Transcription
Hello everybody and welcome back. So up until now, we've looked at how we generate signal. We've looked at how we localize signal within a specific slice, and we've seen how we store that data in order to ultimately generate an MRI image.
In this talk, we're going to be looking at a couple more of the nitty-gritties, some of the details in our pulse sequence that we haven't touched on so far. Now, so far, we've seen that we can select a specific slice, use a frequency encoding gradient to delineate the frequencies along the x-axis, and use multiple phase encoding gradients to figure out where the signal is coming from along our y-axis.
Now, in this talk, we're going to be looking at this data acquisition period, the time when we're sampling that analog signal and converting it into a digital signal. We're going to look at how strong the gradient along the frequency encoding direction needs to be, as well as how much time we have to take each one of those samples along that analog signal. Remember, as we're reading out the signal, we're applying a frequency encoding gradient along the x-axis, and we need to apply that gradient long enough in order to sample the signal enough times. And we've seen that the number of times that we sample the signal determines the number of pixels that we can put in our x-axis of our image. It determines x-axis resolution.
And so far, when we've been looking at a specific slice that we are selecting from the patient, we've been thinking of that slice as one large homogeneous piece of tissue. And you'd see that if we were imaging a patient like this, there's only a small area of that entire slice that we're interested in. And if we've got a set resolution along the x-axis, surely we want to narrow down just onto the tissue of interest. So if we were to take this slice here and look at it in more detail, we'd see that there's lots of spare space around the anatomy that we're interested in.
Now, we can define the boundaries of this slice using a distance value here. For example, we're using 50 centimeters by 50 centimeters. Now, it turns out these dimensions, we can actually select when operating the MRI machine, and these dimensions are what is known as the field of view. Now, you can see here for this specific field of view, 50 centimeters by 50 centimeters, there are regions where no signal is going to be coming back, no useful information is going to be stored.
When we look at the y-axis here, we know that in order to delineate signal along that y-axis, we need to add multiple phase encoding steps. And each phase encoding step means a repeat of the entire time to repetition interval. It adds time to our acquisition. And along the x-axis here, in order to get the number of x-axis pixels that we have available to us, we need to sample the analog signal a certain number of times.
It would make sense to reduce that field of view so that we only had the anatomy of interest within this region here. You can see now we've changed this field of view to 25 centimeters by 25 centimeters, using the same number of phase encoding steps and the same number of data acquisition samples during the frequency encoding gradient. You can see we would get much better resolution in this field of view than we would in this field of view.
Now, how exactly do we define resolution in an image? Well, the resolution is the dimension of individual pixels along the X and Y axis within the image that we're generating. So if we were to divide this field of view into an 8x8 pixel matrix, here we've got 64 pixels, eight along the X direction and eight along the phase encoding or Y direction. In this particular instance, we would need at least eight phase encoding steps, and we would need to acquire at least eight signals from our analog signal during that frequency encoding gradient.
If we were to use that same pulse sequence in this image here, we would still create an 8x8 matrix, but with a smaller field of view. We've used the same number of phase encoding steps, and we've used the same number of data acquisition samples during our frequency encoding gradient, but our resolution is much better. The pixels are smaller.
Both of these images here are said to have the same matrix size. Now, the matrix of an image is the number of pixels within that image, and here we've got an 8x8 matrix in both of these different field of views. Now, although the matrix is the same, we can see that the resolution is different, and that resolution difference is happening because of that decreasing in field of view size.
Now, we can change the matrix here. If we wanted the same resolution within these two images, we could change the matrix in this smaller field of view to have a 4x4 matrix. You can see now the pixel sizes are the same, the resolution between these two images are the same. However, to acquire this image here would take much less time than it would to acquire this image here because we've reduced the number of phase encoding steps to produce this image. So although this takes quicker time, we get the same resolution between these two images.
Now let's look at this particular slice here and discuss a topic that's known as bandwidth. Now, when we are imaging a particular slice, we apply a frequency encoding direction along the x-axis, and we then sample the signal that's coming from that slice. Now, when we apply a frequency encoding gradient along the x-axis, we get a range of frequencies depending on where the tissue or the signal lies along that x-axis. And the frequencies, as we've seen multiple times now, correspond to a specific x-axis location.
