Considering New Regularization Parameter-Choice Techniques for the Tikhonov Method to Improve the Accuracy of Electrocardiographic Imaging

The electrocardiographic imaging (ECGI) inverse problem highly relies on adding constraints, a process called regularization, as the problem is ill-posed. When there are no prior information provided about the unknown epicardial potentials, the Tikhonov regularization method seems to be the most commonly used technique. In the Tikhonov approach the weight of the constraints is determined by the regularization parameter. However, the regularization parameter is problem and data dependent, meaning that different numerical models or different clinical data may require different regularization parameters. Then, we need to have as many regularization parameter-choice methods as techniques to validate them. In this work, we addressed this issue by showing that the Discrete Picard Condition (DPC) can guide a good regularization parameter choice for the two-norm Tikhonov method. We also studied the feasibility of two techniques: The U-curve method (not yet used in the cardiac field) and a novel automatic method, called ADPC due its basis on the DPC. Both techniques were tested with simulated and experimental data when using the method of fundamental solutions as a numerical model. Their efficacy was compared with the efficacy of two widely used techniques in the literature, the L-curve and the CRESO methods. These solutions showed the feasibility of the new techniques in the cardiac setting, an improvement of the morphology of the reconstructed epicardial potentials, and in most of the cases of their amplitude

Using state-of-the-art inverse problem techniques to develop reconstruction methods for fluorescence diffuse optical

