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Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, USA
Mathematics 2018, 6(2), 17; https://doi.org/10.3390/math6020017
Received: 12 December 2017 / Revised: 17 January 2018 / Accepted: 22 January 2018 / Published: 29 January 2018
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion. View Full-Text
Keywords: pulsed-field gradient (PFG) anomalous diffusion; fractional derivative; nuclear magnetic resonance (NMR); magnetic resonance imaging (MRI) pulsed-field gradient (PFG) anomalous diffusion; fractional derivative; nuclear magnetic resonance (NMR); magnetic resonance imaging (MRI)
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Lin, G. Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches. Mathematics 2018, 6, 17.

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