## Frequency And Phase Encoding

In practice MR imaging is most frequently performed by using the approach suggested by Kumar et al. . This approach employs a sequence of pulsed magnetic field gradients which are applied during a free induction decay or a spin echo to ensure that the signal is given by Fourier transform of the transverse magnetization in the sample. At this stage we are ready to discuss two basic imaging techniques for Fourier encoding: frequency encoding and phase encoding.

### Frequency Encoding

Frequency encoding is implemented by acquiring signal in the presence of an external magnetic field gradient. The purpose of the gradient, known as the frequency-encoding or readout gradient, is to make the Larmor frequency of nuclei spatially dependent during signal acquisition.

The signal acquired in the presence of a readout gradient is composed of components with frequencies from a narrow range, known as the signal bandwidth, around the Larmor frequency at the center of the field-of-view. The frequency lo0 is also known as the reference frequency. Each of the signal components is produced by spins from a certain location in the object. Because the typical signal bandwidth is much smaller than io0,3 it is important to remove the reference frequency from the signal in order to distinguish between the spectral components within the bandwidth. In order to do this a special technique, referred to as the phase-sensitive detection (see Appendix), is used to shift the signal down in frequency by iv() As a result the output signal consists of primarily low frequency components.4

At this stage we consider signal acquisition in the presence of a readout gradient, Gx = dBz/dx, given by

—Gotx, t0 < t < t0 + Ts/2 Gr = G0iX, t0 + TJ2 <t<t0 + 3TJ2 (3.2.1)

0, elsewhere where Gij x is a constant (Figure 3.2). Suppose that the signal acquisition starts at t0 + TJ2 and ends at t0 + 3Ts/2. The phase of the transverse magnetization accumulated during the interval [£0, t0 + Ts/2] in the presence of the first (defocusing) lobe of the readout gradient can be written as ft0 + TJ2

The readout gradient reversal at t = t0 + Ts/2 initially causes refocusing of spins, which in turn leads to a signal increase, known as the gradient echo, such that the maximum signal occurs in the center of the acquisition interval (Figure 3.2). It can easily be shown that during the acquisition the phase of the transverse magnetization is given by

</>(x, t) = >yG0jc(t ~ k)x - jG0)XTsx. (3.2.3)

Suppose that N samples of the signal are collected at t = t0 + Ts/2 + prw, where an integer p changes from 0 to N — 1. The sampling interval tw — Ts/N is known as the dwell time and Ts is referred to as the acquisition or readout time. Using Eq. (3.2.3) we can express the phase of magnetization as

3 For example, the Larmor frequency of 1H nuclei at 1.5 Tesla is about 63.86 MHz, whereas typical signal bandwidth in MRI is less than 100 Khz.

4 Strictly speaking, during phase-sensitive detection the signal is shifted in frequency by +/ — loq. Therefore, the output signal also includes high-frequency components that are centered at 2lj0. However, these components are subsequently removed from the signal by passing them through a low-pass filter.  form; (b) accumulated phase; (c) NMR signal.

To understand how NMR signal obtained with frequency encoding can be used for image reconstruction we need to consider the equation describing the relationship between the signal and magnetization in the object. We will assume for simplicity that Ts <C T2 and that the effect of magnetic field inhomogeneities during data acquisition can be neglected. We will also assume that the transmitter and receiver coils used for excitation and signal detection, respectively, generate uniform r.f. fields. From the principle of reciprocity (see Chapter 1), it then follows that the sampled signal in a one-dimensional case can be written as

S(n) = S{t{n)) = J Mxy(x)emx't{n]) dx = Mxy{x)ejk'{n)x dx. (3.2.5)

In this equation £ is a constant that, for simplicity, will be given the value of one in the derivations below, Mxy is the (complex) transverse magnetization in the object (see Appendix), and kx{n) = iG0tXTwn.

From Eq. (3.2.5) it follows that the acquired signal is defined by Fourier transform of the magnetization in the object. Therefore, according to the results obtained in the previous section, inverse discrete FT of the signal in Eq. (3.2.5) will reconstruct Mxy with spatial resolution Lx/N, where

Sampling

To understand better the important relationship between the field-of-view, readout gradient, and dwell time in Eq. (3.2.7), we need to recall that according to the sampling theorem  a continuous function f(t) can be reconstructed fully from a series of discrete samples provided that: a) FT{f(t)} is zero for all frequencies \v\ > umax; and b) sampling interval does not exceed l/2vmax. Conversely, if sampling interval exceeds 1/2umax then some of the high-frequency components of f(t) cannot be determined. The insufficiently high sampling rate in the latter case, often referred to as under sampling, causes aliasing, which generally makes it impossible to reconstruct a continuous function faithfully from its discrete samples.

The signal bandwidth in MR imaging is defined by the readout gradient and field-of-view

The signal is digitized at a rate of r^1 by using a special device called analog-to-digital converter (ADC). By examining Eq. (3.2.7) in terms of the sampling theorem, we realize that this equation ensures that the sampling rate is sufficient (i.e., tw = 1/2vmax) to avoid aliasing when reconstructing a continuous NMR signal from its discrete samples.

### Phase Encoding

Two-dimensional spatial encoding in MRI is normally achieved through the use of an additional gradient, known as the phase-encoding gradient, which is perpendicular to the frequency-encoding gradient. In the presence of the phase-encoding gradient, Gy = dBz/dy, applied prior to readout, the transverse magnetization acquires a phase

where tph is the duration of the gradient. As a result of frequency and phase encodings the NMR signal in a two-dimensional case is given by

2D Fourier transform of the magnetization:

S = J J Mxy(x,y)ejk'ln,x+J'My> dxdy = J J Mxy(x,y)ejk*[n)x+Jk>ydxdy, (3.2.10)

Phase encoding requires repetitive excitations of the transverse magnetization in the object in order to collect signals at different values of ky. In the initially proposed phase-encoding scheme, changes in ky were achieved by varying the duration of the phase-encoding gradient while keeping its strength constant . A more common approach is to vary the strength of the phase-encoding gradient in a step-like fashion; that is,

while its duration, tph, is kept constant . In Eq. (3.2.11) G0 y is a constant, and an integer m changes from —M/2 to Mj2 - 1 (M is the total number of phase-encoding steps).

Using Eq. (3.2.11), the signal obtained with frequency and phase encodings can be written as

S(n,m) = Jj Mxy(x,y)ejkAn)x+jk>{m)ydxdy, (3.2.12)

where kx(n) = ^G0xrwn and ky(m) = ^Gq ytpyjn. After all phase encodings are implemented, image reconstruction is performed by computing inverse discrete FT of S(n,m). Using results from the previous section we find that the reconstructed image intensity is given by the convolution of the transverse magnetization in the sample and a two-dimensional PSF. The resultant spatial resolution in the x-y plane is (.Lx/N) x (Ly/M), where Ly is the field-of-view in the direction of the phase-encoding gradient:

Note that the reconstructed image intensity is a complex quantity which can be written in polar form where A and 9 are the amplitude and phase of the intensity, respectively. It is the amplitude of the intensity that is normally used to display MR images (often referred to as magnitude images), while the phase information is neglected. However, in several instances (e.g., imaging of flow, imaging of static magnetic field inhomogeneities etc.) the reconstructed phase of the intensity may also be used. Basic aspects of image display are discussed in the Appendix.

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