A Calderon Regularized Symmetric Formulation for the Electroencephalography Forward Problem

The symmetric formulation of the electroencephalography (EEG) forward problem is a well-known and widespread equation thanks to the high level of accuracy that it delivers. However, this equation is first kind in nature and gives rise to ill-conditioned problems when the discretization density or the brain conductivity contrast increases, resulting in numerical instabilities and increasingly slow solutions. This work addresses and solves this problem by proposing a new regularized symmetric formulation. The new scheme is obtained by leveraging on Calderon identities which allow to introduce a dual symmetric equation that, combined with the standard one, results in a second kind operator which is both stable and well-conditioned under all the above mentioned conditions. The new formulation presented here can be easily integrated into existing EEG imaging packages since it can be obtained with the same computational technology required by the standard symmetric formulation. The performance of the new scheme is substantiated by both theoretical developments and numerical results which corroborate the theory and show the practical impact of the new technique

A 3D Finite-Difference BiCG Iterative Solver with the Fourier-Jacobi Preconditioner for the Anisotropic EIT/EEG Forward Problem

The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique

Forward volumetric modeling framework for realistic head models towards accurate EEG source localization

Synergetic effects connecting spatial and functional neuroimaging techniques allows reduction of the weakness for single method analysis. Specifically, Electroencephalographic (EEG) Source Imaging (ESI) relating structural head models and distributed source localization techniques improves the time and spatial resolution of single MRI or EEG analysis. The construction of more accurate forward models for ESI solutions, holding better precision and less computational burden is an important task for investigative purposes, but also for surgery planning and disorder treatments. In this regard, we present a novel finite-difference EEG forward problem solution that we called ghost-filling finite difference anisotropic reciprocity method (GFDARM). First, we introduce a finite difference numerical solution for the conservative form of the Poisson equation, using an asymmetric volumetric stencil, together with the transition layer technique to formulate finite differences that properly deal with the considered Newman and Dirichlet boundary conditions. Later, we formulate a solution for an irregular free-form boundary domain, based on a second-order accuracy ghost-filling approximation for the homogeneous Newman flux condition, allowing us to solve the discretized finite differences volume only for the significant potential unknowns. Then we analyze the linear equation system solution and the considerations for a reciprocity solution over the electrodes space. Further, we test our method using a multilayer spherical head model that can include anisotropy and can admit an analytical solution of the Poisson equation. Finally, we analyze a noisy linear equation system to study the numerical stability of the technique in the presence of perturbations. Our results show stability and super-linear convergence. Moreover, validation against an analytical solution shows high correspondence in the potential distribution for a wide range of dipole positions and orientations. As a final stage, we introduce a realistic patient-specific EEG forward modeling pipeline, including anisotropy in the skull and the white matter; MRI segmentation; electrode co-register; voxelwise conductivity definitions; reciprocity space solution; and GFDARM numeric EEG forward solver. Our results using Bayesian model selection for group studies in a random fixed effect analysis show strong evidence in favor of more complex head models, including anisotropic skull and white matter modelingResumen: Los efectos conjuntos conectando técnicas espaciales y funcionales de neuro-imagen permiten el mejoramiento de las características de un solo método. Específicamente, la generación de imágenes de fuentes de activación (ESI) mediante electroencefalografía (EEG) que relaciona modelos estructurales de conductividad y técnicas de localización de fuentes distribuidas, permite un mejoramiento en la resolución espacial, conservando la resolución temporal del EEG. La construcción de modelos de conductividad más precisos, con una mayor precisión y menos carga computacional es una tarea importante para soluciones que emplean ESI, así como para fines de investigación, planificación de cirugía y/o los tratamientos de trastornos neurológicos en general. En este trabajo presentamos una nueva solución del problema directo empleando diferencias finitas, a la que llamamos método de diferencias finitas empleando llenado-fantasma, reciprocidad y anisotropía (GFDARM). Primero, nosotros presentamos una solución numérica de diferencias finitas para la forma conservativa de la ecuación de Poisson, utilizando una plantilla volumétrica asimétrica, junto con la técnica de transición de capas, para formular diferencias finitas que aborden adecuadamente las condiciones de contorno de Newman y Dirichlet. Más adelante, formulamos la solución para una frontera irregular y de forma libre basada en una aproximación de segundo orden de llenado-fantasma que permite cumplir la condición de flujo homogéneo de Newman, lo que nos permite resolver el volumen discretizado solo para las incógnitas de potencial diferentes de cero (significativas). Posteriormente se analiza la solución del sistema de ecuaciones lineales y las consideraciones para una solución de reciprocidad sobre el espacio de los electrodos. Además, realizamos pruebas utilizando un modelo de cabeza esférico multicapa que puede incluir anisotropía y del cual se puede obtener una solución analítica. Finalmente, se analiza la solución del sistema lineal de ecuaciones en presencia de ruido estudiando la estabilidad numérica de la técnica. Nuestros resultados muestran estabilidad y convergencia súper lineal y una alta correspondencia en la distribución de potenciales para una amplia gama de posiciones y orientaciones de dipolos comparando contra una solución analítica esférica. Finalmente se una metodología para el modelado directo de EEG empleando modelos realistas y paciente-específicos, que incluye anisotropía en el cráneo y la materia blanca; segmentación de MRI; co-registro de electrodos; definiciones de conductividad voxel a voxel; solución de espacio de reciprocidad; y solución numérica del problema directo en EEG empleando GFDARM. El desempeño de la técnica y la influencia de los modelos directos realísticos son analizados empleando selección de modelos para estudios de grupos en un marco Bayesiano, los cuales muestran fuerte evidencia a favor de modelos de conductividad más complejos, que incluyan modelado anisótropo del cráneo y la materia blancaDoctorad

