As has been shown in Section 2.2, the fundamental equation of magnetic resonance is the Larmor equation, . In an NMR experiment a measurement of the frequency of precession of the magnetisation gives information on the field experienced by that group of spins. By manipulating the spatial variation of the field in a known way, this frequency information now yields spatial information.

Consider a linear field gradient in **B**
which increases along the x axis, such that

**(2.36)**

where G is the gradient strength. This makes the Larmor equation

**(2.37)**

or in its more general three dimensional form

**(2.38)**

Under a linear field gradient along the x axis, all the spins which lie at a particular value of x will precess at the same frequency. The FID from such a sample will contain components from each of the x values represented by the sample, and the frequency spectrum will therefore represent the number of spins that lie along that plane

**(2.39)**

as shown in Figure 2.7c.

This simple spectrum therefore gives the spatial information about the object being imaged along one dimension. To build up the complete 3D image it is necessary to apply time varying field gradients. A number of methods for doing this are described later in this section, but first the notion of k-space is introduced, which is useful in describing all these techniques.

Having denoted the number
of spins at a particular location **r**, as the spin density,
r(**r**), the
signal from the sample can be written

**(2.40)**

The reciprocal space vector is defined

**(2.41)**

and the Fourier relationship between signal and spin density becomes obvious

**(2.41)**

Thus **k** is the conjugate variable for **r**. The
resolution of the image depends on the extent of k-space that is
sampled, and it is by looking at how different imaging techniques
cover k-space that they can usefully be compared
[8]. For example in the
previous section a gradient along the x-axis was applied to a
cylindrical sample. This meant that values in k_{x} could be
sampled, but not in k_{y}
and so the complete 2D structure could not be obtained.

There are imaging techniques which sample all three dimensions of k space, but most techniques reduce the problem to two dimensions by applying slice selection.

Slice selection is a technique to isolate a single plane in the object being imaged, by only exciting the spins in that plane. To do this an r.f. pulse which only affects a limited part of the NMR spectrum is applied, in the presence of a linear field gradient along the direction along which the slice is to be selected (Figure 2.8). This results in the excitation of only those spins whose Larmor frequency, which is dictated by their position, is the same as the frequency of the applied r.f. pulse.

Consider an r.f. pulse of duration 2t
which is applied in the
presence of a gradient along some axis, say z, such that
2tgB_{1}=
p/2 (a 90 degree pulse).
The spins where B=B_{0}
(i.e. z=0) will precess into the transverse plane,
whilst those where B>>B_{0} (i.e. z>>0) will precess
about B_{eff}=Dw/g
**k**+B_{1}**i**, and have little effect on the transverse
magnetisation (Figure 2.9).

To tailor the
shape of slice selected it is necessary to modulate the pulse profile.
For small flip angles the relationship between the pulse modulation
and slice profile can be derived. To do this consider applying an
r.f. pulse lasting from t=-T to +T, to a sample in a gradient
G_{z}. The
Bloch equations as shown earlier (equation. 2.29) in the rotating
frame, neglecting T_{1} and T_{2} relaxation,
become

**(2.43)**

Now these are transformed to another frame of reference
which rotates at angular frequency gG_{z}z. Each slice will have a different
reference frame, depending on the value of z, but considering just one
of these gives

**(2.44)**

with the two
reference frames coinciding at t=-T. If the assumption is made that
the flip angles are small then it is possible to say that
M_{z'}
varies very little, and takes the value M_{0}.
So equations 2.44 become

**(2.45)**

If M_{x} and M_{y} are treated as real
and imaginary parts of the complex magnetisation , the equation of motion becomes

**(2.46)**

By integrating and converting back to the original (rotating) frame of reference, since

gives

**(2.47)**

This is the Fourier transform
of the r.f. pulse profile, so a long pulse gives a narrow slice and
vice-versa. This equation also shows one other thing about the
magnetisation in the slice after the pulse. There is a phase shift
through the slice of gG_{z}*zT*
which will cause problems for imaging later. This
can be removed by reversing the gradient G_{z} for a duration T,
which is known as slice refocusing.

