IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH

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1 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH Array Response Kernels for EEG and MEG in Multilayer Ellipsoidal Geometry David Gutiérrez*, Member, IEEE, and Arye Nehorai, Fellow, IEEE Abstract We present forward modeling solutions in the form of array response kernels for electroencephalography (EEG) and magnetoencephalography (MEG), assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a dipole current models the source. The use of an ellipsoidal geometry is useful in cases for which incorporating the anisotropy of the head is important but a better model cannot be defined. The structure of our forward solutions facilitates the analysis of the inverse problem by factoring the lead field into a product of the current dipole source and a kernel containing the information corresponding to the head geometry and location of the source and sensors. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search. Our forward solutions have the potential of facilitating the solution of the inverse problem, as they provide algebraic representations suitable for numerical implementation. The applicability of our models is illustrated with numerical examples on real EEG/MEG data of N20 responses. Our results show that the residual data after modeling the N20 response using a dipole for the source and an ellipsoidal geometry for the head is in average lower than the residual remaining when a spherical geometry is used for the same estimated dipole. Index Terms Dipole source signal, electroencephalography (EEG), ellipsoidal head model, magnetoencephalography (MEG), N20 response, sensor array processing. I. INTRODUCTION ARRAY processing methods have been developed to solve problems related to the localization of brain activity sources using electroencephalography (EEG) and magnetoencephalography (MEG) arrays, and the solutions are useful in neurosciences and clinical applications [1], [2]. Solutions to the forward modeling problem in EEG/MEG consist of computing the electric potentials over the scalp and the magnetic field outside the head, respectively, given a current source within the brain. The forward model is necessary for solving the inverse problem (i.e., finding the current distributions using EEG/MEG measurements). Since solving the inverse problem often involves an iterative solution of the forward problem, it Manuscript received February 19, 2007; revised June 11, Asterisk indicates corresponding author *D. Gutiérrez is with Centro de Investigación y Estudios Avanzados (CINVESTAV), Unidad Monterrey, Apodaca, NL 66600, México ( davidgtz@cinvestav.mx). A. Nehorai is with the Department of Electrical and Computer Engineering, Washington University, St. Louis, MO USA ( nehorai@ese.wustl. edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TBME is important to have an efficient form for the analytical and numerical solutions of the forward problem in order to minimize the computational burden [3]. The use of an ellipsoidal geometry to model the head is useful in cases for which incorporating the anisotropy of the head is important but a better model cannot be defined. This is the case of fetal MEG studies [4], where the inaccessibility to the fetal head as well as health issues do not permit the use of tomographic techniques to obtain more realistic head models. Furthermore, the ellipsoidal model is useful in MEG studies in adults, as it decouples not only the source location from the dipole moment, but also the source location from the sensor location, allowing for further simplification in the computation. Recently, second-order approximations for the electric potentials and magnetic fields in multilayer ellipsoidal geometry have been developed [5], [6]. However, the mathematical expressions for those approximations are not suitable for direct use in the inverse neuroelectromagnetic problem. In this paper (see also [7] and [8]), we present forward modeling solutions in the form of array response kernels for EEG/ MEG, assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a dipole current models the source. The structure of our solution facilitates the analysis of the inverse problem by decoupling the dipole source signal (linear parameter) from the source location (nonlinear parameter). We factor the lead field into a product of the current dipole source and kernel. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search [9]. In Section II, we introduce the original formulations of the forward solutions for both EEG and MEG. In Section III, we present the algebraic steps necessary to manipulate the EEG/MEG forward solutions and take them to their factored forms, while in Section IV, we extend those solutions to an array response representation for the case in which measurements are obtained from an array of detectors. In Section V, experiments with real data are used to demonstrate the applicability of our methods to the solution of practical EEG/MEG forward and inverse problems. Finally, our results and future work are discussed in Section VI. II. FORWARD MODELING SOLUTIONS In this section, we present the derivation of the solution to the forward problem of computing the magnetic field outside an ellipsoidal conductor and the electric potential over the surface due to a current dipole source /$ IEEE

