This is the second video in a small series
discussing how MRI works - delivered in a very timely fashion. If you haven't seen Part 1,
you really should watch that first or you may get lost here - link in the description. In this video
I'm hoping to go a little deeper into Boltzmann Magnetism and our magnetic moment ensemble, the
NMR experiment, and the effects of inhomogeneous magnetic fields. This will all lead us to the
primary topic of the video: the spin echo. On our last episode we discussed the basics of
the NMR phenomenon including precession, T1 and T2 relaxation, and the signal equation following a
90 degree pulse. Then we asked the question: what is Boltzmann Magnetization., M0? Well we touched
on it last time, but never actually defined it. Instead we examined a related quantity: the
polarization - which describes the fraction of spins which contribute to NMR signal. So lets see
how polarization and magnetization are related. Continuing with where we left off in the last
video, Boltzmann magnetism or magnetization is the equilibrium nuclear magnetism of a spin
sample in a magnetic field. Every proton has the same magnetic moment, mu which is equal to gamma
(it's gyromagnetic ratio) times hbar (the reduced planks constant) times its spin quantum number
which in the case of protons is 1/2. As we know, our sample is composed of a great many protons
each with the same magnetic moment. Boltzmann magnetism is simply the vector sum of all of these
magnetic moments when they are in equilibrium. Of course, in the absence of a magnetic field
these spins lack a defined orientation, but in the presence of the magnetic field, these
spins orient themselves in 1 of 2 possible states - spin up or spin down. The spin up state
(where the spin is aligned with the field) is the low energy state and is thus preferred, but as we
saw in the last video, the energy difference is so small compared to the temperature of the sample
that the spins are nearly equally distributed between these states - only a few spins out of
every million are in excess in the spin up state. In the last video, this led us to a discussion of
polarization: the Fraction of spins which are in excess in the aligned, low energy state - which
we saw simplified to this expression for samples in thermal equilibrium - as a reminder this
polarization is the spins' gyromagnetic ratio gamma, times the reduced planks constant hbar,
times the external magnetic field strength B0, divided by 2 times boltzmann's constant
k times the sample temperature T in kelvin. We were able to work this out directly
using concepts from Boltzmann statistics (again, link in the description for a refresher)...but
then we just kinda stopped without actually defining Boltzmann Magnetism. But here's
a bit more detail to help out. If we want to actually calculate Equilibrium
magnetization of our spin sample, like we said, that should just be the vector sum
of our magnetic moments. Since we know that the spins exist in only 2 states - up or down - we
know every pair of up and down spins will result in a net cancellation of magnetization. Thus
only the unpaired spins contribute to sample's nuclear magnetism. And now we can intuitively
write an expression for Boltzmann magnetism: it's simply the magnetic moment of a single spin
times the number of spins we have in our sample, then times the polarization (the fraction
of those spins which are unpaired). Plugging in these values we find
the total magnetization of the sample is given by this expression for protons
(ie hydrogen nuclei) in a magnetic field B0. We often compile all of these terms
into a single variable, chi naught, which is the static nuclear susceptibility
of a sample - it essentially tells you how big of a magnetic moment you'll get in a given
magnetic field strength - and it is inversely proportional to temperature. This is called 'curie
law' (after Pierre Curie, husband of Marie Curie). As we've said a number of times Boltzmann
magnetism is the nuclear magnetism of our sample in thermal equilibrium, so I find
myself using those terms interchangably: Boltzmann magnetization/equilibrium magnetization.
