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Home arrow Computer Science arrow Computational Diffusion MRI: MICCAI Workshop, Athens, Greece, October 2016

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Fractional Diffusion

We have already seen that the stretched exponential can be derived from the assumption of a space-fractional process. A more general form of this approach is to consider time-fractional transport as well. Here, the transport process makes use of a Continuous-time Random Walk (CTRW) model, in which spins have a waiting time associated with successive steps in their random walks. This model describes a more general class of transport processes than conventional Fickian approaches, including subdiffusion, in which mean squared-displacement of diffusing spins increases more slowly than linearly with time. The CTRW model predicts a signal curve described by the Mittag-Leffler function

where Г(-) is the gamma function and a and are temporal and spatial scaling exponents respectively. Note that when a = 1 this reduces to the series expression for the exponential.

Again, this model assumes a single continuum and make no strong assumptions about tissue geometry. The continuous time random walk provides a very flexible model of an effective diffusion process and describes the observed signal in very few parameters, but making microstructural inferences from the model is more difficult. Strictly speaking, this model requires measurements over a range of diffusion times, although additional assumptions make it possible to apply to multiple b-values at a single diffusion time. This model has been applied to rat and human brain [19, 31, 46] and also in muscle [18].

 
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