Dipole current

R.C. Barr, T.C. Pilkington, J.P. Boineau, and M.S. Spach. Determining surface potentials from current dipoles, with application to electrocardiography. IEEE Trans. Biomed. Eng., 13 88-92, 1966. This is the first paper on the application of the boundary element method to problems in electrocardiography. 

If suddenly the channels (Figure 5.5) open so that the intracellular potential changes abruptly, the ions must also supply a transient discharge current of the membrane capacitance. At the extracellular side, the current is not with respect to a far-away reference electrode, but concentrated to an interstitial fluid zone near the cell. The current flow can be modeled with local current dipoles and is clearly measurable with unipolar or bipolar pickup electrodes in the interstitial liquid. When the cell is depolarizing, cations... 

Figure 5.11 Signal transfer from deep current dipole to surface pickup electrodes and the effect of layers, (a) Without layer, (b) With poorly conducting layer, (c) With well-conducting layer on top of the poorly conducting layer. Arrows represent direction and magnitude of E-field.

An ideal CC dipole mathematically defined is a dipole in an infinite homogeneous medium where I oo, a 0 and Lee 0, keeping m constant. However, in physics and engineering, the concept of infinitely small CC points with infinite current density is not ideal. We will therefore define the current dipole as technically ideal also with a>0 if Lee a and the dimensions of the homogeneous volume are much larger than Lee- The technically ideal current carrying dipole uses the same equations as the mathematically ideal dipoles. 

Figure 6.4 Equipotential lines in any plane through a current dipole axis, not referred to as a Cartesian coordinate system. High density of equipotential lines in the center is only due to the...

Equation (6.13) gave the potential difference A > between two points in the field of a technically ideal current dipole. This potential difference can be picked up by an ideal recording dipole. Such an ideal dipole is not CC and the dipole spheres may therefore approach points. Let the distance r between the two dipoles be large and the PU dipole length Lp be small so that ri = t2 = r. ri — F2 = Lp and therefore ... 

A technically ideal (Section 6.2.3) current dipole with constant moment moves with constant velocity along the jc-axis in an ideal volume. Figure 6.10. The monopolar and bipolar potentials are calculated at a fixed point on the y-axis at distance 5Lcc and lOLcc. The monopolar and bipolar potentials are calculated using Eq. (6.11) with Ip = 2(KX)tc. 

Figure 6.10 Moving current dipole ( ) with fixed position unipolar recording electrode.

Figure 6.11 Unipolar potentials from a current dipole moving along the horizontal x-axis. Dipole length is L c) the unit of the x-axis is L c- The recording electrode is at distance SL c and lOLcc (a) horizontal dipole orientation, biphasic waveforms, and high spatial resolution (b) vertical dipole orientation, monophasic waveforms, and largest signal amplitude of...

Figure 6.12 Bipolar potentials from a current dipole source. Conditions as for Figure.6.11 with...

The lead vector defines a transfer factor between the CC and PU dipoles. It depends on the resistivity p of the medium, the distance r between the dipoles, and the PU dipole length vector Lp . The scalar signal u created by a current dipole m = ILcc is ... 

Vector H has a free starting point but fixed magnitude and direction according to Eq. (6.30). The current dipole m is bound, but may vary in current I and length Lee-... 

An in vitro example with a liquid filled tube is shown (Figure 7.26). Let us first analyze the tube as a two-electrode system with the two-tube endplates as PU electrodes and a dipole moment m positioned somewhere in the volume. The PU voltage can be found with Eq. (6.31) m = H-m. The lead vector H can be found from Eq. (6.30), Jj-ed is uniform all over the tube and equal to 1/A, where A is the cross-sectional area of the tube. Integrating Eq. (6.30) H = zp/A, where z is the unity vector in direction of the tube axis. Erom Eq. (6.30), the recorded voltage from the current dipole m is m = p/Az-m. 

A dipolar current injecting electrode pair uses two equal electrodes, each electrode contributing in the same way (current dipole). The dipole is the most fundamental model of a bioelectric endogenous signal source. 

With an ideal current dipole in an infinite volume, zero potential will exist in all directions if the distance to the dipole is large enough. Unfortunately, an indifferent (neutral) ECG electrode does not exist because the human body is not large enough. If we go out in one direction along a limb, the limb proper is isoelectric with respect to the heart activity, but not with respect to other sources (e.g., respiration). If we go out along a second limb that too will be isoelectric however, the potential will not be equal to that of the first limb. Therefore none of them represents a true indifferent electrode. 

FIGURE 17.16 Illustration showing the simplified concept of a traveling band of depolarization creating a net current dipole in the heart. 