To excite a sharp edged slice, the r.f. pulse modulation required is a sinc function. For practical implementation of this kind of pulse it is necessary to limit the length of the pulse, and so usually a five lobe sinc function is used. The above approximation is only correct for small pulse angles. For larger flip angles, it is hard to analytically determine the pulse modulation required for a desired slice profile, and so it is usually necessary to use some form of iterative method to optimise pulses [9]. Using such techniques it is possible to design a range of pulses which are able to select different slice profiles and carry out other spin manipulation techniques such as fat or water suppression. It is also possible to design pulses which do not require refocusing [10]. Usually the limit in such novel designs is their practical implementation, and so the most common three pulse modulations used are a 'hard' pulse, with top-hat modulation, a Gaussian modulation or a 'soft' pulse, with truncated sinc modulation.

If any slice other than the central (B=B_{0})
slice is required, then the frequency of oscillation
can be altered and the slice selected will be at the position along
the z-axis given by

**(2.48)**

The extensively used techniques in MRI are all Fourier based, that is the 'spin-warp' technique and Echo Planar Imaging (EPI) [11]. However the early MR images used point and line methods and these are described here, along with the technique of projection reconstruction.

The f.o.n.a.r. technique (*f*ield f*o*cused *n*uclear
m*a*gnetic *r*esonance) was proposed by Damadian, and was
used to produce the first whole body image in 1977
[12]. The basis of the
technique is to create a shaped B_{0} field which has a central
homogeneous region, surrounded by a largely inhomogeneous region. The
signals from the inhomogeneous regions will have a very short
T_{2}^{*} and can thus be distinguished from
the signal from the homogeneous region. By scanning the homogeneous
field region across the whole of the sample an image can be made up.
This is a time consuming procedure taking tens of minutes for a single
slice. The two advantages of the technique are its conceptual
simplicity and the lack of a requirement for the static field to be
homogeneous over a large area. Shaped r.f. pulses can also be used to
isolate a small region, and such methods form the basis of localised
spectroscopy.

As was shown in section 2.3.1, if we have only a one dimensional object then a single linear gradient is sufficient to locate position directly from the FID by Fourier transformation. If then the three dimensional sample can be reduced to a set of one dimensional components then the whole sample can be imaged. This can be done by selective irradiation, or slice selection, in two of the dimensions. First, in the presence of a gradient along the z-axis, a selective pulse is applied which saturates all the spins outside the plane of interest (Figure 2.10a). Then a gradient is applied along the x-axis, and those spins not saturated are tipped into the transverse plane by a selective 90 degree pulse (Figure 2.10b). Immediately after the second r.f pulse the only region with any coherent transverse magnetisation is the line of intersection of the two selected planes. A gradient is now applied along the y-axis, and the evolution of the FID recorded. Fourier transform of the FID gives the proton densities along that line. By repeating the line selection for all the lines in the plane, an image of the whole of the plane can be built up. There are many variations of the line scan technique, some of which utilise 180 degree pulses, but they are inefficient in comparison to the Fourier methods discussed in the next section.

(a) (b)

One final method of interest is projection reconstruction. This is the method used to build up X-ray CT scans [13], and was the method used to acquire the early MR images. Following slice selection, a gradient is applied along the x-axis and the projections of the spin densities onto that axis obtained by Fourier transformation of the FID. Then a linear gradient is applied along an axis at some angle to the x-axis, q. This can be achieved by using a combination of the x and y gradients

.

**(2.49)**

Projections are taken as q is incremented up to 180 degrees (Figure 2.11a). The set of projections can then be put together using back projection. This distributes the measured spin densities evenly along the line normal to the axis it was acquired on. By reconstructing all the angled projections the image appears. There is however a blurred artefact across the whole image, where an attempt was made to assign spin density to areas where there is in fact none, and star artefacts where a finite number of projections have been used to define point structures. This can be corrected using a technique called filtered back projection, which convolves each of the profiles with a filter

**(2.50)**

This is done in
the Fourier domain by dividing the Fourier projections by the modulus
of the vector **k** as defined in section 2.3.2.

