# Basic Physics of Nuclear Medicine/MRI & Nuclear Medicine

This is a developing chapter of a wikibook entitled Basic Physics of
Nuclear Medicine.

Magnetic resonance imaging (**MRI**) is widely used to provide
colocalization information for correlative applications with nuclear
medicine images. An introduction to this imaging method is provided in
this chapter. We\'ll consider fundamental features to a depth sufficient
to describe the contrast and spatial characteristics of MR image data so
that spatial correspondence features with nuclear medicine data can be
appreciated. Since MRI is such a vast field, we\'ll limit our treatment
to a description of essential features only and you should refer to the
Bibliography
for more comprehensive accounts.

We\'ll start with an overview of the imaging process and then describe
magnetic resonance features of the simplest atomic nucleus known, i.e.
^1^H. From there we\'ll describe phenomena which generate contrast in MR
images and finish with a description of image formation using spatial
encoding techniques.

## MR Imaging Process

In principle, MRI is quite a simple imaging technique, as illustrated in
the animated graphic below. A patient is first of all placed inside a
large magnet -- the poles of which are indicated by the N and S boxes in
the figure. Radio waves are then used to excite the nuclei of hydrogen
atoms within the patient\'s body using an external coil and these are
subsequently re-emitted by the patient in a manner which is
characteristic of the NMR properties of the tissues involved. These
radio waves are detected using an external coil, digitized, processed by
a computer and displayed as tomographic slices revealing the
distribution of different tissues.

```{=html}
<div class="center">
```
  ----------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  !Illustration of the MR imaging process.   !`<small>`{=html}Representation of the water molecule with the oxygen atom in blue and the two hydrogen atoms in red. The human body contains a substantial amount of water and therefore has a large number of hydrogen nuclei distributed throughout its volume to generate NMR signals for imaging purposes.`</small>`{=html}
                                                                                                        
  ----------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

```{=html}
</div>
```
The imaging process is a bit more complex in practice, as indicated in
the following figure:

```{=html}
<div class="center">
```
  -------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  !Flow diagram for the MR imaging process.   !Illustration of the principal components of MRI scanner. Tx and Rx refer to RF transmitter and receiver circuitry, respectively. The gradient coils are used to establish gradients in the magnetic field generated by the magnet.
                                                                                                           
  -------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

```{=html}
</div>
```
The complexity arises from the use of gradient magnetic fields and
sophisticated radio wave excitations so as to provide spatial
localization information for the subsequent detection phase, and to
isolate a particular NMR characteristic of interest. One powerful
feature of MRI is that the method can generate images depicting a number
of different contrast characteristics. Three such image types are shown
as examples below:

!Three types of MR image: the T1 weighted image depicts relatively
bright grey matter and dark CSF; the T2 weighted image highlights the
CSF, while the PD weighted image shows little contrast between
tissues.

The physics behind the production of these types of images is briefly
described in this chapter. We\'ll start with some insights into the
nature of nucleons which we have not considered previously in this
wikibook.

#top

## Nuclear Spin

Our earlier
consideration
of the atomic nucleus concentrated on its composition in terms of the
mass, electrical charge and the relative number of its neutrons and
protons -- the intention being to introduce the phenomenon of
radioactivity. We neglected to mention, for the sake of simplicity, that
nucleons can be considered to spin in space around a central axis.
We\'ll now concentrate on this spinning characteristic in order to
introduce the phenomenon of nuclear magnetic
resonance (**NMR**), and
we\'ll see later how images can be formed on this basis.

Let\'s take an isolated proton as an example -- see the following
figure:

!A spinning proton produces a magnetic field similar to a bar
magnet.

Besides having an electrical charge, the proton also spins on its axis.
Since this spinning constitutes a moving electrical charge (i.e.
rotating around a central axis), a magnetic field is generated, as
illustrated, similar to that generated by a bar magnet -- but on a much
smaller scale, of course. This spinning magnet aligns itself with any
external magnetic field.

There\'s an additional feature to this rotation, however, called
`<b>`{=html}precession`</b>`{=html}, which
can be seen in many rotating objects, as illustrated by the spinning
gyroscope below:

!The precession of a
gyroscope.