2 1104 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 A. Biot-Savart-Maxwell Solution We model both the neuroelectric and neuromagnetic phenomena using the quasi-static approximation of Maxwell s equations given that the time-derivatives of the associated electric and magnetic fields are sufficiently small to be ignored [10]. Under this condition, the static electromagnetic field equations can be written as (1) (2) (3) (4) where is the magnetic field, is the electric field, is the observation point, is the magnetic permeability (assumed to be the same inside and outside the brain), and is the current density. Since is irrotational, it can be represented in terms of the electric potential as The current density can be divided into passive and primary components. The passive currents result from the macroscopic electric field in the conducting medium and are described by the following expression: where is the electric conductivity at. The primary currents can be considered as the sum of the impressed neural current and the microscopic passive cellular currents and are given by (5) (6) Furthermore, assume that the source is modeled by an equivalent current dipole (ECD); i.e., (10) where is the dipole moment, and is the source location. The ECD model is a common simplification whose use is justified in cases where the source dimensions are relatively small compared with the distances from the source to measurement sensors [12], as is often true for evoked response and event-related experiments. Hence, substituting the ECD model in (9) and (8), and through simple vector identities, we can rewrite the volume integrals as a sum of surface integrals. For the case of the magnetic field, we have [13] (11) where and are the conductivities on the inner and outer sides of, respectively, and is the outward unit vector normal to the surface at a point. Meanwhile, the electric potential on the boundary of, where, is given by [14] (12) (7) Under these conditions, the equation that relates and is the integral form of the Biot Savart Maxwell law [11]: (8) where is the source point and indicates the space interior to the volume. Similarly, and are related by B. Forward Solutions for an Ellipsoidal Volume Assume that our regions are bounded by concentric ellipsoids, each defined by the following equation: (13) where are the semi-axes of the th ellipsoid. Suppose, without loss of generality, that (9) In the typical head model, we assume that the head may be represented by different regions (typically four for the brain, cerebrospinal fluid, skull, and scalp, or three when the cerebrospinal fluid is not considered). Another common assumption is that the conductivity is constant and isotropic within these regions. Therefore, the gradient of the conductivity is zero except at the surfaces between regions, which allows the volume integrals to be reworked into surface integrals. Under these conditions, we can assume the regions of our head model are bounded by surfaces, for, going from the inner to the outer region, each with conductivity. Then, (13) defines an ellipsoidal system [15] with coordinates such that,, and, where indicates that any of the ellipsoids can be considered, as all are confocal. The equations connecting the ellipsoidal coordinates to the Cartesian coordinates are given in the Appendix A.

3 GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY ) Multilayer MEG Forward Solution: A multilayer model is assumed so that different layers composing the head are included. Under this condition, using (13) in the evaluation of (11), the solution for the magnetic field in Cartesian coordinates becomes [5] In (15), are the conductivity constants of order and degree. These conductivity constants reflect the effects of the different paths that currents could possibly take within the various conductivity layers, as well as the effect of the geometry in the volumetric currents. The conductivity constants in (15) for layers can be computed by (14) where refer to the components of in the ellipsoidal coordinate system; and are the second-degree exterior solid ellipsoidal harmonics of orders 1 and 2, respectively; is the second-degree elliptical integral of order ; is the 3-D vector with 1 in the th position and zero elsewhere; and are the roots of the quadratic equation with ; denotes ellipsoidal terms of degrees greater than or equal to three; and is the dipole moment modified by the spatial and conductivity effects of the anisotropy imposed by the multilayer ellipsoidal geometry. For the case of layers, is derived in [5] and it is given by (16) (17) a (18) (19) where,,, and are the layer conductivities (from inner to outer layer), and (20) (15) where is the interior Lamé function of order and degree, and is a nuisance variable. Note that the elliptical integrals,

4 1106 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 exterior solid ellipsoidal harmonics, and interior Lamé functions used in (14) (20) are defined in the Appendix B. In a similar way, we can write for the case of layers by and (24) 2) Multilayer EEG Forward Solution: We evaluate (12) for the case when to obtain the electric potential measured at the outer layer. The forward solution under those conditions has been derived in [6] and it is given by (25), which is shown at the bottom of the page. In (25), and are the second-degree interior solid ellipsoidal harmonics (defined in the Appendix B) of orders 1 and 2, respectively. For the case of, constants are given by (16), while for the case of, constants are defined in (22). III. RESPONSE KERNELS Clearly, (14) and (25) are not suitable for a numerical solution of the inverse problem in EEG/MEG. Therefore, in this section we develop novel reformulations to the forward solutions based on algebraic factorizations. Our goal is then to represent the magnetic field as (26) and its corresponding conductivity constants will be given by (21) where is the 3 3 kernel matrix for MEG; and the electric potential as (27) where is the 3 1 kernel vector for EEG. A. MEG Kernel Matrix In order to reach the form of (26), we first need to factor (15) and (21) in a matrix form. We start by defining the following auxiliary matrices: (22) (28) (23) (29) (25)

