# Physics of Nuclear Magnetic Resonance Imaging

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```					         BASIC PHYSICS OF MAGNETIC RESONANCE IMAGING:

DRAW-A-LONG LECTURE NOTES.

by James Thompson, PhD

1. PROTONS & PRECESSION

   Atoms are made up of different combinations of protons, neutrons, and electrons.

For example, the most common isotope of the most abundant element in the

universe, hydrogen (1H), is made up of a single proton.

   Protons have a positive charge

   They spin around on an axis, like the way the earth spins around on its North-

South axis

   The positive charge on a

proton spins around with

it, creating an electrical

current. As we all know,

an electrical current also

generates a magnetic

field.

   Protons are normally aligned randomly. However, if you place them in an

external magnetic field (also called B1), they will align themselves with it. Some

will be in a low energy state, and align themselves parallel to the external
magnetic field. Others will be in a high energy state, and align themselves anti-

parallel to the external magnetic field.

   At the normal temperature of the Earth’s surface, there will be close to even

numbers aligned parallel

and anti-parallel. Slightly

more will be parallel than

will be anti-parallel:

100,007 parallel for every

100,000 anti-parallel.

   For every anti-parallel

proton, there is a parallel

one that cancels it out. As

there are slightly more parallel protons, the net magnetization of all these

protons will be in the direction parallel to the external magnetic field.

   I lied a little – protons don’t quite line up with the external magnetic field.

Instead, they move around the axis of the external magnetic field like a spinning

top. This is called precession.

   The frequency () at which they spin around the direction of B1 depends its

strength (B0) (measured in T or Tesla) and the gyromagnetic ratio () for protons.

This is the Lamour equation:

B0

   Protons in a 1 Tesla magnetic field precess at around 42MHz (that is, about 42

million times per second).
2. LONGITUDINAL MAGNETIZATION

       It helps to describe the

precession of protons in an

external magnetic field using a

3-dimensional co-ordinate

system. In this, the z direction

is the direction of the external

magnetic field.

       All of the spinning protons

have a magnetic field, which is

aligned in the z-direction, either parallel or anti-parallel. As the parallel magnetic

fields cancel out the anti-parallel ones, we end up with net magnetization in the z-

direction.

   In theory, we can get this by summing the vectors that represent the magnetic

fields of all the protons that

are positive in the z-

direction.

     At this point, all the protons

are precessing out of phase

with each other. This means

if we sum all the vectors in

the x- and y-directions we get

a big fat zero.
   The magnetic field in the z-direction is known as longitudinal magnetization.

It’s just like longitudinal co-ordinates on a globe (remember, lat is flat, long goes

from North to South).

   Unfortunately, we can’t measure longitudinal magnetization directly, because it is

in the same direction as the external magnetic field. Instead we need to perturb the

system in some way and then see how the field changes.

3. TRANSVERSE MAGNETIZATON

   To perturb protons, we add a high-frequency pulse of energy, known as a

radio frequency (RF) pulse. This RF pulse exchanges energy with the protons,

causing some of the low energy (parallel) protons to go into a high energy

(anti-parallel) state. We call this excitation.

   In order to transfer energy to the protons, the frequency of the RF pulse must

be as close as possible to the frequency at which they are precessing around

the z-axis. This is called resonance.

   One effect of making some of the parallel protons go anti-parallel is to reduce

the longitudinal magnetization
   Another effect is to make all the protons precess in phase with each other.

Now if we sum the x- and y-vectors we get something more than zero. This is

called an increase in transverse magnetization.

   If we pick a point

on the x- and y-

plane and measure

the transverse

magnetic field of

the protons as they

all precess past in

phase at the

precession

frequency. This signal is one of the things that contributes to our images.

4. LONGITUDINAL RELAXATION

   However, just turning the RF signal on and leaving it on doesn’t give us a lot

of information, because all the protons will act similar to each other,

regardless of what environment they are in.

   Instead, we send a pulse of RF. When it is turned off, the protons will start to

go back to their original state.
   First, those protons

that switched to the

high-energy state

when the RF pulse

was switched on

will start to go back

to their original

low-energy state.

Thus, the

longitudinal magnetization in the z-direction will start to increase. This is

called longitudinal relaxation. If we plot longitudinal magnetization as a

function of time, we get a curve.

   When longitudinal magnetization reaches 63% we get a value called T1

   The process of going from high-energy back to low-energy involves emitting

energy to the surrounding environment. This environment is called the lattice.

