Ensemble Average Propagator Reconstruction Research Articles

A geometric framework for ensemble average propagator reconstruction from diffusion MRI.

Diffusion-weighted magnetic resonance imaging (dMRI) is a non-invasive technique to probe the complex micro-architecture of the tissue being imaged. The diffusional properties of the tissue at the imaged resolution are well captured by the ensemble average propagator (EAP), which is a probability density function characterizing the probability of water molecule diffusion. Many properties in the form of imaging 'stains' can then be computed from the EAP that can serve as bio-markers for a variety of diseases. This motivates the development of methods for the accurate estimation of the EAPs from dMRI, which is an actively researched area in dMRI analysis. To this end, in the recent past, dictionary learning (DL) techniques have been applied by many researchers for accurate reconstruction of the EAP fields from dMRI scans of the central nervous system (CNS). However, most of the DL-based methods did not exploit the geometry of the space of the EAPs, which are probability density functions. By exploiting the geometry of the space of probability density functions, it is possible to reconstruct EAPs that satisfy the mathematical properties of a density function and hence yield better accuracy in the EAP field reconstruction. Using a square root density parameterization, the EAPs can be mapped to a unit Hilbert sphere, which is a smooth manifold with well known geometry that we will exploit in our formulation of the DL problem. Thus, in this paper, we present a general formulation of the DL problem for data residing on smooth manifolds and in particular the manifold of EAPs, along with a numerical solution using an alternating minimization method. We then showcase the properties and the performance of our algorithm on the reconstruction of the EAP field in a patch-wise manner from the dMRI data. Through several synthetic, phantom and real data examples, we demonstrate that our non-linear DL-based approach produces accurate and spatially smooth estimates of the EAP field from dMRI in comparison to the state-of-the-art EAP reconstruction method called the MAPL method, as well as the linear DL-based EAP reconstruction approaches. To further demonstrate the accuracy and utility of our approach, we compute an entropic anisotropy measure (HA), that is a function of the well known Rényi entropy, from the EAP fields of control and injured rat spinal cords respectively. We demonstrate its utility as an imaging 'stain' via a quantitative comparison of HA maps computed from EAP fields estimated using our method and competing methods. The quantitative comparison is achieved using a two sample t-test and the results of significance are displayed for a visualization of regions of the spinal cord affected most by the injury.

Joint 6D k-q Space Compressed Sensing for Accelerated High Angular Resolution Diffusion MRI.

High Angular Resolution Diffusion Imaging (HARDI) avoids the Gaussian. diffusion assumption that is inherent in Diffusion Tensor Imaging (DTI), and is capable of characterizing complex white matter micro-structure with greater precision. However, HARDI methods such as Diffusion Spectrum Imaging (DSI) typically require significantly more signal measurements than DTI, resulting in prohibitively long scanning times. One of the goals in HARDI research is therefore to improve estimation of quantities such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF) with a limited number of diffusion-weighted measurements. A popular approach to this problem, Compressed Sensing (CS), affords highly accurate signal reconstruction using significantly fewer (sub-Nyquist) data points than required traditionally. Existing approaches to CS diffusion MRI (CS-dMRI) mainly focus on applying CS in the q-space of diffusion signal measurements and fail to take into consideration information redundancy in the k-space. In this paper, we propose a framework, called 6-Dimensional Compressed Sensing diffusion MRI (6D-CS-dMRI), for reconstruction of the diffusion signal and the EAP from data sub-sampled in both 3D k-space and 3D q-space. To our knowledge, 6D-CS-dMRI is the first work that applies compressed sensing in the full 6D k-q space and reconstructs the diffusion signal in the full continuous q-space and the EAP in continuous displacement space. Experimental results on synthetic and real data demonstrate that, compared with full DSI sampling in k-q space, 6D-CS-dMRI yields excellent diffusion signal and EAP reconstruction with low root-mean-square error (RMSE) using 11 times less samples (3-fold reduction in k-space and 3.7-fold reduction in q-space).

Regularized spherical polar fourier diffusion MRI with optimal dictionary learning.

