Behavior of the magnetization in spin-locking magnetic resonance imaging using numerical solutions to the time-dependent Bloch equations.

We present a simple method for calculating the magnetization in spin-locking (SL) magnetic resonance imaging (MRI), in which a simple matrix equation was derived for solving the time-dependent Bloch equations in the 2-pool chemical exchange model. We also present a method for visualizing the trajectory of a magnetization vector in a three-dimensional (3D) space. The longitudinal relaxation time in the rotating frame (T(1ρ)) was calculated by fitting the z component of magnetization for a duration of SL (t(SL)) (M(z)(t(SL))) to M(z)(t(SL)) = (M(0) - M(zss))exp ( - t(SL)/T(1ρ)) + M(zss), where M(0) and M(z)(ss) denote the thermal equilibrium and steady-state z component of magnetization, respectively, and was compared with that calculated from the solution given by Trott and Palmer. Our 3D plots clearly visualized the effect of SL. When the population of the two pools was highly asymmetric, there was good agreement between the T(1ρ) values obtained by our method and Trott and Palmer's solutions. The difference between them increased with decreasing asymmetry in the population of the two pools. Our method will be useful for better understanding and optimization of SL MRI, because it allows us to calculate the magnetization vector and to visualize its trajectory simply and quickly.