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General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

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  • Lin, Guoxing

Abstract

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion {0<α,β≤2} based on the fractional derivative, has not been obtained, where α and β are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. The theory and the CTRW simulation agree with each other. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:497:y:2018:i:c:p:86-100
    DOI: 10.1016/j.physa.2018.01.008
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    References listed on IDEAS

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    1. Balescu, R., 2007. "V-Langevin equations, continuous time random walks and fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 62-80.
    2. Lin, Guoxing, 2017. "Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 277-288.
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    Cited by:

    1. Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.

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