Influence of atlas-based and patient dependent forward models in EEG source reconstruction

Abstract: Electroencephalography Source Imaging (ESI) techniques have become the most attractive alternative to support the estimation of neuronal activity through the mapping of electrical potentials measured over the scalp. It takes advantage of the low implementation cost, the high temporal resolution, and non-invasiveness in the patient. ESI techniques require a volumetric conductor model (commonly named Electroencephalography (EEG) Forward Model), including information about the physiological and geometrical properties of the head, and modeling the electromagnetic field propagation of the neuronal activity throughout the head tissues to reach the scalp. In this regard, the accuracy of ESI solutions depends partially on the capabilities of the forward model to correctly describe the structural information provided by a Magnetic Resonance Image (MRI). However, acquiring MRIs for generating personalized head models is expensive, slow, and in some cases unpractical. In this work, we investigate how the head model influences the source reconstruction based on EEG when progressively including different levels of prior structural information. Hence, we evaluate two approaches to enhance the model of brain structure in the EEG forward problem formulation. First, the incorporation of different brain tissue morphology, mainly, based on a Generic MRI, based on a target population Atlas, or based on a patient-specific MRI. Second, the variation of the tissue model complexity in the number of segmented brain layers. All the head models are build using the Finite Difference Reciprocity Method (FDRM). Model comparison is carried out under a Parametric Empirical Bayesian (PEB) framework using Event-Related Potentials (ERPs) taken from the studied population. Obtained results show that the more realistic and subject dependent model, the better performance of the ESI solutionResumen: Las técnicas de reconstrucción de fuentes basadas en Electroencefalografía (ESI) son la alternativa más interesante para la estimación de fuentes mediante los potenciales eléctricos medidos sobre el cuero cabelludo, aprovechando el bajo costo de implementación, la alta resolución temporal, y la poca invasión que requiere en el paciente. Es por esto, que estas técnicas requieren un modelo de conducción volumétrico (comúnmente llamado modelo directo), que incluye información de las propiedades físicas y geométricas de la cabeza, además de modelar la propagación del campo electromagnético generado por la actividad neuronal a través de los diferentes tejidos de la cabeza hasta alcanzar el cuero cabelludo. En este sentido, el correcto desempeño de las técnicas ESI depende directamente de las capacidades del modelo directo para describir de manera apropiada la información estructural aportada por una Imagen por Resonancia Magnética (MRI). Sin embargo, adquirir MRIs para generar modelos de la cabeza personalizados, es costoso, lento, y en algunos casos poco práctico. En este trabajo, se investiga la manera en que el modelo directo influencia la tarea de reconstrucción de fuentes basada en EEG, incluyendo de manera progresiva diferentes niveles de información estructural relacionada al paciente. Así, se evaluan dos enfoques específicos para mejorar el modelo de la estructura cerebral en la formulación del problema directo de EEG. El primer enfoque está relacionado con la incorporación de diferentes morfologías de tejido cerebral, principalmente, basadas en un MRI genérico, un atlas de la población estudiada, ó en el MRI específico del paciente. El segundo enfoque es la variación de la complejidad del modelo en términos de el número de tejidos segmentados en el cerebro. En este trabajo el modelo directo se soluciona usando el Método de Diferencias Finitas con Reciprocidad (FDRM). La comparación de modelos se realiza bajo un framework Bayesiano Empírico Paramétrico (PEB), que permite contrastar los diferentes enfoques del modelo directo, usando datos reales. En general, los resultados obtenidos muestran que usar modelos más realistas y más dependientes de la población de estudio, mejora significativamente el desempeño de las técnicas ESIMaestrí

