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Model of extended curved current source
Vojko Jazbinšek and ZvonkoTrontelj
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Ljubljana, Slovenia
INTRODUCTION
RESULTS
MODEL DESCRIPTION
In studying the relation between the electric and the magnetic field produced by the current
sources of different forms there is a challenging question[1]: Which are possible current
distributions in the brain and/or the heart that require either magnetic or electric measurements
to obtain the relevant information? A feasible example of a current source, which can be
detected only by the magnetic measurements, is a vortical or curved current distribution.
Here we approximated an extended curved current source with a model assuming a shape of
a circular arc and a constant current along the arc (Fig. 1).
FITTING PROCEDURE
We applied Levenberg-Marquardt least square procedure to find parameters of the arc source.
For comparison, we have also included in the inverse procedure three other simple models
(current dipole, magnetic dipole and extended linear source).
We have fitted these models to measured data obtained from four healthy volunteers [2]:
magnetic field in 119 Bz - channels above the chest and electric potential in 148 leads (Figs. 2,
3). The magnetic data were fitted using all four models, while for the electric potential data only
the current dipole source model was applied. The equivalent current sources were calculated
for several time instants during the QRS complex (Fig 4). All localization results were projected
on axial, frontal and sagittal cross-sectional planes of the torso model (Fig. 2) to check if the
obtained equivalent current source fits into the heart region (Figs. 5, 6). Goodness of fit was
evaluated by the root mean square (RMS) error and relative differences (RD) between
measured maps and maps obtained from a model in the fitting procedure (Table 1). The
influence of measurement noise on the stability of the inverse problem solution was also
studied (Fig. 7).
Fig. 1: Schematic view of the curved current source in the sensor coordinate system S and the source coordinate
system S’. The transformation from S to S’ is defined with the translation by r’, rotation by  around z-axis and
rotation by  around y'-axis.
The current J is approximated by N uniformly distributed current dipoles positioned tangentially
along the arc.
Fig. 2: Anterior view of the torso model with the
epicardial surface, obtained from MRI images.
Electric leads (148) are denoted with •.
Magnetic sensors (119 Bz gradio-meters) were
arranged above the chest in a planar hexagonal
lattice of 30 mm spacing and of 37 cm diameter.
Fig. 3: MCG isofield and ECG isopotential maps
for two data sets A and B, measured 15 and 20
ms after the QRS onset, respectively.
Notation of models
Magnetic data:
• dip - current dipole
• mag - magnetic
dipole
• arc - extended
curved source
• lin - extended
linear source
•Electric data:
• ele - current dipole
Fig. 4: Fitting results for the two data sets A and B (Fig. 3) obtained by different models.
DISCUSSION
The curved current source model gives the best result (the smallest RD and RMS error) for the
selected data sets A and B (Figs. 3-6, Table 1). For instance, RDs for the arc source are only
9.1 % and 6.3%, RMS errors 162 fT and 350fT in the cases A and B, respectively (Table 1).
The peak-to-peak values for these two maps are around 9 and 25 pT (Fig. 3). The equivalent
curved current sources are situated inside the heart region (Fig 5, 6). In the case B, where the
radius of the curved source is around 3 cm and the whole arc of length is around 300°, which
Fig. 5: Projection of curved current source on the
forms almost a closed circular current loop, also the magnetic dipole has RD less then 10%
torso model (Fig. 1) for the two data sets A and B.
Curved arrow represents current flow and the straight
arrow indicates the normal to the source plane. The
radius and the length of the arc are 41 mm, 204° and
29 mm, 303° in the cases A and B, respectively.
and RMS error slightly above 500 fT. For comparison, the current dipole and the linear source
have for this map RD around 36% and RMS error almost 2 pT.
Our study shows that the proposed model of a curved current source can be successfully
applied in the localization procedure particularly in cases where the magnetic isofield map
indicates the presence of possible vortex currents. On the other hand, the curved current
source cannot be identified by the electric measurements since the electric field generated by
the curved current source is equivalent to the field of the straight linear source connecting the
starting and the ending point of the source.
References
[1] Wikswo JP, Theoretical Aspects of the ECG-MCG Relationship, In: Williamson SJ, Romani G-L, Kaufman L, Modena I, Eds.,
Biomagnetism: An Interdisciplinary Approach, New York, Plenum Press, 1983.
[2] Jazbinsek V, Kosch O, Meindl P, Steinhoff U, Trontelj Z, Trahms L. Multichannel vector MFM and BSPM of chest and back. In:
Nenonen J, Ilmoniemi RJ, Katila T, Eds., 12th Int. Conf. on Biomagnetism, Espoo, Helsinki Univ. of Technology, 2001.
Fig. 7: Influence of random noise on the inverse solution. On the measured data we superimposed Gaussian noise
of different levels: 2, 4, 8, 16 and 32 % of the measured field RMS values (1769 fT and 5452 fT for the data sets A
and B from the Fig. 3, respectively). For each noise level and each data set we generated 100 noise distributions.
Graphs display average values of fitting results and their standard deviations obtained by the arc source:
r’ - position,  - radius, 0 - starting angle,  - length, orientation ( and  ) and goodness of fit (RD and RMS).
Table 1: Localization results and goodness of fit for
the two data sets A and B (Figs. 3,4) for different fitting
functions F (dip, lin, mag, arc, ele). Locations (x,y,z)
are presented in the MCG sensor coordinate system
with the origin in the center of the measuring xy-plane,
4 cm above the chest, where x is pointed to the left
side of the thorax, y toward the head and z (depth of
the source) away from the chest.
Fig. 6: Projection of different current sources on
the epicardium model :
× - arc source,
× - magnetic dipole,
× - current dipole (magnetic data),
× - current dipole (electric data).