Optimizing Fractional Bloch Equation Anomalous Diffusion Parameters

Abstract

In NMR, the Bloch equations are phenomenological equations that describe the nuclear magnetization as a function of time when relaxation times T1 and T2 are present. MRI is important due to its ability to reveal details about the complex, heterogeneous structure and function of human tissue. Studies on MRI signal behavior at high or ultra-high gradient fields show an increased deviation from the classically mono-exponential relaxation model to the stretched exponential relaxation model in a form of exp[-(t /τ)^β]. This result can be derived from solving the generalize fractional-order Bloch equations using fractional calculus. The generalization fractional-order Bloch equations for anomalous relaxation can contribute to a better understanding of the interaction of spins and magnetization vector with their surroundings. Several recent studies have investigated the socalled anomalous diffusion stretched exponential model exp [-(bD) ^α], where α is a measure of tissue heterogeneity, 0 < α < 1, D is the apparent diffusion coefficient mm^2/s and b depends on the specific gradient pulse sequence parameters. This equation can be used to correlate developing pathology with localized diffusion. In this paper we study different factors that influence the strength, clarity, and contrast of MRI signal relaxation by manipulating the anomalous diffusion parameters of generalization fractional-order Bloch equations by studying the maximum and minimum of the resulting stretched exponential relaxation function. Further developments of this approach may be useful for characterizing anomalous diffusion in tissues with neurodegenerative, malignant and ischemic diseases.