Analysis of the multi-echo spin-echo pulse sequence

The multi-echo spin-echo sequence is a series of operators, referred to as periodic operators. Each periodic operator consists of a free rotation (no RF), a refocusing RF pulse, and another free rotation, identical to the first one. A preparation operator that precedes the periodic operators converts the equilibrium magnetization M_z into an initial magnetization M_i. It is shown that a multi-echo sequence is equivalent to a simple rotation of the magnetization about a tilted axis. The component of M_i along the rotation axis is stationary and provides a stable signal, denoted pseudo steady-state. The perpendicular component rotates and eventually de-phases. Using this model, we derive analytic expressions to the signal for different preparation operators, and show how to align M_i with the rotation axis such that the signal is maximized. A simple and efficient algorithm is presented to calculate the Fourier coefficients of the magnetization during the sequence using the discrete Fourier transform. Finally, formulas of the echo signal when unavoidable phase errors are generated are derived. We show how to eliminate artifacts caused by these errors and restore the original image.