Finsler Streamline Tracking with Single Tensor Orientation Distribution Function for High Angular Resolution Diffusion Imaging

Q-Ball imaging (QBI) is a successful and widely used high angular resolution diffusion imaging (HARDI) technique which can compute orientation distribution function (ODF). This technique only needs single shell HARDI data and does not require any assumption about the diffusion signal outside the sampling sphere. However the originally proposed ODF (the radial project of the probability density function (PDF)) is not a true ODF. In contrast the marginal ODF with solid angle consideration, represents a true ODF with a correct probabilistic interpretation. In this paper a novel model-free and single-shell HARDI method for analytical reconstruction of the marginal ODF based on Funk-Radon transform (FRT) is proposed. In other words, a transformation of QBI for marginal ODF is proposed in this paper. While there's also a proportional factor introduced by FRT. Its complexity is comparable to that of the original QBI. Finally the present paper verify the algorithm and show the improved result of my method compared to the original QBI on synthetic data, biological phantom data and real human brain data.

We consider the problem of tracking white matter fibers in high angular resolution diffusion imaging (HARDI) data while simultaneously estimating the local fiber orientation profile. Prior work showed that an unscented Kalman filter (UKF) can be used for this problem, yet existing algorithms employ parametric mixture models to represent water diffusion and to define the state space. To address this restrictive model dependency, we propose to extend the UKF to HARDI data modeled by orientation distribution functions (ODFs), a more generic diffusion model. We consider the spherical harmonic representation of the HARDI signal as the state, enforce nonnegativity of the ODFs, and perform tractography using the directions at which the ODFs attain their peaks. In simulations, our method outperforms filtered two-tensor tractography at different levels of noise by achieving a reduction in mean Chamfer error of 0.05 to 0.27 voxels; it also produced in vivo fiber tracking that is consistent with the neuroanatomy.

High angular resolution diffusion imaging (HARDI) improves upon more traditional diffusion tensor imaging (DTI) in its ability to resolve the orientations of crossing and branching neural fibre tracts. The HARDI signals are measured over a spherical shell in q-space, and are usually used as an input to q-ball imaging (QBI) which allows estimation of the diffusion orientation distribution functions (ODFs) associated with a given region-of interest. Unfortunately, the partial nature of single-shell sampling imposes limits on the estimation accuracy. As a result, the recovered ODFs may not possess sufficient resolution to reveal the orientations of fibre tracts which cross each other at acute angles. A possible solution to the problem of limited resolution of QBI is provided by means of spherical deconvolution, a particular instance of which is sparse deconvolution. However, while capable of yielding high-resolution reconstructions over spacial locations corresponding to white matter, such methods tend to become unstable when applied to anatomical regions with a substantial content of isotropic diffusion. To resolve this problem, a new deconvolution approach is proposed in this paper. Apart from being uniformly stable across the whole brain, the proposed method allows one to quantify the isotropic component of cerebral diffusion, which is known to be a useful diagnostic measure by itself.

Diffusion tensor imaging (DTI) accurately delineates white matter pathways when the Gaussian model of diffusion is valid. However, DTI yields erroneous results when diffusion takes on a more complex distribution, as is the case in the brain when fiber tracts cross. High angular resolution diffusion imaging (HARDI) overcomes this limitation of DTI by more fully characterizing the angular dependence of intravoxel diffusion. Among the various HARDI methods that have been proposed, QBI offers advantages such as linearity, model independence, and relatively easy implementation. In this work, reconstruction of the q-ball orientation distribution function (ODF) is reformulated in terms of spherical harmonic basis functions, yielding an analytic solution with useful properties of a frequency domain representation. The harmonic basis is parsimonious for typical b-values, which enables the ODF to be synthesized from a relatively small number of noisy measurements and thus brings the technique closer to clinical feasibility from the standpoint of total imaging time. The proposed method is assessed using Monte Carlo computer simulations and compared with conventional q-ball reconstruction using spherical RBFs. In vivo results from 3T whole-brain HARDI of adult volunteers are also provided to verify the underlying mathematical theory.

