Transcription
Hello, everybody, and welcome back! Today's talk is the first in a three-part series where we're going to be looking at different types of MRI pulse sequences. We'll start off today by looking at spin Echo pulse sequences before then moving on to gradient Echo and inversion recovery sequences.
Now, the focus of today's talk is first going to be to understand what exactly a spin Echo is and why we would want to go about generating an echo within our sequence. And we'll see that the generation of an echo is going to help us to recover some of that signal that's been lost during free induction Decay, or T2* decay. Once we understand what exactly a spin Echo is, then we're going to move on and look at three different pulse sequences that utilize this spin Echo phenomenon.
Now, if you look at this pulse sequence here, this is the kind of pulse sequence we've been looking at throughout this module. Now, the first thing we do is apply a radio frequency pulse that will tip the net magnetization vector out of the longitudinal plane, and it will allow that vector to gain some transverse magnetization. As well as gaining transverse magnetization, it's simultaneously losing longitudinal magnetization.
Now, why is this important? The signal that we can measure in MRI is transverse magnetization. We can't measure longitudinal magnetization because of that large magnetic field that we've applied along the patient that is always on. There's no way that we can place a receiver coil and manage to somehow measure this longitudinal magnetization. We can only measure what we flip into the transverse plane.
Now, we've applied a slice selection gradient at the same time that we apply that radio frequency pulse, and that just allows us to select a specific slice. And if you don't understand that concept, go back to the slice selection talk. We've then seen that we've applied a phase encoding gradient, one gradient per complete cycle here, to the entire slice, and that phase encoding gradient will cause the spins to de-phase along the y-axis.
We've then previously touched on why we would apply a 180-degree RF pulse and how that goes about generating an echo within our sequence, and we're going to revise that in a little bit more depth here today. Then, when that echo occurs, when there's that rephasing of those spins, we sample the signal at a time known as TE, the time to Echo. And it's named TE because that's where this echo, the spin Echo, occurs, and that's what we're going to be looking at today.
We sample over a period of time, and we sample while we are applying the frequency encoding gradients. As that frequency encoding gradient is being applied, we're taking multiple discrete data points and then placing those data points within a single line on K-space that corresponds to the specific phase encoding gradient that we applied for that specific pulse sequence. We then wait a long period of time until we repeat the process again, the time to repetition, until we then flip those spins back into the 90-degree plane.
Now, TE is a certain period of time after we have flipped those spins, in this example, to 90 degrees. Now, what causes these spins to lose transverse magnetization? It's not those spins returning into the longitudinal plane that contributes ever so slightly to signal loss. The main contribution to loss of that transverse signal is the dephasing of those spins. If we were to take two spins here that we have flipped from the longitudinal plane into the transverse plane, we then stop our 90-degree RF pulse. Those spins are going to de-phase; they're going to become out of phase with one another based on their local environments.
Now, that dephasing is causing loss of transverse signal because they are no longer in phase with one another. And that loss of transverse signal, we can measure that rate, and that rate is what's known as T2. And in an ideal world, that loss of transverse signal from the dephasing would only be due to spin-spin interactions, with spins interacting with one another causing dephasing. But as we'll see throughout this talk, there are local magnetic field inhomogeneities that are actually responsible for the majority of the dephasing of those spins and the loss of transverse magnetization. And that loss of transverse magnetization because of the local magnetic field inhomogeneities is what's known as T2* or free induction Decay, and that happens much more rapidly than T2.
Now, these spins, they've lost phase with one another, so there's no longer any transverse magnetization. The net magnetic moments of those spins, though, they're still in the 90-degree plane, or they may have lost some of that angle. There is still magnitude to those vectors. If there's some way that we can cause those spins to re-phase, they will start to regain some of that transverse magnetization. That free induction Decay is a loss of transverse signal purely because of de-phasing. We are not losing it because those spins have now gone into the longitudinal plane. That's a really important concept to remember. That's why T2 decay happens so much faster than the longitudinal relaxation or T1 relaxation. T1 relaxation takes a lot longer for the spins to ultimately lie back in parallel with that main magnetic field. That's why TR is often so much longer than TE.
