Mri english letter

Introduction to Magnetic Resonance
Imaging Techniques
Lars G. Hanson, larsh@drcmr.dk
Danish Research Centre for Magnetic Resonance (DRCMR),
Copenhagen University Hospital Hvidovre
Latest document version: http://www.drcmr.dk/
Translation to English: Theis Groth
Revised: August, 2009
It is quite possible to acquire images with an MR scanner without understanding the principles
behind it, but choosing the best parameters and methods, and interpreting images and artifacts,
requires understanding. This text serves as an introduction to magnetic resonance imaging techniques.
It is aimed at beginners in possession of only a minimal level of technical expertise, yet it introduces
aspects of MR that are typically considered technically challenging. The notes were written in
connection with teaching of audiences with mixed backgrounds.
Contents
1 Introduction 2
1.1 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Recommended books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Magnetic resonance 4
3 The magnetism of the body 8
4 The rotating frame of reference 12
5 Relaxation 12
5.1 Weightings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Causes of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Inhomogeneity as a source of signal loss, T∗
2 . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Sequences 16
7 Signal-to-noise and contrast-to-noise ratios 17
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8 Quantum Mechanics and the MR phenomenon 18
8.1 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9 Imaging 20
9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.3 Slice selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9.4 Spatial localization within a slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9.5 Extension to more dimension – k-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9.6 Similarity and image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9.7 Moving in k-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9.8 Image acquisition and echo-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9.9 Frequency and phase encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9.10 Spatial distortions and related artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.11 Slice-selective (2D-) versus 3D-sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.12 Aliasing and parallel imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.13 Finishing remarks on the subject of imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10 Noise 36
11 Scanning at high field 37
12 MR Safety 38
1 Introduction
This text was initially written as lecture notes in Danish. Quite a few of the students are English
speaking, however. Moreover, the approach taken is somewhat different than for most other
introductory texts. Hence it was deemed worth the effort to do a translation. The task was taken on
by Theis Groth (July 2009 version) whose efforts are much appreciated.
The goal of the text is to explain the technical aspects accurately without the use of mathematics.
Practical uses are not covered in the text, but it introduces the prerequisites and, as such, seeks to
provide a sturdy base for further studies. While reading, you may find the glossary of help (located
towards the end of the notes). It elaborates on some concepts, and introduces others.
Material has been added along the way whenever a need was felt. As a result, the text covers
diverse subjects. During the accompanying lectures, software that illustrates important aspects
of basic MR physics, including popular sequences, contrast and imaging, was used. It is highly
recommended that you download and experiment with these programs, which are available at no
charge. Corrections, comments and inspiring questions concerning the text and software are always
appreciated.
The text is concerned only with general aspects that are discussed in pretty much any technical
textbook on MR. Therefore, very few specific references are given.
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Figure 1: For teaching purposes, a free, interactive simulator, that illustrates a large number of important
concepts and techniques, has been developed. The user chooses the initial conditions for the simulation and
can thereafter manipulate the components of the magnetization with radio waves and gradients, as is done
during a real scanning session. A screenshot from the simulator is shown above. It is described in detail at
http://www.drcmr.dk/bloch
1.1 Supplemental material
http://www.drcmr.dk/bloch Interactive software that can illustrate a broad spectrum of
MRI concepts and methods, and that can contribute significantly to the understanding. Anima-
tions that illustrate the use of the software and selected MR phenomena are on the homepage.
http://www.drcmr.dk/MR Text and animations that briefly describes MR in English. The
page was created as a supplement to an article that discusses the connection between the
classical and the quantum mechanical approach to the MR phenomenon, and which also
describes problems that appear in many basic text books.
http://www.drcmr.dk/MRbasics.html Links to this text and an example of accompany-
ing “slides”.
1.2 Recommended books
Even technically oriented readers can benefit from reading a relatively non-technical introduction.
Having a basic understanding makes it much easier to understand the formalism in later stages.
Once a basic understanding is acquired, the Internet is a good source of more specific texts. Some
books deserve special mention:
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“Magnetic Resonance Imaging: Physical and Biological Principles” by Stewart C. Bushong. A
good introduction that does not demand a technical background from its readers.
“Clinical Magnetic Resonance Imaging” by Edelman, Hesselink and Zlatkin. Three volumes
featuring a good mixture of technique and use. Not an intro, but a good follow-up (according
to people who have read it. I haven’t).
‘Magnetic Resonance Imaging – Physical Principles and Sequence Design” by Haacke, Brown,
Thompson and Venkantesan. Broadly oriented textbook with plenty of physics, techniques
and sequences. Not an easily read introduction, but suitable for physicists and similar people.
“Principles of Nuclear Magnetic Resonance Microscopy” by Paul T. Callaghan. A classic within
the field of probing molecular dynamics. Technically demanding. Should only be read in the
company of a grown-up.
“Spin Dynamics: Basics of Nuclear Magnetic Resonance” by Malcolm H. Levitt. Covers theoretical
aspects of MR spectroscopy as used in chemical analysis and is thus irrelevant to most who
work with MR image analysis. Excels in clear, coherent writing and a description of quantum
mechanics devoid of typical misconceptions.
Please note that these books have been published in several editions.
2 Magnetic resonance
Initially it is described how magnetic resonance can be demonstrated with a pair of magnets and a
compass. The level of abstraction already increases towards the end of this first section, but despair
thee not: As the complexity rises, so shall it fall again. A complete understanding of earlier sections
is not a prerequisite for future gain.
If a compass happens to find itself near a powerful magnet, the compass needle will align with
the field. In a normal pocket compass, the needle is embedded in liquid to dampen its oscillations.
Without liquid, the needle will vibrate through the north direction for a period before coming to rest.
The frequency of the oscillations depend on the magnetic field and of the strength of the magnetic
needle. The more powerful these are, the faster the vibrations will be.
Radio waves are magnetic fields that change in time (oscillate1) and as long as the needle vibrates,
weak radio waves will be emitted at the same frequency as that of the needle. The frequency is
typically reported as the number of oscillations per second, known as Hertz (Hz). If the needle
oscillates 3 times per second, radio waves will be emitted at a frequency of 3 Hz, for example. The
1Throughout this text, the term “radio waves” is used for rapidly oscillating magnetic fields. It has been pointed out by
David Hoult in particular, that the wave nature of the oscillating magnetic field is not important for MR except at
ultra-high field. The typical choice of wording is therefore unfortunate (“oscillating magnetic field” or “B1 field” is
preferred over “radio wave field”). I agree, but will nevertheless comply with the somewhat unfortunate standard as it
facilitates reading and correctly highlights the oscillatory nature of the B1-field. The difference lies in the unimportant
electrical component of the field, and in the spatial distribution not felt by individual nuclei.
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(a) (b) (c)
Figure 2: (a) A magnetic needle in equilibrium in a magnetic field. The needle orients itself along the field
lines that go from the magnets northern pole to the southern pole (and yes, the magnetic south pole of the
earth is close to the geographical North Pole). (b) A weak perpendicular magnetic field can push the magnet a
tad away from equilibrium. If the magnetic field is changed rhythmically in synchrony with the oscillations
of the needle, many small pushes can result in considerable oscillation. (c) The same effect can be achieved
by replacing the weak magnet with an electromagnet. An alternating current will shift between pushing and
pulling the north pole of the magnetic needle (opposite effect on the south pole). Since the field is most
powerful inside the coil, the greatest effect is achieved by placing the needle there.
strength of the radio waves (known as the amplitude) is weakened as the oscillations of the needle
gradually vanish.
Imagine the following situation: A compass is placed in a magnetic field created by one or more
powerful magnets. After a short period of time, the needle has settled and is pointing in the direction
of the magnetic field, figure 2(a). If the needle is given a small push perpendicular to the magnetic
field (a rotation), it will vibrate through north, but gradually settle again. The oscillations will occur
at a frequency that will hereafter be referred to as the resonance frequency. As long as the magnetic
needle is oscillating, radio waves with the same frequency as the oscillation will be emitted. The
radio waves die out together with the oscillations, and these can in principle be measured with an
antenna (coil, figure 2(c)). The measurement can, for example, tell us about the strength of the
magnetic needle and the damping rate of its oscillations.
As will be made clear, there are also “magnetic needles” in the human body, as hydrogen nuclei
are slightly magnetic. We can also manipulate these so they oscillate and emit radio waves, but
it is less clear how we may give the needles the necessary “push.” In order to make an ordinary
compass needle swing, we may simply give it a small push perpendicular to the magnetic field with
a finger, but this is critically less feasible in the body. Instead, we may take advantage of the fact that
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magnets influence other magnets, so that the weak push perpendicular to the magnetic field can be
delivered by bringing a weak magnet near the needle, as shown in figure 2(b). With this method, we
may push the needle from a distance by moving a weak magnet towards the needle and away again.
In an MR scanner, the powerful magnet is extremely powerful for reasons that will be explained
below. The magnetic field across is quite weak in comparison. The push that the weak magnet can
deliver is therefore too weak to produce radio waves of any significance, if the magnetic needle is
only “pushed” once.
If, on the other hand, many pushes are delivered in synchrony with the mentioned oscillations of
the magnetic needle, even waving a weak magnet can produce strong oscillations of the magnetic
needle. This is achieved if the small magnet is moved back and forth at a frequency identical to the
natural oscillation frequency, the resonance frequency, as described above.
What is described here is a classical resonance phenomenon where even a small manipulation can
have a great effect, if it happens in synchrony with the natural oscillation of the system in question.
Pushing a child on a swing provides another example: Even weak pushes can result in considerable
motion, if pushes are repeated many times and delivered in synchrony with the natural oscillations
of the swing.
Let us focus on what made the needle oscillate: It was the small movements of the magnet, back
and forth, or more precisely the oscillation of a weak magnetic field perpendicular to the powerful
stationary magnetic field caused by the movement of the magnet.
But oscillating magnetic fields is what we understand by “radio waves”, which means that in
reality, we could replace the weak magnet with other types of radio wave emitters. This could, for
example, be a small coil subject to an alternating current, as shown in figure 2(c). Such a coil will
create a magnetic field perpendicular to the magnetic needle.
The field changes direction in synchrony with the oscillation of the alternating current, so if the
frequency of the current is adjusted to the resonance frequency of the magnetic needle, the current
will set the needle in motion. When the current is then turned off, the needle will continue to swing
for some time. As long as this happens, radio waves will be emitted by the compass needle. The coil
now functions as an antenna for these radio waves. In other words, a voltage is induced over the coil
because of the oscillations of the magnetic field that the vibration of the magnetic needle causes.
In summary, the needle can be set in motion from a distance by either waving a magnet or by
applying an alternating current to a coil. In both situations, magnetic resonance is achieved when
the magnetic field that motion or alternating currents produce, oscillates at the resonance frequency.
When the waving or the alternating current is stopped, the radio waves that are subsequently
produced by the oscillating needle, will induce a voltage over the coil, as shown. The voltage
oscillates at the resonance frequency and the amplitude decreases over time. A measurement of the
voltage will reflect aspects of the oscillating system, e.g. “the relaxation time”, meaning the time it
takes for the magnetic needle to come to rest.
The above mentioned experiment is easily demonstrated, as nothing beyond basic school physics
is involved. Nobody acquainted with magnets and electromagnetism will be surprised during the
experiment. Nonetheless, the experiment reflects the most important aspects of the basic MR
phenomenon, as it is used in scanning.
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Figure 3: The spin of the nuclei
(rotation) makes them magnetic.
B0
M
Figure 4: A magnetization M
that precess around the magnetic
field B0 because of spin (rotation
around M).
There are, however, differences between MR undertaken with
compasses and nuclei. These are due to the fact that the nuclei
are not only magnetic, but also rotate around an axis of their own,
as shown in figure 3 to the right (a twist to this is discussed in
section 8). The rotation, called spin, makes the nuclei magnetic
along the rotational axis. This is equivalent to a situation where
the compass needle described above constantly and quickly rotates
around its own length direction (has spin or angular momentum).
Such a needle would swing around north in a conical motion if
the mounting allowed for this, rather than swing in a plane through
north. This movement is called precession and it is illustrated in
figure 4. Think of it as a variant of the normal needle oscillation.
Similarly, a pendulum can swing around a vertical axis rather than
in a vertical plane, but basically there is no great difference. In the
rest of this section, the consequences of spin are elaborated on. The
finer nuances are not important for understanding MR scanning.
Precession around the direction of a field is also known from a
spinning top (the gravitational field rather than the magnetic field,
in this example): A spinning top that rotates quickly will fall slowly,
meaning that it will gradually align with the direction of the grav-
itational field. Rather than simply turning downwards, it will slowly
rotate around the direction of gravity (precess) while falling over.
These slow movements are a consequence of the fast spin of the
spinning top.
The same applies to a hypothetical magnetic needle that rotates quickly around its own length
axis, as the atomic nuclei in the body do. In the experiment described above, such a magnetic
needle with spin would rotate around north (precess) after having received a push. It would spiral
around north until it eventually pointed in that direction, rather than simply vibrate in a plane as a
normal compass needle. It would meanwhile emit radio waves with the oscillation frequency, which
we may now also call the precession frequency. This frequency is independent of the amplitude,
i.e., independent of the size of the oscillations. This consequence of spin is another difference to
the situation shown in figure 2 since the natural oscillation frequency of a normal compass is only
independent of amplitude for relatively weak oscillations.
Like before, it is possible to push the magnetic needle with a weak, perpendicular magnetic
field that oscillates in synchrony with the precession (meaning that it oscillates at the resonance
frequency). Just as the magnetic needle precess around the stationary magnetic field, it will be
rotated around the weak, rotating magnetic field, rather than (as before) moving directly in the
direction of it. In practice, this means that a perpendicular magnetic field, that rotates in synchrony
with the precession of the needle, will rotate it slowly around the rotating fields direction (meaning
a slow precession around the weak rotating magnetic field).
We have now in detail described the magnetic resonance phenomenon, as employed for scanning:
The influence of spin on the movement of the magnetic axis can, at first glance, be difficult to
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understand: That the force in one direction can result in pushes towards another direction may
appear odd (that the pull in a magnet needle towards north will make it rotate around north, if the
needle rotates around its own length axis (has spin)). It does, however, fall in the realms of classical
mechanics.
