Geodesic tractography segmentation for directional medical image analysis

Acknowledgements page removed per author's request, 01/06/2014.Geodesic Tractography Segmentation is the two component approach presented in this thesis for the analysis of imagery in oriented domains, with emphasis on the application to diffusion-weighted magnetic resonance imagery (DW-MRI). The computeraided analysis of DW-MRI data presents a new set of problems and opportunities for the application of mathematical and computer vision techniques. The goal is to develop a set of tools that enable clinicians to better understand DW-MRI data and ultimately shed new light on biological processes. This thesis presents a few techniques and tools which may be used to automatically find and segment major neural fiber bundles from DW-MRI data. For each technique, we provide a brief overview of the advantages and limitations of our approach relative to other available approaches.Ph.D.Committee Chair: Tannenbaum, Allen; Committee Member: Barnes, Christopher F.; Committee Member: Niethammer, Marc; Committee Member: Shamma, Jeff; Committee Member: Vela, Patrici

Tubular Surface Evolution for Segmentation of the Cingulum Bundle From DW-MRI

Presented at the 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical and Statistical Methods for Biological Shape Variability Modeling, September 6th, 2008, Kimmel Center, New York, USA.This work provides a framework for modeling and extracting the Cingulum Bundle (CB) from Diffusion-Weighted Imagery (DW-MRI) of the brain. The CB is a tube-like structure in the brain that is of potentially of tremendous importance to clinicians since it may be helpful in diagnosing Schizophrenia. This structure consists of a collection of fibers in the brain that have locally similar diffusion patterns, but vary globally. Standard region-based segmentation techniques adapted to DW-MRI are not suitable here because the diffusion pattern of the CB cannot be described by a global set of simple statistics. Active surface models extended to DW-MRI are not suitable since they allow for arbitrary deformations that give rise to unlikely shapes, which do not respect the tubular geometry of the CB. In this work, we explicitly model the CB as a tube-like surface and construct a general class of energies defined on tube-like surfaces. An example energy of our framework is optimized by a tube that encloses a region that has locally similar diffusion patterns, which differ from the diffusion patterns immediately outside. Modeling the CB as a tube-like surface is a natural shape prior. Since a tube is characterized by a center-line and a radius function, the method is reduced to a 4D (center-line plus radius) curve evolution that is computationally much less costly than an arbitrary surface evolution. The method also provides the center-line of CB, which is potentially of clinical significance

Measures for validation of DTI tractography

pre-printThe evaluation of analysis methods for diffusion tensor imaging (DTI) remains challenging due to the lack of gold standards and validation frameworks. Significant work remains in developing metrics for comparing fiber bundles generated from streamline tractography. We propose a set of volumetric and tract oriented measures for evaluating tract differences. The different methods developed for this assessment work are: an overlap measurement, a point cloud distance and a quantification of the diffusion properties at similar locations between fiber bundles. The application of the measures in this paper is a comparison of atlas generated tractography to tractography generated in individual images. For the validation we used a database of 37 subject DTIs, and applied the measurements on five specific fiber bundles: uncinate, cingulum (left and right for both bundles) and genu. Each measurments is interesting for specific use: the overlap measure presents a simple and comprehensive metric but is sensitive to partial voluming and does not give consistent values depending on the bundle geometry. The point cloud distance associated with a quantile interpretation of the distribution gives a good intuition of how close and similar the bundles are. Finally, the functional difference is useful for a comparison of the diffusion properties since it is the focus of many DTI analysis to compare scalar invariants. The comparison demonstrated reasonable similarity of results. The tract difference measures are also applicable to comparison of tractography algorithms, quality control, reproducibility studies, and other validation problems

