Although it has been recognized that diffusion in biological tissues does not follow the classical Gaussian model, there has not been a consensus on which model best characterizes the diffusion process in tissues. Over the past few years, a fractional motion (FM) diffusion model has been actively pursued by the biophysics community [1-6], challenging the validity of other anomalous diffusion models, including the continuous-time random-walk (CTRW) model, which was recently introduced [7-12]. Publications on the FM model, particularly those focusing on evaluation and validation, have been performed in cell culture or at molecular level [3-6]. Under these conditions, it has been shown that the FM model can better describe anomalous diffusion than the CTRW model [3-6]. However, this may not be valid in diffusion-weighted MRI where the diffusion behavior within an imaging voxel must be considered. In this study on healthy human brain, we investigate (a) whether the FM model can effectively characterize diffusion behavior at a scale of an imaging voxel, and (b) whether the FM model offers an advantage over the CTRW model.

The anomalous diffusion-induced signal attenuation, for a Stejskal-Tanner diffusion gradient with a pulse width of δ and a lobe separation of ∆, is described by Eqs. (1) and (2) for the CTRW [11,12] and FM models (with simplifications on the expression in [1]), respectively. $$S/S_{0}=E_{\alpha}\left(-\left(bD_{m}\right)^{\beta}\right)(1)$$ $$S/S_{0}=exp\left(-\eta^{'} D_{fm}b^{\varphi/2}\left(\triangle-\frac{\delta}{3}\right)^{-\frac{\varphi}{2}}\delta^{\varphi+\psi}\delta^{-\varphi}\right) (2)$$ In Eqs. (1) and (2), D_m and D_fm are the anomalous diffusion coefficient for the corresponding model. The parameters α and β are temporal and spatial diffusion heterogeneity parameters, and φ and ψ are the parameters governing the variance and correlation properties of the diffusion process’ increments. Eα is a Mittag-Leffler function and the dimensionless is a function of φ, ψ, δ, ∆, etc. to express D_fm in mm²/sec. In this general formulation, the FM model has four distinct cases, Brownian, Levy, fractional Brownian, and fractional Levy motions, corresponding to different φ and ψ values [2].

Subjects and image acquisition: This study involved two groups of healthy subjects: young group (YG) (n=6; 29-44 years-old) and elderly group (EG) (n=5; 65-76 years-old). All subjects underwent diffusion MRI examination with 14 b-values (0-4000 sec/mm2) at 3T using a 32-channel head coil. The key acquisition parameters were: TR/TE=4200/86ms, slice thickness=3mm, Δ=47ms, δ=32.2ms, FOV=24cm×24cm, and matrix size=128×128 (reconstructed to 256×256). Analysis: Trace-weighted images were used to minimize the effect of diffusion anisotropy. The CTRW parameters (D_m, α, β) and FM parameters (D_fm, φ, ψ) were estimated by respectively fitting Eq. (1) and Eq. (2) to the multi-b-value diffusion images using a nonlinear least-squares estimation. Both D_fm and D_m were estimated by a mono-exponential model using the images at b≤1200 s/mm². The other parameters, (φ, ψ) or (α,β), were simultaneously estimated from all b-values with the fixed D_fm or D_m. Two representative gray matter (GM) regions from the thalamus (THAL) and putamen (PUT), and two white matter (WM) regions from the splenium and genu of the corpus callosum (SCC and GCC) were selected and the mean values of (D_m,α,β) and (D_fm,φ,ψ) over the region-of-interests (ROIs; Fig.1) were computed. The parameter maps of the CTRW (D_m,α,β) and FM (D_fm,φ,ψ) models for one representative subject from each of the two groups are given in Figs.2 and 3, respectively. The values of D_fm and D_m are lower in WM than GM, indicating slower anomalous diffusion in WM. The CTRW parameter β and the FM parameters φ and ψ showed substantially similar GM/WM contrast. Although the FM parameters, φ and ψ, theoretically represent two different statistical properties of diffusion process, the GM/WM contrasts in φ and ψ maps were similar, possibly due to correlation between them in Eq.(2). The CTRW parameters, α and β, however, can reveal distinct diffusion information, temporal and spatial heterogeneity, as they are independent parameters in Eq.(1). The boxplots of the mean β (of the CTRW) and φ or ψ (of the FM) showing the difference between the GM (THAL and PUT) and WM (SCC and GCC) are given in Figs. 4 and 5 for the YG and EG, respectively. We have shown that the FM model can be successfully applied to in vivo human studies at 3T and provide comparable, but not superior, results to characterize healthy brain tissue to the CTRW model. The CTRW parameters, α and β, and FM parameters, φ and ψ, all exhibit characteristic contrasts in healthy brain tissue, reflecting different aspects of the diffusion process. The exact nature of these aspects remains intriguing and may require high resolution in vitro studies to provide more insights.