Analysis of PFG anomalous diffusion via real-space and phase-space approaches. (English) Zbl 1459.78010
Summary: 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.
MSC:
| 78A55 | Technical applications of optics and electromagnetic theory |
| 78A70 | Biological applications of optics and electromagnetic theory |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 92C55 | Biomedical imaging and signal processing |
| 26A33 | Fractional derivatives and integrals |
| 33E12 | Mittag-Leffler functions and generalizations |
| 35R11 | Fractional partial differential equations |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
pulsed-field gradient (PFG) anomalous diffusion; fractional derivative; nuclear magnetic resonance (NMR); magnetic resonance imaging (MRI)References:
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