Multi-valued geodesic tractography for diffusion weighted imaging

Diffusion-Weighted Imaging (DWI) is a Magnetic Resonance(MR) technique that measures water diffusion characteristics in tissue for a given direction. The diffusion profile in a specific location can be obtained by combining the DWI measurements of different directions. The diffusion profile gives information about the underlying fibrous structure, e.g., in human brain white matter, based on the assumption that water molecules are moving less freely perpendicularly to the fibrous structure. From the DW-MRI measurements often a positive definite second-order tensor is defined, the so-called diffusion tensor (DT). Neuroscientists have begun using diffusion tensor images (DTI) to study a host of various disorders and neurodegenerative diseases including Parkinson, Alzheimer and Huntington. The techniques for reconstructing the fiber tracts based on diffusion profiles are known as tractography or fiber tracking. There are several ways to extract fibers from the raw diffusion data. In this thesis, we explain and apply geodesic-based tractography techniques specifically, where the assumption is that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. A shortest path in this manifold is one with the strongest diffusion along this path. Therefore geodesics (i.e., shortest paths) on this manifold follow the most efficient diffusion paths. The geodesics are often computed from the stationary Hamilton-Jacobi equation (HJ). One characteristic of solving the HJ equation is that it gives only the single-valued viscosity solution corresponding to the minimizer of the length functional. It is also well known that the solution of the HJ equation can develop discontinuities in the gradient space, i.e., cusps. Cusps occur when the correct solution should become multi-valued. HJ methods are not able to handle this situation. To solve this, we developed a multi-valued solution algorithm for geodesic tractography in a metric space defined by given by diffusion tensor imaging data. The algorithm can capture all possible geodesics arriving at a single voxel instead of only computing the first arrival. Our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points. In this thesis, we first focus on the mathematical and numerical model for analytic and synthetic fields in twodimensional domains. Later, we present the algorithm in three-dimensions with examples of synthetic and brain data. Despite the simplicity of the DTI model, the tractography techniques using DT are shown to be very promising to reveal the structure of brain white matter. However, DTI assumes that each voxel contains fibers with only one main orientation and it is known that brain white matter has multiple fiber orientations, which can be arbitrary many in arbitrary directions. Recently, High Angular Resolution Diffusion Imaging (HARDI) acquisition and its modeling techniques have been developed to overcome this limitation. As a next contribution we propose an extension of the multi-valued geodesic algorithm to HARDI data. First we introduce the mathematical model for more complex geometries using Finsler geometry. Next, we propose, justify and exploit the numerical methods for computing the multi-valued solution of these equations

A simplified algorithm for inverting higher order diffusion tensors

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann-Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann-Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices.</p

Radiation therapy effects on white matter fiber tracts of the limbic circuit

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/134834/1/mp5560.pd

Brain connectivity using geodesics in HARDI

International audienceWe develop an algorithm for brain connectivity assessment using geodesics in HARDI (high angular resolution diffusion imaging). We propose to recast the problem of finding fibers bundles and connectivity maps to the calculation of shortest paths on a Riemannian manifold defined from fiber ODFs computed from HARDI measurements. Several experiments on real data show that out method is able to segment fibers bundles that are not easily recovered by other existing methods

Classification of Tensors and Fiber Tracts Using Mercer-Kernels Encoding Soft Probabilistic Spatial and Diffusion Information

In this paper, we present a kernel-based approach to the clustering of diffusion tensors and fiber tracts. We propose to use a Mercer kernel over the tensor space where both spatial and diffusion information are taken into account. This kernel highlights implicitly the connectivity along fiber tracts. Tensor segmentation is performed using kernel-PCA compounded with a landmark-Isomap embedding and k-means clustering. Based on a soft fiber representation, we extend the tensor kernel to deal with fiber tracts using the multi-instance kernel that reflects not only interactions between points along fiber tracts, but also the interactions between diffusion tensors. This unsupervised method is further extended by way of an atlas-based registration of diffusion-free images, followed by a classification of fibers based on nonlinear kernel Support Vector Machines (SVMs). Promising experimental results of tensor and fiber classification of the human skeletal muscle over a significant set of healthy and diseased subjects demonstrate the potential of our approach

Segmenting Fiber Bundles in Diffusion Tensor Images

Abstract. We consider the problem of segmenting fiber bundles in diffusion tensor images. We cast this problem as a manifold clustering problem in which different fiber bundles correspond to different submanifolds of the space of diffu-sion tensors. We first learn a local representation of the diffusion tensor data using a generalization of the locally linear embedding (LLE) algorithm from Euclidean to diffusion tensor data. Such a generalization exploits geometric properties of the space of symmetric positive semi-definite matrices, particularly its Riemannian metric. Then, under the assumption that different fiber bundles are physically distinct, we show that the null space of a matrix built from the local representation gives the segmentation of the fiber bundles. Our method is computationally simple, can handle large deformations of the principal direction along the fiber tracts, and performs automatic segmentation without requiring previous fiber tracking. Results on synthetic and real diffusion tensor images are also presented.