Inverse skull conductivity estimation problems from EEG data

International audienceA fundamental problem in theoretical neurosciences is the inverse problem of source localization, which aims at locating the sources of the electric activity of the functioning human brain using measurements usually acquired by non-invasive imaging techniques, such as the electroencephalography (EEG). EEG measures the effect of the electric activity of active brain regions through values of the electric potential furnished by a set of electrodes placed at the surface of the scalp and serves for clinical (location of epilepsy foci) and functional brain investigation. The inverse source localization problem in EEG is influenced by the electric conductivities of the several head tissues and mostly by the conductivity of the skull. The human skull isa bony tissue consisting of compact and spongy bone compartments, whose shape and size vary over the age and the individual’s anatomy making difficult to accurately model the skull conductivity

Dictionary learning for M/EEG multidimensional data

International audienceSignals obtained from magneto- or electroencephalography (M/EEG) are very noisy and inherently multi-dimensional, i.e. provide a vector of measurements at each single time instant. To cope with noise, researchers traditionally acquire measurements over multiple repetitions (trials) and average them to classify various patterns of activity. This is not optimal because of trial-to-trial variability (waveform variation, jitters). The jitter-adaptivedictionary learning method (JADL) has been developed to better handle for this variability (with a particular emphasis on jitters). JADL is a data-driven method that learns a dictionary (prototype pieces) from a set of signals, but is currently limited to a single channel, which restricts its capacity to work with very noisy data such as M/EEG. We propose an extension to the jitter-adaptive dictionary learning method, that is able to handle multidimensional measurements such as M/EEG

Modèles de conductivité patient-spécifiques : caractérisation de l’os du crâne.

One of the major issues related to electroencephalography (EEG) is to localize where in the brain signals are generated, this is so called inverse problem of source localization. The quality of the source localization depends on the accuracy of the geometry and the electrical conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of the source localization,due to its low value. Moreover, the human skull is a bony tissue consisting of compact and spongy bone layers, whose thickness vary across the skull. As the skull tissue composition has strong inter-individual variability both in terms of geometry and of individual conductivity, conductivity estimation techniques are required in order to determine the unknown skull conductivity. The aim of this thesis is to reduce the uncertainty on the skull conductivity both in spherical and realistic head geometries in order to increase the quality of the inverse source localization problem. Therefore, conductivity estimation is first performed on a 3-layered spherical head model. Existence, uniqueness and stability of the conductivity in the intermediate skull layer are discussed, together with a constructive recovery scheme. Then a simulation study is performed comparing two realistic head models, a bulk model where the skull is modelled as a single compartment and a detailed one accounting for the compact and spongy bone layers, in order to determine the importance of the internal skull structure for conductivity estimation in EEG.Les problèmes inverses de localisation de sources en électroencéphalographie (EEG) consistent à retrouver le lieu d'origine dans le cerveau des signaux mesurés sur le scalp. La qualité du résultat de localisation dépend des modèles géométriques et de conductivité électrique utilisés pour la résolution du problème. Parmi les tissus composant la tête, le crâne est celui dont la conductivité est la plus influente, en particulier à cause de sa faible valeur. De plus, le crâne humain est un tissu osseux comportant des parties dures et spongieuses, d'épaisseurs variables. Sa composition est très variable selon les individus, en termes de géométrie et de valeurs des conductivités, d'où la nécessité de développer des technique d'estimation de conductivités inconnues dans le crâne. Le but de cette thèse est de réduire l'incertitude sur la conductivité du crâne, pour des géométries sphériques et réalistes, en particulier en vue d’améliorer les résultats d'estimation des sources dans le problème inverse EEG. Dans le cas d'un domaine sphérique à 3 couches, l'existence, l'unicité et la stabilité de la conductivité dans la couche intermédiaire (crâne) sont discutées, et une procédure de reconstruction est proposée. Puis deux modèles plus réalistes de tête sont étudiés, l'un pour lequel le crâne est modelisé par un seul compartiment, l'autre dans lequel les parties spongieuse et dure sont distinguées. Des simulations numériques mettent en évidence le rôle de la structure interne du crâne pour la détermination de sa conductivité

