The aim of this study was to investigate the composition of an acoustic scattering layer in the North Sea that is particularly strong at 38 kHz. A full definition of the biological composition of the layer, along with its acoustic properties, would allow for it to be confidently removed from data collected during acoustic fish surveys, where it presents a potential source of bias. The layer, traditionally and informally referred to as consisting of zooplankton, appears similar to others observed internationally. The methodology utilised in this study consisted of biological and acoustic sampling, followed by application of forward and inverse acoustic modelling techniques. Acoustic data was collected at 38, 120 and 200 kHz in July 2003, with the addition of 18 kHz in July 2004. Net samples were collected in layers of relatively strong 38 kHz acoustic scattering using a U-tow vehicle (2003) and a MIKT net (2004). Acoustic data were scrutinised to determine actual backscattering, expressed as mean volume backscattering strength (MVBS) (dB). This observed MVBS (MVBSobs) was compared with backscattering predicted by applying the forward problem solution (MVBSpred) to sampled animal densities in order to determine whether those animals were responsible for the enhanced 38 kHz scattering. In most instances, MVBSobs > MVBSpred, more pronounced at 38 kHz. It was found that MVBSpred approached MVBSobs more closely with MIKT than with U-tow samples, but that the 38 kHz mismatch was present in both. Inversion of candidate acoustic models predicted gas-bearing scatterers, which are strong at 38 kHz, as most likely to be responsible for this. Potential sources of inconsistencies between MVBSpred and MVBSobs were identified. The presented forward and inverse solutions infer that although the layer often contains large numbers of common zooplankton types, such as copepods and euphausiids, these are not the dominant acoustic scatterer at 38 kHz. Rather, there remains an unidentified, probably gas-bearing scatterer that contributes significantly to observed scattering levels at this frequency. This study identifies and considerably narrows the list of candidates that are most likely to be responsible for enhanced 38 kHz scattering in the North Sea layer, and recommendations are made for potential future studies.

One of the goals of Electroencephalography (EEG) is to correctly localize brain activities by the help of voltage measurements taken on scalp. However, due to computational difficulties of the problem and technological limitations, the accuracy level of the activity localization is not perfect and should be improved. To increase accuracy level of the solution, realistic, i.e. patient dependent, head models should be created. Such head models are created via assigning realistic conductivity values of head tissues onto realistic tissue positions. This study initially focuses on obtaining patient dependent spatial information from T1-weighted Magnetic Resonance (MR) head images. Existing segmentation algorithms are modified according to our needs for classifying eye tissues, white matter, gray matter, cerebrospinal fluid, skull and scalp from volumetric MR head images. Determination of patient dependent conductivity values, on the other hand, is not considered as a part of this study, and isotropic conductivity values anticipated in literature are assigned to each segmented MR-voxel accordingly. Upon completion of the tissue classification, forward problem of EEG is solved using the Finite Difference (FD) method employing a realistic head model. Utilization of the FD method aims to lower computational complexity and to simplify the process of mesh creation for brain, which has a very complex boundary. Accuracy of the employed numerical method is investigated both on Electrical Impedance Tomography (EIT) and EEG forward problems, for which analytical solutions are available. The purpose of EIT forward problem integration into this study is to evaluate reciprocal solution of the EEG forward problem.

