Line- and Surface- Source Estimation using MagnetoencephalographyI. S. Yetik1, A. Nehorai11, C. Muravchik2, J. Haueisen3, M. Eiselt3

1University of Illinois at Chicago, USA,2Universidad Nacional de La Plata, Argentina,3Friedrich-Schiller-University, Germany.

ABSTRACT

We propose a number of current source models that are spatially distributed on a line or surface using magnetoencephalography (MEG). We develop suchmodels with increasing degrees of freedom, derive forward solutions, maximum likelihood (ML) estimates, and Cramer-Rao bound (CRB) expressions for theunknown source parameters. We compare the proposed line- and surface-source models with existing focal source models, and show their usefulness for certainbiomagnetic experiments. We apply our line-source models to N20 response after electric stimulation of the median nerve known to be an extended source andour surface-source models to KCl (potassium-chloride) induced spreading depression.KEYWORDS

Magnetoencephalography, Extended source modeling, N20 responses.INTRODUCTION

We propose several line- and surface-source dipole models for MEG and investigate their aspects. For each of the proposed models, we give the forwardsolution, derive the maximum likelihood (ML) estimates and Cramer-Rao bounds (CRBs) for the unknown parameters. We also discuss the special case ofthe spherical head model and radial sensors which results in more efficient calculations. We apply our models to N20 responses and KCl induced spreadingdepression.METHODSLet b(r, t) be the magnetic field induced by a focal electrical sourcej(r, t) = q(t)δ(r − p) using a realistic head model (Muravchik [2001]), whereq(t)denotes the dipole moment,p the source position,r the position, andt the time. Assumingm MEG sensors the measurement model can be written asy(t) = A(�p)q(t) + e(t), wherey(t) is a vector of dimensionm × 1 of the measured magnetic fields,e(t) is additive noise,�p = [ϕ, φ, p] is the vectorof source position parameters withϕ denoting the azimuth,φ the elevationp the distance from the origin, andA(�p) is a gain matrix of dimensionm × 3.Considering this model withK independent trials (e.g. evoked responses),N temporal samples, and assuming the source position is fixed in time, the MLestimate of�p is �p = arg min�p

[∑N

t=1−y(t)TP (�p)y(t)

], wherey(t) = (1/K)

∑K

k=1yk(t), yk(t) denotes the measurement vector for thekth trial,

andP (�p) = A(�p)[A(�p)TA(�p)]−1A(�p)T. The ML estimate ofq(t) can be calculated using the Moore-Penrose pseudoinverse (Dogandzic [2000]) asq(t) = [A(�p)TA(�p)]−1A(�p)Ty(t). We derive the CRB which is a lower bound on the covariance of any unbiased estimator, and is asymptotically achievedby the ML estimator. It is an important performance measure that can be used to evaluate the statistical efficiency of estimation algorithms, to determine themain regions where good and poor estimates are expected, and to optimize the sensor system design. See (Yetik [2004]) for details for the line-source case.Line-Source Models: We give the source currents for the three line-source models that we propose: variable-azimuth constant moment (VACM), variable-position constant moment (VPCM), and variable-position variable-moment (VPVM).VACM: The source current isq(t)δ(p − p0)δ(φ − φ0)[u(ϕ − ϕ1) − u(ϕ − ϕ2)] with q(t) = [qx(t), qy(t), qz(t)]

T, whereφ0 is the fixed elevation of thesource,ϕ1 andϕ2 the limits of the azimuth extent of the source, andu(·) the unit step function.VPCM: The source position is[px(s), py(s), pz(s)]

T = WΨ(s) resulting in the source currentq(t) for r = [px(s0), py(s0), pz(s0)], s0 ε [s1, s2] and0otherwise, whereW is the matrix of unknown coefficients that determine the position of the source,Ψ(s) is the set of known basis functions,px,y,z(s) denotesthe position of the source in cartesian coordinates, ands1 ands2 determine the extent of the source.VPVM: The source moment density varies with position resulting in the source currentq(s, t) = X(t) (s) for r = [px(s), py(s), pz(s)], s ε [s1, s2] and0otherwise, whereX is the matrix of unknown coefficients that determine the spatial variation of the dipole moment and (s) is the set of known basis functions.Surface-Source Models: We present the source currents for the three surface-source models that we propose: constant-radius constant moment (CRCM),variable-position constant moment (VPCM), and variable-position variable-moment (VPVM).CRCM: The source current isq(t)δ(p− p0)[u(φ− φ1)− u(φ− φ2)][u(ϕ− ϕ1)− u(ϕ− ϕ2)], with q(t) = [qx(t), qy(t), qz(t)]

T.VPCM: The source position is[px(sa, sb), py(sa, sb), pz(sa, sb)]