Patient specific conductivity models : characterization of the skull bones

Les problèmes inverses de localisation de sources en électroencéphalographie (EEG) consistent à retrouver le lieu d'origine dans le cerveau des signaux mesurés sur le scalp. La qualité du résultat de localisation dépend des modèles géométriques et de conductivité électrique utilisés pour la résolution du problème. Parmi les tissus composant la tête, le crâne est celui dont la conductivité est la plus influente, en particulier à cause de sa faible valeur. De plus, le crâne humain est un tissu osseux comportant des parties dures et spongieuses, d'épaisseurs variables. Sa composition est très variable selon les individus, en termes de géométrie et de valeurs des conductivités, d'où la nécessité de développer des technique d'estimation de conductivités inconnues dans le crâne. Le but de cette thèse est de réduire l'incertitude sur la conductivité du crâne, pour des géométries sphériques et réalistes, en particulier en vue d’améliorer les résultats d'estimation des sources dans le problème inverse EEG. Dans le cas d'un domaine sphérique à 3 couches, l'existence, l'unicité et la stabilité de la conductivité dans la couche intermédiaire (crâne) sont discutées, et une procédure de reconstruction est proposée. Puis deux modèles plus réalistes de tête sont étudiés, l'un pour lequel le crâne est modelisé par un seul compartiment, l'autre dans lequel les parties spongieuses et dure sont distinguées. Des simulations numériques mettent en évidence le rôle de la structure interne du crâne pour la détermination de sa conductivité.One of the major issues related to electroencephalography (EEG) is to localize where in the brain signals are generated, this is so called inverse problem of source localization. The quality of the source localization depends on the accuracy of the geometry and the electrical conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of the source localization, due to its low value. Moreover, the human skull is a bony tissue consisting of compact and spongy bone layers, whose thickness vary across the skull. As the skull tissue composition has strong inter-individual variability both in terms of geometry and of individual conductivity, conductivity estimation techniques are required in order to determine the unknown skull conductivity. The aim of this thesis is to reduce the uncertainty on the skull conductivity both in spherical and realistic head geometries in order to increase the quality of the inverse source localization problem. Therefore, conductivity estimation is first performed on a 3-layered spherical head model. Existence, uniqueness and stability of the conductivity in the intermediate skull layer are discussed, together with a constructive recovery scheme. Then a simulation study is performed comparing two realistic head models, a bulk model where the skull is modelled as a single compartment and a detailed one accounting for the compact and spongy bone layers, in order to determine the importance of the internal skull structure for conductivity estimation in EEG

Turbulence modeling of gaseous injection and mixing in DI engines.

With increasing interest in alternative fuel technology, natural gas has become an attractive fuel for reciprocating engines. In this work, a modified version of the Los Alamos KIVA3 code has been used to model gaseous injection and mixing, by establishing the necessary boundary conditions at the injector/cylinder interface. It has been shown that gaseous injection models have to be accompanied by sufficient grid refinement to capture length scales of the order of the injector diameter (Abraham, 1997). In addition, high fidelity turbulence models are needed to monitor the plume evolution away from the injector. Turbulence modeling is explored in this work by considering variants of the k-$\epsilon$ and LES models. The standard k-$\epsilon$ model (Launder and Spalding, 1972) was found to underpredict the recirculation length in the backward-facing step geometry, overpredict dissipation in the confined co-flow jet geometry and underpredict penetration histories in all three gaseous jet configurations considered. These jet configurations included downward injection of methane in a pressurized vessel (Aesoy, 1996), horizontal injection of hydrogen in a confined chamber (Tomita et al., 1997) and upward impinging transient acetylene jet in a confined box (Fujimoto et al., 1997). The implementation of the nonlinear k-$\epsilon$ model (Speziale, 1986) either provided moderate corrections in the geometries of the backward-facing step and confined co-flow jets, or produced convergence problems in engine calculations. The implementation of the RNG based k-$\epsilon$ model (Han and Reitz, 1995) provided better agreement with experiments for both the recirculating (backward-facing step) and gaseous injection cases. The implementation of the LES model was based on the formulation of Moin et al. (1991), and the evaluation of the sub-grid scale stresses proposed by Lily, (1992). It was shown that this model was significantly affected by the high numerical diffusivity of KTVA-3. Based on the better predictions for recirculating flows and closer agreement with measured penetration data, the RNG k-$\epsilon$ model was selected for engine calculations. Parametric studies of injector and chamber geometries indicated that an optimum injection strategy could lead to as much as 100% flammable mixture near top dead center.Ph.D.Applied SciencesAutomotive engineeringMechanical engineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130792/2/9811154.pd

