Cardiac motion estimation using covariant derivatives and Helmholtz decomposition

The investigation and quantification of cardiac movement is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider new aperture problem-free methods to track cardiac motion from 2-dimensional MR tagged images and corresponding sine-phase images. Tracking is achieved by following the movement of scale-space maxima, yielding a sparse set of linear features of the unknown optic flow vector field. Interpolation/reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev-norm expressed in covariant derivatives (rather than standard derivatives). These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. They are defined on a fiber bundle where sections coincide with vector fields. Furthermore, the optic flow vector field is decomposed in a divergence free and a rotation free part, using our multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in a single non-singular analytic kernel operator. Finally, we combine this multi-scale Helmholtz decomposition with vector field reconstruction (based on covariant derivatives) in a single algorithm and present some experiments of cardiac motion estimation. Further experiments on phantom data with ground truth show that both the inclusion of covariant derivatives and the inclusion of the multi-scale Helmholtz decomposition improves the optic flow reconstruction

Feature based estimation of myocardial motion from tagged MR images

In the past few years we witnessed an increase in mortality due to cancer relative to mortality due to cardiovascular diseases. In 2008, the Netherlands Statistics Agency reports that 33.900 people died of cancer against 33.100 deaths due to cardiovascular diseases, making cancer the number one cause of death in the Netherlands [33]. Even if the rate of people affected by heart diseases is continually rising, they "simply don’t die of it", according to the research director Prof. Mat Daemen of research institute CARIM of the University of Maastricht [50]. The reason for this is the early diagnosis, and the treatment of people with identified risk factors for diseases like ischemic heart disease, hypertrophic cardiomyopathy, thoracic aortic disease, pericardial (sac around the heart) disease, cardiac tumors, pulmonary artery disease, valvular disease, and congenital heart disease before and after surgical repair. Cardiac imaging plays a crucial role in the early diagnosis, since it allows the accurate investigation of a large amount of imaging data in a small amount of time. Moreover, cardiac imaging reduces costs of inpatient care, as has been shown in recent studies [77]. With this in mind, in this work we have provided several tools with the aim to help the investigation of the cardiac motion. In chapters 2 and 3 we have explored a novel variational optic flow methodology based on multi-scale feature points to extract cardiac motion from tagged MR images. Compared to constant brightness methods, this new approach exhibits several advantages. Although the intensity of critical points is also influenced by fading, critical points do retain their characteristic even in the presence of intensity changes, such as in MR imaging. In an experiment in section 5.4 we have applied this optic flow approach directly on tagged MR images. A visual inspection confirmed that the extracted motion fields realistically depicted the cardiac wall motion. The method exploits also the advantages from the multiscale framework. Because sparse velocity formulas 2.9, 3.7, 6.21, and 7.5 provide a number of equations equal to the number of unknowns, the method does not suffer from the aperture problem in retrieving velocities associated to the critical points. In chapters 2 and 3 we have moreover introduced a smoothness component of the optic flow equation described by means of covariant derivatives. This is a novelty in the optic flow literature. Many variational optic flow methods present a smoothness component that penalizes for changes from global assumptions such as isotropic or anisotropic smoothness. In the smoothness term proposed deviations from a predefined motion model are penalized. Moreover, the proposed optic flow equation has been decomposed in rotation-free and divergence-free components. This decomposition allows independent tuning of the two components during the vector field reconstruction. The experiments and the Table of errors provided in 3.8 showed that the combination of the smoothness term, influenced by a predefined motion model, and the Helmholtz decomposition in the optic flow equation reduces the average angular error substantially (20%-25%) with respect to a similar technique that employs only standard derivatives in the smoothness term. In section 5.3 we extracted the motion field of a phantom of which we know the ground truth of and compared the performance of this optic flow method with the performance of other optic flow methods well known in the literature, such as the Horn and Schunck [76] approach, the Lucas and Kanade [111] technique and the tuple image multi-scale optic flow constraint equation of Van Assen et al. [163]. Tests showed that the proposed optic flow methodology provides the smallest average angular error (AAE = 3.84 degrees) and L2 norm = 0.1. In this work we employed the Helmholtz decomposition also to study the cardiac behavior, since the vector field decomposition allows to investigate cardiac contraction and cardiac rotation independently. In chapter 4 we carried out an analysis of cardiac motion of ten volunteers and one patient where we estimated the kinetic energy for the different components. This decomposition is useful since it allows to visualize and quantify the contributions of each single vector field component to the heart beat. Local measurements of the kinetic energy have also been used to detect areas of the cardiac walls with little movement. Experiments on a patient and a comparison between a late enhancement cardiac image and an illustration of the cardiac kinetic energy on a bull’s eye plot illustrated that a correspondence between an infarcted area and an area with very small kinetic energy exists. With the aim to extend in the future the proposed optic flow equation to a 3D approach, in chapter 6 we investigated the 3D winding number approach as a tool to locate critical points in volume images. We simplified the mathematics involved with respect to a previous work [150] and we provided several examples and applications such as cardiac motion estimation from 3-dimensional tagged images, follicle and neuronal cell counting. Finally in chapter 7 we continued our investigation on volume tagged MR images, by retrieving the cardiac motion field using a 3-dimensional and simple version of the proposed optic flow equation based on standard derivatives. We showed that the retrieved motion fields display the contracting and rotating behavior of the cardiac muscle. We moreover extracted the through-plane component, which provides a realistic illustration of the vector field and is missed by 2-dimensional approaches

