Langevin equation approach to diffusion magnetic resonance imaging

The normal phase diffusion problem in magnetic resonance imaging (MRI) is treated by means of the Langevin equation for the phase variable using only the properties of the characteristic function of Gaussian random variables. The calculation may be simply extended to anomalous diffusion using a fractional generalization of the Langevin equation proposed by Lutz [E. Lutz, Phys. Rev. E 64, 051106 (2001)] pertaining to the fractional Brownian motion of a free particle coupled to a fractal heat bath. The results compare favorably with diffusion-weighted experiments acquired in human neuronal tissue using a 3 T MRI scanner.

Figure 1

Plots of data (symbol) and fit (solid line) for (a) gray matter and (b) white matter regions of interest for subject 1. The image intensity was normalized by dividing by the $b_0$ image. The fractional diffusion model [Eq. (44), solid line] was fitted to the experimental data to determine $D$ and $\alpha$ (Tables ). The experimental data are plotted against $t^3$.

Figure 2

Plots of data (symbol) and fit (solid line) for (a) gray matter and (b) white matter regions of interest for subject 2.

Figure 3

Plots of data (symbol) and fit (solid line) for (a) gray matter and (b) white matter regions of interest for subject 3.

Figure 4

The data plotted against diffusion-weighting gradient strength squared, with the fit overlaid as a solid line. The region of interest for this plot is chosen from gray matter.

Figure 5

(Color online) These images illustrate the success of the fit of the fractional model to the experimental in vivo data, for each voxel in a selected slice. (a) The image acquired with no diffusion weighting, or the $b_0$ image. (b)–(d) show maps of the fitting parameters $A_0$, $D$, and $\alpha$ respectively, for each voxel in a given anatomical slice. (b) The representation of each $A_0$ produced during a fit of each voxel is the predicted $b_0$ image from the fit. (e) This image is an example of an inversion recovery image, which was used to select the regions of interest in gray and white matters for fitting.