A theoretical and numerical consideration of the longitudinal and transverse relaxations in the rotating frame

We previously derived a simple equation for solving time-dependent Bloch equations by a matrix operation. The purpose of this study was to present a theoretical and numerical consideration of the longitudinal (R_1ρ = 1/T_1ρ) and transverse relaxation rates in the rotating frame (R_2ρ = 1/T_2ρ), based on this method. First, we derived an equation describing the time evolution of the magnetization vector (M(t)) by expanding the matrix exponential into the eigenvalues and the corresponding eigenvectors using diagonalization. Second, we obtained the longitudinal magnetization vector in the rotating frame (M_1ρ(t)) by taking the inner product of M(t) and the eigenvector with the smallest eigenvalue in modulus, and then we obtained the transverse magnetization vector in the rotating frame (M_2ρ(t)) by subtracting M_1ρ(t) from M(t). For comparison, we also computed the spin-locked magnetization vector. We derived the exact solutions for R_1ρ and R_2ρ from the eigenvalues, and compared them with those obtained numerically from M_1ρ(t) and M_2ρ(t), respectively. There was excellent agreement between them. From the exact solutions for R_1ρ and R_2ρ, R_2ρ was found to be given by R_2ρ = (2R₂ + R₁)/2 − R_1ρ/2, where R₁ and R₂ denote the conventional longitudinal and transverse relaxation rates, respectively. We also derived M_1ρ(t) and M_2ρ(t) for bulk water protons, in which the effect of chemical exchange was taken into account using a 2-pool chemical exchange model, and we compared the R_1ρ and R_2ρ values obtained from the eigenvalues and those obtained numerically from M_1ρ(t) and M_2ρ(t). There was also excellent agreement between them. In conclusion, this study will be useful for better understanding of the longitudinal and transverse relaxations in the rotating frame and for analyzing the contrast mechanisms in T_1ρ- and T_2ρ-weighted MRI.