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 Realisc 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 potenals – Solving the matrix for every source is slow – We only need potenals at electrode locaons

 Use the reciprocal formulaon: – Inject current at electrodes, solve volume

potenals 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 acvaon in number of dipoles given locaons.

 Nonlinear opmizaon  Linear opmizaon techniques techniques

 May converge to local minima  Needs addional constraints

 Non-linear least squares,  Bayesian methods, MNE, beamforming, MUSIC, LORETA, LAURA, etc. simulated annealing, genec 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 Mul-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 wrien 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 realisc 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 – Fied 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. Corcal 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

 Shi all electrodes and re-register – 5° backwards – 5° le

 Localize using shied electrodes

 Plot locaon and orientaon errors Locaon Errors with 5° electrode shi Observaons

 Errors increase close to the surface near electrode locaons.

 Changing or incorrectly registering electrodes may cause 5-10 mm localizaon error. Head ssue conducvies

 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 conducvity measurement

Measurement of skull conductivity MREIT Magnetic stimulation In vivo In vitro Current injection

Hoekama et al, 2003

He et al, 2005 Skull conducvity

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) rao!

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 Conducvity

 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 conducvity mis-esmaon 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 conducvity esmaon 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. Electrocorcography

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: Posive (seizure free)

16 min of data, 2 seizures  Provided by Dr. Greg Worrell, Mayo Clinic ) 2 Seizure 1 Seizure 2 Channelvoltage (uV 5 10 15 Time (min) Akalin Acar et al, 2008 ICA decomposion

Extended Infomax ICA Decomposition 16 seizure components (ICs) selected Independent Components

IC 1

Potentials on scalp Potentials on plastic sheet On the brain surface

IC 2 Source Localizaon Results Dipole source localization Distributed source localization - SBL

IC 1

Radial source Gyral source

IC 2

Tangential source Sulcal source Corcal acvity of seizure components

Activations of 13 seizure components

Cortical activity of Seizure components 13

Movie(t) = ∑Si × Acti (t) i=1

€ Thank you…

Swartz Center for Computaonal Neuroscience Algebraic formulaon of the FP Scalp potentials for N electrodes and p dipoles: p p V(r) = g(r,r ,d ) = g(r,r ,e )d ∑ dip i ∑ dip d i i i i ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ V (r1) ⎤ g(r1,rdip ,ed1)  g(r1,rdip ,edp ) d1 d1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V =  =     = G({r ,r ,e })  ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ j dipi d i ⎢ ⎥ € ⎢V (r )⎥ ⎢ g(r ,r ,e )  g(r ,r ,e )⎥ ⎢ d ⎥ ⎢ d ⎥ ⎣ N ⎦ ⎣ N dip d1 N dip dp ⎦ ⎣ p ⎦ ⎣ p ⎦ For N electrodes and p dipoles and T discrete time samples: ⎡ ⎤ ⎡ V (r1,1)  V (r1,T) ⎤ d1,1  d1,T € ⎢ ⎥ ⎢ ⎥ V =    = G({r ,r ,e })    ⎢ ⎥ j dipi d i ⎢ ⎥ ⎢V (r ,1)  V (r ,T)⎥ ⎢ d  d ⎥ ⎣ N N ⎦ ⎣ p,1 p,T ⎦ V = GD + n

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€ To Solve the Forward Problem WE NEED Spherical  Head Model Model – Conducvity values – Geometry

 Source distribuon – Magnitude – Locaon – Direcon

 Field Locaons  Solver Numerical Model Anisotropy

 Direconal conducvity for skull and WM.

 WM anisotropy can be obtained from diffusion tensor imaging (DTI).  WM anisotropy rao = 9:1

 Skull rao = 10:1 Anisotropy

Return currents for a left thalamic source on a coronal cut Wolters et al, 2006 Potenal fields on the scalp Shallow tangential source Deep tangential source

right left

front top view of head front Potenal fields on the scalp Shallow radial source Deep radial source

right left

front top view of head front