Giving energy from spinning protons to the lattice is called spin-lattice

relaxation.

5. T1

   We mentioned above that in order for the RF pulse to transmit energy to

protons it must be at the same frequency as the protons are precessing. We

called this resonance. Well, it doesn’t need to be exactly the same frequency,
but the further away it is, the less efficiently is the energy transfer. The same

goes for protons giving off energy during spin-lattice relaxation.

   If the lattice is spinning at the same frequency as the protons, then the energy

transfer will be fast and the protons will return to a low-energy state faster and

have a short T1. For example, carbon bonds in fat have a frequency close to

the precession frequency. Thus, the T1 of fat is relatively fast.

   If the lattice is spinning at a different frequency to the protons, then the energy

transfer will be slow. Water spins much faster than the protons, so energy

transfer is less

efficient and the

relaxation to the

low energy state

is slower,

resulting in a

slower T1.

   Therefore, at the

T1 for fat we will

have more longitudinal magnetization coming from protons in fat than we will

from protons in water.

   At higher external magnetic field strengths, protons spin faster and energy

transfer is less efficient. This leads to longer T1’s than at lower field strengths.
6. TRANSVERSE RELAXATION

    As well as going from high-energy to low-energy, the protons do something else.

Remember that when the RF pulse was switched on, the protons all started

precessing in phase. Now, if we switch the RF pulse off, the protons will all start

to go back to being out of phase with each other. This is called dephasing.

   Just like how putting

the protons in phase

led to an increase in

transverse

magnetization,

dephasing leads to a

decrease in

transverse

magnetization in the

x- and y-direction. This is called transverse relaxation.

    As it involves spins going out of phase with each other, we also call it spin-spin

relaxation.

7. T2

    Like T1, we can also plot transverse relaxation as a function of time. In this case,

it decreases as a function of time. The point at which the transverse magnetization

reaches 37% is called T2.
   T2 relaxation is determined by two variables. One is minor variations in field

strength across the external magnetic field – called B0 inhomogeneity. The other

is local differences in the magnetic properties of different types of tissue.

   If local magnetic fields are homogenous, such as water molecules that spin fast

and in a random fashion,

then it takes a long time

for the protons to go out

of phase. In these

circumstances, T2 will be

relatively longer.

   If local magnetic fields

are inhomogenous, such

as viscous, impure liquids

like blood, or fat, then protons go out of phase more quickly. This leads to

relatively shorter T2.

8. GETTING AN MRI SIGNAL

   When we send in an RF pulse, we don’t just send in any type of pulse. To flip

some of the protons over to the high-energy state, we need flip the vector that

makes up longitudinal magnetization over by 90o. To do this, we need a pulse

with a flip angle of 90o.
   Imagine we have 6 protons in the low-energy state contributing to longitudinal

magnetization. These are precessing out of phase with each other, giving us zero

transverse magnetization.

   We send in an RF pulse

that flips 3 of these protons

into the high-energy state

and also makes them

precess in phase with each

other. This reduces our

longitudinal magnetization

to zero, but gives us

transverse magnetization.

The vector that gives us transverse magnetization looks like we have just flipped

the z-axis by 90o.

    Now when we switch off the RF pulse, two things happen. The protons in the

high-energy state return to their low-energy state (longitudinal relaxation), and all

the protons start to go out of phase with each other (transverse relaxation. These

things happen simultaneously but independently from each other.

   While the increase in longitudinal magnetization and the decrease in transverse

magnetization happen independently, we can represent the sum of longitudinal

magnetization and transverse magnetization as a sum vector. The horizontal

plane of this vector is determined by transverse magnetization, and the vertical

plane is determined by longitudinal magnetization.
   This sum vector is what gives

us our MR signal. If we have

an antenna (or receiver coil)

nearby, we can record a

signal that oscillates at the

precession frequency.

9. TR AND T1 CONTRAST

   We know from earlier that different properties of the environment (or lattice)

affect the T1. So, we just wait for the longitudinal relaxation to reach the point of

maximum difference between two environments (e.g., tissue types), right? Well,

not quite.

   If we just send a pulse in and wait, we are going to get both longitudinal (T1) and

transverse (T2) relaxation contributing to our signal. Remember also that we can’t

just measure longitudinal magnetization – we need transverse magnetization to

get a signal.

