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

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  • Guoxing Lin

    (Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, USA)

Abstract

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.

Suggested Citation

  • Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:2:p:17-:d:129236
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    References listed on IDEAS

    as
    1. Lin, Guoxing, 2018. "General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 86-100.
    2. Balescu, R., 2007. "V-Langevin equations, continuous time random walks and fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 62-80.
    3. Farida Grinberg & Ezequiel Farrher & Luisa Ciobanu & Françoise Geffroy & Denis Le Bihan & N Jon Shah, 2014. "Non-Gaussian Diffusion Imaging for Enhanced Contrast of Brain Tissue Affected by Ischemic Stroke," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-15, February.
    4. Guido Germano & Mauro Politi & Enrico Scalas & Ren'e L. Schilling, 2008. "Stochastic calculus for uncoupled continuous-time random walks," Papers 0802.3769, arXiv.org, revised Jan 2009.
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    Cited by:

    1. Richard L. Magin & Hamid Karani & Shuhong Wang & Yingjie Liang, 2019. "Fractional Order Complexity Model of the Diffusion Signal Decay in MRI," Mathematics, MDPI, vol. 7(4), pages 1-16, April.

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