T = WΨ(sa, sb), (sa, sb) ε [sa1, sa2] ∩ [sb1, sb2], resulting in the source currentq(t) forr = [px(sa, sb), py(sa, sb), pz(sa, sb)], (sa, sb) ε [sa1, sa2] ∩ [sb1, sb2] and0 otherwise.VPVM: The source moment density varies over the surface resulting in the source currentq(sa, sb, t) = X(t) (sa, sb)for r = [px(sa, sb), py(sa, sb), pz(sa, sb)], (sa, sb) ε [sa1, sa2] ∩ [sb1, sb2] and0 otherwise. Using these source currents the magnetic field can be calculatedby integrating the magnetic field induced by a dipole. This integral will be a line integral for the line-source models and surface integral for the surface-sourcemodels, the details can be found in (Yetik [2004]).Spherical Head Model: The special case of a spherical head and radial sensors results in more compact forms of induced magnetic field involving ellipticintegrals (Byrd [1954]). See (Yetik [2004]) for more details for the line-source case.Low-rank Gain Matrices: The radial components of the dipole sources do not produce magnetic fields outside a spherical head. Therefore, the gain matrixfor the focal source model has a rank equal to two when a spherically symmetric head model is used. A similar situation exists for the proposed line- andsurface-source models under certain conditions. These conditions are derived in (Yetik [2004]) for the line-source case.RESULTSDistinguishing Between Line-Source and Focal Source Models: We first investigate when it is possible to distinguish between a line source and a focal sourceusing the Neyman-Pearson hypothesis test. A collection of ROC’s for different source lengths is given in Figure 1a using VAVM. A similar plot is given forVPVM in Figure 1b. SelectingPD = 0.9 andPF = 0.1 as the boundary of a confident decision, the minimum source length which is distinguishable from afocal source is1.81cm for VAVM, and1.69cm for VPVM.Application to N20 response: We applied the VPVM model to real data of N20 response after electric stimulation of the median nerve. The resulting N20

generator is known to be an extended source along the wall of the central sulcus, which is mainly one-dimensional and a good example where line-source models

1The work of I. S. Yetik and A. Nehorai was supported by the National Science Foundation Grants CCR-0105334 and CCR-0330342. The work of Carlos H. Muravchik wassupported by CIC-PBA, UNLP and ANPCTIP of Argentina.

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Figure 1:Parts (a)-(b): Receiver operating characteristics for different source lengths (denoted by∆) for VAVM (part (a)), VACM (part(b)) and focal sourcemodels. Part (c): Estimated source positions using the focal source model and VPVM. The symbol “+” shows the estimated position of the focal source, andthe line shows the estimated position and extent of the line-source.

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Figure 2:Spreading depression on the rat brain. Part (a) is fort = 10sec after spreading depression started, (b)250 sec, (c)500 sec, and (d)750 sec.

are needed (Nolte [2000]). The experiment was performed in eight healthy volunteers. The MEG data were recorded over the contralateral somatosensorycortex with a 31 channel biomagnetometer (Philips, Hamburg, Germany). Constant current of0.2 ms square-wave pulses were delivered to the right or left wristat a stimulation rate of4 Hz. We have used realistic head models when applying our models to N20 data. The estimated source positions (for the focal sourceand VPVM models) are plotted in Figure 1c for one of the subjects. We also compare the source extents (path length of the source curve) and mean-squarederrors for all eight subjects in Table 1.Estimating source extents using different noise realizations: We conducted another simulation to show that the estimated source extents are the real extents ofthe sources rather than a misleading result of the brain noise. We considered a straight line-source (s) = [1, s]T (which is close to the estimated line-source ofone of the subjects), and calculated the exact induced magnetic field intensity using this source. We then added different noise realizations using the real brainnoise samples before the stimulation. We observe that reasonably close estimates for the source extent are obtained and the standard deviation of the estimatedsource extent is0.47cm with a mean of1.73cm.Application to KCl induced spreading depression: We applied the surface VPCM model to real data of KCl induced spreading depression using five date

sets taken from three rats. The MEG data were recorded with a4 × 4 array of first order gradiometers. During spreading depression a depolarisation wave istraveling over the cortex with an increasing extent providing an example where surface source models are necessary. The estimated source positions at differenttime instants for one of the data sets is given in Figure 2.

Subject A B C D E F G HFocal MSE(×10−25T2) 1.74 3.19 1.91 4.41 0.98 1.25 1.54 2.26

VPVM MSE (×10−25T2) 1.24 2.61 1.24 3.58 0.70 1.03 1.43 1.34Error Decrease 29% 18.3% 35.4% 18.8% 28.7% 17.1% 6.9% 40.5%

Est. Source Length (cm) 1.56 0.94 1.99 1.07 2.27 1.32 0.62 2.44

Table 1:Estimated source lengths and estimation performance results for real MEG data of eight subjects.

CONCLUSIONWe have proposed several source models for MEG which are spatially distributed on a line or surface, and applied these models to real MEG data for N20response in humans and spreading depression in rats. We refer the reader to (Yetik [2004]) for a more comprehensive discussion of our line-source models andresults. As a future work it is possible to extend our models to EEG.REFERENCESMuravchik C and Nehorai A. EEG/MEG error bounds for a static dipole source with a realistic head model. IEEE Tran. Biomed. Eng. 2001;49:470-484.Dogandzic A and Nehorai A. Estimating evoked dipole responses in unknown spatially correlated noise with EEG/MEG arrays. IEEE Tran. Biomed. Eng.

2000;48:13-25.Yetik IS, Nehorai A, Muravchik CH, and Haueisen J. Line-source modeling and estimation with magnetoencephalography. IEEE Tran. Biomed. Eng. 2004;in

revision.Byrd PF and Friedman MD. Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer, 1954.Nolte G and Curio G. Current multipole expansion to estimate lateral extent of neuronal activity: a theoretical analysis. IEEE Tran. Biomed. Eng. 2000;47:1347-

1355.