So, let's look at transverse Decay. Take three different tissues: CSF, fat, and muscle. Now, if this was pure spin-spin interaction, this would be known as T2 relaxation or transverse Decay. Now, once the transverse signal has lost 63 percent of its magnetization, that is what's known as the T2 time constant. And we can see that the T2 values for these three different tissues will have different time-based values. That T2 value is a constant: how quickly are those tissues losing their transverse magnetization?
Now, that loss of transverse magnetization is because of the dephasing of those spins. These spin-spin interactions in CSF are much less than the spin-spin interactions in fat and in muscle. And it's this that provides us contrast within our image. If we do a TE that's based, say, at this period of time, we can see that the signal from muscle will be much less than that of fat, and the signal from fat will be less than that of CSF.
Now, the problem I've mentioned is that local magnetic field inhomogeneities cause the signal to be lost much quicker than the T2 constant, and that's what's known as T2* free induction Decay. And you can see here on this graph just how much quicker that happens. And this again occurs because of local magnetic field inhomogeneities, either from our machine (it's hard to make a machine that has a perfectly uniform magnetic field) and when you place anything into the magnetic field, like a patient, there are different molecules and atoms within that patient that are going to ever so slightly distort that magnetic field, like we saw in our chemical shift talk.
Now, you might think, how big do these magnetic field differences need to be in order to cause such a drastic loss of transverse magnetization? Why did the spins de-phase so quickly? Well, if we look at the gyromagnetic ratio of hydrogen, if we were to place hydrogen within a one Tesla main magnetic field, those hydrogens will precess at a frequency of 42.58 million Hertz, 42.58 megahertz. And we can use this equation here to look at two separate spins. The first is experiencing the main magnetic field, and the second is experiencing a magnetic field that is one millionth of a Tesla difference.
If this spin is precessing at 42.58 million Hertz per second, this spin, instead of precessing at 42.58 million Hertz, it's going to have an extra 42.58 precessions per second. It's going to be spinning, or it's going to be precessing, at a frequency that's 42.58 Hertz more. Now, we're talking about million Hertz, so that difference is ever so small. We can use the rotational frame to compare how is this spin spinning in comparison to this spin. If this spin stayed still as a rotational frame, and we just looked at this spin relative to this spin, this spin will precess 42.58 Hertz more per second.
Now, all it has to do is precess 180 degrees more to be completely out of phase with the spin. We flip them into the transverse plane; they're both precessing in the megahertz, millions of Hertz range. But we know that every second this spin is precessing 42.58 Hertz more than the spin. It only takes a very small amount of time for the first time for these two spins to be completely out of phase from one another. And it turns out, if you use these equations, we can see that the time taken for them to be out of phase with one another is less than 12 milliseconds. It's an extremely small amount of time, and that is caused by the most minute difference in main magnetic field. That's why we get this free induction Decay, and it's very difficult to measure that signal so quickly, in less than 12 milliseconds, at a TE that's going to give us accurate measurements. And not only that, you can see now that the contrast between these two tissues has been greatly reduced because of this free induction decay. These spins have de-phased; we need a way of rephasing them and measuring it at TE, and that is the basis of spin Echo pulse sequences.
So, let's go to an example here where we're looking at some form of tissue. We've applied a B1, we've flipped those spins into the 90-degree plane, and we allow them now to lose phase over time. Now, this loss of phase is the T2 relaxation, spin-spin interactions causing this dephasing, and that actually happens over a very long period of time, especially if we're looking at water in CSF. That dephasing, that loss of transverse magnetization, is very slow.
Now, I've said to you that that loss of transverse magnetization happens much quicker in the real world. Free induction decay occurs as local magnetic field inhomogeneities cause that signal to be lost very quickly because of that rapid dephasing of those spins.
Now, if we were to look at two separate examples here, the one on the left here is the laboratory frame, how we've been looking at these spins spinning from the outside. The one on the right is the rotational frame, which is sometimes more easy to conceptualize when looking at spin Echo. In the laboratory frame, as we've seen, now those spins are going to de-phase over time, mainly due to those local magnetic field inhomogeneities. The rotational frame is going to show you how the spins de-phase relative to the Larmor frequency, the Larmor frequency at this specific location. How are some of the spins going to de-phase slower, some of them de-phase faster, because of those slight differences in precessional frequencies based on those local magnetic field inhomogeneities?