One does not need to know why spinning tops, gyros, nuclei and other rotating objects act queer
in order to understand MR, but it is worth bearing in mind that the spin axis rotates around the
directions of the magnetic fields. The precession concept is thus of importance, an it is also worth
remembering that spin and precession are rotations around two different axes.
3 The magnetism of the body
Equipped with a level of understanding of how magnet needles with and without spin are affected
by radio waves, we now turn to the “compass needles” in our very own bodies.
• Most frequently, the MR signal is derived from hydrogen nuclei (meaning the atomic nuclei
in the hydrogen atoms). Most of the body’s hydrogen is found in the water molecules. Few
other nuclei are used for MR.
• Hydrogen nuclei (also called protons) behave as small compass needles that align themselves
parallel to the field. This is caused by an intrinsic property called nuclear spin (the nuclei
each rotate as shown in figure 3). By the “direction of the nuclear spins” we mean the axis of
rotation and hence the direction of the individual “compass needles”.
• The compass needles (the spins) are aligned in the field, but due to movements and nuclear
interactions in the soup, the alignment only happens partially, as shown in figure 5 – very little,
actually. There is only a weak tendency for the spins to point along the field. The interactions
affect the nuclei more than the field we put on, so the nuclear spins are still largely pointing
randomly, even after the patient has been put in the scanner. An analogy: If you leave a bunch
of compasses resting, they will all eventually point towards north. However, if you instead
put them into a running tumble dryer, they will point in all directions, and the directions of
individual compasses will change often, but there will still be a weak tendency for them to
point towards north. In the same manner, the nuclei in the body move among each other and
often collide, as expressed by the temperature. At body temperature there is only a weak
tendency for the nuclei to point towards the scanners north direction.
Together, the many nuclei form a total magnetization (compass needle) called the net magneti-
zation. It is found, in principle, by combining all the many contributions to the magnetization,
putting arrows one after another. If an equal number of arrows point in all directions, the net
magnetization will thus be zero. Since it is generally the sum of many contributions that swing
in synchrony as compass needles, the net magnetization itself swings as a compass needle.
It is therefore adequate to keep track of the net magnetization rather than each individual
contribution to it.
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B0
Figure 5: The figure shows the same situations in two and three dimensions (top and bottom,
respectively). The nuclear spins are shown as numerous arrows (vectors). In the lower graphs, they
are all drawn as beginning in the same point, so that the distribution over directions is clear (implicit
coordinate system (Mx,My,Mz)). When a patient arrives in the ward, the situation is as shown in the
two graphs to the left: The spins are oriented randomly, with a uniform distribution over directions,
meaning that there is about an equal number of spins pointing in all directions. The net magnetization
is near zero and the nuclei do not precess. When a magnetic field B0 is added, as shown in the two
figures to the right, a degree of alignment (order) is established. The field is only shown explicitly
in the top right figure, but the effect is visible in both: The direction distribution becomes “skewed”
so that a majority of the nuclei point along the field. In the lower right figure, the net magnetization
(thick vertical arrow) and the precession (the rotation of the entire ball caused by the magnetic field)
are shown. The lower figures appear in the article Is Quantum Mechanics necessary for understanding
Magnetic Resonance? Concepts in Magnetic Resonance Part A, 32A (5), 2008.
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As mentioned above, the nuclei in the body move among each other (thermal motion) and
the net magnetization in equilibrium is thus temperature dependent. Interaction between
neighboring nuclei obviously happens often in liquids, but they are quite weak due to the small
magnetization of the nuclei. Depending on the character and frequency of the interaction,
the nuclei precess relatively undisturbed over periods of, for example, 100 ms duration. At
any time, there is a certain probability that a nucleus takes part in a dramatic clash with other
nuclei, and thus will point in a new direction, but this happens rather infrequently.
The net magnetization is equivalent to only around 3 per million nuclear spins oriented
along the direction of the field (3 ppm at 1 tesla). This means that the magnetization of a
million partially aligned hydrogen nuclei in the scanner equals a total magnetization of only 3
completely aligned nuclei.
With the gigantic number of hydrogen nuclei (about 1027) found in the body, the net magneti-
zation still becomes measurable. It is proportional to the field: A large field produces a high
degree of alignment and thus a large magnetization and better signal-to-noise ratio.
• If the net magnetization has been brought away from equilibrium, so it no longer points along
the magnetic field, it will subsequently precess around the field with a frequency of 42 million
rotations/second at 1 tesla (42 MHz, megahertz). This is illustrated in figure 4. Eventually it
will return to equilibrium (relaxation), but it takes a relatively long time on this timescale (e.g.
100 ms as described above). Meanwhile, radio waves at this frequency are emitted from the
body. We measure and analyze those. Notice: The position of the nuclei in the body does not
change - only their axis of rotation.
• The precession frequency is known as the Larmor frequency in the MR community. The
Larmor equation expresses a connection between the resonance frequency and the magnetic
field, and it is said to be the most important equation in MR:
f = γB0
The equation tells us that the frequency f is proportional to the magnetic field, B0. The
proportionality factor is 42 MHz/T for protons. It is called “the gyromagnetic ratio” or simply
“gamma”. Thus, the resonance frequency for protons in a 1.5 tesla scanner is 63 MHz, for
example.
The Larmor equation is mainly important for MR since it expresses the possibility of designing
techniques based on the frequency differences observed in inhomogeneous fields. Examples
of such techniques are imaging, motion encoding and spectroscopy.
• But how is the magnetization rotated away from its starting point? It happens by applying
radio waves at the above mentioned frequency.
Radio waves are magnetic fields that change direction in time. The powerful stationary field
pushes the magnetization so that it precesses. Likewise the radio waves push the magnetization
around the radio wave field, but since the radio wave field is many thousand times weaker
than the static field, the pushes normally amount to nothing.
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Figure 6: Scene from animation found at http://www.drcmr.dk/MR that shows how radio
waves affect a collection of nuclear spins precessing around B0 (vertical axis) at the Larmor frequency.
The radio wave field that rotates around the same axis at the same frequency, induces simultaneous
rotation around a horizontal axis, as symbolized by the circular arrow. The relative orientation of the
nuclei does not change, and it is therefore adequate to realize how the net magnetization (here shown
by a thick arrow) is affected by the magnetic field.
Because of this, we will exploit a resonance phenomenon: By affecting a system rhythmically
at an appropriate frequency (the systems resonance frequency), a large effect can be achieved
even if the force is relatively weak. A well-known example: Pushing a child sitting on a
swing. If we push in synchrony with the swing rhythm, we can achieve considerable effect
through a series of rather weak pushes. If, on the other hand, we push against the rhythm (too
often or too rarely) we achieve very little, even after many pushes.
With radio waves at an appropriate frequency (a resonant radio wave field), we can slowly
rotate the magnetization away from equilibrium. “Slowly” here means about one millisecond
for a 90 degree turn, which is a relatively long time as the magnetization precesses 42 million
turns per second at 1 tesla (the magnetization rotates 42 thousand full turns in the time it takes
to carry out a 90 degree turn, i.e., quite a lot faster).
Figure 6 is a single scene from an animation found at http://www.drcmr.dk/MR that
shows how a collection of nuclei each precessing around both B0 and a rotating radio wave
field as described earlier, together form a net magnetization that likewise moves as described.
The strength of the radio waves that are emitted from the body depends on the size of the net
magnetization and on the orientation. The greater the oscillations of the net magnetization,
the more powerful the emitted radio waves will be. The signal strength is proportional to the
component of the magnetization, that is perpendicular to the magnetic field (the transversal
magnetization), while the parallel component does not contribute (known as the longitudinal
magnetization). In figure 4, the size of the transversal magnetization is the circle radius.
If the net magnetization points along the magnetic field (as in equilibrium, to give an example)
no measurable radio waves are emitted, even if the nuclei do precess individually. This is
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because the radio wave signals from the individual nuclei are not in phase, meaning that they
do not oscillate in synchrony perpendicular to the field. The contributions thereby cancel
in accordance with the net magnetization being stationary along the B0-field (there is no
transversal magnetization).
• The frequency of the radio waves is in the FM-band so if the door to a scanner room is open,
you will see TV and radio communication as artifacts in the images. At lower frequencies we
find line frequencies and AM radio. At higher frequencies, we find more TV, mobile phones
and (far higher) light, X-ray and gamma radiation. From ultra-violet light and upwards, the
radiation becomes “ionizing”, meaning that it has sufficient energy to break molecules into
pieces. MR scanning uses radio waves very far from such energies. Heating, however, is
unavoidable, but does not surpass what the body is used to.
4 The rotating frame of reference
Confusion would arise if the descriptions to come continued to involve both a magnetization that
precesses and radio waves that push this in synchrony with the precession. It is simply not practical
to keep track of both the Larmor-precession and the varying radio wave field at the same time, and
this is necessary to establish the direction of the pushes, and thus how the magnetization changes.
Instead, we will now change our perspective.
An analogy: We describe how the horse moves on a merry-go-round in action. Seen by an
observer next to the merry-go-round, the horse moves in a relatively complicated pattern. However,
if the merry-go-round is mounted, the movement of the horse appears limited to a linear up-and-
down movement. We say that we have changed from the stationary frame of reference to a rotating
frame of reference.
In the same vein, we can simplify the description of the resonance phenomenon by moving to
a rotating frame of reference and getting rid of the Larmor-precession. We mount the merry-go-
round that rotates with the Larmor frequency and recognize that in this frame the magnetization is
stationary until we employ radio waves. The effect of B0 therefore appears to be gone, as shown in
figure 7.
5 Relaxation
Interactions happening at near-collisions between nuclei give rise to the magnetization constantly
approaching the equilibrium size. This is called relaxation. The speed at which relaxation occurs
depends on the protons interactions with their neighbors, which in turn depends on the firmness
of the substance (the consistency). It is the difference in consistency and the presence of large
molecules that limit the waters free movement, which causes most of the contrast we see in MR
images.
The relaxation occurs on two different time scales: The magnetization perpendicular to the
magnetic field (the transversal magnetization) often decreases relatively rapidly, while it can take
considerably longer to recover the magnetization along the field (the longitudinal magnetization).
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B0
M M
B1
M
Figure 7: In the stationary frame of reference, a rapid precession occurs (left). In the rotating frame of
reference (middle) the magnetization usually only moves slowly due to relaxation and smaller differences
in frequency. When resonant radio waves are transmitted, the effect of these is to induce precession around
an axis perpendicular to the B0 field (right). Descriptions in the rotating and stationary frames of reference
are equally precise, but explanations are almost always given in the rotating frame of reference, since the
movement of the magnetization is more simple there. Furthermore, the scanner always presents measurements
as if they were obtained in the rotating frame of reference. This happens because the scanner removes the
rapid variation in the signals before they are stored in the computer (the signals are modulated down from the
Larmor frequency to lower frequencies).
• The transversal magnetization (Mxy) decreases exponentially on a timescale T2 (e.g. around
100 ms for brain tissue. Several seconds for pure water).
• The longitudinal magnetization (Mz) approaches equilibrium M0 on a timescale T1 (for
example approximately 1 s for brain tissue. Several seconds for pure water).
The relaxation times depend on the mobility of the molecules and the strength of the magnetic field,
as discussed in section 5.2.
5.1 Weightings
The contrast in an MR-image is controlled by the choice of measuring method (sequence and
sequence parameters, which will be discussed later). For example, we call an image T2-weighted if
the acquisition parameters are chosen so the image contrast mainly reflect T2-variations. One must
understand, however, that even in a heavily T2-weighted image, the contrast will often reflect more
than just T2-variation. To provide an example, variation in water content always results in some
contrast.
The echo time, TE, is the period from we rotate the magnetization into the transversal plane until
we decide to measure the radio waves (a more precise definition will follow later). Meanwhile, a
loss of magnetization and signal will occur due to T2-relaxation. The echo time is thus the period
within the measurement which gives T2-weighting in the images. A long TE compared to T2 will
thus result in considerable T2-contrast, but only little signal. The greatest sensitivity to T2-variation
will be achieved when TE T2.
Often, we will repeat similar measurements several times, e.g. once per line in an image. The
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repetition-time, TR, is the time between these repetitions. Every time we make such a measurement,
we (partially) use the longitudinal magnetization present (the magnetization is rotated into the
transversal plane which results in emission of radio waves while the transversal component gradually
disappears). If we use the magnetization often (short TR), every repeat will therefore only produce a
small signal. If we, on the other hand, wait longer between repetitions (long TR), the magnetization
will nearly recover to equilibrium between repetitions. What is meant by short and long TR? It is
relative to T1 that is the time scale on which the longitudinal magnetization is rebuilt.
If the magnetization is completely rebuilt between measurements for all types of tissue in the
scanner, meaning if TR is significantly longer than the maximum T1, the T1-contrast will disappear.
In this case, the transversal magnetization immediately following the rotation of the nuclei reflects
the equilibrium magnetization. The radio waves do so, as well. The equilibrium magnetization is
governed by the hydrogen concentration, also known as the proton density (PD).
Thus, we may conclude that using a long TR results in limited T1-weighting but a strong signal.
If we apply a shorter TR, the signal is reduced for all types of tissue, but the signal becomes
more T1-weighted, meaning that the images will be less intense, but with a relatively greater signal
variation between tissues with differing T1.
Finally, we can minimize both T1- and T2-contrast, which results in a PD-weighted image. In such
an image, variation in the water content is the primary source of contrast, since the proton density is
the density of MR-visible hydrogen that is largely proportional to the water content.
Summarized, the following apply to weightings in a simple excitation-wait-measure-wait-repeat
sequence.
• T1-weighted images are made by applying short TR and short TE, since T1 contrast is thereby
maximized and T2 contrast is minimized.
• T2-weighted images are made by applying long TR and long TE, since T1 contrast is thereby
minimized and T2 contrast maximized.
• PD-weighted images are made by applying long TR and short TE, since both T1- and T2-
contrast are thereby minimized. Meanwhile, the signal is maximized, but it is of no use if the
contrast disappears, since only small variations exist in the water content of the tissue.
The “lacking” combination of long TE and short TR results in a weak signal and mixed T1- and T2
contrast. Images of bottles with jello in picture 8 illustrates how the contrast and signal in an image
varies with TE, TR and consistency.
5.2 Causes of relaxation
The difference between T1 and T2 is due to a difference in the causes of relaxation. For the protons
in firm matter, the spins will rapidly be dephased after excitation, meaning that they will point
in all directions perpendicular to the field (have all kinds of phases). This is caused by the small
local contributions to the magnetic field that the individual spins are making, and that make the
neighboring nuclei precess at altered frequencies. In firm matter, these interactions are constant in
time, while they vary in fluids, since nuclei are constantly experiencing new neighbors. Thus, the
spins can remain in phase for relatively long (seconds) in fluids, while they loose their common
14