5 GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY 1107 and then (39) Using (28) (30), we can factor as (30) (31) where (16) (19) are used in the calculation of (29) for the case of defined by (15), and (22) (24) are used for the case of defined by (21). Equation (31) can be further simplified by defining matrix as Then, (31) reduces to (32) (33) Similarly, let us define the support matrices,, and as (34) (35) This result has the advantage of decoupling not only and, but also and, while the effects of the geometry and conductivities of the layers are contained in matrix. B. EEG Kernel Vector In a similar way as in Section III-A, we define the auxiliary vector, and the auxiliary matrices, and as (40) (42), shown at the bottom of the page, where (16) is used to compute for the case of, and (22) is used to compute for the case of, respectively. Therefore, using (40) (42) in (25), and discarding the higher order terms, is expressed as (43) where. From (43) we note that decoupling and in the EEG forward solution is not possible. Finally, a summary of the solution kernels developed in this section is presented in Table I. IV. ARRAY RESPONSE MATRIX In this section, we consider the case in which EEG/MEG measurements are obtained from an array of detectors located at. Then, we can extend (26) and (27) to an array representation that contains all kernel solutions. and (36) A. MEG Array Response Let represent the forward solution of all the array as and. Then, we can Using (33) (36) in (14), and discarding the higher order terms, we can express as We can further simplify (37) by defining as (37) (38) (44) where is the 3 array response matrix. Note that can be written as with (45) This last representation is especially useful for the computer implementation of the forward model. (40) (41) (42)

6 1108 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 TABLE I SUMMARY OF THE SOLUTION KERNELS FOR THE MULTILAYER ELLIPSOIDAL HEAD MODEL In practical situations only some of the components of the full magnetic field are available. Note that each component of can be expressed as (46) (47) (48) where denotes the Kronecker product and is an identity matrix. For any of these components, we can express the array response matrix as (49) Finally, we can easily extend (44) to a spatio-temporal model if we allow to change in time; i.e.,. B. EEG Array Response In a similar way as in Section IV-A, let and define the 3 array response matrix as. Then, we can extend (27) to an array representation that contains all kernel solutions; i.e., while the spatio-temporal model is given by. V. NUMERICAL EXAMPLES (50) We conducted a series of experiments using real EEG/MEG data. The goal of our experiments is to show the applicability of our methods by determining the goodness-of-fit in modeling the data using the proposed multilayer ellipsoidal geometry. We will use a current dipole to model the source. Then, we will compare the unexplained (residual) data obtained from using the ellipsoidal geometry against the residuals produced when a classical multilayer spherical geometry is used. A. Measurements and Models The measurements used in our experiments correspond to real EEG/MEG data of the N20 response from six healthy subjects. The EEG and MEG data were recorded simultaneously over the contralateral somatosensory cortex using a bimodal array with 32 EEG and 31 MEG channels (Philips, Hamburg, Germany). Constant current of 0.2 ms square-wave pulses were delivered to the right or left wrist at a stimulation rate of 4 Hz. The data were sampled at 5000 Hz with a 1500 Hz antialiasing lowpass filter, resulting in 250 time samples for each subject. Even though N20 response is more extended in one direction (from superior-posterior of the wall of the central sulcus to inferior-anterior) and then is a good example where line-source models can be applied [16], [17], we decided to use a single dipole to model the source in order to compare the head models, which is the main interest in this paper. For the head, we used two different models: The first model corresponds to a three-layer ellipsoid where the semi-axes of the outer layer were chosen to fit the scalp of each subject as closely as possible. The dimensions of the inner layers were chosen to be and. The second head model corresponds to a three-layer sphere with radii, where is the radius of the outer sphere that provides the best fit to each subject s scalp. Finally, for the layer conductivities, we used the values of in both head geometries. B. Experiments and Results We applied the spatial filtering method described in [18] to estimate the source location from the measured EEG/MEG data. The search of the source was constrained to the region over the central sulcus, where the N20 generator is know to be located [19]. For the ellipsoidal model we used the forward models proposed in Sections IV-A and IV-B, while the forward solutions for EEG/MEG in [3] were used for the spherical model. Then, we computed the residual data as (51)