Doctor of Philosophy

dissertationThe statistical study of anatomy is one of the primary focuses of medical image analysis. It is well-established that the appropriate mathematical settings for such analyses are Riemannian manifolds and Lie group actions. Statistically defined atlases, in which a mean anatomical image is computed from a collection of static three-dimensional (3D) scans, have become commonplace. Within the past few decades, these efforts, which constitute the field of computational anatomy, have seen great success in enabling quantitative analysis. However, most of the analysis within computational anatomy has focused on collections of static images in population studies. The recent emergence of large-scale longitudinal imaging studies and four-dimensional (4D) imaging technology presents new opportunities for studying dynamic anatomical processes such as motion, growth, and degeneration. In order to make use of this new data, it is imperative that computational anatomy be extended with methods for the statistical analysis of longitudinal and dynamic medical imaging. In this dissertation, the deformable template framework is used for the development of 4D statistical shape analysis, with applications in motion analysis for individualized medicine and the study of growth and disease progression. A new method for estimating organ motion directly from raw imaging data is introduced and tested extensively. Polynomial regression, the staple of curve regression in Euclidean spaces, is extended to the setting of Riemannian manifolds. This polynomial regression framework enables rigorous statistical analysis of longitudinal imaging data. Finally, a new diffeomorphic model of irrotational shape change is presented. This new model presents striking practical advantages over standard diffeomorphic methods, while the study of this new space promises to illuminate aspects of the structure of the diffeomorphism group

Fluid flow estimation with multiscale ensemble filters based on motion measurements under location uncertainty

International audienceThis paper proposes a novel multi-scale fluid flow data assimilation approach, which integrates and complements the advantages of a Bayesian sequential assimilation technique, the Weighted Ensemble Kalman filter (WEnKF). The data assimilation proposed in this work incorporates measurement brought by an efficient multiscale stochastic formulation of the well-known Lucas-Kanade (LK) estimator. This estimator has the great advantage to provide uncertainties associated to the motion measurements at different scales. The proposed assimilation scheme benefits from this multiscale uncertainty information and enables to enforce a physically plausible dynamical consistency of the estimated motion fields along the image sequence. Experimental evaluations are presented on synthetic and real fluid flow sequences

A constitutive model of human esophagus tissue with application for the treatment of stenosis