If you want to test your understanding a bit further, work out the units of this expression -
does it match the SI unit of a magnetic moments? With this foundation, I'm going to abandon the
Energy state picture of nuclear magnetism and replace it with a single vector representing
our Boltzmann Equilibrium magnetization. Since we always align the z axis of our frame
of reference with the magnetic field B0, our Boltmann magnetization vector
is created along the same axis. Any NMR experiment essentially involves knocking
our sample out of equilibrium whereupon it delivers detectable signal as it returns
to Boltzmann - which we detect and analyze. Here I'm illustrating an NMR experiment where
we might knock the nuclear magnetism our of alignment with the B0 field by flip angle
theta. The key to understanding NMR dynamics is that our magnetization vector will return to
Boltzmann following the relaxation principles we discussed in the last video: it's transverse
magnetization will decay to 0 at the T2 rate and its longitudinal magnetization will
return to Boltzmann at the T1 rate. Because of the precession phenomenon, the amount
of our magnetization in the transverse plane dictates the amount of signal we detect, which
we saw last time increases as the precession frequency omega increases. And thus we report
an expression describing NMR signal of a sample: the signal is proportional to the number of spins
in our sample times the magnetic moment of an individual spin, times the fraction of
those spins contributing to the NMR signal, times omega, which is gamma B0. Notice the
effect our magnetic field strength B0 has our detected signal: Let's say we double our external
field strength B0, what happens to our signal? It quadrupels! Remember for samples in thermal
equilibrium, there's another B0 buried in the polarization expression. Increasing our magnetic
field increases our precession frequency AND our Boltzmann magnetism - thus signal is proportional
to B0 squared. Neato Huh! Note however, that if our sample is not in thermal equilibrium, such as
in hyperpolarized gas mri, then the expression we derived for polarization no longer applies and our
signal will scale linearly with field strength. Before moving on, all of these magnetization
dynamics are typically summed up into a tidy expression called the Bloch equation
named after Felix Bloch, pioneer of NMR. I'm not going to go into details about it
here because there's other stuff to do, but it would be remiss not to at least mention
such a fundamental equation here. Perhaps, in a future video we'll build and solve the
Bloch equation. In its simplest form, the Bloch equation says that in the presence
of a magnetic field B, a magnetization vector M will precess about B in sort of conical
trajectory whose cone angle is essentially twice the angle between B and M. In the mean
time, try working out the math for yourself! Ok, this may be a good time to clarify something. On the last video I drew this ball and arrow
representation of a proton, which I think helps to conceptualize the spin magnetic moment,
precession, flip angle, and relaxation phenomena. Just to clarify: individual protons can only exist
in one of 2 Zeeman states within a magnetic field: spin-up or spin-down. The dynamics we're examining
pertain to ensembles of many, many protons. For example, the longitudinal magnetization
of our sample (ie the magnetization pointing in the direction of our field), is actually
representative of the number of protons in each state: higher if more spins are in the
spin-up state, negative if more spins are in the spin-down state, and zero if there are
equal numbers of spins in each state. Thus, longitudinal magnetization returning to Boltzmann
equilibrium, is simply the return of spins to their equilibrium state populations which we saw
was contained within the polarization equation. The signal detected during precession is the
energy released during state transitions. For those familiar with electronic enegy transitions
which occur in fluorescent lamps and lasers, this nuclear energy transition follows
nearly identical quantum mechanics. The difference here is that the energy
separation of the two states is proportional to the external magnetic field strength
B0, as opposed to just atomic structure. Someday we'll do a proper video of NMR quantum
mechanics, but for now we'll stick with the tried and true classical description. From
now on I will use a single arrow vector to describe the nuclear magnetization vector
of our spins within a particular space. Ok, so the simplest NMR experiment we can devise
is to place a sample of spins in a magnetic field, B0, and deliver a separate magnetic
field B1 perpendicular to B0 about which our magnetization will precess. In a
reference frame which rotates at the larmor frequency of our magnetization M in B0, the
B1 pulse is stationary. In the laboratory frame of reference the B1 pulse rotates at the
larmor frequency, as does the magnetization M. When our magnetization has been rotated say 90
degrees it is entirely perpendicular to the static B0 field maximizing our detectable signal during
its precession. We then detect the resultant free induction decay as the magnetization works its
way back to Boltzmann equilibrium. This is often called a 'ping' experiment or pulse/acquire. Note
that my vector cartoon is showing the rotating frame picture - right, so it's animated in a