7 GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY 1109 Fig. 1. Interpolated iso-contour maps of the original, fitted, and residual potentials over the scalp for one subject at t =20ms. The black dots indicate the position of the electrodes. Maps (a) (c) represent the original potential, the fitted potential using a spherical model, and the fitted potential using an ellipsoidal model, respectively. Maps (d) and (e) represent the residual data for spherical and ellipsoidal model, respectively. Fig. 3. Standard deviation of the residual data over all channels as a function of time for the EEG data of one subject. The dashed dotted line indicates the average value. Fig. 4. Standard deviation of the residual data over all channels as a function of time for the MEG data of one subject. The dashed dotted line indicates the average value. Fig. 2. Interpolated iso-contour maps of the original, fitted, and residual tangential components of the magnetic field outside the head for one subject at t = 20 ms. Maps (a) (c) represent the original field, the fitted field using a spherical model, and the fitted field using an ellipsoidal model, respectively. Maps (d) and (e) represent the residual data for spherical and ellipsoidal model, respectively. where is the vector containing the measured data in all channels at a single time, and is the fitted data using either the ellipsoidal or spherical head model. Note that and are used instead of and, respectively, for notational convenience. Examples of and for one subject at the activation of the N20 response are shown in Figs. 1 and 2 for the EEG and MEG data, respectively. The quantity (51) is essentially the data which is orthogonal to the space spanned by the electric potential or magnetic field induced by a dipole. Once we have computed the residual, we calculated the standard deviation of this residual over all channels and plotted it as a function of time. An example of such plot for one subject is shown in Figs. 3 and 4 for the EEG and MEG data, respectively. In those figures we can note that the standard deviation is, in average over time, lower when we use an ellipsoidal head model than the case when a spherical model is used, and this is true for both the EEG and MEG cases. We repeated this experiment for all six subjects and the average value of the standard deviation, as well as the value at, are shown in Tables II and III for the EEG and MEG data, respectively. Again, our results show that a better fit (indicated by the lower average standard deviation) is achieved in all cases when an ellipsoidal model is used, with an average improvement of up to 45.63% less standard deviation in the residual data (Table II, subject 2).

8 1110 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 TABLE II COMPARISON OF GOODNESS-OF-FIT OF THE EEG DATA USING SPHERICAL AND ELLIPSOIDAL HEAD MODELS APPENDIX A ELLIPSOID COORDINATE SYSTEM The ellipsoidal coordinates the Cartesian coordinates relationships: are connected to through the following (A.1) (A.2) (A.3) TABLE III COMPARISON OF GOODNESS-OF-FIT OF THE MEG DATA USING SPHERICAL AND ELLIPSOIDAL HEAD MODELS where the subscript of the semi-axes that indicates the layer has been dropped for notational convenience as all ellipsoids are assumed confocal. Most of the time we want to go from the Cartesian to the Ellipsoidal coordinates. Therefore, we need to solve the nonlinear system formed by (A.1) (A.3) constrained to the range in values of,, and as described in Section II-B. APPENDIX B ELLIPSOIDAL HARMONICS VI. CONCLUSION We presented a solution to the EEG/MEG forward problem for a multilayer ellipsoidal head model in the form of an array response kernel. This matrix structure has the potential of facilitating the solution of the inverse problem as it provides an algebraic representation suitable for numerical implementations. The simplification is greater in the case of MEG where all the signal and location parameters are decoupled from each other. A series of numerical examples with real EEG/MEG data showed the applicability of our forward solutions in the inverse problem of estimating the source location, and in the direct problem of modeling the data. Our experiments showed that a better fit of the original data may be achieved by using the proposed multilayer ellipsoidal model instead of the classical spherical model. Still, many other factors should be considered in the modeling problem, like the model used for the source and the accuracy of the source localization. However, under similar conditions, the anisotropy introduced by the ellipsoidal model provides advantageous conditions for more accurate data modeling. In future work we will evaluate the accuracy of higher order approximations to the full ellipsoidal solutions, such as those described in [20], in order to determine the optimal expansion size required for the EEG/MEG inverse problem without compromising the overall accuracy of the computation. Future work will also focus on solving this issue by approximations similar to those described in [21] for the spherical head model, as well as the application of our models to the estimation of conductivities in ellipsoidal geometries. In order to evaluate (14) and subsequent equations, we first need to compute the values of the elliptic integrals, the interior solid ellipsoidal harmonics, and the exterior solid ellipsoidal harmonics of different orders and degrees. The elliptic integral of order and degree is given by (B.1) where is the interior Lamé function of order and degree, and is a nuisance variable. In our computations, we are required to evaluate only for degrees less or equal than 2. Under this condition, the interior Lamé functions [15] are given by (B.2) (B.3) (B.4) (B.5) (B.6) Furthermore, the interior solid ellipsoidal harmonics are defined as (B.7) Then, we can express the exterior solid ellipsoidal harmonics as (B.8)

