Derivation of Fredholm Integrals of the First Kind in terms of Analytical Solutions to Bloch Nuclear Magnetic Resonance (NMR) Flow Equation and Relaxometric Tissue Profiles

Abstract

The Bloch NMR equation is fundamental to understanding nuclear magnetization in the presence of diffusion, relaxation, and magnetic fields. However, it becomes particularly complex when applied to fluid mixtures with varying diffusion coefficients and relaxation properties. This complexity necessitates advanced mathematical techniques for deriving accurate analytical solutions. In this study, we derive the Fredholm integrals of the first kind based on these analytical solutions to the Bloch MRI equation for relaxation time. Using these solutions as a modified kernel distribution function and Tikhonov regularization providing accurate estimates of the T1 and T2 maps, a Python algorithm has been developed for interpreting NMR signals in heterogeneous systems with useful outputs. This is particularly useful in developing improved methods for multidimensional correlation magnetic resonance imaging (MRI) for general tissue diagnosis. Furthermore, this method offers a new perspective on the inverse problem of reconstructing molecular dynamics and interactions from observed NMR relaxometric data distribution which provides vital insights into tissue microstructure that enhances diagnostic capabilities in clinical MRI.