Anomalous relaxation in diffusion - weighted MRI

for the time dependence of the macoscopic magnetisation M(t) of a sample under the influence of an external field H(t) where γ =μ / Ih is the gyromagnetic ratio of the nuclei under consideration with magnetic moment μ and spin I, i, j, and k are the usual triad of unit vectors along the Cartesian axes. The external field has the form H(t) = H0k + H1(t), where H0 is strong and constant while H1 is relatively weak and an arbitrary function of the time. M0 is the equilibrium magnetisation in the field H0 and the establishment of thermal equilibrium described by two relaxation time constants T1 and T2, the longitudinal and transverse relaxation times respectively. Bloch’s equations form the basis for MRI techniques. To study relaxation we suppose that H1 is zero; in this case the transverse and longitudinal components of M separate. The decay of the transverse magnetization Mxy, is described by the stochastic partial differential equation