A FRACTIONAL LANGEVIN EQUATION APPROACH TO DIFFUSION MAGNETIC RESONANCE IMAGING - Advances in Chemical Physics - page 351

1.04

1.02

1.00

0.98

0.96

Rect(t) Rect(G) Tri(t) Ramp(t) Bennett(t)

Figure 21.

A box plot illustrating the values of α as listed in Table I.

10 − 3 mm 2 s − 1 . The average chi-squared goodness-of-fit value for these
results (Tables I and II) was 4.6

2 . 4 ×

10 − 5 .

×

B. Investigation of Fractional Diffusion Equation In Vivo

In diffusion weighted MRI, we assume that the diffusion coefficient of the sample
under investigation is constant throughout the experiment. In neuronal tissue, we
can be reasonably confident that while on a microscopic level, small changes in
D are occurring, however, when averaged over the voxel dimensions, the changes
may be negligible. It follows that the only cause of the variation in the magnitude
of the acquired signal are the experimental parameters, that is, the magnitude,

TABLE II
Fitted Values of D (mm 2 s − 1 ) for the 5 ROIs in the Water Phantom

ROI 1 2 3 4 5 Average

2.11 × 10 − 3 1.97 × 10 − 3 2.11 × 10 − 3 1.99 × 10 − 3 2.12 × 10 − 3 2.06 × 10 − 3 Rectangular

10 − 3 10 − 3 10 − 3 10 − 3 10 − 3 10 − 3

× × × × × ×

Triangular 2.14 2.15 2.14 2.15 2.15 2.146

10 − 3 10 − 3 10 − 3 10 − 3 10 − 3 10 − 3 Ramped 2.14 × 2.14 × 2.12 × 2.13 × 2.12 × 2.13 ×

10 − 3 10 − 3 10 − 3 10 − 3 10 − 3 10 − 3 Bennett 2.18 × 2.08 × 2.25 × 2.07 × 2.15 × 2.15 ×