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.