The rate of proton precession is what\'s important in NMR and has an
angular frequency referred to as the
Larmor frequency. Its specified by the
gyromagnetic ratio of the nucleus in
question and for hydrogen it has a value of 2.68x10^8^ rad/s/T. This
ratio implies a rate of 42.58 million precessions per second when a
hydrogen nucleus is placed in a magnetic field of strength 1
Tesla "wikilink"), i.e. rather higher than that for the
precession shown in the animation above!

The following equation is of relevance here:

```{=html}
<div class="center">
```
`<span style="color:green;">`{=html}`<big>`{=html}f = γ
B~o~`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
where `<span style="color:green;">`{=html}f`</span>`{=html} is the
precessional frequency,
`<span style="color:green;">`{=html}γ`</span>`{=html} is the
gyromagnetic ratio and
`<span style="color:green;">`{=html}B~o~`</span>`{=html} is the strength
of the magnetic field. This is a very important relationship in NMR and
is exploited in numerous ways in MRI, as you will see below.

Clinical MRI scanners typically use field strengths between 0.1 and 3 T
depending on the application. The strength of the earth\'s magnetic
field in comparison is about 50
μT, i.e. about 2,000 to 60,000 times smaller, while that of a magnetic
door lock is about 0.1 T.

We\'ve just considered the situation for an isolated proton in a
magnetic field. An interesting phenomenon occurs when many billions of
protons, in a sample of tissue for example, are placed in a magnet:

:   

    :   `<span style="color:blue;">`{=html}some of the protons line up
        in parallel with the field and some line up against
        it.`</span>`{=html}

These two states are referred to as **parallel** and **anti-parallel**
states, as illustrated in the following figure:

!The protons align themselves either parallel or anti-parallel to an
external magnetic field, B~o~. The anti-parallel state is slightly lower
in energy than the parallel state. Transitions from the lower energy to
the higher energy state can occur if an exact amount of energy ΔE is
supplied.

The above figure also indicates that the two alignments are equivalent
to two quantum energy states separated by an energy difference, ΔE. More
nuclei occupy the lower energy than the higher energy state, but their
excess is incredibly small -- less than 0.001%. Its enough, however, to
exploit in NMR!

This energy difference is directly related to the precessional frequency
through the following equation:

```{=html}
<div class="center">
```
`<span style="color:green;">`{=html}`<big>`{=html}ΔE = h
f`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
where `<span style="color:green;">`{=html}h`</span>`{=html} is Planck\'s
Constant and `<span style="color:green;">`{=html}f`</span>`{=html} is
the precessional frequency, related to the Larmor frequency,
`<span style="color:green;">`{=html}ω`</span>`{=html}, through:

```{=html}
<div class="center">
```
`<span style="color:green;">`{=html}`<big>`{=html}ω = 2 π
f`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
The important point here is that:

:   

    :   `<span style="color:blue;">`{=html}transitions from the parallel
        to the anti-parallel state can be induced in the sample when it
        is excited with electromagnetic radiation of energy,
        ΔE.`</span>`{=html}

This energy is about 1.75x10^-7^ eV for a proton in a 1 T field, i.e. a
tiny amount of energy compared with electron binding energies, for
instance. Compared to a 140 keV gamma-ray emitted by ^99m^Tc, its energy
is over a million, million times smaller.

The energy difference is equivalent to the energy of electromagnetic
radiation in the radio frequency
(**RF**) region of the electromagnetic
spectrum, as illustrated below:

!The electromagnetic spectrum indicating regions exploited in nuclear
medicine, radiology and
MRI.

So, if we had a tissue sample placed in a magnetic field of 1 T and we
beamed in radio waves at 42.58 MHz, we can expect to excite protons from
the parallel to the anti-parallel state. This phenomenon is the
`<b>`{=html}*resonance*`</b>`{=html} feature of NMR.

Its what happens following excitation that\'s exploited in MRI:

:   

    :   `<span style="color:blue;">`{=html}transitions back to the
        parallel state occur spontaneously over a time period which is
        characteristic of individual tissues and their various
        pathological conditions.`</span>`{=html}

These transitions cause electromagnetic radiation to be emitted from our
tissue sample whose amplitude decreases with time. The detected
emissions form what is called a Free Induction
Decay, as illustrated below:

!A free induction decay characterized by a time constant,
T2\*.

Its the time constant of signal decays
such as these which is determined in MRI and which can be encoded into
MR images -- once we\'ve figured out where within the patient the
signals have originated from.