Compressed Sensing (CS) takes advantage of signal sparsity or compressibility and allows superb signal reconstruction from relatively few measurements. Based on CS theory, a suitable dictionary for sparse representation of the signal is required. In diffusion MRI (dMRI), CS methods proposed for reconstruction of diffusion-weighted signal and the Ensemble Average Propagator (EAP) utilize two kinds of Dictionary Learning (DL) methods: 1) Discrete Representation DL (DR-DL), and 2) Continuous Representation DL (CR-DL). DR-DL is susceptible to numerical inaccuracy owing to interpolation and regridding errors in a discretized q-space. In this paper, we propose a novel CR-DL approach, called Dictionary Learning - Spherical Polar Fourier Imaging (DL-SPFI) for effective compressed-sensing reconstruction of the q-space diffusion-weighted signal and the EAP. In DL-SPFI, a dictionary that sparsifies the signal is learned from the space of continuous Gaussian diffusion signals. The learned dictionary is then adaptively applied to different voxels using a weighted LASSO framework for robust signal reconstruction. Compared with the start-of-the-art CR-DL and DR-DL methods proposed by Merlet et al. and Bilgic et al., respectively, our work offers the following advantages. First, the learned dictionary is proved to be optimal for Gaussian diffusion signals. Second, to our knowledge, this is the first work to learn a voxel-adaptive dictionary. The importance of the adaptive dictionary in EAP reconstruction will be demonstrated theoretically and empirically. Third, optimization in DL-SPFI is only performed in a small subspace resided by the SPF coefficients, as opposed to the q-space approach utilized by Merlet et al. We experimentally evaluated DL-SPFI with respect to L1-norm regularized SPFI (L1-SPFI), which uses the original SPF basis, and the DR-DL method proposed by Bilgic et al. The experiment results on synthetic and real data indicate that the learned dictionary produces sparser coefficients than the original SPF basis and results in significantly lower reconstruction error than Bilgic et al.'s method.

Bessel Fourier Orientation Reconstruction (BFOR): An analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices

The ensemble average propagator (EAP) describes the 3D average diffusion process of water molecules, capturing both its radial and angular contents. The EAP can thus provide richer information about complex tissue microstructure properties than the orientation distribution function (ODF), an angular feature of the EAP. Recently, several analytical EAP reconstruction schemes for multiple q-shell acquisitions have been proposed, such as diffusion propagator imaging (DPI) and spherical polar Fourier imaging (SPFI). In this study, a new analytical EAP reconstruction method is proposed, called Bessel Fourier Orientation Reconstruction (BFOR), whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition, and is validated on both synthetic and real datasets. A significant portion of the paper is dedicated to comparing BFOR, SPFI, and DPI using hybrid, non-Cartesian sampling for multiple b-value acquisitions. Ways to mitigate the effects of Gibbs ringing on EAP reconstruction are also explored. In addition to analytical EAP reconstruction, the aforementioned modeling bases can be used to obtain rotationally invariant q-space indices of potential clinical value, an avenue which has not yet been thoroughly explored. Three such measures are computed: zero-displacement probability (Po), mean squared displacement (MSD), and generalized fractional anisotropy (GFA).

Multiple q-shell diffusion propagator imaging

Many recent high angular resolution diffusion imaging (HARDI) reconstruction techniques have been introduced to infer an orientation distribution function (ODF) of the underlying tissue structure. These methods are more often based on a single-shell (one b-value) acquisition and can only recover angular structure information contained in the ensemble average propagator (EAP) describing the three-dimensional (3D) average diffusion process of water molecules. The EAP can thus provide richer information about complex tissue microstructure properties than the ODF by also considering the radial part of the diffusion signal. In this paper, we present a novel technique for analytical EAP reconstruction from multiple q-shell acquisitions. The solution is based on a Laplace equation by part estimation between the diffusion signal for each shell acquisition. This simplifies greatly the Fourier integral relating diffusion signal and EAP, which leads to an analytical, linear and compact EAP reconstruction. An important part of the paper is dedicated to validate the diffusion signal estimation and EAP reconstruction on real datasets from ex vivo phantoms. We also illustrate multiple q-shell diffusion propagator imaging (mq-DPI) on a real in vivo human brain and perform a qualitative comparison against state-of-the-art diffusion spectrum imaging (DSI) on the same subject. mq-DPI is shown to reconstruct robust EAP from only several different b-value shells and less diffusion measurements than DSI. This opens interesting perspectives for new q-space sampling schemes and tissue microstructure investigation.