Now, as we play this now with those local magnetic field inhomogeneities, we are losing signal rapidly compared to how we were losing signal when it was only spin-to-spin interaction. See now how we've got fanning out of that rotational frame; those spins are de-phasing relative to one another. The laboratory frame, those spins have either gone faster or slower depending on the local magnetic field. Now, we need a way to re-phase those spins to allow re-accumulation of phase and give us a better transverse magnetization signal.
Now, this is what's called a spin Echo, where we regain signal, and that signal is ever so slightly less than our true T2 signal. What we've done here is accounted for those local magnetic field inhomogeneities, and the loss of signal at this time period is generally only due to spin-spin interactions or molecules that have moved within the tissue during this period of time and have experienced a slightly different magnetic field strength. Then, if we were to allow this to carry on for a further period of time, we will again lose signal at the free induction Decay.
So, let's look at exactly how this happens. We're going to be looking at the rotational frame when we look at these examples because it's much more easy to conceptualize, at least for me. So, let's now look: we've flipped our spins into the 90 degrees. Here we're looking from the side; here we're looking end-on. As we allow those spins to then de-phase, we are losing phase, we're losing transverse magnetization because of those local magnetic field inhomogeneities. We've said that this happens really quickly, in a matter of milliseconds, and it's very difficult to measure any good signal here at TE. We saw how rapidly that dropped off, and even if we did measure it, there wouldn't be much contrast between the tissues.
What we can do is, as we look at this end-on, we can apply the 180-degree RF pulse. Now, when we are flipping spins into the 90 degrees, we actually need to think of the sample as containing spins in the parallel and spins in the anti-parallel direction. And as we apply that RF pulse, those spins can gain transverse magnetization. Applying an RF pulse that's either stronger than this 90-degree RF pulse, or applying the same strength but for double the period of time, will allow that net magnetization vector to surpass the 90-degree angle and actually form in a higher energy state in that anti-parallel side of our longitudinal magnetization. That's how we can get past the 90 degrees in our 90-degree RF pulse.
If we were to apply the 180-degree RF pulse here, look what happens to these spins that have de-phased relative to our Larmor frequency. Some have de-phased slower, and some have de-phased faster, like this. These spins down here are our leading spins; they've de-phased faster. These here are lagging spins. Let's apply a 180-degree RF pulse, and what that does is it spins that magnetization vector 180 degrees along that XY plane. Now, these were our leading spins; they're de-phasing faster. These were our lagging spins. Also, because we have now flipped these spins a full 180 degrees, spins that were precessing in, say, the clockwise direction, as we've now flipped them, are going to be spinning in the anti-clockwise direction. So, these leading spins are now going to be lagging behind the lagging spins. But because they are dephasing faster, they're experiencing a higher local magnetic field strength, they are going to re-phase again as time passes by. You can see that those leading spins now became the lagging spins and then re-phased.
We can now measure that signal at TE, our time to Echo, and we can see how much higher that signal is. And that signal now is much closer to the true T2 signal, the signal loss because of spin-spin interaction. We've accounted for those local magnetic fields inhomogeneities. We're not reproducing signals here; we're not adding more energy into the system. Technically, what we are doing is allowing those magnetization vectors to re-phase, and then that accumulation of phase is what's giving us that signal. I hope this makes sense as to how we are accounting for those local magnetic field inhomogeneities. After all, it's those inhomogeneities that are causing this rapid dephasing. This spin here, based on its location, is experiencing a different magnetic field strength than this spin here. And because that magnetic field strength is different, it's rapidly dephasing with the other spin. Once we apply that 180-degree RF pulse, as long as these spins are in the exact same location, this spin is still going to rapidly de-phase, but now the dephasing because of this change in orientation actually happens to be a re-phasing, and that is the basis for spin Echo production.
Now, once we've generated that spin Echo and we've measured this signal here, this analog signal that we've measured, we've converted it to a digital signal. We then place that digital signal within one line of K-space. We then need to wait till TR so we can redo this entire process and fill another line of K-space. We've got many lines of K-space to fill, and not only that, for each line of K-space, we repeat it multiple times and get multiple signal averages. And then K-space only represents one slice of our patient. We still need to do that for all the slices within the patient. You can see how if we're waiting a second to three seconds for TR, this is going to start taking a very long time to take our MRI image.
