Fractal Bloch model to characterize stretched magnetization relaxation in magnetic resonance imaging

Abstract

This study proposes a fractal Bloch model to describe the stretched magnetization relaxation in magnetic resonance imaging (MRI). The fractal Bloch model extends the relaxation terms and preserves the linear Larmor precession terms under a constant magnetic field. The solutions of the fractal Bloch model have the form of stretched exponential and converge to the exponential form for the classical Bloch equation when the orders of the fractal derivative α=1, and β=1. The results show that the damped oscillations in Mx(t) or My(t) have a faster decay at short times, and slower decay at long times with the increasing values of α. The relaxation of Mz(t) with the stretched-exponential form increases more quickly than the exponential case at first, but then converges more slowly to the equilibrium magnetization, which needs a longer time to return the equilibrium in the cases of smaller order β. The fractal Bloch model is verified to characterize the signal decays in rat brain and bovine nasal cartilage. The fitting results show that the fractal Bloch model is effective to describe the stretched magnetization relaxation in MRI data. Thus, the fractal derivative is an alternative tool to use to understand MRI in complex systems, and its order can serve as an index to distinguish different relaxation processes in magnetization.