(a) Four images, all obtained
with a common TR=5 seconds
and TE=90, 50, 20, 15 ms (shown
in reading order).
(b) Six images obtained with a common TE=15 ms
and TR=500, 1000, 2000, 3000, 4000, 5000 ms
(shown in reading order).
Figure 8: Phantom data which illustrates signal intensity and contrast for bottles filled with jello af varying
consistency. Where is T1 long/short? How long, how short? The same for T2? Which bottles might be pure
water? Which jello is most firm? What pictures are the most T1-, T2- and PD-weighted?
orientation in a matter of milliseconds or less in firm matter. We can therefore conclude that T2 is
short in firm matter.
The described process affects the individual spins Larmor-frequency, but does not give rise to a
change of the longitudinal magnetization, since the interaction of just two nuclei cannot alter the
combined energy, which is proportional to the longitudinal magnetization. Therefore this type of
nuclear interaction does not contribute T1-relaxation – that requires more drastic nuclear interactions
that involve an exchange of energy with the surroundings. All processes that result in T1-relaxation
also result in T2-relaxation, ensuring that T1 is never shorter than T2.
Generally T2 becomes shorter with increasing firmness of the matter, but this does not apply to
T1, which is long in very firm matter and very fluid matter (e.g., several seconds), but is short for
semi-firm matter.
The shortest T1 is achieved precisely when random interactions with the neighbors fulfill the
resonance condition, meaning, for example, that at 1 tesla, T1 is shortest for matter where a proton
meets about 42 million other nuclei in 1 second (the Larmor frequency is 42 MHz). This is
understandable – even random pushes to a swing can result in considerable oscillations if the
frequency is approximately correct. If we instead push way too frequently or rarely, very little will
be accomplished regardless of whether the pushes are random or not.
5.3 Inhomogeneity as a source of signal loss, T∗
2
Interactions between nuclei in the ever changing environment of molecules is the cause of radio
signal loss on a timescale T2 and longitudinal magnetization recovery on a longer timescale T1.
However, a loss of transversal magnetization is also observed because of inhomogeneity in the
field, meaning variation in B0. As expressed through the Larmor-equation, the nuclei precess at
a frequency depending on the magnetic field. If this varies over the investigated area, the nuclei
15

will after some time, point in all directions transversally, even if they had some degree of alignment
after excitation. This process is called dephasing. Since the measured signal is proportional to
the transversal net magnetization, inhomogeneity gives rise to a loss of signal. The larger the field
inhomogeneity, the faster the dephasing. How quickly this happens in denoted by the time constant
T∗
2 (pronounced “T2-star”). The degree of inhomogeneity depends partly on the scanners ability to
deliver a uniform field (called a good shim). T1 and T2, are tissue-dependent parameters for which
normal values are published. This does not apply to T∗
2 values, since they depend, for example, on
the size of the voxel since the inhomogeneity increases with this.
Loss of signal due to interactions of nuclei is irreversible, but through a sly trick, signal lost due
to inhomogeneity can be recovered. The inverse process of dephasing is called refocusing, and it
involves gradual re-alignment of nuclei in the transversal plane. Refocusing is triggered by repeated
use of radio waves as described below. The recovered signal is known as an “echo”.
6 Sequences
A “sequence” is an MR measurement method and there exists an extraordinary number of these. A
measurement is characterized by the sequence of events that play out during the measurements. In a
classic “spin-echo-sequence” as shown in figure 9, the magnetization is first rotated 90 degrees away
from equilibrium through the use of radio waves. A short waiting period follows with dephasing
caused by inhomogeneity. Subsequently the magnetization is turned an additional 180 degrees
with radio waves. After a little more waiting time and refocusing, the signal is measured. This
course is usually repeated. In the sequence description above, a couple of sequence parameters were
involved, e.g. the waiting time from excitation to measurement. A measurement method can thus be
described by a sequence name and the associated sequence parameters. Together they control the
measurement’s sensitivity to different tissue parameters, and the choice is therefore made on the
background of the actual clinical situation. Sequences include elements such as
excitation Turning of the magnetization away from equilibrium.
dephasing Field inhomogeneity causes the nuclei to precess at different speeds, so that the align-
ment of the nuclei – and thus the signal – is lost.
refocusing pulse After excitation and dephasing, part of the signal that has been lost due to
inhomogeneity can be recovered. This is achieved by sending a 180-degree radio wave
pulse (in this context known as a refocusing pulse) that turns the magnetization 180 degrees
around the radio wave field. The effect is that those spins which precess most rapidly during
dephasing are put the furthest back in evolution and vice versa (mirroring around the radio
wave field causes the sign of the phase angle to be reversed). In the following “refocusing
period” where the nuclei are still experiencing the same inhomogeneity, they will therefore
gradually align again (come into phase). The lost signal is recovered, so that an “echo” is
measured.
readout Measurement of MR signal from the body.
16

Dephasing Refocusing
180◦
x
Readout
90◦
y
.
Figure 9: Sequence diagrams illustrate with an implicit time axis from left to right, the events during a
measurement. The first blocks in this spin-echo sequence ilustrate radio wave pulses, meaning periods
wherein the radio wave sender is turned on. The diagram is read as follows: Initially radio waves are sent
for the duration needed to turn the magnetization by 90 degrees around the y-axis in the rotating frame of
reference. After a short waiting period wherein the spins will dephase, radio waves are again sent for a period
appropriate for turning the magnetization 180 degrees around the x-axis. This gives rise to gradual refocusing.
The signal is read out (measured) when this is complete, which happens after a brief waiting period equal
to the duration of the dephasing. Sequence diagrams most often contain several events. An example with
gradients is found in figure 17
.
waiting times Periods wherein the relaxation contributes to the desired weighting.
Further sequence elements (building blocks) that can be mentioned include inversion, bipolar
gradients, spoiling and imaging gradients. Some of these are described later in the main text or in
the glossary.
7 Signal-to-noise and contrast-to-noise ratios
In section 5.1, it was hinted that PD-weighted images are rarely the best at reflecting pathology. This
is in spite of the fact that the signal-to-noise ratio is often good in such images, because imaging
with fully relaxed magnetization and short echo-time increases the signal strength. This exemplifies
that the contrast-to-noise ratio always is more important for diagnostics than the signal-to-noise
ratio, meaning that a good image is characterized by the signal difference between conditions or
tissues that we wish to distinguish (the contrast) being large compared to the relevant noise (that
is, the signal variations that make the distinction difficult, whether these are random or not). If the
contrast-to-noise ratio is small, on the other hand, the relevant signal differences drown in noise.
An image can be quite noisy, meaning that is has a low signal-to-noise ratio, but still be good
at reflecting a given pathology. One example is diffusion-weighted images in connection with
ischemia, since diffusion weighting lowers the signal for all types of tissue, but improves contrast
between normal and ischemic tissue. Something similar is often true for images acquired with long
echo-time.
In rough terms, the signal-to-noise ratio is an expression for how nice-looking an image is,
while the contrast-to-noise ratio is a measure of the usefulness insofar as the contrast is defined
meaningfully from all the parameters that can help answering the clinically relevant questions. It is
thus not trivial to define the contrast for a given clinical situation (a metric), but it is in principle
simple to optimize it. In practice the optimization can easily be the most difficult part, however, as it
may require many measurements to be performed with different sets of parameters.
17

