Pseudo current density maps of electrophysiological heart, nerve or brain function and their physical basis

1477-044X-4-5 1477-044X Research Pseudo current density maps of electrophysiological heart, nerve or brain function and their physical basis Haberkorn Wolfgang wolfgang.haberkorn@ptb.de Steinhoff Uwe uwe.steinhoff@ptb.de Burghoff Martin martin.burghoff@ptb.de Kosch Olaf olaf.kosch@socratec-pharma.de Morguet Andreas morguet@medizin.fu-berlin.de Koch Hans hans.koch@ptb.de

Physikalisch-Technische Bundesanstalt, Berlin, Germany

Charité Campus Benjamin Franklin, Clinic II, Berlin, Germany

BioMagnetic Research and Technology 1477-044X 2006 4 1 5 http://www.biomagres.com/content/4/1/5 17040559 10.1186/1477-044X-4-5 04 8 2006 13 10 2006 13 10 2006 2006 Haberkorn et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

In recent years the visualization of biomagnetic measurement data by so-called pseudo current density maps or Hosaka-Cohen (HC) transformations became popular.

Methods

The physical basis of these intuitive maps is clarified by means of analytically solvable problems.

Results

Examples in magnetocardiography, magnetoencephalography and magnetoneurography demonstrate the usefulness of this method.

Conclusion

Hardware realizations of the HC-transformation and some similar transformations are discussed which could advantageously support cross-platform comparability of biomagnetic measurements.

Background

In 1976 Cohen et al. introduced in a sequence of publications a method to construct so-called pseudo current density- or arrow-maps from multichannel biomagnetic signals obtained by magnetocardiography (MCG) 1234. The purpose was to transform the measured magnetic field values in a way that the resulting maps could be more easily related to the underlying current density distribution. Later this method was frequently referred to as the Hosaka-Cohen (HC) transformation and its performance was analyzed in some detail 56. However, it did not find widespread application until recent years, when a kind of renaissance of this method occurred. Recently, the HC-transformation is used in MCG 789101112131415161718192021, fetal MCG 222324, magnetoencephalography (MEG) 252627 and magnetoneurography (MNG) 28.

A reason for this new development may be the advance of computing power and visualization tools. In addition, in former times system designers preferred to display magnetic field maps (MFM), since they were interested in the measured physical quantity. However, for the end-user -the physicians- MFMs are not very instructive, as the MFM maximum values do not occur above those positions where the generating currents are flowing.

Figs. 1, 2, 3 illustrate this point: it shows two instants of the atrial excitation marked by the cursors in the MCG-butterfly-plot in Fig. 1 (a butterfly-plot is obtained by superpositioning the MCG-Signals of all channels in one display). The respective pseudo current density (PCD-) plots show very clearly and intuitively the preceding activation over the right atrium (Fig. 2, right) followed by that over the left atrium (Fig. 3, right), whereas the MFMs in Fig. 2 (left) and in Fig. 3 (left) require expert knowledge to interpret them in the same way.

Figure 1

Butterfly plot of a multichannel magnetocardiogramm

Butterfly plot of a multichannel magnetocardiogramm. The two cursors mark the time instants related to the visualizations shown in Fig. 2 and 3 respectively.

Figure 2

Visualization of the atrial activation for the first time instant marked by a cursor in Fig.1

Visualization of the atrial activation for the first time instant marked by a cursor in Fig.1. Left: B_z-map drawn as isocontour maps with a magnetic flux density difference of 0.5pT between adjacent contour lines (red: positive, i.e. directed towards the subject; blue: negative; black: B_z= 0). Right: the corresponding pseudo current density map.

Figure 3

Visualization of the atrial activation for the second time instant marked by the respective cursor in Fig.1

Visualization of the atrial activation for the second time instant marked by the respective cursor in Fig.1.

Other features of modern pseudo current density maps helped to spur interest:

i) while Hosaka and Cohen coded the information of the pseudo current density amplitude into the size of the arrows the recent display techniques added an underlying false-colour scaling to the maps.

ii) visually attractive results are achieved, if a sequence of maps is presented as an animated clip. Then the spatio-temporal dynamics of the electrophysiological function are more easily perceptible.

