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Using the sequence showed in the previous page we only measure the positive {EQUATION()}$k_z${EQUATION} values at the Fourier space, although the integral in in the Fourier Transform is an improper one having limits in the positive and negative infinity. Therefore it would be intuitively desirable to cover negative {EQUATION()}$k_z${EQUATION} points as well. The necessity of such coverage and the symmetric properties of the Fourier space will be detailed in a later chapter.
The coverage of negative {EQUATION()}$k${EQUATION}-space can be achieved by bipolar gradients, like the one in the pulse sequence shown in Figure 1. The time- and space-dependent phase of spins will be the following:
::{EQUATION()}
\label{1D_imaging6}
\phi(z,t) =
\begin{cases}
\gamma G z (t-t_1) & t_1 < t < t_2 \\
\gamma G z (t_2 - t_1) z - \gamma G z (t-t_3) & t_3 < t < t_4
\end{cases}
{EQUATION}(1)::
From (1) we can calculate the time when the spin phase will be zero everywhere, in other words, the echo time: {EQUATION()}$T_E = t_3 + t_2 - t_1${EQUATION}. This solution is somewhat trivial in the graphic interpretation: the echo comes when the phase is space-independent, i. e. when {EQUATION}$\mathbf{k}=0${EQUATION}. This happens at the time until which the signed integral of the gradient is zero. This echo achieved by rephasing the spins with gradient fields rather than a refocusing pulse is called a "gradient echo".
{img fileId="3178" thumb="y" rel="box[g]" width="720" imalign="center" align="center" desc=Figure 1. Left: FID with bipolar gradient to cover both positive and negative k-values. Right: Trajectory in k-space.}
To have a same positive and negative coverage of the {EQUATION()}$k${EQUATION}-space with same gradient amplitude in both directions one should apply the readout gradient twice as long as the negative gradient before the acquisition, that is, {EQUATION()}$t_4-t_3 = 2 (t_2 - t_1)${EQUATION}.
Nonetheles, there is another method to reach negative {EQUATION()}$k${EQUATION}-values: applying refocusing pulses. As mentioned at the spin echo secquence a 180° RF pulses negates the spin phases. If we use the expression for the phase we got in a previous section, the negation gains a new meaning:
::{EQUATION()}
\label{phase_negation1}
\phi(\mathbf r, t) = -2\pi\mathbf{k}\mathbf{r} \to - \phi(\mathbf r, t) = 2\pi\mathbf{k}\mathbf{r} = -2\pi(-\mathbf{k})\mathbf{r}
{EQUATION}::
::{EQUATION()}
\label{phase_negation2}
\mathbf k \to - \mathbf k
{EQUATION}(2)::
What we get as a result is extremely profitable in MRI: a 180° RF pulse operates as a centered mirroring in the {EQUATION()}$k${EQUATION}-space, in other words, it negates the {EQUATION()}$\mathbf k${EQUATION} vector of the system. For a graphical illustration we sould go back to the figure with the schematic helix of the spins in the firts imaging page: by rotating the spins folded into a helix by 180° the helicity of the helix becomes the opposite, i. e. the helix will twist to the other direction.
Therefore we can cover both positive and negative {EQUATION()}$k${EQUATION}-values by applying a gradient after the 90° pulse just like in the FID sequence, then "throw" the system from the reached positive {EQUATION()}$k${EQUATION} value to its negation by a 180° pulse, and finally apply the same gradient while measuring the signal. Such a method is called "spin echo imaging", the sequence diagram is shown in Figure 2.
{img fileId="3179" thumb="y" rel="box[g]" width="720" imalign="center" align="center" desc=Figure 2. Spin echo imaging sequence. Right: Trajectory in k-space.}
In Figure 2. we have introduced some new notations. First, the pulses are denoted by radians as instead of degrees, as usual in the literature. Second, we indicated the direction of the pulses in the rotating frame. In this case these directions does not have great importance but in a more complicated sequence they can be crucial.
As we have seen the {EQUATION()}$k${EQUATION}-space is properly measured in this way, however, we still need to deal with the refocusing property of the {EQUATION()}$\pi${EQUATION} pulse. That is, with a sequence like this we will generally have two echoes: one from the gradient refocusing called gradient echo and one from the spins dephased by field inhomogeneities and refocused by the {EQUATION()}$\pi${EQUATION} pulse, i. e. a traditional spin echo. We take into account the space-dependent field inhomogeneities by an additional term {EQUATION()}$\Delta B(z)${EQUATION} in the field and in spin phase. It is important to notice that field inhomogeneities alter the spin phase at all time, not just when the gradient fields are switched on.
::{EQUATION()}
\label{spin_echo_imaging1}
B = B_0 + \Delta B (z) + G(t)z
{EQUATION}(3)::
::{EQUATION()}
\label{spin_echo_imaging2}
\phi(z,t) =
\begin{cases}
-\gamma \Delta B(z)t - \gamma G z (t-t_1) & t_1 < t < t_2 \\
\gamma \Delta B(z) \tau + \gamma G z(t_2-t_1) - \gamma \Delta B(z) (t-\tau) - \gamma G z(t-t_3) & t_3 < t < t_4
\end{cases}
{EQUATION}(4)::
As we have seen it previously the gradient echo will come at {EQUATION()}$T_E = t_3 + t_2 - t_1${EQUATION} while the spin echo will arise at {EQUATION()}$2\tau${EQUATION}. Usually the gradient fields scheduled in such a way that these two echoes come in the same time, meaning that {EQUATION()}$2\tau = t_3 + t_2 - t_1${EQUATION}.