In the human brain, electrical activities are created due to the body functions. These electrical activities create potentials and magnetic fields which can be monitored elec- trically (Electroencephalography - EEG) or magnetically (Magnetoencephalography - MEG). Electrical activities in human brain are usually modeled by electrical dipoles. The purpose of Electro-magnetic source imaging (EMSI) is to determine the position, orientation and strength of dipoles. The first stage of EMSI is to model the human head numerically. In this study, The Finite Element Method (FEM) is chosen to han- dle anisotropy in the brain. The second stage of EMSI is to solve the potentials and magnetic fields for an assumed dipole configuration (forward problem). Realistic con- ductivity distribution of human head is required for more accurate forward problem solutions. However, to our knowledge, conductivity distribution for an individual has not been computed yet. The aim of this thesis study is to investigate the feasibility of a new approach to update the initially assumed conductivity distribution by using the evoked potentials and fields acquired during EMSI studies. This will increase the success of source localization problem, since more realistic conductivity distribution of the head will be used in the forward problem. This new method can also be used as a new imaging modality, especially for inhomogeneities where the conductivity value deviates. In this thesis study, to investigate the sensitivity of measurements to conductivity perturbations, a FEM based sensitivity matrix approach is used. The performance of the proposed method is tested using three different head models - homogeneous spherical, 4 layer concentric sphere and realistic head model. For spherical head models rectangular grids are preferred in the middle and curved elements are used nearby the head boundary. For realistic cases, head models are developed using uniform grids. Tissue boundary information is obtained by applying segmentation algorithms to the Magnetic Resonance (MR) images. A paralel computer cluster is employed to assess the feasibility of this new approach. PETSc library is used for forward problem calculations and linear system solutions. The performance of this novel approach depends on many factors such as the head model, number of dipoles and sensors used in the calculation, noise in the measure- ments, etc. In this thesis study, a number of simulations are performed to investigate the effects of each of these parameters. Increase in the number of elements in the head model leads to the increase in the number of unknows for linear system solu- tions. Then, accuracy of the solution is improved with increased number of dipoles or sensors. The performance of the adopted approach is investigated using noise-free measurements as well as noisy measurements. For EEG, measurement noise decreases the accuracy of the approach. For MEG, the effect of measurement noise is more pronounced and may lead to a larger error in tissue conductivity calculation.

The electrical currents generated in the heart propagate to the outward direction of the body by means of conductive tissues and these currents yield a potential distribution on the body surface. This potential distribution is recorded and analyzed by a tool called electrocardiogram. It is not a problem, if this process continues normally / however, when it is distorted by some abnormalities, the results will be fatal. Electrocardiography (ECG) is the technique dealing with the acquisition and interpretation of the electrical potentials recorded at the body surface due to the electrical activity of the heart. This can be realized by using one of the two approaches utilized in ECG namely / forward and inverse problems. The former one entails the calculation potentials on the body surface from known electrical activity of the heart and the latter one does the reverse. In this thesis, we will construct the body surface potentials in a realistic torso model starting from the epicardial potentials. In order to solve the forward problem, one needs a geometric model that includes the torso and the heart surfaces, as well as the intermediate surfaces or the intervening volume, and some assumptions about the electrical conductivity inside the enclosed volume. A realistic torso model has a complex geometry and this complexity makes it impossible to solve the forward problem analytically. In this study, Boundary Element Method (BEM) will be applied to solve the forward problem numerically. Furthermore, the effect of torso inhomogeneities such as lungs, muscles and skin to the body surface potentials will be analyzed numerically.

