LEICI   25638
INSTITUTO DE INVESTIGACIONES EN ELECTRONICA, CONTROL Y PROCESAMIENTO DE SEÑALES
Unidad Ejecutora - UE
congresos y reuniones científicas
Título:
Validation of forward solvers accuracy in EEG, EIT and TES
Autor/es:
MARIANO FERNÁNDEZ CORAZZA; SERGEI TUROVETS; PHAN LUU; CARLOS MURAVCHIK; NICK PRICE; DON TUCKER
Lugar:
Vancouver
Reunión:
Conferencia; OHBM 2017; 2017
Institución organizadora:
Organization for Human Brain Mapping
Resumen:
1 IntroductionAccurate electrical head models are needed in electroencephalography (EEG) [1], electrical impedance tomography (EIT) [2] and in transcranial electrical stimulation (TES) [3]. The forward problem (FP) consists in computing the electric potential in the head given the current sources: set of monopoles on the scalp in EIT and TES or dipoles on the cortex in EEG. EIT/TES and EEG FPs are related by the reciprocity principle [4]. The FP can be handled semi-analytically for spherical [5, 6] and numerically for realistically shaped head models using the finite element method (FEM) [3], the finite difference/volume method (FDM) [7], and the boundary element method (BEM) [2]. However, determining the model accuracy is not a straightforward task.In this work the numerical and semi-analytical solutions in a three layer spherical model of the head are compared. The novelty is in cross-validation of FDM and FEM versus semi-analytics formulated for a practical use with more than ten thousand Legendre polynomial expansion terms and therefore very accurate.2 MethodsMatrices used in this work: i) ψ, the potential scalar field impressed in the head by each independent pair of electrodes on the scalp (L-1 pair, where L is the total number of electrodes); ii) T, vector electric fields (gradients of ψ) at each voxel, this is the transfer matrix from the scalp to the brain; iii) M, the lead field matrix from the brain to the scalp computed using reciprocity from T.FEM assumes a discretization of the volume into elements (f.e., tetrahedrons) and the integrals are solved assuming base functions for each element. We used our own implementation of linear FEM with the Galerkin approach [8].FDM takes the original segmented volumetric image and solves the volume integrals in the native voxel space [9].The analytical formulation involves summation of infinite number of terms containing Legendre polynomials and the ratio of natural powers of real numbers [1, 6]. In addition to an increase of computational time to include higher order terms, some coefficients tend to diverge very fast and the algorithm becomes unstable typically after hundreds of terms. In a direct formulation for EEG this problem is avoided by approximating the terms dependence from their index analytically [1, 6]. In the TES/EIT formulation we managed to rearrange many internal variables in such a way that the stability extends up to thousands of terms included in our Matlab implementation of the ?infinite? summation.3 ResultsFor a three layer sphere with radii 9.2, 8.5, and 7.5 cm we compared the potentials computed semi-analytically with 100-10000 terms and with the FE and FD methods and derived from those the matrices T and M by applying the gradient operation and the reciprocity principle. We assumed 256 sensors following the EGI geodesic standard and 2400 dipoles at the brain layer with normal to cortex orientation. Fig. 2 shows examples of the potentials computed using the three different methods. The squared errors in the potentials were 0.37mV for the FEM, 11.3mV for the FDM, 0.45mV for analytics with 1000 terms, and 5.4mV for the analytics with 100 terms, in all cases compared to the analytic solution with 1000 terms.4 ConclusionsThe use of a large number of terms in the analytical solution reduced the difference between the numerical methods and the analytical solution, indicating that higher order terms might be still relevant. The differences between the three methods are negligible with respect to modelling errors and other sources of noise. Thus, our in-house FDM and FEM implementations are valid for solving the EEG, EIT, and TES electromagnetic FPs in realistic head models.