Geodesic regression on orientation distribution functions with its application to an aging study

The detection of white matter microstructural changes using diffusion magnetic resonance imaging (dMRI) often involves extracting a small set of scalar features, such as fractional anisotropy (FA) and mean diffusivity (MD) in diffusion tensor imaging (DTI). With the advent of more advanced dMRI techniques, such as high angular resolution diffusion imaging (HARDI), a number of mathematically inspired new scalar features have been proposed. However, it is unclear how to select the most biologically informative combinations of features for biomarker discovery. This paper proposes an automatic HARDI feature selection algorithm which is based on registering HARDI features to feature atlases for optimal clinical usability in population studies. We apply our framework to the characterization of beta-amyloid (Aβ) pathology for the early detection of Alzheimer’s disease (AD) to better understand the relationship between Aβ pathology and degenerative changes in neuroanatomy.

We introduce a geodesic based tractography method for High Angular Resolution Diffusion Imaging (HARDI). The concepts used are similar to the ones in geodesic based tractography for Diffusion Tensor Imaging (DTI). In DTI, the inverse of the second-order diffusion tensor is used to define the manifold where the geodesics are traced. HARDI models have been developed to resolve complex fiber populations within a voxel, and higher order tensors (HOT) are possible representations for HARDI data. In our framework, we apply Finsler geometry, which extends Riemannian geometry to a directionally dependent metric. A Finsler metric is defined in terms of HARDI higher order tensors. Furthermore, the Euler-Lagrange geodesic equations are derived based on the Finsler geometry. In contrast to other geodesic based tractography algorithms, the multi-valued numerical solution of the geodesic equations can be obtained. This gives the possibility to capture all geodesics arriving at a single voxel instead of only computing the shortest one. Results are analyzed to show the potential and characteristics of our algorithm.

This work aimed to demonstrate an essential phase shift for better quantifying and in human brain white matter (WM), and to further elucidate its origin related to the directional diffusivities from standard diffusion tensor imaging (DTI). was integrated into a proposed generalized transverse relaxation model for characterizing previously published and orientation dependence profiles in brain WM, and then comparisons were made with those without . It was theorized that anisotropic diffusivity direction was collinear with an axon fiber subject to all eigenvalues and eigenvectors from an apparent diffusion tensor. To corroborate the origin of , orientation dependences referenced by were compared with those referenced by the standard principal diffusivity direction at b-values of 1000 and 2500 (s/mm2 ). These orientation dependences were obtained from -weighted images (b =0) of ultrahigh-resolution Connectome DTI datasets in the public domain. A normalized root-mean-square error ( ) and an -test were used for evaluating curve-fittings, and statistical significance was considered to be a p of 0.05 or less. A phase-shifted model resulted in significantly reduced compared with that without in quantifying various and profiles, both in vivo and ex vivo at multiple fields. The profiles based on manifested a right-shifted phase ( ) at two b-values, while those based on became free from . For all phase-shifted and profiles, generally depended on the directional diffusivities by , as predicted. In summary, a ubiquitous phase shift has been demonstrated as a prerequisite for better quantifying transverse relaxation orientation dependences in human brain WM. Furthermore, the origin of associated with the directional diffusivities from DTI has been elucidated. These findings could have a significant impact on interpretations of prior and datasets and on future research.

High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over the sphere. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel model for representing this diffusion ODF namely, a mixture of von Mises-Fisher (vMF) distributions. Our model is compact in that it requires very few parameters to represent complicated ODF geometries which occur specifically in the presence of heterogeneous nerve fiber orientations. We present a Riemannian geometric framework for computing intrinsic distances (in closed-form) and for performing interpolation between ODFs represented by vMF mixtures. We also present closed-form equations for entropy and variance based anisotropy measures that are then computed and illustrated for real HARDI data from a rat brain.

