In configuration t2, let rA be the location of the source electrode, which injects current into the head by establishing a positive potential at that point, and let rB be the location of the sink electrode, which extracts current from the head by establishing a negative potential at that point. The normal component of the current density on the surface may then be written formally
Inserting (6.4) and (6.5) into (6.3) and performing the integrals trivially over the delta functions gives
I2 (ra) — $1^3)] = —I1 [$2(r+) — $2(r_)] (6.6)
Expanding the difference $2(r±) in powers of d and taking the usual dipole limit as d ^ 0 gives
where the lead field vector is defined
Thus the lead field vector L for a particular electrode pair (A,B) is proportional to the current density J2 which would be created in V at the dipole position r1 if unit current I2 were injected through the electrode pair. The proportionality constant is the reciprocal of the local conductivity a at the dipole location r1 .
The lead field L has the content of the usual forward problem, but is interpreted somewhat differently. It is computed as a function of the dipole position for fixed electrode positions. That is opposite the normal formulation of the forward solution, in which the potential at any point is computed for fixed dipole location. In this way the lead field gives a measure of the sensitivity of a particular electrode pair to dipoles are arbitrary locations in the volume. This may be used to reduce the computational demand of the forward problem for a fixed electrode array.
The lead field vector L is the proportionality constant between p and and is a measure of the sensitivity of an electrode pair to dipoles at various locations. Since the orientation dependence implemented by the dot product is rather trivial, the magnitude of the lead field vector L = |L| may be defined as the sensitivity of an electrode pair (Rush and Driscoll 1968). The amount of tissue probed by a particular pair may be quantified through the concept of half-sensitivity volume (Malmivuo and Plonsey 1995; Malmivuo et al. 1997).
The half-sensitivity volume (HSV) is defined as follows. For a given electrode pair, we compute the scalar sensitivity L(r) for many 104) points r inside the brain volume, and determine the maximum sensitivity Lmax for this pair. We then identify all points in the brain volume whose sensitivity is at least Lmax/2. The HSV is the volume filled by these points. The threshold of 1/2 is certainly arbitrary, but does give some indication of the volume in which the largest sensitivities occur. We further define the depth D of the sensitivity distribution as the maximum depth of all points included in the HSV. Using a four-sphere model of the human head, the outer radii of the four tissue layers are 8.0 cm (brain), 8.2 cm (CSF), 8.7 cm (skull) and 9.2 cm (scalp).
Figure 5 shows L in a two-dimensional plane including the electrodes (A,B) and the origin. The vector nature of L is retained to illustrate its dependence
Fig. 5. The EEG lead field vector L(r) shown only within the HSV, for a four-sphere head model with 03/04 = 1/24. The electrode separation angles 6 are: (a) 10, (b) 30, (c) 60 and (d) 90 degrees. Axes are in cm on orientation, but only its magnitude L = |L| is used to define the sensitivity and the HSV. In such a simple head model, the HSV is seen to be a single contiguous volume for nearby electrode pairs, which bifurcates near 60 degrees into two separate volumes for more distant pairs. Like the potential difference
— the lead field L changes only by a minus sign under interchange of A and B; the geometric pattern of sensitivity is unaffected.
The vector direction of L shows how the direction sensitivity of EEG bipolar recordings changes as a function of angle 0 between the electrodes. Between nearby electrodes the sensitivity is primarily tangential to the sphere, while under each electrode the sensitivity is more radial. This observation refines the intuition that nearby electrodes are primarily sensitive to tangential dipoles between them. In fact, the greatest sensitivity lies not between the electrodes, but under each electrode, and has a significant radial component. For distant electrodes, the sensitivity is localized under each electrode separately. It is primarily radial, yet on the periphery of each lobe of the HSV there is some tangential component. This observation refines the intuition that distant electrodes are primarily sensitive to radial dipoles. In summary, both nearby and distant electrodes are sensitive to both radial and tangential dipoles. In both cases, the location of maximum sensitivity is directly under the electrodes, where the lead field L is oriented nearly radially. Thus EEG is predominantly but not exclusively sensitive to radial dipoles. This effect is enhanced by the fact that cortical gyri are populated with radial dipoles and are located closer to the detectors than are sulci.
Figures 6 and 7 show summarizations of the HSV results as a function of the angle 0 between electrodes in the visualization plane. Intuitively, the smaller the HSV, the more refined an estimate of dipole position can be made
Fig. 6. Half-sensitivity volume (HSV) as a function of electrode separation angle 0, for 03/04 = 1/24 (solid) and 03/04 = 1/80 (dashed). Figure (a) is expanded in (b) for small 0
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