NDLI: The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

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The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

Content Provider	MDPI
Author	Barrick, Thomas Spilling, Catherine Hall, Matt Howe, Franklyn
Copyright Year	2021
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 mm² 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.
Starting Page	1763
e-ISSN	22277390
DOI	10.3390/math9151763
Journal	Mathematics
Issue Number	15
Volume Number	9
Language	English
Publisher	MDPI
Publisher Date	2021-07-26
Access Restriction	Open
Subject Keyword	Mathematics Neurosciences Fractional Calculus Continuous Time Random Walk Diffusion Magnetic Resonance Imaging Non-gaussian Diffusion Quasi-diffusion Imaging Quasi-diffusion Model
Content Type	Text
Resource Type	Article

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