IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 5, MAY 2005 839 Line-Source Modeling and Estimation With Magnetoencephalography ˙ Imam S ¸amil Yetik*, Arye Nehorai, Fellow, IEEE, Carlos H. Muravchik, Senior Member, IEEE, and Jens Haueisen, Member, IEEE Abstract—We propose a number of source models that are spa- tially distributed on a line for magnetoencephalography (MEG) using both a spherical head with radial sensors for more efficient computation and a realistic head model for more accurate results. We develop these models with increasing degrees of freedom, de- rive forward solutions, maximum-likelihood (ML) estimates, and Cramér-Rao bound (CRB) expressions for the unknown source pa- rameters. A model selection method is applied to select the most ap- propriate model. We also present numerical examples to compare the performances and computational costs of the different models, to determine the regions where better estimates are possible and when it is possible to distinguish between line and focal sources. We demonstrate the usefulness of the proposed line-source models over the previously available focal source model in certain distributed source cases. Finally, we apply our methods to real MEG data, the N20 response after electric stimulation of the median nerve known to be an extended source. Index Terms—Line-source models, magnetoencephalography, N20 responses, source localization. I. INTRODUCTION M AGNETOENCEPHALOGRAPHY (MEG) and elec- troencephalography (EEG) are noninvasive techniques for analyzing spatial and temporal electrical activities in the brain with high temporal resolution. Locating electrical sources using MEG/EEG has broad applications ranging from clinical (for instance finding the position of the abnormality before a surgery in epilepsy) to neuroscientific (such as locating parts of the brain that control certain physical activities). In neuroscience typically a stimulus is applied and MEG/EEG measurements are recorded during the course of the activation, whereas in the case of epilepsy measurements of spontaneous activities are taken. These data are then used to infer certain properties of the source. For some applications, models with a single focal source [1], [2], a moving source [3] or multiple focal sources [4] are suffi- Manuscript received January 12, 2004; revised October 10, 2004. The work of ˙ I. S ¸. Yetik and A. Nehorai was supported by the the National Science Foundation (NSF) under Grant CCR-0105334 and Grant CCR-0330342. The work of C. H. Muravchik was supported by the CIC-PBA, UNLP, and ANPCTIP of Argentina. Asterisk indicates corresponding author. * ˙ I. S ¸. Yetik is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, 851 S. Morgan, Room 1020 SEO, Chicago, IL 60607 USA (e-mail: syetik@ece.uic.edu). A. Nehorai is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA. C. H. Muravchik is with Universidad Nacional de La Plata, 1900 La Plata, Argentina. J. Haueisen is with Friedrich-Schiller-University, 07743 Jena, Germany. Digital Object Identifier 10.1109/TBME.2005.844276 cient. Single focal source models are valid only when the elec- trical activity is confined to a very small area, and multiple focal sources may be useful for multiple, individually concentrated, separated sources. The performances of estimating these models degrade when the region of activity gets larger, as a result there is a need to apply distributed source models. One method for studying distributed sources is to divide the brain into many small voxels using a discrete grid. Each of these voxels is assumed to potentially carry a dipole source as in [5]–[8], and then the parameters of these sources are estimated. However, there are several problems with this approach. First, estimating large number of parameters is an ill-posed problem without a unique solution. Secondly, the computational com- plexity can render such methods unattractive. The ill-posed problem is often tackled using regularization techniques [6], [9] to obtain a unique solution. There are fundamental differences between our proposed models and these distributed source models. In particular, the number of parameters that define the spatial distribution of the source are much smaller compared with the distributed source models. The distributed source models are useful for estimating the electrical activity in the whole brain, and in general do not have spatial restrictions. The proposed models have spatial restrictions making them more useful for electrical sources concentrated in a certain area. A different approach for reducing the number of parame- ters is considered in [10]–[13] where the distance between a sensor and the source is presented as a function. This function is then expressed using a Taylor series expansion, and the third or higher order terms are truncated to decrease the number of unknown parameters. However, in [10] for example, the extent of the source is defined as the standard deviation of the support function of the source rather than the actual physical limits. The methods proposed in this paper directly estimate the limits of the source which is a more direct way to find its extent and are more accurate than the truncated Taylor series expansion. We propose four different line-source dipole models for MEG and investigate their several aspects. In Section II we develop the line-source models in an increasing order of complexity. For each of the proposed models, we give the forward solution, derive the maximum-likelihood (ML) estimates and Cramér-Rao bounds (CRBs) for the unknown parameters assuming a spherical head and radial sensors since the resulting integrals cannot be solved analytically using known functions otherwise. We also discuss how these models are interrelated. Section III explains the cases of low-rank gain matrices, i.e., the cases where we cannot uniquely estimate certain parameters. Section V is devoted to numerical 0018-9294/$20.00 © 2005 IEEE