In the previous section you saw how a simple two dimensional imaging procedure could be performed using the backprojection technique. In this section we will introduce the concept of a third category of magnetic field gradient called a phase encoding gradient and incorporate it plus the slice selection gradient and frequency encoding gradient, to see how present day tomographic, Fourier transform MRI is performed.

The phase encoding gradient is a gradient in the magnetic field
B_{o}. The phase encoding gradient is used to impart a specific phase angle to a transverse magnetization vector.
The specific phase angle depends on the location of the transverse magnetization vector.

For example, lets imagine we have three regions with spin. The transverse magnetization vector from each spin has been rotated to a position along the X axis. The three vectors have the same chemical shift and hence in a uniform magnetic field they will possess the same Larmor frequency.

If a gradient in the magnetic field is applied along the X direction the three vectors will precess about the direction of the applied magnetic field at a frequency given by the resonance equation.

While the phase encoding gradient is on, each transverse magnetization vector has its own unique Larmor frequency. Thus far, the description of phase encoding is the same as frequency encoding. Now for the difference. If the gradient in the X direction is turned off, the external magnetic field experienced by each spin vector is, for all practical purposes, identical. Therefore the Larmor frequency of each transverse magnetization vector is identical.

The phase angle, , of each vector, on the other hand, is not identical. The phase angle being the angle between a reference axis, say the Y axis, and the magnetization vector at the time the phase encoding gradient has been turned off. There are three distinct phase angles in this example.

Just as in the examples of the frequency encoding gradient, if we had some way of measuring the frequency (in this case phase) of the spin vectors we could assign them a position along the X axis. We are now ready to explain the simple Fourier transform tomographic imaging sequence.

One of the best ways to understand a new imaging sequence is to examine a timing diagram for the sequence. The timing diagram for an imaging sequence has entries for the radio frequency, magnetic field gradients, and signal as a function of time.
The simplest FT imaging sequence contains a
90^{o} slice selective pulse,
a slice selection gradient pulse,
a phase encoding gradient pulse,
a frequency encoding gradient pulse,
and a signal.
The pulses for the three gradients represent the magnitude and duration of the magnetic field gradients.
The actual timing diagram for this sequence is a bit more complicated, this one has been simplified for introductory purposes.
The first event to occur in this imaging sequence is to turn on the slice selection gradient.
The slice selection RF pulse is applied at the same time.
The slice selective RF pulse is an apodized sinc function shaped burst of RF energy.
Once the RF pulse is complete the slice selection gradient is turned off and a phase encoding gradient is turned on.
Once the phase encoding gradient has been turned off a frequency encoding gradient is turned on and a signal is recorded.
The signal is in the form of a free induction decay.
This sequence of pulses is usually repeated 128 or 256 times to collect all the data needed to produce an image.
The time between the repetitions of the sequence is called the repetition time, TR.
Each time the sequence is repeated the magnitude of the phase encoding gradient is changed.
The magnitude is changed in equal steps between the maximum amplitude of the gradient and the minimum value.
Here is a quick example of what eight phase encoding steps worth of the sequence would look like.

The slice selection gradient is always applied perpendicular to the slice plane. The phase encoding gradient is applied along one of the sides of the image plane. The frequency encoding gradient is applied along the remaining edge of the image plane. The following table indicates the possible combination of the slice, phase, and frequency encoding gradient.

Gradient | |||
---|---|---|---|

Slice Plane | Slice | Phase | Frequency |

XY | Z | X or Y | Y or X |

XZ | Y | X or Z | Z or X |

YZ | X | Y or Z | Z or Y |

Now we will examine the sequence from a macroscopic perspective of the spin vectors.
Imagine a cube of spins placed in a magnetic field.
The cube is composed of several volume elements each with its own net magnetization vector.
Suppose we wish to image a slice in the XY plane. The
B_{o} magnetic field is along the Z axis.
The Slice selection gradient is applied along the Z axis.
The RF pulse rotates only those spins packets within the cube which satisfy the resonance condition.
These spin packets are located within an XY plane in this example.
The location of the plane along the Z axis with respect to the isocenter is given by

where Δν is the frequency offset from ν_{o}
( *i.e.* ν - ν_{o} ),
G_{s} the magnitude of the slice selection gradient,
and γ the gyromagnetic ratio.
Spins located above and below this plane are not affected by the RF pulse.
They will therefore be neglected for purposes of this presentation. To simplify the remainder of the presentation, we shall concentrate on a 3x3 subset of the net magnetization vectors.
The picture of these spins in this plane looks like this.
Once rotated into the XY plane these vectors would precess at the Larmor frequency given by the magnetic field each was experiencing. If the magnetic field was uniform, each of the nine precessional rates would be equal.
In the imaging sequence a phase encoding gradient is applied after the slice selection gradient. Assuming this is applied along the X axis, the spins at different locations along the X axis begin to precess at different Larmor frequencies.
When the phase encoding gradient is turned off the net magnetization vectors precess at the same rate, but possess different phases.
The phase being determined by the duration and magnitude of the phase encoding gradient pulse.

