Approximating Dipoles from Human EEG Activity: The Effect of Dipole Source Configuration on Dipolarity Using Single Dipole Models

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1 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 2, FEBRUARY Approximating Dipoles from Human EEG Activity: The Effect of Dipole Source Configuration on Dipolarity Using Single Dipole Models Junko Hara,* Toshimitsu Musha, Member, IEEE, and William Rodman Shankle Abstract Dipolarity is the goodness-of-fit of the observed potential distribution with one calculated using specific assumptions about the source of the electrical potential distribution. We used computer simulations to examine the effect of different distributions of sources on their resulting dipolarity values. Electric dipoles were placed in a head-shaped model with uniform conductivity using four different dipole configurations (randomly oriented dipoles, a uniform dipole disk layer, a dipole disk layer with uniformly distributed holes, or one with randomly oriented dipoles). The best-fitting single dipole for each configuration was calculated and the dipolarity was computed as the mean squared error of the electrical potential distributions generated by the actual dipole configuration and by the best-fitting single dipole. The simulations show that: 1) a smooth dipole layer with or without holes gives dipolarities above 99.5% even when extended over areas as large as 1256 mm 2 ; 2) randomly oriented dipoles under a smooth dipole layer also give dipolarities above 99.5%; and 3) randomly oriented and distributed dipoles, even if contained in a small portion of the total area, give dipolarities below 93.0%. These simulations show that inhomogeneity (holes) within a dipole disk layer per se do not lower dipolarity; rather, it is the random orientation and distribution of these dipoles which lowers dipolarity. Furthermore, dipolarity is not lowered by such randomly oriented and distributed dipoles when they are beneath a dipole disk layer. Index Terms Dipolarity, electroencephalogram (EEG), randomly oriented dipoles, source configuration, source localization. I. INTRODUCTION ELECTRIC generators in the cerebral cortex are often approximated by several equivalent current dipoles through electroencephalographic (EEG) measurement. The term, equivalent dipole, is widely used, sometimes without mention. Homma et al. [1] have defined equivalent dipole as that which best approximates the electrical phenomenon and minimizes the mean squared error between the observed and modeled potential distributions. In this paper, we use the term, equivalent dipole, as defined by Homma et al. Dipole approximations have been used, for example, to localize epileptic foci from scalp potential distributions of Manuscript received May 14, 1998; revised July 23, Asterisk indicates corresponding author. *J. Hara is with the Graduate School of Media and Governance, Keio University, 5322 Endo, Fujisawa Kanagawa , Japan ( junkoh@mag.keio.ac.jp). T. Musha is with the Brain Functions Laboratory, Inc., Kawasaki, Kanagawa , Japan. W. R. Shankle is with the Department of Cognitive Science, University of California, Irvine, CA USA. Publisher Item Identifier S (99) interictal spikes [2], and to localize the spatial and temporal sequence of electric generators during the neural processing of visually-presented letters [3]. These localization estimates are made by finding the equivalent current dipoles which best fit the observed electrical potential distribution over the scalp electrodes (the inverse solution). The goodness-of-fit or dipolarity,, of this solution is expressed as where and are observed and calculated potentials at electrode sites, respectively, and is the number of scalp electrodes. Interictal spikes have a high dipolarity because the epileptic focus is very concentrated in space. In cognitive process, dipolarity is large at certain latencies. Musha and Homma [4] showed that dipole localization is accurate when dipolarity is larger than 98% in the single-dipole model and larger than 99% in the two-dipole model. When signal-to-noise ratio is low, dipolarity will also be low regardless of the head model and electric generator conditions [5]. To improve source localization accuracy and reliability, several studies solved the inverse problem using different head models [6] [9], among which the most realistic one used uniformly conducting electrical compartments for scalp, skull, cerebrospinal fluid, and brain tissue with different electric conductivity. Dipole localization studies using electrodes implanted in the human brain have shown that electric current generators closer to the cortical surface give much