9 GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY 1111 ACKNOWLEDGMENT The authors are thankful to Prof. J. Haueisen of the Neurological University Hospital at Jena, Germany, for providing the real EEG/MEG data used in the paper. REFERENCES [1] C. D. Binnie and P. F. Prior, Electroencephalography, J. Neurol., Neurosurg., Psych., vol. 57, no. 11, pp , [2] C. Baumgartner, Clinical applications of magnetoencephalography, J. Clin. Neurophysiol., vol. 17, no. 2, pp , [3] J. Mosher, R. Leahy, and P. Lewis, EEG and MEG: Forward solutions for inverse methods, IEEE Trans. Biomed. Eng., vol. 46, no. 3, pp , Mar [4] D. Gutiérrez, A. Nehorai, and H. Preissl, Ellipsoid head model for fetal magnetoencephalography: Forward and inverse solutions, Phys. Med. Biol., vol. 50, no. 9, pp , [5] G. Dassios, S. N. Giapalaki, A. N. Kandili, and F. Kariotou, The exterior magnetic field for the multilayer ellipsoidal model of the brain, Quart. J. Mech. Appl. 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Scherg, A fast method for forward computation of multiple-shell spherical head models, Electroencephalography and Clinical Neurophysiology, vol. 90, no. 1, pp , David Gutiérrez (M 05) received the B.Sc. degree (hons) in electrical engineering from the National Autonomous University of Mexico (UNAM), Mexico, in 1997, the M.Sc. degree in electrical engineering and computer sciences, and the Ph.D. degree in bioengineering from the University of Illinois at Chicago (UIC), in 2000 and 2005, respectively. From March 2005 to May 2006, he held a postdoctoral fellowship at the Department of Computer Systems Engineering and Automation, Institute of Research in Applied Mathematics and Systems (IIMAS), UNAM. In June 2006, he joined the Center of Research and Advanced Studies (CINVESTAV) at Monterrey, Mexico. There, he works as Researcher and Academic in the area of medical sciences. His research interests are in statistical signal processing and its applications to biomedicine. He is also interested in image processing, neurosciences, and real-time algorithms. Dr. Gutiérrez is a former Fulbright Scholar and he has been a fellow of the National Council for Science and Technology (CONACYT), Mexico, since Arye Nehorai (S 80 M 83 SM 90 F 94) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion, Haifa, Israel, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA. From 1985 to 1995, he was a faculty member with the Department of Electrical Engineering, Yale University, Hartford, CT. In 1995, he joined as Full Professor the Department of Electrical Engineering and Computer Science at The University of Illinois at Chicago (UIC). From 2000 to 2001, he was Chair of the department s Electrical and Computer Engineering (ECE) Division, which then became a new department. In 2001 he was named University Scholar of the University of Illinois. In 2006, he became Chairman of the Department of Electrical and Systems Engineering at Washington University in St. Louis. He is the inaugural holder of the Eugene and Martha Lohman Professorship and the Director of the Center for Sensor Signal and Information Processing (CSSIP) at WUSTL since Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during the years 2000 to In the years 2003 to 2005 he was Vice President (Publications) of the IEEE Signal Processing Society, Chair of the Publications Board, member of the Board of Governors, and member of the Executive Committee of this Society. From 2003 to 2006 he was the founding editor of the special columns on Leadership Reflections in the IEEE Signal Processing Magazine. He was co-recipient of the IEEE SPS 1989 Senior Award for Best Paper with P. Stoica, coauthor of the 2003 Young Author Best Paper Award and co-recipient of the 2004 Magazine Paper Award with A. Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005 and received the 2006 IEEE SPS Technical Achievement Award. He is the Principal Investigator of the new multidisciplinary university research initiative (MURI) project entitled Adaptive Waveform Diversity for Full Spectral Dominance. He has been a Fellow of the Royal Statistical Society since 1996.