An inverse problem is a mathematical framework that is used to obtain info about a physical object or system from observed measurements. It usually appears when we wish to obtain information about internal data from outside measurements and has many applications in science and technology such as medical imaging, geophysical imaging, image deblurring, image inpainting, electromagnetic scattering, acoustics, machine learning, mathematical finance, physics, etc. The main goal of this PhD thesis was to use state-of-the-art inverse problem techniques to develop modern reconstruction methods for solving the fluorescence diffuse optical tomography (fDOT) problem. fDOT is a molecular imaging technique that enables the quantification of tomographic (3D) bio-distributions of fluorescent tracers in small animals. One of the main difficulties in fDOT is that the high absorption and scattering properties of biological tissues lead to an ill-posed inverse problem, yielding multiple nonunique and unstable solutions to the reconstruction problem. Thus, the problem requires regularization to achieve a stable solution. The so called “non-contact fDOT scanners” are based on using CCDs as virtual detectors instead of optic fibers in contact with the sample. These non-contact systems generate huge datasets that lead to computationally demanding inverse problem. Therefore, techniques to minimize the size of the acquired datasets without losing image performance are highly advisable. The first part of this thesis addresses the optimization of experimental setups to reduce the dataset size, by using l₂–based regularization techniques. The second part, based on the success of l₁ regularization techniques for denoising and image reconstruction, is devoted to advanced regularization problem using l₁–based techniques, and the last part introduces compressed sensing (CS) theory, which enables further reduction of the acquired dataset size. The main contributions of this thesis are: 1) A feasibility study (the first one for fDOT to our knowledge) of the automatic Ucurve method to select the regularization parameter (l₂–norm). The U-curve method has shown to be an excellent automatic method to deal with large datasets because it reduces the regularization parameter search to a suitable interval. 2) Once we found an automatic method to choose the l₂ regularization parameter for fDOT, singular value analysis (SVA) of fDOT forward matrix was used to maximize the information content in acquired measurements and minimize the computational cost. It was shown for the first time that large meshes can be reduced in the z direction, without any loss in imaging performance but reducing computational times and memory requirements. 3) Dealing with l₁–based regularization techniques, we presented a novel iterative algorithm, ART-SB, that combines the advantage of Algebraic reconstruction method (ART) in handling large datasets with Split Bregman (SB) denoising, an approach which has been shown to be optimum for Total Variation (TV) denoising. SB has been implemented in a cost-efficient way to handle large datasets. This makes ART-SB more computationally efficient than previous TV-based reconstruction algorithms and most splitting approaches. 4) Finally, we proposed a novel approach to CS for fDOT, named the SB-SVA iterative method. This approach is based on the analysis-based co-sparse representation model, where an analysis operator multiplies the image transforming it in a sparse one. Taking advantage of the CS-SB algorithm, we restrict the solution reached at each CS-SB iteration to a certain space where the singular values of the forward matrix and the sparsity structure combine in beneficial manner. In this way, SB-SVA forces indirectly the wellconditioninig of the forward matrix while designing (learning) the analysis operator and finding the solution. Furthermore, SB-SVA outperforms the CS-SB algorithm in terms of image quality and needs fewer acquisition parameters. The approaches presented here have been validated with experimental. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------El problema inverso consiste en un conjunto de técnicas matemáticas para obtener información sobre un fenómeno físico a partir de una serie de observaciones, medidas o datos. Dicho problema aparece en muchas aplicaciones científicas y tecnológicas como pueden ser imagen médica, imagen geofísica, acústica, aprendizaje máquina, física, etc. El principal objetivo de esta tesis doctoral fue utilizar la teoría del problema inverso para desarrollar nuevos métodos de reconstrucción para el problema de tomografía óptica difusiva por fluorescencia (fDOT), también llamada tomografía molecular de fluorescencia (FMT). fDOT es una modalidad de imagen médica que permite obtener de manera noinvasiva la distribución espacial 3D de la concentración de sondas moleculares fluorescentes en animales pequeños in-vivo. Una de las dificultades principales del problema inverso en fDOT, es que, debido a la alta difusión y absorción de los tejidos biológicos, es un problema fuertemente mal condicionado. Su solución no es única y presenta fuertes inestabilidades, por lo que el problema debe ser regularizado para obtener una solución estable. Los llamados escáneres fDOT “sin contacto” se basan en utilizar cámaras CCD como detectores virtuales, en vez de fibras ópticas en contacto con la muestras. Estos sistemas, necesitan un volumen de datos muy elevado para obtener una buena calidad de imagen y el coste computacional de hallar la solución llega a ser muy grande. Por esta razón, es importante optimizar el sistema, es decir, maximizar la información contenida en los datos adquiridos a la vez que minimizamos el coste computacional. La primera parte de esta tesis se centra en optimizar el sistema de adquisición, reduciendo el volumen de datos necesario usando técnicas de regularización basadas en la norma l₂. La segunda parte se inspira en el gran éxito de las técnicas de regularización basadas en la norma l₁ para la reconstrucción de imagen, y se centra en regularizar el problema fDOT mediante dichas técnicas. El trabajo finaliza introduciendo la técnica de “compressed sensing” (CS), que permite también reducir el número de datos necesarios sin por ello perder calidad de imagen. Las contribuciones principales de esta tesis son: 1) Realización de un estudio de viabilidad, por primera vez en fDOT, del método automático U-curva para seleccionar el parámetro de regularización (norma l₂). U-curva mostró ser un método óptimo para problemas con un volumen elevado de datos, ya que dicho método ofrece un intervalo donde encontrar el parámetro de regularización. 2) Una vez encontrado el método automático de selección de parámetro de regularización se realizó un estudio de la matriz del sistema de fDOT basado en el análisis de valores singulares (SVA), con la finalidad de maximizar la información contenida en los datos adquiridos y minimizar el coste computacional. Por primera vez se demostró que el uso de un mallado con menor densidad en la dirección perpendicular al plano obtiene mejores resultados que el uso convencional de una distribución isotrópica del mismo. 3) En la segunda parte de esta tesis, usando técnicas de regularización basadas en la norma l₁, se presenta un nuevo algoritmo iterativo, ART-SB, que combina la capacidad de la técnica de reconstrucción algebraica (ART) para lidiar con problemas con muchos datos con la efectividad del método Split Bregman (SB) para reducir ruido en la imagen mediante su variación total (TV). SB fue implementado de forma eficiente para procesar un elevado volumen de datos, de manera que ART-SB es computacionalmente más eficiente que otros algoritmos de reconstrucción presentados previamente en la literatura, basados en la TV de la imagen y que la mayoría de las técnicas llamadas de “splitting”. 4) Finalmente, proponemos una nueva aproximación iterativa a CS para fDOT, llamada SB-SVA. Esta aproximación se basa en el llamado modelo analítico co-disperso (co-sparse), donde un operador analítico multiplica la imagen convirtiéndola en una imagen dispersa. Este método aprovecha el método SB para CS (CS-SB) para restringir la solución alcanzada en cada iteración a un espacio determinado, donde los valores singulares de la matriz del sistema y la dispersión (“sparsity”) de la solución en dicha iteración combinen beneficiosamente; es decir, donde valores singulares muy pequeños no estén asociados a valores distintos de cero de la solución “sparse”. SB-SVA mejora el mal condicionamiento de la matriz del sistema a la vez que diseña el operador apropiado a través del cual la imagen se puede representar de forma dispersa y soluciona el problema de CS. Además, SB-SVA mostró mejores resultados que CS-SB en cuanto a calidad de imagen, requiriendo menor número de parámetros de adquisición. Todas las aproximaciones que presentamos en esta tesis fueron validadas con datos experimentales

