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Fig. 3. After an initial 90° radiofrequency excitation pulse, all magnetic moments are in phase, but slowly drift out of phase because of a distribution of Larmor frequencies. The figure illustrates this situation in a rotating frame of reference in which the Larmor frequency of one magnetic moment exactly matches the rotation frequency of the reference frame. By applying a 180° pulse at t = t, the magnetic moment precessing more rapidly than the average (light gray arrow) has a negative phase after the 180° pulse and vice versa for the magnetic moment precessing more slowly (solid black arrow). The precession frequency of the magnetic moments remains unchanged, and they regain phase coherence at t = 2t. The initial free induction decay and the echo signal after the 180° pulse are shown in the graph. By applying another 180° pulse at t = 3%, a second spin-echo can be produced at t = 2%, and this can be repeated while t is not much larger than T2. This example also illustrates the fact that decay of the echo amplitudes (with time constant T2) is considerably slower than the free induction decay (with time constant T2*).

### 2.6. The Spin-Echo

A loss of phase coherence because of any spread in Larmor frequencies, such as that caused by magnetic field inhomogene-ities, can be (at least partially) reversed by applying a 180° pulse that flips the magnetization in the x-y plane over such that the faster precessing spins now lag behind, and the more slowly precessing spins are ahead, compared to spins precessing at the mean Larmor frequency.

Figure 3 illustrates how the spin-echo is a very effective means of rephasing the spins, such that after the 180° pulse an attenuated mirror image of the free induction decay is observed. Once the echo amplitude peaks, the spread of Larmor frequencies again causes a loss of phase coherence. Multiple 180° pulses can be applied to reverse the loss of phase coherence repeatedly and thereby produce a train of spin-echoes. The decay of the spin-echo amplitudes is governed by the decay constant T2; a free induction decays with a characteristic time constant T2*, with T2* < T2.

Fig. 4. (A) An illustration of the principle of magnetic resonance imaging (MRI) for a one-dimensional (1D) distribution of spins with a density distribution as a function of position. (B) The variation of the Larmor frequency in the presence of a linear magnetic field gradient.

(C) The free induction decay recorded after application of a short radiofrequency pulse and in the presence of a magnetic field gradient.

(D) The frequency spectrum of the signal in (C) is calculated by Fourier analysis with a fast Fourier transform (FFT). The frequency profile reproduces the shape of the spin density distribution shown in (A), and this is a result of the linear variation of the Larmor frequency as a function of the position during application of a magnetic field gradient. MR, magnetic resonance.

Fig. 4. (A) An illustration of the principle of magnetic resonance imaging (MRI) for a one-dimensional (1D) distribution of spins with a density distribution as a function of position. (B) The variation of the Larmor frequency in the presence of a linear magnetic field gradient.

(C) The free induction decay recorded after application of a short radiofrequency pulse and in the presence of a magnetic field gradient.

(D) The frequency spectrum of the signal in (C) is calculated by Fourier analysis with a fast Fourier transform (FFT). The frequency profile reproduces the shape of the spin density distribution shown in (A), and this is a result of the linear variation of the Larmor frequency as a function of the position during application of a magnetic field gradient. MR, magnetic resonance.

Importantly, for cardiac imaging applications, it is useful to note that the spin-echo and spin-echo trains in particular provide a method to attenuate the signal from flowing blood while obtaining "normal" spin-echoes from stationary or slow-moving tissue.

### 2.7. Magnetic Resonance Imaging

The basic concept of MRI consists of producing a spatial variation of the magnetic field, which gives rise to a distribution of Larmor frequencies. A linear variation of the magnetic field strength will cause a one-to-one correspondence between the Larmor frequency and a spatial position. A gradient that is applied during the recording of the MR signal is called a readout gradient. The readout gradient creates a distribution of frequencies of width proportional to the readout gradient strength Gr and to the extent in the readout direction of the object that is imaged. The decay of transverse magnetization is recorded while this magnetic field gradient is left on. The frequency contents of the free induction decay are analyzed by Fourier transformation. As shown in Fig. 4, the 1D frequency spectrum will be proportional to the density distribution of nuclear magnetic moments along the direction of the applied magnetic field gradient.

The MR signal needs to be sampled at a frequency that takes into account the extremes of this frequency distribution. If the desired field of view dimension in the direction of the readout gradient is Lx, then the sampling frequency needs to be equal to or greater than the Nyquist frequency fNyq; that is, the sampling frequency should at least be as large as the Nyquist frequency:

2.8. Slice-Selective Excitation

As a tomographic imaging modality, MRI relies on the selective excitation of spins within a slice or slab of arbitrary, user-specified orientation, position, and thickness. The selective excitation of the nuclear spins requires the application of a radiofrequency pulse, which produces the desired response (e.g., the creation of a transverse magnetization component) only within a well-defined region, typically a slice of thickness Ax. For small tip angles, this creation of a transverse magnetization component can be described in terms of the frequency spectrum of the applied radiofrequency pulse.

For a slice-selective radiofrequency excitation, the desired slice orientation determines the direction of the magnetic field gradient. A radiofrequency excitation of limited bandwidth excites spins in a slice that is perpendicular to the direction of the applied magnetic field gradient, and the slice width is proportional to the bandwidth of the radiofrequency pulse (Fig. 5).

### 2.9. Spatial Phase Encoding

The basic idea of phase encoding consists of applying a gradient pulse that gives rise to a position-dependent phase angle change for the transverse magnetization. We denote the angle of the transverse component in the rotating frame by 9 and refer to it as the phase of the transverse magnetization. Under ideal conditions, the phase angle 9 is the same for all spins in the slice right after a radiofrequency excitation.

Application of a gradient pulse Gx(t) in the x direction for a duration T causes the phase to vary as a function of the x-coor-dinate, and the phase angle at the end of the gradient pulse is given by

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