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.
Funder
Innovate UK
St George's, Univeristy of London
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference88 articles.
1. Quasi-diffusion magnetic resonance imaging (QDI): A fast, high b-value diffusion imaging technique
2. The random walk's guide to anomalous diffusion: a fractional dynamics approach
3. Continuous Time Random Walk, Mittag–Leffler Waiting Time and Fractional Diffusion: Mathematical Aspects;Gorenflo,2008
4. Continuous time random walks and space-time fractional differential equations;Meerschaert,2019
5. Applications to Stochastic Models;Gorenflo,2020