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2015 | OriginalPaper | Buchkapitel

Finslerian Diffusion and the Bloch–Torrey Equation

verfasst von : T. C. J. Dela Haije, A. Fuster, L. M. J. Florack

Erschienen in: Visualization and Processing of Higher Order Descriptors for Multi-Valued Data

Verlag: Springer International Publishing

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Abstract

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings.An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownian motion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal.

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Vorheriges Kapitel Diffusion-Weighted Magnetic Resonance Signal for General Gradient Waveforms: Multiple Correlation Function Framework, Path Integrals, and Parallels Between Them

Nächstes Kapitel Fiber Orientation Distribution Functions and Orientation Tensors for Different Material Symmetries

We only consider reversible Finsler functions, meaning that for all \((\mathbf{x},\mathbf{y}) \in \mathit{TM}\), we have \(F(\mathbf{x},-\mathbf{y}) = F(\mathbf{x},\mathbf{y})\).

Different gradient sequences can be used to find the coefficients {D ^ij}, based on modified (but similar) versions of Eq. (8).

In conformity with diffusion MRI literature we omit the diacritical mark above the covectors \(\mathbf{q}\) and \(\mathbf{G}\).

There are two diffusion MRI models called generalized DTI, one by Özarslan et al. [34] and one by Liu et al. [27]. Whenever we refer to GDTI we mean the former.

Since we only report results of a single experiment we provide only the b-value. A more extensive analysis should consider the influence of the different parameters δ, Δ, and \(\|\mathbf{G}\|\) separately.

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Titel: Finslerian Diffusion and the Bloch–Torrey Equation
verfasst von: T. C. J. Dela Haije
A. Fuster
L. M. J. Florack
Verlag: Springer International Publishing
Buch: Visualization and Processing of Higher Order Descriptors for Multi-Valued Data
Print ISBN: 978-3-319-15089-5

Electronic ISBN: 978-3-319-15090-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-15090-1_2

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