Читать книгу Essential Concepts in MRI - Yang Xia - Страница 29

In contrast to the situation in CW NMR (Section 2.9) where each particular type of nucleus in a chemical is brought into the resonance individually and sequentially, the use of an rf pulse excites simultaneously all NMR-active nuclei in the chemical. This method of NMR experiments is called FT NMR, which is much more efficient than CW NMR and offers the opportunity of SNR improvement by co-addition of many spectral responses.

In an FT NMR experiment, a resonant rf pulse results in a non-zero transverse component of the magnetization M, which precesses in the transverse plane. This precessional motion of M can be detected by means of a receiver coil (often called an rf coil, cf. Chapte 5.2), which can sense the transverse electromotive force (emf) and produces an FID signal.

Immediately following a 90°|x_′ pulse (i.e., after B₁(t) has been turned off), M is given by

(2.28a)

(2.28b)

Subsequently, assuming a uniform field B₀, the evolution of magnetization can be obtained by solving the Bloch equation, as

(2.29a)

(2.29b)

(2.29c)

Since there is a 90˚ phase difference between Mx(t) and My(t), we can combine the two transverse components (the FID) expressed in Eqs. (2.29a) and (2.29b) into a complex term, as

(2.30)

The complex nature of the NMR signal means that we can measure not only the amplitude of the signal but also its phase, which is one of the remarkable advantages of NMR and MRI. In comparison, several other types of spectroscopy and imaging (e.g., Fourier transform infrared spectroscopy and imaging, computer tomography) can only measure the amplitude of the signal.

The time evolution of the Mx(t) and My(t) components are illustrated in Figure 2.14a and Figure 2.14b, where at t = 0 (the end of the 90˚ rf pulse), Mx(t = 0) = 0, and My(t = 0) = M₀. The FT of the FID signal is a Lorentzian in the frequency domain with a line width of (πT₂)^-1, shown in Figure 2.14c and Figure 2.14d, which are commonly termed as the absorption signal and the dispersion signal, respectively.

Figure 2.14 (a) and (b) The time-domain NMR signal in the transverse plane (the FID) is complex and contains real and imaginary components. By the way of Fourier transform, the time-domain NMR signal can be represented by the equivalent signals in the frequency domain, as the absorption and dispersion components, shown in (c) and (d). The peak shift f₀ in the frequency domain corresponds to the oscillation of the FID in the time domain. The recovery of the longitudinal magnetization is shown in (e). As noted in Figure 2.10, since T₂ is commonly much shorter than T₁, the transverse magnetization in (a) and (b) should decay to zero long before the longitudinal magnetization (e) returns to its thermal equilibrium [i.e., the time scales in (a) and (b) are comparable but are different from the time scale in (e)].

The time evolution of the Mz(t) component expressed by Eq. (2.29c) is illustrated in Figure 2.14e. Note that since T₁ > T₂ in most liquid-containing specimens, it takes much longer for Mz(t) to return to its thermal equilibrium than for My(t) and Mx(t) to decay to zero; that is, the time axes in the schematics in Figure 2.14 between (a) and (b) are scaled but between (a) and (e) or (b) and (e) are not scaled.

When a 90˚|y’ pulse is used in the excitation, the solutions of the Bloch equation take the form

(2.31a)

(2.31b)

(2.31c)

which only switches the oscillation terms between Mx(t) and My(t), or in other words the phase of the signal.