Transcription
Hello everybody and welcome back. So let's pick up from where we left off in the last talk and then expand on some of the concepts that we looked at there.
Now, as we've seen, we can select a field of view, a predetermined area that we want to accurately image within our MRI slice. And we can further subdivide that field of view into what's known as a matrix. Now, the matrix size determines the number of pixels within our image. And as we've seen, the x-axis pixels, or the number of pixels along the x-axis, determines the number of times that we need to sample the signal during the frequency encoding gradient during that data acquisition period, so that we can do that one-dimensional Fourier transform and separate that signal into x-axis location. We've seen that the number of pixels in the y-axis determines the number of phase encoding steps that we need to include in our sequence, and that gives us our y-axis resolution.
Now, once we've selected the field of view and the matrix size, we then select a value known as bandwidth. Now, bandwidth represents the range of frequencies along the x-axis of our slice, and it's a function of the field size as well as the gradient strength. Now, previously we looked at bandwidth, but we didn't really look at how it ultimately affects our image, and that's what I want to focus on here today. We're going to look at how bandwidth ultimately affects the signal-to-noise ratio, and what are the positives and negatives for changing the bandwidth in our image.
Now, if we take an example like this of 30,000 Hertz, we can represent that bandwidth in both negative and positive values here. We can represent it as positive 15,000 Hertz and negative 15,000 Hertz. And we've seen that when we're measuring a signal from an entire slice, that signal is said to be a complex signal. It's made up of both real and imaginary vectors, and that just means that we are detecting change in frequencies that are both positive and negative, depending on the gradient that is applied along the x-axis of our slice. And when we're measuring these frequencies, we're not measuring the absolute frequency, we're measuring frequency changes relative to other frequencies within the x-axis.
Now that we've selected our field of view and the bandwidth, we can use this equation, or at least the MRI machine can use this equation, to ultimately calculate the gradient strength that would create this bandwidth across this 25 centimeters in this slice. Now, that gradient strength is then applied to our slice, and it will create what's known as the Nyquist limit, the frequency, the maximum change in frequency that we need to be able to accurately sample. Remember, we're turning an analog signal into a digital signal, and we need to sample that analog signal frequently enough in order to accurately represent what the actual frequency of that sample is.
Now, the Nyquist limit can help us determine the sampling rate, the rate at which we need to sample that analog signal to accurately represent the frequency of that analog signal. And as we can see from this equation here, our sampling rate needs to be twice that of the Nyquist limit. We need to sample that frequency at a rate that is twice the frequency itself. So our sampling rate is twice the Nyquist limit. It's twice this value here, 15,000 Hertz in this case. And as we can see, sampling rate and bandwidth have the same numerical value. And as we looked at in the previous talk, although they have the same numerical value, they're not describing the same thing. But when we use them in calculations, we can use them interchangeably because of that same numerical value.
Now that we know our sampling rate, how many times we need to sample per second, we can work out how much time we have for each one of those samples. If we take one second and divide it by our sampling rate, we'll get what's known as the sampling interval, the amount of time we have for each one of those data acquisition points within our sample acquisition. Now, we also have selected a matrix, so we know how many pixels we need along the x-axis of our image. So if we times our sampling interval, the time it takes to take one sample, by the number of pixels that we need, we'll get the total amount of time that it's going to take to sample the image within the frequency encoding gradient.
Now, all of these concepts are really important when we look at how changing bandwidth is going to change our signal-to-noise ratio later on. Now, when we talk about bandwidth, we often talk about it as a narrow or a broad bandwidth. The narrower the bandwidth, the less the difference in frequency along the x-axis here. So a lower value in Hertz of bandwidth means a narrower bandwidth. As that range of frequencies get bigger, as our gradient gets steeper, or as the field of view gets bigger, our bandwidth increases, this Hertz value increases. And as the bandwidth changes, we'll see that the sampling rate, the rate at which we need to sample to accurately represent where those frequencies are on the x-axis, will also change proportionally to that bandwidth.
So let's have a look at an example here. We've seen these sequences now. You'll be accustomed with what we're doing, what these lines represent. Now, this blue line represents our frequency encoding gradient that we apply along the x-axis of our slice. Now, until now, we've been looking at the net magnetization vectors processing at a specific frequency. And the frequency that they've been processing at, we're looking at them in what's known as the laboratory frame, what's the actual value of the precessional frequency. And those are in the realms of 64 million or 41 million Hertz. It's a really rapid rate. If we were looking at what's the absolute precessional frequency of these spins, what we can do is look at these precessional frequencies relative to one another. We can look at it in what's known as the rotating frame. We can pick a specific subset of spins and use that as our baseline, and then see what the difference in frequencies are depending on where they are along the x-axis.
