Estimation de paramètres électrophysiologiques pour

Transcription

1 Introduction Estimation de paramètres électrophysiologiques pour l imagerie cérébrale Maureen Clerc Inria Sophia Antipolis, France Paris, 20 jan 2017 Séminaire Jacques-Louis Lions Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 1 / 33

2 Introduction Introduction schematic organization variability of cortical foldings subject-dependent localization of activity Brain activity can be localized: invasively: brain stimulation, depth electrodes non-invasively: neuroimaging Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 1 / 33

3 Introduction Non-invasive recordings: electric potential 1924: Hans Berger measures electrical potential variations on the scalp. birth of Electro-Encephalography (EEG) several types of oscillations detected (alpha 10 Hz, beta 15 Hz) origin of the signal unclear at the time scalp topographies ressemble dipolar field patterns Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 2 / 33

4 Introduction Noninvasive recordings: from electric to magnetic field A dipole generates both an electric and a magnetic field electric field 1963: Magnetocardiography, 1972: Magneto-Encephalography (MEG) D. Cohen, MIT, measures alpha waves, 40 years after EEG Superconductive QUantum Interference Device Magnetic shielding Advantage of MEG over EEG: spatially more focal magnetic field [Badier, Bartolomei et al, Brain Topography 2015] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 3 / 33

5 Introduction Origin of brain activity measured in EEG and MEG [Baillet et al., IEEE Signal Processing Mag, 2001] Pyramidal neurons Current perpendicular Neurons in a post-synaptic currents to cortical surface macrocolumn co-activate Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 4 / 33

6 Introduction Influence of orientation (spherical geometry) [courtesy of S.Baillet] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 5 / 33

7 Introduction Influence of depth (realistic geometry) [courtesy of S.Baillet] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 6 / 33

8 Forward problem: from Sources to Sensors Outline 1 Introduction 2 Forward problem: from Sources to Sensors Solving the Forward problem A nested geometry problem Boundary Element Method 3 Inverse Source Reconstruction Source Models Uniqueness Results Current Source Density Mapping 4 Conductivity estimation A transmission problem Semirealistic geometry Spherical geometry with Toufic Abboud and Théo Papadopoulo Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 7 / 33

9 Forward problem: from Sources to Sensors A nested geometry problem Nested domains Ω 1... Ω N, constant conductivity σ 1... σ N Outside domain: air Ω N+1 with vanishing conductivity σ N+1 = 0. Sources limited to domain Ω 1 : J p = {(p k, q k )} k=1...k K dipoles Quasistatic regime: electric Ω2 potential V satisfies Poisson Ω1 σ2 σ1 equation: ΩN+1 σn+1 Laplace fundamental solution: E = δ 0 dans R 3 In infinite homogeneous region β p J = T (Jp ) S1 S2 SN ΩN σn div σ V = K β p J (r) = q k E(r p k ) k=1 u = V σ 1 1 βp J harmonic in R3 \ {p k }. K q k δ pk = T (J p ), k=1 (E(r) = (4π r ) 1 Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 8 / 33

10 Forward problem: from Sources to Sensors Solving the forward problem simplest model: overlapping spheres no meshing required analytical methods crude approximation of head conduction, especially for EEG Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 9 / 33

11 Forward problem: from Sources to Sensors Solving the forward problem simplest model: overlapping spheres no meshing required analytical methods crude approximation of head conduction, especially for EEG surface-based-model: piecewise constant conductivity only surfaces need to be meshed Boundary Element Method (BEM) only isotropic conductivities Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 9 / 33

12 Forward problem: from Sources to Sensors Solving the forward problem simplest model: overlapping spheres no meshing required analytical methods crude approximation of head conduction, especially for EEG surface-based-model: piecewise constant conductivity only surfaces need to be meshed Boundary Element Method (BEM) only isotropic conductivities most sophisticated model: volume-based conductivity detailed conductivity model, (anisotropic: tensor at each voxel) Finite Element Method (FEM), huge meshes, difficult to handle Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 9 / 33

13 Forward problem: from Sources to Sensors Transmission principle through integral operators + V harmonic in each Ω i, i 2 + V (x) = [E(x y) V (y) V (y) E(x y)] dy Ω S i i 1 = v ext S i i 1(x) vi int (x) vi 1 ext (x) = ( ) S i 1 E(x y) V + n y (y) V + (y) E n y (y x) ds(y) harmonic outside of S i 1 vi int (x) = ( ) S i E(x y) V n y (y) V (y) E n y (y x) ds(y) harmonic inside of S i. Jumps across S i : ϕ i = v int vi ext q i = i int vi n ext v n i Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 10 / 33

