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August  2018, 15(4): 905-932. doi: 10.3934/mbe.2018041

## EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity

 1 GRAMFC INSERM U1105, Department of Medicine, Amiens University Hospital, 80000 Amiens, France 2 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France 3 Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Université, Université de Technologie de Compiègne, 60205 Compiègne, France 4 Laboratoire de Mathématiques de Reims, EA4535, Université de Reims Champagne-Ardenne, 51687 Reims cedex 2, France

* Corresponding authorr: marion.darbas@u-picardie.fr.

Received  July 20, 2017 Accepted  October 13, 2017 Published  March 2018

The paper is devoted to the analysis of electroencephalography (EEG) in neonates. The goal is to investigate the impact of fontanels on EEG measurements, i.e. on the values of the electric potential on the scalp. In order to answer this clinical issue, a complete mathematical study (modeling, existence and uniqueness result, realistic simulations) is carried out. A model for the forward problem in EEG source localization is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. From a mathematical point of view, the model consists in solving an elliptic problem with a singular source term in an inhomogeneous medium. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy norm ($H^1$-norm). Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model including or not the fontanels have been obtained and corroborate the theoretical results. A mathematical tool related to the concept of Gâteau derivatives is introduced which is able to measure the sensitivity of the electric potential with respect to small variations in the fontanel conductivity. This study attests that the presence of fontanels in neonates does have an impact on EEG measurements.

Citation: Hamed Azizollahi, Marion Darbas, Mohamadou M. Diallo, Abdellatif El Badia, Stephanie Lohrengel. EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity. Mathematical Biosciences & Engineering, 2018, 15 (4) : 905-932. doi: 10.3934/mbe.2018041
##### References:

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##### References:
Fontanels and skull of a neonate.
Behavior of factors RDM and MAG with respect to the eccentricity of the dipole. Different mesh sizes (finest mesh $M_3$). Neonatal three-layer spherical head model without fontanels. Exact reference solution.
A spherical head model with the main fontanel.
Errors in $H^1$-norm with respect to the mesh size $h$ in logarithm scale. Three-layer spherical head model with the anterior fontanel (Gaussian behavior for the fontanel conductivity). Numerical reference solution computed on $M_{\tiny{\mbox{ref}}}$. Left: one single source $S = (0, 0, 40mm)$, $\mathbf{q} = (0, 0, J)$. Right: two sources $S_1 = (0, 0, 10mm)$, $S_2 = (0, 10mm, 0)$ with moments $\mathbf{q}_1 = (0, 0, J)$, $\mathbf{q}_2 = (0, J, 0)$. Intensity $J = 10^{-6} A.m^{-2}$.
Behavior of factors RDM and MAG with respect to the eccentricity dipole position for different meshes. Three-layer spherical model with the anterior fontanel (Gaussian behavior for the fontanel conductivity). Numerical reference solution computed on $M_{\tiny{\mbox{ref}}}$.
Realistic head model of a neonate. Left: skull and fontanels. Right: mesh of the fontanels.
The coronal, sagittal and axial plane of the head model and its 3D reconstruction.
Variations of factors RDM and MAG with respect to different conductivities $(\sigma_{\!f}, \sigma_{skull})$. Four-layer realistic head model. Reference solution computed with the model without fontanels.
Sensitivity of the electric potential on the scalp with respect to eccentricity. Distance source-interface brain/CSF $\approx 5$mm (left) and $\approx 15$mm (right).
Sensitivity of the electric potential on the scalp with respect to orientation. Distance source-interface brain/CSF $\approx 15$mm. Left: moment $\mathbf{q} = (0, J, J)$. Right: moment $\mathbf{q} = (J, J, 0)$.
Sensitivity of the electric potential on the scalp for a deep source.
Definition of meshes (neonatal three-layer spherical head model).
 Mesh Nodes Tetrahedra Boundary nodes $h_{min}$ [m] $h_{max}$ [m] $M_1$ $102 540$ $594 907$ $16 936$ $8.16 10^{-4}$ $4.81 10^{-3}$ $M_2$ $302 140$ $1\ 855 005$ $23 339$ $6.35 10^{-4}$ $3.07 10^{-3}$ $M_3$ $596 197$ $3 632 996$ $54 290$ $4.1 10^{-4}$ $2.46 10^{-3}$ $M_{\rm ref}$ $2 754 393$ $17 263 316$ $124 847$ $2.5 10^{-4}$ $1.51 10^{-3}$
 Mesh Nodes Tetrahedra Boundary nodes $h_{min}$ [m] $h_{max}$ [m] $M_1$ $102 540$ $594 907$ $16 936$ $8.16 10^{-4}$ $4.81 10^{-3}$ $M_2$ $302 140$ $1\ 855 005$ $23 339$ $6.35 10^{-4}$ $3.07 10^{-3}$ $M_3$ $596 197$ $3 632 996$ $54 290$ $4.1 10^{-4}$ $2.46 10^{-3}$ $M_{\rm ref}$ $2 754 393$ $17 263 316$ $124 847$ $2.5 10^{-4}$ $1.51 10^{-3}$
 Mesh Nodes Tetrahedra Boundary faces $h_{min}$ [m] $h_{max}$ [m] $M_{real}$ 108 669 590 878 55 660 $3.4\ 10^{-4}$ $14\ 10^{-3}$
 Mesh Nodes Tetrahedra Boundary faces $h_{min}$ [m] $h_{max}$ [m] $M_{real}$ 108 669 590 878 55 660 $3.4\ 10^{-4}$ $14\ 10^{-3}$
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