ON THE NUMBER OF SURFACE EXTREMA OF THE POTENTIAL GENERATED BY A SINGLE DIPOLE IN AN ELLIPTICAL CYLINDER

J. Troquet¹, R. Evrard²
¹Institut L. Frédéricq de Physiologie L1, Université de Liège,
17 Place Delcour, B-4020 Liège, BELGIUM
²Institut de Physique B5, Université de Liège,
Sart Tilman, B-4000 Liège, BELGIUM

Abstract: We have developed an analytical method based on Mathieu functions to solve Poisson's equation in homogeneous ohmic conductors with the shape of a right elliptical cylinder. In this communication, we describe the results obtained for the electric-potential distribution on the surface of such a conductor fed by an inside current dipole. The potential contour map on the surface often has more than one maximum and one minimum. This is the case for a dipole parallel to the symmetry plane containing the ellipse major axis and the cylinder axis, at least if it lies not too far from this plane and if the cylinder ellipticity is large enough. It is also the case of dipoles perpendicular to the axis of short cylinders.

INTRODUCTION

In electrophysiology, it is often believed that the overall aspect of the surface potential in a homogeneous body directly reflects the nature, symmetry, and orientation of the source independently of its position and of the body shape. This led Okada to claim in one of his papers [1] that a single dipolar current source cannot give rise to an electric potential with more than two local extrema on the surface. Indications of the existence of more than two extrema can be found in several papers. However, either these results were overlooked or the calculations were not detailed and precise enough to draw indisputable conclusions. In particular, Berry [2] was the first to obtain by calculation more than one pair of surface potential extrema. It was in the case of an oblate spheroid fed by an electric dipole off-centered along the major axis.

The aim of this communication is to show in a more systematical study that the surface electric potential of a dipolar source has more than a pair of surface extrema on some conditions and to discuss these conditions. To fulfill this goal, we solve Poisson’s equation in the case of a single dipolar current source feeding a homogeneous right elliptical cylinder with insulating surface and we draw the potential contour map on this surface.

METHOD

To obtain the electric potential distribution, we solve Poisson’s equation using Green functions. The method is similar to that developed by Lambin and Troquet [3] in the case of circular cylinders. In our case, the Green function appears as a sum of products of Mathieu functions. This sum is easily performed, even on a personal computer. It gives convergent results except at points whose position along the cylinder axis is close to that of the source. At these points, we obtain the potential by interpolation between the values that we have previously calculated on both sides of the source.

RESULTS AND DISCUSSION

Figure 1 shows the contour map of the electric potential produced by a centered longitudinal dipolar source on the half surface of an elliptic cylinder. The cylinder axis ratio is a/b = 2, where a and b respectively denote the semimajor- and semiminor-axis lengths. The ratio l/a of the cylinder length to the semimajor axis is 3.25. We use the notation s for z/l where z is the longitudinal coordinate of the point under consideration and l, the cylinder length. As for s, it denotes the curvilinear coordinate measured along the elliptic cylinder base and s₀ is the length of its perimeter. We use p/gd² as unit of electric potential. Here, p, g, and d respectively denote the current dipole moment of the source, the electrical conductivity, and half the distance between the ellipse foci. Therefore, the electric potential appears as a dimensionless quantity.

Figure 1. Contour map of the electric potential on half the lateral surface of a cylinder with a/b = 2 and l/a = 3.25. The source is a centered and vertical dipole.

The electric-potential distribution on the whole lateral surface has two maxima and two minima. The positions of the two of them on the half surface represented in Fig. 1 are indicated by crosses. All four extrema remain if the dipole is displaced along the minor ellipse axis by as much as 17% of its half length. The extrema which are located farthest from the source disappear in the case of larger shifts. The asymmetry between the anteroposterior and lateral directions explains the existence of four extrema. Indeed, due to the relatively strong cylinder ellipticity, the points where the extrema are located are closer to the source than most of the other points on the surface. In the case of circular cylinders, the electric potential of an off-centered dipolar source parallel to the axis has only a surface maximum and a surface minimum.

In the case of a centered dipole lying along the ellipse major axis, the potential has again two maxima and two minima on the lateral surface. When the dipole is moved far enough towards the end of the axis, the extrema near this end merge into a single one. This is illustrated in Fig. 2, which shows the potential contour map on the lateral surface due to a dipole located at a point such that x/a = 0.6 where x is the point coordinate along the ellipse major axis. In this case, the two maxima on the lateral surface have merged into a single one at the end of the major axis, which was occupied by a saddle point for source positions nearer the center. A similar saddle point with a potential of opposite sign is still present at the other axis end. Notice that this situation and the resulting potential distribution bear some similarity with that met by Berry [2] in his study of the potential distribution on the surface of an oblate spheroid.

Figure 2. Contour map of the electric potential on the lateral surface of a cylinder with a/b = 2 and l/a = 3.25. The dipolar source is at s = 0.5 and lies on the ellipse major axis at a distance x = 0.6 a from the cylinder center.

A case which is not realistic in electrocardiology, but useful for understanding the reasons of the existence of multiple pairs of extrema, is that of cylinders with l/d £ 1. As an illustration of this case, Fig. 3 shows the contour map on the upper base of an elliptical cylinder with again a/b = 2, but now, l/d = 0.8. The dipolar source is at the cylinder center and lies along the ellipse major axis. We see that the pairs of extrema have moved from the lateral surface to the bases. These pairs of extrema on the bases are also present in the case of short circular cylinders. This is the only case where the circular cylinder has more than a pair of surface potential extrema. For shortening cylinders, i.e., decreasing values of the ratio l/d, the extrema on each base come closer to each other. They merge with the position of the source at the limit of a two-dimensional plate, either an ellipse or a circle.

Figure 3. Contour map of the electric potential on the base of a cylinder with a/b = 2 and l/d = 0.8. The dipolar source is at the cylinder center and aligned with the ellipse major axis.

CONCLUSION

We have systematically studied the effects of the body asymmetry and dipolar-source position on the presence of more than a pair of surface potential extrema. Our results show that, even in the case of a homogeneous conductor, the presence of more than two local extrema on the surface does not exclude the possibility that the potential is due to a single localized dipolar current source. Therefore, Okada’s state-ment about the maximum number of surface potential extrema is not valid under all circumstances. We have shown that the lack of symmetry of the body under consideration and the position of the source are the factors limiting the validity of this statement.

REFERENCES

[1] Robert H. Okada, “A critical review of vector electrocardiography,” IEEE Trans. BME, vol.10, pp. 95-98, 1963.

[2] P. M. Berry, “N, M space harmonics of the oblate spheroid,” Ann. NY Ac. Sc., vol. 65, pp. 1126-1134, 1957.

[3] Ph. Lambin and J. Troquet, “Complete calculation of the electric potential produced by a pair of current source and sink energizing a circular finite length cylinder,” J. Appl. Phys., vol. 54, pp. 4174-4184, 1983.