#top

## Magnetic Sum Vector

Its useful before proceeding to consider the situation from a magnetic
vector perspective, as illustrated in the following figure:

!Illustration of the cancelling effect feature of parallel and
anti-parallel
vectors.

The figure shows the situation for a hypothetical tissue sample placed
in a magnetic field such that seven hydrogen nuclei are in the parallel
state and five are in the anti-parallel state. Five of the parallel
vectors cancel the effects of the five anti-parallel vectors and we\'re
left with a net condition of two parallel vectors. We can simplify the
situation further by considering the summation of these two
vectors to give us a magnetic sum vector along
the Z-axis, as illustrated in the following diagram:

!Illustration of the magnetic sum vector,
M~z~.

Since the magnetic field direction is generally along the Z-axis in MRI,
our magnetic sum vector is referred to as **Longitudinal
Magnetization**. As a result, a longitudinal magnetization becomes
established when a patient is placed inside the magnet of an MRI
scanner.

#top

## RF Excitation

Let\'s consider the situation once longitudinal magnetization has become
established and when a pulse of RF radiation at the resonant frequency
is beamed into our tissue sample, as illustrated in the following
diagram:

!Illustration of RF
excitation.

Two processes occur simultaneously during absorption of the radiant
energy:

:\***Reduction in M~z~**: Some of the protons resonate and move to the
anti-parallel state, resulting in a reduction in the longitudinal
magnetization, as illustrated below:

!Illustration of the reduction in M~z~ during RF
excitation.

:   

    :   Notice that our initial situation with five protons aligned
        parallel, which gives rise to a longitudinal magnetization,
        M~z~, of relatively large magnitude, changes to one where three
        are left with this alignment and two with the anti-parallel
        alignment. The magnitude of M~z~ is reduced as a result.

:\***Phase Coherence**: the vectors align with each other in phase, as
illustrated below:

!Illustration of the establishment of transverse
magnetization.

:   

    :   The outcome of phase coherence is the establishment of a
        magnetic sum vector in the X-Y plane, called the **Transverse
        Magnetization**.

So, once longitudinal magnetization is established by placing a patient
in the magnet of an MRI scanner and radio waves at the resonant
frequency are generated, we in effect establish a transverse
magnetization while reducing the longitudinal magnetization. Its what
happens next to these two magnetic sum vectors, M~z~ and M~xy~, that\'s
of interest in MRI.

#top

## RF Emission

We can consider the emission phase of the MR imaging process from both
the longitudinal as well as the transverse magnetization perspectives.
Since the protons in our tissue sample have been excited by the
absorption of RF energy at the resonant frequency, the emission phase
corresponds with what is called **relaxation** as protons return to
their lower energy state.

-   **Longitudinal Relaxation**

:   

    :   The situation following excitation is illustrated in the top
        left panel of the following figure:

!Illustration of longitudinal
relaxation.

:   

    :   A hypothetical situation is shown where four protons are aligned
        parallel to the external magnetic field, and four excited
        protons are aligned anti-parallel. The magnetic sum vector is
        zero at this stage. Excited protons then return progressively to
        their lower energy state, as shown by the other panels in the
        above figure. The result is a re-establishment of the
        longitudinal magnetization, M~z~, as illustrated in the
        following figure:

!A T1 curve.

:   

    :   It is seen in this figure that the longitudinal magnetization
        becomes re-established in an exponential manner characterized by
        a time constant, called **T1**.

-   **Transverse Relaxation**

:   

    :   The situation following excitation is illustrated in the top
        left panel of the following figure, with phase coherence taken
        into account. As a result, a transverse magnetization, M~xy~
        becomes established:

!Illustration of transverse
relaxation.

:   

    :   A hypothetical situation is illustrated where three protons are
        aligned parallel and three anti-parallel, in phase coherence wih
        each other. The other panels in the figure illustrate a
        progressive loss of coherence that is characterized by this form
        of relaxation, where the transverse magnetization decreases to
        zero, as shown in the following figure:

!A T2\* curve.

:   

    :   It is seen in this figure that the transverse magnetization
        decreases in an exponential manner characterized by a time
        constant, called **T2\*** -- also called *T2 Star*.

