Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T₂*-weighted magnetic resonance imaging at 7 T

Abstract

We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain. Magn Reson Med 77:1485-1494, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

“…Furthermore, when the m i are not one, so that the individual components are not scalar, then there is no simple representation such as (19), and new approaches are needed. Before we consider this new approach, we note the converse, namely that (19) can always be written in the form of (16) for a suitable matrix A with a special structure. In particular, we can write (19) in the form of (16) with p = r − 1 as an r dimensional, r index problem with α = (β 0 , β 1 , · · · , β p ), and:…”

“…Before we consider this new approach, we note the converse, namely that (19) can always be written in the form of (16) for a suitable matrix A with a special structure. In particular, we can write (19) in the form of (16) with p = r − 1 as an r dimensional, r index problem with α = (β 0 , β 1 , · · · , β p ), and:…”

“…We note that scalar linear sequential fractional problems have been considered whose solution can be described by multi-indexed Mittag-Leffler functions [16], and there are a number of articles on the numerical solution of multi-term fractional differential equations [17][18][19][20][21][22]. While mixed index problems can, in some cases, be written in the form of linear sequential problems, namely ∑ R i=1 D β i t y = f (y), we claim that it is inappropriate to do so in many cases.…”

A fractional motion diffusion model for a twice‐refocused spin‐echo pulse sequence

“…Among numerous anomalous diffusion models, the continuous‐time random‐walk (CTRW) model and the fractional motion (FM) model are actively pursued by the biophysics community and regarded as the major “archrivals” . The CTRW model, which recognizes intra‐voxel diffusion heterogeneity in both time and space, has already been introduced to the diffusion MRI community and successfully demonstrated in a number of clinical applications . The FM model, on the other hand, describes the complex diffusion process based on the intricate statistical properties, such as variances and correlations .…”

“…17 The CTRW model, which recognizes intra-voxel diffusion heterogeneity in both time and space, has already been introduced to the diffusion MRI community and successfully demonstrated in a number of clinical applications. [3][4][5][6][7][8][18][19][20] The FM model, on the other hand, describes the complex diffusion process based on the intricate statistical properties, such as variances and correlations. 15,17 Based on the Langevin equation, 21 one of the fundamental differential equations to describe stochastic dynamics of motion in a randomly fluctuating environment, 22,23 the FM model provides an alternative avenue to probing the complexity and the stochastic characteristics of diffusion in biological tissues.…”