Sasaki et al, (1996) review their MEG studies on the human frontal association cortex. The no-go potential was first found at go/ no-go reaction-time hand movement tasks with discrimination between different colour light stimuli in the prefrontal cortex of monkeys. In humans it s current dipoles could be localized by use of MEG in the dorsolateral part of the frontal association cortex in both hemispheres. The function for no-go decision and subsequent suppressor action was thus substantiated in the human frontal cortex,... 

There was no significant difference in latency (before 102ms 7.65, after 102.8ms 8.07) and amplitude (before 39.2 ft/cm 9.65, after 37.7 ft/cm 17.2) of visual evoked fields before and after Miswak induced short duration tooth brushing exercise. Equivalent Current Dipole (ECD) was in the back of the head (occipital lobe, arrow. Fig 2). 

Fig. 2 PlOOm latency and amplitude (upper left for before and lower left for aftCT ) and an equivalent current dipole (ECD, arrows, uppa- right for before and lower right fa- after ) computed at the peak of the PlOOm at occipital area (contour map of VEF from a representative subject, right)...

Fig. 4 An equivalent current dipole (ECD) computed at the peak of the epileptiform discharges (PLEDs) at left temporal area (isocontour map of the MEG data, with the ECD as a green arrow). F- Frontal, LLeft...

Example Current dipole within an arbitrary fiber 445... 

In the previous chapter we examined the excitation of modes of a fiber by illumination of the endface with beams and diffuse sources, i.e. by sources external to the fiber. Here we investigate the power of bound modes and the power radiated due to current sources distributed within the fiber, as shown in Fig. 21-1. Our interest in such problems is mainly motivated by the following chapter, where we show that fiber nonuniformities can be modelled by current sources radiating within the uniform fiber. Thus, isolated nonuniformities radiate like current dipoles and surface roughness, which occurs at the core-cladding interface, can be modelled by a tubular current source. 

The vector current density J for a point current dipole of length d, small compared with a wavelength, and located at position (r, Zj) in Fig. 12-3 is given by... 

Fig. 21-2 A current dipole is located at radius and azimuthal angle in the fiber cross-section so that its direction is parallel to (a) the x-axis and (b) the z-axis. The direction along contours of constant electric field amplitude for the two cases, when = 0, are shown in (c) and (d), respectively, which ignore the slight effect of the fiber.

Within the free-space approximation, the total power radiated by the current dipole of Sections 21-4 and 21-5 is independent of both the position and orientation of the dipole. Thus, for simplicity, we can assume the dipole is parallel to the z-axis at the origin in Fig. 21-4. Substituting Eq. (21-7) into Eq. (21-20), we deduce that the vector potential has only a z-component. Hence... 

We can now determine the efficiency with which a current dipole excites the fundamental modes for various profiles. For a dipole at radius from the axis in the core of a step-profile fiber, Eqs. (21-8), (21-10), (21-23b) and Table 14-3, page 313, show that the efficiency Po/F,ad given by... 

The tubular current source was described in Section 21-6, where we showed that it is ineffective in exciting bound modes unless either of the resonance conditions of Eq. (21-15) is satisfied. A similar conclusion holds for the radiation fields. If the tube length 2L is large compared to the spatial period 2n/Sl, where 2 is the frequency in Eq. (21-13), it is intuitive that power will be radiated essentially at a fixed angle to the fiber axis. This is also a consequence of Floquets theorem [7]. However, unlike the current dipole, radiation now depends on the orientation of the currents on the tube. 

The fundamental mode on a weakly guiding, bent fiber has been replaced by an antenna, or thin wire, carrying the current distribution of Eq. (23-4). Because the currents are orthogonal to the antenna, it may be helpful to think of them as a continuous distribution of Z-directed current dipoles. To calculate the power radiated, we assume, for simplicity, that the antenna is a closed loop of radius as shown in Fig. 23-2(a). If (s. O, < ) are spherical polar coordinates... 

To give a simple example, consider the z-directed current dipole of Section 21-9 when located on the axis of a step-profile fiber. The angular dependence of the free-space radiation pattern of Eq. (21-24b) is proportional to sin 0, where 0 is the angle between the field point and the fiber axis. In this situation, we set Tq = 0 and thus Cj(0) = 1 in Eq. (21-39), whence the radiation pattern in the presence of the fiber varies as... 

Fig. 24-7 Peaks in the far-field radiation pattern of a current dipole on the axis of a step-profile fiber correspond to leaky-mode resonances. The dashed curve is the free-space pattern.

An x-directed current dipole of strength / and length d is located in an infinite uniform medium of refractive index k, at the origin of coordinates, as shown in Fig. 25-1 (a). The current can be represented by... 

Fig. 25-1 (a) A current dipole of strength I and length d is located at the origin of coordinates and is parallel to the x-direction. (b) Cross-section of the tubular source showing the orientation of axes and the current direction parallel to the x-axis. 

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