[EN] The behavior of the heart is governed by electrical currents generated in the myocardium, and therefore, the study of the cardiac electrical activity is essential for the diagnosis of cardiac diseases. The forward problem of the electrocardiography (FP) entails the calculation of the torso potentials from the electrical activity of the heart and the 3D body model, while the inverse problem (IP) resolution allows the noninvasive reconstruction of the electrical activity of the heart from surface potentials. The IP is of great importance in clinical applications since it allows estimating the electrical activity of the myocardium with only noninvasive recordings. However, IP resolution is still a big challenge in electrocardiography since it is ill-posed, very unstable and has multiple solutions. In this thesis different algorithms and strategies based on the IP resolution were developed and applied in the noninvasive diagnosis of ventricular and atrial arrhythmias and evaluated with mathematical cellular models and clinical data bases. The thesis focuses on the IP resolution for the noninvasive reconstruction of the myocardial electrical activity for different diseases and propagation patterns, implementing a novel system for complex propagation patterns. The obtained results and propagation patterns were evaluated and classified with the corresponding optimal resolution strategy that minimizes the error and increases the stability of the system, proving its advantages and disadvantages depending on the different diseases and their activation pattern. A novel iterative method was implemented for the IP dipolar resolution optimized for representing simple propagation patterns, achieving a high stability and robustness against noise by constraining the solution to a limited number of dipoles. However, propagation patterns not representable by few dipoles need to be computed with the IP in terms of epicardial solutions which provide a more detailed estimation of the myocardial activity. IP resolution in the voltage and phase domains showed a good accuracy for simple and organized propagation patterns. This method allowed the noninvasive diagnosis of the Brugada syndrome or the location of ectopic focus in atrial arrhythmias by performing a parametric analysis of the electrograms morphology or the activation map reconstruction. However, mathematical and patient results presented in this thesis proved that, for complex propagation patterns like atrial fibrillation (AF), inverse solutions in the voltage and phase domains are over-smoothed and over-optimistic, simplifying the complex AF activity, leading to non-physiological results that do not match with the complex intracardiac electrograms recorded in AF patients. In this thesis, we proposed a novel technique for the noninvasive identification and location of high dominant frequency AF sources, based on the assumption that in many cases atrial drivers present the highest activation rate with an intermittent propagation to the rest of the tissue that activates at a slower rate. Although, voltage and phase inverse solutions for AF complex propagation patterns were over smoothed and inaccurate, the noninvasive estimation of frequency maps was significantly more accurate, allowing the identification of the AF frequency gradient and location of high frequency sources. This technique may help in planning ablation procedures, avoiding unnecessary interseptal punctures for right-to-left frequency gradients cases and facilitating the targeting of the AF drivers, reducing risk and time of the clinical procedure. / [ES] El comportamiento del corazón se rige por corrientes eléctricas generadas en el miocardio y, por lo tanto, el estudio de su actividad eléctrica es esencial para el diagnóstico de enfermedades cardíacas. El problema directo (PD) de la electrocardiografía implica el cálculo de los potenciales del torso a partir de la actividad eléctrica del corazón y el modelo 3D del cuerpo, mientras que la resolución del problema inverso (PI) permite la reconstrucción no invasiva de la actividad eléctrica del corazón a partir de los potenciales de superficie, cobrando una gran importancia en la práctica clínica. Sin embargo, sigue siendo un gran desafío para la electrocardiografía ya que está mal planteado, es muy inestable y tiene múltiples soluciones. A lo largo de esta tesis se han desarrollado diferentes estrategias para la resolución del PI, aplicándolas en el diagnóstico no invasivo de arritmias ventriculares y auriculares, verificándolas mediante modelos celulares matemáticos y bases de datos clínicas. La tesis se centra en la resolución del PI para la reconstrucción no invasiva de la actividad eléctrica del miocardio para diferentes enfermedades cardiacas con diferentes patrones de propagación, implementando un novedoso sistema para patrones de propagación complejos. Además, se han validado los resultados obtenidos y se han clasificado los diferentes patrones de propagación con la estrategia de resolución del PI óptima que minimice el error y aumente la estabilidad del sistema. Un nuevo método iterativo fue implementado para la resolución del PI para fuentes dipolares, siendo óptimo para representar patrones de propagación simples, logrando una alta estabilidad e inmunidad al ruido al restringir la solución a un número limitado de dipolos. Sin embargo, los patrones de propagación que no pueden ser representados por un número limitado de dipolos deben calcularse mediante la resolución del PI en términos de potenciales epicárdicos, proporcionando una estimación más detallada de la actividad del miocardio. La resolución del PI en el dominio de la tensión y fase mostró ser muy preciso para patrones