Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging

We present a Bayesian probabilistic model to estimate the brain white matter atlas from high angular resolution diffusion imaging (HARDI) data. This model incorporates a shape prior of the white matter anatomy and the likelihood of individual observed HARDI datasets. We first assume that the atlas is generated from a known hyperatlas through a flow of diffeomorphisms and its shape prior can be constructed based on the framework of large deformation diffeomorphic metric mapping (LDDMM). LDDMM characterizes a nonlinear diffeomorphic shape space in a linear space of initial momentum uniquely determining diffeomorphic geodesic flows from the hyperatlas. Therefore, the shape prior of the HARDI atlas can be modeled using a centered Gaussian random field (GRF) model of the initial momentum. In order to construct the likelihood of observed HARDI datasets, it is necessary to study the diffeomorphic transformation of individual observations relative to the atlas and the probabilistic distribution of orientation distribution functions (ODFs). To this end, we construct the likelihood related to the transformation using the same construction as discussed for the shape prior of the atlas. The probabilistic distribution of ODFs is then constructed based on the ODF Riemannian manifold. We assume that the observed ODFs are generated by an exponential map of random tangent vectors at the deformed atlas ODF. Hence, the likelihood of the ODFs can be modeled using a GRF of their tangent vectors in the ODF Riemannian manifold. We solve for the maximum a posteriori using the Expectation-Maximization algorithm and derive the corresponding update equations. Finally, we illustrate the HARDI atlas constructed based on a Chinese aging cohort of 94 adults and compare it with that generated by averaging the coefficients of spherical harmonics of the ODF across subjects

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

Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI

We develop a general analytical and numerical framework for estimating intra- and extra-neurite water fractions and diffusion coefficients, as well as neurite orientational dispersion, in each imaging voxel. By employing a set of rotational invariants and their expansion in the powers of diffusion weighting, we analytically uncover the nontrivial topology of the parameter estimation landscape, showing that multiple branches of parameters describe the measurement almost equally well, with only one of them corresponding to the biophysical reality. A comprehensive acquisition shows that the branch choice varies across the brain. Our framework reveals hidden degeneracies in MRI parameter estimation for neuronal tissue, provides microstructural and orientational maps in the whole brain without constraints or priors, and connects modern biophysical modeling with clinical MRI.Comment: 25 pages, 12 figures, elsarticle two-colum

MODELING AND QUANTITATIVE ANALYSIS OF WHITE MATTER FIBER TRACTS IN DIFFUSION TENSOR IMAGING

Diffusion tensor imaging (DTI) is a structural magnetic resonance imaging (MRI) technique to record incoherent motion of water molecules and has been used to detect micro structural white matter alterations in clinical studies to explore certain brain disorders. A variety of DTI based techniques for detecting brain disorders and facilitating clinical group analysis have been developed in the past few years. However, there are two crucial issues that have great impacts on the performance of those algorithms. One is that brain neural pathways appear in complicated 3D structures which are inappropriate and inaccurate to be approximated by simple 2D structures, while the other involves the computational efficiency in classifying white matter tracts. The first key area that this dissertation focuses on is to implement a novel computing scheme for estimating regional white matter alterations along neural pathways in 3D space. The mechanism of the proposed method relies on white matter tractography and geodesic distance mapping. We propose a mask scheme to overcome the difficulty to reconstruct thin tract bundles. Real DTI data are employed to demonstrate the performance of the pro- posed technique. Experimental results show that the proposed method bears great potential to provide a sensitive approach for determining the white matter integrity in human brain. Another core objective of this work is to develop a class of new modeling and clustering techniques with improved performance and noise resistance for separating reconstructed white matter tracts to facilitate clinical group analysis. Different strategies are presented to handle different scenarios. For whole brain tractography reconstructed white matter tracts, a Fourier descriptor model and a clustering algorithm based on multivariate Gaussian mixture model and expectation maximization are proposed. Outliers are easily handled in this framework. Real DTI data experimental results show that the proposed algorithm is relatively effective and may offer an alternative for existing white matter fiber clustering methods. For a small amount of white matter fibers, a modeling and clustering algorithm with the capability of handling white matter fibers with unequal length and sharing no common starting region is also proposed and evaluated with real DTI data

