Forward and Inverse Problem of EEG Zeynep AKALIN ACAR Swartz Center for Computaonal Neuroscience November, 2012 Cortical surface Generators of EEG Baillet et al, 2001 Forward and inverse problem Forward Problem X = LS € EEG/ MEG S = L−1X Inverse Problem = 1000s of FP solution € Source localizaon is ill-posed X = LS + n X: scalp recorded potentials S: current density vector L: transfer matrix ‘the head volume conductor model’ € The inverse problem refers to finding S given known X. 2 O(S) = min X − LS Infinite solutions! Apply electrophysiological neuroanatomical constraints 1. The electrical head model used, 2. The inverse solution itself € Head volume conductor model Simple Head Models Realis/c Head Models Single layer sphere, Boundary Element (BEM) spheroid Finite Element (FEM) 3-4 layer sphere Finite Difference (FDM) ANALYTICAL SOLVER NUMERICAL SOLVER Simple, fast, but not Represents head shape better, accurate but computationally complex Numerical Head Models BEM FEM Generated using Tetgen NFT BEM mesh from NFT BEM mesh Formulaon of the FP S P σ(x,y,z) ∇⋅ (σ∇Φ) = −∇⋅ J inside V φ ∂Φ V σ = 0 on S ∂n σ (x,y,z) : conductivity distribution : current source p € € Reference: Gulrajani, R., Bioelectricity and biomagnetism BEM Formulaon Integral equation for Potential Field: − + 1 n ⎛ σ −σ ⎞ R φ(r ) = 2g(r ) + k k φ(r ' ) ⋅ dS (r ' ) ∑k=1⎜ − + ⎟ ∫ 3 k 2π ⎝ σ i +σ i ⎠ S R k Primary sources € Secondary sources kth surface BEM Formulaon Integrating the previous integral equation over all elements a set of equations are obtained. In matrix notation for the potential field we obtain M: number of nodes The expression for the magnetic field: n: number of magnetic sensors Transfer matrix Electrode potentials −1 Φe = DA g Φe mx1 vector of electrode potentials € D is an mxM sparse matrix to select m rows of A −1 € € Let the transfer matrix E be defined as: E = DA −1 € Taking the transpose of both sides, and multiplying by AT AT e d € i = i € € FEM transfer matrix FEM computes volume potenAals – Solving the matrix for every source is slow – We only need potenAals at electrode locaons Use the reciprocal formulaon: – Inject current at electrodes, solve volume potenAals V I I A B A B a b a b I I V Inverse Problem Equivalent dipole Methods Linear distributed Methods Overdetermined Underdetermined Searches for parameters of a Searches for acAvaon in number of dipoles given locaons. Nonlinear opAmizaon Linear opAmizaon techniques techniques May converge to local minima Needs addiAonal constraints Non-linear least squares, Bayesian methods, MNE, beamforming, MUSIC, LORETA, LAURA, etc. simulated annealing, geneAc algorithms, etc. Equivalent current dipole (ECD) 2 O(S) = min X − LS 6 parameters are estimated for each dipole: Location, orientation and strength € Linear distributed methods X = LS L is the lead field matrix: Potential vectors of all possible solutions € Anatomical constraint: Sources are on the cortex perpendicular to the cortex Daunizeau, 2009 MulA-scale patch-basis source localizaon with Sparse Bayesian Learning X = LS Dij = geodesic_distance(i, j) L := [m × v] Lead field matrix Dij = Inf if Dij > scale (1) (3) 2 L˜ = LW LW ⎛ ⎞ [ ]m 3v (k) 1 Dij × W = gauss(D ,σ ) = exp⎜ − ⎟ ij ij k 2 2 2 2πσk ⎝ σk ⎠ € σk = scale /3 X = ASˆ € ˆ th Three truncated Gaussian patches of different scales (radii) S q := [1× T] q IC activation € radius 10 mm 6 mm 3 mm € ˜ ˜ ˙ Aq = LM q +U q σk 3.33 mm 2 mm 1 mm ˜ −1 ˜ € L = SBL(Aq,L ) ˜ ˜ −1 M q = L Aq [ ]3v×1 ˜ Mq = reshape(M q,v × 3) 3 M = M˜ (:,i) q ∑i=1 q Akalin Acar, et al (2008a,2009) IEEE EMBC [v x T] cortical surface ˆ th Ramirez, et al, HBM, 2007 Pq = Mq S q potentials for q IC € SBL Simulaon Study with MNI model (SNR=50) Three examples: Source (x 15) Max. dis. (mm) Energy dif. DF (%) Type Scale (mm) original reconstructed Gyral 10 0 1.5 103.8 Sulcul 10 1.01 29.8 101.4 Gyral Sulcul 5 2.12 4.1 37.6 Dual 10 11.6 29.3 89.2 Gyral 5 1.01 4.7 41.3 Sulcal Sulcul 12 1.8 10.6 125.5 Term Definition geodesic distance between original and max displacement reconstructed patch centers Sulcal energy difference |original energy - reconstructed energy| degree of focalization (DF) reconstructed energy / original energy Akalin Acar, et al (2009) IEEE EMBC Neuroelectromagnetic Forward Head Modeling Toolbox T1-weighted http://sccn.ucsd.edu/nft NFT - Segmentaon NFT – Mesh generaon NFT – source space generaon Generates a simple source space: Regular Grid inside the brain With a given spacing and distance to the mesh NFT – electrode co-registraon NFT – Template warping NFT – Forward problem solver MATLAB interface to numerical solvers Boundary Element Method or Finite Element Method – EEG Only (for now) – Interfaces to the Matrix generator executable