de propagación simples y organizados. Este método permite el diagnóstico no invasivo del síndrome de Brugada o la ubicación de focos ectópicos en arritmias auriculares mediante un análisis paramétrico de la morfología de los electrogramas o la reconstrucción de los mapas de activación. Sin embargo, los resultados matemáticos y clínicos presentados en esta tesis demostraron que, para patrones de propagación complejos como la fibrilación auricular (FA), los resultados obtenidos mediante la resolución del PI en el dominio de la tensión y fase son demasiado suaves y optimistas, simplificando enormemente la complejidad de la FA, llevando a resultados no fisiológicos que no coinciden con la actividad compleja de los electrogramas intracardiacos registrados en pacientes con FA. En esta tesis, se ha propuesto una novedosa técnica para la identificación y localización no invasiva de fuentes con una frecuencia dominante alta, basado en la suposición de que en muchos casos las fuentes eléctricas que generan y mantienen la FA presentan una tasa de activación más alta, con una propagación intermitente hacia el resto del tejido auricular cuya frecuencia de activación es más lenta. Aunque las soluciones en el dominio de la tensión y fase para patrones de propagación complejos fueron más suaves y menos precisas, la estimación no invasiva de los mapas de frecuencia fue significativamente más precisa, permitiendo la identificación del gradiente de frecuencia y ubicación de las fuentes de FA de alta frecuencia. Esta técnica puede ser de gran ayuda en la planificación de los procedimientos de ablación, evitando punciones interseptales innecesarias para casos con un gradiente de frecuencia de derecha a izquierda y facilitando la localización de las fuentes de alta frecuencia / [CAT] El comportament del cor es regeix per corrents elèctrics generades en el miocardi i, per tant, l'estudi de la seua activitat elèctrica és essencial per al diagnòstic de malalties cardíaques. El problema directe (PD) de l'electrocardiografia implica el càlcul dels potencials del tors a partir de l'activitat elèctrica del cor i el model 3D del cos, mentre que la resolució del problema invers (PI) permet la reconstrucció no invasiva de l'activitat elèctrica del cor a partir de els potencials de superfície. La resolució del PI de l'electrocardiografia té una gran importància en la pràctica clínica atès que fa possible una estimació de l'activitat elèctrica del miocardi únicament a partir de registres no invasius. No obstant això, la resolució del PI segueix sent un gran desafiament per a la electrocardiografia ja que està mal plantejat, és molt inestable i té múltiples solucions. Al llarg d'aquesta tesi s'han desenvolupat diferents estratègies basades en la resolució PI, aplicant-les en el diagnòstic no invasiu d'arítmies ventriculars i auriculars, verificant mitjançant models cel·lulars matemàtics i bases de dades clíniques. La tesi se centra en la resolució del PI per a la reconstrucció no invasiva de l'activitat elèctrica del miocardi per a diferents malalties cardíaques amb diferents patrons de propagació, implementant un nou sistema per a patrons de propagació complexos. A més se han validat els resultats obtinguts i se han classificat els diferents patrons de propagació amb l'estratègia de resolució del PI òptima que minimitze l'error i augmente l'estabilitat del sistema. Un nou mètode iteratiu va ser implementat per a la resolució del PI per fonts dipolars, sent òptim per representar patrons de propagació simples, aconseguint una alta estabilitat i immunitat al soroll en restringir la solució a un nombre limitat de dipols. No obstant això, els patrons de propagació que no poden ser representats per un nombre limitat de dipols s'han de calcular mitjançant la resolució del PI en termes de potencials epicàrdics, proporcionant una estimació més detallada de l'activitat del miocardi. La resolució del PI en el domini de la tensió i fase va mostrar ser molt precís per a patrons de propagació simples i organitzats. Aquest mètode permet el diagnòstic no invasiu de la síndrome de Brugada o la ubicació de focus ectòpics en arítmies auriculars mitjançant una anàlisi paramètric de la morfologia dels electrogrames o la reconstrucció dels mapes d'activació. No obstant això, els resultats matemàtics i clínics presentats en aquesta tesi van demostrar que, per patrons de propagació complexos com la fibril·lació auricular (FA), els resultats obtinguts mitjançant la resolució del PI en el domini de la tensió i fase són massa suaus i optimistes, simplificant enormement la complexitat de la FA, obtenint resultats no fisiològics que no coincideixen amb l'activitat complexa dels electrogrames intracardiacos registrats en pacients amb FA. En aquesta tesi, s'ha proposat una nova tècnica per a la identificació i localització no invasiva de fonts amb una freqüència dominant alta, basat en la suposició que en molts casos les fonts elèctriques que generen i mantenen la FA presenten una taxa d'activació més alta, amb una propagació intermitent cap a la resta del teixit auricular on la freqüència d'activació és més lenta. Encara que, les solucions en el domini de la tensió i fase per patrons de propagació complexos van ser més suaus i menys precises, l'estimació no invasiva dels mapes de freqüència va ser significativament més precisa, permetent la identificació del gradient de freqüència i ubicació de les fonts de FA d'alta freqüència. Aquesta tècnica pot ser de gran ajuda en la planificació dels procediments d'ablació, evitant puncions interseptales innecessaris per a casos amb un gradient de freqüència de dreta a esquerra i facilitant la / Pedrón Torrecilla, J. (2015). Non-invasive Reconstruction of the Myocardial Electrical Activity from Body Surface Potential Recordings [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/58268 / TESIS

