Derivation of Fredholm Integrals of the First Kind Based on Bloch NMR Flow Equation for Digital Molecular Magnetic Resonance Imaging

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 Tikhonov regularization, the ill-posed equation is turned into a well-posed integral form, allowing for stable solution, providing a robust mathematical framework for interpreting NMR signals in heterogeneous systems. Numerical simulations show that using regularized least-squares approach help to stabilize the solution and help to reconstruct high-quality MRI images. These integrals offer a new perspective on the inverse problem of reconstructing molecular dynamics and interactions from observed NMR data. This solution is crucial for generating detailed relaxation time distribution which provides vital insights into tissue microstructure that enhances diagnostic capabilities in clinical MRI.