wriden in C++ Other computaon done in MATLAB Generated matrices are stored on disk for future use. NFT - Forward Problem Solver (BEM) NFT – Forward Problem Solver (FEM) NFT – Dipole fing Requires EEGLAB integraon to access Component indices. Uses FieldTrip in EEGLAB for dipole fing. http://www.sccn.ucsd.edu/nft Effects of Forward Model Errors on EEG Source Localizaon MODELING ERRORS Head Model Generaon Reference Head Model – From whole head T1 weighted MR of subject – 4-layer realisAc BEM model MNI Head model – From the MNI head – 3-layer and 4-layer template BEM model Warped MNI Head Model – Warp MNI template to EEG sensors Spherical Head model – 3-layer concentric spheres – Fided to EEG sensor locaons The Reference Head Model 18541 nodes 37090 elements – 6928 Scalp – 6914 Skull – 11764 CSF – 11484 Brain CSF Brain Scalp Skull The MNI Head Model 4-layer – 16856 nodes – 33696 elements 3-layer – 12730 nodes – 25448 elements Brain CSF Skull Scalp The Warped MNI Head Model Registered MNI template Warped MNI mesh The Spherical Head Model 3-Layer model Outer layer is fitted to electrode positions Head Modeling Errors Solve FP with reference model – 3D grid inside the brain. – 3 Orthogonal dipoles at each point – ~7000 dipoles total – 4 subjects Localize using other head models – Single dipole search. Plot locaon and orientaon errors Spherical Model Locaon Errors Localization errors may go up to 4 cm when spherical head models are used for source localization. The errors are largest in the inferior regions where the spherical models diverged most from the 4-layer realistic model. 3-Layer MNI Locaon Errors 3-Layer MNI 3-Layer Warped MNI 4-Layer MNI Locaon Errors 4-Layer MNI 4-Layer Warped MNI Observaons Spherical Model – Locaon errors up to 3.5 cm. CorAcal areas up to 1.5 cm. 3-Layer MNI – Large errors where models do not agree. – Higher around chin and the neck regions. 4-Layer MNI – Similar to 3-Layer MNI. – Smaller in magnitude. Electrode co-registraon errors Solve FP with reference model Shil all electrodes and re-register – 5° backwards – 5° le Localize using shiled electrodes Plot locaon and orientaon errors Locaon Errors with 5° electrode shil Observaons Errors increase close to the surface near electrode locaons. Changing or incorrectly registering electrodes may cause 5-10 mm localizaon error. Head Assue conducAviAes Scalp : 0.33 S/m Skull: 0.0032 S/m (0.08-0.0073 S/m) CSF: 1.79 S/m Brain: 0.33 S/m Skull conducAvity measurement Measurement of skull conductivity MREIT Magnetic stimulation In vivo In vitro Current injection Hoekama et al, 2003 He et al, 2005 Skull conducAvity Brain to skull rao Rush and Driscoll 1968 80 Cohen and Cuffin 1983 80 Oostendorp et al 2000 15 Lai et al 2005 25 Measurement Age σ (mS/m) ra/o! Agar-agar phantom – 43.6 7.5 Paent 1 11 80.1 4 Paent 2 25 71.2 Skull4.6 conductivity by age Paent 3 36 53.7 6.2 Paent 4 46 34.4 9.7 Paent 5 50 32.0 Hoekama10.3 et al, 2003 Post mortem skull 68 21.4 15.7 Effect of Skull ConducAvity Solve FP with reference model – Brain-to-Skull rao: 25 Generate test models – Same geometry – Brain-to-Skull rao: 80 and 15 Localize using test model Plot locaon and orientaon errors Skull conducAvity mis-esAmaon Effect of white maer White matter conductivity: 0.14 S/m White matter surface obtained Simplified WM BEM model using Freesurfer Effect of white maer Number of electrodes and coverage Locaon errors Summary If we have MRI of the subject: – Subject specific head model – Distributed source localizaon If we don’t have MRIs – Warped 4-layer MNI model – Dipole source localizaon Skull conducAvity esAmaon is as important as the head model used. WM modeling does not have much effect on source localizaon. Epilepsy Head Modeling CASE STUDY Epilepsy ♦ Neurological disorder characterized by seizures. ♦ 50 million people worldwide ♦ 30 % can not be controlled with medical therapy ♦ 4.5 % potential candidate for surgical treatment. ElectrocorAcography ECoG recording ECoG grid and strips Macro and micro Depth electrodes Electrodes (Mayo Clinic) iEEG data Project Summary Solution of the forward Sparse problem using Bayesian realistic head Learning models (BEM, Component open skull) Cluster Sources Sources Data AMICA PMI Clusters co-registration decomposition Mesh Generation Source space Multiscale EEG data MRI/CT Segmentation head image Forward modeling Cortex (Freesurfer) BEM model: Plastic sheet 80 000 source vertices Skull with craniotomy hole Scalp Z. Akalin Acar - Head Modeling and Cortical Source Localization in Epilepsy Analyzing Epilepsy Recordings Pre-Surgical Evaluaon Rest Data 78 ECoG (subdural EEG) electrodes 29 scalp electrodes Surgical Outcome: PosiAve (seizure free) 16 min of data, 2 seizures Provided by Dr.