Optical Tomography is an imaging modality that enhances early diagnosis of disease through use of harmless Near-Infrared rays instead of conventional x-rays. The subsequent images are used to reconstruct the object. However Optical Tomography has not been effectively utilized due to the complicated photon scattering phenomenon and ill-posed nature of the corresponding image reconstruction scheme. The major method for reconstruction of the object is based on an iterative loop that constantly minimizes the difference between the predicted model of photon scattering with acquired images. Currently the most effective method of predicting the photon scattering pattern is the solution of the Radiative Transfer Equation (RTE) using the Finite Elements Method (FEM). However, the conventional FEM uses classical C0 interpolation functions, which have shortcomings in terms of continuity of the solution over the domain as well as proper representation of geometry. Hence higher discretization is necessary to maintain accuracy of gradient-based results which may significantly increase the computational cost in each iteration. This research implements the recently developed Isogeometric Approach (IGA) and particularly IGA-based FEM to address the aforementioned issues. The IGA-based FEM has the potential to enhance adaptivity and reduce the computational cost of discretization schemes. The research in this study applies the IGA method to solve the RTE with the diffusion approximation and studies its behavior in comparison to conventional FEM. The results show comparison of the IGA-based solution with analytical and conventional FEM solutions in terms of accuracy and efficiency. While both methods show high levels of accuracy in reference to the analytical solution, the IGA results clearly excel in accuracy. Furthermore, FE solutions tend to have shorter runtimes in low accuracy results. However, in higher accuracy solutions, where it matters the most, the IGA proves to be considerably faster. / Doctor of Philosophy / CT scans can save lives by allowing medical practitioners observe inside the patient's body without use of invasive surgery. However, they use high energy, potentially harmful x-rays to penetrate the organs. Due to limits of the mathematical algorithm used to reconstruct the 3D figure of the organs from the 2D x-ray images, many such images are required. Thus, a high level of x-ray exposure is necessary, which in periodic use can be harmful. Optical Tomography is a promising alternative which replaces x-rays with harmless Near-infrared (NIR) visible light. However, NIR photons have lower energy and tend to scatter before leaving the organs. Therefore, an additional algorithm is required to predict the distribution of light photons inside the body and their resulting 2D images. This is called the forward problem of Optical Tomography. Only then, like conventional CT scans, can another algorithm, called the inverse solution, reconstruct the 3D image by diminishing the difference between the predicted and registered images. Currently Optical Tomography cannot replace x-ray CT scans for most cases, due to shortcomings in the forward and inverse algorithms to handle real life usages. One obstacle stems from the fact that the forward problem must be solved numerous times for the inverse solution to reach the correct visualization. However, the current numerical method, Finite Element Method (FEM), has limitations in generating accurate solutions fast enough using economically viable computers. This limitation is mostly caused by the FEM's use of a simpler mathematical construct that requires more computations and is limited in accurately modelling the geometry and shape. This research implements the recently developed Isogeometric Analysis (IGA) and particularly IGA-based FEM to address this issue. The IGA-based FEM uses the same mathematical construct that is used to visualize the geometry for complicated applications such as some animations and computer games. They are also less complicated to apply due to much lower need for partitioning the domain. This study applies the IGA method to solve the forward problem of diffuse Optical Tomography and compare the accuracy and speed of IGA solution to the conventional FEM solution. The comparison reveals that while both methods can reach high accuracy, the IGA solutions are relatively more accurate. Also, while low accuracy FEM solutions have shorter runtimes, in solutions with required higher accuracy levels, the IGA proves to be considerably faster.

