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Chapter 6

MEDRADSC 2Z03 Chapter Notes - Chapter 6: Larmor Precession, Fourier Transform, Sinc Function

Medical Radiation Sciences
Course Code
Dawn Danko

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MRI – Spatial Encoding
Firstly, the use of a frequency selective RF pulse in combination with a field
gradient can produce signal from a slice of spins through the subject.
Then, orthogonal read and phase encoding gradients are applied to spatially
encode the spins within the plane of the slice.
A series of echoes is acquired using a ramp of phase encoding gradients to fully
characterise the frequencies present in the slice.
Having acquired the array of echo data, a two-dimensional Fourier transformation
is applied to produce a spatial map of signal, representing the distribution of water
within the subject.
The frequency bandwidth excited by a pulse is inversely proportional to the length
of the pulse, or pulse time.
So a short pulse will excite a broad range of frequencies and as the pulse length is
increased, the excitation frequency bandwidth decreases.
Fourier pairs are waveforms that are interrelated by the Fourier transform.
The Fourier transform of a rectangular waveform is a sinc function with the form
sin( x)/ x. π π
Therefore, an RF waveform amplitude modulated with a sinc shape will generate
a rectangular shaped RF excitation profile.
Shaped pulses of this type are frequently used for selective excitation of a specific
bandwidth of frequencies.
The magnetic field strength B0 determines the precessional frequency of the
nuclei according to the Larmor equation.
We have considered the magnetic field to be constant over the sample and so all
spins have the same Larmor frequency, unless there is a variation due to field
We consider the Larmor frequency of spins along a particular axis through the
magnet, say along the B0 direction.
All spins will have the same Larmor frequency at every position along that
direction (blue lines).
However, if a variation of the B0 field is applied in that direction, such that the
field strength is linearly increased from one end to the other (green lines) then the
Larmor frequency will vary as a function of position along that direction.
In the centre, the frequency will not have changed.
This linearly changing B0 field is called a field gradient and is generated in an
MRI magnet by applying current through specially designed coils of wire.
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The linear field gradient adds to the main field. If we apply a steeper field
gradient, then the difference in Larmor frequencies at the same two points along
the B0 direction will increase (orange lines).
So we now have a means by which we can spatially encode the frequency of spins
along a particular direction in space.
A combination of a linear field gradient and its associated frequency gradient,
together with a frequency selective RF pulse can be used to excite a specific slice
of spins.
In this animation (below), the vertical axis indicates an RF pulse that has a
2000Hz excitation bandwidth.
If a field gradient is applied which produces a frequency variation of 2000 Hz per
centimetre along its direction, then it follows that only a section of the sample will
be excited by the pulse creating a slice of excited spins with a thickness of 1 cm.
This slice will be orientated in the plane perpendicular to the direction of the field
gradient and will be centred at the central frequency of the RF pulse.
The thickness of this slice can be altered by changing the bandwidth of the RF
pulse: increasing the pulse length will decrease the bandwidth and therefore make
the slice thinner; decreasing the pulse length will increase the bandwidth,
selecting a thicker slice.
Alternatively, we can change the steepness of the field gradient.
As the gradient is increased, that is, increasing the frequency spread per
centimetre, the slice of spins will become narrower,
for example if the gradient strength was doubled to 4000 Hz/cm then the slice of
excited spins would be 5 mm thick.
Those spins that reside within a slice where the Larmor frequency matches the
central frequency of the RF pulse will be excited, with a slice thickness according
to the pulse bandwidth and gradient strength.
The position of the slice can be moved, simply by changing the central RF
frequency of the selective pulse.
We can now scan along the direction of the slice gradient creating a slice of
excited spins at any position we choose.
This use of selective RF pulses in the presence of a field gradient is the basis for
slice selection in MRI and this gradient is referred to as the slice gradient.
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