Mathematical description of q‐space in spherical coordinates: Exact q‐ball imaging

Abstract

Novel methodologies have been recently developed to characterize the microgeometry of neural tissues and porous structures via diffusion MRI data. In line with these previous works, this article provides a detailed mathematical description of q-space in spherical coordinates that helps to highlight the differences and similarities between various related q-space methodologies proposed to date such as q-ball imaging (QBI), diffusion spectrum imaging (DSI), and diffusion orientation transform imaging (DOT). This formulation provides a direct relationship between the orientation distribution function (ODF) and the diffusion data without using any approximation. Under this relationship, the exact ODF can be computed by means of the Radon transform of the radial projection (in q-space) of the diffusion MRI signal. This new methodology, termed exact q-ball imaging (EQBI), was put into practice using an analytical ODF estimation in terms of spherical harmonics that allows obtaining model-free and model-based reconstructions. This work provides a new framework for combining information coming from diffusion data recorded on multiple spherical shells in q-space (hybrid diffusion imaging encoding scheme), which is capable of mapping ODF to a high accuracy. This represents a step toward a more efficient development of diffusion MRI experiments for obtaining better ODF estimates. Diffusion MRI (dMRI) data are routinely used for studying molecular diffusion in fluids (1). When the diffusion is constrained by the presence of obstacles, dMRI experiments yield information about the confining geometry (2). Recently, there has been a large amount of interest in using dMRI techniques to obtain information about the microgeometry of porous structures and biological cells, which is supported by the noninvasive nature of these measurements. Also, dMRI techniques are able to probe the pore geometry over a range of length scales not accessible to standard techniques like x-ray and small-angle neutron scattering.Diffusion tensor imaging (DTI) is the first dMRI technique that has been proposed to quantify in vivo the anisotropy of the water self-diffusion process in fibrous tissues (3). This approach allows estimating a second-order self-diffusion tensor D from a series of dMRI data without requiring exogenous contrast agents. DTI provides relevant information that is not available from other imaging modalities; however, this technique can only resolve a single fiber direction within each voxel.In the last few years a number of novel methodologies have been developed to characterize more accurately the microgeometry of neural tissues (for a partial list of such publications, see . Among these advanced methods and algorithms, q-ball imaging (QBI) (6) and diffusion spectrum imaging (DSI) (11) have generated considerable interest due to a number of benefits such as model independence, theoretical soundness, and the ability to resolve intravoxel orientational heterogeneity.In DSI the molecular displacement probability density function (also ...

Quantitative Comparison of Reconstruction Methods for Intra-Voxel Fiber Recovery From Diffusion MRI

“…Diagonal elements of D are the diffusivities along the main axis of the fiber (λ 1 ) and in the plane perpendicular to it (λ 2 , λ 3 ). In this contest, the diffusivities were generated from the following ranges typically observed in the human brain [60]:…”

“…Interestingly, Tournier et al [48] have shown that numerical simulations performed using these models agree well with physical phantoms but, as already mentioned, they represent an over-simplification of what happens in the white matter. Even though the level of complexity they offer is way too simplistic with respect to the real architecture of the nervous system, they actually constitute the validation framework of choice for the majority of the algorithms proposed in the literature, as in [8], [17], [20], [23], [27], [28], [33], [34], [37], [59], [60] to mention a few. Unfortunately, when a new algorithm is proposed, the performances are normally assessed with ad-hoc synthetic data and evaluation criteria, and comparing different approaches can be difficult.…”

Beyond fractional anisotropy: Extraction of bundle-specific structural metrics from crossing fiber models

“…These methods do not require any prior assumptions on the number of underlying fiber bundles. They include Q-Ball Imaging (Tuch, 2004, Barnett, 2009, Canales-Rodríguez et al, 2009, Tristán-Vega et al, 2009, Aganj et al, 2010 approximating the diffusion orientation density function (dODF) and Spherical Deconvolution (SD) (Tournier et al, 2004, Dell'Acqua et al, 2007, Kaden et al, 2007, Tournier et al, 2007 modeling the fiber orientation density function (fODF). The fODF represents the direction dependent density of fibers in every voxel and therefore is an angular spatial fiber density.…”

Denoising and fast diffusion imaging with physically constrained sparse dictionary learning

“…In every voxel of the dataset, the diffusion signal corresponding to the underlying fiber configuration is simulated according to the same gradient list as our real DSI data. The diffusion signal, S(q), is simulated using the classical Gaussian mixture model (Tuch, 2004;Descoteaux et al, 2007;Canales-Rodríguez et al, 2009):…”

“…In the contest data, E 0 = 1 without loss of generality and diffusivities were generated using symmetric tensors, D i = diag(λ 1 , λ 2 , λ 3 ), in the range of λ 1 ∈ [1, 2] × 10 −3 mm 2 /s and λ 2 = λ 3 ∈ [0.1, 0.6] × 10 −3 mm 2 /s, as done in (Canales-Rodríguez et al, 2009). The datasets were corrupted with additive Rician noise, .…”