8 Quantum Mechanics and the MR phenomenon
This section is non-essential for understanding the rest. Here, some common misconceptions found
in many introducing textbooks (even good ones) are pointed out. These misconceptions have been
recognized by a minority for a long time, but they are sadly still appearing in many new books,
as authors often quote each other without a deep understanding of MR phenomena. The problem
derives mainly from the fact that few authors of MR textbooks have a good background in quantum
mechanics, which is what the misconceptions center around. Such a background is needed, if
authors present explanations based on quantum mechanics, but luckily such explanations are not
needed.
The fundamental problem is that quantum mechanics is often presented as necessary in order to
understand MR, where a classical description is fully adequate. If the two are combined incorrectly,
one easily ends up with something that is contrary to both common sense and experimental evidence.
First of all, a little background: In physics, there exists a distinction between classical phenomena
and quantum phenomena. The first are those that can be described by classical physics, such as
Newtons laws. The vast majority of the experiences we gain from everyday life belong in the
classical category. In certain situations, however, the classical description is inadequate, so that
predictions derived from it turns out to be inconsistent with observations. In such cases, one needs
quantum mechanics that is believed to be able to describe all of the currently observed physical
phenomena, but which is difficult to understand, since we make very few unclassical observations
in our everyday life. One example of a quantum phenomenon is superconductivity (resistance
free electrical conductance), that we use when creating powerful magnetic fields in the scanner.
Superconductivity would not be possible if classical physics were accurate on an atomic level.
Quantum mechanics, on the other hand, describes superconductivity, meaning that if one takes
the quantum mechanical equivalents to Newtons fundamental laws as a basis, it can be shown
that a broad range of surprising phenomena are predictable, including superconductivity. All
phenomena that can be described using classical mechanics can also be described with quantum
mechanics, although the complexity grows tremendously, since explanations require familiarity with
the quantum mechanical terminology rather than simply experience from everyday life.
Magnetic resonance is not a quantum phenomenon, although it is often presented this way. It
can be described with quantum mechanics, but the approach necessitates considerable background
knowledge. MR can alternative be described classically, since it can be shown that the classical
description of magnetic dipoles behavior in the magnetic field is a direct consequence of quantum
mechanics.
The existence of the proton spin and magnetic moment is, on the contrary, a quantum mechanical
phenomenon, and the classical figure 3 should therefore not be taken too literally (it has its limitations,
although these are not evident in MR experiments). If one accepts the existence of the magnetic
needle quality of nuclei, the rest follows from classical physics.
For a number of introductions to MR, the classical approach is not chosen. On the other hand, the
authors typically do not posses the necessary background in order to use the heavy (and in this case
superfluous) quantum mechanical formalism in a meaningful way. Instead, we are often presented
with a semi-classical description with elements of classical physics and elements of quantum physics.
18

Such a description can be precise and useful, but easily gets to include elements of nonsense, which
is the case for much literature on the subject. Examples are presented below.
Although quantum mechanics is not needed to understand the magnetic resonance phenomenon
or pretty much anything else going on in scanners, physicists and chemists can still benefit from
learning about it. Quantum mechanics is, for example, necessary in order to determine the size
of T1 and T2 correctly from theoretical assumptions alone (without actual measurements), while
the existence of T1 and T2 is purely classical (they can be calculated classically, but the results will
be wrong). In the same vein, quantum mechanics is practical in connection with certain types of
spectroscopy.
There are consequently good reasons to use quantum mechanics in descriptions aimed at chemists
and physicists with the necessary quantum background, while an incorrect pseudo-quantum intro-
duction is never helpful. Finally, quantum mechanics is necessary to describe measurements carried
out on single protons. Such measurements are never performed, however, as the signal always
comes from a large collection of protons, a so-called statistical ensemble. For such an ensemble
of non-interacting protons, it is a relatively easy to show, quantum mechanically, that the classical
description is exactly right.
Quantum mechanics is most often employed in the following contexts in introductions to magnetic
resonance:
• Quantum mechanics is very convenient for “explaining” the resonance phenomenon. This is
often done by drawing two lines, calling them energy levels and say that quantum mechanics
tell us that the photon energy must match the energy difference between the energy levels.
A problem with this approach is that it is non-intuitive and raises more questions than it
answers, unless the reader is already schooled in quantum mechanics (why are almost half
of the nuclei pointing opposite the field? Why doesn’t continuous use of radio waves level
out the population differences? What happens if the photon-energy isn’t quite precise?
Alternatively, it can be intuitively argued (as attempted above) that the radio waves must push
the magnetization in synchrony with the precession in order to rotate the magnetization away
from equilibrium.
• Quantum mechanics can conveniently be used to calculate the size of the equilibrium mag-
netization. In equilibrium, the coherence terms average to zero and the partition function
consequently is reduced to a sum of just two terms. Classically, more states contribute, and
the calculation is therefore more complicated. The results of the two calculations are equal,
however.
8.1 Corrections
After this tirade, time has come to comment on some of the myths and incorrect figures often
found in literature. It is more important to recognize and ignore the erroneous figures than it is
to understand why they are wrong. The details can be found in the article Is Quantum Mechanics
necessary for understanding Magnetic Resonance? Concepts in Magnetic Resonance Part A, 32A
(5), 2008, that contains detailed accounts.
19

MR is a quantum mechanical phenomenon No, the resonance phenomenon is classical. See
above. The existence of nuclear spin and aspects of certain experiments are, however,
quantum mechanical.
Protons can only be in two conditions: Near-parallel or anti-parallel with the field.
This situation, as illustrated in figure 10, does not occur. There are two conditions that are
characterized quantum mechanically by having a well-defined energy. As expected, however,
the spins can point in all directions both classically and quantum mechanically.
The radio waves align the spins on the two cones The illustration is the result of the erroneous
belief that the protons can only be in one of two conditions, combined with the fact that
we can rotate the magnetization. The result is pure hocus-pocus: Why are the individual
spins not rotated the same way by the radio waves? How do the protons “know” where they
are each supposed to be pointing? According to this view, it appears possible to change the
magnetization with radio waves. Why don’t we see this happening in practise? The questions
are numerous, simply because the drawing is wrong. It is relatively simple to show (classically
as well as quantum mechanically) that the locally homogeneous fields that magnet and RF
coils generate, can never change the relative orientation of spins locally. Only inhomogeneous
fields such as those the nuclei affect each other with, can do that. They are the source of
relaxation.
N
S Mx
My
Mz
Figure 10: These figures are seen often, but they are misleading. They reflect a factually wrong perception
of spins as only being capable of being in spin-up or spin-down conditions.
9 Imaging
So far, it has been discussed how the body can be brought to emit radio waves, but it has not been
mentioned how radio waves from one position in the body can be distinguished from radio waves
20

RF
MxMx
MyMy
MzMz
M
Figure 11: The spins are not aligned on two cones with a net magnetization in the transversal plane, as this
figure claims. Homogeneous radio wave fields cannot change the relative orientation of spins.
emitted at another position, which is rather necessary for imaging. In this section, techniques for
spatial localization and imaging are described. Initially, it is explained why these techniques are
fundamentally different from more easily understood techniques known from everyday life.
9.1 Background
The most obvious methods for MR imaging could be imagined to be projection or the usage
of antennas that can detect where in the body the radio waves are emitted. X-ray and normal
microscopy are examples of such “optical” techniques, and it would appear obvious to extend this
type of imaging to MR. Optical techniques are, however, “wavelength-limited”, which means that
they cannot be used to acquire images more detailed than approximately one wavelength.
In other words: Due to fundamental causes, one cannot localize the source of radio waves more
precisely than about one wavelength when using lenses or other direction-sensitive antennas. The
radio waves used in MR scanning are typically several meters long, so with optical techniques we
can hardly determine whether the patient is in or outside the scanner (this argument is really only
valid in the far field, which is the background for parallel imaging. More on this later).
Optical techniques as we know them from binoculars, eyesight, CT, X-ray, ultrasound and
microscopes, are thus practically useless for MR-imaging, and a fundamentally different principle is
necessary. This principle was introduced by Paul Lauterbur in 1973, and it resulted in the Nobel
Prize in Medicine in 2003. Basically, Lauterbur made the protons give their own locations away by
making the frequency of the emitted radio waves reflect the position. Lauterbur shared the prize
with Sir Peter Mansfield, who also contributed greatly to the development of techniques used in
magnetic resonance imaging (MRI).
9.2 Principles
A requirement for MR imaging is that the scanner is equipped with extra electromagnets called
“gradient coils” that cause linear field variations. Direction and strength can be altered as desired.
21

The spatial localization occurs according to different principles, of which the most simple is slice
selection. Other forms of coding involve the so-called k-space, which will be introduced in a later
section.
9.3 Slice selection
By using gradient coils, the magnetic field strength can be controlled so that it, for example, increases
from left to right ear, while the direction is the same everywhere (along the body). This is called
a field gradient from left to right. By making the field inhomogeneous in this way, the resonance
frequency varies in the direction of the field gradient. If we then push the protons with radio waves
at a certain frequency, the resonance condition will be fulfilled in a plane perpendicular to the
gradient as shown in figure 12. The spins in the plane have thus been rotated significantly, while
spins in other positions simply vibrate slightly. Thus we have achieved slice selective excitation of
the protons and a sagittal slice has been chosen.
Figure 12: Spin is influenced selectively in a sagital slice, if a gradient from left to right is applied while
radio waves are transmitted.
9.4 Spatial localization within a slice
After the protons in a slice are excited, they will all emit radio waves. In order to create images
of the slice, we must introduce a way to distinguish the signals from different positions within the
slice. The fundamental principle can appear a bit foreign, but will be explained in detail at a later
stage. Briefly told, different patterns in the magnetization throughout the slice are created with the
help of gradients. The strength of the radio signals that are returned tell us how much the object in
the scanner “looks like” the applied pattern. By combining patterns according to their measured
similarity to the object, the well-know MR-images are created.
22

What is meant by “patterns” is first illustrated in one dimension, meaning that we consider spins
placed on a line (e.g. between the ears) and watch their location and direction immediately after
excitation.
As illustrated, immediately after excitation the spins all point in the same direction perpendicular
to the magnetic field, which points out of the paper. They will thereafter precess around the magnetic
field, that is, they will rotate in the plane of the paper at a frequency that is dependent on the
magnetic field. Insofar as the field is made to increase from left to right by applying a field gradient
briefly, the spins will each turn an angle that depends linearly on the nucleus’ position:
This so-called “phase roll” is an example of the above mentioned spin patterns, that can be created
in the patient by use of gradients. The word “phase” expresses the direction that the spins point in.
It is seen that the neighboring spins point in nearly the same direction, but throughout the object, the
magnetization has rotated several turns. The longer time a gradient is turned on, and the greater the
field variation that arises, the more “phase roll” is accumulated (more rotations per unit length).
We have through use of gradients, made the spins point in all directions in a controlled fashion
and have thus simultaneously lost the signal. This is seen by comparing the two situations above,
since the measured magnetization is the sum of all the contributions from the individual spins. When
the spins are in phase (that is, pointing in the same direction), they jointly create a considerable
magnetization that gives rise to radio waves being emitted. When the spins point in all directions
as when a gradient has been applied, their sum is quite small. As a result, comparably weak radio
waves are emitted.
The gain from using the gradient can thus appear quite small: We have simply lost the signal.
That does not, however, have to be the case. Look, for example, at the situation illustrated below
where there are not (as above) protons uniformly distributed from left to right, but where there is a
regular variation in water content instead.
Before gradient:
After gradient:
In the figure we see how the spins point before and after applying the same gradient as above. The
only difference between the two earlier figures are “holes” that indicate a lack of protons (locally
water-free environment – bone, for example).
23

We now consider the total magnetization and thus the signal in this new situation. As long as
the spins all point in the same direction, i.e. before the gradient is applied, we receive less signal
than before, as fewer spins are contributing to the magnetization. The signal before the gradient is
applied is thus a measure of the total water content.
After the gradient is applied, on the other hand, we receive more signal than in the comparable
situation above (homogeneous water distribution). This may appear odd, since there are fewer
protons that contribute to the magnetization. The remaining protons all largely point in the same
direction, however, with the result that their combined magnetization is relatively large.
That the object with structure (varying water content) emits radio waves even after the gradient
has been applied, is caused by the match of the phase roll and the structure in the object, understood
in the sense that the “wave length” of the phase roll is the same as the distance between the water
puddles in the example.
If this is not the case, there will be tendency for the different contributions to the magnetization to
cancel each other. The signal is thus a measure of “similarity” between the phase roll pattern and the
structure in the object: After a phase roll has been introduced in the object by applying a gradient,
the signal we receive is a measure of whether there is structure in the object matching the phase roll.
Thus we now have a basis for understanding MR-imaging: Different phase roll patterns in the
body are drawn one after the other. The resulting radio wave signals are recorded for each of the
patterns. The size of the signal for a particular phase roll pattern tells us whether there is similarity
to the structure of the body.
In few cases there is an obvious similarity between structure and phase roll pattern. Most often,
the similarity is very limited. Generally, the images are obtained by combining patterns weighted
by the appropriate measured signals. This principle is actually known from the game MasterMind,
which was popular long ago (ﬁgure 13).
9.5 Extension to more dimension – k-space
So far, we have considered a situation with hydrogen nuclei placed as pearls on a string. We will
now extend to two dimension and consider the situation in a slice of tissue with uniform water
content after a gradient has been applied:
The ﬁgure show a phase roll in the gradient direction (upwards and right). The “thick arrow” on
the middle of the drawing shows the net magnetization, that is, the sum of all the small magneti-
24

