Parametric spherical deconvolution: Inferring anatomical connectivity using diffusion MR imaging

Abstract

The human brain forms a complex neural network with a connectional architecture that is still far from being known in full detail, even at the macroscopic level. The advent of diffusion MR imaging has enabled the exploration of the structural properties of white matter in vivo. In this article we propose a new forward model that maps the microscopic geometry of nervous tissue onto the water diffusion process and further onto the measured MR signals. Our spherical deconvolution approach completely parameterizes the fiber orientation density by a finite mixture of Bingham distributions. In addition, we define the term anatomical connectivity, taking the underlying image modality into account. This neurophysiological metric may represent the proportion of the nerve fibers originating in the source area which intersect a given target region. The specified inverse problem is solved by Bayesian statistics. Posterior probability maps denote the probability that the connectivity value exceeds a chosen threshold, conditional upon the noisy observations. These maps allow us to draw inferences about the structural organization of the cerebral cortex. Moreover, we will demonstrate the proposed approach with diffusion-weighted data sets featuring high angular resolution.

Quantitative mapping of the per‐axon diffusion coefficients in brain white matter

“…For example, the centrum semiovale has occasionally a fractional anisotropy lower than 0.2, suggesting almost isotropic diffusion at the voxel‐resolution level. In fact, this brain region has a complex orientational structure with crossings of the pyramidal tract, the callosal fibers, and the superior longitudinal fasciculus 10, 35. The mean diffusivity of an axonal segment and the quantity derived from the classical tensor model share a similar image contrast, but from a theoretical viewpoint the per‐axon and fiber‐population mean diffusivities are in general two different indices.…”

“…Eventually, the spherical mean of the diffusion signal with the parametric response model (7) reads $\begin{array}{l}{\overline{e}}_b({\lambda }_{\parallel },{\lambda }_{\perp })=\exp {(-b{\lambda }_{\perp })}_1{F}_1(1/2;3/2;-b({\lambda }_{\parallel }-{\lambda }_{\perp }))\\ =\exp (-b{\lambda }_{\perp })\frac{\sqrt{\pi }\text{erf}(\sqrt{b({\lambda }_{\parallel }-{\lambda }_{\perp })})}{2\sqrt{b({\lambda }_{\parallel }-{\lambda }_{\perp })}},\end{array}$where ${}_1{F}_1$ denotes the confluent hypergeometric function and erf is the error function. Equation [8] has been derived before 10, 14, 35, but was used in a different context, that is, for the recovery of the fiber orientation distribution.…”

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

“…They seek to decompose the diffusion propagator into a number of sub-components, each associated with an underlying roughly collinear fiber bundle. Alternatively, one can also try to model the fiber density of the bundles directly, for example by representing the angular fiber density of each bundle by a Bingham or 4 Watson distribution and then convolve it with the (assumed) signal of a single fiber (Kaden et al, 2007;Zhang et al, 2012;Sotiropoulos et al, 2012).…”

“…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.…”

Studying neuroanatomy using MRI

“…Nevertheless, fiber dispersion is represented explicitly in more recent multi-compartment white matter models (Fig 6D) that relax the assumption of orientation homogeneity (e.g. 145,146 ). Dispersion models are also increasingly being used to make inferences about grey matter compartments that collectively represent neurites, i.e.…”