Line- and Surface- Source Estimation using Magnetoencephalography I. S ¸. Yetik 1 , A. Nehorai 11 , C. Muravchik 2 , J. Haueisen 3 , M. Eiselt 3 1 University of Illinois at Chicago, USA, 2 Universidad Nacional de La Plata, Argentina, 3 Friedrich-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 such models with increasing degrees of freedom, derive forward solutions, maximum likelihood (ML) estimates, and Cramer-Rao bound (CRB) expressions for the unknown source parameters. We compare the proposed line- and surface-source models with existing focal source models, and show their usefulness for certain biomagnetic experiments. We apply our line-source models to N20 response after electric stimulation of the median nerve known to be an extended source and our 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 forward solution, derive the maximum likelihood (ML) estimates and Cram´ er-Rao bounds (CRBs) for the unknown parameters. We also discuss the special case of the spherical head model and radial sensors which results in more efficient calculations. We apply our models to N20 responses and KCl induced spreading depression. METHODS Let ( ,t) be the magnetic field induced by a focal electrical source ( ,t)= (t)δ( - ) using a realistic head model (Muravchik [2001]), where (t) denotes the dipole moment, the source position, the position, and t the time. Assuming m MEG sensors the measurement model can be written as (t)= A( p )(t)+ (t), where (t) is a vector of dimension m × 1 of the measured magnetic fields, (t) is additive noise, p =[ϕ, φ, p] is the vector of source position parameters with ϕ denoting the azimuth, φ the elevation p the distance from the origin, and A( p ) is a gain matrix of dimension m × 3. Considering this model with K independent trials (e.g. evoked responses), N temporal samples, and assuming the source position is fixed in time, the ML estimate of p is ˆ p = arg min p £∑ N t=1 -¯(t) T P ( p )¯(t) / , where ¯(t) = (1/K) ∑ K k=1 k (t), k (t) denotes the measurement vector for the kth trial, and P ( p )= A( p )[A( p ) T A( p )] -1 A( p ) T . The ML estimate of (t) can be calculated using the Moore-Penrose pseudoinverse (Dogandzic [2000]) as ˆ(t)=[A( ˆ p) T A( ˆ p)] -1 A( ˆ p) T ¯(t). We derive the CRB which is a lower bound on the covariance of any unbiased estimator, and is asymptotically achieved by the ML estimator. It is an important performance measure that can be used to evaluate the statistical efficiency of estimation algorithms, to determine the main 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 is (t)δ(p - p 0 )δ(φ - φ 0 )[u(ϕ - ϕ1) - u(ϕ - ϕ2)] with (t)=[qx(t),qy (t),qz (t)] T , where φ0 is the fixed elevation of the source, ϕ 1 and ϕ 2 the limits of the azimuth extent of the source, and u(·) the unit step function. VPCM: The source position is [px(s),py (s),pz (s)] T = W Ψ(s) resulting in the source current (t) for =[px(s0),py (s0),pz (s0)],s0 ² [s1,s2] and 0 otherwise, where W is the matrix of unknown coefficients that determine the position of the source, Ψ(s) is the set of known basis functions, x,y,z (s) denotes the position of the source in cartesian coordinates, and s 1 and s 2 determine the extent of the source. VPVM: The source moment density varies with position resulting in the source current (s, t)= X(t) (s) for =[px(s),py (s),pz (s)],s² [s1,s2] and 0 otherwise, where X 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 is (t)δ(p - p 0 )[u(φ - φ 1 ) - u(φ - φ 2 )][u(ϕ - ϕ 1 ) - u(ϕ - ϕ 2 )], with (t)=[q x (t),q y (t),q z (t)] T . VPCM: The source position is [px(sa,s b ),py (sa,s b ),pz (sa,s b )] T = W Ψ(sa,s b ), (sa,s b ) ² [sa1,sa2] ∩ [s b1 ,s b2 ], resulting in the source current (t) for =[p x (s a ,s b ),p y (s a ,s b ),p z (s a ,s b )], (s a ,s b ) ² [s a1 ,s a2 ] ∩ [s b1 ,s b2 ] and 0 otherwise. VPVM: The source moment density varies over the surface resulting in the source current (sa,s b ,t)= X(t) (sa,s b ) for =[p x (s a ,s b ),p y (s a ,s b ),p z (s a ,s b )], (s a ,s b ) ² [s a1 ,s a2 ] ∩ [s b1 ,s b2 ] and 0 otherwise. Using these source currents the magnetic field can be calculated by 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-source models, 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 elliptic integrals (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 matrix for 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- and surface-source models under certain conditions. These conditions are derived in (Yetik [2004]) for the line-source case. RESULTS Distinguishing Between Line-Source and Focal Source Models: We first investigate when it is possible to distinguish between a line source and a focal source using 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 for VPVM in Figure 1b. Selecting PD =0.9 and PF =0.1 as the boundary of a confident decision, the minimum source length which is distinguishable from a focal source is 1.81cm for VAVM, and 1.69cm 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 1 The 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 was supported by CIC-PBA, UNLP and ANPCTIP of Argentina.