Maximizing the information content in acquired measurements of a parallel plate non-contact FDOT while minimizing the computational cost: singular value analysis

[Poster] 4th European Molecular Imaging Meeting, Barcelona, Spain, May 27 - 30, 2009This work assesses the effect of different settings of the acquisition parameters (distribution of mesh points, density of sources and detectors) of a parallel-plate non-contact fdoT, in order to achieve the best possible imaging performance, i.e. using the minimum number of singular values of W to maximize the information content in acquired measurements while minimizing the computational costThis work is supported in part by fundación Caja navarra (#12180), Ministerio de Ciencia e innovación (TeC2008-06715 and TeC2007-64731) and fP7 project fMTxCT-201792Publicad

An automatic method to select a noise threshold in the singular-value domain for reconstrucion of parallel plate non-contact FDOT images

[Poster] 4th European Molecular Imaging Meeting, Barcelona, Spain, May 27 - 30, 2009This work is supported in part by fundación Caja navarra (#12180), Ministerio de Ciencia e innovación (TeC2008-06715 and TeC2007-64731) and EU-FP7 project FMTXCT-201792Publicad

Uso del método de Split Bregman para la resolución del problema de compressed sensing en imagen de resonancia magnética dinámica cardiaca para pequeño animal

Actas de: XXIX Congreso Anual de la Sociedad Espñaola de Ingeniería Biomédica (CASEIB 2011). Cáceres, 16-18 Noviembre 2011.La imagen dinámica de resonancia magnética en pequeño animal es una herramienta muy importante en el estudio de enfermedades cardiovasculares. La reducción de los tiempos de adquisición de este tipo de imágenes es especialmente relevante para la obtención de imágenes de calidad con una buena resolución espacial y temporal. Actualmente existen diversas técnicas de aceleración que permiten reducir estos tiempos de adquisición, entre ellas la técnica de 'compressed sensing', en auge en los últimos años. Ésta técnica permite la reconstrucción de una imagen a partir de datos submuestreados mediante el uso de métodos de reconstrucción no lineales que minimizan la variación total de la imagen. Recientemente el método de Split Bregman ha demostrado ser computacionalmente eficiente para resolver este problema en imágenes de resonancia magnética. En este trabajo se amplía la metodología de Split Bregman para minimizar la variación total espacial y temporal en imágenes dinámicas, y se aplica a imágenes cardiacas de pequeño animal. Los resultados preliminares muestran que con la metodología propuesta es posible reducir el tiempo de adquisición hasta 5 veces manteniendo la calidad de imagen.Este trabajo ha sido financiado parcialmente por el Ministerio de Ciencia e Innovación (Red RECAVA) y la Comunidad de Madrid y los Fondos FEDER (proyecto ARTEMIS-S2009DPI-1802)Publicad

Do we need to enforce the homogeneous Neumann condition on the torso for solving the inverse electrocardiographic problem?