better dipole approximations than deep generators [10] [12]. Also, using a larger number of electrodes slightly improves the dipole approximation of the source generator [13] [15]. Even though dipolarity has been used to measure the accuracy of dipole approximations of electric source generators [3], [6], [16], we know of no studies of how dipolarity is affected by different electric source generator configurations in the human brain. In the present study, we simulated four different configurations of noise-free electric generators in a human head model with uniform conductivity and calculated their dipolarities. II. METHOD Throughout the present computer simulation, a head model with uniform electric conductivity has been adopted, which /99$ IEEE

2 126 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 2, FEBRUARY 1999 Fig. 2. Illustration of the third electric source configuration used in the present study. The disk dipole layer has holes of uniform diameter (inhomogeneity) with up to 1000 randomly oriented single dipoles distributed randomly beneath the dipole disk layer within a cylindrical volume of radius = 15 mm and height = 10 mm. For different simulations of this electric source configuration, hole diameter was set to different values such that the proportion of the disk layer occupied by inhomogeneity varied between 0% and 45%. Fig. 1. The first electric source generator configuration used in the present study is shown as a disk dipole layer with radius rd (rd ranges from 0 to 20 mm) with no holes (homogeneous). The disk layer is decomposed into 1 mm 2 units with a dipole (moment = 1) placed on each unit. Each dipole is oriented perpendicular to the disk layer. The disk layer is oriented parallel to the x y plane at elevations of z =53or z =68mm. The maximum surface area of the dipole disk layer is 1256 mm 2 (rd =20mm). simplifies computation without loss of generality in studying the relation between the electric generator distribution and the dipolarity. Several electric source generator configurations are specified in a realistic head-shaped model, and the boundary element method [17] is used to calculate the surface potential at each scalp electrode. From this calculated potential distribution, an equivalent dipole location and vector moment are determined using the simplex method [18]. This equivalent dipole is then compared to the actual electric source generator configuration to determine the goodness-of-fit (dipolarity). Throughout the remainder of this article, dipolarity will be used instead of goodness-of-fit. A. Head Model The surface of the head model is constructed from MR Images taken at 3 mm intervals. The head is covered with 328 nodes and 652 triangular elements for the boundary element method for forward calculation of the surface potentials. The -axis connects the nasion and the inion, and the -axis is perpendicular to the -axis at its midpoint and parallel to the line connection the two auditory meati. The -axis is perpendicular to the - and the -axis through their intersection point. B. Source Generator In the present simulation, four different dipole configurations are used. All dipoles in all simulations are given a moment 1. In the first configuration (Fig. 1), dipoles, uniformly distributed on a disk layer with radius (ranging from 0 to 20 mm), each occupy 1 mm of the disk and are oriented perpendicular to the surface. The second configuration is identical to the first except that the disk layer has radius 15 mm and that holes (moment zero) of constant radius are uniformly spaced throughout the dipole disk layer. The hole was determined so that the hole replaced either 4, 13, or 28 dipoles for different simulations. The degree of inhomogeneity of the dipole disk layer was varied between 0% and 45% for different simulations by altering, the ratio between the number of dipoles perpendicular to the surface and the sum of dipoles forming the holes. The third configuration (Fig. 2) used the structure of the second configuration with a disk layer radius, mm, and added randomly distributed and oriented dipoles up to 10 mm beneath the dipole disk layer. The third configuration, therefore, occupied a cylinder with height mm and radius mm. For different simulations, was varied between 0% and 45%, and, the ratio between and the number of dipoles with random orientation to the surface was varied between 32 and 700. The fourth configuration has no dipole disk layer but randomly distributes up to 1000 randomly oriented dipoles into a cylinder with radius 15 mm and height 10 mm. For each value of, the dipoles were randomly distributed 200 times and the dipolarity was calculated for each distribution. Fig. 4 shows the dipolarity as a function of and the variance of dipolarity among the 200 simulations per value of. C. Calculation of Equivalent Dipoles Potentials generated by a specific dipole configuration in the head model are defined as. The values of are calculated using the boundary element method at 21 electrode sites arranged according to the International standard system. is, therefore, a twenty-one-dimensional (21-D) column vector where and are the dipole moment and location, respectively, is the number of dipoles in the configuration, and is the transfer matrix. To calculate, the expected value of, in an iterative fashion, an initial moment and position are arbitrarily selected