14 Forward problem: from Sources to Sensors Calderon operators Sq(y) = E(x y) q(x) ds(x) S D Dϕ(y) = E S n x (x y) ϕ(x) ds(x) ( ) D S Calderon operator: H = N D q adjoint Nϕ hypersingular Associated projectors: C ext = I 2 H and C int = I 2 + H [ ] int [ ] [ ] V = C int ϕi ϕi 1 n V i + H q i,i 1 S i q i 1 i [ ] ext [ ] [ ] V = C ext ϕi ϕi+1 n V i + H q i,i+1 S i q i+1 i Continuity of V and σ n V across surfaces: transmission equations boundary element approximation Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 11 / 33

15 Forward problem: from Sources to Sensors Boundary Elements Electric potential generated by sources J p approximated through linear system: A V = B(J p ) Note: potential only defined up to a constant, to be fixed for system inversion Main steps involved for Boundary Element Method (BEM) 1 Geometry: segment Magnetic Resonance Image to recover surfaces S i and source space 2 Discretize surfaces S i (conforming trianglations) 3 Assemble matrices A et B(J p ) (mixed P 1 /P 0 elements) 4 Solve A V = B(J p ) Magnetic field: apply a matrix representing Biot-Savart to V. Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 12 / 33

16 Forward problem: from Sources to Sensors Symmetric BEM system [Kybic, Clerc, Abboud et al, 2005] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 13 / 33

17 Forward problem: from Sources to Sensors Symmetric BEM software: OpenMEEG [Gramfort, Papadopoulo, Olivi, Clerc, 2010] EEG and MEG Electrical Impedance Tomography Cortical Mapping ElectroCorticography Functional Electrical Stimulation Intracortical electrodes Cochlear Implants Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 14 / 33

18 Inverse source reconstruction Outline 1 Introduction 2 Forward problem: from Sources to Sensors Solving the Forward problem A nested geometry problem Boundary Element Method 3 Inverse Source Reconstruction Source Models Uniqueness results Current Source Density Mapping 4 Conductivity estimation A transmission problem Semirealistic geometry Spherical geometry Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 15 / 33

19 Inverse source reconstruction Inverse Problems Use measurements and prior knowledge to recover hidden information: source reconstruction Cortical Mapping conductivity estimation Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 16 / 33

20 Inverse source reconstruction Source reconstruction Two types of source models considered: isolated distributed unknowns measurements unknowns measurements sensitivity to model order indeterminacy regularization necessary Uniqueness of reconstruction: proven for each model. Ill-posedness, due to instability. Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 17 / 33

21 Inverse source reconstruction Uniqueness problem Is it possible to determine J p by measuring (V, n V ) on the scalp? or the magnetic field B on magnetometers? Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 18 / 33

22 Inverse source reconstruction Uniqueness problem Is it possible to determine J p by measuring (V, n V ) on the scalp? or the magnetic field B on magnetometers? Silent sources (Helmholtz, 1853): spherically symmetric conductor + radial dipole, B = 0 but V 0 current loop J p = 0, V = 0 but B 0 J p distributed to S, with constant magnitude, V = 0, B = 0 Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 18 / 33

23 Inverse source reconstruction Uniqueness results for EEG if V perfectly measured on the scalp: Nested volume model, with homog. conductivities σ i On S N : σ n V = 0. By linearity: uniqueness proved by proving that if V = 0 on scalp, then J p = 0. In each volume with no sources, σ i V = σ i V = 0. Continuity across S i : σ i ( n V ) = σ i+1 ( n V ) +. Holmgren theorem: unique continuation of harmonic potential whose Cauchy data(neumann and Dirichlet b.c.) are known on a dense portion of boundary Iteratively apply Holmgren: V = 0 in each volume, up to innermost surface containing sources. Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 19 / 33

24 Inverse source reconstruction Uniqueness results Distributed source model J p (r) = q(r) n(r) δ S (r) with n(r) S at position r Assumption S Ω 1 with constant conductivity Result q(r) can be determined up to a constant. (classical elliptical operator theory) Isolated dipole model moments and positions of the sources uniquely determined [El Badia - Ha Duong 2000] In spite of uniqueness, the inverse problem remains ill-posed (instability). Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 20 / 33

25 Inverse source reconstruction Current Source Density mapping Cortical Source reconstruction : sometimes cumbersome cortical surface highly convoluted, difficult to segment high number of vertices Alternative approach: mapping current sources on a simpler surface Recall that electric potential satisfies so outside of supp(j p ), div σ V = 0. Cortical Mapping principle div σ V = J p Reconstruct normal current on inner skull surface, given that V is known on sensors, div σ V = 0 outside the brain. Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 21 / 33

26 Inverse source reconstruction (rappel): éléments finis de surface symétriques Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 22 / 33

27 Inverse source reconstruction Current Source Density mapping true (simulated) reconstructed reconstructed (with noise) [Clerc Kybic Physics Med Biol 2007] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 23 / 33

28 Inverse source reconstruction Current Source Density mapping [He Neuroimage 2002] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 24 / 33