Projection reconstruction has been largely superseded by methods which sample k-space more uniformly.

Quadrature detection of the FID means that the phase as well as the frequency of the signal can be recorded. This is utilised in the Fourier techniques described in this section.

The 'spin-warp'
method (often called 2DFT) as described by Edelstein *et. al.*
[14] is
a development of the earlier technique of Fourier zeugmatography
proposed by Kumar, Welti, and Ernst [15].
The Fourier zeugmatography
sequence can be split into three distinct sections, namely slice
selection, phase encoding, and readout. The pulse sequence diagram
for the sequence is shown in Figure 2.12. Such diagrams are commonly
used to describe the implementation of a particular MR sequence, and
show the waveform of the signal sent to the three orthogonal gradient
coils and the r.f. coil.

Having excited only those spins which lie in one plane, a gradient is applied along the y-axis. This will cause the spins to precess at a frequency determined by their y position, and is called phase encoding. Next a gradient is applied along the x-axis and the FID is collected. The frequency components of the FID give information of the x-position and the phase values give information of the y-position.

More
specifically, if a gradient of strength G_{y} is applied for
a time t_{y} during the phase
encoding stage, and then a gradient G_{x} is applied for a duration
t_{x} the
signal recorded in the FID is given by

**(2.51)**

By writing equation (2.51) becomes

**(2.52)**

so the single step is
equivalent to sampling one line in the k_{x} direction of k-space.
To cover the
whole of k-space it is necessary to repeat the sequence with slightly
longer periods of phase encoding each time (Figure 2.13).

Having acquired data for all values of k_{x} and k_{y},
a 2D Fourier transform recovers the spin density function

**(2.53)**

Figure 2.14 illustrates how the magnetisation evolves under these two gradients.

One drawback of this technique is that the time between exciting the
spins, and recording the FID varies throughout the experiment. This
means that the different lines in the k_{y} direction will
have different
weighting from T_{2}^{*} magnetisation decay.
This is overcome in spin-warp imaging by keeping the length of the y
gradient constant for each acquisition, and varying k_{y} by
changing the gradient strength. The pulse diagram for this technique
is shown in Figure 2.15.

It
is desirable to have as much signal as possible for each FID, and a
necessity that the amount of transverse magnetisation available
immediately after the r.f. pulse is the same for each line. This can
be a problem since the recovery of longitudinal magnetisation is
dependent on spin-lattice relaxation, and T_{1} values in
biomedical imaging are of
the order of seconds. Keeping the time between adjacent spin
excitations, often known as TR, the same throughout the image
acquisition will keep the transverse magnetisation the same for each
FID, provided the first few samples are discarded to allow the system
to come to a steady state. Leaving the magnetisation to recover
fully, however, would be very costly in time, so it is usually
necessary to have a TR which is less than T_{1}. To maximise
the signal received for small TR values it is possible to use a smaller
flip angle than 90 degrees. The transverse magnetisation that is
available after such a
pulse is less than it would be after a 90 degree pulse, but there is
more longitudinal magnetisation available prior to the pulse. To
optimise the flip angle q, for a particular
TR, we first assume that the steady
state has been reached, that is that

**(2.54)**

Now the magnetisation is flipped by a q degree pulse, and the z-magnetisation becomes

.

**(2.55)**

The recovery of the magnetisation is governed by the equation

**(2.56)**

which can be integrated to find M',

**(2.57)**

The transverse magnetisation following the pulse, which we want to maximise, is given by

**(2.58)**

which has its maximum value when

**(2.59)**

The angle this occurs at is known as the Ernst angle
[16]. The amount of signal available is
very dependent on the repetition time TR. For example, if a sample has a
T_{1} of 1s,
then at a TR of 4s M'=0.98M_{0}, however as TR is reduced to 500ms,
M'=0.62M_{0}.

