Langevin equation approach to diffusion magnetic resonance imaging

Abstract

The normal phase diffusion problem in magnetic resonance imaging ͑MRI͒ 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 ͑2001͔͒ 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.

Fractional Langevin Equation Model for Characterization of Anomalous Brownian Motion from NMR Signals

“…We have obtained a very good agreement with these measurements but, since in [12] an erroneous formula for S (t) was used (see [3]), the parameters ε and the generalized diffusion coefficient D ε = k B T/γ ε extracted from the experiments are very different. For example, instead of ε = 0.31 determined in [12] (Table 1), the best correspondence with the measurements is obtained for ε about 0.42.…”

“…We have obtained a very good agreement with these measurements but, since in [12] an erroneous formula for S (t) was used (see [3]), the parameters ε and the generalized diffusion coefficient D ε = k B T/γ ε extracted from the experiments are very different. For example, instead of ε = 0.31 determined in [12] (Table 1), the best correspondence with the measurements is obtained for ε about 0.42. Even if ε was correctly extracted from the experiments, D ε found in [12] would have to be divided by 2(2 1−ε − 1)/(2 − ε), which, for ε ∈ [0, 1] changes from 0 to 1 (e.g., if ε = 0.5, see Table II [12], D ε should be about 1.8 times larger).…”

“…S (t) is used in the form obtained by the method of accumulation of phases in the frame rotating with the resonance frequency [3] and applicable for any time and kind of the stochastic motion od spins. Although our work was mainly aimed to contribute to adequate interpretations of the NMR experiments on the BM in complex fluids, in the limit of free particles coupled to a fractal heat bath our results correct the description and give favorable comparison with experiments acquired in human neuronal tissues [12].…”

“…(42) [10]. The limit of (3) for a free particle was used in [12] to calculate the NMR signals from human neuronal tissues, however, with an incorrect spin-echo attenuation S (t) (see [3]). The basic formulas used to calculate S (t) are presented in the following section.…”

“…The function S (t) calculated from the first term in (5) at this limit can be compared with the results [12] of the attenuation of steady-gradient (∆ = δ = t) spin-echo signal from water molecules anomalously diffusing in a human brain tissue. We have obtained a very good agreement with these measurements but, since in [12] an erroneous formula for S (t) was used (see [3]), the parameters ε and the generalized diffusion coefficient D ε = k B T/γ ε extracted from the experiments are very different.…”