International audienceRobust calculations of the inverse electrocardiographicproblem may require accurate specification of boundaryconditions at the torso and cardiac surfaces. In particular,the numerical specification of the no-flux condition onthe torso is difficult because surface normals must be computed,and because the torso may alternatively be consideredinfinitely far away from the heart. Using the method offundamental solutions (MFS), this boundaryconditions can be taken into account in different manners.Specifically, the no-flux condition on the torso can beignored, or weighted with respect to the Dirichlet boundarycondition associated to the torso data, or can bestrongly enforced through a saddle-point problem.In this work we provide a preliminary comparison ofthese different strategies

Use of IBASPM atlas-based automatic segmentation toolbox in pathological brains: effect of template selection

Proceeding of: 2008 IEEE Nuclear Science Symposium Conference Record (NSS '08), Dresden, Germany, 19-25 Oct. 2008IBASPM software is an atlas-based method for automatic segmentation of brain structures, available as a freeware toolbox for the SPM package. To test the influence of the atlas when segmenting normal and pathologic brains, manual segmentation of the caudate nucleus head was compared to automatic segmentations using four different atlases: the default MNI AAL atlas; a customized atlas created from a combined sample of patients (n=20) and controls (n=18); and a customized atlas obtained separately for each group. Maximum average ratio of overlapping voxels (dice overlap) between manual and automatic segmentation was 71 o~ for controls and 52% for patients. In both groups, overlap ratios were better when using the customized atlases, instead of the standard MNI AAL atlas. Accuracy of the method was biased between left and right hemispheres, and also between groups, individual variability being higher in patients than in controls. Volumetric measurements using the customized atlases were also more accurate than using the MNI AAL atlas. Volume data were closer to manual segmentation values than dice overlap ratio (average differences ranging from 22.7°~ for MNI AAL atlas to 10.1 for customized atlas of patients and controls combined). Results suggests a low overaU performance of IBASPM as an automatic segmentation method for the head of the caudate nucleus. Because of the biases observed, the use of this method for analyzing caudate nucleus in patients presenting anatomical abnormalities should be cautiously carried out.This work is partially funded by the following projects: CD-TEAM Project, CENIT Program (Spanish Ministerio de Industria)~ FIS PI052271 (Spanish Ministerio de Sanidad y Consumo)~ and Fundación Mutua Madrilen

Deep Learning Formulation of ECGI for Data-driven Integration of Spatiotemporal Correlations and Imaging Information

International audienceThe challenge of non-invasive Electrocardiographic Imaging (ECGI) is to recreate the electrical activity of the heart using body surface potentials. Specifically, there are numerical difficulties due to the ill-posed nature of the problem. We propose a novel method based on Conditional Variational Autoencoders using Deep generative Neural Networks to overcome this challenge. By conditioning the electrical activity on heart shape and electrical potentials, our model is able to generate activation maps with good accuracy on simulated data (mean square error, MSE = 0.095). This method differs from other formulations because it naturally takes into account spatio-temporal correlations as well as the imaging substrate through convolutions and conditioning. We believe these features can help improving ECGI results

Validation and Opportunities of Electrocardiographic Imaging: From Technical chievements to Clinical Applications