frame of reference that rotates at the larmor frequency - but my Transmit and Receive plots here
show both the rotating frame signal (in white) and the laboratory frame signals in yellow for
transmit and cyan for receive. We'll discuss the mathematics behind transitioning between
rotating and laboratory reference frames in a later lecture, but for the curious this is most
often achieved electronically by heterodyning. Oh, and notice that my rotating frame cartoon
shows B1 along the x hat axis which excites our magnetization into the y hat axis. Turns out
you can precisely deliver B1 pulses along any direction within the rotating frame's x-y plane
and thus place M along any direction you'd like. In order to do this, you need to take note of
the phase of the larmor oscillator - the phase of the oscillator within your excitation pulse will
dictate the excitation axis for your magnetization and thus the phase of your detected signal as
well. If we look at our cartoon immediately at the point in time immediately following the B1 pulse,
we can see that the choice of laboratory frame phase in our pulse diagram dictates the precise
alignment of our B1 field within the rotating frame and thus the detected signal. In the ping
experiment it may not seem like that matters too much, but we'll see later that many NMR/MRI pulse
sequences are designed to leverage this fact, especially in cases of multiple B1 excitations
so just keep that in mind as we move forward. So in order to perform this experiment IRL you
go to the NMR physics store and buy yourself an NMR - a device which can deliver these
excitation pulses and display the acquired signal on your oscilloscope. You prepare a
sample of plain water (plenty of protons - ie hydrogen nuclei - to examine). You place
your sample in a strong magnetic field B0 and wrap the sample with wire which will be an
rf coil for transmission of excitation pulses and subsequent reception of NMR signals. The NMR
is plugged into your oscilloscope and you window the display to show 1 second of received signal
because you know from a simple google search that the T2 of water is about 1 second, so you're
expecting the received signal to look something like this - after 1 second the signal will have
decayed to about 1/e or 37% of its initial value. In exhilerating anticipation you execute the ping
sequence in your NMR AAND.....you get this.... You triple check your equipment, read all the
manuals, retry the experiment, but for some reason the NMR signal you detect decays much
much more quickly than you expected. Why does your signal decay so fast??? Perhaps your
water sample is contaminated with minerals which reduce T2? Or perhaps the internet was wrong
when it said water has T2 of about 1 second? Well actually it Turns out there's an
assumption we've made throughout our tour of NMR which is now being challenged:
our B0 field isn't perfectly homogeneous. We've assumed that B0 at every point in space is
exactly the B0 field strength in the z direction, but of course no field is so
perfect, and even slight variations in field strength and direction can have a
substantial effect on our NMR experiments. Consider two proton spins: One experiences a B0
field of 1 Tesla and the other experiences a B0 field of 1.000001 Tesla - exactly 1 microtesla
higher than his buddy. Given that the gyromagnetic ratio of protons is 267.5 times 10^6 radians
per tesla per second (that's 42.6 MHz/T) how long will it take before these spins are 180
degrees our of phase with each other and thus will produce no NMR signal? Pause the
video to work it out if you'd like. This difference in B0 leads to a difference
in omega following the Larmor equation. The difference in B0 field is one millionth of
a tesla, and thus the difference in frequency is 267.5 radians (or 42.6 revolutions) which
means the spin in the higher field will complete 42.6 more precession cycles than the first
spin every second. The spins will therefore be pi radians or 180 degrees out of phase in
11.7 milliseconds! A difference in magnetic field of one part per million reduces the
coherence of spins down to milliseconds! This is what's happening in our NMR
experiment - tiny imperfections in our magnetic field homogeneity can have
a dramatic effect on our detected signal! This is a problem, because now the
rate at which our signal decays does not reflect properties of the sample - it
reflects the external conditions of our experimental setup - namely how good (or
bad) of a magnetic field we can produce. So last we saw that signal decays at rate T2 due
to spin-spin interaction s, but when relaxation due to field inhomogeneity is considered as well
our observed signal decays according to a new time constant which we call T2*. That means our ability
to contrast tissue is substantially reduced. So now what? Can we fix this? Oh, and you
might also be thinking 'wait a minute, we saw on the last video that spins will
only experience excitation if the frequency of our B1 excitation pulse is matched to the
spins' larmor frequency - well if the spins are in different field strengths then they have
different Larmore frequencies. Could you still excite both spins with a single-frequency B1
pulse? I guess their larmor frequencies are close enough? what if the second spin was in a B0
field of 1.001 Tesla - a part per thousand off?' If you're thinking along those lines, good!
We will absolutely examine all the intricate details of excitation - in a later video.
