Recent advances in the analysis of biomagnetic signals

Transcription

1 Recent advances in the analysis of biomagnetic signals Kensuke Sekihara Mind Articulation Project, Japan Science and echnology Corporation okyo Metropolitan Institute of echnology

2 Application of spatial filter techniques to biomagnetic source localization

3 Definitions b () t 1 b () t 2 data vector: b(t) = b () t M b () t 1 b () t 2 b () t M b (): t the jth sensor recording at t j data covariance matrix: D = b() tb () t represents time average

4 Source moment magnitude at r = [ xyz,, ] orientation: and at t: s( r,) t h( r,t ) = [ η x ( r,t ), η y ( r,t ), η z ( r,t)] source moment vector: ηx( r,t) s x( r,t) sr (,) t = s ( r,t ) ηy( r,t) = s y( r,t) ηz( r,t) s z( r,t) z θ y ϕ x η = sinθ cosϕ x η = sinθ sinϕ η y x = cosθ

5 Lead field vector for the j th sensor l ( r) = [ l x ( r), l y ( r), l z ( r)] j j j j Lead field matrix for the whole sensor array x y z l ( r) ( ) ( ) ( ) 1 l r l r l r x y z l ( r) ( ) ( ) ( ) 2 l r l r l r Lr ( ) = = x y z l ( r) l ( ) l ( ) l ( ) M r r r M M M

6 Basic relationship b () t = ( ) ( ) j l rsr,t dr j or b() t = Lrsr ( ) (,t) dr Problem of source localization: Estimate sr (,) t from the measurement b() t Estimate

7 Spatial filter bt () 1 sr ˆ(,) t = w ()() rb t = [ w (), r, w ()] r 1 M b () t M estimate of sr (,) t weight vector (neglecting the explicit time notation) b = Lrsr ( ) ( ) dr } ˆ sr ( ) = ( ) ( ') ( ') d ' w rlr sr r ˆs( r)= w ( rb ) R( r, r' ) Resolution kernel

8 Resolution kernel: sr ˆ( ) = R(, rrsr ') ( ') dr' he weight w(r) must be chosen so that the resolution kernel has a reasonable shape. What is reasonable shape? peak at r small mail-lobe width low side-lobe amplitude r main-lobe width side-lobe amplitude

9 Non-adaptive weight wr () is data independent Adaptive weight wr () is data dependent

10 Data-independent (non-adaptive) weight Spatial resolution is limited by sensor-array configuration. Final results are not influenced by source temporal correlation. Data-dependent (adaptive) weight Spatial resolution can exceed the limit imposed by the sensor-array configuration. Strong temporal correlation among source activities severely degrade the quality of final estimation results

11 Data-independent (non-adaptive) weight minimum-norm estimate (Hamalainen et al.) he weight wr ( ) is obtained by [ R ] 2 min (, rr') δ ( r r ) dr 1 w () r = L ( rg ), where G = lrl ( ) ( r) dr ij, i j 1 Inverse solution: sr ˆ( ) = L ( rgb )

12 Property of G matrix G = l ( rl ) ( r) dr i,j i j Overlaps of sensor lead fields is large Overlaps of sensor lead fields is small G is poorly conditioned G has a small condition number

13 G is usually calculated by introducing pixel grid r j b = Lrsr ( ) ( ) dr = Lrsr ( ) ( ) j = 1 sr ( ) 1 = Lr ( ),, Lr ( ) = Ls 1 L ( ) N sr N sn herefore G = LL and w () r = L ()( r LL) or N N N N N j j N N N 1 w () r = L ()( r LL + I) N N γ 1 [regularized versi on]

14 Minimum-norm weight with normalized lead field wr LL I Lr Lr 1 () = ( + γ ) ()/ () N N where L N Lr ( ) Lr ( ) 1 N =,, Lr ( ) Lr ( ) 1 N Minimum-norm estimate with normalized weight Dale et al., Valdes et al. Calculate 2 2 q ˆ( r) = w () rb / wr ()