Figure 13: The game Mastermind. One player places a row of colored pieces hidden behind a screen at one
end of the board. The opposing player must then guess the combination of pieces. After each guess (pieces
placed in the opposite end) the ﬁrst player informs to which degree the guess was correct, that is, how many
pieces were placed correctly (though not which). The guessing player will after a few rounds be able to
predict the hidden combination based on responses to earlier guesses. The same principle is in action in MR
imaging: The inner parts of the patient are hidden, but through gradients we may “draw” spin patterns in the
patient. In return, we receive radio waves whose strength reveals the similarity between patient structure and
the drawn pattern.
zation vectors (a vector is an arrow). It is seen that the sum is relatively small, since there is an
approximately equally many arrows pointing in all directions.
As in the one-dimensional case, there may still be a considerable net magnetization even after a
gradient has been applied. This happens if there, as shown in the following ﬁgure, is structure in the
object that matches the phase roll (the dots mark places where there are no protons).
25

For the sake of clarity, the spin patterns in the following will be shown in two different ways.
Rather than showing arrows pointing in different directions, phase rolls will be shown as an intensity
variation as in the figure in the middle, where the color reflects the direction of the arrows2:
kx
ky
k
For each such pattern, a vector k is assigned, whose direction indicates the direction of change
and whose length indicates the density of stripes as shown to the right.
By applying gradients, we “draw” patterns in the patient, one by one, in a predetermined order
specific for the chosen sequence. All directions of stripes and all densities of stripes are drawn
up until a certain limit. As said, the returned radio wave signal depends on the similarity between
object and pattern and it is registered as a function of k, as shown in figure 14. Example patterns are
shown, but for each single point in k-space there is a corresponding pattern.
9.6 Similarity and image reconstruction
Above, it has been described how the similarity between object and phase roll is measured, and
it has been said that the image is calculated by adding patterns together that are weighted by the
measured MR-signal. This section elaborates on what this actually means.
The concept of “similarity” is illustrated in this case with an example.
To the left is shown part of a city map (Manhattan). To the right is shown the unique stripe pattern
that looks most like the map in an MR sense. The similarity may seem small, but it is evident when
2The figure is not brilliant: the left situation corresponds to a stripe pattern with only 2 stripes rather than the 6 shown in
the middle. The direction, however, is spot on in all three illustrations.
26

Figure 14: The structure of k-space. The signal is measured as a function of the k-vector (kx,ky), which
changes when ﬁeld gradients are applied. The lighter region in the middle of k-space tells us that the object
in the scanner (a head) has greater similarity to slowly variating patterns, or expresed differently, the most
powerful radio waves are emitted from the body when low-density phase roll patterns are created.
27

it is realized that the stripes are going in the same directions as the north-south roads, that they have
about the same distance as these and that the pattern is shifted, so that it is bright where the map is
bright.
We find almost equally high similarity with a similar east-west oriented pattern. If the two patterns
are put on top of each other (stacked as two overheads), one would almost have a usable road map.
This is how image reconstruction is done, with the exception that it usually takes a lot of patterns
to create a useful image. This is caused by the fact that the similarity between object and each
individual pattern is typically extremely small, but not zero.
It may appear surprising that very complex pictures can be created by simply adding different
patterns. There is also an important difference from the overhead example, which would provide
darker and darker pictures the more patterns are added: The difference lies in the fact that one
must also be able to subtract patterns in order to produce usable images, meaning that the sign of
the MR-signal (the phase) is important. Figure 15 shows how patterns can be added to form brain
images.
9.7 Moving in k-space
The phase roll is controlled by applying gradients with different directions, strengths and durations.
As long as a gradient is active, and the field thus varies linearly with position, the phase roll will
change, but it will always have the shape of a stripe pattern as shown earlier. Meanwhile, there will
continuously be emitted radio waves from the body which tell us how well the pattern matches the
structure of the object.
Example: If a gradient going from ear to ear (the x-direction) is applied, the result is a phase roll
in this direction corresponding to the k-vector growing along the kx-axis. As long as the gradient
is “on”, the stripes in the pattern are getting closer as the phase rolls accumulate (the wavelength,
defined as the distance between stripes, becomes shorter). If another gradient in the direction from
scalp to foot is applied, a phase roll in this direction will accumulate and the k-vector grows along
the ky-axis.
If both gradients are applied, a tilted phase roll is achieved. This is true regardless of whether the
gradients are applied one at a time or simultaneously: By applying them one at a time, the k-vector
moves along the kx-axis first and then along ky. By applying them simultaneously, it moves directly
to the same point and the final result is the same.
One example of the path through k-space is shown for the spin-echo sequence in figure 16. The
gradient, which is itself described by a vector, determines at all times where the measurement is
headed in k-space and is thus the speed of the k-vectors movement in k-space.
That the emitted signal depends entirely of the position in k-space and not on the path there,
is important for the success of the k-space description of MR-imaging. Many sequences differ
by the path through k-space (the order in which patterns are created), but image reconstruction is
fundamentally identical.
28

Figure 15: Image reconstruction. The ﬁgure shows how simple patterns (line 1) can be summed up to
complex images (line 2). The reconstructed images are, in this case, created from the patterns that have the
greatest similarity with the subject in the scanner, meaning the areas of k-space where the strongest radio
waves were measured (line 3). Still more patterns are included in the image reconstruction (indicated by the
dark areas on the k-space images and the reconstructions are likewise becoming more and more detailed (the
number of included patterns is doubled in each step). The ﬁnal reconstructed image (lower right) has been
created on the background of the thousand patterns appearing most similar to the excited slice. The top line
shows the “most recently added” pattern. It is seen that the slow signal variations (intensities) are measured
in the middle of k-space, while the sharpness of the edges are derived from the measurements further out in
the k-space.
29

Preparation Imaging
90◦
Output
Gy
Gx
RF
Gz
180◦
TE/2TE/2
Ny
TR2
TR
PHy
ky
kx
Figure 16: Here, the spin-echo sequence illustrates how gradients can be used to maneuver in k-space.
As in figure 9, the top line indicates the use of radio wave pulses, while the bottom three show gradients
in perpendicular directions. Initially, the magnetization is rotated 90 degrees into the transversal plane.
Midway between excitation and output, an RF refocusing pulse is applied so that the signal at time TE after
excitation is unaffected by field inhomogeneity, if such is present. A gradient along the z-direction is applied
simultaneously with the two radio wave pulses, so that these become slice-selective as shown in figure 12.
The gradient direction is inverted shortly after the first RF pulse in order to remove the phase roll that is
accumulated in the period immediately following excitation. There is consequently no phase roll immediately
prior to application of the imaging gradients in the x- and y-directions: The local net magnetization point in
the same direction everywhere and we are at this point located in the middle of k-space. The field is now made
inhomogeneous in both the x- and y-directions (positive Gy, negative Gx) so that a phase roll is accumulated
and (kx,ky) is likewise changed toward the upper left corner of k-space. When the y-gradient is turned off
and the x-gradient inverted, readout of a line in k-space is begun. The process is repeated after a period TR
with a new value of the y-gradient so that a new line of k-space is read out. This is indicated in the sequence
diagram with a gradient table (PHy) and an outer bracket that indicates repetition for each line in k-space.
30

9.8 Image acquisition and echo-time
Earlier in the text, the echo time was described as the duration from excitation until the point where
radio waves are measured. Since the transversal magnetization in this period is lost on a timescale of
T2 (or T∗
2 , if a refocusing pulse is not used), the echo time TE is thus determining the corresponding
relaxation time weighting of the measurement.
After the introduction of imaging, we now need to reconsider the definition of echo time, since
several positions in k-space are typically being measured after each excitation. A typical approach
to imaging is, for example, to measure individual points along a line in k-space one by one after
each excitation.
The single points (corresponding to different stripe patterns) are consequently not measured at the
same time after excitation, and the echo-time definition has therefore become blurry. For echo-planar
imaging (EPI), this problem is extreme, since the entire image is measured after a single excitation,
meaning that some points in k-space are measured milliseconds after excitation, while others are
measured, for example, 100 ms later and thus with a completely different T2 weighting.
Is the possibility to characterize T2-weighting by a single parameter therefore lost? No! It turns
out that the echo-time definition can be adjusted, so that it still can be interpreted as above. A
surprising characteristic comes into play here: Even though parts of k-space are acquired shortly
after excitation and other parts a long time after, the reconstructed image looks (contrast-wise) as if
it has been acquired at a very certain time after excitation, that being the time where the middle of
k-space has been acquired. As such, it makes good sense to define the echo-time as the duration
from excitation until the time where the middle of k-space is measured (see an example in figure 16).
9.9 Frequency and phase encoding
Introductions to MR imaging methodology often takes their outset in “frequency encoding” that
is readout of signal in presence of a gradient so that the frequency of the radio waves reflect the
position. It is intuitively easy to understand how a frequency analysis of the measured signal reveals
how the body water is distributed along the direction of the gradient and how frequency encoding
thereby offers simple 1D imaging. Even though frequency encoding is easier to understand, a direct
k-space approach was chosen above. The reason is partly that frequency encoding leaves a need
to explain “phase encoding” which is a technique necessary to obtain information about the water
distribution in the direction perpendicular to the readout and slice selection gradient (figure 16).
Phase encoding is most easily understood from phase roll and similarity considerations (nuclei
are not given a unique phase along the phase encoding direction though attempts to explain the
technique that way are often made – look at figure 14 for confirmation). Another reason to choose
the direct k-space approach is that the frequency and phase encoding approaches are inadequate for
understanding sequences such as spiral EPI and to some extent also normal “blipped” EPI (even
the necessity of the pre-winding gradient for normal frequency encoding appears non-obvious). In
contrast, the k-space approach gives a far more general understanding of MR imaging (see glossary).
A related advantage is that frequency and phase encoding in k-space appears as simple variations on
a general theme. Even sequences such as spiral EPI appear conceptually similar to other sequences.
31

It is important to know of frequency and phase encoding for several reasons, however. Firstly, the
corresponding directions are often referred to in articles and in scanner software as the scan duration,
for example, often depends on the number of phase encoding directions and steps. Secondly, the
artifacts in the phase and frequency encoding directions are very different. Examples are given
in the following sections, and it will be advantageous to have frequency encoding in mind when
reading them. Whether readout gradients were used or not, it is often fruitful to consider in terms of
frequency encoding how a particular artifact would appear, had k-space along a particular direction
been filled using a readout gradient respecting the timing of the actual filling.
9.10 Spatial distortions and related artifacts
As described above in section 9.8, MR images appear contrast-wise to have been captured at a
certain time after excitation (the echo time) although measurements are far from always performed
that way. The cause of this is that the central parts of k-space primarily contains information about
the image contrast whereas the outer regions contain information about the finer structural details
(edges). That the acquisition of the images occurs during a longer period is therefore not reflected in
the contrast that is determined in the relatively short time it takes to pass the middle of k-space. The
duration is instead primarily reflected in spatial distortions (displacements) and other oddities.
The so-called “chemical shift” artifact and the EPI distortion artifact are both such consequences
of different parts of k-space being acquired at different times after excitation. The artifacts appear
as relative displacements of nuclei in the image plane, happening if these precess at different
frequencies, even in absence of imaging gradients. It is a fundamental assumption in imaging that
all frequency differences are caused by linear imaging gradients, and violations cause artifacts.
For the distortion artifacts, it is differences between phase rather than intensity that cause the
displacements. Intensity differences also have an effect, however, but it is hard to see at first glance:
Because the outer parts of k-space contain information about finer structural details, and because
variation in signal over k-space is T2-dependent, the spatial resolution of a measurement is dependent
on T2.
Among other things, this has the surprising effect that a blood vessel under a bolus-passage of
contrast agent will not only change intensity on EPI images, but will also appear to be “smeared”
over a larger area of the image. Because of the phase differences that induced field changes give rise
to, the vessel will even appear displaced under the bolus passage.
Displacements in the image plane of two substances that oscillate at different frequencies (fat and
water, for example) is proportional to the frequency difference. In order to calculate the displacement
in pixels, the frequency difference must be divided by the “bandwidth per pixel” that can often
be examined or even controlled from the scanner software. In accordance with the above, the
bandwidth is inversely proportional to the time it takes to traverse k-space in a given direction. The
displacement in a given direction is thus simply the product of the frequency difference and the time
it takes to pass k-space in this direction. The clock is “reset” at each new excitation.
32