For now we will assume all spins magnetic moments experience excitation - even if their
larmor frequencies are not exactly equal. Let's examine explore this more closely in our
sample of spins here - I'll draw a single plane of spins to make it easier to see, but note
that of course spins exist in the full 3D space of our sample. I'll go ahead and add
a coordinate system to orient ourselves. We begin with a spin sample which is initially
in equilibrium with a magnetic field pointing in the positive z direction - that is, they are
all at Boltzmann magnetizm. We'll work in the rotating frame here - remember this means we
spin ourselves around at the larmor frequency, so from this perspective the spins do
not precess about the B0 field - which essentially means I won't animate the
spins precession about the z axis here. Ok, let's deliver our B1 excitation magnetic
field in on the negative y axis. Our spins then precess about the B1 field, and when
they are perpendicular to the B0 field, i.e at a 90 degree flip angle, we turn off the B1
field. Now the total magnetization of our spins is in the positive x direction of the transverse
plane and thus delivering detectable signal, so let's think about the dynamics of
their subsequent dephasing and T2 decay. I'll orient the camera so the magnetic field
is coming out of the screen here, add I'll also add in another vector here to represent
the magnetic moment vector sum here as well. As we said in the last video, the randomly
distributed spins are experiencing the external B0 field of our magnet, but also the
magnetic fields of each of their neighbors. If we try to describe this for a single spin, we would
say its precession frequency, omega sub k for the k'th spin, is equal to the larmor frequency plus
some Delta omega which of course is just gamma times some delta B. And remember even though I'm
not animating it here, the spins are also bouncing around all over the place by Brownian diffusion
so the delta B here can fluctuate in time as well. This Delta B is what causes our spins to dephase
and our transverse magnetization to decrease over time - the T2 decay we know and love. Now
we're working in the rotating frame, right, which means we're spinning our frame of reference
at omega 0 - thus we drop the omega 0 term here. The spins only exhibit precession in this frame
if their delta omega is non-zero - clockwise if it's greater than 0, counterclockwise if its
less than 0. So let's get our magnetization plot ready and play the animation. Hopefully
this seems familiar from the last lecture: the spins dephase and total magnetization
decreases following e to the minus t/T2. Rotating frame's kinda nice isn't it? Conveys all
the information we need without making me dizzy. Now most textbooks will illustrate
this using a 'stacked' view of the spins rather than a messy image
of all the spins all over the place. Since the NMR coil is assumed to detect all spins
with equal sensitivity this illustration makes perfect sense and is much cleaner to visualize,
especially for our upcoming discussion on echoes. So lets put our total Magnetization
vector back in and play the T2 animation. This 'fanning out' of spins represents T2
decay which is manifest in the reduction of the Magnetization vector length over time. With this picture available to us, perhaps
we can develop a mathematical expression for the total magnetization vector M. This
shouldn't be too bad, right, after all, the magnetization M is just the vector sum of
the individual magnetic moments mu sub k. Hmm, ok, so what is mu sub k - the vector
describing the kth magnetic moment? Well it's probably easiest to describe
in terms of its frequency: omega sub k. If we do that we can see that the x-component
of mu is cos( omega sub k times time) and the y component would be minus sin (omega
sub k times time). So that makes sense and it suggests that if omega is positive, mu will
precess clockwise in this orientation which is indeed how spins with positive gyromagnetic ratio
- like protons - precess about magnetic fields. Well, I don't know about you but whenever
I see cos and sin arranged like this in a vector expression, I immediately think of Euler's
relation. With this, we can abandon the Cartesian x hat and y hat expression in favor of real and
imaginary axes. Thus I can express my vector as a complex number e to the minus i omega sub k t.
This is so much cleaner because complex numbers by design contain two pieces of information
- real and imaginary - in a single expression with the added bonus that rotations are
much easier to describe mathematically. If you're not used to doing this I would encourage
you to give it a try next time you're dealing with rotations or sinusiodal motion - it truly makes
life much easier. Now, with each moment being mu times e to the minus i omega t, we express
our transverse magnetization in time as such. Note that in this model, the individual spin
magnetic moments all have the same magnitude. This makes mu a constant here so we can pull
that out front. Now summing over omega sub k isn't super helpful unless we know what all the
mu's frequencies actually are. Well what we can do then is examine the distribution of spins
with a given frequency - that is, the number of spins which have a given omega - kind of like a
histogram for omegas but since it's continuous we call it a frequency distribution (and the title
omega frequency distribution is not redundant, this is a frequency distribution of omegas).