3 HARA et al.: APPROXIMATING DIPOLES FROM HUMAN EEG ACTIVITY 127 The squared error between and is defined as An equivalent dipole which best fits is obtained by minimizing. Let the values for and which minimize be defined as and. To minimize, we first find as a function of Substituting the formula for into gives where is a 21-D unit matrix. Since is a function of, becomes a function of. The location of an equivalent dipole,, which minimizes is searched for by means of the simplex method [18], and the vector moment of the equivalent dipole, is given as. The iterative solution of finally converges to. D. Dipolarity In the present simulation, dipolarity,, is calculated as III. RESULTS A. Spreading Electric Generators Fig. 3 shows the dipolarity values of the equivalent dipole for the first electric source configuration. Dipolarity is high for all disk radii tested (at mm, % for and 99.60% for mm; at mm, % for mm and 99.72% for mm). In the case of mm, only one electric dipole is assigned in the head model. For distributed electric source generators (i.e., mm), the accuracy of the equivalent dipole s location decreases because a widely distributed surface potential with uniform amplitude (as is generated by a uniform dipole disk layer at any depth) will produce a single equivalent dipole located deeper than the true electric source generator. B. Perforated Disk Source Table I gives dipolarity values for the second electric source generator configuration (see Method Section), in which dipole disk layer inhomogeneity (holes) of up to 45% ( values) were simulated. This source configuration simulates EEG potential activity generated by several smooth, loosely connected dipole layers. Dipolarity is high for all values of. C. Perforated Disk Source and Randomly Oriented Dipoles Simulations using the third electric source generator configuration (Fig. 3), show that randomly oriented and distributed single dipoles beneath a dipole disk layer have little effect on dipolarity (dipolarity values are 99.5%). Furthermore, the effect of these single dipoles on dipolarity does not Fig. 3. Dipolarity and the estimated depth of an equivalent dipole (ED) for the first electric source configuration. The dipole disk layer is placed at elevations of z =53mm and z =68mm. The dipolarity of these disk layers is shown by the black squares (z =68mm) and the white squares (z =53mm) as a function of the disk radius, r d. The estimated depth of the equivalent dipole of these disk layers is shown by the black triangles (z =68 mm) and by the white triangles (z =53mm) as a function of the radius, r d. The accuracy in estimating the depth of the equivalent dipole is much poorer for the deeper dipole disk layer (z =53mm). Dipolarity is essentially unaffected by the radius of the dipole disk layer. depend upon whether the dipole disk layer has inhomogeneity of up to 45%. For comparison, when the radius of the perforated disk layer is reduced to 5 mm, dipolarity decreases as either or as cylinder height increases. The results of such narrow, perforated disk dipole layers with randomly oriented dipoles underneath approach those obtained by the fourth configuration of source electric generators. D. Randomly Oriented Electric Source Generators Fig. 4(a) shows the effect on dipolarity due to dipoles that are randomly oriented and distributed within a cylinder without an overlying dipole disk layer (the fourth electric source generator configuration). As shown by the large variance [Fig. 4(b)], dipolarity is sensitive to the number of randomly oriented dipoles within the cylindrical volume when no disk dipole layer is present above. A large variance of dipolarity is also obtained when the randomly oriented single dipoles are replaced with small dipole disks of radius 5 mm. IV. DISCUSSION AND CONCLUSIONS Results of the present simulation are summarized as follows. 1) Uniform or inhomogeneous (perforated) electric source generators with areas as large as 1256 mm ( mm) give dipolarities 99.5%.

