1692 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 4,JULY 2000 Equivalent Ellipsoid as an Interpretation Tool of Extended Current Distributions in Biomagnetic Inverse Problems Marek Ziolkowski, Jens Haueisen, Hannes Nowak, and Hartmut Brauer, Member, IEEE Abstract—The paper presents an equivalent ellipsoid approach for interpretation and visualization of extended current distribu- tions in biomagnetic inverse problems. The example of simulations performed with physical thorax phantom is also given. Index Terms—Biomagnetics, inverse problems, modeling, visu- alization. I. INTRODUCTION I N the last decade the number of biomagnetic measurement systems available in clinical centers is rapidly growing up. Biomagnetic measurements can provide information of the actual behavior of electric active organs, like heart or brain, in noninvasive way. The interpretation of measured, extremely weak ( T), magnetic fields generated by human organs requires the application of special algorithms for local- ization/reconstruction of the sources. One class of methods for finding a distribution of extended current sources is based on a minimum norm approach. It is assumed that the total length of the vector that represents the best fitting current dipoles is mini- mized. Minimum norm solutions of the inverse problems in bio- magnetism have been usually presented as color coded current density maps or current dipoles sets on brain/heart surfaces or in other defined regions like planes, volumes, etc. [1] These rep- resentations are usually interpreted as activation maps/volumes and the data analysis is often broken at this point. However, for statistical data analysis a method is needed which enables us to compare current density distributions for different formula- tions/hypotheses and within groups of patients or volunteers. To achieve this goal we propose to use a new technique based on a parameterization of current density distributions by means of equivalent ellipsoids. In this paper we would like to define the equivalent ellipsoid technique and to concentrate on the visual- ization aspects of proposed method. The results received during simulations with the models of extended sources placed in a re- Manuscript received October 25, 1999. M. Ziolkowski is Chair of Theoretical Electrotechnics and Computer Science, Technical University of Szczecin, Al. Piastów 19, 70–310 Szczecin, Poland (e-mail: mz@ps.pl). J. Haueisen is with the Biomagnetic Center, Friedrich-Schiller-University Jena, Philosophenweg 3, D-07740 Jena, Germany (e-mail: haueisen@ biomag.uni-jena.de). H. Nowak is with the JENASENSORIC e.V., Jena, Germany (e-mail: hnowak@biomag.uni-jena.de). H. Brauer is with the Ilmenau University of Technology, P.O.Box 100565, D-98684 Ilmenau/Thür., Germany (e-mail: brauer@e-technik.tu-ilmenau.de). Publisher Item Identifier S 0018-9464(00)04968-2. Fig. 1. Kurtosis values for different 1D distributions. alistically shaped torso-phantom are used as an illustration of the method. II. DATA VALIDATION PROCEDURES A. Statistical Description of Data For a statistical description of reconstructed extended cur- rent distributions we have used well known parameters [2] such as: mean value, which enables to estimate the value around which central clustering occurs, variance, a measure of “vari- ability” around mean value, standard deviation —a measure of “width” of distribution around the mean value, and also less popular like: skewness (third central moment), which character- izes the degree of asymmetry of the distribution around its mean and kurtosis—a measure of the relative peakness/flatness of the distribution, defined below as (Fig. 1): (1) The above parameters are used as the first step in valida- tions of different extended current distributions. For more pre- cise comparisons, we introduce an equivalent ellipsoid approach which enables not only a fast visualization but also a better in- terpretation of reconstructed currents. B. Equivalent Ellipsoid The equivalent ellipsoid is defined as an object connected with a certain concentration of current dipoles representing the current density distribution respectively to a predefined threshold. It is chosen for its geometrical simplicity (easy to draw) and straightforward interpretation of axes connected with 0018–9464/00$10.00 © 2000 IEEE