Patient-specific modellling of cortical spreading depression applied to migraine studies.

254 p.-La migraña es un trastorno neurológico muy común. Un tercio de los pacientes que sufren migraña experimentan lo que se denomina aura, una serie de alteraciones sensoriales que preceden al típico dolor de cabeza unilateral. Diversos estudios apuntan a la existencia de una correlación entre el aura visual y la depresión cortical propagada (DCP), una onda de despolarización que tiene su origen en el córtex visual para propagarse, a continuación, por todo el córtex hacia las zonas periféricas. La complejidad y la elevada especificidad de las características del córtex cerebral sugieren que la geometría podría tener un impacto significativo en la propagación de la DCP. En esta tesis hemos combinado dos modelos existentes: un modelo neurológico pormenorizado para el componente electrofisiológico de la DCP y un modelo de reacción-difusión que tiene en consideración la difusión del potasio, el impulsor de la propagación de la DCP. Durante el proceso, hemos integrado dos aspectos de la DCP que tienen lugar en diferentes escalas de tiempo: la dinámica electrofisiológica seguiría un patrón temporal del orden de milisegundos, mientras que la dinámica del potasio extracelular que acciona las funciones de propagación de la DCP se mediría en una escala de minutos. Como resultado, obtendremos un modelo multiescalar EDP-EDO. Asimismo, hemos incorporado los datos específicos del paciente en el modelo DCP: (i) la geometría cerebral específica de un paciente obtenida a través de resonancia magnética, y (ii) los tensores de conductividad personalizados obtenidos a través de diffusion tensor images. A fin de estudiar el papel que desempeña la geometría en la propagación de la DCP, hemos definido las cantidades de interés (CdI) relacionadas con la geometría y las que dependen de la DCP y las hemos evaluado en dos casos prácticos. Si bien la geometría no parece tener un impacto significativo en la propagación de la DCP, algunas CdI han resultado ser unas candidatas muy prometedoras para facilitar la clasificación de individuos sanos y pacientes con migraña. Finalmente, para justificar la carencia de datos experimentales para la validación y selección de los parámetros del modelo, hemos aplicado diversas técnicas de cuantificación de la incertidumbre al modelo DCP y hemos analizado el impacto de las diversas elecciones de parámetros en el resultado del modelo.Migraine is a common neurological disorder and one-third of migraine patients suffer from migraine aura, a perceptual disturbance preceding the typically unilateral headache. Cortical spreading depression (CSD), a depolarisation wave that originates in the visual cortex and propagates across the cortex to the peripheral areas, has been suggested as a correlate of visual aura by several studies. The complex and highly individual-specific characteristics of the brain cortex suggest that the geometry might have a significant impact on CSD propagation. In this thesis, we combine two existing models, a detailed neurological model for the electrophysiological component of CSD and a reaction-diffusion model accounting for the potassium diffusion, the driving force of CSD propagation. In the process, we integrate two aspects of CSD that occur at different time scales: the electrophysiological dynamics features a temporal scale in the order of milliseconds, while the extracellular potassium dynamics that triggers CSD propagation features is on the scale of minutes. As a result we obtain a multi-scale PDE-ODE model. In addition, we incorporate patient-specific data in the CSD model: (i) a patient-specific brain geometry obtained from magnetic resonance imaging, and (ii) personalised conductivity tensors derived from diffusion tensor imaging data. To study the role of the geometry in CSD propagation, we define geometric and CSD-dependent quantities of interest (QoI) that we evaluate in two case studies. Even though the geometry does not seem to have a major impact on the CSD propagation, some QoI are promising candidates to aid in the classification of healthy individuals and migraine patients. Finally, to account for the lack of experimental data for validation and selection of the model parameters, we apply different techniques of uncertainty quantification to the CSD model and analyse the impact of various parameter choices on the model outcom

