Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation.
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.
Magin
Richard L
RL
Department of Bioengineering, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7052, USA. rmagin@uic.edu
Abdullah
Osama
O
Baleanu
Dumitru
D
Zhou
Xiaohong Joe
XJ
D000465:Algorithms; D001931:Brain Mapping / Q000379:methods*; D002102:Cadaver; D002358:Cartilage, Articular / Q000033:anatomy & histology*; D003911:Dextrans; D038524:Diffusion Magnetic Resonance Imaging / Q000379:methods*; D017709:Fractals; D006801:Humans; D007091:Image Processing, Computer-Assisted; D008962:Models, Theoretical
D003911:Dextrans; C025614:sephadex