If a 3D volume is to be imaged then it is possible to acquire extra slices with no time penalty. This is because it is possible to excite a separate slice, and acquire one line of k-space, whilst waiting for the longitudinal magnetisation of the previous slice to recover. This technique is called multi-slicing.

The common implementation of the spin-warp technique, FLASH (Fast Low-Angle SHot imaging) [17], uses very small flip angles (~5 degrees) to run at with a fast repetition rate, acquiring an entire image in the order of seconds.

In Echo Planar Imaging [18,
19] (EPI) the whole of k-space is
acquired from one FID. This is possible because, having acquired one
set of frequency information, the sign of the readout gradients can be
reversed and the spins will precess in the opposite direction in the
rotating frame (Figure 2.16) and subsequently rephase causing a
regrowth of the NMR signal. This is a called a gradient echo. By
switching the readout gradient rapidly, the whole of k-space can be
sampled before spin-spin (T_{2}) relaxation attenuates the
transverse magnetisation. Phase encoding is again used in order to
sample k_{y}.

The three gradients in EPI are usually labelled the slice select (z), blipped (y) and switched (x), because of their respective waveforms. Echo planar imaging is a technically demanding form of MRI, usually requiring specialised hardware, however it has the advantage of being a very rapid imaging technique, capable of capturing moving organs like the heart, and dynamically imaging brain activation. This is only a brief introduction to EPI, since its strengths and limitations are discussed in more detail in the following chapters.

There are numerous variations on the basic MRI sequences described above. Several of them, notably interleaved EPI, are explained in other chapters. Other important sequences are outlined here.

It was shown in the previous section how phase encoding enabled the information on the second dimension to be added to the one dimensional line profile. It is possible to extrapolate this procedure to the third dimension by introducing phase encoding along the z axis. Thus the 2D-FT technique is extended to a 3D-FT technique [20]. All such volumar imaging sequences first involve the selection of a thick slice, or slab. Then phase encoding is applied in the z-direction and the y-direction, followed by a readout gradient in the x-direction, during which the FID is sampled. The assembled FID's are then subject to a three dimensional Fourier transform yielding the volume image. Phase encoding can also be used in EPI instead of multi-slicing. The slice select gradient and r.f. pulse being replaced by a slab selective pulse and a phase encoding gradient along the z-axis.

Going even further, it is possible to acquire all the
data to reconstruct a 3D volume from one FID, in the technique called
Echo Volumar Imaging (EVI) [21].
This uses another blipped gradient in the
z direction, as shown in Figure 2.18. The limitation in EVI is the
need to switch the gradients fast enough to acquire all the data,
before T_{2}^{*} destroys the signal.

Often the main limitation in implementing fast imaging
sequences such as EPI is switching the gradients at the fast rates
required. A sequence which is similar to EPI, but slightly easier to
implement is spiral imaging. This covers k-space in a spiral from the
centre outwards, which requires sinusoidal gradients in x and y,
increasing in amplitude with time. Such gradient waveforms are easier
to produce than the gradients required for EPI. Spiral imaging also
has the advantage of sampling the centre of k-space first, and so the
low spatial frequencies, that affect the image the most are sampled
first, when the signal has not been eroded by T_{2}^{*}.
The pulse sequence diagram and coverage of k-space for
spiral imaging are shown in Figure 2.19.

In general imaging, the chemical shifts of the protons are ignored, and usually seen only as an artefact. However it is possible to image the chemical shifts, which gives not only spatial information but also spectral information. The technique, called Chemical Shift Imaging (CSI) [22], treats the chemical shift as an extra imaging gradient in the fourth dimension. By introducing a variable delay between the excitation pulse and imaging gradients, the chemical shift 'gradient' will phase encode in this direction. Fourier transformation in this case gives the conventional NMR spectrum.

Finally, it is possible to image nuclei other than the proton.
Sodium, phosphorus and carbon-13 have all been used to form biomedical
images. In the case of ^{13}C, its low natural abundance makes
it useful for tracer studies.