Patient-specific modelling of cortical spreading depression applied to migraine studies

Migraine is a common neurological disorder and one-third of migraine patients suffer from migraine aura, a perceptual disturbance preceding the typically unilateral headache. Cortical spreading depression (CSD), a depolarisation wave that originates in the visual cortex and propagates across the cortex to the peripheral areas, has been suggested as a correlate of visual aura by several studies. The complex and highly individual-specific characteristics of the brain cortex suggest that the geometry might have a significant impact on CSD propagation. In this thesis, we combine two existing models, a detailed neurological model for the electrophysiological component of CSD and a reaction-diffusion model accounting for the potassium diffusion, the driving force of CSD propagation. In the process, we integrate two aspects of CSD that occur at different time scales: the electrophysiological dynamics features a temporal scale in the order of milliseconds, while the extracellular potassium dynamics that triggers CSD propagation features is on the scale of minutes. As a result we obtain a multi-scale PDE-ODE model. In addition, we incorporate patient-specific data in the CSD model: (i) a patient-specific brain geometry obtained from magnetic resonance imaging, and (ii) personalised conductivity tensors derived from diffusion tensor imaging data. To study the role of the geometry in CSD propagation, we define geometric and CSD-dependent quantities of interest (QoI) that we evaluate in two case studies. Even though the geometry does not seem to have a major impact on the CSD propagation, some QoI are promising candidates to aid in the classification of healthy individuals and migraine patients. Finally, to account for the lack of experimental data for validation and selection of the model parameters, we apply different techniques of uncertainty quantification to the CSD model and analyse the impact of various parameter choices on the model outcome

Investigation of general-purpose computing on graphics processing units and its application to the finite element analysis of electromagnetic problems

In this dissertation, the hardware and API architectures of GPUs are investigated, and the corresponding acceleration techniques are applied on the traditional frequency domain finite element method (FEM), the element-level time-domain methods, and the nonlinear discontinuous Galerkin method. First, the assembly and the solution phases of the FEM are parallelized and mapped onto the granular GPU processors. Efficient parallelization strategies for the finite element matrix assembly on a single GPU and on multiple GPUs are proposed. The parallelization strategies for the finite element matrix solution, in conjunction with parallelizable preconditioners are investigated to reduce the total solution time. Second, the element-level dual-field domain decomposition (DFDD-ELD) method is parallelized on GPU. The element-level algorithms treat each finite element as a subdomain, where the elements march the fields in time by exchanging fields and fluxes on the element boundary interfaces with the neighboring elements. The proposed parallelization framework is readily applicable to similar element-level algorithms, where the application to the discontinuous Galerkin time-domain (DGTD) methods show good acceleration results. Third, the element-level parallelization framework is further adapted to the acceleration of nonlinear DGTD algorithm, which has potential applications in the field of optics. The proposed nonlinear DGTD algorithm describes the third-order instantaneous nonlinear effect between the electromagnetic field and the medium permittivity. The Newton-Raphson method is incorporated to reduce the number of nonlinear iterations through its quadratic convergence. Various nonlinear examples are presented to show the different Kerr effects observed through the third-order nonlinearity. With the acceleration using MPI+GPU under large cluster environments, the solution times for the various linear and nonlinear examples are significantly reduced

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds