The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

Thomas R. Barrick, Catherine A. Spilling, Matt G. Hall and Franklyn A. Howe
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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
Franklyn A. Howe: Neurosciences Research Centre, Molecular and Clinical Sciences Research Institute, St George’s University of London, London SW17 0RE, UK

Mathematics, 2021, vol. 9, issue 15, 1-23

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 ( ? ) and spatial ( ? ) fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, D (in mm 2 s ?1 ), and a fractional exponent, ? . 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.

Keywords: fractional calculus; continuous time random walk; diffusion magnetic resonance imaging; non-Gaussian diffusion; quasi-diffusion imaging; quasi-diffusion model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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