So as these spins here are spinning at the Larmor frequency, we can then see if spins to the right of it here are spinning faster than it, how much faster are they spinning, how many more revolutions or precessions are they having per second. And that's what this value is here. Relative to the center of our image, these spins on this side of the image are going at 15,000 Hertz faster, 15,000 precessions more than the Larmor frequency here. Now, why are we comparing it to the middle of our slice? But when we apply the frequency encoding gradient, you can see the center of the frequency encoding gradient has no positive or negative additional magnetization. It's processing at the Larmor frequency.
Now, we've also seen that applying a frequency encoding gradient, we first apply a dephasing frequency encoding gradient. Now, the amplitude of these frequency encoding gradients shows us how strong that frequency encoding gradient is. And as we've seen, as we change the bandwidth, the gradient needs to change. The steeper that gradient, the higher this amplitude of this visual representation of the frequency encoding gradient will be. And we're going to look at that in this talk. The x-axis value determines the amount of time that we are applying this frequency encoding gradient. We first start by dephasing with a frequency encoding gradient, by applying an equal and opposite frequency encoding gradient here. That allows us, when we measure the signal out during our rephasing frequency encoding gradient, to have signal that is building up as those spins start to precess more and more in phase. Because we've allowed them to dephase first, they build up to a maximum at TE, and then they start to dephase again because of those differences in frequencies.
Now, if we apply this dephasing gradient here, these spins on the left are going to spin faster relative to these spins at the no point in the x-axis here. These spins on the right are going to spin slower relative to these spins. Now, watch what happens. We are going to illustrate this slightly different to what we've been illustrating it before. Now, I'm showing you the relative spins, not the absolute spins. We're looking at the rotating frame, not the laboratory frame. So let's apply this frequency encoding gradient, the dephasing frequency encoding gradient. You see how these spins are spinning faster than these central spins? Now, the central spins are staying still because that's what we are comparing the changes in frequencies to. We're comparing it to the central Larmor frequency. So compared to itself, it's not moving. We are using that Larmor frequency as our net zero spin, and we're comparing the changes in frequency along the x-axis.
Now, as we apply the rephasing frequency encoding gradient, it's going to be an equal but opposite frequency encoding gradient, but we're going to apply it for twice as long. Now, if we were to just apply this rephasing frequency encoding gradient without this dephasing gradient here, we would start in phase with one another, and they would rapidly dephase as we applied this frequency encoding gradient, and our signal would rapidly decline here. Applying this dephasing gradient first allows for the signal to generate up before dephasing again. Instead of giving us half the signal, we are now measuring the signal as it goes up and measuring the signal as it goes down. It gives us more time to sample the signal coming from this slice.
Now, we know the bandwidth here, it's 30,000 Hertz. We've selected that bandwidth. It's a relatively narrow bandwidth. We can use that bandwidth then to determine the sampling time, the amount of time we need to sample during this frequency encoding gradient. The bandwidth will allow us to determine how long we need to apply this frequency encoding gradient. Now, how is that the case? Well, think of each one of these squares as a data acquisition point. Now, I haven't done 256 data acquisition points because I can't fit them in here. This is just a visual representation. Now, we need to sample the signal coming from the entire slice 256 times if we want 256 pixels within the x-axis. We need to work out then how much time we have for each one of these samples. And the way we work that out is by figuring out what sampling rate do we need.
Now, the sampling rate we need will determine whether we can accurately represent this 15,000 Hertz, the Nyquist limit. Now, we've seen that the sampling rate needs to be twice that of the Nyquist limit, or the sampling rate equals our bandwidth. So to figure out how much time we have for each one of these points, the sampling interval, we can take one second and divide it by our sampling rate. The sampling rate is how many times we are sampling in one second. So if you divide one by that number, we'll get how much time we have for each one of those samples.
