High Angular Resolution Diffusion Imaging

We present a regularized method for solving an inverse problem in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. In the case of brain images, DT-MR imagery technique produces a tensor field that indicates the local orientation of nerve bundles. Now days, the spatial resolution of this technique is limited by the partial volume effect produced in voxels that contain fiber crossings or bifurcations. In this paper, we proposed a method for recovering the intra-voxel information and inferring the brain connectivity. We assume that the observed tensor is a linear combination of a given tensor basis, therefore, the aim of our approach is the computation of the unknown coefficients of this linear combination. By regularizing the problem, we introduce the needed prior information about the piecewise smoothness of nerve bundles orientation. As a result, we recover a multi-tensor field. Moreover, for estimating the nerve bundles trajectory, we propose a method based on stochastic walks of particles through the computed multi-tensor field. The performance of the method is demonstrated by experiments in both synthetic and real data.

Diffusion MRI (dMRI) is the unique technique to infer the microstructure of the white matter in vivo and noninvasively, by modeling the diffusion of water molecules. Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF) are two important Probability Density Functions (PDFs) which reflect the water diffusion. Estimation and processing of EAP and ODF is the central problem in dMRI, and is also the first step for tractography. Diffusion Tensor Imaging (DTI) is the most widely used estimation method which assumes EAP as a Gaussian distribution parameterized by a tensor. Riemannian framework for tensors has been proposed successfully in tensor estimation and processing. However, since the Gaussian EAP assumption is oversimplified, DTI can not reflect complex microstructure like fiber crossing. High Angular Resolution Diffusion Imaging (HARDI) is a category of methods proposed to avoid the limitations of DTI. Most HARDI methods like Q-Ball Imaging (QBI) need some assumptions and only can handle the data from single shell (single $b$ value), which are called as single shell HARDI (sHARDI) methods. However, with the development of scanners and acquisition methods, multiple shell data becomes more and more practical and popular. This thesis focuses on the estimation and processing methods in multiple shell HARDI (mHARDI) which can handle the diffusion data from arbitrary sampling scheme. There are many original contributions in this thesis. -First, we develop the analytical Spherical Polar Fourier Imaging (SPFI), which represents the signal using SPF basis and obtains EAP and its various features including ODFs and some scalar indices like Generalized Fractional Anisotropy (GFA) from analytical linear transforms. In the implementation of SPFI, we present two ways for scale estimation and propose to consider the prior $E(0)=1$ in estimation process. -Second, a novel Analytical Fourier Transform in Spherical Coordinate (AFT-SC) framework is proposed to incorporate many sHARDI and mHARDI methods, explore their relation and devise new analytical EAP/ODF estimation methods. -Third, we present some important criteria to compare different HARDI methods and illustrate their advantages and limitations. -Fourth, we propose a novel diffeomorphism invariant Riemannian framework for ODF and EAP processing, which is a natural generalization of previous Riemannian framework for tensors, and can be used for general PDF computing by representing the square root of the PDF called wavefunction with orthonormal basis. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms, the weighted Riemannian mean and median uniquely exist and can be estimated from an efficient gradient descent. Log-Euclidean framework and Affine-Euclidean framework are developed for fast data processing. -Fifth, we theoretically and experimentally compare the Euclidean metric and Riemannian metric for tensors, ODFs and EAPs. -Finally, we propose the Geodesic Anisotropy (GA) to measure the anisotropy of EAPs, Square Root Parameterized Estimation (SRPE) for nonnegative definite ODF/EAP estimation, weighted Riemannian mean/median for ODF/EAP interpolation, smoothing, atlas estimation. The concept of \emph{reasonable mean value interpolation} is presented for interpolation of general PDF data.

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulae and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diff...

Abstract: High angular resolution diffusion imaging (HARDI) has recently been of great interest in characterizing non-Gaussian diffusion processes. In the white matter of the brain, non-Gaussian diffusion occurs when fiber bundles cross, kiss or diverge within the same voxel. One important goal in current research is to obtain more accurate fits of the apparent diffusion processes in these multi- ple

High angular resolution diffusion imaging (HARDI) has proven to better characterize complex intra-voxel structures compared to its predecessor diffusion tensor imaging (DTI). However, the benefits from the modest acquisition costs and significantly higher signal-to-noise ratios (SNRs) of DTI make it more attractive for use in clinical research. In this work we use contextual information derived from DTI data, to obtain similar crossing information as from HARDI data. We conduct a synthetic phantom study under different angles of crossing and different SNRs. We compare the extrapolated crossings from contextual information with HARDI data. We qualitatively corroborate our findings from the phantom study to real human data. We show that with extrapolation of the contextual information, the obtained crossings are similar to the ones from the HARDI data, and the robustness to noise is significantly better.

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green's functions. We provide analytic approximation formulae and explicit sharp Gaussian esti- mates for these Green's functions. In our design and analysis for appropriate (non-linear) con...

The analysis and visualisation of Diffusion Tensor Images (DTI) is still a challenge since it is multi-valued and exploratory in na- ture: tensors, fiber tracts, bundles. This quickly leads to clutter problems in visualisation but also in analysis. In this paper, a new framework for the multi-resolution analysis of DTI is proposed. Based on fast and greedy watersheds operating on a multi-scale representation of a DTI image, a hierarchical depiction of a DTI image is determined conveying a global-to-local view of the fibrous structure of the analysed tissue. The multi-resolution watershed transform provides a coarse to fine partitioning of the data based on the (in)homogeneity of the gradient field. With a transversal cross scale linking of the basins (regions), a hierarchical representation is established. This framework besides providing a novel hierarchical way to analyse DTI data, allows a simple and interactive segmentation tool where dif- ferent bundles can be segmented at different resolutions. We present preliminary experimental results supporting the validity of the proposed method.

