Given the heterogeneous nature of biological tissues, it has been well-recognized that water diffusion in tissues does not follow Gaussian statistics, but instead exhibits “anomalous” diffusion behavior in which the mean-square displacement shows nonlinear time dependence. Several empirical [1-5] and theoretical [6-11] models have been proposed to account for this anomalous phenomenon. Among these models, the fractional motion (FM) model is actively pursued by the biophysics community because of its intricate statistical properties, such as fluctuations (mild vs. wild) and correlations (short-range vs. long-range vs. no memory) [12], to describe the diffusion process. These properties can be directly related to the underlying microstructural and topological features of the tissue. Although the FM model has been increasingly used to characterize molecular diffusion in cell culture [13], analysis of its voxel-level behavior, especially in the context of addressing a clinical question, remains unexplored. In this study, we investigate whether the FM model can be applied to differentiating low-grade and high-grade pediatric brain tumors.Based on the analytical expression given in [11], we have simplified the anomalous diffusion-induced signal attenuation as, $$S/S_{0}=exp\left(-\eta^{'} D_{fm}b^{\varphi/2}\left(\triangle-\frac{\delta}{3}\right)^{-\frac{\varphi}{2}}\delta^{\varphi+\psi}\delta^{-\varphi}\right)$$ for a Stejskal-Tanner diffusion gradient with a pulse width of δ, and a lobe separation of Δ. In this equation, D_fm is an anomalous diffusion coefficient, and φ and ψ are the parameters governing the variance and correlation properties of the increments of diffusion process, respectively. The dimensionless quantity $$$\eta^{'}$$$ is formulated as the combination of the FM parameters (φ and ψ), imaging parameters (δ and Δ), and the space and time parameters needed to express D_fm in mm²/sec.Patients: This study involved 70 children (20 girls from 4 months to 9 years old and 50 boys from 4 months to 13 years old) with histopathologically proven brain tumors. According to the 2007 WHO classification, 30 tumors were low grade (I or II) and 40 high grade (III or IV). Image acquisition: All patients underwent MRI examination on a 3T GE scanner with an 8-channel head coil. In addition to FLAIR, T2, and contrast-enhanced T1 (T1+C) sequences, the imaging protocol included a multi-b-value DWI sequence with 12 b-values (0-4000 sec/mm²). The key acquisition parameters were: TR/TE=4700/100ms, slice thickness=5mm, Δ=38.6ms, δ=32.2ms, FOV=22cm×22cm, matrix size=128×128 (reconstructed to 256×256), total scan time~3 minutes. Trace-weighted images were obtained to minimize the effect of diffusion anisotropy. Analysis: The FM parameters (D_fm, φ, ψ) were simultaneously estimated by fitting the FM model to the multi-b-value diffusion images on a voxel-by-voxel basis using nonlinear least-squares estimation. For comparison, the conventional ADC values were also computed. The statistical analysis was performed on the mean D_fm, φ, and ψ values in the tumor regions of interest (ROIs). The performance of the FM parameters for differentiating low- and high-grade tumors was evaluated using receiver operating characteristic (ROC) curves with a multivariate logistic regression algorithm for the following combinations: (D_fm, φ), (D_fm, ψ), (φ, ψ), and (D_fm, φ, ψ).

Figure 1 shows D_fm, φ, and ψ maps from one representative patient in the low-grade and high-grade group, respectively. The high-grade tumor (upper row) showed substantially lower values in all three FM parameters than the low-grade tumor (lower row). This trend was preserved with the group analysis as presented in the boxplots of the mean D_fm, φ, and ψ (Fig.2a) using histopathology as a gold standard. The FM parameters exhibited a significant difference between low- and high-grade tumors (p-values<0.001 from t-test) as summarized in Fig.2b. Figures 3a and 3b show the results of the ROC analysis for different combinations of the FM parameters as well as the ADC value. The combination of (D_fm, φ, ψ) yielded the largest AUC (0.910) and the highest specificity cut-off (0.925), considerably outperforming ADC. Figure 4 shows the close resemblance between the FM-parameter-based classification and histopathology-based true classification for differentiating low- and high-grade tumors.

Our results showed that the FM model can be successfully applied to in vivo human studies and provide a set of new diffusion parameters that govern the micro- or macro-level statistics of water diffusion through complex tissue structures. Using an ROC analysis, we showed that the FM parameters offer better performance than conventional ADC for differentiating low- and high-grade pediatric brain tumors. Comparing to another anomalous diffusion model based on the continuous-time random-walk theory [9,10,14], the FM diffusion model provided comparable, but not superior, performance. Although the validity of the FM model for voxel-based (as opposed to cell culture) diffusion analysis requires further investigation, FM model has been shown useful for non-invasive differentiation of low- and high-grade pediatric brain tumors.