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Barrick, T R; Spilling, C A; Hall, M G; Howe, F A (2021) The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging. Mathematics, 9 (15). 1763
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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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| Keywords: | fractional calculus; continuous time random walk; diffusion magnetic resonance imaging; non-Gaussian diffusion; quasi-diffusion imaging; quasi-diffusion model |
| Subjects: | Mathematics and Scientific Computing > Modelling |
| Divisions: | Medical, Marine & Nuclear |
| Identification number/DOI: | 10.3390/math9151763 |
| Last Modified: | 24 May 2023 13:13 |
| URI: | http://eprintspublications.npl.co.uk/id/eprint/9473 |
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