In case you\'re wondering about phase coherence at this stage, a good
analogy to think of is that of analogue time clocks in different parts
of the world. You may have seen such clocks on international TV news
programmes showing the time in different parts of the world or in
foreign exchange banks, for instance. Dublin, Sydney, New York and
Mumbai, for example, all have clocks which take 60 minutes for one
revolution of the big pointer. Their frequencies are identical, in other
words. But they each show times separated by several hours because of
their geographic location and daylight saving factors. We can conclude
that these clocks are out of phase with each other, even though their
frequencies are the same. However, all clocks located in Sydney, for
example, would have the same phase and we can say that they are in
*phase coherence*. In a similar manner, phase coherence of proton
precessions results during RF excitation in NMR and its the loss of this
coherence which happens in transverse relaxation.

So, what do these two time constants, T1 and T2\*, mean in terms of our
tissue sample?

The protons which generate NMR signals are primarily those in cellular
fluids and lipids, i.e. nuclei of hydrogen atoms that are relatively
free to move within their environment. Hydrogen protons in tightly-bound
environments such as within proteins or DNA, however, usually do not
contribute to NMR signals and the same situation exists for those in
solid structures such as bone.

We can first of all consider a water molecule moving through its
environment within our tissue sample as a result of local chemical and
physical forces. The magnetic properties of its two hydrogen protons
generate a small magnetic field of about 1 mT and the molecule\'s motion
is therefore also influenced by the magnetic properties of other water
molecules in its vicinity -- and it in turn influences their motion.
When excited protons are involved following RF excitation, it is
interactions with their local environment which cause them to loose
their excess energy and return to the lower energy state with the
emission of RF radiation. This can be considered to be the origin of the
re-establishment of longitudinal magnetization during relaxation.
Chemistry calls this phenomenon **spin-lattice relaxation** -- *spin*
referring to the spinning proton and *lattice* to its local environment.

The rate at which molecules can move within their environment is related
to their size and hence small molecules have a low probability for
interaction. This is why fluids such as CSF have long T1 values, for
instance -- see the table below. Medium-sized molecules (e.g. lipids),
in contrast, move more slowly, have a greater probability for
interaction as a result, and exhibit relatively short T1 values.

The T2\* phenomenon results from the compounding effect of imperfections
in the external magnetic field and from so-called **spin-spin**
interactions. These latter are interactions where two nearby protons can
cause each other to flip so that one changes from anti-parallel to
parallel alignment, while the other changes from parallel to
anti-parallel, i.e. one gains the excitation energy from the other.
Phase coherence with other excited protons is lost during this exchange
and the end result is a relaxation of the transverse magnetization. This
spin-spin interaction is also called T2 relaxation. It tends to happen
much more rapidly than T1 relaxation and T2 values are therefore
generally less than T1 values, as shown in the following table:

+--------------+----------------------+----------------------+
| Tissue       | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | **T1 (ms)**          | **T2 (ms)**          |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+==============+======================+======================+
| Muscle       | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 870                  | 47                   |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| Liver        | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 490                  | 43                   |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| Kidney       | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 650                  | 58                   |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| Grey Matter  | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 920                  | 100                  |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| White Matter | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 790                  | 92                   |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| Lung         | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 830                  | 80                   |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+
| CSF          | ```{=html}           | ```{=html}           |
|              | <div class="center"> | <div class="center"> |
|              | ```                  | ```                  |
|              | 2,400                | 160                  |
|              |                      |                      |
|              | ```{=html}           | ```{=html}           |
|              | </div>               | </div>               |
|              | ```                  | ```                  |
+--------------+----------------------+----------------------+

: `<small>`{=html}T1 and T2 values at 1.5 Tesla.`</small>`{=html}

Since T2 arises mainly from neighbouring protons, a higher interaction
probability exists with larger than with smaller molecules.
Macromolecular environments will therefore display shorter T2 values
than water-based fluids, e.g. CSF.

A final point to note is that both T1 and T2 measurements from a small
volume of tissue result from the integrated motional effects of all
molecular species which contain hydrogen protons in that volume, be they
those of small molecules, lipids or macromolecules.

#top

## Inversion Recovery

A practical problem arises when we try to measure T1 because M~z~ exists
within the Z-axis of the patient. If we could flip the longitudinal
magnetization vector into the XY plane, then T1 could be measured using
external coils. This is what the Inversion Recovery imaging technique is
designed to do.