Once the phase encoding gradient pulse is turned off a frequency encoding gradient pulse is turned on. In this example the frequency encoding gradient is in the -Y direction. The frequency encoding gradient causes spin packets to precess at rates dependent on their Y location. Please note that now each of the nine net magnetization vectors is characterized by a unique phase angle and precessional frequency. If we had a means of determining the phase and frequency of the signal from a net magnetization vector we could position it within one of the nine elements.

A simple Fourier transform is capable of this task for a single net magnetization vector located somewhere within the 3x3 space. For example, if a single vector was located at (X,Y) = 2,2, its FID would contain a sine wave of frequency 2 and phase 2. A Fourier transform of this signal would yield one peak at frequency 2 and phase 2. Unfortunately a one dimensional Fourier transform is incapable of this task when more than one vector is located within the 3x3 matrix at a different phase encoding direction location. There needs to be one phase encoding gradient step for each location in the phase encoding gradient direction. The point is you need one equation for each unknown you are trying to solve for. Therefore if there are three phase encoding direction locations we will need three unique phase encoding gradient amplitudes and have three unique free induction decays. If we wish to resolve 256 locations in the phase encoding direction we will need 256 different magnitudes of the phase encoding gradient and will record 256 different free induction decays.

The free induction decays or signals described above must be Fourier transformed to obtain an image or picture of the location of spins. The signals are first Fourier transformed in the frequency encoding direction (X in this example) to extract the frequency domain information, and then in the phase encoding direction (Y in this example) to extract information about the locations in the phase encoding gradient direction.

To see the relationship between the signals (also called the raw data and k-space data), the Fourier transforms in the phase and frequency encoding directions, and the resultant image, several examples are presented.

There is a single voxel with net magnetization. The time and phase domain data, often referred to as the raw data, will look like this. Notice there is one frequency of oscillation in the time domain. You may also be able to see one frequency of oscillation in the phase direction. Fourier transforming first in the frequency encoding direction yields a series of peaks at the frequency corresponding to the X location of the voxel with spin.

Notice how the amplitude of the peaks are oscillating as you look from top to bottom in the phase encoding direction. We can readjust our perspective of the data to make this more obvious. Fourier transforming down in the phase encoding direction yields a single peak. The frequency and phase of this peak correspond to the location of the voxel with spins

There is a single voxel with net magnetization at a new frequency encoding location but the same phase encoding location. The raw data will look like this. Notice there is still one frequency of oscillation in the time domain but it is different than in the first example. You may also be able to see one frequency of oscillation in the phase direction. Fourier transforming first in the frequency encoding direction yields a series of peaks at the frequency corresponding to the new X location of the voxel with spin.

Notice how the amplitude of the peaks are oscillating as you look from top to bottom in the phase encoding direction. We can readjust our perspective of the data to make this more obvious. Fourier transforming down in the phase encoding direction yields a single peak. The frequency and phase of this peak correspond to the location of the voxel with

There is a single voxel with net magnetization. The frequency encoding location is unchanged but the phase encoding location has been changed. The raw data will look like this. Notice there is still one frequency of oscillation in the time domain. You may also be able to see one frequency of oscillation in the phase direction. Fourier transforming first in the frequency encoding direction yields a series of peaks at the frequency corresponding to the X location of the voxel with spin.

Notice how the amplitude of the peaks are oscillating as you look from top to bottom in the phase encoding direction. We can readjust our perspective of the data to make this more obvious. Fourier transforming down in the phase encoding direction yields a single peak. The frequency and phase of this peak correspond to the location of the voxel with spins

There are now two voxels with net magnetization in the imaged plane. The raw data will look like this. Notice there is a beat pattern to the frequency of oscillation in the time domain indicating more than one frequency. You may also be able to see a beat frequency in the oscillation in the phase direction, also indicating two frequencies. Fourier transforming first in the frequency encoding direction yields a series of peaks at two frequencies corresponding to the X locations of the voxels with spin.

Notice how the amplitude of the peaks are oscillating as you look from top to bottom in the phase encoding direction. We can readjust our perspective of the data to make this more obvious. Fourier transforming down in the phase encoding direction yields two peaks. The frequency and phase of these peaks correspond to the location of the voxels with spin.

The Fourier transformed data is displayed as an image by converting the intensities of the peaks to intensities of pixels representing the tomographic image.