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 ac...

In this work we present a method for non-rigid registration of high angular resolution diffusion imaging (HARDI) datasets that are modeled by a field of antipodally symmetric spherical functions, represented by their expansion in the real spherical harmonic (RSH) basis. We use a multichannel demons algorithm which utilizes a computationally simple, rotationally invariant similarity function defined in the RSH space.

Background: The aim of this study was to investigate potential differences in neural structure in individuals with Asperger syndrome (AS), high-functioning individuals with autism spectrum disorder (ASD). The main symptoms of AS are severe impairments in social interactions and restricted or repetitive patterns of behaviors, interests or activities. Methods: Diffusion weighted magnetic resonance imaging data were acquired for 14 adult males with AS and 19 age, sex and IQ-matched controls. Voxelwise group differences in fractional anisotropy (FA) were studied with tract-based spatial statistics (TBSS). Based on the results of TBSS, a tract-level comparison was performed with constrained spherical deconvolution (CSD)-based tractography, which is able to detect complex (for example, crossing) fiber configurations. In addition, to investigate the relationship between the microstructural changes and the severity of symptoms, we looked for correlations between FA and the Autism Spectrum Quotient (AQ), Empathy Quotient and Systemizing Quotient. Results: TBSS revealed widely distributed local increases in FA bilaterally in individuals with AS, most prominent in the temporal part of the superior longitudinal fasciculus, corticospinal tract, splenium of corpus callosum, anterior thalamic radiation, inferior fronto-occipital fasciculus (IFO), posterior thalamic radiation, uncinate fasciculus and inferior longitudinal fasciculus (ILF). CSD-based tractography also showed increases in the FA in multiple tracts. However, only the difference in the left ILF was significant after a Bonferroni correction. These results were not explained by the complexity of microstructural organization, measured using the planar diffusion coefficient. In addition, we found a correlation between AQ and FA in the right IFO in the whole group. Conclusions: Our results suggest that there are local and tract-level abnormalities in white matter (WM) microstructure in our homogenous and carefully characterized group of adults with AS, most prominent in the left ILF.

In this article we develop a new method to segment Q-Ball imaging (QBI) data. We first estimate the orientation distribution function (ODF) using a fast and robust spherical harmonic (SH) method. Then, we use a region-based statistical surface evolution on this image of ODFs to efficiently find coherent white matter fiber bundles. We show that our method is appropriate to propagate through regions of fiber crossings and we show that our results outperform state-of-the-art diffusion tensor (DT) imaging segmentation methods, inherently limited by the DT model. Results obtained on synthetic data, on a biological phantom, on real datasets and on all 13 subjects of a public QBI database show that our method is reproducible, automatic and brings a strong added value to diffusion MRI segmentation.

We denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging. Diffusion imaging is a relatively new and powerful method to measure the 3D profile of water diffusion at each point. This can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. However, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter preselected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. We propose a variational meth...

We denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging. Diffusion imaging is a relatively new and powerful method to measure the 3D profile of water diffusion at each point. This can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. However, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter pre-selected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. We propose a variational method to denoise HARDI data by denoising the spherical Apparent Diffusion Coefficient (sADC), a field of radial functions derived from the data. We use vectorial total variation regularization, an L 1 data fidelity term and the logarithmic barrier function in the minimization. We present experiments of denoising synthetic and real HARDI data.

We present a regularized method for solving an inverse problem in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. In the case of brain images, DT-MR imagery technique produces a tensor field that indicates the local orientation of nerve bundles. Now days, the spatial resolution of this technique is limited by the partial volume effect produced in voxels that contain fiber crossings or bifurcations. In this paper, we proposed a method for recovering the intra-voxel information and inferring the brain connectivity. We assume that the observed tensor is a linear combination of a given tensor basis, therefore, the aim of our approach is the computation of the unknown coefficients of this linear combination. By regularizing the problem, we introduce the needed prior information about the piecewise smoothness of nerve bundles orientation. As a result, we recover a multi-tensor field. Moreover, for estimating the nerve bundles trajectory, we propose a method based on stochastic walks of particles through the computed multi-tensor field. The performance of the method is demonstrated by experiments in both synthetic and real data.

Human brain connectivity is disrupted in a wide range of disorders - from Alzheimer's disease to autism - but little is known about which specific genes affect it. Here we conducted a genome-wide association for connectivity matrices that capture information on the density of fiber connections between 70 brain regions. We scanned a large twin cohort (N=366) with 4-Tesla high angular resolution diffusion imaging (105-gradient HARDI). Using whole brain HARDI tractography, we extracted a relatively sparse 70×70 matrix representing fiber density between all pairs of cortical regions automatically labeled in co-registered anatomical scans. Additive genetic factors accounted for 1-58% of the variance in connectivity between 90 (of 122) tested nodes. We discovered genome-wide significant associations between variants and connectivity. GWAS permutations at various levels of heritability, and split-sample replication, validated our genetic findings. The resulting genes may offer new lead...