Now, the bandwidth is the range of frequencies that occur across this entire slice in the field of view that we've selected. And that range of frequencies occurs in the frequency encoding direction because that is where we are creating a difference in frequencies based on x-axis location. Now, in the center of this frequency encoding gradient, these spins here are going to resonate at the Larmor frequency. We haven't added or subtracted any additional magnetic field strength at the center of the frequency encoding gradient. And we've seen that the Larmor frequency is equal to the gyromagnetic ratio of the hydrogen protons, as well as the magnetic field strength.
In a 1.5 Tesla machine, we are going to have spins resonating at roughly 64 megahertz or 64 million Hertz at the center of our frequency encoding gradient. Now, the rates of precessional frequencies are going to change depending on the magnetic field strength along this x-axis. And as we apply that frequency encoding gradient, we'll see that these spins will resonate at a frequency higher at this end of the slice and lower at this end of the slice. And we can see that we've added an additional 50,000 Hertz or 50 kilohertz along the x-axis direction here, and we've subtracted 50 kilohertz or 50,000 Hertz along the x-axis as we head this way along the slice.
Now, the total bandwidth across this entire slice is going to be this 50 kilohertz plus this 50 kilohertz. Our total bandwidth, our total range of frequencies along the x-axis, is a hundred kilohertz or a hundred thousand Hertz, and that is what's known as the bandwidth. Now, I'm going to show you later that we actually select the bandwidth that we want, and the MRI machine calculates the gradient that's needed in order to generate this bandwidth.
If we were to reduce the field of view size here, we've now reduced that 50 centimeter by 50 centimeter slice to a 25 centimeter by 25 centimeter slice. What has that done to the bandwidth along the x-axis of the slice that we're interested in? You can see now the range of frequencies from the center of the frequency encoding gradient to the edge of our slice is now 25,000 Hertz, 25 kilohertz. Our bandwidth now is only 50,000 Hertz. We have reduced the field of view, and that reduction in field of view has reduced our bandwidth from a hundred thousand Hertz to 50,000 Hertz. So you can see that reducing the field of view will also reduce the bandwidth, the range of frequencies along the x-axis. And you can see hopefully that bandwidth now is somehow proportional to field of view. As the field of view increases, our bandwidth increases, and as the field of view decreases, our bandwidth will decrease.
Now, what would happen if we were to decrease the severity of this frequency encoding gradient, decrease the strength of the gradient along that x-axis? Look what happens to this slope of the gradient. As we decrease the gradient, you see how that slope has got certain more gradual across the x-axis, and look what's happened to the frequencies. Now, in the previous example, we had 25,000 Hertz difference between the center of our slice and the periphery. Now we've got a 12 and a half thousand Hertz difference. Decreasing the gradient or decreasing the magnetic field strength along the x-axis of our slice has also decreased the bandwidth. Now, the bandwidth from one end of our slice to the other end of our slice is only 25,000 Hertz. So we can see a reduction in field of view results in a reduction in bandwidth, and so does a reduction in gradient. So our bandwidth is also somehow proportional to gradient field strength. The stronger our gradient, the more the bandwidth is across that slice, the more the range of frequencies along the x-axis of our slice.
Now, it turns out that bandwidth is a function of both gradient strength and field of view size. As we increase gradients and as we increase field of view, we are going to increase the bandwidth, the range of frequencies along the x-axis. And it's this function here, times by a constant, that is going to determine the exact bandwidth across the slice.
Now, let's have a look at our initial example where we had a bandwidth of 50,000 Hertz. We had a field of view that was narrowed down to the anatomy that we wanted, and we had a corresponding gradient that would allow us to have this 50,000 Hertz gradient. If we were to look at that slice like we've been looking at it in the previous talks, we know that the gradients applied along the x-axis here is going to cause these net magnetization vectors to precess at different frequencies dependent on that gradient. Now, here, when we look at this 64 million 25,000 Hertz and the 63,975,000 Hertz, we are looking at the absolute precessional frequencies. And previously, when I've illustrated that, I've used the absolute precessional frequencies to show you how these net magnetization vectors are spinning.