The question that remains open is: what do pseudo current density maps really show? Already the term "pseudo" indicates that the real current density distribution is different and may deviate considerably. This is already evident when considering the fact that the PCD-maps are only 2D-projections of a 3D reality. The initial papers of Hosaka and Cohen just gave an empirical explanation, why their maps produce an approximate image of the underlying current density distribution. Later explanations e.g. by other authors 7 relating the curl of the measured magnetic induction curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ with the current density j→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGQbGAgaWcaaaa@2E1B@ were incorrect and misleading. Therefore in the following chapters an analytically based calculation is presented that illustrates the physical justification and the limitations of this visualization method.

This paper will not deal with minimum norm estimates or other inverse methods calculating the current density from field maps. Rather, the Hosaka-Cohen transformation provides just another representation of the measured magnetic field by a postprocessing of the magnetic field data. The underlying current distribution does not enter in the calculation of the HC-transformation. We intend to clarify in which way certain features of the PCD-maps can nevertheless be related to the underlying current distribution. Some common fallacies in the interpretation of PCD maps are elucidated.

Finally we would like to stress the utility of PCD maps to provide a visualization of measurement results obtained by biomagnetic systems with very different sensor configurations. Whether magnetometers, planar gradiometers or vector magnetometers are used always similar PCD maps may be computed and will thus allow a simple cross-platform, i.e. multicentre comparability of biomagnetic investigations.

Methods

Construction of pseudo current density maps

PCD-maps are obtained from magnetic field values at a number of points in space 2930. Multichannel measurement systems, containing a number of SQUIDs (superconducting quantum interference devices) as magnetic field sensors, are used to measure the magnetic fields generated by electrophysiological functions in the heart (MCG = magnetocardiography), the brain (MEG = magnetoencephalography) or in other muscles or nerves (MNG = magnetoneurography). In MEG, helmet systems are used, where the SQUIDs are arranged on the surface of a sphere. For other applications, the SQUIDs are distributed more or less in a plane.

For the following discussion a simple current dipole source with current dipole moment

p → = I s → ( 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaiabg2da9iabdMeajjqbdohaZzaalaGaaCzcaiaaxMaadaqadaqaaiabigdaXaGaayjkaiaawMcaaaaa@3586@

may be considered, i.e. a source-drain configuration with vanishing source-drain distance with unit vector s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@ and a source strength of I. This current element generates a magnetic flux density B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@) which – according to the law of Biot-Savart – is expressed as

B → ( r → ) = μ 0 4 π p → × r → r 3 . ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGcbGqgaWcaiabcIcaOiqbdkhaYzaalaGaeiykaKIaeyypa0ZaaSaaaeaaiiGacqWF8oqBdaWgaaWcbaGaeGimaadabeaaaOqaaiabisda0iab=b8aWbaadaWcaaqaaiqbdchaWzaalaGaey41aqRafmOCaiNbaSaaaeaacqWGYbGCdaahaaWcbeqaaiabiodaZaaaaaGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGOmaidacaGLOaGaayzkaaaaaa@43FD@

Often, planar SQUID-systems measure only one component of the B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@-field, e.g.B_z. Fig. 4 shows the B_z-distribution calculated with (2) for a measurement plane of size 40 cm × 40 cm which is positioned 10 cm above a current dipole with a dipole moment of 1 μAm. The direction of the dipole within the x-y-plane is diagonal to the coordinate system and the magnetic flux density distribution is presented as an isocontour plot. The difference between two adjacent contour lines corresponds to a magnetic flux density difference of 0.5 pT. Red lines correspond to positive B_z-values and blue lines to negative B_z-values and the black line marks B_z= 0. The pseudo current density c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@(x, y) in the map in Fig. 5 is gained from the B_z-values by the following transformation

Figure 4

B_z-map of the magnetic flux density calculated from Biot-Savart's law for a current dipole (|p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@| = 1 μAm) at a position 10 cm below the map's plane with a x-y-projection as indicated by the arrow

Figure 5

Psydo current density map corresponding to Fig.4

Psydo current density map corresponding to Fig. 4.

