22][23][24] have been validated using finite element method modeling.…”

“…More importantly, the Laplacian estimate equation used in [12] corresponded to a constant inter-ring distances (CIRD) TCRE with 1:1 ratio of inter-ring distances. Furthermore, an inherent limitation of [22][23][24] is the use of the negligible dimensions model (NDM) of a CRE where the widths of concentric rings and the radius of the central disc are assumed to be negligible. In order to optimize all of the CRE parameters simultaneously, these parameters need to be included in the general optimization problem similar to the NDM based one in [24] along with the number of rings and the inter-ring distances.…”

“…In order to optimize all of the CRE parameters simultaneously, these parameters need to be included in the general optimization problem similar to the NDM based one in [24] along with the number of rings and the inter-ring distances. Such a comprehensive finite dimensions model (FDM) of a CRE would build and improve upon all the previous NDM based findings from [22][23][24] and could be used to optimize future CRE designs maximizing the accuracy of the surface Laplacian estimation. The first step toward FDM has been taken in [25] by deriving a Laplacian estimate for a proof of concept TCRE with nonnegligible radius of the central disc and widths of the concentric rings.…”