Now, when we go about measuring the net magnetization vector from the entire slice, that analog signal, and we convert it into a digital signal, what we are measuring is not the absolute precessional frequencies, what is known as the laboratory frame of reference, as if we were looking at those spins from the laboratory outside of the MRI machine and measuring their absolute spin values. What we're measuring is the relative spin value changes along the x-axis slice. So here, we've got our bandwidth of 50,000 Hertz. From the center of our slice, where there is no frequency encoding gradient, out to the periphery, we get a change in precessional frequency in this example of 25,000 Hertz. When we compare a spin that is in the center of our x-axis and we compare that spin to one that is right at the edge of our slice, the precessional frequency differences will be 25,000 Hertz. This net magnetization vector is going to precess 25,000 times more in a second than the net magnetization vector at the center of our slice.
Now, when we are converting that analog signal to a digital signal and delineating those frequencies, and then plotting those x-axis values based on those frequencies, what we're actually measuring is this change in frequency. We're not measuring the absolute values, we're measuring the relative change in frequency. Here, it's as if if we were sitting on top of that net magnetization vector in the center of our slice, and we were holding that net magnetization vector on our arms and we were spinning, but we were only looking at the net magnetization vector, we wouldn't see that vector as being spinning because we were spinning with that magnetization vector. If we were to look across the slice, add another spin, and it was spinning slightly faster than us, we would think that that spin is precessing at 25,000 Hertz, and that's what's called the rotational frame of reference. We're measuring the differences in frequency here, and because we're only measuring the differences in precessional frequency, we can sample that signal at a much slower rate than we would have to sample if we were trying to measure 64 million Hertz within one second.
So let's take the slice here and look what's actually happening when we measure that analog signal coming from the entire slice. Remember, we are measuring this complex signal once we are applying the frequency encoding gradient, and we are taking that analog signal and converting it into a digital signal. The computer, the MRI machine, can't store an analog signal. It needs to store discrete numerical values over time. So each one of these squares here represents a particular period in time whilst we're measuring the net magnetization vector of the entire slice, and it corresponds to the data point at that analog signal, a numerical value, a net magnetization vector of the whole slice. And we've seen that we can do a one-dimensional inverse Fourier transform to delineate all the frequencies that are contributing to this complex signal.
Previously, when I showed you this, I wasn't giving you the whole truth. I was showing you the absolute frequencies, and we use those absolute frequencies to figure out where that signal was coming from along the x-axis of our slice. Now, I've just shown you that we are actually comparing the relative frequency changes along the x-axis, and this is what's called a complex signal. Now, we haven't gone into the mathematics behind this in great detail, but what you need to know about this signal, what a complex signal means, is that we are able to measure both real and imaginary vectors coming from this slice.
Now, what do real and imaginary vectors mean? We are able to measure both positive frequency changes and negative frequency changes because the frequency change at equal distance away from the center of this slice is going to be the same. We need some way of delineating whether that comes from the positive frequency change or whether it comes from the negative frequency change, and that's what we're able to do in an MRI signal because we're measuring both real and imaginary components of the vector coming from this slice.
Now, don't worry if that doesn't make sense to you. What that essentially means is that the frequencies that we delineate from this analog signal, we can plot away from the midline in both positive and negative values. So although these frequencies match each other, we are able to mathematically calculate whether this frequency comes from the left-hand side of the slice or whether this frequency comes from the right-hand side of our slice.
So let's take these frequencies and overlay them onto the MRI image based on their x-axis location. Now, what we've done here is taken the data acquisition time here and looked at each point that we are sampling that analog signal. Now, what happens is the analog signal is coming from the slice, and the frequency of that analog signal depends where it's coming from the slice along the x-axis location. Now, the MRI machine itself is unable to see that analog signal. It's only able to take the discrete points in time that it's measuring that signal, and then it has to calculate what it thinks that frequency of the signal is.
So let's take some examples of this. Let's look at this signal here and look at the analog signal that would be coming from our entire slice. Now, this analog signal is sampled by the MRI machine at different intervals, the sampling interval, which we're going to look at data. And what it can calculate is the data point at which it sampled that signal. Now, you can see at this frequency, the MRI machine is very accurately able to determine the frequency of the signal.
