In Depth Analysis of Stretch Function Resulting from Solving the Generalize Fractional-Order Bloch Equations Using Fractional Calculus

Abstract

In this paper the stretch function resulting from solving the fractional-order Bloch equations using fractional calculus was discussed. This function has promising results to represent diffusion signal decay from MRI images. Conventional analyses of (DWI) measurements resolve the normalized magnetization decay profiles in terms of discrete and mono-exponential components with distinct lifetimes. In complex, heterogeneous biological and biophysical samples such as tissue, multi-exponential decay functions can appear to provide truer representation to normalized magnetization decay profile than the assumption of a mono-exponential decay, but the assumption of multiple discrete components is arbitrary and is often erroneous. Moreover, interactions, between both normalized magnetization and with their environment, can result in complex normalized magnetization decay profiles that represent a continuous distribution of lifetimes. The purpose in this paper is to study different factors that influence the stretch function strength, clarity, and contrast of MRI magnetization signal relaxation by manipulating the anomalous diffusion parameters Δ,δ,Gz,β and μ. of Bloch equations. Through this study, it was found that complex normalized magnetization decay profiles behave like stretch exponential function inside power law. Further developments of this study may be useful in optimizing anomalous diffusion in tissues with neurodegenerative, and ischemic diseases.