It involves a two stage excitation process, where an RF pulse of
sufficient duration to excite all the lower energy protons is applied
initially and this is followed by a second RF pulse after a time
interval of duration sufficient for similar numbers of protons to exist
in the two alignments. The first stage is illustrated in the following
figure:

!Illustration of the application of a 180^o^ RF
pulse.

The RF pulse in this case is referred to as a 180^o^ pulse since the
longitudinal magnetization, M~z~, has been inverted during the
excitation process.

Then a short time interval is used to allow some longitudinal relaxation
to occur, as illustrated in the following figure:

!Recovery of M~z~ following a 180^o^ RF pulse in cases where T1 is long
(left panel) and short (right
panel). and short (right panel).")

Notice that M~z~ recovers relatively slowly when the T1 of the tissue in
question is relatively long, and that it recovers quite quickly when T1
is short. We can conclude that the extent of the recovery is related to
T1.

A 90^o^ RF pulse is then used to move M~z~ into the XY plane so that its
amplitude can be measured, as shown in the next figure:

!The recovered M~z~ is moved onto the XY plane using a 90^o^ RF
pulse.

The sequence of RF pulses is shown in the left panel of the figure
below, separated by the time interval, TI, and the resultant Free
Induction Decay (FID) is shown below it. The amplitude of the FID is
dependent on the relative values of T1 and the time interval, TI. For
instance, when the time interval is less than T1, the longitudinal
magnetization can still be negative when the 90^o^ is applied -- as we
saw in an earlier figure. At the other extreme, when the time interval
is much greater than T1, the longitudinal magnetization has sufficient
time to fully recover and the amplitude of the FID is dependent
primarily on the **proton density**.

The figure in the right panel below shows the dependence of the
longitudinal magnetization measured from the FIDs of grey matter and CSF
against the time interval, TI. It is seen that both substances generate
a negative value of M~z~ when the time interval is short, as we might
expect. It is also seen that M~z~ increases in each case when the time
interval is increased and a region is reached where the values are
strongly dependent on T1. It is further seen that a large difference can
exist between the measured values for our two substances in this region.
This feature gives rise to the generation of contrast between these two
substances in T1-weighted images. Typically a time interval of 400 ms is
used for such tissue discrimination.

!The inversion recovery pulse sequence and resultant FID (left) and the
dependence of M~z~ on the time interval, TI
(right). and the dependence of Mz on the time interval, TI (right).")

A final point to note is the proton density-weighting that results when
long time intervals are used, with little tissue discrimination.
Typically an interval of 700 ms is used for such studies.

There are numerous other pulse sequences used besides Inversion Recovery
in MRI. These include:

:\*Spin Echo

:\*Saturation Recovery

:\*Short TI IR (STIR)

:\*Echo Planar Imaging

:\*Small Angle Gradient Echo (SAGE).

The Spin Echo sequence, for instance, can be used to generate a
T2-weighting in images. You\'ll remember from earlier that an FID decays
exponentially with a time constant, T2\*. This parameter results from
the combined effect of two phenomena:

:\*static and oscillating magnetic fields induced by neighbouring
protons (and unpaired
electrons!) which we\'ve
discussed earlier and which generate the T2 information;

:\*small imperfections in the primary magnetic field, B~o~, which are
referred to as field inhomogeneities, and generate a time constant,
T~inh~.

These two time constants can be considered to add, such that:

```{=html}
<div class="center">
```
$\frac{1}{T2^*} = \frac{1}{T2} + \frac{1}{T_{inh}}$

```{=html}
</div>
```
and the function of the Spin Echo pulse sequence is to disentangle this
relationship to extract the T2 information. Note that this pulse
sequence can also be used to generate images with a T1 or a proton
density weighting depending on various time intervals used in its pulse
sequence.

The further treatment of Spin Echo and other pulse sequences is beyond
the introductory level of this chapter and you are referred to the
Bibliography
for additional learning resources. However, you\'re likely to find that
the treatment above will provide a sufficient foundation for developing
your understanding of these other pulse sequences.

You might also like to check out the superb e-MRI online learning
resource

#top

## Spatial Localization of NMR Signals

Since we\'ve seen above how contrast between tissues can be generated in
MRI we\'re now in a position to consider how images are formed. In other
words, we\'ll consider how to localize the origin of the FIDs generated
within the patient following RF excitation.