As we head out more to the peripheries, the frequency is going to increase because of this increased gradient here. At an increased frequency, we are still sampling at the same interval rate. Our sampling rate remains the same, but the frequency of that signal we're trying to delineate is getting quicker. As that frequency increases, we can still plot those discrete numerical values, and you can see here that the computer or the MRI machine is still able to accurately represent the frequency coming from this location on the x-axis.
Now let's look what happens when we take the limit of the frequencies that we want to accurately sample within the x-axis of our slice. If we were to take this frequency, which is 25,000 Hertz more than the Larmor frequency at the center of our slice, and plot that on this graph here, this is the analog signal that's coming out. We sample that analog signal rapidly. The computer then takes those numerical values that it's stored, those are the values that we're going to store in k-space, and then it calculates that frequency. And you can see we are still able to accurately calculate that frequency. So the bandwidth here is determining the frequency that we have to be able to calculate accurately.
Now, what happens when we extend past the borders of our field of view? What happens if we were trying to measure a frequency that came from outside of the slice here, say 30,000 Hertz? Now, in this example, there's no tissue lying outside of our field of view, so technically there wouldn't be any signal. But say the patient's arm was next to their head here and it was giving some signal away here, which signal would we measure? What's the analog signal that will be coming out shot from the 30,000 kilohertz here? Now, we are going to sample that signal rapidly at a specific sampling rate, and you can see that we then digitize that sample. And when the computer tries to calculate the frequency of that sample, you can see now the frequency doesn't match up with the frequencies that we are actually measuring. And this is what's known as aliasing. Here you can see how if there was signal coming from this part of the image, we would falsely represent that signal at a lower frequency change within our image, and we'll get an aliasing artifact, which we're going to look at in a future talk. As that frequency increases further, the accuracy of the MRI machine gets worse and worse, and we are not able to sample these signals quick enough in order to generate an accurate signal. We are now calculating frequencies that are much lower than the actual frequencies, and you can see how the bandwidth determines what this frequency is going to be at the edge of that slice, and ultimately it's going to determine how quickly we have to sample that analog signal.
Now, how do we go about calculating how quickly we need to sample this signal in order to accurately represent it along our slice? Well, let's have a look at that signal that was generated on the edge of our slice. It turns out that in order to accurately sample a specific frequency, we need to sample that frequency at least twice during one wavelength. And that limit, that rate limit that allows us to accurately represent these frequencies, is what's known as the Nyquist limit.
So let's take this example here where our bandwidth is 50,000 Hertz. The maximum frequency that we're trying to calculate here is 25,000 Hertz. That's the maximum change in frequency from the center of this frequency encoding gradient to the edge of our slice. Now, what rate do we need to sample that frequency in order to accurately represent this frequency? Well, we've seen that the Nyquist limit is the sampling rate divided by two, and our Nyquist limit here is 25,000 Hertz. So our sampling rate needs to be 50,000 Hertz. We need to sample that signal 50,000 times in a second in order to accurately calculate that that frequency is 25,000 Hertz.
So we can rearrange this equation to get our sampling rate. Now, you might notice here that the sampling rate equals the bandwidth that we've selected. Our bandwidth in this example is 50,000 Hertz, and the sampling rate in this example is 50,000 Hertz. And no matter what our bandwidth is, it turns out our sampling rate is going to equal our bandwidth because the bandwidth here determines the maximum signal that we're trying to accurately represent, and that maximum signal is half of the bandwidth. We can see that this maximum signal that's equal to half of the bandwidth is also equal to the Nyquist limit, the maximum frequency that we're trying to accurately sample. And the Nyquist limit is equal to the sample rate divided by two. We saw that bandwidth divided by two equals Nyquist limit, sampling rate divided by two equals Nyquist limit. That's why the sampling rate and the bandwidth are the same values. They don't represent the same thing, but they will have the same numerical values.