Recall from
Chapter 5
that relationship between the sampling rate, f_{s}, and the spectral width. This same relationship applies here and determines the field of view (FOV), or distance across the image, in the frequency encoding direction. This relationship assumes quadrature detection of the transverse magnetization.

To avoid the wrap around problem, the field of view must be greater than the width of the imaged object. More information on the wrap around problem will be presented in the section on imaging artifacts.

The phase encoding gradient is typically varied from a maximum value of G_{max} and a minimum value of - G_{max} in 128 or 256 equal steps.
The relationship between the FOV and G_{max} is

where N is the number of phase encoding steps. The integral
G_{max} dt is over the time the phase encoding gradient is turned on. The shape of the phase encoding gradient pulse is immaterial as long as the area under the pulse is appropriate.
.

In conclusion, during the MRI signal acquisition the phase and frequency encoding gradients are varied and a signal recorded. The signal is used to fill k-space. The order in which k-space is filled depends on the timing and order in which the phase and frequency encoding gradients are applied. The phase encoding gradient is used to position the spin system at a specific line in k-space. Application of the frequency encoding gradient and recording signal as a function of time moves the spin system across a line in k-space. This process fills k-space with data about the spins in the image which is Fourier transformed to produce the image.

When two features in an image are distinguishable, they are said to be resolved. The ability to resolve two features in an image is a function of many variables; T_{2}, signal-to-noise ratio, sampling rate, slice thickness, and image matrix size, to name a few. Resolution is a measure of image quality. When two features 1 mm apart are resolvable in an image, the image is said to be a higher resolution image than one where two features are not resolvable.
Resolution is inversely proportional to the distance of two resolvable features.

It is easy to see the relationship between resolution, FOV, and number of data points, N, across an image. We will never resolve two features located less than FOV/N, or a pixel, apart. You might think that increasing the number of data points across an image would improve resolution.
Increasing the number of data points will decrease the pixel size, but not improve the resolution. Even with a noiseless image and optimal contrast, you may not be able to resolve two features the size of a pixel because T_{2}* comes into play.

A magnetic resonance image can be thought of as a convolution of the NMR spectrum of the spins with their spatial concentration map. This will be easier to describe if we assume a one-dimensional image, h(x), consisting of a single type of spin. If g(x) is the distribution of the spins, and
f(ν)is the NMR spectrum of the spins, and
f(ν G_{x}^{-1} γ^{-1})is the NMR spectrum in distance units in the presence of the magnetic field gradient G_{x}, then

Based on the discussion of Fourier pairs in Chapter 5, the full line width in Hz at half height, Γ, is

Compare the result, h(x), of the convolution of the NMR spectrum f(x) from a type of spin with a distribution g(x) for a short T_{2}* (wide Γ)
,
with that of a long T_{2}* (narrow Γ)
.

Therefore, the pixel size should be chosen to be approximately equal to

Here are two images of an infinitely small point source of NMR signal.
One has a long T_{2}* and the other a short T_{2}*.
Both images were recorded with a pixel size much less than (π G_{x} γ T_{2}*)^{-1}.

- Two samples are located in a magnetic field at x=0 cm and x=5 cm. A 1 G/cm phase encoding gradient is applied in the +X-direction for 10 ms. How much phase will be acquired by the sample located at x=5 cm relative to that located at x=0 cm?
- You wish to produce an image of hydrogen nuclei in the zx-plane. What directions should the slice, phase, and frequency encoding gradients be applied in?
- A particular magnetic resonance imager uses a 1 G/cm frequency encoding gradient to produce an image with an 8 cm FOV. What quadrature sampling rate should be used to produce this FOV when imaging hydrogen?
- You wish to produce an image with a 8 cm FOV and 256 phase encoding gradient steps. The maximum phase encoding gradient you can produce is 1 G/cm. What should the width of the phase encoding gradient be?
- Show that: FOV = f
_{s}/ (γ G_{f}) . - Two samples are located in a magnetic field at x=0 cm and x=-4 cm. A 2 G/cm phase encoding gradient is applied in the +X-direction for 5 ms.
How much phase will be acquired by the sample located at x=-4 cm relative to that located at x=0 cm?
- You are using an imaging sequence that applies a slice selection gradient in the x direction, a phase encoding gradient in the z direction, and a frequency encoding gradient in the y direction. What type of imaging plane will be produced?
- A particular magnetic resonance imager uses a 2 G/cm frequency encoding gradient and a quadrature sampling rate of 32 kHz. What field of view will be obtained when imaging hydrogen?
- You wish to produce an image with a 4 cm FOV and 512 phase encoding gradient steps. The maximum phase encoding gradient you can produce is 2 G/cm. What should the width of the phase encoding gradient be?

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