Its helpful to consider an analogy at this stage because this imaging
situation is distinctly different to that encountered in nuclear
medicine, radiography or sonography where the radiation travels in
straight lines and imaging techniques based on emission, transmission
and reflection can be readily conceptualized. Its a little bit more
difficult in MRI because the radiation is emitted in all directions
simultaneously, so that there\'s little or no chance in knowing its
origin from an FID measurement. In other words, the FID only allows the
amplitude, frequency and phase of the radio waves to be measured.

!A grand piano with a keyboard and scales
overlaid.{width="128"}

The analogy we\'ll use is that of a piano keyboard
and the playing thereof. Let\'s imagine a concert pianist, for instance,
on stage and playing the keyboard of a grand piano. Notice that the
pianist\'s fingers hit keys to the left of the keyboard in order to
produce a relatively low frequency (i.e. low pitch) sound, keys on the
right for high frequency notes and keys in the middle for intermediate
frequencies. So, besides hearing the music produced while the pianist
plays, we also receive information about where in space the pianist\'s
fingers were located at any individual moment. In technical jargon we
can say that we are using a method of spatial localization through
frequency discrimination of audio signals to determine locations along
the piano\'s keyboard. This same approach can be applied in MRI using
frequency discrimination techniques for the spatial localization of
radio signals.

The 3D situation is of course a bit more complex than the 1D piano
keyboard, but the principle is basically the same. Frequency
discrimination can be achieved by arranging for RF emissions from the
patient to have slightly different frequencies depending on their
origin. Furthermore, phase discrimination can also be applied so that
the phase of the RF emissions can be related to locations within the
patient. The fidelity of the frequency and phase discrimination should
of course be sufficient so that the corresponding voxels within the
patient are sufficiently small enough for clinically-useful views of the
anatomy to be generated. Afterall, the fidelity of our frequency
discrimination in the case of a piano keyboard is limited at an extreme
by the width of the keys!

The magnetic field of the magnet in an MRI scanner is therefore
supplemented by small fields introduced using gradient coils so that a
linear gradient, for instance, in the field can be established. With
such a gradient applied along the patient\'s Z-axis, we can expect the
resonant frequency of hydrogen protons to be different for planes along
this axis, as illustrated in the following figure:

!Illustration of the application of a linear gradient (in blue) in the
magnetic field along a patient\'s Z-axis so that the resonant frequency
can be related to planes along this
axis. in the magnetic field along a patient's Z-axis so that the resonant frequency can be related to planes along this axis.")

Should we wish to excite a single slice along this axis, for instance,
all we need to do is generate radio waves of the relevant frequency, as
dictated by our earlier equation:

```{=html}
<div class="center">
```
`<big>`{=html}`<span style="color:green;">`{=html}f = γ
B~o~`</span>`{=html}.`</big>`{=html}

```{=html}
</div>
```
This approach is called **slice selective excitation**. Both the
strength of the gradient field and the
bandwidth of the RF pulse produced by the
transmitter can influence the thickness of the selected slice as
indicated in the following figure:

!Effect of gradient field (left) and bandwidth (right) on slice
thickness. and bandwidth (right) on slice thickness.")

We can see in the left panel above that increasing the gradient in the
magnetic field from G~a~ to G~b~, for instance, decreases the slice
thickness, ΔZ, when the bandwidth, Δf, is kept constant. In comparison,
we can see in the right panel that increasing the bandwidth of the RF
pulse generated by the transmitter circuitry, from Δf~a~ to Δf~b~ for
instance, increases the slice thickness.

It is important to note that the slice selection gradient is applied
during the application of the RF excitation pulses. The protons in any
given plane can then be made to resonate by changing the RF pulse centre
frequency, f~o~, during the scan.

The G~z~ gradient and the RF bandwidth control the slice thickness and
position in this manner. Note that although the slice selection gradient
has been described here in terms of the Z-axis, there is no reason in
principle why any other directions cannot be chosen for the application
of the gradient. The slice selection is therefore not governed by
mechanical considerations, as is the case in X-ray CT, SPECT and PET.

One method of tomographic imaging based on this approach is considered
next. Let\'s assume for simplicity that our patient can be represented
by an elongated cube, as shown in the following figure:

!Illustration of slice selection during the RF excitation phase using a
longitudinal magnetic field
gradient.

