Approximate analytical expressions for the Carr–Purcell–Meiboom–Gill sequences: Decay rates and modulation zeros of the echo train and the relation between the [formula omitted] and [formula omitted] relaxation times

Abstract

Methods of multi-pulse sequences are widely used in magnetic resonance to study the local properties of magnetic particles and to increase the lifetime of spin coherence. Imperfect refocusing pulses lead to non-exponential signal decay due to the contribution of the coherence pathways in which T1 and T2 relaxation segments are mixed. Here, we present analytical approximations for echoes generated in the Carr-Purcell-Meiboom-Gill (CPMG) sequence. They provide simple expressions for the leading terms of the echo train decay and allow the relaxation times to be estimated for sequences with a relatively small number of pulses. For a given refocusing angle α, the decay times for the fixed phase and alternating phase CPMG sequences can be approximated as (T2-1+T1-1)/2 and T2O, respectively. The ability to estimate relaxation times from short pulse sequences can reduce the acquisition time, which is essential for the methods used in magnetic resonance imaging. In the case of a CPMG sequence with the fixed phase, the relaxation times can also be obtained from the points in the sequence at which the echo changes sign. Numerical comparison of the exact and approximate expressions shows the practical limits of the analytical formulas obtained. It is also shown that a double echo sequence in which the interval between the first two pulses is not equal to half the interval of the subsequent refocusing pulses provides the same information as two separate CPMG (or CP) sequences with fixed and alternating phases of the refocusing pulses. In addition, the two double-echo sequences differ in the parity of the number of intervals with longitudinal magnetization evolution (relaxation), i.e. the echo in one sequence is formed only from those coherence pathways that have an even number of intervals with longitudinal magnetization evolution, while the other sequence has an odd number of such intervals.