A new anisotropic fractional model of diffusion suitable for applications of diffusion tensor imaging in biological tissues

Abstract

An anomalous anisotropic diffusion equation is constructed in which the order of the spatial pseudodifferential operator is generalized to be distributed with a directionally dependent distribution. A time fractional version of this equation is also considered. First, it is proved that the equation is positivitypreserving and properly normalized. Second, the existence of a smooth Green's function solution is proved. Finally, an expression for the diffusive flux density for this new fractional order process is calculated. This approach may find utility in modelling diffusion tensor imaging data in the white matter of the human brain where both the apparent diffusion coefficient and the order of the pseudo-differential operator are anisotropic.

Anisotropic fractional diffusion tensor imaging

“…Then one defines Q f ( x ) to be the function with Fourier transform Q̂ ( k ) f̂ ( k ). Hanyga and Magin (2014) continue to prove that solutions to (5.1) remain nonnegative for a nonnegative initial condition p ( x , 0) = f ( x ) ≥ 0. Next we provide an alternative proof of this fact, by showing that the solutions to (5.1) are the probability densities of a Lévy process (Sato, 1999; Meerschaert and Sikorskii, 2012, Section 4.3).…”

“…The paper of Hanyga and Magin (2014) also proposed a time-fractional version of their anisotropic fractional diffusion model. Define the pseudo-differential operator Q on the space C 0 (ℝ d ) of smooth functions with compact support (i.e.…”

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

“…On the contrary, analytical solutions to the time-fractional setting have only been presented in the absence of diffusion [31], under the simplest case of an applied constant magnetic gradient and with no-crossed terms [7], or for modified versions of the Bloch-Torrey equation [8,32]. Even in the latter case, Stejskal-Tanner formulas for the acquired signals were not presented given their complexity.…”

“…The non-local nature of the Caputo operator poses a number of challenges in the solution of this problem. For solutions of the form S (r, t) = S 0 A(t)ϕ(r, t) as defined by (2), the definition of the fractional Laplacian given by (8) still holds, and therefore −(−∇ · D∇) α/2 S = −w(t) α/2 S . However, an immediate difficulty arises in the calculation of the Caputo derivative of the product of two functions, which is given by an infinite series of fractional Riemann integral operators [19, p. 59].…”

“…In recent years, fractional calculus has arisen as a powerful and robust theoretical framework in order to account for the effects of anomalous diffusion in MRI [7][8][9][10][11][12]. The main advantage of such an approach is that insights can A C C E P T E D M A N U S C R I P T be derived from generalisations of the physical principles describing the magnetisation of water protons in MRI: the Bloch-Torrey equation [13].…”