A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging

Liang, Yingjie; Ye, Allen Q.; Chen, Wen; Gatto, Rodolfo G.; Colon-Perez, Luis M.; Mareci, Thomas H.

doi:10.1016/j.cnsns.2016.04.006

Cited by 103 publications

(45 citation statements)

References 27 publications

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“…Differentiation and/or integration on fractals are reported in [7,12,20,27]. In conjunction with the main result of the present work, it is noteworthy that [27] introduces a local non-integer order calculus on Cantor set, which (in its general topological definition) is a prototype of a famous and mostly observed pattern.…”

Section: Introductionsupporting

confidence: 52%

Non-differentiability and fractional differentiability on timescales

2017

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This work deals with concepts of non-differentiability and a non-integer order differential on timescales. Through an investigation of a local non-integer order derivative on timescales, a mean value theorem (a fractional analog of the mean value theorem on timescales) is presented. Then, by illustrating a vanishing property of this derivative, its objectivity is discussed. As a first-hand result, the potentials and capability of this fractional derivative connected to nonsmooth analysis, including non-differentiable paths and a class of self-similar fractals, are stated. It is stated that the non-integer order derivative never vanishes almost everywhere. It has been shown that with the help of changing the order of differentiability on a q-timescale, the non-differentiability disappears.

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Section: Introductionsupporting

confidence: 52%

Non-differentiability and fractional differentiability on timescales

2017

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“…In recent years, fractional calculus has arisen as a powerful and robust theoretical framework in order to account for the effects of anomalous diffusion in MRI [7][8][9][10][11][12]. The main advantage of such an approach is that insights can A C C E P T E D M A N U S C R I P T be derived from generalisations of the physical principles describing the magnetisation of water protons in MRI: the Bloch-Torrey equation [13].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

Bueno‐Orovio

Burrage

2017

Communications in Nonlinear Science and Numerical Simulation

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Highlights• The full fractional Bloch-Torrey setting for magnetic resonance imaging is solved.• Solutions are given by convolution of extended Mittag-Leffler/Lauricella functions.• Analytical solutions replicate super-and sub-diffusive regimes in signal decay.• Residual signal phase shifts due to incomplete spin refocusing are also captured.• This allows an estimation of tissue properties based on exact diffusive processes. AbstractThe quantification of anomalous diffusion is increasingly being recognised as an advanced modality of analysis for the evaluation of tissue microstructure in magnetic resonance imaging (MRI). One powerful framework to account for anomalous diffusion in biological and structurally heterogeneous tissues is the use of diffusion operators based on fractional calculus theory, which generalises the physical principles of standard diffusion in homogeneous media. However, their non-locality makes analytical solutions often unavailable, limiting the applicability of these modelling and analysis techniques. In this paper, we derive compact analytical signal decays for practical MRI sequences in the anisotropic fractional Bloch-Torrey setting, as described by the space fractional Laplacian and importantly the time Caputo derivative. The attained solutions convey relevant characteristics of MRI in biological tissues not replicated by standard diffusion, including super-diffusive and sub-diffusive regimes in signal decay and the diffusion-driven incomplete refocusing of spins at the end of the sequence. These results may therefore have significant implications for advancing the current interpretation of MRI, and for the estimation of tissue properties based on exact solutions to underlying diffusive processes.

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“…Classical Brownian motion theory was extended by generalizing the Gaussian probability density function for situations where a mismatch between the scale at which measurements are taken and the scale at which changes occur is present. Fractional calculus (calculus using non‐integer derivatives) provides a mechanistic approach for formulating equations for such systems and equations representing anomalous diffusion have been developed using non‐integer derivatives [Khanafer et al, ; Magin et al, ; Norris, ] allowing anomalous dynamics in MRI to be characterised [Ingo et al, ; Kimmich, ; Köpf et al, ; Liang et al, ; Magin et al, ; Magin et al, ; Magin et al, ; Stapf et al, ; Stapf, ].…”

Section: Introductionmentioning

confidence: 99%