The optimal dimensions of thin coil systems of three and four square coils for producing uniform magnetic fields are calculated. We find that for three square coils, of side d and separation s between the outer coils, the most uniform field distribution occurs with s/ d = 0.821 116 and with l' /1 = 0.512 797. 1'/1 is the ratio of the currents in the center coil to that of the outer coils. With four square coils, the best uniformity is obtained when a/ d = 0.128 106 and b / d = 0.505 492, where a is the distance from the center to the inner coils and b is the distance from the center to the outer coils. The ratio of the current in the inner pair of coils to that in the outer pair must be 1'/ 1 = 0.423514. We compare the uniformity of the field produced by these coil systems with each other and with Rubens' five-coil system, both on and offaxis. It is shown that the optimal four-coil design is superior to the three-and five-coil systems. The sensitivity of the uniformity to the precision of construction is discussed. Dimensions of regions around the center of the coil systems, uniform to 1 part in 10 6 to 1 part in 10 2 , are given.

An analytic solution is derived for the magnetic field generated by current sources located in a piecewise homogeneous volume conductor. A linear discretization approach is used, whereby the surface potential is assumed to be a piecewise linear function over each tessellated surface defining the regions of differing conductivity. The magnetic field is shown to be a linear combination of vector functions which are strictly dependent on the geometry of the problem, the surface tesselation, and the observation point.

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BackgroundThe electroencephalogram (EEG) reflects the electrical activity in the brain on the surface of scalp. A major challenge in this field is the localization of sources in the brain responsible for eliciting the EEG signal measured at the scalp. In order to estimate the location of these sources, one must correctly model the sources, i.e., dipoles, as well as the volume conductor in which the resulting currents flow. In this study, we investigate the effects of dipole depth and orientation on source localization with varying sets of simulated random noise in 4 realistic head models.MethodsDipole simulations were performed using realistic head models and using the boundary element method (BEM). In all, 92 dipole locations placed in temporal and parietal regions of the head with varying depth and orientation were investigated along with 6 different levels of simulated random noise. Localization errors due to dipole depth, orientation and noise were investigated.ResultsThe results indicate that there are no significant differences in localization error due tangential and radial dipoles. With high levels of simulated Gaussian noise, localization errors are depth-dependant. For low levels of added noise, errors are similar for both deep and superficial sources.ConclusionIt was found that if the signal-to-noise ratio is above a certain threshold, localization errors in realistic head models are, on average the same for deep and superficial sources. As the noise increases, localization errors increase, particularly for deep sources.

We describe a fast and numerically effective biomagnetic inverse solution using a moving dipole in a realistic homogeneous torso. We applied the localization model and high-resolution magnetocardiographic mapping to localize noninvasively the ventricular preexcitation site in ten patients suffering from Wolff-Parkinson-White syndrome. In all cases, the computed localization results were compared to the results obtained by invasive catheter technique. Using a standard-size torso model in all cases, the average 3-D distance between the computed noninvasive locations and the invasively obtained results was 2.8 +/- 1.4 cm. When the torso was rescaled to better match the true shape of the subject in five cases, the 3-D average was improved to 2.2 +/- 1.0 cm. This accuracy is very satisfactory, suggesting that the method would be clinically useful.