Lead vector

Concepts of Lead Vector and Lead Field Lead Fields of... 

The vector c is a three-dimensional transfer coefficient that describes how a dipole source p at a fixed point Q inside a volume conductor influences the voltage measured from the lead and is called the lead vector. 

Mathematically, the voltage V is the scalar product of dipole p and the lead vector c... 

FIGURE 27.1 The concepts of (a) lead vector and (b) lead field. (See the text for more details.)... 

The lead vector c describes what is the sensitivity of the lead to a source locating at the source location. It is self-evident that for another source location the sensitivity may have another value. Thus the sensitivity, that is, the lead vector, varies as a function of the location, and we may say that it has a certain distribution in the volume conductor. This is called the sensitivity distribution. 

We may define the value of the lead vector at every point in the volume conductor. If we then place the lead vectors to the points for which they are defined, we have a field of lead vectors throughout the volume conductor. This field of lead vectors is called the lead field /l. The lead field illustrates the behavior of the sensitivity in the volume conductor and is a very powerful tool in analyzing the properties of electric and magnetic leads (see Figure 27.1b). 

Because of reciprocity, the field of lead vectors is the same as the current field raised by feeding a reciprocal current of 1A to the lead. 

The lead field may be illustrated either with lead vectors in certain locations in the volume conductor or as the flow lines of the distribution of the reciprocal current in the volume conductor. This is called the lead current field. In the latter presentation, the lead field current flow lines are oriented in the direction of the sensitivity, and their density is proportional to the magnitude of the sensitivity. 

It is clear from this equation that it does not matter which dipole is CC and which is PU, in accordance with the principle of reciprocity. We separate the part of Eq. (6.29) associated with the PU dipole and call it the lead vector H ... 

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 ... 

Zt is a scalar in space, but may be a time vector Zt related to a frequency dependence of p. Both the lead vector H and file scalar transfer impedance Zt are transfer factors between the CC and PU dipoles. The relationship between them is Zt = H-Lcc [il]. 

The Einthoven triangle is an example of a recorded voltage u (the electrocardiogram signal) modeled as the dot product of the lead vector of a bipolar PU electrode pair and the bound heart dipole vector m (see Section 10.1). 

Schmitt (1957) called the lead vector H transfer impedance. This may he misleading hoth because impedance is a spatial scalar, whereas H is a spatial vector and with dimension [Q/meter]. 

Figure 6.15 illustrates a situation with a finite and inhomogeneous volume where m is the dipole source, symbolizing for instance the heart vector. Under such conditions, the dipole equations are not exact solutions, and a more realistic approach according to limited conductor volumes and inhomogeneous tissue must be found. To find the lead vector H under more realistic conditions, the concept of reciprocal excitation was introduced (McFee and Johnston, 1953). Reciprocal excitation is to let the PU electrodes be CC with... 

It is still possible to use the lead vector concept (e.g., from the example shown in Figure 6.14). The lead vector of the PU dipole is not a vector field it is the total... 

A lead vector H is determined by the geometry and conductivity distribution of the signal PU system, and is valid for one tissue volume and one set-up with all electrode and source positions and distributions defined and constant. Changing one PU electrode or source position, leaving all other factors constant, results iu another lead vector H. 

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. 

Figure 10.6 Locus of the heart vector m given each 10 ms in the QRS diastole. Derived from the Einthoven triangle (Figure 10.5). The graph is in the frontal plane and the x-axis has the direction of the lead vector H (a = 0).

It was a long way to go before the genesis of the surface potential differences caused by action potentials deep in the thorax was understood. It was the work of Einthoven and the lead concept that paved the way. It was Burger and van Milaan who introduced the lead vector, making it possible to find the direction to go. Richard McFee replaced the lead vector by the lead field, defined as the electric field set up in the body by a unit current applied to the pick up electrode pair. Otto Schmitt reintroduced the old Helmholtz concept about reciprocity and introduced the concept of transfer impedance already known from the use of four-electrode technique. And it was David Geselowitz who finally put it in the elegant mathematical form. A certain similarity with the Faraday—Maxwell intellectual process runs in our minds. 

FIGURE t7J8 The ECG lead vectors are based on die Eiiidioven triangle placement of electrodes. 

Other Lead systems are also possible and have been standardized into what is known as the 12-lead system that constitutes the clinical ECG. In addition to leads I, II, and m, there are three others known as the augmented leads aVR, aVL, and aVF. These leads use an electric addition process of the limb-electrode signals to create a virtual signal reference point in the center of the chest known as the Wilson central terminal (WCT). From the WCT reference point, the augmented lead vectors point to the right arm, left arm, and left leg, respectively. 

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