So, if we think about acquisition time, we've seen that the time to take the total scan is the time from our first RF pulse all the way to TR. What is our TR value? Then each phase encoding step is going to fill a different line of K-space, and the number of phase encoding steps is going to give us our resolution in the y-axis, but each one requires a repetition of this entire cycle. The NEX represents the number of excitations, or the number of signal averages, that we do for each line in K-space. Every time we add another excitation, we add a full TR onto our scan time, so we need to account for the number of excitations. Then we need to do all of these steps for every single slice, so we need to multiply that by the number of slices. That's going to give us our total acquisition time.
Now, the three different pulse sequences that we're going to look at here that utilize spin Echo pulse sequences, we're going to see how this downtime between TE and TR is going to be utilized to reduce this total scan time.
Now, before we move on to these pulse sequences, you may have noticed some subtle changes here. Our dephasing frequency encoding gradient now lies before the 180-degree pulse, and this is a common site for this frequency encoding gradient to be placed. But you see now it's no longer below this line; it's above the line here. Now, why is it here? Previously, we placed the frequency encoding gradient below this line, a dephasing gradient prior to our data acquisition time when we applied the frequency encoding gradients along the x-axis. Now, we apply that dephasing frequency encoding gradient to allow those spins, despite having different frequencies along the x-axis, to re-phase at TE and de-phase again, allow us to increase signal and decrease signal, something we covered in the frequency encoding talk. If we were to do that before the 180-degree RF pulse, we would require that frequency encoding gradient to be in the same direction along the x-axis as when we reapply it here at our time to Echo. And that's because this 180-degree RF pulse causes a flipping of those net magnetization vectors. We can apply our phase encoding gradient prior or after this 180-degree RF pulse, and I've illustrated it here to show you that we can use those two separate timings.
So, let's get into our first pulse sequences, what is known as the multi-echo spin Echo Imaging. We've done exactly what we've looked at in this talk so far: flip the spins to 90, allow them to de-phase, flip them 180 degrees, allow them to re-phase, giving us an echo at TE, and we read out that echo during the frequency encode gradients. We then got all this time to wait until our TR. What we can do here is apply another 180-degree RF pulse.
Those spins, when we first flipped them to 90 degrees, started to de-phase. We flipped them 180 degrees, and they started to re-phase again. Boom, we read out our signal at this point. Now, they will de-phase at that free induction Decay again. They've still got magnitude; we haven't lost the transverse signal. It hasn't flipped into the longitudinal plane. We've seen that takes a very long time for T1 relaxation to occur. What's happened after that first 180-degree RF pulse? Once they've re-phased, given us our Echo, they are dephasing again at T2*. We can apply another 180-degree pulse, and that again is going to cause that rephasing. We can generate another Echo here and measure out a separate TE. Now, this TE is going to have slightly less signal than our first TE because the spin Echo is only accounting for the local magnetic field inhomogeneities. We can't do anything to recover more than T2 decay. T2 decay, true T2 decay, is going to happen no matter what. That spin-spin interaction is going to cause loss of transverse magnetization that is unrecoverable, not even by spin Echo Imaging.
Now, what have we done here? We've created two separate data acquisition points that have different TEs: one a very short TE and one a slightly longer TE. Now, which sequences use a very short TE? Well, a very short TE is used in proton density imaging because we haven't allowed those differences in T2 relaxation to occur, and in T1 weighted imaging because we don't want differences in T2 to provide contrast to our image. The second TE here is a longer TE, like we're using T2 imaging. We've allowed a longer period of time for that true T2 relaxation to occur, and we've separated the contrast in our image based on those T2 values. We can now place this data acquisition into one K-space and this data acquisition into a separate K-space during this one pulse sequence.
Now, we've created one K-space for this specific slice at this phase encoding gradient. We filled one part of K-space here. Our TR is going to be long; we are waiting a long TR, and that's what we want in proton density imaging. We want those spins to regain their full longitudinal relaxation before then flipping them again into the transverse plane. So, a short TE and a long TR is going to give us a proton density weighted image in this K-space. And when we fill this K-space with our second data acquisition, we've filled now this line here. It's the same slice that we've selected with the same phase encoding gradient. The only thing that's changed is our TE. That slightly longer TE is going to bring out those T2 differences in tissue, and we are going to generate a slice that has T2 weighting, or more T2 weighting, in that image. You can see how we've utilized this downtime to now create two separate images.
