Abstract
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease.
Highlights
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials
The pigments can be identified by a UV-Vis spectrophotometer, but in painting color is described by hue, value, tone, tint, shade, and saturation, all of which are processed by the eye as characteristics of a picture
Due to the mismatch between the sub-millimeter resolution of magnetic resonance imaging (MRI) and the sub-micron architecture of biological tissues, mathematical models are needed to describe the mesoscale complexity of living systems
Summary
Diffusion-weighted MRI (DW-MRI), based on the Bloch-Torrey equation, is implemented by selective phase encoding within each imaging voxel This typically involves using a pair of rectangular gradient pulsed (Stejskal-Tanner pulses [5]) to capture the diffusion of water in tissue over the distance of several microns. In the Discussion we will describe how these models expand the available modeling tools for DW-MRI data and describe how they can be used to fit animal and clinical data Overall, this approach provides a way to extend the heuristic stretched exponential approach toward more complete multiple-compartment models. This approach provides a way to extend the heuristic stretched exponential approach toward more complete multiple-compartment models This bridge uses fractional order derivatives and varying diffusion coefficients as connecting links
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