Now that we know how long it takes for each one of these samples, one over sampling rate or one over bandwidth, one over 30,000 Hertz, we can see that the sampling interval, the amount of time we have, is going to be 33 microseconds. Now, we need to take 256 different samples. So we can then work out how long it will take to acquire all of the data that we need. Take 33 microseconds and times it by the number of samples we need, 256. We get 8,448 microseconds, or 8.448 milliseconds. Now, that's quite a long time when we're looking at specific pulse sequences. We've seen TEs that are much shorter than this value here. So you can kind of see that there are certain bandwidths that won't allow us to use this long sampling period.
Now, when we are measuring a signal from an entire slice, some of that signal is going to be known as background noise. If you think of noise, your environmental noise, there's a certain level of background noise going on all the time. And if you are 10 meters away from me and I whisper, and I don't whisper loud enough to overcome that background noise, you're not going to be able to hear what I'm saying. I will need to raise my voice to a specific level for you to be able to hear my voice over that background noise. The same thing is happening when we are measuring MRI signal. Some of that signal will be true signal coming from our slice, and some of it will be noise generated from the patient, from the inhomogeneities in the magnetic field, from the machine itself. Some of the signal is just going to be background noise.
So when we measure the signal over time, we apply this frequency encoding gradient for this period of time. This is what the signal is going to look like. See how the spins are rephasing because of their differences in frequencies, and then dephasing again because now these spins are spinning faster relative to these spins because of this encoding gradients along the x-axis. Now, this orange line here represents the background noise, and this blue line represents the true signal coming from our slice. The time it took us to acquire the signal, we've calculated here, and it's based on the sampling rate, which is a function of the bandwidth that we've selected. Now, we know this period of time, so we only apply that frequency encoding gradient for that period of time. And because this is a narrow bandwidth, the range of frequencies isn't that great along the x-axis of the slice, those frequencies are more in phase for a longer period of time. There's a shallow rise to our signal and a shallow decline.
Now, let's look what happens when we make a larger frequency encoding gradient and we increase the bandwidth. We have a broader bandwidth. Our bandwidth here is 60,000 Hertz. We've doubled the bandwidth. Now, as you can see, doubling the bandwidth means we need a stronger gradient. The amplitude here is much higher. We're applying a stronger gradient, and we need to apply the stronger gradient for a shorter period of time. And we're going to work out exactly how long we need to apply that frequency encoding gradient. So again, we apply the dephasing frequency encoding gradient. It's stronger now because we've selected a larger bandwidth. These spins are going to spin faster than the central spins, relatively speaking, and these spins are going to spin slower. So again, we're going to get this dephasing of the spins based on this frequency encoding gradient, causing these spins to precess at different frequencies.
Now, we're going to apply the frequency encoding gradient, equal and opposite in direction. These spins are going to catch up, they're going to rephase with all the spins before dephasing again. But this frequency encoding gradient is much stronger. That rephasing is going to happen quicker, and the dephasing is going to happen quicker. Again, we're acquiring 256 samples, but we can work out how much time we have to acquire each one of those samples. And that time is going to be shorter. We've seen that this time it takes, the amount of time we have for each digitization of that analog signal, is one over the sampling rate, and our sampling rate is equal to our bandwidth.
Now, in this example, the sampling rate is 60,000. In the previous example, it was 30,000. We've gone much less time to take each sample here, and it makes sense. We need to sample more rapidly because we're trying to detect a higher frequency, a higher Nyquist limit. So the time taken here is 16.6 microseconds, and we can times that by the number of pixels in our x-axis and see that the time now that we need to apply this frequency encoding gradient is 4.295 milliseconds, a much shorter period of time. So in sequences where we need this TE to be really short, and we can't allow this data acquisition to overlap with a 180-degree pulse, we might need to use a broader bandwidth so that we can use a smaller or a shorter sampling time.
Let's look at the signal that's going to be generated from this slice. So background noise is going to be the same, but the signal is going to be different because the frequencies in the slice are different. The rate of those frequencies rephasing and dephasing is going to be different. So let's look now as they rephase, we get an increase in signal and then a sharp decrease in signal. Now, hopefully you can see that firstly, the data acquisition period is much shorter because our sampling rate needs to be much faster. We're still taking the same number of samples as we did in the previous example, but we're taking them much more quickly. And we can also see that when we start taking those samples, the noise predominates, then signal takes over, and at the end of this data acquisition, noise predominates again. If this is me whispering here, and this is background noise, you can't really hear me over that background noise. In the previous example, our signal matched that of the noise. There's a better signal-to-noise ratio.