4 128 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 2, FEBRUARY 1999 TABLE I EFFECT ON DIPOLARITY DUE TO INHOMOGENEITY (HOLES) WITHIN THE DIPOLE DISK LAYER USING THE SECOND ELECTRIC SOURCE GENERATOR CONFIGURATION (SEE METHOD SECTION). DIFFERENT SIMULATIONS USING THIS ELECTRIC SOURCE GENERATOR CONFIGURATION USED A CONSTANT HOLE RADIUS SUCH THAT THE HOLE OCCUPIED EITHER 4, 13, OR 28 DIPOLES. SIMULATIONS ALSO VARIED THE PROPORTION OF THE TOTAL DISK AREA OCCUPIED BY INHOMOGENEITY BETWEEN 0% AND 45%. THE RESULTS SHOW THAT THERE IS NO EFFECT ON DIPOLARITY DUE TO HOLE SIZE OR DEGREE OF INHOMOGENEITY WITHIN THE DISK DIPOLE LAYER (a) 2) Randomly oriented dipole generators under a uniform or inhomogeneous dipole disk layer do not lower dipolarity. 3) In the absence of an overlying dipole disk layer, randomly oriented and distributed electric source generators within cylindrical volumes (7068 mm ) remarkably reduce dipolarity. In order to investigate the relation between dipolarity and the configuration of randomly oriented dipoles, we define as a measure of randomness of the dipole configuration is the magnitude of the total dipole moment, normalized by ( 1), the average dipole moment where individual vector dipole moments are,, and. is the summed Euclidean distances of all possible dipole pairs, normalized by, the square root of the dimensions of the cylinder,,,,, and, are the coordinates for each pair of dipoles, and, respectively, and equals the number Fig. 4. (b) (a) The effect on dipolarity using the fourth electric source configuration of the present study. In this configuration, there is no overlying dipole disk layer. Dipolarity and its variance are plotted against the number of randomly oriented and distributed dipoles (Nr) within a cylindrical volume of radius = 15 mm and height = 10 mm. For each value of Nr, the dipoles were randomly distributed 200 times and the dipolarity calculated for each distribution. (b) The large variance at each value of Nr indicates that dipolarity is sensitive to the distribution of the randomly oriented dipoles. Dipolarity is strongly affected by the number of single dipoles when they are not masked by an overlying disk dipole layer. of possible pairs among dipoles. denotes the degree of spatial scatter of the individual dipoles within the specified volume, such that large amounts of spatial scatter reduce. Relatively small values of occur when the dipoles are randomly oriented because is a vector sum., therefore, decreases as the degree of randomness of individual dipole orientation increases. A large occurs when individual dipole moments are coherently oriented in a certain direction (high ) and they are relatively concentrated in space (low ). Electric source generators consisting of a single dipole will, therefore, have large values of both and of dipolarity. The relation between the dipolarity and is shown in Fig. 5. In conclusion, our simulations show that dipolarity, a measure of the goodness-of-fit between electric source generators and their estimated equivalent dipoles, is not affected by inhomogeneity within a dipole disk layer and is not affected by randomly oriented electric source generators beneath a dipole disk layer. However, randomly oriented and distributed

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Hagbarth, Conductivity ratio of the scalp-skull-brain model in estimating equivalent dipole sources in human brain, Neurosci. Res., vol. 22, pp , Junko Hara was born in Tokushima, Japan, on November 15, She received the B.E. degree in information science and intelligent systems from the University of Tokushima, Tokushima, Japan, in 1994, and the M.S. degree in media and governance from Keio University, Kanagawa, Japan, in She is working towards the the Ph.D. degree in media and governance at Keio University. She is working at the Department of Information and Computer Science, the University of California, Irvine, as a Research Specialist. Her research interests include computational modeling for electrical activities of a cortex with Alzheimer s disease and for human cortical development. Toshimitsu Musha (M 86) was born on June 29, 1931 in Tokyo, Japan. He graduated from the Department of Physics, the University of Tokyo, Tokyo, Japan. He worked for the Electrical Communication Laboratory of NTT, the Research Laboratory Electronics of Massachusetts Institite of Technology, Cambridge, MA, the Royal Instititute of Technology, Stockholm, Sweden, and Tokyo Institute of Technology, Tokyo, Japan. He retired from TIT in 1992 and established Brain Functions Laboratory, where he has developed a technique to estimate the degree of impairment of human brain cortex in terms of EEG and numerical analysis of the human emotions. William Rodman Shankle was born in Pasadena, CA, on January 28, He received the B.S. degree from Stanford University, Stanford, CA, in biology, the M.S. degree from the University of Southern California, Los Angeles, CA, in biomedical statistics, and the M.D. degree from Brown University, Providence, RI. He did postgraduate work in statistics at Harvard University, Cambridge, MA. He trained in internal medicine at Cook County Hospital in Chicago, IL. He did his residency in neurology at the Los Angeles County/University of Southern California Medical Center, Los Angeles, CA, and did a fellowship in computational modeling at the University of California, Irvine, in 1987, then founded and developed the Alzheimer s Clinic and Research Program there, from 1988 to During this time, he developed expertise in medical informatics and artificial intelligence research, and established the first web site for Alzheimer s disease, and the largest single database for Alzheimer s disease in the U.S. In 1997, he joined the Cognitive Science Department in order to learn mathematical methods of modeling cognition. He has also established the largest community dementia assessment program in Orange County, CA, and collaborates with Japanese and American researchers in Alzheimer s disease. His other research interest has been brain development, and he recently discovered that the human brain doubles its number of neurons after term birth.