Now, this is a little bit of an old technique. With the development of FLAIR, which we're going to look at when we look at inversion recovery sequences, we generally don't use proton density weighted imaging, especially when we're looking at brain imaging. But this was a good way to see if there were lesions close to water within the brain. We could acquire a T2 image, and if the lesion was T2 bright, it's quite difficult to see the difference between the CSF and the lesion. And this proton density image would allow us to tease out some of those differences. No longer really used because of FLAIR imaging, which we're going to look at later, but this is the first way that we can utilize some of that spare time. We can then repeat the sequence over and over again. The only thing we're changing is the phase encoding gradient here, and as we change the phase encoding gradient, we're going to fill the rest of K-space.
Now, the next pulse sequence we're going to look at is a much more common pulse sequence. We start again with exactly the same sequence here. What are we going to do with this extra period of time? Now, the slice that we have selected here, we've only caused a very specific slice along our patient to then be flipped into the transverse plane. All the other protons within our patient are still just experiencing the main magnetic field. They weren't precessing at the frequency of this RF pulse that we flipped into the 90 degrees. We've then generated our spin Echo, we've measured that spin Echo from that specific slice. The rest of the tissue is just waiting around, precessing at the Larmor frequency. Again here, the only magnetic field that's on is the main magnetic field.
What we can do in multi-slice spin Echo Imaging is apply a different RF pulse with the same slice selection gradient. We are going to create a gradient along the z-axis that causes the spins to precess at different frequencies along that z-axis. We can select a different RF pulse with a different RF bandwidth to select a different slice. While this is happening, that first slice is still relaxing, waiting for it to get all its longitudinal relaxation before we flip it again. What we're doing here is, firstly, we have selected one specific slice, we have excited that slice, we flip those spins into the 90 degrees, we've generated our spin Echo, and we've measured that signal. In the second signal here, we are using a different RF pulse, a different radio frequency pulse. That different frequency is going to then account for a different slice within our patient. These spins in that first slice are still regaining their longitudinal relaxation, and we can now, at the same TE (this TE here is the same as this TE here), so we've got the same TE but on different slices. We can now fill two separate K-spaces with these two Echoes, and we can repeat this process for multiple different slices while we're waiting for that next TR to occur. The TR here is going to correspond to this first slice. Once we repeat this process again, we are going to get a separate TR that's going to correspond to this slice. The TE and the TR for each slice will be exactly the same, and they are filling different K-spaces. Remember, K-space encodes for a specific slice in our patients.
So, let's look at what that looks like. We are again filling two separate K-spaces. In multi-echo imaging, those two K-spaces represent a different weighting. In multi-slice spin Echo, we are representing the same weighting here. We've got the same TE per slice and the same TR per slice, but these two K-spaces are encoding for different slices. Now, in multi-echo imaging, with different weighting for the same slice, in multi-slice spin Echo Imaging, we've got different slices with the same weighting. There's a subtle difference between the two.
Now, we don't want to excite slices that are exactly next to each other because when we looked at that RF bandwidth, remember there was a range of frequencies in that RF bandwidth, and there's some overlap between slices here. So, what we can do is, as we're looking at our patient, we can select slices that are further apart from one another. And as we select slices that are further apart from one another, when we repeat the cycle again, we can choose slices that lie between those slices, and afterwards, we can combine those signals. This is what's known as interleaving of our signals. We take all those separate K-space data points and put them in their correct positions corresponding to the correct slices in our patient. You can see how this is going to drastically reduce the amount of time it takes to acquire our MRI signal, and the number of slices that we can put into a single TR is going to correspond to the total reduction within our scan time. So, multi-slice Imaging allows us to use that downtime to then image other slices within our patient before combining all of that to generate our whole MRI image that we can scroll through from slice to slice.
Now, the last spin Echo sequence that we're going to look at is what's known as fast spin Echo imaging. Now, again, we've started all these sequences the same: apply the 90-degree RF pulse, allow them to de-phase, flip them 180 degrees, allowing them to re-phase, creating an echo, and that echo is then sampled and placed in K-space. We've now created a sample that's going to correspond to this region in K-space that corresponds to this specific phase encoding gradient. We can then apply another 180-degree RF pulse, like we did in our multi-echo Imaging, but we can see that something different is happening here in our phase encode gradients.