I here present a combined AVO analysis of P-P and P-S reflection data whose objective is to improve the identification of lithology by estimating the specific values of Poisson's ratio, [sigma], for each rock formation in a given geological model, rather than a contrast between formations. Limited knowledge on the elastic parameters of a given rock formation and difficulty regarding the availability and processing of P-S data constitute hindrances of lithology identification. Considering that ocean bottom seismology (OBS) has aided in solving the problem of P-S data availability, limited information on elastic parameters is still a challenge, and the focus of this thesis. The present analysis is based on Zoeppritz' solution for the P-P and P-S reflection coefficients, RPP and RPS, with a slight modification. We used the normalized P-S reflection coefficient; i.e., R'PS = RPS / sin [theta] for [theta] > 0, instead of RPS, where [theta] is the incident angle. By normalizing RPS, we avoid dealing with the absence of converted S-waves at small incident angles and enhance the similar linear behavior of the P-P and normalized P-S reflection coefficients at small angles of incidence. We have used the linearity of RPP and R'PS at angles smaller than 35 degrees to simultaneously estimate the average VP/VS ratio, the contrasts of P- and S-wave velocities, and the contrast of density. Using this information, we solve for Poisson's ratio of each formation, which may enable lithology discrimination. The feasibility of this analysis was demonstrated using nonlinear synthetic data (i.e., finite-difference data). The results in estimating Poisson's ratio yielded less than 5 percent error. We generalize this new combined P-P and P-S AVO analysis for dipping interfaces. Similarly to the nondipping interface case, our derivations show that the amplitude variation with offset (AVO) of P-P and P-S for a dipping interface can be cast into intercepts and gradients. However, these intercepts and gradients depend on the angle of the dipping interface. Therefore, we further generalize our analysis by including a migration step that allows us to find the dipping angle. Because seismic data is not available in terms of RPP and R'PS, this process includes recovery of reflection coefficients after migrating the data and correcting for geometrical spreading, as done by Ikelle et al. (1986 and 1988). The combination of all of these steps, namely geometrical-spreading correction, migration, and AVO analysis, is another novelty of this thesis, which leads to finding the specific values of Poisson's ratio of each rock formation directly from the seismic data.

Electro-Magnetic Source Imaging (EMSI) is the estimation of the position, orientation and strength of active electrical sources within the brain from electrical and magnetic measurements. For an accurate source localization, the head model must correctly represent the electrical and geometrical properties of the head. To solve the forward problem using realistic head models numerical techniques must be used. This work uses the Boundary Element Method (BEM) for solving the forward problem. The accuracy of the existing BEM formulation is improved by using second order elements, recursive integration and the isolated problem approach (IPA). Two new formulations are developed to improve the solution speed by computing transfer matrices for EEG and MEG solutions. The IPA formulation is generalized and integrated into the accelerated BEM algorithm. Once the transfer matrices are computed, the forward solutions take about 300 ms for a 256 sensor EEG and MEG system. The head model used in the BEM solutions is constructed by segmenting three dimensional multimodal magnetic resonance images. For segmentation, a semi-automatic hybrid algorithm is developed that makes use of snakes, morphological operations, thresholding and region growing. The mesh generation algorithm allows intersecting tissue compartments. For the inverse problem solution genetic algorithm (GA) is used to search for a given number of dipoles. Source localization with simulated data show that the localization error is within 1.1 mm for EEG and 1.2 mm for MEG when SNR is 10 on a realistic model with 7 compartments. When a single-dipole source in a realistic model is explored using a best-fit spherical model, the localization errors increase up to 8.5 mm for EEG and 7 mm for MEG. Similar tests are also performed with multiple dipoles. It was observed that realistic models provide definitely more accurate results compared to spherical models. The EMSI approach is also tested using experimental EEG data to localize the sources of auditory evoked potentials. The reconstructed source locations are correctly found in the Heschl&#039 / s gyrus. In conclusion, this thesis presents a complete source localization framework for future brain research using the EMSI.