9.11 Slice-selective (2D-) versus 3D-sequences
A distinction is made between multi-slice and 3D-sequences. This is confusing since the result is a
3D-dataset in both cases (most often represented as a series of slices). The somewhat unfortunate
choice of words does not refer to the result but rather to the method of acquisition: For slice-selective
sequences, protons in one slice at a time are affected as described above (section 9.3), while in
3D sequences all spins in the coil are affected identically and simultaneously. Subsequently the
k-space methods described in chapter 9.4 are used to distinguish signals from different positions
(3D k-space is similar to 2D k-space). A frequently used 3D sequence is MPRAGE (Magnetization
Prepared Rapid Acquisition Gradient Echo sequence). This is shown in figure 17 and is described in
appendix A.
3D sequences are almost always applied with a short repetition time TR, since the measurements
will otherwise become very time consuming. Therefore, they are practically always T1-weighted, but
can furthermore contain significant T2-weighting, as is the case for the CISS sequence, for example,
that has mixed contrast. Sometimes, the T1 weighting is isolated in the edges of k-space, so that the
image appears T2 weighted. Because of the short TR, there is much less room for contrast variation
compared to multi-slice sequences.
High spatial resolution in all directions simultaneously can be achieved for 3D measurements, e.g.
isotropic (1 mm)3 resolution (=1 microliter). For regular 2D sequences, the case is different since
slice thicknesses under 3 mm are rarely employed. The resolution in the image plane is often set
high for 2D sequences, which results in good-looking images but quite oblong voxels. This may
cause problems when distinguishing small or flat structures such as grey matter. In summary:
3D sequences Allows for high spatial resolution and isotropic voxels, but most often only T1
weighting and mixed weighting turn out well for short measurement time.
Multi slice sequences High “in plane” spatial resolution, but rarely under 3 mm slice-thickness.
Flexible contrast.
33

Preparation 3D-FLASH Recovery
180◦
PHx
Output
Gy
Gz
RF
Gx
α
TDTETI
Nx Ny
TR
PHy
Figure 17: The MPRAGE sequence is an example of a 3D sequence where excitation is non-selective or
weakly selective, and where the 3D k-space must be traversed in order to achieve spatial resolution in all
directions. Initially, the magnetization is rotated by radio waves so that it points opposite the magnetic field in
the following period TI (the “inversion time”). This is done to increase the T1-weighting of the magnetization
and thus also of the image. In order to detect the magnetization, it is tilted slightly into the transversal plane by
an RF pulse with a small flip-angle α. The signal is coded spatially by the green gradients and it is measured
in the readout period. The inner, stippled bracket indicates that the enclosed part of the sequence is repeated
once for each kx value. By applying small flip-angles, the longitudinal magnetization is only affected slightly
each time, so that many lines along the kz axis can be measured before the magnetization must once again be
inverted. This happens after a recovery period TD. The entire sequence is repeated for each ky value, so that
all of k-space is covered. The gradient applied simultaneously with the α pulse is not strictly necessary, but
can be used to choose a thick slice (slab) within which phase encoding can be carried out. Thus, aliasing may
be avoided, even if only relatively few phase encoding steps are employed. Further information regarding
MPRAGE and FLASH can be found in the appendix.
34

9.12 Aliasing and parallel imaging
Figure 18: Aliasing: The image dis-
plays a brain in profile, but since the
sampling density in k-space is too low,
the nose ends up in the neck and vice
versa. Thus, pathology can be hidden.
That k-space imaging is a Fourier technique (meaning that it is
based on the addition of phase rolls) is reflected in the artifacts
that may appear in images, if the underlying assumptions are
not met. If we do not have knowledge about the subject in
the scanner a priori, the similarity with all possible phase roll
patterns must in principle be measured in order to reconstruct
a perfect MR image. It does, however, take time to create
patterns, and in practise only a central, rectangular part of
k-space is covered with finite resolution, as illustrated in
figure 15. Usually, it is the number of measured lines that
determines the duration of a measurement. A limited density
in k-space of these is therefore desired.
If the chosen density is too low, it results in a characteristic
“aliasing artifact”: If sampling in k-space is not dense enough
to catch signal variations (i.e., if the Nyquist-criteria is not
met), errors will appear in the reconstructed images. The
errors consist of signals being misplaced and overlapping, as
seen to the right.
In 1997, however, Sodickson and Manning brought atten-
tion to the fact that if several antennas with different sensitivity profiles are measuring the same
k-space, the “lacking” k-space samples can be calculated, thus avoiding aliasing. Simultaneously
applying two antennas for signal reception can in principle therefore accelerate the measurement
by a factor of two, although in practice it requires several more antennas to get reliable images.
Pruessmann et al soon proved that sets of reconstructed, aliased images acquired with different
antennas can intuitively be “un-aliased”, which is principally equivalent. The techniques are today
known under the common name “parallel imaging”.
That parallel imaging is an independent technique, even though it was first proposed as a modifica-
tion of the k-space technique, is most easily understood through a thought experiment: In the extreme
limit “each-pixel-has-its-own-antenna” (e.g. many tiny antennas spread equidistantly through the
imaged object), the gradients are not necessary to measure detailed images. It is simultaneously
realized that parallel imaging is not a Fourier technique, meaning that reconstruction is not based on
the summation of stripe patterns.
9.13 Finishing remarks on the subject of imaging
Typically, relaxation times are relatively long which is responsible for the enormous flexibility in
MR scanning. In 100 ms, for example, series of RF and gradient pulses can be applied to prepare
the magnetization in a way that allows its size or direction to reflect physiologically interesting
parameters, in other words manipulating the contrast in the images. Additionally, long T2’s allows
for the generation of a large number of phase roll patterns and for measurement of the corresponding
35

radio wave signals before the system’s memory of the preparation has been erased. Actually, the
whole k-space can be covered after a single excitation (echo-planar imaging, EPI).
It is interesting to note how the wavelength limitation known from optical techniques was
avoided. In the case of k-space techniques, this happened by using frequency differences rather
than far field properties, for spatial source localization of radio waves. The traditional wavelength
limitation is thus replaced with a requirement that the neighboring nuclei’s frequencies must be
distinguishable during the measurement period, if the positions are to be distinguished. This can be
expressed alternatively: No structural details considerably less than the shortest wavelength of the
phase roll can be observed. Thus, a new wavelength limitation is introduced, which is intuitively
understandable from figure 15. Since it takes time to move in k-space, the ultimate limitations for
the spatial resolution are determined by gradient strengths, relaxation times and movement of the
water molecules, including diffusion.
In parallel imaging, the limited and different sensitivities of the coil elements are used to avoid
the wavelength limitation. Since the sensitivity area’s dimensions are more or less equal to those of
the coil elements, the spatial resolution is determined by the coil elements’ size for “pure” parallel
imaging. This resolution is usually inadequate for practical purposes.
Parallel imaging is, however, completely compatible with k-space imaging, in that the two
techniques can be combined and images acquired faster than normally. In some situations, speed
is more important than spatial resolution, and therefore experimentation is now done with images
acquired without applying gradients. Such images are acquired with sub-millisecond time resolution
of up to hundreds of antennas simultaneously. These images are of interest for imaging of substances
with short relaxation times, for example.
10 Noise
The limiting factor for many MR examinations is noise. We can, for example, not directly detect
substances in vivo in concentrations below a few millimolars on a a reasonable timescale, because
the signal is drowned by noise.
The noise can be physiological (pulse, respiration, movement), but even if the patient is lying
completely still, there exists an upper limit for the image quality achievable in a given period. In the
absence of physiological noise and under a couple of other assumptions (which are rarely completely
fulfilled), the signal to noise ratio is proportional to the voxel size and the square root of the time it
has taken to acquire the image.
It is essential to realize that the signal-to-noise ratio is not dependent on the number of voxels in
the image. If you double the matrix (points along one side of the image) and the field-of-view it will
typically imply that the signal-to-noise ratio is increased by a factor of
√
2 because the measuring
time is thereby typically doubled while the voxel size remains unaltered. That it may only be extra
air that is included by the expanded field-of-view is irrelevant, and the noise level is not affected by
the fact that 4 times as many voxels are being measured.
Insofar as the scanner is well functioning, the electronics is not the primary source of noise.
Instead, that is the random motion of charged particles (ions) in the patient. When charged particles
36

diffuse, they emit random radio waves as they change their direction of motion. The higher the
temperature and conductivity of a material, the more noise it emits.
Thermal noise is evenly distributed despite the fact that it is emitted almost only from within the
patient, who may only fill part of the image area. This is caused by the noise not being an MR signal,
and the gradient-induced spatial coding of the signal is therefore not affecting the noise. Instead, the
noise is received in a steady stream during the entire measurement and it is therefore evenly spread
over k-space and consequently also evenly over the MR image.
The noise from patients cannot be avoided, but we can, to some extent, avoid measuring it. The
idea is to use a small coil that only detects noise from a small area of the patient (a surface coil,
for example). It is a common (but non-essential) misconception that using a small coil primarily
enhances the signal-to-noise-ratio through its improved sensitivity to signal. Improved signal
sensitivity, however, also increases sensitivity to noise that is generated in the same regions of the
body. Improving the sensitivity therefore does not by itself improve the signal-to-noise ratio. Instead,
the surface coil limits the noise by being sensitive to a smaller part of the body, of size similar to
that of the surface coil. A small surface coil only detects the noise (and signal) from a small part of
the body, and this noise appears evenly distributed over the entire image. The surface coil therefore
improves the signal-to-noise ratio by reducing the sensitivity to distant noise sources. These effects
are well described by Redpath in The British Journal of Radiology, 71:704-7, 1998, for example.
11 Scanning at high field
Three tesla is the largest, clinically available field strength at present.
The high field is especially beneficial for imaging with a high spatial
resolution, as well as for performing measures that normally suffer
under a low signal-to-noise-ratio, including perfusion measurements
and spectroscopy. The image to the right depicts a newborn child and
demonstrates high resolution and fine contrast after only 3 minutes
acquisition at 3T. A small periventricular bleeding is visible in
the right hemisphere (left on the image, according to radiological
convention). The relaxation times are changed at high field (typically
longer T1 and shorter T2). Sequences and parameters must therefore
be adjusted to the field strength in order to obtain good images.
For functional imaging, fMRI, the high field strength provides
enhanced contrast upon brain activation, so it can be measured in which areas activation occurs in a
given situation (see fMRI in the vocabulary). The methods can, for example, be used clinically for
planning of surgery preserving essential functions such as language. There are, however, several
pitfalls doing fMRI and the quality of the results must thus be evaluated carefully before conclusions
can be drawn.
37

