Dependence of Fractional Order Diffusion Model Parameters on Diffusion Time

INTRODUCTION At high b-values (e.g., ≥1,500 s/mm), it is well known that diffusion-induced MR signal loss in the brain tissue exhibits anomalous behavior that cannot be described by a mono-exponential function. This phenomenon has been attributed to tissue heterogeneity manifested by cellular structures, cell membranes, and/or intraand extra-cellular spaces [1]. Over the past few years, a number of groups have proposed various models [2-7] to describe diffusion signals at high b-values with a common goal to establish the interrelation between diffusion measurements and tissue microstructures. Recently, a diffusion model employing fractional order calculus was reported [6, 7]. By generalizing the Bloch-Torrey equation using fractional order derivatives in space, this model describes anomalous diffusion by three parameters: diffusion coefficient (D), fractional order spatial derivative β (0<β<1), and a spatial variable μ (in units of μm). Although it has been suggested that β and μ can be correlated with the degree of tissue heterogeneity [6], the biophysical basis of this correlation is yet to be established. In this study, we have investigated the effects of diffusion times on the three parameters obtained from the fractional order diffusion model, and report experimental findings that will help establish the interrelation between the diffusion parameters and tissue microstructures.