This dissertation is a research about the mechanical behavior of the human esophagus. This work is intended to be applied in the treatment of stenosis and other esophageal diseases that frequently require a procedure of forced dilation, that involves high pressures on the esophagus wall. This study proposes a constitutive model to simulate forced dilatations. This study includes the experimental characterization of the mechanical behavior of human esophagus. In addition, some theoretical questions and experimental issues were addressed and solved. This dissertation summarizes the previous work on esophageal tissues by other authors. Additionally, a short explanation about the microcontinuum theory developed in the last decades is given, as well as a summary of the general theory of nonlinear hyperelastic constitutive models (with large deformation). This study required extensive testing of esophageal tissue in order to characterize the in vitro mechanical behavior. The testing included mainly tensile tests and complementary inflation tests. Optical motion track analysis was used for accurate computation of the strains in the tissue. The results of the tests were used for adjusting the mechanical properties that characterize the mechanical behavior of esophagus in the proposed models of the literature. The statistical analysis of the data revealed, some significant correlations between anthropometric factors, such as the body mass index, and some mechanical properties were found in the analysis of the data. The typical values of the mechanical properties were used to perform some numerical finite element simulations based on the proposed models. In addition, a number of theoretical results were obtained concerning the residual stress and the predictions of statistical mechanics for a system of collagenous fibers inside a soft tissue. The main result is a constitutive non-linear microstretch anisotropic hyperelastic constitutive model with large deformations (and with residual stresses) to characterize the multi-layered tissue of human esophagus. This model is suitable for numerical simulation.Esta tesis es una investigación sobre el comportamiento mecánico del esófago humano. Este trabajo se pensó para ser aplicado al tratamiento de la estenosis y otras afecciones esofágicas que, con frecuencia, requieren dilatación forzada, lo que supone altas presiones sobre la pared esofágica. Este estudio propone un modelo constitutivo para simular estas dilataciones forzadas. Este estudio incluye la caracterización experimental del comportamiento mecánico del esófago. Además, se han planteado algunas cuestiones teóricas y experimentales que han sido resueltas. Esta tesis resume el trabajo previo sobre tejido esofágico de otros investigadores. Además, se da una pequeña explicación sobre la teoría del medio microntinuo desarrollada en las últimas décadas y un breve resumen de la teoría general de modelos constitutivos hiperelásticos no-lineales (con grandes deformaciones). El estudio requirió experimentación de tejido para caracterizar el comportamiento in vitro. Los test incluyeron test de tracción y test de inflado. Se empleó rastreo óptico para un cálculo adecuado de la deformación. Los resultados de los test permitieron encontrar las propiedades mecánicas según dos modelos de esófago de la literatura. El análisis de los datos rebeló, algunas correlaciones significativas entre factores como el índice de masa corporal y algunas propiedades mecánicas. Los valores típicos de las propiedades fueron usados para algunas simulaciones numéricas basadas en los modelos propuestos. Además se han obtenido algunos resultados teóricos sobre la tensión residual y las predicciones de la mecánica estadística para un sistema de fibras de colágeno del tejido. El principal resultado es un modelo constitutivo de microestiramiento anisótropo no-lineal e hiperelástico para caracterizar el esófago Este modelo es adecuado para la computación numérica.Postprint (published version

Tensors and second quantization

Starting from a pair of vector spaces (formula) an inner product space and (formula), the space of linear mappings (formula), we construct a six-tuple (formula). Here (formula) is again an inner product space and (formula) the space of its linear mappings. It is required that (formula), as linear subspaces. (formula) Further, (formula) and (formula) denotes a lifting map (formula) such that, whenever (formula) solves an evolution equation (formula) then any product of operator valued functions (formula) solves the associated commutator equation in (formula), (formula) Furthermore, (formula). We also note that (formula) represents the state of k identical systems ’living apart together’. Cf. the free field ’formalism’ in physics. Such constructions can be realized in many different ways (section 2). However in Quantum Field Theory one requires additional relations between the creation operator C and its adjoint (formula), the annihilation operator. These are the so called Canonical (Anti-)Commutation Relations, (section 3). Here, unlike in books on theoretical physics, the combinatorial aspects of those 1This note is meant to be Appendix K in the lecture notes ’Tensorrekening en Differentiaalmeetkunde’. restrictions are dealt with in full detail. Annihilation/Creation operators don’t grow on trees! However, apart from the way of presentation, nothing new is claimed here. This note is completely algebraic. For topological extensions of the maps C; A to distribution spaces we refer to Part III in [EG], where a mathematical interpretation of Dirac’s formalism has been presented