Anyways, Maybe it looks something like this: most spins are pretty darn close to an omega
of 0 (which, in the laboratory frame would be the larmor frequency). Some spins are a little
faster, some are a little slower, but as we get further and further from w=0 the number of spins
with those frequencies goes down substantially. I wonder what kind of distribution this is?
Well it actually doesn't matter, the point is that this distribution, f(omega), describes
how many spins have a given frequency: at there are f(omega 1) spins with frequency omega 1,
omega 2 would have f(omega 2) spins etc, etc. So we can just rewrite our sum over k omegas as a
sum over all omegas where e^i omega t describes a particular spin frequency omega, and f(omega)
is how many spins we have at that frequency. Now of course we're dealing with septillions of
magnetic moments over a continuous spectrum of possible frequencies - so this sum is really
an integral - and we have the result that our Magnetization as a function of time (and thus our
detected signal as a function of time) is just mu times the integral over all frequencies w of the frequency distribution times
e to the minus i omega t d omega. O. M. G.
This is the definition of the Fourier Transform of the frequency distribution, f(w)!
This means that the rate at which our signal decays in time, reflects the distribution of
frequencies our spins possess - and vice versa. The way I drew this frequency distribution
here was not an accident - it is called a Lorentzian distribution, and we'll do a deep
dive into the actual math in a later lecture. My dear friends, Fourier Theory is THE
mathematical foundation of MRI - truly, you cannot say you understand how MRI works if you can't
also say you know how the Fourier Transform works. For some of you, the beauty of seeing
a Fourier transform appear here makes perfect sense and you probably saw it coming
from miles away, but perhaps for others just the phrase 'Fourier Transform' incites feelings
of dread and panic. Fear not! The next video in our journey through MRI will be dedicated entirely
to what the Fourier Transform is and how it works. Until then, let's just develop a conceptual
understanding here. A sample of slowly dephasing spins demonstrates a corresponding slow decay of
Magnetization and a narrow frequency distribution showing a relatively tight grouping of spins
near omega=0 compared with more rapid dephasing and quicker decay. This forms the basis
of T2 contrast weighting we saw last time. Now because we know that omega equals gamma B0, we
know our omega distribution is actually indicative of a distribution of regional magnetic field
B since all protons have the same gamma. This distribution of the magnetic field strengths
experienced by our spins dictates our FID decay. So returning to our picture of the many spins
in space - we know that each spin experiences some delta B due to the tissue environment, but
also some delta B due to our imperfect field. To help us visualize, I'm going to show the magnetic
field in space as a color map. Spins in the purple region experience a slightly lower field, and
spins in the red region a slightly higher field and a full range of delta B in between.
The larmor equation tells us that field inhomogeneity will broaden our frequency
distribution, thus reducing our decay time. So instead of placing our spins on a field
color map, I'll just color each spin according to it's experienced field strength here. Now
we can stack them and observe the accelerated dephasing. We'll use this stacked picture again in
a minute, but let's walk through the math a bit. Immediately following excitation all our spins
are aligned in the x direction and we have a macroscopic ensemble moment to detect. If our
spins were all at the exact same frequency, our signal would never decay and our
distribution would be perfectly narrow. Our signal in time would just be a constant.
But of course our spins do dephase due to the intrinsic molecular dynamics of our
system or tissue and dephase - our signal is thus modulated by e^-Rt. R is the decay
rate and is characterized by time constant T2 which we've seen before. Our signal
thus decays and our distribution widens. Now we have another source of decay,
so our already decaying function gets a second modulation term e^-Rb times time. Rb
is the decay rate due to field inhomogeneity. The nice thing about exponentials is that
the decay constants multiplying time just add, and we can write a a decay rate
R* comprehensive of all sources of spin dephasing. R* by this model is clearly just
equal to 1/T2 plus R sub B which we can write as gamma times delta B which is just delta omega.
