1. INTRODUCTION 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). Model of extended curved current source Vojko Jazbinšek and ZvonkoTrontelj Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Ljubljana, Slovenia 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. RESULTS 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). 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. 5: Projection of curved current source on the 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. Fig. 6: Projection of different current sources on the epicardium model : × - arc source, × - magnetic dipole, × - current dipole (magnetic data), × - current dipole (electric data). 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. 2: Anterior view of the torso model with the epicardial surface, obtained from MRI images. Electric leads (148) are denoted with. Magnetic sensors (119 B z 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 forms almost a closed circular current loop, also the magnetic dipole has RD less then 10% 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. 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 B z - 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). MODEL DESCRIPTION