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

Spectral Clustering en IRM de diffusion pour Retrouver les Faisceaux de la Matière Blanche

White matter fiber clustering allows to get insight about anatomical structures in order to generate atlases, perform clear visualizations and compute statistics across subjects, all important and current neuroimaging problems. In this work, we present a Diffusion Maps clustering method applied to diffusion MRI in order to cluster and segment complex white matter fiber bundles. It is well-known that Diffusion Tensor Imaging (DTI) is restricted in complex fiber regions with crossings and this is why recent High Angular Resolution Diffusion Imaging (HARDI) such has Q-Ball Imaging (QBI) have been introduced to overcome these limitations. QBI reconstructs the diffusion orientation distribution function (ODF), a spherical function that has its maxima agreeing with the underlying fiber populations. In this paper, we introduce the usage of the Diffusion Maps technique and show how it can be used to directly cluster set of fiber tracts, that could be obtained through a streamline tractography for instance, and how it can also help in segmenting fields of ODF images, obtained through a linear and regularized ODF estimation algorithm based on a spherical harmonics representation of the Q-Ball data. We first show the advantage of using Diffusion Maps clustering over classical methods such as N-Cuts and Laplacian Eigenmaps in both cases. In particular, our Diffusion Maps requires a smaller number of hypothesis from the input data, reduces the number of artifacts in fiber tract clustering and ODF image segmentation and automatically exhibits the number of clusters in both cases by using an adaptive scale-space parameter. We also show that our ODF Diffusion Maps clustering can reproduce published results using the diffusion tensor (DT) clustering with N-Cuts on simple synthetic images without crossings. On more complex data with crossings, we show that our ODF-based method succeeds to separate fiber bundles and crossing regions whereas the DT-based methods generate artifacts and exhibit wrong number of clusters. Finally, we illustrate the potential of our approach on a real brain dataset where we successfully segment well-known fiber bundles

Doctor of Philosophy

dissertationRecent developments in magnetic resonance imaging (MRI) provide an in vivo and noninvasive tool for studying the human brain. In particular, the detection of anisotropic diffusion in biological tissues provides the foundation for diffusion-weighted imaging (DWI), an MRI modality. This modality opens new opportunities for discoveries of the brain's structural connections. Clinically, DWI is often used to analyze white matter tracts to understand neuropsychiatric disorders and the connectivity of the central nervous system. However, due to imaging time required, DWI used in clinical studies has a low angular resolution. In this dissertation, we aim to accurately track and segment the white matter tracts and estimate more representative models from low angular DWI. We first present a novel geodesic approach to segmentation of white matter tracts from diffusion tensor imaging (DTI), estimated from DWI. Geodesic approaches treat the geometry of brain white matter as a manifold, often using the inverse tensor field as a Riemannian metric. The white matter pathways are then inferred from the resulting geodesics. A serious drawback of current geodesic methods is that geodesics tend to deviate from the major eigenvectors in high-curvature areas in order to achieve the shortest path. We propose a method for learning an adaptive Riemannian metric from the DTI data, where the resulting geodesics more closely follow the principal eigenvector of the diffusion tensors even in high-curvature regions. Using the computed geodesics, we develop an automatic way to compute binary segmentations of the white matter tracts. We demonstrate that our method is robust to noise and results in improved geodesics and segmentations. Then, based on binary segmentations, we present a novel Bayesian approach for fractional segmentation of white matter tracts and simultaneous estimation of a multitensor diffusion model. By incorporating a prior that assumes the tensor fields inside each tract are spatially correlated, we are able to reliably estimate multiple tensor compartments in fiber crossing regions, even with low angular diffusion-weighted imaging. This reduces the effects of partial voluming and achieves a more reliable analysis of diffusion measurements