Similar to T2, R* can be characterized by its own time constant which we designate T2* which
is the observed signal decay due to all the dephasing mechanisms present in our sample. This
is problematic for us, because now our signal decay is more representative of our magnetic
enviroment, not molecular properties intrinsic to tissue. Thus our ability to contrast
tissue in an image is drastically reduced. To illustrate this, consider
3 tissues of various T2's and equal Boltzmann magnetizations. Here are their
T2 decau plots in a perfectly homogeneous field, delta b = 0. At an echo time of say 40
ms we can contrast the tissue very well. If our field inhomogeneity increases to just 1
part per million, our observed decay time T2* is reduced to milliseconds. Not only do we have
to acquire our signals faster, but our ability to contrast tissue is dramatically reduced, if
not gone depending on our signal to noise ratio. Best case scenario is we'd be imaging a
tissue with infinite T2 - that would drop this term out - but notice we can't
do better than 1 over gamma delta B. And this part per million example I keep using
does indeed reflect most commercial MR scanner limits. Manufacturers boast field inhomogeneity of
about 1ppm within the volume of the scanner bore. Of course field homogeneity gets better
as you image a smaller and smaller volume. Fields can be 10 to 100 parts per billion if
your field of view is down to 10s of centimeters, which is quite impressive, but of course is
a large percentage of a scanner's price tag. But, it's important to remember that even if
you can manufacture a magnet with absolutely perfect field homogeneity, Once you put
something to image inside - especially living, physiologically active tissue - the
field will be distorted and delta B will increase. So you can only go so
far perfecting a field anyways. Oh and you might have noticed that somewhere
along our discussion, we made a jump from having a frequency distribution describing
our field inhomogeneity to a single number delta B. What happened there? Shouldn't
it be, idk, some sort of integral or something? Well, it so happens that if we model our
distribution using the Lorentzian lineshape I mentioned earlier, delta B can in fact be
represented by a single number. This number is most often related to the Lorentzian's full
width at half maximum (or half width at half maximum depending on which textbook you read).
In our Fourier Theory video, we'll develop the relevant math, but for now rest assured this
is a common, mathematically sound practice. Ok quick recap: the ping NMR experiment
involves exciting nuclear magnetization into the transverse plane where its transverse
decay reflects our detected signaland intrinsic properties of the tissue. But, because
our field is imperfect, this decay is accelerated, and our ability to contrast tissue is reduced.
We're about to discuss a revolutionary method for reversing the effect of field inhomogeneity
on our FID. But before we do, can you think of a way get our spins back in phase after dephasing
due to field inhomogeneity? Perhaps our cartoon of the color spins dephasing might help? Pause the
video if you'd like to ponder your own solution. Well...
Consider the NMR experiment. I'll draw our signal plot here and I'll use these
pink bars to indicate excitation pulses. Of course the 90 degree pulse knocks our magnetization out
of the longitudinal plane and into the transverse plane so we can detect the FID. A simple,
yet clever idea was suggested by Felix Bloch, but first experimentally verified by
Erwin Hahn: just deliver a second pulse. We will deliver this pulse soon after the FID has
decayed, and we'll make it a 180 degree pulse as was suggested by Carr. Can you predict
what might happen to our detected signal? Here's our 3D magnetization
vector cartoon to help. As expected, the FID is seen following the
90, but the addition of the 180 results in a signal echo of sorts. This is called
the spin echo. So, what just happened? Well let's return to our dephasing spin cartoon
to help. The B0 field is coming out of the screen here and I'll add our signal plot for us to
follow. Remember that the spin colors represent their associated external field strengths:
red is in a high field, purple in a low field. Notice what the 180 pulse does to our
dephased spins. After they're rotated 180 degrees their precession continue s, but
they eventually rephase with each other! We can deliver another 180 if you'd like - the
result is another echo. Notice that after the 180, the spins continue to precess at a rate dictated
by their field strength and represented by their color: purples in low field are always precessing
counterclockwise, reds always clockwise. The 180 pulse doesn't change where the spins are in space
and so does not change their precession frequency. Perhaps this is easier to see if we look at the
animation of the spins unstacked. If you want you can focus on an individual spin to verify
that its precession frequency does not change following excitation. The spins dephase, then
following the 180 they rephase into a peak and then continue to dephase. There's two important
things to notice about the plot I've drawn here: first, you might notice that the dephase/rephase
parts of the plot seem to be mirrored around the 180 pulses. Second, you see that our
echo peaks get smaller in time. What's going on? So as we've seen the detected FID follows
T2* decay. If we deliver out 180 at time tau, then in that time the spins will
have dephased by a certain amount. We should therefore expect that following the 180
the spins will take exactly that amount of time to rephase. The echo peaks decay because unlike
dephasing due to field inhomogeneity, intrinsic T2 decay is not reversible. Spins are still
losing coherence due to their self-interactions. Here I'll draw the plot of T2 decay. Notice
that our peak comes right up to the T2 plot. And in fact, this is the take-home.