So let's compare these two signals that we've generated, one with a narrower bandwidth and one with a broader bandwidth. You can see that the narrower bandwidth took longer to acquire those signals because our sampling rate didn't have to be as high. We had more time to sample that analog signal, and because the range of frequencies wasn't as great, they stayed in phase with one another for longer and took longer to dephase. So our signal relative to the background noise over time was relatively high. We had a relatively high signal-to-noise ratio, and that's going to give us an image that looks like this. Now, some of the best places to look for background noise is in areas where there shouldn't be any signal being generated. Now, when there's no signal being generated, that should be purely black. The more noise we get in an image, the more mottling we're going to get in that black background. Here, that noise is going to be providing some sort of signal that we then use and plot onto our image.
Now, as we increase the bandwidth, we can see that the signal-to-noise ratio gets worse. The amount of signal here is far less than the noise, and we only have a short period of time where the signal is higher than the background noise. And it turns out that increasing the bandwidth, getting a broader bandwidth, reduces the signal-to-noise ratio. The narrower the bandwidth, the better our signal-to-noise ratio. So if we were to acquire this sample like this, we would have an image that started to look a little bit more like this. It had some background noise, see the mottling now within the black in the background here.
Now, we may be required to use a broader bandwidth because we want to take pulse sequences that have very short TEs. The shorter the TE, the shorter the sampling time we may need. In some sequences, that data acquisition period can't be too long because our TEs need to be very short. So when we increase our bandwidth, what exactly does that do to our signal to noise? Well, we can use this formula here. Signal-to-noise ratio equals one over the square root of the bandwidth. So as bandwidth increases, our signal-to-noise ratio will decrease. The amount of signal that we have to the amount of noise we have will decrease. Our image quality will get a little bit worse. As we narrow the bandwidth, as we reduce the range of frequencies along that x-axis, our signal relative to the noise will get better. Our signal-to-noise ratio will increase. So if we halve our bandwidth, we are going to get a 40% increase in our signal-to-noise ratio.
Now, why don't we use really narrow bandwidths all the time? Surely we just want to use the narrowest bandwidth possible, get the best signal-to-noise ratio possible? Well, we've touched on one factor. We can't do very short TEs because this will become way too long when trying to acquire the signal. The second reason is when we have a high gradient, a high bandwidth, and think about one pixel along that x-axis, we have a really high gradient. The frequency differences just along one pixel will be quite big. The narrower our bandwidth, or the lower the range of frequencies along that x-axis, the frequency differences along one pixel will become less and less. And there's an artifact that we're going to look at later called the chemical shift artifact, where we look at water and we look at fat, and we see that because of the local conditions around water and around fat, the hydrogens in water and fat actually precess at slightly different frequencies despite experiencing the same external magnetic field.
Now, if we have a high gradient, we've got quite a large range of frequencies within one pixel, so we can cover those differences in frequencies and still plot them in the correct pixel. When we have a narrow bandwidth, a low range of frequencies, each pixel represents only a very small change in frequency value. And because of those differences in fat and water, we will start to misrepresent where water and where fat lie on the x-axis. Now, don't worry about getting into this. We're going to dedicate a whole talk looking at the chemical shift artifact. It also turns out that the lower, or the narrower, our bandwidth, the worse that metal artifact is within our image. So it comes as a trade-off. Selecting the bandwidth, reducing it or narrowing it, is going to give us better signal-to-noise ratio, but that comes at a cost: more metal artifact, we're going to have more chemical shift artifact, both of which we're going to look at later, and we can't use very short TE acquisition sequences.
So I hope these talks have been useful to you. There are a lot of points to go over here, but they're very practical points. These are things that we actually need to select. How does changing our field of view change the gradient and change the bandwidth? How does choosing the matrix size change the number of samples that we need to take, or the number of phase encoding steps that we need? Why do we select specific bandwidths, and what does selecting that bandwidth do to our image? Ultimately, these are all questions that hopefully you can answer now, and these are the type of questions that come up over and over again in exams. And I've looked at multiple different past papers and seen which questions in MRI come up over and over again. I've collated all of those questions, ranked them by the frequency that they come up, and then I've scooped off the most frequent questions, curated them in a question bank, and then answered them in video format. So if that's something that will interest you, go and check out the link in the description below. Test yourself on those questions. Otherwise, I'll see you in the next talk where we're going to look at aliasing artifact, which is closely related to the Nyquist limit that we've been looking at in the previous two talks. So until then, goodbye everybody.