c → = ∂ B z ∂ y e → x − ∂ B z ∂ x e → y . ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaiabg2da9maalaaabaGaeyOaIyRaemOqai0aaSbaaSqaaiabdQha6bqabaaakeaacqGHciITcqWG5bqEaaGafmyzauMbaSaadaWgaaWcbaGaemiEaGhabeaakiabgkHiTmaalaaabaGaeyOaIyRaemOqai0aaSbaaSqaaiabdQha6bqabaaakeaacqGHciITcqWG4baEaaGafmyzauMbaSaadaWgaaWcbaGaemyEaKhabeaakiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIZaWmaiaawIcacaGLPaaaaaa@48FB@

Thus the slopes of the B_z-surface function determine the amplitude and the direction of the pseudo current arrows c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@(x, y). e→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGLbqzgaWcaaaa@2E11@_xand e→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGLbqzgaWcaaaa@2E11@_yare the unit vectors in x- and y-direction.

In practice the partial differential ratios ∂Bz∂x MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdkeacnaaBaaaleaacqWG6bGEaeqaaaGcbaGaeyOaIyRaemiEaGhaaaaa@33C1@ and ∂Bz∂y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdkeacnaaBaaaleaacqWG6bGEaeqaaaGcbaGaeyOaIyRaemyEaKhaaaaa@33C3@ are approximated by the difference ratios ΔBzΔx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabfs5aejabdkeacnaaBaaaleaacqWG6bGEaeqaaaGcbaGaeuiLdqKaemiEaGhaaaaa@33C1@ and ΔBzΔy MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabfs5aejabdkeacnaaBaaaleaacqWG6bGEaeqaaaGcbaGaeuiLdqKaemyEaKhaaaaa@33C3@. They in turn may be easily obtained by utilizing the smooth surface function Bz(xi,yj)|i=0,...,n|j=0,...,m MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaabcaqaamaaeiaabaGaemOqai0aaSbaaSqaaiabdQha6bqabaGcdaqadaqaaiabdIha4naaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaemyEaK3aaSbaaSqaaiabdQgaQbqabaaakiaawIcacaGLPaaaaiaawIa7amaaBaaaleaacqWGPbqAcqGH9aqpcqaIWaamcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGUbGBaeqaaaGccaGLiWoadaWgaaWcbaGaemOAaOMaeyypa0JaeGimaaJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemyBa0gabeaaaaa@4DBB@ that has been used to construct the map in Fig. 4.

The arrows drawn in Fig. 5 represent the c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@-vectors at the respective coordinates. However, only the strongest c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@-vectors are drawn to obtain a clearer picture. Although the amplitude of c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ is coded as the arrow length, a map with just those arrows is not as intuitive as the image shown. By underlying a false-color map scaled by the amplitude |c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@| a considerable visual enhancement of the information is achieved.

As to be anticipated the maximum amplitude occurs just above the source and also the directions of the central strongest arrowc→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ and that of the current dipole p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@ coincide. On the other hand the PCD-map does not reproduce the point-like character of the current dipole! It is rather a characteristic point-spread function of the source.

Another interesting point to mention: the Hosaka-Cohen transformation utilizes two terms of

curl B → = ( ∂ B z ∂ y − ∂ B y ∂ z ) e → x + ( ∂ B x ∂ z − ∂ B z ∂ x ) e → y + ( ∂ B y ∂ x − ∂ B x ∂ y ) e → z . ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@76AD@

As – according to Maxwell – curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ = μ₀j→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGQbGAgaWcaaaa@2E1B@ some authors concluded that this is the rationale for the pseudo-current density maps. However, at the sites where B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ is measured the current density is zero and thus also curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ = 0 holds. Hence no direct relation between j→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGQbGAgaWcaaaa@2E1B@ and c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ and curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ exists at the location of the sensors. But since curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ = 0 everywhere in the measurement space, the two terms that represents c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ must exactly compensate the remaining terms of curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ (from equation 4). This in turn leads to the conclusion that also the remaining terms lead to equivalent arrow maps, as it is discussed later in this paper.

Results

Pseudo current density maps of analytically solvable models

A single current dipole alone (i.e. without considering return currents) and the application of Biot-Savart's law describe a too artificial, non-physical situation. The physical background of the PCD-maps may be evaluated by:

i) modeling the MCG by a current dipole in a conductive half space, and

ii) modeling the MNG by an extended linear or curved source 2831 or by a train of current dipoles in a conducting half space and

iii) modeling the MEG by a current dipole in a conductive sphere.