12 MR Safety
Is the MR technique harmless, and just how certain are we of this? It is a natural question, since
we do use radio waves that have been suspected of being carcinogenic in connection with high
voltage wiring and mobile telephones. Furthermore, magnetic fields of quite significant strength are
employed, from 25.000 to 75.000 times the average field of the earth. By nature, we cannot know
with total certainty whether MR is completely harmless. At this time, no harmful effects have been
documented, although the question has been explored in depth.
It is known that possible harmful effects are rare or have very limited effect. This knowledge is
partially derived from animal experiments with long term exposure. Population studies have been
conducted of radiographers and others working close to magnets or radio wave fields (chemists and
physicists have worked in high magnetic fields since the 40’s and many people working with radar
have been exposed to monstrous amounts of radio waves).
The area is regulated by international standards and the scanners are equipped with monitoring
equipment that ensures that no radio waves more powerful than allowed are emitted, even by
accident.
The most significant risks associated with MR scanning are proba-
bly resulting from misinterpretations of the images, as well as purely
practical issues concerning handling. Reported injuries include hear-
ing damage, burns as a result of wire loops (in jewelry, for exam-
ple), as well as serious incidents with metal objects such as the flying
chair shown to the right. There are scary examples of oxygen cannis-
ters, floor waxers and other things going astray, found in anecdotes at
http://www.simplyphysics.com/flying objects.html
Even though the scanning itself is harmless, it has been shown that MR contrast agents injected
intravenously in connection with certain types of scanning, can be dangerous to patients with kidney
problems. Because of differences in the contents and chemical stability of the contrast agents used,
the risk varies. In the case of patients with serious kidney problems, the use of gadolinium based
contrast agents should be avoided when possible, in accordance with the guidelines found on the
websites of the European and International MR Societies.
http://www.esmrmb.org/ and http://www.ismrm.org/
Usually, the contrast agents are excreted from the body through the kidneys shortly after scanning,
but in the case of kidney patients, it can be trapped in the body. It must be emphasized that the
problem only occurs in very special cases (exclusively when combining a high dose of contrast
agent, a heavily impaired kidney function and the use of certain substances). At Hvidovre Hospitals
MR department, there are no suspected cases, although contrast have been used for 20 years for
thousands of patients. The same is true of most other departments with MR scanners.
38

Appendices
Appendix A: The educational software mprage
This appendix is the online help for the software “mprage”. It also describes the design and
important qualities of this often used 3D-sequence. The program is unfortunately not in a form
where it can be widely spread, but it can be started in Hvidovre from the linux command line by
writing mprage. It can also be distributed on demand.
Introduction
This is the online manual for a simulation program for the sequences MPRAGE and FLASH. The program
was developed in connection with a research project (inhomogeneity correction) and has since been adapted
for sequence optimization and educational use.
The MPRAGE sequence
The MPRAGE sequence as shown in figure 17, is an often used T1-weighted 3D sequence that includes two
repetitions (loops): An inner loop repeating an excitation pulse with a small flip angle, and an outer loop that
begins with a 180 degree inversion pulse.
The sequence is a 3D FLASH with phase encoding in two directions, meaning that the magnetization is
rotated away from the direction of the magnetic field by the use of many RF pulses with small flip angles in
quick succession (typically over a hundred with about 10ms interval). In contrast to the spin echo sequence
40

where the magnetization is used up completely at each excitation, a typical FLASH flip angle is small, e.g. 10
degrees.
Each pulse only affects the magnetization slightly, but when many pulses are employed, the longitudinal
magnetization approach a level that depends on T1 and the strength and frequency of the RF pulses.
A FLASH sequence usually results in T1 weighted images, since the degree of saturation is T1 dependent.
Said in another way: The many RF pulses eventually drive the magnetization to a new equilibrium level
(“driven equilibrium”), that reflects how quickly the magnetization is rebuilt. For the purpose of providing
extra T1-weighting, the MPRAGE sequence has a built-in inversion pulse, employed before each FLASH
module.
After each excitation, the fresh transversal magnetization is affected by gradients that creates a phase roll
corresponding to a new position in k-space. A readout gradient causes movement through k-space along a
straight line while signal is acquired for a few milliseconds. When the inner and outer loops are finished, the
entire 3D k-space is covered and an image of the whole volume can be reconstructed. Each line in k-space is
weighted differently since the preceding FLASH pulses influence the magnetization.
T2 does not play a significant role concerning the contrast in FLASH and MPRAGE, since the echo time
typically is a few milliseconds and thus much shorter than T2.
Using the program
The parameters from MPRAGE can be varied in the program, and the equivalent sequence and evolution of
the longitudinal magnetization, Mz, can be followed (left graphs). The parameters occurring are
TI The inversion time. The period from the inversion pulse until the first FLASH pulse.
Beta The flip angle of the inversion pulse (180◦ for MPRAGE, 0◦ for FLASH, 90◦ for saturation-recovery-
FLASH).
Alpha The FLASH flip-angle.
TD Pause from last FLASH pulse until the next inversion pulse.
Nlin The number of lines that are acquired in one of the k-space phase encoding directions. The other
directions (phase encoding and readout) are not so interesting in this context since they do not influence
contrast or voxel shape. The uninteresting phase encoding direction is here chosen perpendicular to
the image plane.
EchoLin% A percent rate that indicates when in the FLASH period the middle of k-space is passed. The
contrast is primarily determined here. As an example, a choice of 50% indicates that the center of
k-space is measured after half of the FLASH pulses are played out. Often, the center of k-space is
measured asymmetrically to increase the effect of the inversion pulse.
The naming of the parameters are in accordance with older Siemens scanners. On more recent scanners, the
parameters are defined and named differently. Ask.
On the graph seen to the left, a single iteration of the outer loop is illustrated. The evolution of the
magnetization is shown at the top, and the RF pulse timings are shown at the bottom. The horizontal red line
in the top graph shows the timing of the FLASH module and the vertical line illustrates when the middle of
k-space is reached.
The outer loop is typically repeated many times (e.g. 200) and after a few iterations, the same curve is
achieved every time (the one shown). The magnetizations shown in the right and left side of the graph are
therefore connected in the sense that only the sign differs between the first and last point for beta equal to 180
degrees, for example (an inversion pulse inverts the magnetization, hence Mz changes sign).
41

Along with the contrast evolution, simulated k-space images (raw data) and reconstructed images are shown.
The raw data are generated by adding k-space images for the single tissue types, but weighted line-wise with
the relevant Mz during the FLASH period. The reconstructed images are produced by Fourier-transforming
the k-space images.
When the parameters are changed, the corresponding changes in contrast and efficiency are shown. The
latter is a contrast measure corrected for sequence duration (achieving a better contrast is easy, if the duration
of the measuring is increased). The contrast is calculated as the signal difference between grey and white
matter (GM, WM) at the time when the middle of k-space is passed. This contrast measure is reasonable, if the
voxel shape is not bizarre for any tissue type, i.e., if the k-space weighting is reasonably uniform. Efficiency
is defined as the contrast per
√
duration (so that it is independent of the number of repeated measures (signal
averages)).
Important points
The program illustrates a number of important aspects:
• There are many parameters to adjust in a sequence like MPRAGE. The contrast and image quality
depends strongly on these.
• It is not possible to provide easy rules for how the contrast changes when a given parameter is adjusted
– it depends on the other parameters. Simulation tools like this one are thus required. The software also
contains optimization tools. Ask, if this interests you.
• Contrast and spatial resolution (matrix) are not independent.
• During the pauses, Mz approaches the equilibrium magnetization (proportional to proton density) on a
timescale of T1.
• During the FLASH period, when pulsing is frequent, the magnetization approaches another level
dictated by T1, pulse interval and strength. The greater the flip angle alpha, the faster “driven
equilibrium” is achieved, and the less the influence of the inversion pulse.
• The signal from one type of tissue can be “nulled” in the middle of k-space through appropriate
parameter choices. The signal from the tissue type in question is thereby suppressed. The tissue still
contributes with signal in other regions in k-space, however, and bright edges may therefore result.
• The voxel shape and spatial resolution depend on the type of tissue, since the k-space weighting
(meaning the signal variation over k-space) depends on the type of tissue. This is most obvious when a
null-crossing appears close to the k-space center (often producing ugly images).
• Although the GM/WM contrast, as defined above, can be good for a given set of parameters, the image
may be rubbish. This occurs if the k-space weighting produces an unfortunate voxel shape for one
or more types of tissue (see above). Thus it is difficult to automatically optimize the sequence in a
meaningful way (tradeoff between voxel shape and contrast).
• The importance of the inversion pulse for the contrast, is dependent on the duration and number of
FLASH pulses that occur between inversion and passage of the k-space center.
Precautions
The simulation is not more precise than the parameters entering the simulation, including
• the assumed relaxation times (shown in the window - these depend on the field strength),
• the assumed proton densities,
42

• sequence parameters, including FLASH-TR, which at the moment can not be varied without altering
the program slightly,
• the assumed noise level,
• the choice of phase encoding directions,
• the lack of inﬂuence from RF-inhomogeneity that is not simulated. The program has “hidden” graphics,
that illustrate this sensitivity (press Print). Ask, if your interest has been aroused.
43

Appendix B: Glossary
This glossary can be used for repetition and for learning more about individual topics.
The BOLD effect This effect is the basis for most fMRI. The rate at which the MR signal dies out is
weakly dependent on the concentration of oxygen in the blood, since oxygenation changes the
magnetic qualities of the blood. Deoxyhemoglobin is a paramagnetic contrast agent contrary
to oxygenated hemoglobin that is diamagnetic like the tissue.
The signal intensity is usually increased by neuronal activation, since the blood supply
(perfusion) increases more than the oxygen consumption. This is the so-called BOLD-effect
(blood oxygenation level dependent effect) that results in signal variation of a few percent in
gradient-echo sequences. The greatest sensitivity occurs at TE T∗
2 (consider why this is).
Coil This word is used for several concepts in MR. It is typically used for the antennas that send
and receive radio waves. The coil is usually fitted to the body part that is being scanned, so it
is near the area of investigation. A head coil surrounds the entire head, for example, and not
much else. The choice of coil involves a trade-off of sensitivity, homogeneity and coverage.
In contrast, gradient coils, which are electromagnets, are not used for transmitting and
receiving radiowaves, but for creating linear field gradients (see Gradients). Shim coils are
similarly used to to eliminate non-linear, background field gradients.
Contrast agents Substances that affect the contrast in images. Contrary to X-ray contrast agents,
for example, the MR contrast agents are usually not seen directly. Rather, we see their effect
on the protons (increased relaxation – see gadolinium). Contrast agents are, for example,
used for detecting defects in the blood-brain barrier: Gd-DTPA usually stays in the blood
stream, but is temporarily trapped in the tissue in the case of blood-brain barrier breakdown.
Magnetically tagged blood can also be used as a contrast agent (spin labelling).
Diamagnetism Magnetic fields deform the electronic orbits so that the magnetic field is weakened
slightly. This phenomenon is called diamagnetism and it is exhibited by all substances.
However, there are substances where the diamagnetic quality is completely overshadowed by
far more powerful influences (paramagnetic and ferromagnetic substances). The so-called
magnetic susceptibility expresses how much the field is strengthened or weakened by a
material.
Diffusion Diffusion is measured by employing a powerful gradient that introduces a short-wave
phase roll over the object. When the gradient is then inverted (or equivalently, if a refocusing
pulse is applied), the spins will be rotated back in phase, forming an echo insofar as the spins
have not moved meanwhile. If the spins have diffused, meaning that they moved randomly
among each other, the signal will only be partially restored. The signal intensity therefore
reflects the mobility of the protons (the diffusion coefficient).
Echo-planar imaging, EPI A rapid imaging method where k-space is traversed using oscillating
gradients after a single excitation. The acquisition time for a single EPI image is typically
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below 100 ms. EPI often exhibits low signal-to-noise ratio and pronounced artifacts, including
spatial distortions and signal voids near air-filled spaces. It is used frequently, however, when
time resolution is crucial.
Eddy current See induction.
Electrons Negatively charged particles that orbit the positive atomic nuclei. The electron cloud
is responsible for the chemical properties of atoms and molecules. Its appearance is weakly
reflected in the MR signal, since the electron cloud partly screens off the magnetic field. This
is exploited in spectroscopy. Electrons also have spin, and ESR is the term used for Electron
Spin Resonance. Unfortunately, the short relaxation times of electrons make ESR useless for
medical imaging.
Excitation The magnetization cannot be measured before it is rotated and has acquired a component
perpendicular to the magnetic field (a transversal component). The rotation is caused by
transmitted resonant radio waves, a process called excitation.
Exponential function When “something” decreases exponentially (the transversal magnetization,
for example), it means that a specific fraction is lost when we wait for a certain period of
time: When we say that T2 is 100 ms it means that about 60% of the signal is lost in 100 ms.
The same fraction of the remaining signal is lost again if we wait another 100 ms. The T1
relaxation too involves exponential functions, since the longitudinal magnetization approaches
equilibrium exponentially after excitation (the difference decreases exponentially).
Ferromagnetism Diamagnetism provides a minimal weakening of the field, and paramagnetism
typically provides a more powerful increase of the field. Such changes are in the range of
per mille or less. There are, however, a few agents that have far more powerful magnetic
attributes, among them iron, nickel and cobalt that are ferromagnetic. These materials may not
be introduced in the scanner room without special precautions. Be aware that only metallic
iron is ferromagnetic – ionic iron is paramagnetic.
Flow Velocity and direction of uniform movement can be measured with a technique similar to that
described above for measuring diffusion. Yet, the employed gradients are typically weaker
and the measurements are interpreted differently, since it is the net rotation rather than the
signal loss that reflects the speed. After employing the bipolar gradients, the spins have
achieved a total phase change that is proportional to the velocity along the gradient. The
technique is therefore called phase contrast MRI and the velocity is determined by comparing
phase images.
fMRI Functional MRI. Mapping of the areas of the brain that are activated in a given situation.
Usually based on the BOLD-effect.
Frequency encoding If signal is read out while a gradient coil is active, the frequency content
will directly reflect the distribution of nuclei along the gradient direction. A one-dimensional
image (a projection) can consequently be acquired by doing a frequency analysis of the
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measured signal. Since the gradient is active during signal readout, its direction is called
the readout direction. Even though frequency encoding is easy to understand, it has proven
fruitful to think instead of how the phase roll that the gradient is creating, is causing the signal
to reflect sample structure. Frequency encoding is consequently a special case of k-space
imaging. See also “phase encoding”.
Gadolinium Gadolinium (Gd) is one of the less well-known elements, a heavy metal. It is charac-
terized by having unpaired electrons in inner shells, which results in the atom having magnetic
qualities that are far stronger than that of protons. The field near a gadolinium atom is strongly
inhomogeneous. Protons that pass near gadolinium therefore receive a powerful magnetic
influence which gives rise to increased relaxation. Free gadolinium is toxic and it is therefore
placed in a larger organic structure – DTPA is typically used. Contrary to X-ray contrast
agents, among others, it is not the actual agent that is seen with MR, but rather the protons
that come near gadolinium (or those that do not, for T2-weighted images). Thus, there exists
long range effects of gadolinium (e.g. over the blood-brain barrier, which water can pass but
Gd-DTPA cannot).
Gradients Gradients are variations in the magnetic field. The scanner is equipped with gradient
coils that give linear field variations whose direction and strength can be altered as needed.
This is essential for slice selection, imaging and more. Field alterations result in mechanical
forces on the scanner, vibrations, and thus acoustic noise.
Non-linear gradients also appear due to imperfect shimming. On a microscopic scale, the field
is inhomogeneous, e.g., near paramagnetic particles such as gadolinium and deoxyhemoglobin.
For this reason, T∗
2 is always shorter than T2.
Gradient echo A gradient creates a phase roll, i.e. a position-dependent rotation of the spins. If
the gradient is subsequently inverted, the phase roll will gradually rewind until the spins
are back in phase. Thus the magnetization is recovered (in fact, it has only been hidden).
Signal recovered in this manner is called a gradient-echo, and it is employed for imaging
and for flow- and diffusion measurements. The related spin-echo removes the effect of
inhomogeneities caused by variation in the magnetic properties of the tissue. This does not
apply to the gradient echo, since we can only actively invert the gradient direction for the
gradients we control, i.e. the ones that are created with electromagnets.
Induction If the magnetic field through a wire loop is changed, a current in the wire will be induced
(an eddy current). This may, for example, happen to electrodes that form a noose in the
scanner, which may result in burns. Radio waves are typically the most significant source of
heating. Eddy currents also appear in the scanner metal. Magnetic fields formed in this way
can cause distortions of the MR-signals.
Inversion Often, the equilibrium magnetization is inverted for a period by use of a 180-degree pulse
before it is flipped into the transversal plane to be measured. The inversion time, TI, is this
period with T1-weighting, i.e., the time from inversion until the excitation pulse. Inversion is
46

