Development of a Tri-polar Concentric Ring Electrode for Acquiring Accurate Laplacian Body Surface Potentials

Abstract

Potentials recorded on the body surface from the heart are of a spatial and temporal function. The 12-lead electrocardiogram (ECG) provides a useful means of global temporal assessment; however, it yields limited spatial information due to the smoothing effect caused by the volume conductor. In an attempt to circumvent the smoothing problem, researchers have used the five-point method (FPM) to numerically estimate the analytical solution of the Laplacian with an array of monopolar electrodes. Researchers have also developed a bipolar concentric ring electrode system to estimate the analytical Laplacian, and others have used a quasi-bipolar electrode configuration. In a search to find an electrode configuration with a close approximation to the analytical Laplacian, development of a tri-polar concentric ring electrode based on the nine-point method (NPM) was conducted. A comparison of the NPM, FPM, and discrete form of the quasi-bipolar configuration was performed over a 400 × 400 mesh with 1/400 spacing by computer modeling. Different properties of bipolar, quasi-bipolar and tri-polar concentric ring electrodes were evaluated and compared, and verified with tank experiments. One-way analysis of variance (ANOVA) with post hoc t-test and Bonferroni corrections were performed to compare the performance of the various methods and electrode configurations. It was found that the tri-polar electrode has significantly improved accuracy and local sensitivity. This paper also discusses the development of an active sensor using the tri-polar electrode configuration. A 1-cm active Laplacian tri-polar sensor based on the NPM was tested and deemed feasible for acquiring Laplacian cardiac surface potentials.

APPENDIX

Since the proposed quasi-bipolar electrode shown in Fig. 2 has equal interelectrode distance, the analysis of the nine-point arrangement formed by points p ₁ through p ₈ and p ₀ of Fig. 1, with the same interpoint distance, is considered. Voltages v ₀ through v ₈ are the potentials at these points. Each potential in this nine-point arrangement can be written in the form of a Taylor series expansion as explained by Huiskamp.6

The average of the outer potentials v ₅, v ₆, v ₇, and v ₈ after the Taylor series expansion becomes

$$\frac{{v_5 + v_6 + v_7 + v_8 }}{4} = v_0 + r^2 \left. {\left( {\frac{{\partial ^2 v}}{{\partial x^2 }} + \frac{{\partial ^2 v}}{{\partial y^2 }}} \right)} \right|_{p_0 }\\ +\, \frac{{r^4 }}{3}\left. {\left( {\frac{{\partial ^4 v}}{{\partial x^4 }} + \frac{{\partial ^4 v}}{{\partial y^4 }}} \right)} \right|_{p_0 } + \cdots$$

(A.1)

The average of the potentials v ₁, v ₂, v ₃, and v ₄ after the Taylor series expansion becomes

$$\frac{{v_1 + v_2 + v_3 + v_4 }}{4} = v_0 + \frac{{r^2 }}{4}\left. {\left( {\frac{{\partial ^2 v}}{{\partial x^2 }} + \frac{{\partial ^2 v}}{{\partial y^2 }}} \right)} \right|_{p_0 }\\ +\, \frac{{r^4 }}{{48}}\left. {\left( {\frac{{\partial ^4 v}}{{\partial x^4 }} + \frac{{\partial ^4 v}}{{\partial y^4 }}} \right)} \right|_{p_0 } + \cdots$$

(A.2)

Adding v ₀ to both sides of (A.1) and then dividing by 2 gives

$$\frac{{\frac{1}{4}\sum\limits_{i = 5}^8 {v_i } + v_0 }}{2} = v_0 + \frac{{r^2 }}{2}\left. {\left( {\frac{{\partial ^2 v}}{{\partial x^2 }} + \frac{{\partial ^2 v}}{{\partial y^2 }}} \right)} \right|_{p_0 }\\ +\, \frac{{r^4 }}{6}\left. {\left( {\frac{{\partial ^4 v}}{{\partial x^4 }} + \frac{{\partial ^4 v}}{{\partial y^4 }}} \right)} \right|_{p_0 } + \cdots$$

(A.3)

Subtracting (A.2) from (A.3) results in

$$\frac{1}{2}\left( {\frac{1}{4}\sum\limits_{i = 5}^8 {\mathop v\nolimits_i } + \mathop v\nolimits_0 } \right) - \frac{1}{4}\sum\limits_{j = 1}^4 {\mathop v\nolimits_j } = \frac{{\mathop r\nolimits^2 }}{4}\left. {\left( {\frac{{\mathop \partial \nolimits^2 v}}{{\partial \mathop x\nolimits^2 }} + \frac{{\mathop \partial \nolimits^2 v}}{{\partial \mathop y\nolimits^2 }}} \right)} \right|_{P0}\\ + \frac{{7\mathop r\nolimits^4 }}{{48}}\left. {\left( {\frac{{\mathop \partial \nolimits^4 v}}{{\partial \mathop x\nolimits^4 }} + \frac{{\mathop \partial \nolimits^4 v}}{{\partial \mathop y\nolimits^4 }}} \right)} \right|_{\mathop p\nolimits_0 } + \ldots$$

(A.4)

