### Synopsis

**Over the past decades, several non-Gaussian diffusion
models have been developed to reveal the underlying structures of the complex
and heterogeneous tissue by measuring the anomalous diffusion behavior. Within
these, continuous-time random-walk (CTRW) and fractional motion (FM) models
have actively been studied by several groups to compare them at ***cellular* or *molecular* level. In this study, we provide a comparison between the
CTRW and FM models on human brain *in vivo*
at the voxel level using high *b*-value
diffusion MRI. It was observed that all CTRW and FM parameters exhibit characteristic
contrasts, reflecting different aspects of the complex diffusion process.### Introduction

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.

### Theory

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 mm^{2}/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].

### Methods

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/mm

^{2}. 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.

### Results

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.

### Discussion and Conclusion

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.

### Acknowledgements

This work was supported in part by NIH 1S10RR028898 and
3R01MH081019. We thank Drs. Frederick C. Damen, Richard L. Magin, and Rong-Wen
Tain, and Zheng Zhong for valuable discussions. ### References

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