[EN] Electrocardiographic imaging (ECGI) reconstructs the electrical activity of the heart from a dense array of body-surface electrocardiograms and a patient-specific heart-torso geometry. Depending on how it is formulated, ECGI allows the reconstruction of the activation and recovery sequence of the heart, the origin of premature beats or tachycardia, the anchors/hotspots of re-entrant arrhythmias and other electrophysiological quantities of interest. Importantly, these quantities are directly and non-invasively reconstructed in a digitized model of the patient's three-dimensional heart, which has led to clinical interest in ECGI's ability to personalize diagnosis and guide therapy. Despite considerable development over the last decades, validation of ECGI is challenging. Firstly, results depend considerably on implementation choices, which are necessary to deal with ECGI's ill-posed character. Secondly, it is challenging to obtain (invasive) ground truth data of high quality. In this review, we discuss the current status of ECGI validation as well as the major challenges remaining for complete adoption of ECGI in clinical practice. Specifically, showing clinical benefit is essential for the adoption of ECGI. Such benefit may lie in patient outcome improvement, workflow improvement, or cost reduction. Future studies should focus on these aspects to achieve broad adoption of ECGI, but only after the technical challenges have been solved for that specific application/pathology. We propose 'best' practices for technical validation and highlight collaborative efforts recently organized in this field. Continued interaction between engineers, basic scientists, and physicians remains essential to find a hybrid between technical achievements, pathological mechanisms insights, and clinical benefit, to evolve this powerful technique toward a useful role in clinical practice.This study received financial support from the Hein Wellens Fonds, the Cardiovascular Research and Training Institute (CVRTI), the Nora Eccles Treadwell Foundation, the National Institute of General Medical Sciences of the National Institutes of Health (P41GM103545), the National Institutes of Health (NIH HL080093), the French government as part of the Investments of the Future program managed by the National Research Agency (ANR-10-IAHU-04), from the VEGA Grant Agency in Slovakia (2/0071/16), from the Slovak Research and Development Agency (APVV-14-0875), the Fondo Europeo de Desarrollo Regional (FEDER), the Instituto de Salud Carlos III (PI17/01106) and from Conselleria d'Educacio, Investigacio, Cultura i Esport de la Generalitat Valenciana (AICO/2018/267) and NIH grant (HL125998) and National Science Foundation (ACI-1350374).Cluitmans, M.; Brooks, D.; Macleod, RS.; Dossel, O.; Guillem Sánchez, MS.; Van Dam, P.; Svehlikova, J.... 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Exploring different approaches to improve the inverse problem solutions in electrocardiography

Abstract for the talk given 2017-08-23 at PCH60: Computational Inverse Problems - Insight and Algorithms.A workshop on the occasion of Per Christian Hansen's 60th birthdayCopenhagen, Denmark, August 23–25, 2017Cardiovascular diseases are the leading cause of death in the EU. It causes 11000 death per day in Europe and 5200 death per day in the EU. Non-invasive techniques that identify patients at risk, provide accurate diagnosis, offer a better understanding of the cardiac electrophisilogy and guide therapy still fail. These include electrocardiographic imaging (ECGI), an approach in which inverse methods are used to reconstruct heart electrical activity from potentials measured on the body surface. Despite all the success of ECGI, the understanding and treatment of many cardiac diseases is not feasible yet without an improvement of the solution of its inverse problem. A homogeneous meshless scheme based on the method of fundamental solution (MFS) was adapted to ECGI. In the MFS, the potentials are expressed as a linear combination of Laplace fundamental solution over a discrete set of virtual point sources placed outside of the domain of interest. This formulation yields to a linear system, which involves contributions of the Dirichlet and the homogenous Neumann conditions (HNCs) at torso surface (or zero-flux) in an equivalent manner. In this work, we first used the singular value analysis and discrete picard condition (DPC) to optimize our setup (in terms of the respective boundary conditions and virtual sources placement) making it less sensible to the regularization chosen. Then, we reconstruct the potentials, by using a new regularization parameter choice method for the MFS ECGI problem based on DPC. Results demonstrate that: 1. An optimal placement of the sources and/or neglecting the homogeneous Neumann conditions reduces the ill-posedness of the problem, making the solution more robust (less sensible to the regularization chosen). This is especially significant when noise/artifact is present. Furthermore, the computational burden is reduced. 2. The new regularization parameter choice method provided higher correlation coefficients and lower relative errors than the current one in terms of potentials and activation maps, especially for the spiral data. 3. A spatio-temporal solution seems advisable. To conclude, novel inverse problem methods, some adapted from quite different fields of computer science and mathematics, seems to give a hope to improve the performance of the MFS ECGI solution