   To get a signal that gives us the best T1 contrast, we want to do things that

maximize the contribution of longitudinal relaxation differences. First, we send a

90o RF pulse. This flips the longitudinal magnetization vector by 90o. Then we

wait for some relaxation to occur.
   At some point, the longitudinal magnetization vectors from the different tissue

types will be different. Then we send another 90o pulse, again flipping the

longitudinal magnetization vector by 90o. The difference in magnetization

between the two tissue types that was reflected in longitudinal magnetization will

now be reflected in the transverse magnetization, giving us MR signal differences

between the two tissue types. This is called T1-dependent contrast, and is the

basis of most anatomical MR scans.

   If we wait to long

before repeating the 90o

pulse, both tissue types

will have reached

relaxation and we wont

be able to tell them

apart. Thus, for T1-

dependent contrast you

need to use a short

repetition time, or TR. As a rule of thumb, a long TR is around 1-2 seconds (the

time it takes for recovery of longitudinal magnetization) while a short TR is

around 500ms.
10. TE AND T2 CONTRAST

   T2 –dependent contrast is a little more difficult to explain. Remember, if we send

in a 90o pulse, not only do we flip the longitudinal magnetization by 90o, we also

make all the protons precess in phase.

   Once we turn the pulse off, they start to go out of phase. The time it takes them to

get out of phase depends on B0 inhomogeneities and local magnetic

inhomogeneities depending on the properties of the tissue the protons are part of.

So different tissue types will show different rates of dephasing, this different rates

of transverse relaxation.

   Protons that are

dephasing quickly

will be showing

more transverse

relaxation than

those dephasing

slowly.

   Now, to get a signal

in which the

contribution comes from transverse magnetization and not longitudinal

magnetization, we need to be clever. Rather than sending in a second 90o pulse,

which gives us the T1 signal, we send in an 180o pulse.
   There are no net effects to the MR signal from longitudinal relaxation, as it has

just been flipped upside down and doesn’t contribute to the transverse

magnetization.

   The 180o pulse acts like a rubber wall, as it sends the protons spinning in the

opposite direction – back from where they came. At some point they will go back

in phase with each other and give us transverse magnetization.

   The effect of this 180o pulse on spins is like an echo, that’s why we call it the

spin-echo. The time from the 90o being switched off until the 180o pulse is

switched on is one half of the echo time or TE. The full echo time is the time it

takes for all the protons to come back into phase after the 180o pulse.

   Transverse relaxation

occurs at a much faster

time scale than

longitudinal relaxation.

Its on the order of a

hundred milliseconds.

Therefore, TEs tend to be

around 10 to 40msecs.

   We can send as many 180o pulses as we want. One reason we might want to send

lots is because the effects of these pulses only acts to reverse dephasing due to B0

inhomogeneities. It doesn’t reverse the dephasing due to local tissue effects.

   Therefore, while the 180o pulses will send the dephasing protons back into phase

and restore the transverse magnetization, eventually the signal will decrease after
multiple echos over an extended period of time. This is called T2 effects.

Different tissue types will show different T2 effects, giving us T2-dependent

contrast.

11. T2* (pronounced T2-star)

   We don’t need to send in a 180o pulse to view effects on transverse

magnetization. The dephasing that occurs after the 90o pulse, which is due to both

B0 inhomogeneity and local tissue inhomogeneity will contribute to this

dephasing. This is called the T2* –dependent contrast. T2* effects occur at a

much faster time scale than T2 effects – in the order of tens of milliseconds.

   It’s not very good at

distinguishing between

different tissue types, but is

good at measuring changes to

the local magnetic properties

(susceptibility) of a location.

So, just say you measure the

T2* from one point in space,

and then something happens

to change the local magnetic susceptibility of that point (like, for example, an

influx of oxygenated blood that has different magnetic susceptibility to the

deoxygenated blood). You will get a change in T2* signal. This forms the basis of

the blood oxygen level-dependent (BOLD) signal.
12. SLICE SELECTION

   OK, so now we can get our MR signals from our different tissue types and

separate T1 and T2 effects. But at the moment we only have a 1-dimensional

signal. How can we localize things in the brain?

   First thing is to be able to select brain slices. These are slices across which we are

going to form a 2 dimensional image describing the magnetization at every

horizontal and vertical location. We call these locations that make up an image

voxels.

   Selecting a slice is actually

pretty easy. Remember

that protons precess at a

frequency that is described

by Lamour’s equation. In

essence, this tells us that

the precession frequency

will vary as a function of

the strength of the external

magnetic field B0.

   And remember that the transfer of energy from the RF pulse, called excitation,

depends on the match between the precession frequency and the RF frequency.