Influence of skull modelling on conductivity estimation for EEG source analysis

International audienceThe skull conductivity strongly influences the accuracy of EEG source localization methods. As the conductivity of the skull has strong inter-individual variability, conductivity estimation techniques are required. Typically, conductivity estimation is performed on data from a single event-related stimulation paradigm, which can be explained by one dipole source. A conductivity value for the skull can be estimated as the value for which the single dipole source provides the best goodness of fit to the data. This conductivity value is then used to analyse the actual data of interest. It is known that the optimal local skull conductivity when modelling the skull as one compartment depends on the amount of spongiosa present locally. The research question arising is: Is conductivity estimation based on data from a single paradigm meaningful without accounting for the internal skull structure

Dictionary Learning for Multidimensional Data

International audienceElectroencephalography(EEG) and magnetoencephalography (MEG) measure the electrical activity of the functioning brain usinga set of sensors placed on the scalp (electrodes and magnetometers). Magneto- or electroencephalography (M/EEG) have the same biological origin, the activity of the pyramidal neurones within the cortex. The signals obtained from M/EEG are very noisy and inherently multi-dimensional,i.e. provide a vector of measurements at each single time instant. To cope with the noise, researchers, traditionally acquire measurements overmultiple repetitions (trials) and average them to classify various patterns of activity. This is not optimal because of trial to trial variability. Thejitter-adaptive dictionary learning method (JADL) [1] has been developed to better handle for this variability. JADL is a data-based method thatlearns a dictionary from a set of signals, but is currently limited to a single channel, which restricts its capacity with very noisy data such asM/EEG. In this paper, we propose an extension to the jitter-adaptive dictionary learning method, in order to handle multidimensional measurements such as M/EEG. A modified model is developed and tested using synthetically generated data set as well as real M/EEG signals. The results obtained using our model look promising, and show superior performance compared to the original single-channel JADL framework

Inverse conductivity recovery problem in a spherical geometry from EEG data: uniqueness, reconstruction and stability results

International audienceElectroencephalography (EEG) is a non invasive imaging technique that measures the effect of the electric activity of active brain regions, called sources, through values of the electric potential furnished by a set of electrodes placed at the surface of the scalp. A fundamental problem there is the inverse problem of source localization which aims at locating the sources of the electric activity using the acquired EEG measurements [2]. The quality of the source estimation depends on the accuracy of the conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of EEG source localization [3]. Often, conductivity estimation is performed prior to the source estimation to determine the unknown conductivity using either supplementary EEG measurements or even measurements acquired by other imaging techniques. Such a technique is electrical impedance tomography (EIT) where current is injected through a pair of EEG electrodes while the unknown conductivities can be estimated by the resulting measurements at the rest electrode locations. We examine the inverse skull conductivity estimation problem, which aims at recovering the electrical conductivity properties of the skull from measurements given at the surface of the head by EEG measurements. Our goal is to show uniqueness and a constructive scheme for the inverse skull conductivity estimation problem using partial boundary EEG data from a single experiment, in the preliminary case of an homogeneous skull conductivity. This is a version of the many inverse conductivity issues still under study nowadays [1]. The head is assumed to be an isotropic piecewise homogeneous medium and we examine a layered spherical head model made of three concentric nested spheres, each of them modelling scalp, skull and brain tissues (from the outermost to the innermost layer). Each of the three layers is supposed to have a constant conductivity. We also assume that the conductivities of the brain and the scalp are known, while the conductivity to be recovered is the one of the intermediate spherical layer (skull). We solve the above conductivity estimation problem from the available EEG partial boundary data, expanded on the spherical harmonics basis, and transmitted over the spherical interfaces by transfer functions, while we consider that the source term is already estimated (through a number of coefficients of its spherical harmonics expansion). Linear algebra computations then allow us to find polynomials that possess a root which should coincide with the unknown skull conductivity, thus solving the estimation problem. We derive uniqueness properties and a reconstruction algorithm for the skull conductivity. A numerical study shows that the algorithm is able to accurately estimate the skull conductivity, with good robustness properties with respect to various levels of noise. The properties of the inverse conductivity estimation problem are also examined with various source configurations (partially known sources) and EIT measurements. This work was supported by the Région Provence-Alpes-Côte d'Azur, France, and BESA GmbH, Germany