From (A.4) the approximate solution for the Laplacian of the potential at p ₀ can be estimated. Therefore, the Laplacian at p ₀ is

$$\Delta p_0 = \left. {\left( {\frac{{\partial ^2 v}}{{\partial x^2 }} + \frac{{\partial ^2 v}}{{\partial y^2 }}} \right)} \right|_{p_0 } = \frac{4}{{r^2 }}\Bigg[ {\frac{1}{2}\left( {\frac{1}{4}\sum\limits_{i = 5}^8 {v_i + v_0 } } \right)}\\ {- \frac{1}{4}\sum\limits_{j = 1}^4 {v_j }} \Bigg] + O(r^2 )\ldots$$

(A.5)

where $O(r^2 ) = \frac{2}{{r^2 }}[O(r^4 )] = \frac{{7r^2 }}{{48}}\left. {\left( {\frac{{\partial ^4 v}}{{\partial x^4 }} + \frac{{\partial ^4 v}}{{\partial v^4 }}} \right)} \right|_{_{p_0 } } + \cdots$ is the truncation error.

Equation (A.5) can also be generalized to the quasi-bipolar concentric ring electrodes, neglecting the truncation error O(r ²). By applying a similar procedure as was used for the bipolar electrode configuration, performing the integral along a circle of radius r around point p ₀ of the Taylor expansion and defining X = r sin(θ) and Y = r cos(θ)6 result in (A.6), which is the potential on the middle ring.

$$\int\nolimits_0^{2\pi } {(v(r,\theta )} d\theta = \int\nolimits_0^{2\pi } {v_0 } \,d\theta + \frac{{r^2 }}{4}2\pi \Delta v_0\\ +\, \frac{{r^4 }}{{24}}\int\nolimits_0^{2\pi } {\sum\limits_{j = 0}^4 {(\sin \theta )^{4 - j} (\cos \theta )^j d\theta \left. {\left( {\frac{{\partial ^4 v}}{{\partial x^{4 - j} \partial y^j }}} \right)} \right|_{p_0 } } }\\ +\, \frac{{(2r)^6 }}{{6!}}\int\nolimits_0^{2\pi } {\sum\limits_{j = 0}^6 {(\sin \theta )^{6 - j} (\cos \theta )^j \theta \left. {\left( {\frac{{\partial ^6 v}}{{\partial x^{6 - j} \partial y^j }}} \right)} \right|_{p_0 } } }\\ +\, \cdots \cdots$$

(A.6)

Similarly performing the integral along a circle of radius 2r around p ₀ and defining X = 2r sin(θ) and Y = 2r cos(θ)6 result in (A.7), which is the potential on the outer ring.

$$\int\limits_0^{2\pi } {v(2r,\theta )} d\theta = \int\nolimits_0^{2\pi } {v_0 } d\theta + r^2 2\pi \Delta v_0\\ +\, \frac{{2r^4 }}{3}\int\nolimits_0^{2\pi } {\sum\limits_{j = 0}^4 {(\sin \theta )^{4 - j} (\cos \theta )^j d\theta \left. {\left( {\frac{{\partial ^4 v}}{{\partial x^{4 - j} \partial y^j }}} \right)} \right|_{p_0 } } }\\ +\, \frac{{(2r)^6 }}{{6!}}\int\nolimits_0^{2\pi } {\sum\limits_{j = 0}^6 {(\sin \theta )^{6 - j} (\cos \theta )^j d\theta \left. {\left( {\frac{{\partial ^6 v}}{{\partial x^{6 - j} \partial y^j }}} \right)} \right|_{p_0 } } }\\ +\, \cdots$$

(A.7)

Adding the potential of the disc, $\int\nolimits_0^{2\pi } {v_0 \,d\theta }$, to both sides of (A.7) and then dividing by 2 results in the average of outer ring and center disc potentials, representing the short in the quasi-bipolar method. This results in

$$\frac{{\int\nolimits_0^{2\pi } {(v(2r,\theta ) + v_0 )} d\theta }}{2} = \int\nolimits_0^{2\pi } {v_0 \,d\theta } + r^2 \pi \Delta v_0\\ +\, \frac{{r^4 }}{3}\int\nolimits_0^{2\pi } {\sum\limits_{j = 0}^4 {(\sin \theta )^{4 - j} (\cos \theta )^j d\theta \left. {\left( {\frac{{\partial ^4 v}}{{\partial x^{4 - j} \partial y^j }}} \right)} \right|_{p_0 } } }\\ +\, \cdots$$

(A.8)

Neglecting the truncation error and subtracting Equation (A.6) from (A.8) result in a proportionate approximation (A.9) to the Laplacian at point p ₀ using the quasi-bipolar concentric ring electrode.

$$\Delta v_0 \cong \frac{4}{{r^2 }}\Bigg( {\frac{{\frac{1}{{2\pi }}\int\limits_0^{2\pi } {(v(2r,\theta ) + v_0 )} d\theta }}{2} - \frac{1}{{2\pi }}\int\nolimits_0^{2\pi } {v(r,\theta )d\theta } } \Bigg)$$

(A.9)

where $\frac{1}{{2\pi }}\int\nolimits_0^{2\pi } {v(r,\theta )\,d\theta }$ and $\frac{1}{{2\pi }}\int\nolimits_0^{2\pi } {v(2r,\theta )d\theta }$ represent the average potentials on the middle ring and outer ring, respectively.