Of course also those are quite crude models of the reality but they represent basic models of sound physics and can be treated completely analytical. Thus the relations between source and PCD-map and the role of curl B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ are exactly traceable.

Current dipole in a conductive half space

To a first approximation the MCG may be modeled by a current dipole p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@ = (p_x, p_y, p_z) at r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@₀= (x₀, y₀, z₀), representing the heart's electrical activity, in a conductive half space, representing the torso. The coordinate system is chosen such that z = 0 at the boundary between the "torso" with constant conductivity and the non-conducting space containing the measurement sites.

The magnetic flux density at coordinate r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaWcaaaa@2E2B@ = (x, y, z) above the half space (z > 0) is according to 32 given by

B → ( r → ) = μ 0 4 π K 2 { [ ( p → × R → ) e → z ] ∇ K − K e → z × p → } ( 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGcbGqgaWcaiabcIcaOiqbdkhaYzaalaGaeiykaKIaeyypa0ZaaSaaaeaaiiGacqWF8oqBdaWgaaWcbaGaeGimaadabeaaaOqaaiabisda0iab=b8aWjabbccaGiabdUealnaaCaaaleqabaGaeGOmaidaaaaakmaacmqabaWaamWaaeaadaqadaqaaiqbdchaWzaalaGaey41aqRafmOuaiLbaSaaaiaawIcacaGLPaaacuWGLbqzgaWcamaaBaaaleaacqWG6bGEaeqaaaGccaGLBbGaayzxaaGaey4bIeTaem4saSKaeyOeI0Iaem4saSKaeeiiaaIafmyzauMbaSaadaWgaaWcbaGaemOEaOhabeaakiabgEna0kqbdchaWzaalaaacaGL7bGaayzFaaGaaCzcaiaaxMaadaqadaqaaiabiwda1aGaayjkaiaawMcaaaaa@582D@

with R→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGsbGugaWcaaaa@2DEB@ = r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaWcaaaa@2E2B@ - r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaWcaaaa@2E2B@₀, R = |R→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGsbGugaWcaaaa@2DEB@|, K = R (R + R→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGsbGugaWcaaaa@2DEB@e→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGLbqzgaWcaaaa@2E11@_z), and ∇ K=(2+R→ e→zR)R→+Re→z MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHhis0cqqGGaaicqWGlbWscqGH9aqpcqGGOaakcqaIYaGmcqGHRaWkdaWcaaqaaiqbdkfaszaalaGaeeiiaaIafmyzauMbaSaadaWgaaWcbaGaemOEaOhabeaaaOqaaiabdkfasbaacqGGPaqkcuWGsbGugaWcaiabgUcaRiabdkfasjqbcwgaLzaalaWaaSbaaSqaaiabdQha6bqabaaaaa@415A@, where ∇ is the nabla operator. In Cartesian coordinates B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ can be explicitly written as

B x ( r → ) = μ 0 4 π p x X Y X 2 + Y 2 [ 2 X 2 + Y 2 ( 1 − Z R ) − Z R 3 ] + μ 0 4 π p y 1 X 2 + Y 2 [ Y 2 − X 2 X 2 + Y 2 ( 1 − Z R ) + X 2 Z R 3 ] , ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=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@96B9@

B y ( r → ) = μ 0 4 π p x 1 X 2 + Y 2 [ Y 2 − X 2 X 2 + Y 2 ( 1 − Z R ) − Y 2 Z R 3 ] − μ 0 4 π p y X Y X 2 + Y 2 [ 2 X 2 + Y 2 ( 1 − Z R ) − Z R 3 ] , ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=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@9B73@

B z ( r → ) = μ 0 4 π Y p x − X p y R 3 ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGcbGqdaWgaaWcbaGaemOEaOhabeaakmaabmaabaGafmOCaiNbaSaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaGGaciab=X7aTnaaBaaaleaacqaIWaamaeqaaaGcbaGaeGinaqJae8hWdahaamaalaaabaGaemywaKLaemiCaa3aaSbaaSqaaiabdIha4bqabaGccqGHsislcqWGybawcqWGWbaCdaWgaaWcbaGaemyEaKhabeaaaOqaaiabdkfasnaaCaaaleqabaGaeG4mamdaaaaakiaaxMaacaWLjaWaaeWaaeaacqaI4aaoaiaawIcacaGLPaaaaaa@48DF@

with X = (x - x₀), Y = (y - y₀) and Z = (z - z₀).

