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The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

Author

Listed:
  • Thomas R. Barrick

    (Neurosciences Research Centre, Molecular and Clinical Sciences Research Institute, St George’s University of London, London SW17 0RE, UK)

  • Catherine A. Spilling

    (Neurosciences Research Centre, Molecular and Clinical Sciences Research Institute, St George’s University of London, London SW17 0RE, UK)

  • Matt G. Hall

    (National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK
    UCL GOS Institute of Child Health, University College London, London WC1E 6BT, UK)

  • Franklyn A. Howe

    (Neurosciences Research Centre, Molecular and Clinical Sciences Research Institute, St George’s University of London, London SW17 0RE, UK)

Abstract

Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion dynamics and assumes water diffusion is locally Gaussian within tissue microstructure. By assuming a Gaussian scaling relationship between temporal ( α ) and spatial ( β ) fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, D (in mm 2 s −1 ), and a fractional exponent, α . Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development.

Suggested Citation

  • Thomas R. Barrick & Catherine A. Spilling & Matt G. Hall & Franklyn A. Howe, 2021. "The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging," Mathematics, MDPI, vol. 9(15), pages 1-23, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1763-:d:601618
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    References listed on IDEAS

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    1. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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