La reconstruction des sources de l'activité cérébrale à partir des mesures de potentiel fournies par un électroencéphalographie (EEG) nécessite de résoudre le problème connu sous le nom de « problème inverse de l'EEG ». La solution de ce problème dépend de la solution du « problème direct de l'EEG », qui fournit à partir de sources de courant connues, le potentiel mesuré au niveau des électrodes. Pour des modèles de tête réels, ce problème ne peut être résolut que de manière numérique. En particulier, les équations intégrales de surfaces requièrent uniquement la discrétisation des interfaces entre les différents compartiments constituant le milieu cérébral. Cependant, les formulations intégrales existant actuellement ne prennent pas en comptent l'anisotropie du milieu. Le travail présenté dans cette thèse introduit deux nouvelles formulations intégrales permettant de palier à cette faiblesse. Une formulation indirecte capable de prendre en compte l'anisotropie du cerveau est proposée. Elle est discrétisée à l'aide de fonctions conformes aux propriétés spectrales des opérateurs impliqués. L'effet de cette discrétisation de type mixe lors de la reconstruction des sources cérébrales est aussi étudié. La seconde formulation se concentre sur l'anisotropie due à la matière blanche. Calculer rapidement la solution du système numérique obtenu est aussi très désirable. Le travail est ainsi complémenté d'une preuve de l'applicabilité des stratégies de préconditionnement de type Calderon pour les milieux multicouches. Le théorème proposé est appliqué dans le contexte de la résolution du problème direct de l'EEG. Un préconditionneur de type Calderon est aussi introduit pour l'équation intégrale du champ électrique (EFIE) dans le cas de structures unidimensionnelles. Finalement, des résultats préliminaires sur l'impact d'un solveur rapide direct lors de la résolution rapide du problème direct de l'EEG sont présentés. / Electroencephalography (EEG) is a very useful tool for characterizing epileptic sources. Brain source imaging with EEG necessitates to solve the so-called EEG inverse problem. Its solution depends on the solution of the EEG forward problem that provides from known current sources the potential measured at the electrodes positions. For realistic head shapes, this problem can be solved with different numerical techniques. In particular surface integral equations necessitates to discretize only the interfaces between the brain compartments. However, the existing formulations do not take into account the anisotropy of the media. The work presented in this thesis introduces two new integral formulations to tackle this weakness. An indirect formulation that can handle brain anisotropies is proposed. It is discretized with basis functions conform to the mapping properties of the involved operators. The effect of this mixed discretization on brain source reconstruction is also studied. The second formulation focuses on the white matter fiber anisotropy. Obtaining the solution to the obtained numerical system rapidly is also highly desirable. The work is hence complemented with a proof of the preconditioning effect of Calderon strategies for multilayered media. The proposed theorem is applied in the context of solving the EEG forward problem. A Calderon preconditioner is also introduced for the wire electric field integral equation. Finally, preliminary results on the impact of a fast direct solver in solving the EEG forward problem are presented.

La thérapie de l'épilepsie par résection partielle exige l'identification des structures cérébrales qui sont impliquées dans la genèse des crises d'épilepsie focales. Plusieurs modalités telles que l'IRM, le PET SCAN, la sémiologie de la crise et l'électrophysiologie sont exploitées par les experts pour contribuer à la localisation de la zone épileptogène. L'EEG du scalp est la modalité qui procure la résolution temporelle à l'échelle des processus électrophysiologiques étudiés. Cependant du fait du positionnement des capteurs sur le scalp, sa résolution spatiale et, plus précisément, de profondeur est très médiocre. Dans certain cas (épilepsies pharmaco-résistantes), et pour palier à cette déficience spatiale, il est possible d'avoir recours à la SEEG. La SEEG permet des mesures électrophysiologiques intracérébrales : la résolution spatiale et donc anatomique est excellente dans l'axe de la microélectrode. La définition de la zone épileptogène, comme celle proposée par Talairach et Bancaud, est une définition électro-clinique basée sur les résultats d'enregistrements de SEEG intracérébraux. Elle tient compte non seulement de la localisation anatomique de la décharge épileptique partielle, mais également de l'évolution dynamique de cette décharge, c'est à dire les réseaux neurologiques actifs durant la période intercritique-critique et des symptômes cliniques. Récemment, il a été proposé une technique de diagnostic complémentaire de localisation de la zone épileptogénique