Date: Tue, 7 Feb. 2023 19:08:03 +00:00
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Although we have seen that the signal vanishes really fast even with a very little inhomogeneity in the main field, the situation is not as bad as one may think based on this. It is because Hahn et al. found that the eliminated signal is retrievable by an RF pulse trick.
As mentioned before, the signal elimination is caused by two effects. First, the individual spins experience the small magnetic fields of their neighbours, which adds to the external field causing the local Larmor frequencies to vary spatially and the precessing spins to dephase. These spin-spin interactions also evolve in time really fast, making this {EQUATION()}$T_2${EQUATION} dephasing irreversible. However, the situation differs at the dephasing caused by the external field inhomogeneities. These field variations remain constant in time and thus the dephasing caused by them could be reversed by the negation of the phase of each spin.
The idea is the following. With an RF pulse we rotate the spins with 90° to the transverse field. At this point every spin points at the same direction, or in the complex description, they all have the same phase when they start to precess in the slightly inhomogeneous main field. After that if a spin experiences a larger external field and precesses at a higher frequency for the length of time {EQUATION()}$\tau${EQUATION}, it accumulates a positive relative phase {EQUATION()}$\phi${EQUATION} according to spins preseccing at the center frequency. Then we apply another RF pulse of 180° which will turn the spins around and negates their phase. For example the previously described spin that gained {EQUATION()}$\phi${EQUATION} phase in time interval {EQUATION()}$\tau${EQUATION} will have the relative phase {EQUATION()}$ - \phi${EQUATION} right after the 180° pulse. Of course this spin will continue to experience a larger field regardless of the applied 180° pulse, and therefore its relative phase continues to grow in time. After another time interval {EQUATION()}$\tau${EQUATION} it will have zero relative phase again. Since we did not use any specific parameter of the picked spin nor of the field, this thread is applicable to all spins in the sample. So at the time {EQUATION()}$\tau${EQUATION} after the 180° pulse all spins will have the same phase (will point to the same direction) creating a macroscopic magnetization once again, which can be measured.
What we get as a result was quite astonishing at the time of its invention: the MR signal eliminated by {EQUATION()}$T^*_2${EQUATION} dephasing is retrievable by a 180° RF pulse. The latter is called refocusing pulse, the recurring signal after the 180° pulse is called a spin echo, and the total time between the 90° pulse and the echo is referred to as echo time {EQUATION()}$T_E${EQUATION}. The scheme of the process is shown in Figure 1. (Inserted from Wikipedia, under copyright of the author).
{img fileId="3172" thumb="y" rel="box[g]" width="720" imalign="center" align="center" desc=Figure 1. Scheme of the spin echo. A) The spins before the excitation. B) The spins after the 90° RF pulse. C) Dephased spins due to field inhomogeneities. D) The spins turned around by the 180° pulse that negates their phase. E) spins are rephasing after the refocusing pulse. F) Refocused spins and the spin echo. from Wikipedia, under copyright.}
Note that the above are only valid for the dephasing caused by field inhomogeneities, and are not able to reverse the {EQUATION()}$T_2${EQUATION} relaxation caused by spin-spin interactions. Therefore the recurring signal will be smaller then it was right after the 90° pulse as the signal amplitude is enveloped by the exponential decay {EQUATION()}$T_2${EQUATION}.
Knowing the above it is easy to interpret the sequence diagram of the spin echo experiment shown in Figure 2.
{img fileId="3173" thumb="y" rel="box[g]" width="640" imalign="center" align="center" desc=Figure 2. Sequence diagram of a spin echo experiment. Note how the signal is enveloped by the T2 relaxation.}
Without the detailed deduction the expressions for the perfectly demodulated signal in spin echo sequence are the followings:
::{EQUATION()}
\label{echo_envelopes}
M_{\bot} (t) = M_{\bot} (0) \times
\begin{cases}
\mathrm{e}^{ - \tfrac{t}{T^*_2} } & 0 < t < \tau \\
\mathrm{e}^{ - \tfrac{t}{T_2} } \hspace{2pt} \mathrm{e}^{ - \tfrac{T_E - t}{T'_2} } \hspace{10pt} & \tau < t < 2 \tau = T_E \\
\mathrm{e}^{ - \tfrac{t}{T_2} } \hspace{2pt} \mathrm{e}^{ - \tfrac{t - T_E}{T'_2} } & T_E < t
\end{cases}
{EQUATION}(1)::
These expressions allow the measurement of {EQUATION()}$T_2${EQUATION} relaxation time using two spin echo experiments with different echo times, let's say {EQUATION()}$T_E${EQUATION} and {EQUATION()}$T'_E${EQUATION}. The signal maximum in the echo will be proportional to an exponential decay due to {EQUATION()}$T_2${EQUATION}, from which the {EQUATION()}$T_2${EQUATION} characteristic time can be calculated:
::{EQUATION()}
\label{signal_TE}
S(T_E) \propto \mathrm{e}^{-\tfrac{T_E}{T_2}}
{EQUATION}(2)::
::{EQUATION()}
S(T'_E) \propto \mathrm{e}^{-\tfrac{T'_E}{T_2}}
{EQUATION}(3)::
::{EQUATION()}
\label{T2_from_signal_TE}
T_2 = \frac{T'_E - T_E}{\mathrm{ln}\left( \frac{S(T_E)}{S(T'_E)} \right ) }
{EQUATION}(4)::
Of course a more accurate method is to measure the signal maximum with numerous {EQUATION()}$T_E${EQUATION} values and fit an appropriate exponential curve to the data, or a linear curve to the logarithm of the data.
The spin echo sequence has the significant advantage that the appearance of the signal (the echo) and the RF pulses are separated in time, unlike in FID where the signal appears right after the RF excitation. This is a great benefit since (with a one-channel device) we cannot measure the induced signal during the RF pulse, nor in a certain time interval after it - we would only see the voltage induced in the receiving coil by the RF field itself.