MASiVar: Multisite, Multiscanner, and Multisubject Acquisitions for Studying Variability in Diffusion Weighted Magnetic Resonance Imaging

Diffusion weighted imaging (DWI) allows investigators to identify microstructural differences between subjects, but variability due to session and scanner biases is still a challenge. To investigate DWI variability, we present MASiVar, a multisite dataset consisting of 319 diffusion scans acquired at 3T from b = 1000 to 3000 s/mm2 across 97 different healthy subjects and four different scanners as a publicly available, preprocessed, and de-identified dataset. With these data we characterize variability on the intrasession intrascanner (N = 158), intersession intrascanner (N = 328), intersession interscanner (N = 53), and intersubject intrascanner (N = 80) levels. Our baseline analysis focuses on four common DWI processing approaches: (1) a tensor signal representation, (2) a multi-compartment neurite orientation dispersion and density model, (3) white matter bundle segmentation, and (4) structural connectomics. Respectively, we evaluate region-wise fractional anisotropy (FA), mean diffusivity, and principal eigenvector; region-wise cerebral spinal fluid volume fraction, intracellular volume fraction, and orientation dispersion index; bundle-wise shape, volume, length and FA; and connectome correlation and maximized modularity, global efficiency, and characteristic path length. We plot the scan/re-scan discrepancies in these measures at each level and find that variability generally increases with intrasession to intersession to interscanner to intersubject effects and that sometimes interscanner variability can approach intersubject variability. This baseline study suggests harmonization between scanners for multisite analyses is critical prior to inference of group differences on subjects and demonstrates the potential of MASiVar to investigate DWI variability across multiple levels and processing approaches simultaneously.

Higher spectral flow for Dirac operators with local boundary conditions

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.

An interesting example for spectral invariants

In [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULAS

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have "geometric K-theory", namely the "transmission algebra" introduced by Boutet de Monvel [5], the "zero algebra" introduced by Mazzeo in [9, 10] and the "scattering algebra" from [16], we give explicit formulas for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fiber operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles

When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.