employant la stimulation électrique cérébrale de profondeur (Deep Brain Stimulation). Cette source exogène peut activer les réseaux épileptiques et produire une réaction électrophysiologique telle qu'une crise d'épilepsie. Elle permet également de mettre en exergue les zones fonctionnelles cognitives. Cette source exogène est parfaitement définie spatialement et temporellement. Ainsi, la stimulation, couplée aux mesures SEEG, contribue à la modélisation de la propagation électrique cérébrale et, par voie de conséquence, à la compréhension du processus épileptique. De plus, ce travail sur le modèle de propagation directe apporte une aide à la résolution du problème inverse et donc à la localisation de sources. Les différentes tâches accomplies au cours de cette thèse sont les suivantes : création d'une base de données réelles à partir de 3000 stimulations et mesures SEEG pour 42 patients explorés ; extraction par séparation des signaux de propagation de la stimulation électrique (DBS) des mesures multidimensionnelles SEEG : 5 méthodes ont été développées ou adaptées et ont été validées au cours d'une première phase en simulation puis sur des signaux réels SEEG dans une seconde phase ; localisation des électrodes de SEEG dans le repère anatomique de l'IRM et du CT Scanner en y ajoutant une étape de segmentation de la matière grise et blanche, du liquide céphalorachidien et de l'os ; discussion sur de nombreux modèles de propagation réalistes ou non réalistes proposés dans la littérature, à la fois sur le plan du raffinement du modèle mais également sur les implantations numériques possibles : modèles de milieu, sphériques et réalistes infinis basés sur MRI et CT du patient ; comparaison entre les résultats générés par les modèles de sources et de milieux et les données obtenues après séparation de la stimulation électrique in vivo chez l'homme ; validation des modèles de tête FEM en intégrant les conductivités des milieux (CSF), gris et blancs céphalo-rachidiens et perspectives envisagées / The study of epilepsy requires the identification of cerebral structures which are involved in generation of seizures and connexion processes. Several methods of clinical investigation contributed to these studies : imaging (PET, MRI), electrophysiology (EEG, SEEG, MEG). The EEG provides a temporal resolution enough to analyze these processes. However, the localization of deep sources and their dynamical properties are difficult to understand. SEEG is a modality of intracerebral electrophysiological and anatomical high temporal resolution reserved for some difficult cases of pre-surgical diagnosis : drug-resistant epilepsy. The definition of the epileptogenic zone, as proposed by Talairach and Bancaud is an electro-clinical definition based on the results of intracerebral SEEG recordings. It takes into account not only the anatomical localization of partial epileptic discharge, but also the dynamic evolution of this discharge (active neural networks at the time of seizure) and clinical symptoms. Recently, a novel diagnostic technique allows an accurate localization of the epileptogenic zone using Depth Brain Stimulation (DBS). This exogenous source can activate the epileptic networks and generate an electrophysiological reaction. Therefore, coupling DBS with SEEG measurements is very advantageous : firstly, to contribute to the modeling and understanding of the (epileptic) brain and to help the diagnosis, secondly, to access the estimation of head model as an electrical conductor (conductive properties of tissues). In addition, supplementary information about head model improves the solution to the inverse problem (source localization methods) used in many applications in EEG and SEEG. The inverse solution requires repeated computation of the forward problem, i.e. the simulation of EEG and SEEG fields for a given dipolar source in the brain using a volume-conduction model of the head. As for DBS, the location of source is well defined. Therefore, in this thesis, we search for the best head model for the forward problem from real synchronous measurements of EEG and SEEG with DBS in several patients. So, the work of the thesis breaks up into different parts for which we need to accomplish the following tasks : Creation of database 3000 DBS measurements for 42 patients ; Extraction of DBS signal from SEEG and EEG measurements using multidimensional analysis : 5 methods have been developed or adapted and validate first in a simulation study and, secondly, in a real SEEG application ; Localization of SEEG electrodes in MR and CT images, including segmentation of brain matter ; SEEG forward modeling using infinite medium, spherical and realistic models based on MRI and CT of the patient ; Comparison between different head models and validation with real in vivo DBS measurements ; Validation of realistic 5-compartment FEM head models by incorporating the conductivities of cerebrospinal fluid (CSF), gray and white matters