Barrick, TR; Spilling, CA; Hall, MG; Howe, FA
(2021)
The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging.
Mathematics, 9 (15).
p. 1763.
ISSN 2227-7390
https://doi.org/10.3390/math9151763
SGUL Authors: Barrick, Thomas Richard Howe, Franklyn Arron
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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.
Item Type: | Article | ||||||
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Additional Information: | Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). | ||||||
SGUL Research Institute / Research Centre: | Academic Structure > Molecular and Clinical Sciences Research Institute (MCS) | ||||||
Journal or Publication Title: | Mathematics | ||||||
ISSN: | 2227-7390 | ||||||
Language: | en | ||||||
Publisher License: | Creative Commons: Attribution 4.0 | ||||||
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URI: | https://openaccess.sgul.ac.uk/id/eprint/113489 | ||||||
Publisher's version: | https://doi.org/10.3390/math9151763 |
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