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The Basics of MRI

Chapter 7

FT IMAGING PRINCIPLES

y Introduction

y Phase Encoding Gradient

y FT Tomographic Imaging

y Signal Processing

y Image Resolution

y Problems

Introduction

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 plusthe slice selection gradient and frequency encoding gradient, to see how present day

tomographic, Fourier transform MRI is performed.

Phase Encoding Gradient

The phase encoding gradient is a gradient in the magnetic field Bo. The phase encoding gradient

is used to impart a specific phase angle to a transverse magnetization vector. The specific phaseangle 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 thesame 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.

= ( Bo + x Gx) = o + x Gx

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

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magnetic field experienced by each spin vector is, for all practical purposes, identical. Thereforethe 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 Xaxis. We are now ready to explain the simple Fourier transform tomographic imaging sequence.

FT Tomographic Imaging

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 90o

slice selective pulse, a slice selection gradient pulse, a phase encodinggradient pulse, a frequency encoding gradient pulse, and a signal. The pulses for the threegradients 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 isan 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 phaseencoding 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 usuallyrepeated 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 isrepeated 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 aquick 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 encodinggradient 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 FrequencyXY 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

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each with its own net magnetization vector. Suppose we wish to image a slice in the XY plane.The Bo 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 resonancecondition. 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

Z = / Gs

where is the frequency offset from o ( i.e. - o ), Gs the magnitude of the slice selection

gradient, and 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. Tosimplify 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 rotatedinto 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 precessionalrates 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 locationsalong 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 frequencyencoding 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 angleand 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 locatedsomewhere 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 signalwould 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 ata 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 uniquefree 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 differentfree induction decays.

Signal Processing

The free induction decays or signals described above must be Fourier transformed to obtain animage or picture of the location of spins. The signals are first Fourier transformed in the

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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.

Exampl e 1:

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 timedomain. 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 frequencycorresponding to the X location of the voxel with spin.

( - o ) = x Gf

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 moreobvious. 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 withspins

Exampl e 2:

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 frequencyof oscillation in the time domain but it is different than in the first example. You may also beable 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 newX location of the voxel with spin.

( - o ) = x Gf

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

Exampl e 3:

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. Noticethere 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

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encoding direction yields a series of peaks at the frequency corresponding to the X location of the voxel with spin.

( - o ) = x Gf

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 moreobvious. 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 withspins

Exampl e 4:

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 inthe frequency encoding direction yields a series of peaks at two frequencies corresponding to theX locations of the voxels with spin.

( - o ) = x Gf

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. Thefrequency 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 peaksto 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.

FOV = f s / Gf

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 onimaging artifacts.

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

G max dt = N / (2 FOV)

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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 isfilled 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 thespins in the image which is Fourier transformed to produce the image.

Image Resolution

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; T2, 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 bea 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, acrossan image. We will never resolve two features located less than FOV/N, or a pixel, apart. Youmight 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 improvethe resolution. Even with a noiseless image and optimal contrast, you may not be able to resolve

two features the size of a pixel because T2* comes into play.

A magnetic resonance image can be thought of as a convolution of the NMR spectrum of thespins 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 thespins, and f()is the NMR spectrum of the spins, and f( Gx

-1

-1)is the NMR spectrum in

distance units in the presence of the magnetic field gradient Gx, then

h(x) = g(x) f( Gx-1

-1

).

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

is

= ( T2*)-1

.

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 T2* (wide ) , with that of a long T2* (narrow ) .

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

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( Gx T2*)-1

.

Here are two images of an infinitely small point source of NMR signal. One has a long T2*

and the other a short T2*. Both images were recorded with a pixel size much less than ( Gx

T2*)-1

.

Problems

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 beacquired by the sample located at x=5 cm relative to that located at x=0 cm?

We use the equation derived from the resonance equation

J = 2T K x Gx X

Where:J = acquired phase

K = gyromagnetic ratio = 42.58 MHz/Tx = position of sample

Gx = phase encoding gradient = 0.0001 T/cm

X = duration of the phase encoding gradient = 10 ms

J(x=0cm) = 0 radians

J(x=5cm) = 1337.69 radians = 5.65 radians = 324o

2. 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?

The slice selection gradient must be perpendicular to the imaged plane, or along y. The

phase encoding gradient can be along either z or x. The frequency encoding gradient isalong the remaining direction x or z.

3. 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?

The FOV is related to the quadrature sampling rate by the following equation which was

derived from the resonancce equation.

FOV = f s / (K Gf )f s = FOV K Gf

f s = (8 cm) (42.58 MHz/T) (0.0001 T/cm)

f s = 34.064 kHz

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4. 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?

G max dt = N / (2 K FOV)

where N = number of phase encoding steps

K = gyromagnetic ratio for H = 42.58 MHz/TFOV = field of view

G max = maximum phase encoding gradientIf we assume a square phase encoding gradient pulse the equation reduces to

t = N / (2 K FOV G max)where t is the duration of the gradient.

t = 4 ms

5. Show that: FOV = f s / ( Gf ) .

Assume quadrature sampling of the time domain signal, and adopt the symbol devinitions

used in The Basic s of MRI . Start with the resonance equation.

R = K BIn the presence of a magnetic field gradient the equation becomes

R = K (Bo + x Gx).Relative to the frequency at the isocenter of the magnet, we have

R = Ro + K x Gx

R-R

o =(R

= K x Gx To record a range of frequencies (R a quadrature sampling rate of f s is required. In the presence of the magnetic field gradient, this range of frequencies corresponds to the

distance across the image in the frequency encoding direction, or the FOV. Therefore.

f s = K FOV Gf ,

or

FOV = f s / (K Gf ).

6. 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?

7.

You are using an imaging sequence that applies a slice selection gradient in the xdirection, a phase encoding gradient in the z direction, and a frequency encoding gradientin the y direction. What type of imaging plane will be produced?

8. A particular magnetic resonance imager uses a 2 G/cm frequency encoding gradient and aquadrature sampling rate of 32 kHz. What field of view will be obtained when imaging

hydrogen?

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9. 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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