Fractional Order Complexity Model of the Diffusion Signal Decay in MRI

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.

Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range

“…In the classical mono-exponential model, where water molecules diffuse freely without hinderance and restriction, the MR signal attenuation function is concisely characterized by a commonly used exponential function, exp(− bD ). In the CTRW model, α and β quantitatively describe the deviation of diffusion dynamics from the mono-exponential decay [ 30 ]. At short diffusion times where MSD is much smaller than the obstacle scales, the water molecules can diffuse freely in all directions, leading to a process that follows Gaussian dynamics.…”

Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

“…We start our discussion with the model, represented by Equation ( 1), which will be used to analyze the experimental data analyzed in Refs. [29,30] for the bovine optical nerve. The model is an extension of the standard comb model formed by incorporating the memory effects in space and time through the derivatives of fractional order via convolution kernels.…”

Q-Space Quantitative Diffusion MRI Measures Using a Stretched-Exponential Representation

“…The signal in Eq. ( 1) can be represented by a mono-exponential decay E(g) = exp −bg T Dg with g being a normalized vector g = q/ q and D is a covariance matrix of a Gaussian EAP or a more general Kohlrausch-Williams-Watts function so-called a stretched-exponential representation given by [9,22,33]…”