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Electroencephalography (EEG) from scalp potentials has affirmed itself as one of the primary non-invasive techniques to map and study the brain electric activity. EEG imaging is of crucial importance in the electric characterization of epileptic seizures. This is especially true for patients affected by focal epilepsy, when source characterization and localization is a key step of a pre-surgical protocol that precedes the removal of the patient’s brain area that shows an abnormal electric activity.

State of the Art

A key step in EEG based neuroimaging is represented by the modeling of the forward propagation problem, where the electric current is computationally retrieved given input potential data and volumetric information on the brain medium. The three main techniques currently adopted in literature to accomplish this task are the Finite Difference Method (FDM), the Finite Element Method (FEM), and the surface Boundary Element Method (BEM). Both FDM and FEM are differential equation based methods and they directly attempt at numerically solving the Poisson’s Equation.
In FDM, the differential operators are substituted by finite differences on a grid encompassing the entire head volume and, by doing this, the continuous equation is discretized into a sparse linear system that, due to its high dimensionality (and its being implicit), has to be solved iteratively. Similarly to the FDM, the FEM discretizes the Poisson equation obtaining a sparse linear system. Differently from the FDM, however, this is obtained by subdividing the head volume in voxels and by approximately expanding the unknown as a sum of functions (the finite elements) whose domains encompass only few voxels. Both the FDM and the FEM can handle head models of arbitrary geometry. Moreover even inhomogeneous and anisotropic conductive constants can be handled by both schemes. However these schemes are far from being panacea. In fact when the required resolution from the EEG increases, this translates in the need for handling head models of always increasing detail and complexity. This results in the increasing of the number of grid points and voxels in FDM and FEM respectively. As a consequence the dimensionality of the associated linear system grows unbounded. To this it should be added that since both FDM and FEM are discretizing a differential operator, the resulting matrix is prone to severe ill-conditioning when the system dimensionality increases. In this scenario the solution of the linear system becomes increasingly harder, up to the point of being unfeasible for all practical purposes.

Scientific results

The EU Marie Curie project NEUROIMAGEEG has been targeting the solution of the issues above by investigating an innovative EEG forward problem solver based on a fast surface/volume integral equation formulation discretized with a Boundary Element Method (BEM). The general idea is that such a BEM method, (differently from other BEM methods currently used in literature), is able to handle unhomogeneities and anisotropies like the FEMs do. At the same time the new solver will keep the levels of accuracy and numerical stability ensured by the most advanced BEMs in literature, from which it will borrow the virtuous strategies for ensuring high levels precision. In other words the new technology developed in this project combines the benefits of FEM and BEM approaches without sharing their deficiencies. Moreover, the project resulted also in new tools capable of handling inhomogeneities and anisotropies of the brain tissues within an fully integral equation framework. All these new formulations have also the additional unequalled feature of allowing for a fast direct inverse (in the meaning of “non-iterative”) solution of the EEG direct problem with only a linear computational complexity (the standard complexity is cubic). This makes an extreme difference for large problem dimensionalities, and for simulations where the electrodes’s number and configuration are changing. The new solver has been integrated with an inverse EEG problem solver to obtain a complete EEG imaging tool and after validation on a EEG device, the new tool has been applied to the solution of real case scenarios. The overall research effort resulted in a faster EEG imaging tool, providing better spatial resolutions. This notwithstanding, to further minimize the EEG localization error, this project has used the new tool for investigating the optimization of EEG electrodes’ configuration, in the presence of high resolution and patient specific models, high and varying electrode numbers, and different locations of the epileptogenic area.

Prospects of the research career development and re-integration of the fellow

This Career Integration Grant, played a determinant role in the integration of the Fellow in his new research environment and in its achieving of full scientific independence. During the completion of the project, not only the Fellow was promoted to a Full Professorship at a very young age, but he also succeeded in opening his own laboratory and in consolidating his international reputation. This also resulted in several international awards that the Fellow has collected during the completion of this project, including the 2015 Leopold Felsen Award
For Excellence in Electrodynamics and the URSI Issac Koga Gold Medal of the triennium 2014-2016.

Contact: Francesco P. Andriulli

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Institut Telecom
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