Uniqueness result for an inverse conductivity recovery problem with application to EEG

Abstract. Considering a geometry made of three concentric spherical nested layers, (brain, skull, scalp) each with constant homogeneous conductivity, we establish a uniqueness result in inverse conductivity estimation, from partial boundary data in presence of a known source term. We make use of spherical harmonics and linear algebra computations, that also provide us with stability results and a robust reconstruction algorithm. As an application to electroencephalography (EEG), in a spherical 3-layer head model (brain, skull, scalp), we numerically estimate the skull conductivity from available data (electrical potential at electrodes locations on the scalp, vanishing current flux) and given pointwise dipolar sources in the brain

On some inverse conductivity recovery problem in a sphere: Uniqueness and reconstruction results with applications to EEG

International audienceElectroencephalography (EEG) is a non invasive imaging technique that measures the effect of the electric activity of active brain regions, called sources, through values of the electric potential furnished by a set of electrodes placed at the surface of the scalp. A fundamental problem there is the inverse problem of source localization which aims at locating the sources of the electric activity using the acquired EEG measurements [2]. The quality of the source estimation depends on the accuracy of the conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of EEG source localization [3]. Indeed, modelling the electrical conductivity values for the scalp and the brain are relatively well-known and do not vary much across subjects, but this is not the case for the skull.We examine the inverse skull conductivity estimation problem, which aims at recovering the electrical conductivity properties of the skull from measurements given at the surface of the head by EEG measurements. Our goal is to show uniqueness and a constructive scheme for the inverse skull conductivity estimation problem using partial boundary EEG data from a single experiment, in the preliminary case of an homogeneous skull conductivity. This is a version of the many inverse conductivity issues still under study nowadays [1].The head is assumed to be an isotropic piecewise homogeneous medium and we examine a layered spherical head model made of three concentric nested spheres, each of them modelling scalp, skull and brain tissues (from the outermost to the innermost layer). Each of the three layers is supposed to have a constant conductivity. We also assume that the conductivities of the brain and the scalp are known, while the conductivity to be recovered is the one of the intermediate spherical layer (skull).We solve the above conductivity estimation problem from the available EEG partial boundary data, expanded on the spherical harmonics basis, and transmitted over the spherical interfaces by transfer functions, while we consider that the source term is already estimated (through a number of coefficients of its spherical harmonics expansion). Linear algebra computations then allow us to find polynomials that possess a root which should coincide with the unknown skull conductivity, thus solving the estimation problem. We derive uniqueness properties and a reconstruction algorithm for the skull conductivity. It uses a non-linear least squares minimization scheme applied to the computed spherical harmonics coefficients of the solution in the three layers. A numerical study shows that the algorithm is able to accurately estimate the skull conductivity, with good robustness properties with respect to various levels of noise. This work was supported by the Région Provence-Alpes-Côte d'Azur, France, and BESA GmbH, Germany