typically introduced to add T1-weighting (inversion recovery) or to get rid of signal derived
from fat or CSF (STIR, FLAIR).
Magnetic dipole Spin makes nuclei act as small compass needles. Mere precisely, they act as
magnetic dipoles, meaning that they have a south pole and a north pole. Furthermore the
angular momentum (rotation) makes them precess in the magnetic fields, meaning that the
north/south axis rotate around the direction of the field.
Metabolites Substances that play a part in the metabolism. The most easily detectable metabolites
in vivo are NAA, choline, creatine and lipids. Others are lactate, inositoles, GABA, NAAg,
taurine, glutamine, glutamate and alanine. Some of these are painstakingly difficult to measure
reliably. There are many other agents in the body, of course, but only small, mobile molecules
in high concentrations are detectable in a clinical setting.
MR Magnetic Resonance.
MRI Magnetic Resonance Imaging.
NMR “Nuclear Magnetic Resonance”. The same as MR. The reference to the nuclei is said to be
removed to avoid associations with radioactivity, nuclear power and atomic weapons. The
chemists have maintained the N and they use the technique for identifying agents and mapping
protein structures by spectroscopy, for example.
Paramagnetism All substances have diamagnetic properties, but some are also paramagnetic,
which makes them alter the local field significantly more than the diamagnetic contributions
do. Gadolinium and deoxygenated blood are examples. In pure paramagnetic agents, the field
is typically increased in the range of a few per mille. See Diamagnetism above.
Perfusion Contrary to bulk flow where a large amount of blood is flowing in the same direction,
the blood flow in the smallest vessels (capillaries) is far from uniform within a voxel (there
are millions of capillaries oriented randomly within a voxel of tissue). The blood flow in
these is called perfusion and it is responsible for oxygen supply to the tissue. It is typically
measured by following the passage of a contrast agent through the vasculature on a series of
images. The subsequent analysis is demanding, especially for quantitative measurements.
Phase The word “phase” is used for the direction of the magnetization perpendicular to the B0
field (meaning the direction in the transversal plane). This is affected by the magnetic field
that causes precession. Immediately after excitation, the individual contributions to the net
magnetization point along the same direction perpendicular to the field – the spins are said to
be in phase (or having the same phase). Subsequently, a gradual dephasing and signal loss
follows. This reflects that the common phase is lost, resulting in a loss of net magnetization.
This loss can be caused by random interactions between nuclei (in which case the loss is
irreversible), or it can be caused by field gradients (and is then typically reversible, since the
signal can be recovered after using a refocusing pulse).
47

Phase images Usual MR images show the size (amplitude) of the magnetization in the transversal
plane measured at the echo time, but they do not tell us anything about the direction (the
phase). This must, however, be known to measure flow as the velocity is often reflected in
the phase. For most sequences, one can therefore choose to acquire phase images in addition
to regular amplitude images. They are most frequently zebra-striped, which indicates the
presence of a phase roll: The phase angle moves from 0 to 360 degree, jumps to 0 and
increases steadily again. This is equivalent to the spins being rotated one or more turns.
Phase encoding Both slice selection and frequency encoding are relatively easy to understand
without considering phase rolls, but these techniques are inadequate to achieve spatial resolu-
tion in the third dimension. Phase encoding is often the solution: The sequence is repeated
with different strengths of a gradient inserted immediately after the excitation. The direction
of this is called the phase encoding direction. Phase encoding is most easily explained in term
of k-space imaging: Different phase rolls are induced along the gradient direction, and the
resulting signal strength reflects the degree to which the patient structure “matches” the phase
roll. This signal is subsequently frequency encoded and measured. Even simple imaging
becomes relatively complicated when phase and frequency encoding are combined, but de-
scribed separately. They both appear on equal footing, however, in the k-space description of
imaging, which has proven much more viable and general than phase encoding and frequency
encoding separately. There are still good reasons, however, to get acquainted with frequency
and phase encoding as described in the main text.
Proton Nuclear particle. All atomic nuclei are built of protons and neutrons. The hydrogen nucleus
is the smallest and consists of only one proton.
Precession Spin, which is the rotation of the nuclei around their own axis, gives rise to the
nuclear magnetic property as shown in figure 3. This rotation should not be confused with
“precession”, which is the rotation of the magnetization in a magnetic field as shown in
figure 4), i.e., the rotation of the nuclear north-south axis.
Quantum Mechanics A fascinating subject that you should be wise not to ask me about, unless
you have plenty of time on your hands. Irrelevant for understanding most aspects of MR,
including nearly everything that scanners are used for. Introductions to MRI often contain
misunderstood QM.
Radio waves, RF Radio waves in the relevant frequency range are often simply called ’RF’ (for
radio frequency). The employed frequencies overlap with the ones used in radio communica-
tion. MR must therefore be carried out in an RF shielded room (also known as an RF cabin
or a Faraday cage). The shielding can be tackled a number of ways. In animal experimental
scanners, the “plugs” in the ends of the scanner act as boundaries for the shielded volume.
Refocusing pulse A pulse of radio waves transmitted with the intention of creating a spin echo, is
often called a refocusing pulse. The pulse typically has a flip angle of 180 degrees.
48

Relaxation times T1 and T2 are time constants that describe how quickly the magnetization is
approaching equilibrium. The magnetization is longitudinal in equilibrium, meaning that it is
pointed along the magnetic field (Mxy = 0, Mz = M0). Away from equilibrium, the transversal
magnetization Mxy decreases on a timescale T2, while the longitudinal magnetization Mz
approaches M0 on a timescale T1.
Resonance The magnetization will only be rotated in a useful manner if it is subject to radio waves
at the right frequency. This is a resonance phenomenon as known from the example of a
swing: The amplitude of the oscillations will only change if the swing is pushed in synchrony
with its natural oscillation frequency (i.e., “on resonance”).
Saturation Most often, the purpose of excitation is to create a signal, but not always so. Saturation
pulses are used to get rid of signal in a given position, from a given type of tissue or from a
particular metabolite. Examples include saturation slices and lipid suppression.
Saturation more generally refers to the attenuation of signal caused by reduced magnetization,
e.g. resulting from incomplete recovery of longitudinal magnetization after prior excitation
pulses.
Slice selection When excitation is performed while a gradient is active, the protons in some
positions will precess in synchrony with the radio waves, and the nuclei will hence be rotated.
At other positions, the resonance condition will not be fulfilled and the nuclei are hardly
influenced. Specifically, a slice perpendicular to the direction of the gradient will be excited.
When radio wave transmission and the gradient is subsequently turned off, a signal from the
nuclei in the slice can not immediately be detected, however. Perpendicular to the slice, the
last part of the slice selection gradient will have induced an unwanted phase roll that needs
to be unwind before a signal from the slice can be measured. Slice selection will therefore
typically be followed by a gradient directed opposite to the slice direction gradient (figure 16).
This brings the nuclei in the slice into phase.
Spectroscopy The common term for techniques based on measuring differences in frequency con-
tent (color sight, variance in tone, NIRS. .. ). Pretty much any MR technique is spectroscopic
(imaging, for example), but the term MR-spectroscopy is usually used to refer to techniques
aimed at distinguishing metabolite signals.
Spin Spin is a quantum mechanical quality possessed by protons and neutrons. In brief, this may
be perceived as rotation of individual nuclei around an axis of their own. The magnetic
characteristics of the nuclei (the dipole quality) is caused by spin and the dipole is along the
direction of the spin. In nuclei, protons pair up in entities that seemingly have no spin. The
same is true of neutrons. There are consequently only few nuclei that are suitable for MR (an
unequal number of protons and/or neutrons is required).
Spin-echo If the magnetic field is inhomogeneous, the spins will loose alignment after excitation
since some spins precess faster than others – signal is lost. Using pulses of radio waves, the
dephasing can be reversed so that the lost signal is recovered: A 180 degree pulse inverts
49

the phase of the spins, so that those that have precessed the most are set most back in the
process. Waiting a while longer, the signal can be recovered in the form of a spin-echo since
the inhomogeneity persists.
Tesla The unit for magnetic field strength. The earth magnetic field is on average about 0.04 mT,
while the magnetic fields used in MR scanning are typically 1-3 T.
Voxel Volume element. Typically refers to the volume that a given point in an image corresponds
to. The word pixel (picture element) might be better known, but it does not reflect the depth-
aspect of any MR measurement (2D rather than 3D). The voxel size is the same as the spatial
resolution. The voxel dimensions are the edge-lengths of a voxel.
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