Notice that our patient is placed in a magnet with a longitudinal
gradient in the field, as illustrated by the blue line. The field
strength is relatively low, as a result, towards the left of our cube
and relatively high towards the right. A central slice of the cube can
be excited when an RF pulse of the appropriate frequency is applied.
Suppose the selected slice contains two features which generate FIDs, as
indicated by the red and green objects.

The RF pulse is used to excite protons in this slice so that they all
precess at the same frequency. A transverse gradient in the magnetic
field is then applied during the emission phase, as illustrated in our
next figure, so that the field strength is low, for instance, to the
left of the selected slice and higher to the right of the slice.
Precessional frequencies will therefore increase, decrease or remain the
same depending on their horizontal location across the slice. The
emitted FIDs will therefore contain a range of frequencies which
represent this horizontal location.

!Illustration of the application of a transverse gradient during the RF
emission
phase.

The frequency content of the FIDs can be unscrambled by computing the
one-dimensional Fourier
transform,
as illustrated in the figure above, to derive the horizontal spatial
locations of the emitters. The resultant Fourier spectrum constitutes
one projection, somewhat similar to that acquired in X-ray computed
tomography. The subsequent steps are to generate projections at
different angles around the slice by repeating steps 1 and 2 in the
above figure using different transverse gradients. A series of
projections can then result which can be processed using Filtered Back
Projection
to form an image of the axial slice.

This method is rather slow for clinical application and numerous other
methods have been devised. We\'ve considered it here simply because its
a good exercise in integrating many of the features that you\'ve learnt
about in this chapter and because of its relative simplicity. Faster
image acquisition times can be generated using the so-called Spin Warp
method, for example. Here, a two-dimensional Fourier transform is
applied using repeated slice selective excitations along with
frequency-encoding gradients (as we\'ve just described) as well as
phase-encoding gradients during the emission phases.

#top

## Back to the Nucleus

There are many other features of MRI that are beyond the scope of this
chapter. Issues such as safety in magnetic environments are of vital
importance and features such as magnetic resonance
angiography (**MRA**),
magnetic resonance
spectroscopy
(**MRS**), Diffusion Tensor Imaging
(**DTI**), functional MRI (**fMRI**),
contrast media, image artefacts as well as magnet
design, shielding and siting are
fascinating to consider from a nuclear medicine perspective.

We\'ll conclude this chapter with an important basic feature which we
didn\'t consider earlier in the interests of clarity. This brings us
back to the nucleus and the topic of nuclear spin. You\'ll appreciate
that its not just the hydrogen proton that demonstrates the NMR
phenomenon and that it can occur, at the least, in situations where the
nucleus has an odd number of protons. In other words, situations where
there\'s an isolated proton which does not have a partner to pair with
magnetically in its nucleus.

There\'s actually three types of nucleus where nuclear spin is concerned
-- those with:

::\***Odd Mass Number**, i.e. nuclei with one unpaired nucleon;

::\***Even Mass Number & Even Atomic Number**, i.e. nuclei with no
unpaired nucleons;

::\***Even Mass Number & Odd Atomic Number**, i.e. nuclei with two or
more unpaired nucleons.

Examples of nuclei with a net nuclear spin are given in the following
table:

+----------------------+----------------------+----------------------+
| Nucleus              | ```{=html}           | ```{=html}           |
|                      | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | **Gyromagnetic       | **f~o~ at 1 Telsa    |
|                      | Ratio, γ             | (MHz)**              |
|                      | (10^6^rad/s/T)**     |                      |
|                      |                      | ```{=html}           |
|                      | ```{=html}           | </div>               |
|                      | </div>               | ```                  |
|                      | ```                  |                      |
+======================+======================+======================+
| ^1^H | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 268                  | 42.58                |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^2^H | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 41                   | 6.53                 |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^13^C | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 67                   | 10.66                |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^14^N | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 19                   | 3.02                 |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^19^F | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 253                  | 40.27                |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^23^Na | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 71                   | 11.3                 |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+
| ^31^P | <div class="center"> | <div class="center"> |
|                      | ```                  | ```                  |
|                      | 108                  | 17.19                |
|                      |                      |                      |
|                      | ```{=html}           | ```{=html}           |
|                      | </div>               | </div>               |
|                      | ```                  | ```                  |
+----------------------+----------------------+----------------------+

Only ^1^H has found application in routine medical imaging because of
the vast quantity of water in the human body.

#top