Spatial HARDI: Improved visualization of complex white matter architecture with Bayesian spatial regularization

A number of non-parametrically represented High Angular Resolution Diffusion Imaging (HARDI) spherical diffusion functions have been proposed to infer more and more accurately the heterogeneous and complex tissue microarchitecture of the cerebral white-matter. These spherical functions overcome the limitation of Diffusion Tensor Imaging (DTI) at discerning crossing, merging and fanning axonal fiber bundle configurations inside a voxel. Tractography graphically reconstructs the axonal connectivity of the cerebral white-matter in vivo and non-invasively, by integrating along the direction indicated by the local geometry of the spherical diffusion functions. Tractography is acutely sensitive to the local geometry and its correct estimation. In this paper we first propose a polynomial approach for analytically bracketing and numerically refining with high precision all the maxima, or fiber directions, of any spherical diffusion function represented non-parametrically. This permits an accurate inference of the fiber layout from the spherical diffusion function. Then we propose an extension of the deterministic Streamline tractography to HARDI diffusion functions that clearly discern fiber crossings. We also extend the Tensorline algorithm to these HARDI functions, to improve on the extended Streamline tractography. We illustrate our proposed methods using the Solid Angle diffusion Orientation Distribution Function (ODF-SA). We present results on multi-tensor synthetic data, and real in vivo data of the cerebral white-matter that show markedly improved tractography results.

Tractography of the external capsule and cognition: A diffusion MRI study of cholinergic fibers.

Computing the orientation distribution function (ODF) from high angular resolution diffusion imaging (HARDI) signals makes it possible to determine the orientation of fiber bundles of the brain. The HARDI signals are samples measured from a spherical shell and thus require processing on the sphere. Past work on ODF estimation involved using the spherical harmonics or spherical radial basis functions. In this work, we propose three novel directional functions able to represent the measured signals in a very compact manner, i.e., they require very few parameters to completely describe the measured signal. Analytical expressions are derived for computing the corresponding ODF. The directional functions can represent diffusion in a particular direction and mixture models can be used to represent multi-fiber orientations. We show how to estimate the parameters of this mixture model and elaborate on the differences between these functions. We also compare this general framework with estimation of ODF using spherical harmonics on some real and synthetic data. The proposed method could be particularly useful in applications such as tractography and segmentation. Details are also given on different ways in which interpolation can be performed using directional functions. In particular, we discuss a complete Euclidean as well as a "hybrid" framework, comprising of the Riemannian as well as Euclidean spaces, to perform interpolation and compute geodesic distances between two ODF's.

High angular resolution diffusion imaging (HARDI) allows in vivo analysis of the white matter structure and connectivity. Based on orientation distribution functions (ODFs) that represent the directionality of water diffusion at each point in the brain, tractography methods can recover major axonal pathways. This enables tract-based analysis of fiber integrity and connectivity. For multi-subject comparisons, fibers may be clustered into bundles that are consistently found across subjects. To do this, we scanned 20 young adults with HARDI at 4 T. From the reconstructed ODFs, we performed whole-brain tractography with a novel Hough transform method. We then used measures of agreement between the extracted 3D curves and a co-registered probabilistic DTI atlas to select key pathways. Using median filtering and a shortest path graph search, we derived the maximum density path to compactly represent each tract in the population. With this tract-based method, we performed tract-based analysis of fractional anisotropy, and assessed how the chosen tractography algorithm influenced the results. The resulting method may expedite population-based statistical analysis of HARDI and DTI.

A new method for mapping diffusivity profiles in tissue is presented. The Bloch-Torrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized Stejskal-Tanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higher-rank tensors without using the computationally-difficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank-2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank-2 DT. The relationships between higher- and lower-rank Cartesian DTs are also presented. The inadequacy of the traditional rank-2 tensor model is demonstrated with simulations, and the method is applied to excised rat brain data collected in a spin-echo HARDI experiment.