Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

Abstract

Highlights• The full fractional Bloch-Torrey setting for magnetic resonance imaging is solved.• Solutions are given by convolution of extended Mittag-Leffler/Lauricella functions.• Analytical solutions replicate super-and sub-diffusive regimes in signal decay.• Residual signal phase shifts due to incomplete spin refocusing are also captured.• This allows an estimation of tissue properties based on exact diffusive processes. AbstractThe quantification of anomalous diffusion is increasingly being recognised as an advanced modality of analysis for the evaluation of tissue microstructure in magnetic resonance imaging (MRI). One powerful framework to account for anomalous diffusion in biological and structurally heterogeneous tissues is the use of diffusion operators based on fractional calculus theory, which generalises the physical principles of standard diffusion in homogeneous media. However, their non-locality makes analytical solutions often unavailable, limiting the applicability of these modelling and analysis techniques. In this paper, we derive compact analytical signal decays for practical MRI sequences in the anisotropic fractional Bloch-Torrey setting, as described by the space fractional Laplacian and importantly the time Caputo derivative. The attained solutions convey relevant characteristics of MRI in biological tissues not replicated by standard diffusion, including super-diffusive and sub-diffusive regimes in signal decay and the diffusion-driven incomplete refocusing of spins at the end of the sequence. These results may therefore have significant implications for advancing the current interpretation of MRI, and for the estimation of tissue properties based on exact solutions to underlying diffusive processes.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

“…The authors in [17] have made a series of arguments based on Riesz potential theory and the wide range of heterogeneous cardiac tissues for the use of fractional models in cardiac electrophysiology, while Cusimano et al [18] have explored the effects of fractional diffusion on the cardiac action potential shape and electrical propagation. In addition, Magin et al [34], Hori et al [28], Bueno-Orovio and Burrage [15] have considered the calibration of fractional cardiac models through a fractional Bloch-Torrey equation.…”