29 Conductivity estimation Outline 1 Introduction 2 Forward problem: from Sources to Sensors Solving the Forward problem A nested geometry problem Boundary Element Method 3 Inverse Source Reconstruction Source Models Regularized Source Reconstruction Uniqueness Results Current Source Density Mapping 4 Conductivity estimation A transmission problem Semirealistic geometry Spherical geometry with Juliette Leblond and Jean-Paul Marmorat Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 25 / 33

30 Conductivity estimation Conductivity σ Relation between sources J p and potential V σ V = J p EEG sensitive to ratio σ scalp /σ skull [Vallaghé, Clerc IEEE TBME 2009] σ scalp /σ skull Rush & Driscoll [1968] 80 Cohen & Cuffin [1983] 80 Oostendorp & al. [2000] 15 Gonçalves, de Munck etal. [2003] Challenge: calibrating σ, non-invasively, in vivo: injecting known current on the scalp; multimodal measurements (MEG,EEG). Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 26 / 33

31 Conductivity estimation Influence of conductivity on localization σ scalp /σ skull = 80 σ scalp /σ skull = 40 σ scalp /σ skull = 20 Averaged interictal spike. Inverse reconstruction using MUSIC. [courtesy of J-M Badier, La Timone] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 27 / 33

32 Conductivity estimation Uniqueness of conductivity reconstruction Problem statement: 3-layer problem σ 1 σ 2 σ 3 S 1 S 2 S 3 Known σ 1 and σ 3 : Is conductivity value σ 2 uniquely determined from a set of known sources J p and measurements V (on S 3 )? Denote Σ = [ ] I 0 0 σi Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 28 / 33

33 Conductivity estimation Uniqueness via transmission problem Notations x 1 X = x 2 where x i = x 3 [ ϕi ] and U = q i [ βs n β S ] S 1 Transmission relations lead to: AX = BU (Σ 1 C1 int + Σ 2 C1 ext ) Σ 2 H 12 0 with A = Σ 2 H 21 (Σ 2 C2 int + Σ 3 C2 ext ) Σ 3 H 23 0 Σ 3 H 32 (Σ 3 C3 int + Σ 3 C3 ext ) and B = [ Σ ] T Measurements on S 3 : V = CX with C = [ I /2 D 33 S 33 ] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 29 / 33

34 Conductivity estimation Discriminating source Transmission from internal sources U to external data V : Definitions, V = CA 1 BU σ 2 and σ 2 are U-distinguishable (for a given U) if σ 2 σ 2 V V. U is a discriminating source if any two distinct conductivities σ 2 and σ 2 are U-distinguishable. V V = (σ 2 σ 2)CA 1 σ 2 RA 1 σ 2 BU U is a discriminating source if and only if for all σ 2 σ 2, U / KerΦ(σ 2, σ 2 ). = (σ 2 σ 2)Φ(σ 2, σ 2) U. Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 30 / 33

35 Conductivity estimation Semirealistic geometry U-distinguishability property equivalent to: Φ(σ 2, σ 2 ) U 0 for given (σ 2, σ 2 ). σ 1 σ 2 σ 3 S 1 S 2 S 3 Φ(σ 2, σ 2 ) = CA 1 σ 2 RA 1 σ B can be approximated by Boundary Elements. 2 Re-implementing Symmetric BEM using dual (barycentric) finite element complex [Buffa, Christiansen, 2007] [Andriulli et al, 2008] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 31 / 33

36 Conductivity estimation Spherical geometry Spherical harmonic bases, order k (harmonic frequency): u Si (r, θ, ϕ) = k k=0 m= k [ α ikm r k + β ikm r (k+1)] Y km (θ, ϕ) Ω i α ikm vs β ikm are the spherical harmonic coeffs of harmonic vs anti-harmonic parts of u i ( harmonic inside vs outside S i ). Diagonalization by spherical harmonics lead to: so that Φ(σ 2, σ 2) = B 2 (k)σ 2 β 1k = ( A 3 (k)σ A 2 (k)σ 2 + A 1 (k) ) B 2 (k) (A 3 (k)σ 2 σ 2 A 1(k)) (A 3 (k)σ A 2(k)σ 2 + A 1 (k))(a 3 (k)σ A 2(k)σ 2 + A 1(k)) and uniqueness follows from A1(k) A = σ 3(k) 2σ 2. Stability results also obtained [Clerc, Leblond, Marmorat, Papageorgakis, 2016] Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 32 / 33

37 Conductivity estimation Conclusions and perspectives Inverse conductivity problem harder than expected crucial for EEG because of low skull conductivity important role of Calderon operators to be implemented in BEM antiharmonic projection of infinite potential for inverse source reconstruction geometry simplification? Maureen Clerc (Inria, France) Estimation de paramètres en imagerie cérébrale 33 / 33