We called this resonance.
   If B0 is uniform, then the RF will excite all the protons in whatever is in the

magnetic field equally. However, we can vary the magnetic field slightly

producing a gradient in B0. At one point in this gradient, the B0 will be slightly

lower, making the protons in that area of the field precess slightly slower. At other

points B0 will be higher, making the protons in that area of the field precess

slightly faster.

   Now, if we use an RF pulse that matches the slightly slower precession

frequency, we can excite only those protons spinning at that particular frequency.

This allows us to select that particular area of the field. If we use another, slightly

faster RF frequency, we can select another part of the field.

   The thickness of the slices, and the spatial resolution of the images in the other

two directions, will be

determined by the

steepness of the various

gradients used. Steeper

gradients lead to larger

differences in

frequencies across a

given distance, thus

thinner slices.

13. FREQUENCY ENCODING
   The slice selection gradient allows us to select a slice in one direction. This

occurs at the point of excitation, when the RF pulse is turned on. For example,

for a person this might be across their body or head. A gradient that allows us to

choose this slice would run from head to toe. Now we want to locate the voxels

that make up that slice.

   To do this, we can apply another gradient in a different direction, for example

from left to right. This will make the protons located from left to right spin at

different frequencies. This is called frequency encoding.

   This time, unlike with slice selection, we are not exciting protons that are

spinning at a particular frequency. Instead, we are taking the ones we have

excited (go from low-energy to high-energy) and make them precess at different

frequencies.

   This gradient is applied at the data acquisition stage, after we have done the

fancy things with RF pulses.

   Remember that when we are recording the signal, the antenna (actually the

receiving coil) picks up a

signal being transmitted at

the precession frequency. We

can then use information

about the frequency that the

protons are precessing in

order to localize them in the

left-right direction.
13. PHASE ENCODING

   So we can select a slice in the head to toe direction by selectively exciting only a

slice of voxels by combining a gradient field with the RF pulse. And we can

localize voxels within this slice in the left to right direction by applying a

frequency encoding gradient and localizing these frequencies with the receiving

coils. Now we just need one more piece of information - that is the location of

voxels in the front to back direction.

   Luckily, there is one more piece of information we can manipulate to localize

voxels. If we apply a brief, third gradient field in the front to back direction, the

protons will speed up to different extents depending on were they are in relation

to the gradient.

   When this gradient is turned

off, they will go back to the

frequency determined by the

other gradients but will now

differ in phase across the

direction of the phase

encoding gradient. We can

then use this phase

information to determine

where voxels are in the front to back direction.
   Like with frequency encoding, the phase encoding gradient is turned on and off at

the data acquisition stage.

   It is important to note that the directions I have used here are arbitrary and could

all the swapped around. So you could select slices in the front to back direction,

use frequency encoding in the head to toe direction, and phase encoding in the left

to right direction etc etc…

14. K-SPACE

   We now end up with a slice which is made up of a 2-dimensional image

containing the frequency information on one axis and the phase information on

the other axis. The intensity of each data point in this image reflects the MR

signal strength at each frequency and phase.

   The steepness of the slice

selection gradient

determines the slice

thickness. The steepness of

the frequency and phase

encoding gradients

determines the in-plane

resolution (ie the size of

your voxels).

   The 2-D Fourier transform takes any 2-D image and treats it as a combination of a

series of sinusoids and determines the amplitudes of the various frequencies and
their phases. The Inverse Fourier transform does the opposite, taking a series of

frequencies and phases and transforms them into a series of sinusoids. These can

make an image.

   To get an image of the brain, we take the Inverse Fourier transform of the k-space

image and get a brain!

14. 3-D IMAGING AND GRADIENT ECHO SEQUENCES

   While the procedure described above works great for anatomical images in which

we can collect 2-D data slice by slice over a 7-8 minute period, it’s not so good if

we want to collect data from the whole brain in a second or two.

   More rapid techniques can collect 3-D data essentially all in one go, rather than

slice by slice. For example, methods used to measure changes in BOLD use 3-D

imaging methods. Briefly, rather than selecting a thin slice in the slice selection

stage, a thick slab or volume is selected.

   Then, during the data acquisition stage, a second phase encoding gradient is used

to encode voxels in the slice direction. Because we also use phase encoding for

in-plane spatial encoding, this process can be a little tricky!

   Also, while methods based on T2* effects don’t use the spin-echo methods

described earlier, they often use gradients to generate signal echo. Gradient echo

methods allow you to generate signal echo (send the protons back towards being

in phase) without losing sensitivity to local field susceptibility differences.

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