Fuzzy Fibers: Uncertainty in dMRI Tractography

Fiber tracking based on diffusion weighted Magnetic Resonance Imaging (dMRI) allows for noninvasive reconstruction of fiber bundles in the human brain. In this chapter, we discuss sources of error and uncertainty in this technique, and review strategies that afford a more reliable interpretation of the results. This includes methods for computing and rendering probabilistic tractograms, which estimate precision in the face of measurement noise and artifacts. However, we also address aspects that have received less attention so far, such as model selection, partial voluming, and the impact of parameters, both in preprocessing and in fiber tracking itself. We conclude by giving impulses for future research

Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4

Topological visualization of tensor fields using a generalized Helmholtz decomposition

Analysis and visualization of fluid flow datasets has become increasing important with the development of computer graphics. Even though many direct visualization methods have been applied in the tensor fields, those methods may result in much visual clutter. The Helmholtz decomposition has been widely used to analyze and visualize the vector fields, and it is also a useful application in the topological analysis of vector fields. However, there has been no previous work employing the Helmholtz decomposition of tensor fields. We present a method for computing the Helmholtz decomposition of tensor fields of arbitrary order and demonstrate its application. The Helmholtz decomposition can split a tensor field into divergence-free and curl-free parts. The curl-free part is irrotational, and it is useful to isolate the local maxima and minima of divergence (foci of sources and sinks) in the tensor field without interference from curl-based features. And divergence-free part is solenoidal, and it is useful to isolate centers of vortices in the tensor field. Topological visualization using this decomposition can classify critical points of two-dimensional tensor fields and critical lines of 3D tensor fields. Compared with several other methods, this approach is not dependent on computing eigenvectors, tensor invariants, or hyperstreamlines, but it can be computed by solving a sparse linear system of equations based on finite difference approximation operators. Our approach is an indirect visualization method, unlike the direct visualization which may result in the visual clutter. The topological analysis approach also generates a single separating contour to roughly partition the tensor field into irrotational and solenoidal regions. Our approach will make use of the 2nd order and the 4th order tensor fields. This approach can provide a concise representation of the global structure of the field, and provide intuitive and useful information about the structure of tensor fields. However, this method does not extract the exact locations of critical points and lines

Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved to be an indispensable tool for studying medical conditions as well as for general scientific research. For example, it allows to map fiber bundles, the major neuronal pathways through the brain. But all evaluation of scanned data depends on mathematical signal models that describe the raw signal output and map it to biologically more meaningful values. And here we find the most potential for improvement. In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This gives a more detailed quantification of fibers. These additional values can also be estimated by the Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles. This reduces the effects of fiber crossings and allows a more direct examination of fibers. Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of multi-fiber signal models to a continuous distribution of fiber directions. To this approach we introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally, in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the signal can be decomposed into fibers. This means, tractography and other methods that depend on identifying a discrete set of fiber directions greatly benefit from our constraint. Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry. In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces inside the brain. Although not completely obvious, there is a mathematical obstacle coming from differential topology, that prevents general tangential planes spanned by fiber directions at each point to be connected into consistent surfaces. Research into how well this constraint is fulfilled in our brain is hindered by the high precision and complexity needed by previous evaluation methods. This is why we present a drastically simpler method that negates the need for precisely finding fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use our new method to explore and improve streamsurface visualization.<br /

Identifying Changes of Functional Brain Networks using Graph Theory

This thesis gives an overview on how to estimate changes in functional brain networks using graph theoretical measures. It explains the assessment and definition of functional brain networks derived from fMRI data. More explicitly, this thesis provides examples and newly developed methods on the measurement and visualization of changes due to pathology, external electrical stimulation or ongoing internal thought processes. These changes can occur on long as well as on short time scales and might be a key to understanding brain pathologies and their development. Furthermore, this thesis describes new methods to investigate and visualize these changes on both time scales and provides a more complete picture of the brain as a dynamic and constantly changing network.:1 Introduction 1.1 General Introduction 1.2 Functional Magnetic Resonance Imaging 1.3 Resting-state fMRI 1.4 Brain Networks and Graph Theory 1.5 White-Matter Lesions and Small Vessel Disease 1.6 Transcranial Direct Current Stimulation 1.7 Dynamic Functional Connectivity 2 Publications 2.1 Resting developments: a review of fMRI post-processing methodologies for spontaneous brain activity 2.2 Early small vessel disease affects fronto-parietal and cerebellar hubs in close correlation with clinical symptoms - A resting-state fMRI study 2.3 Dynamic modulation of intrinsic functional connectivity by transcranial direct current stimulation 2.4 Three-dimensional mean-shift edge bundling for the visualization of functional connectivity in the brain 2.5 Dynamic network participation of functional connectivity hubs assessed by resting-state fMRI 3 Summary 4 Bibliography 5. Appendix 5.1 Erklärung über die eigenständige Abfassung der Arbeit 5.2 Curriculum vitae 5.3 Publications 5.4 Acknowledgement

Topological Visualization of Brain Diffusion MRI Data

Abstract—Topological methods give concise and expressive visual representations of flow fields. The present work suggests a comparable method for the visualization of human brain diffusion MRI data. We explore existing techniques for the topological analysis of generic tensor fields, but find them inappropriate for diffusion MRI data. Thus, we propose a novel approach that considers the asymptotic behavior of a probabilistic fiber tracking method and define analogs of the basic concepts of flow topology, like critical points, basins, and faces, with interpretations in terms of brain anatomy. The resulting features are fuzzy, reflecting the uncertainty inherent in any connectivity estimate from diffusion imaging. We describe an algorithm to extract the new type of features, demonstrate its robustness under noise, and present results for two regions in a diffusion MRI dataset to illustrate that the method allows a meaningful visual analysis of probabilistic fiber tracking results. Index Terms—Diffusion tensor, probabilistic fiber tracking, tensor topology, uncertainty visualization.