We've applied an equal and opposite phase encoding gradient to the initial phase encoding gradient that we used prior to the 180-degree RF pulse. Remember, when we apply a phase encoding gradient, that phase encoding has memory throughout the sequence. Once we've de-phased them, we turn off the phase encoding gradient, and those spins have the same frequency because it's exposed to the main magnetic field; they have the Larmor frequency. In fact, dephasing has had memory in our slice. If we apply an equal and opposite phase encoding gradient, that is going to re-phase that phase encoding gradient, cancel out the effect of this initial phase encoding gradient. We can then use a different phase encoding gradient, and the echo that we generate here is going to be a different line of K-space. We haven't selected another slice; we've just created another Echo. They de-phased, flipped the 180, re-phased. Then, after that 180, they've again started to de-phase at T2*. Flip them 180, re-phase, getting this Echo here. This Echo, and when we measure out the signal here, is going to correspond to a different phase encoding gradient, a different part of K-space. We can then repeat this again, an equivalent opposite phase encoding gradient, a new phase encoding gradient, to give us another Echo at TE that's going to correspond to a different part of K-space. You can see now that these Echoes are filling different lines of K-space.
Obviously, the signal is going to decrease over time because of that T2 relaxation, and the contrast throughout K-space is not going to be identical. Now, remember, contrast predominantly comes from the middle of K-space, so we can start filling K-space initially with small phase encoding gradients, allowing us to get good contrast within our image, before then filling out K-space with higher and higher phase encoding gradients to give us that detail in our image.
Now, why would we do this? We're obviously going to lose some of that classical contrast that we want, and we're going to lose signal the more and more TEs we cram in between our first 90-degree RF pulse and the TR. Well, it drastically reduces the amount of time it takes to fill K-space here. And the number of Echoes that we have within our pulse sequence between the first RF pulse and TR is what's known as the echo train length, the ETL. The number of Echoes here, and here I've just included three Echoes. If we had an echo train length of three, that would reduce our total scan time by a factor of three. This can lead to a drastic reduction in total scan time when we use hundreds of Echoes between the first RF pulse and the TR. And that's what we can do in fast spin Echo; that's why it's fast. I've got multiple Echoes within one single TR that's filling one slice, one K-space here.
Now, of course, this is going to come with some consequences. We are going to get a reduction in signal-to-noise ratio, and we are going to get a change of contrast as we are getting this T2 relaxation with time. Each Echo is only going to represent the degree of T2 relaxation at that period of time for that specific tissue. So, the Echoes at the end of our sequence, nearing the TR here, are going to have very little signal, very little transverse magnetization left. Now, there are certain tissues that keep their transverse magnetization for quite a long time, like fluid. And when we are creating images like an MRCP where we're only interested in water, these can be really useful because that water retains its signal for such a long period of time because of its long T2 relaxation times.
Now, we can go one step further and make this even quicker. When we looked at K-space before, we saw that K-space had what's known as conjugate symmetry. We can separate K-space, flip one half of K-space, and we see that they're exactly symmetrical here. This tells us that we could probably get away with only filling half of K-space. So, we can create an image where we only fill half of K-space, and the order in which we fill that K-space will also have consequences for the contrast that we're going to get in our image. We've reduced the time it takes to create that single slice by another factor of two, and we can get really quick images even though we're using spin Echo sequences.
Now, spin Echo sequences can't be used for every single application. When we try and create T1 weighted images, we want really short TE times to negate that T2 differences in tissue. And really short TE times are often quite difficult within a spin Echo sequence because we need time to apply the 90-degree RF pulse and apply the 180-degree RF pulse. There are other sequences that allow us to flip spins at smaller flip angles, not using that full 90-degree flip angle, and allow us to retain that longitudinal magnetization that then gets flipped into the transverse plane and give us better T1 weighted images. And that's what we're going to cover when we look at gradient Echo signals. We're then going to round this off by looking at inversion recovery, seeing how we can null signal coming from a specific tissue because we know that T1 and T2 relaxation rates.
Now, this is a complicated and long talk. I would encourage you to go over these concepts to make sure that you have them in your head. I hope I haven't lost you here. Let me know if you found this useful in the comments below. Let me know if you've learned anything today. And as always, I will have a question bank with curated questions linked in the description. You can go and test your knowledge on these concepts. So, until next time, goodbye, everybody!