If our 180 is delivered at time tau, the echo will appear at time 2 tau and its
peak will reach the T2 decay curve. Thus the peak amplitude can be used to calculate T2! We've
regained our T2 contrast. And this will be true Notice by the way that while I animated the
B1 pulse on the rotating frame's x axis, echoes will appear regardless of the
excitation axis. Here's an example with B1 at 45 degrees to x. We still get
an echo, but it appears along the y axis. Thus we can expect our echo to
have a different phase as our FID. I'm going to simplify the animation here
with just a few spins to illustrate the effect of delivering B1 at different
orientations in the rotating frame. Notice the green spin does not precess here:
his delta B is zero, he's perfectly at larmor resonance. As such we know all his buddies
will aligned with him at each echo. We can use this fact to help with our discussion.
Here's my dephased spins immediately before a 180. Notice the angle our green spin
makes with respect to the B1 field. Follwing the 180, our spin will
be at this angle on the other side of B1. Thus we can expect our echo to
be at twice the phase of our B1 pulse. Ok so why does this work?
Well the reason is that our spins have acquired a certain amount of phase at time tau following
the 90 degree pulse. Let's examine a spin with a negative gamma here so The angle theta of our spin
would just be omega t. In the argand plane, the real component mu x reflects cos omega t and the
imaginary component reflects sin omega t. These components can be visualized on a single plot and
imply a counterclockwise progression in phase. After a 180 degree pulse the imaginary component
becomes negative - but the real component is unchanged. This takes our sin omega t and makes
it negative. The reversal of the imaginary component mathematically switches the phase
we've accumulated from positive to negative: minus theta. Our system will continue to evolve
progressing in phase counterclockwise at rate omega t. The phase total accumulated phase
with thus be 0 when our time t equals tau, and we'll detect and echo.
Pretty neat huh? The math is nice and simple here where
B1 is aligned along the real axis, but perhaps you can extend the math for cases of B1
delivered at an arbitrary angle to the system... Oh and here's a leftover spin echo animation
in the laboratory frame just so you can visualize what's going on. I only did
one of these since it makes me dizzy. So there you have it, the spin
echo. While we've focused on dephasing due to field inhomogeneity as
the being reversible by the spin echo, it will also reverse dephasing due to chemical
shift and heteronuclear spin interactions. Spin echo sequences in MRI do have drawbacks which
we'll look at in later lectures. For one, they take longer since echo times are increased.
And they involve much more rf delivery compared with simpler gradient echo sequences. But
all of these details are for another time. And now you know the story of
how helpless NMR signals are freed from the prisons of inhomogeneous magnetic
fields by the humble 180 degree refocusing pulse, thus allowing for true T2 contrast in tissue. If you really want to put your newfound knowledge
to the test, try a few of these exercises. Post your solutions in comments. See if
others got the same answers. Have fun! That concludes Part 2 of our little
MRI intro. Thanks for watching! And Sorry it took so long to deliver. I do intend to continue this little intro series.
The next video will be all about Fourier Theory and the Fourier Transform. Fourier theory is so
ubiquitous that another YouTube lecture may just be a drop in the ocean, but it's so fundamental to
studying MRI that I can't NOT make one for the MRI fundamentals series. I'm thinking 4 videos
total in this intro series with the fourth being a discussion of how gradients produce
images and the Gradient echo pulse sequence. After that I think we'd be well positioned to
discuss MRI topics in detail without leaving anyone behind - things like pulse sequences,
MRI hardware such as spectrometers and coils, chemical shifts, hyperpolarization, clinical MRI,
reconstruction and acceleration techniques, etc. Maybe you have a question or topic
you'd like to see? Post a comment! I'm definitely not a full-time YouTube
person so I simply can't respond to every comment like some of the greats do, but
if I do see questions, and if they have a lot of thumbs up's that'll let me know what
y'all would like to see in a future video. Cheers, friends.