15 no normalization z (cm) normalized lead field used normalized weight used

16 Source imaging experiments Auditory-evoked field were measured using 148-channel whole-head sensor array (Magnes 2500). Stimulus: 1-kHz pure tone applied to subject s left ear Data acquisition: 1 khz sampling frequency, Hz bandpass filtering, 100 epochs averaged

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20 Linear-estimation-based methods LOREA: impose the maximum-smoothness constraint. (Pascual-Marqui et al.) fmri constraint: constrain solution based on fmri results. (Dale et al.) FOCUSS: obtain a focal solution iteratively. (Gorodnitsky et al.) Bayesian approach: impose prior assumptions. (Schmidt et al.) l 1 -norm approach: use the l 1 -norm, instead of using the l 2 -norm. (Matuura et al., Uutela et al., Beucker et al.)

21 Data dependent (adaptive) weight minimum-variance beamformer min w wdw subject to wlrh (, ) = l (, r h) D l (, r h) Db w (, r h) = and sr ˆ(, h) = 1 1 l (, r h) Dlr (, h) l (, r h) Dlr (, h) Lrh () beamformer pointing orientation

22 Generalized Wiener estimate: ˆ ( ) 1 s = RL LRL + C b N N N N source covariance matrix noise covariance matrix Use R I, and C γi (assume no prior-information) ˆ ( ) 1 s = L LL + γi b N N N N minimum-norm estimate

23 Generalized Wiener estimate: ˆ ( ) 1 s = RL LRL + C b N N N N iuse DLRL = + C, and N N iassume off-diagonals of R are all zero Relashonship, sˆ p 1 = Rl pp Db P 2 1 sˆ = R leads to R = 1/( l D l ) p pp pp P p s 1 1 ˆ ( l p )/( l = Db D l p) P P minimum-variance estimate

24 Problems when applying MV beamformer to MEG source localization (1) How to determine beamformer orientation h. (2) Lr () norm artifact. (3) Output SNR degradation due to the matrix inversion.

25 How to determine beamformer orientation h? he weight wr (, h) is calculated for r and h. search all directions (Robinson and Vrba). use the MUSIC algorithm to determine h (Sekihara et al.). use vector beamformer formulation. (Spencer et al.van Veen et al.) What happens if the beamformer orientation is different from the source orientation? Severe signal-intensity loss

26 Computer simulation for calculating beamformer angular response 37-channel sensor array A single source exists 6-cm below the sensor array 6cm δθ assumed source waveform beamformer orientation different from the source orientation with δθ normalized value δθ degree Calculated angular response

27 A single source exist at r with orientation h and time course st () measured data: b() t = [ ηl () r + ηl () r + ηlr ()] st () xx yy zz When beamformer weight has a wrong orientation h ( h) w (, r h )() b t = w (, r h )[ ηxx l ()() r st + ηyy l ()() r st + ηzlr z()() st] virtual three coherent sources 0

28 Vector beamformer formulation hree weight vectors [ wx(), r wy(), r wr z()] detects the x, y, and z component of the source moment, separately. min wdw subject to wlrf (, ) = 1, wlrf (, ) = 0, wlrf (, ) = 0 x x x x x y x z min wdw subject to wlrf (, ) = 0, wlrf (, ) = 1, wlrf (, ) = 0 y y y x y y y z min wdw subject to wlrf (, ) = 0, wlrf (, ) = 0, wlrf (, ) = 1 z z z x z y z z [ w (), r w (), r w ()] r = DLr ()[ L () rdlr ()] x y z and ˆ( t) = [ () ()] () () t s L rdlr L rdb f = [1,0,0], f = [0,1,0], f = [0,0,1] x y z

29 Calculated beamformer angular response cosine curve normalized value normalized value degree scalar beamformer degree vector beamformer

30 Lr () norm artifact Severe artifacts arise near the center of the sphere, when the spherical conductor model is used. o avoid these artifacts use normalized lead field (Van Veen et al.) use normalized weight (Robinson et al., Sekihara et al.)

