It is well known that diffusion-induced MR signal loss deviates from monoexponential decay, particularly at high b-values (e.g., >1500 sec/mm² for human brain tissues). A number of models have been developed to describe this anomalous diffusion behavior and relate the diffusion measurements to tissue structures. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (Magin et al., J Magn Reson 2008;190:255-270; Zhou et al., Proc Int'l Soc Magn Reson Med 2008). Using a spatial Laplacian ∇^2β, this model yields a new set of parameters to describe anomalous diffusion: diffusion coefficient D, fractional order derivative in space β, and a spatial parameter μ (in units of μm). In this study, we demonstrate that the fractional calculus model can be successfully applied to analyzing diffusion images of healthy human brain tissues in vivo. Five human volunteers were scanned on a commercial 3-T scanner using a customized single-shot echo-planar imaging diffusion sequence with 15 b values ranging from 0 to 4700 sec/mm². The set of images was analyzed using the fractional calculus model, producing spatially resolved maps of D, β, and μ. The β and μ maps showed notable contrast between white and gray matter. The contrast has been attributed to the varying degree of complexity of the underlying tissue structures and microenvironment. Although the biophysical basis of β and μ remains elusive, the potential utility of these parameters to characterize the environment for molecular diffusion, as a complement to apparent diffusion coefficient, may lead to a new way to investigate tissue structural changes in disease progression, intervention, and regression. Magn Reson Med 63:562–569, 2010.

To demonstrate that a new set of parameters (D, β, and μ) from a fractional order calculus (FROC) diffusion model can be used to improve the accuracy of MR imaging for differentiating among low- and high-grade pediatric brain tumors. The institutional review board of the performing hospital approved this study, and written informed consent was obtained from the legal guardians of pediatric patients. Multi-b-value diffusion-weighted magnetic resonance (MR) imaging was performed in 67 pediatric patients with brain tumors. Diffusion coefficient D, fractional order parameter β (which correlates with tissue heterogeneity), and a microstructural quantity μ were calculated by fitting the multi-b-value diffusion-weighted images to an FROC model. D, β, and μ values were measured in solid tumor regions, as well as in normal-appearing gray matter as a control. These values were compared between the low- and high-grade tumor groups by using the Mann-Whitney U test. The performance of FROC parameters for differentiating among patient groups was evaluated with receiver operating characteristic (ROC) analysis. None of the FROC parameters exhibited significant differences in normal-appearing gray matter (P ≥ .24), but all showed a significant difference (P < .002) between low- (D, 1.53 μm(2)/msec ± 0.47; β, 0.87 ± 0.06; μ, 8.67 μm ± 0.95) and high-grade (D, 0.86 μm(2)/msec ± 0.23; β, 0.73 ± 0.06; μ, 7.8 μm ± 0.70) brain tumor groups. The combination of D and β produced the largest area under the ROC curve (0.962) in the ROC analysis compared with individual parameters (β, 0.943; D,0.910; and μ, 0.763), indicating an improved performance for tumor differentiation. The FROC parameters can be used to differentiate between low- and high-grade pediatric brain tumor groups. The combination of FROC parameters or individual parameters may serve as in vivo, noninvasive, and quantitative imaging markers for classifying pediatric brain tumors.

Purpose: To develop an analytical formalism describing how noise and selection of diffusion‐weighting scheme propagate through the diffusion tensor imaging (DTI) computational chain into variances of the diffusion tensor elements, and errors in the relative anisotropy (RA) and fractional anisotropy (FA) indices.Materials and Methods: Singular‐value decomposition (SVD) was used to determine the tensor variances, with diffusion‐weighting scheme and measurement noise incorporated into the design matrix. Anisotropy errors were then derived using propagation of error. To illustrate the applications of the model, 12 data sets were acquired from each human subject, over a range of b‐values (500–2500 seconds/mm²) and diffusion‐weighting gradient directions (N = 6–55). The mean RA and FA values and their respective errors were calculated within a region of interest (ROI) in the splenium. The RA and FA errors as a function of b‐value and N were evaluated, and a number of diffusion‐weighting schemes were assessed based on a new metric, sum of diffusion tensor variances.Results: When the acquisition time was held constant, the sum of the diffusion tensor variances decreased as N increased. The same trend was also observed for several diffusion‐weighting schemes with constant condition number when noise in the diffusion‐weighted (DW) images was assumed unity. Errors in both FA and RA increased with b‐value and decreased with N. The FA error in the splenium was approximately threefold smaller than RA error, irrespective of b‐value or N.Conclusion: The condition number may not adequately characterize the noise sensitivity for a given diffusion‐weighting scheme. Signal averaging may not be as effective as increasing N, especially when N is small (e.g., N < 13). Due to its smaller error, FA is preferred over RA for quantitative DTI applications. J. Magn. Reson. Imaging 2004;19:489–498.

BACKGROUND AND PURPOSE: Patients with type 2 diabetes mellitus have considerably higher risk of developing cognitive impairment and dementia. WM changes in these patients have been reported. Our aim was to demonstrate that gradual and continuous WM change and the associated cognitive decline in patients with type 2 diabetes mellitus can be captured by DTI parameters, which can be used to complement neuropsychological test scores in identifying patients with type 2 diabetes mellitus with and without mild cognitive impairment. MATERIALS AND METHODS: Forty-two patients with type 2 diabetes mellitus, divided into a group with mild cognitive impairment (n = 20) and a group with normal cognition (n = 22), were enrolled with age-, sex-, and education-matched healthy controls (n = 26). 3T DTI followed by Tract-Based Spatial Statistics analysis was used to investigate the differences in fractional anisotropy, mean diffusivity, axial diffusivity (λ₁), and radial diffusivity (λ₂₃) among the groups. A receiver operating characteristic analysis assessed the performance of DTI parameters for separating the 2 groups with type 2 diabetes mellitus. RESULTS: The whole-brain Tract-Based Spatial Statistics analysis revealed that 7.3% and 24.9% of the WM exhibited decreased fractional anisotropy and increased mean diffusivity (P < .05), respectively, between the diabetes mellitus with mild cognitive impairment and the diabetes mellitus with normal cognition groups, while considerably larger WM regions showed fractional anisotropy (36.6%) and mean diffusivity (58.8%) changes between the diabetes mellitus with mild cognitive impairment and the healthy control groups. These changes were caused primarily by an elevated radial diffusivity observed in the patients with diabetes mellitus with mild cognitive impairment. Radial diffusivity also exhibited subtle but statistically significant changes between the diabetes mellitus with normal cognition and the healthy control groups. Analyses on individual fiber tracts showed pronounced fractional anisotropy reduction and mean diffusivity elevation in regions related to cognitive functions. The receiver operating characteristic analysis on the right cingulum (hippocampus) showed that fractional anisotropy produced a larger area under the curve (0.832) than mean diffusivity (0.753) for separating mild cognitive impairment from normal cognition among patients with type 2 diabetes mellitus. When fractional anisotropy was combined with mean diffusivity, the area under the curve was further improved to 0.857. CONCLUSIONS: DTI parameters can show a substantial difference between patients with type 2 diabetes mellitus with and without mild cognitive impairment, suggesting their potential use as an imaging marker for detecting cognitive decline in patients with type 2 diabetes mellitus. More important, DTI parameters may capture gradual and continuous WM changes that can be associated with early stages of cognitive decline in patients with type 2 diabetes mellitus before they can be diagnosed clinically by using conventional neuropsychological tests.