IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v467y2017icp277-288.html
   My bibliography  Save this article

Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model

Author

Listed:
  • Lin, Guoxing

Abstract

Pulsed field gradient (PFG) technique is a noninvasive tool, and has been increasingly employed to study anomalous diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). However, the analysis of PFG anomalous diffusion is much more complicated than normal diffusion. In this paper, a fractal derivative model based modified Gaussian phase distribution method is proposed to describe PFG anomalous diffusion. By using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives, and employing some of the traditional Gaussian phase distribution approximation techniques, a general signal attenuation expression for free fractional diffusion is derived. This expression describes a stretched exponential function based attenuation, which is distinct from both the exponential attenuation for normal diffusion obtained from conventional Gaussian phase distribution approximation, and the Mittag-Leffler function based attenuation for anomalous diffusion obtained from fractional derivative. The obtained signal attenuation expression can analyze the finite gradient pulse width (FGPW) effect. Additionally, it can generally be applied to all three types of PFG fractional diffusions classified based on time derivative order α and space derivative order β. These three types of fractional diffusions include time-fractional diffusion with {0<α≤2,β=2}, space-fractional diffusion with {α=1,0<β≤2}, and general fractional diffusion with {0<α,β≤2}. The results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method. This method provides a new, convenient approximation formalism for analyzing PFG anomalous diffusion experiments. The expression that can simultaneously interpret general fractional diffusion and FGPW effect could be especially important in PFG MRI, where the narrow gradient pulse limit cannot be satisfied.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:467:y:2017:i:c:p:277-288
    DOI: 10.1016/j.physa.2016.10.036
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437116307233
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/j.physa.2016.10.036?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Metzler, Ralf & Barkai, Eli & Klafter, Joseph, 1999. "Anomalous transport in disordered systems under the influence of external fields," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 266(1), pages 343-350.
    2. 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.
    3. Lenzi, E.K. & dos Santos, M.A.F. & Vieira, D.S. & Zola, R.S. & Ribeiro, H.V., 2016. "Solutions for a sorption process governed by a fractional diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 443(C), pages 32-41.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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. Pereira-Alves, Felipe & Soares-Pinto, Diogo O. & Paiva, Fernando F., 2024. "NMR diffusion in restricted environment approached by a fractional Langevin model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 641(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Caicedo, Alejandro & Cuevas, Claudio & Mateus, Éder & Viana, Arlúcio, 2021. "Global solutions for a strongly coupled fractional reaction-diffusion system in Marcinkiewicz spaces," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    2. Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.
    3. Halley Gomes & Arlúcio Viana, 2021. "Existence, symmetries, and asymptotic properties of global solutions for a fractional diffusion equation with gradient nonlinearity," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-30, February.
    4. Georgios C Manikis & Kostas Marias & Doenja M J Lambregts & Katerina Nikiforaki & Miriam M van Heeswijk & Frans C H Bakers & Regina G H Beets-Tan & Nikolaos Papanikolaou, 2017. "Diffusion weighted imaging in patients with rectal cancer: Comparison between Gaussian and non-Gaussian models," PLOS ONE, Public Library of Science, vol. 12(9), pages 1-15, September.
    5. Lenzi, E.K. & Menechini Neto, R. & Tateishi, A.A. & Lenzi, M.K. & Ribeiro, H.V., 2016. "Fractional diffusion equations coupled by reaction terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 9-16.
    6. Satin, Seema E. & Parvate, Abhay & Gangal, A.D., 2013. "Fokker–Planck equation on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 30-35.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:467:y:2017:i:c:p:277-288. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.