TY - JOUR
T1 - A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging
AU - Liang, Yingjie
AU - Ye, Allen Q.
AU - Chen, Wen
AU - Gatto, Rodolfo G.
AU - Colon-Perez, Luis
AU - Mareci, Thomas H.
AU - Magin, Richard L.
N1 - Funding Information:
The MRI data acquisition was supported by a user grant (Grant no. ML-MAGIN-002), from the National High Magnetic Field Laboratory (Tallahassee, FL) and the MRI experiments were performed at the Advanced Magnetic Resonance Imaging and Spectroscopy facility in the McKnight Brain Institute of the University of Florida (Gainesville, FL). The work described in this paper was supported by the National Science Funds for Distinguished Young Scholars of China (Grant no. 11125208 ), 111 project (Grant no. B12032), the China Scholarship Council (CSC) (Grant no. 201406710047 ), the Hohai University Training Program for Excellent Doctoral Dissertations (Grant no. 2013B15020056 ).
Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/10/1
Y1 - 2016/10/1
N2 - Non-Gaussian (anomalous) diffusion is wide spread in biological tissues where its effects modulate chemical reactions and membrane transport. When viewed using magnetic resonance imaging (MRI), anomalous diffusion is characterized by a persistent or 'long tail' behavior in the decay of the diffusion signal. Recent MRI studies have used the fractional derivative to describe diffusion dynamics in normal and post-mortem tissue by connecting the order of the derivative with changes in tissue composition, structure and complexity. In this study we consider an alternative approach by introducing fractal time and space derivatives into Fick's second law of diffusion. This provides a more natural way to link sub-voxel tissue composition with the observed MRI diffusion signal decay following the application of a diffusion-sensitive pulse sequence. Unlike previous studies using fractional order derivatives, here the fractal derivative order is directly connected to the Hausdorff fractal dimension of the diffusion trajectory. The result is a simpler, computationally faster, and more direct way to incorporate tissue complexity and microstructure into the diffusional dynamics. Furthermore, the results are readily expressed in terms of spectral entropy, which provides a quantitative measure of the overall complexity of the heterogeneous and multi-scale structure of biological tissues. As an example, we apply this new model for the characterization of diffusion in fixed samples of the mouse brain. These results are compared with those obtained using the mono-exponential, the stretched exponential, the fractional derivative, and the diffusion kurtosis models. Overall, we find that the order of the fractal time derivative, the diffusion coefficient, and the spectral entropy are potential biomarkers to differentiate between the microstructure of white and gray matter. In addition, we note that the fractal derivative model has practical advantages over the existing models from the perspective of computational accuracy and efficiency.
AB - Non-Gaussian (anomalous) diffusion is wide spread in biological tissues where its effects modulate chemical reactions and membrane transport. When viewed using magnetic resonance imaging (MRI), anomalous diffusion is characterized by a persistent or 'long tail' behavior in the decay of the diffusion signal. Recent MRI studies have used the fractional derivative to describe diffusion dynamics in normal and post-mortem tissue by connecting the order of the derivative with changes in tissue composition, structure and complexity. In this study we consider an alternative approach by introducing fractal time and space derivatives into Fick's second law of diffusion. This provides a more natural way to link sub-voxel tissue composition with the observed MRI diffusion signal decay following the application of a diffusion-sensitive pulse sequence. Unlike previous studies using fractional order derivatives, here the fractal derivative order is directly connected to the Hausdorff fractal dimension of the diffusion trajectory. The result is a simpler, computationally faster, and more direct way to incorporate tissue complexity and microstructure into the diffusional dynamics. Furthermore, the results are readily expressed in terms of spectral entropy, which provides a quantitative measure of the overall complexity of the heterogeneous and multi-scale structure of biological tissues. As an example, we apply this new model for the characterization of diffusion in fixed samples of the mouse brain. These results are compared with those obtained using the mono-exponential, the stretched exponential, the fractional derivative, and the diffusion kurtosis models. Overall, we find that the order of the fractal time derivative, the diffusion coefficient, and the spectral entropy are potential biomarkers to differentiate between the microstructure of white and gray matter. In addition, we note that the fractal derivative model has practical advantages over the existing models from the perspective of computational accuracy and efficiency.
KW - Anomalous diffusion
KW - Fractal derivative
KW - Magnetic resonance imaging
KW - Spectral entropy
UR - http://www.scopus.com/inward/record.url?scp=84963959831&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2016.04.006
DO - 10.1016/j.cnsns.2016.04.006
M3 - Article
AN - SCOPUS:84963959831
SN - 1007-5704
VL - 39
SP - 529
EP - 537
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
ER -