31 assumed source waveform 37-channel sensor array latency (ms) z (cm) } three sources -15 reconstruction region relative magnitude latency (ms) y (cm) sphere for the forward modeling generated magnetic field

32 ime-averaged reconstruction sr ˆ(,) t 2 z (cm) z (cm) z (cm) y (cm) no normalization y (cm) normalized lead field used

33 Borgiotti-Kaplan beamformer min w wdw subject to ww = 1 w (, r h) = l (, r h) D and 1 2 (, h) (, h) l r Dlr sr ˆ(, h) 2 = 1 (, h) Dl(, r h) 2 (, h) Dlr (, h) l r l r

34 Vector extension of Borgiotti-Kaplan beamformer min wdw subject to ww = 1, wlrf (, ) = 0, wlrf (, ) = 0 x x x x x y x z min wdw subject to wlrf (, ) = 0, ww = 1, wlrf (, ) = 0 y y y x y y y z min wdw subject to wlrf (, ) = 0, wlrf (, ) = 0, ww = 1 z z z x z y z z let µ = x, y or z w () r µ = DLr ()[ L () rdlr ()] fµ f Wf µ µ Ω = [ L () rdlr ()] L () rd Lr ()[ L () rdlr ()]

35 ime-averaged reconstruction sr ˆ(,) t 2 z (cm) z (cm) y (cm) normalized lead field used y (cm) normalized weight used (B-K beamformer results)

36 Resolution kernel for BK beamformer

37 Output SNR degradation for spatio-temporal reconstruction Signal-to-noise ratio of the beamformer output is severely degraded even by a small error in the estimated lead field his is caused by the use of direct matrix inversion o avoid this, use regularized inverse (Robinson et al.) use eigenspace projection (Sekihara et al.)

38 Eigendecomposition of D Eigenspace projection λ L 0 S D = U λ P U = U U 0 LN λ M U = [ e,, e e,, e ] = [ E E ] 1 P P+ 1 M S N E E S N Extension to eigenspace projection beamformer w = EEw, where µ= xy, or z S S µ µ

39 SNR consideration Output of eigenspace-projected BK beamformer [ l () r G lr ()] 2 S 1-1 where G = EL E, G = E L E 2 S S S S N N N N G lr S [ l () r ()] Output of BK beamformer [ l () r G lr ()] 2 S 2 2 [ l () r G lr () + ] S e G e N overall error in estimating lr ( ) Even when e is small, e 2 G N e may not be small

40 assumed source waveform 37-channel sensor array latency (ms) z (cm) } three sources -15 reconstruction region relative magnitude latency (ms) y (cm) sphere for the forward modeling generated magnetic field

41 Spatio-tempotal reconstruction by vector-extended BK beamformer

42 Spatio-tempotal reconstruction by vector-extended BK beamformer 1 with regularized inverse, ( D + γi)

43 Spatio-tempotal reconstruction by vector-extended BK beamformer with eigen-space projection

44 Eigen-space projection does not preserve the null constraints hat is, EEwx ly() r 0, EEwx lz() r 0, S S S S EEwy lx() r 0, EEwy lz() r 0, S S S S EEwz lx() r 0, EEwz ly() r 0. S S S S his fact does not cause a problem. (Poster: 167b)

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48 7 mm Poster: 92a

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50 right auditory cortex activation left auditory cortex activation correlation coefficient: 0.97

51 Summary Adaptive spatial filter techniques can provide a spatial resolution higher than that of non adaptive techniques. his is because the spatial resolution for non-adaptive techniques is limited by the sensor configuration but adaptive techniques can exceed this limit. Correlated source activities, however, affect the quality of the results obtained by adaptive techniques. herefore Adaptive techniques may be suited to observe relatively small cortical regions with high spatial resolution, and non-adaptive techniques may be suited to observe whole-brain activities.

52 Collaborators Kanazawa Institute of echnology Dr. Isao Hashimoto Laboratory of Physiological Sciences Dr. Keiji Sakuma Communications Research Lab. Dr. ouru Miyauchi Communications Research Lab. Dr. Norio Fujimaki Siraume University Dr. Ryosuke akino University of Utah Dr. Srikantan S. Nagarajan M. I.. Dr. Alec Marantz University of Maryland Dr. David Poeppel University of California Dr. im Roberts University of okyo Dr. Yasushi Miyashita