An inspection of (6)–(8) shows that

• p_zdoes not contribute to B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@) above z > 0,

• B_x(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@), B_y(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@), and B_z(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@) do not depend on the position of the torso boundary as long as it is between measurement point and current dipole,

• the difference to the B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@ – field calculated by Biot-Savart's law for an isolated current dipole occurs only in (6) and (7),

• B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOqaiKbaSaaaaa@3E0A@(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGafmOCaiNbaSaaaaa@3E6A@) is independent of the value of the constant conductivity in the half space.

Note that the above field properties are also valid for a horizontally layered conductor, i.e. for a conductivity σ = σ (z).

Now the Hosaka-Cohen transformation (3) is applied to (8) and yields

c → = μ 0 4 π [ p x e → x + p y e → y R 3 − 3 { [ p x Y 2 − p y X Y ] e → x + [ p y X 2 − p x X Y ] e → y } R 5 ] . ( 9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7456@

Particularly for X = 0, Y = 0, i.e. directly above the current dipole, one obtains

c → = μ 0 4 π p x e → x + p y e → y Z 3 . ( 10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaiabg2da9maalaaabaacciGae8hVd02aaSbaaSqaaiabicdaWaqabaaakeaacqaI0aancqWFapaCaaWaaSaaaeaacqWGWbaCdaWgaaWcbaGaemiEaGhabeaakiqbdwgaLzaalaWaaSbaaSqaaiabdIha4bqabaGccqGHRaWkcqWGWbaCdaWgaaWcbaGaemyEaKhabeaakiqbdwgaLzaalaWaaSbaaSqaaiabdMha5bqabaaakeaacqWGAbGwdaahaaWcbeqaaiabiodaZaaaaaGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGymaeJaeGimaadacaGLOaGaayzkaaaaaa@49F7@

In this case c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ is directly proportional to the x-y-projection of the current dipole moment p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@.

This supports the argument that the Hosaka-Cohen transformation is really related to the underlying current source. However, it is also evident from (9) that additional terms are blurring and distorting the image.

On first sight the distribution of arrows might suggest that this is an image not only of the current dipole but also of the return currents (also termed: volume currents). And indeed, the model "dipole current in a conductive half space" considers the role of the return currents. However, in this special geometry, the volume currents do not contribute to B_z(r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaWcaaaa@2E2B@)as can be seen above. It becomes also evident, that the spatial distribution of c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ away from X = 0, Y = 0 does not represent the return currents if Z is varied. Without loss in validity of equations (6)–(8) one may consider that p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@ is very close to the half space interface z = 0 and the measurement of B_z(x, y) is performed at different distances approaching p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@. In this theoretical case, the image approximates in the limit (z - z₀) = 0 a point-like distribution with vanishing c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@(x, y) apart from the origin X = 0, Y = 0. However, the volume currents keep their amplitude independently from z as only the measurement device is moved and not the current dipole source. Thus the nature of the c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ – image is a point spread function of non-radial symmetry.

A closer look to just one component (e.g. the x-component) of c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@ reveals that it is composed of two terms

c x = ∂ B z ∂ y = μ 0 4 π [ p x R 3 − 3 Y [ Y p x − X p y ] R 5 ] . (11) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@656A@

The spatial distribution of both terms is shown in Fig. 6 as a solid line. While the first term – shown as a dotted line – is radially symmetric the second term is not. Along the symmetry axis parallel to the direction of the dipole this latter term is vanishing, see the dotted line in Fig. 6. Unfortunately the second term is of the same order of magnitude as the first term. Thus c_xis not directly proportional to p_xas the second term contains mixed terms. However, it contributes a kind of focussing effect.

Figure 6

Spatial dependence of |c→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaWcaaaa@2E0D@_p| along the symmetry axis orthogonal to the dipole direction. Dashed line: refers to the first term in eq. (11); dotted line: refers to the second term in eq. (11); solid line: both terms. Same data as in Fig. 5, however the direction of the dipole is in x-direction.

Current dipole in a conductive sphere

For a current dipole p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@ at r→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaWcaaaa@2E2B@