Now, because we know the sampling rate, we know how quickly per second we need to sample that analog signal, we can figure out how much time we have to acquire each one of these samples of the analog signal. We take one second and we divide it by the number of samples that we're taking in that second. We see that the number of samples we're taking in a second is 50,000, 50,000 Hertz, 50,000 samples per second. We take one second and we divide it by that 50,000 Hertz, and that will give us what's known as a sampling interval, or you'll see it in many texts as the dwell time, the amount of time that we have per sample that we are digitizing that analog signal. And in this example, where we're trying to represent a frequency of 25,000 Hertz, we've got a sampling rate of 50,000 Hertz. Our sampling interval, the dwell time, the amount of time we have to take each sample, is going to be 20 microseconds.
And now that we know how long we have for each sample, and we know how many samples we want to take, the x-axis number of pixels that we want, we can figure out how long we need to apply that frequency encoding gradient in order to get all the data for those x-axis signals.
So let's bring this all together. Let's tie all the loose ends, and hopefully you can see how each one of these parameters relates to one another. Initially, we have the tissue that we are trying to sample, and we define the field of view, a set area of the slice that we want to accurately image. And now that we've determined the field of view, we want to know what resolution do we want in our image, how many pixels do we want in that image, and that's what's known as the matrix size of the field of view that we are trying to represent. And the matrix represents how many phase encoding steps and how many samples we need to take during the frequency encoding gradient. That determines the resolution within our image.
Now that we've determined the resolution we want and the field of view, we set the bandwidth, the specific bandwidth that we want. Now, this may seem fairly arbitrary now, and the next talk is going to be showing us how selecting different bandwidths is going to give us different imaging properties. Now, for this example, I'm using 30,000 Hertz across the x-axis of the slice. So that's a 15,000 Hertz change in either direction along that x-axis of the slice. Now, the maximum frequency that we need to be able to calculate here, the Nyquist limit, is going to be 50,000 Hertz.
Now, because we've set the bandwidth, and we've set the field of view, and we've seen that bandwidth is proportional to both gradient strength and field of view times by some constant, you can see that the MRI machine is able to calculate the specific gradient that's needed across 25 centimeters that's going to give us a 30,000 Hertz bandwidth. So then the MRI machine sets that gradient along the x-axis. We need to calculate now how long do we apply that gradient, and how many samples do we need to take along that gradient, and how long we have for each one of those samples that we are acquiring during that frequency encoding gradient.
Now, we know the Nyquist limit. We're trying to calculate a frequency of 15,000 Hertz, and we're trying to accurately represent that frequency along the x-axis of our slice. That frequency, 15,000 Hertz, is equal to the sampling rate divided by two. What divided by two will give us 15,000 Hertz? 30,000 Hertz divided by two is going to give us 15,000 Hertz. Our sampling rate, as you can see, is equal to the bandwidth here.
Now, if someone asks you, is sampling rate the same as bandwidth? They've got the same numerical values, but they're not representing the same thing. Someone asks you, what is the range of frequencies across the x-axis of our slice? That's known as the bandwidth. The range of frequencies along the x-axis of our slice is not the sampling rate. The same goes with sampling rate. If someone says to you, how frequently, how many times per second do we need to sample that analog signal in order to accurately delineate the x-axis signals? You can't call that the bandwidth. That's the sampling rate. Just because they have the same numerical value doesn't mean that they're the same thing.
And now that we've determined the sampling rate, we know how much time per sample, the dwell time or the sampling interval, that we have for each point along that data acquisition cycle along the frequency encoding gradient. That's when we take one second and we divide it by the sampling rate, 1 divided by 30,000 Hertz in this case.
Now that we know the sampling interval, and we know the resolution that we want, we know how many samples we need to take in order to get that x-axis resolution. Say we wanted 256 pixels along the x-axis of our image, we would need to have 256 samples taken whilst we're digitizing that analog signal. So we can take the sampling interval and times it by the number of samples we need, 256 in this case, and that's going to give us the sampling time, how long we need to have that frequency encoding gradient on in order to accurately represent all of those x-axis signals.
I know this is a long video and there's a lot to cover. And if you understand these concepts, you will understand most of what's going on in MRI, especially now when we go into looking at different MRI pulse sequences.
Now, in the next talk, we're going to expand on these concepts a little bit further, and I'm going to show you why we will choose different bandwidths. Here you can see we had no rhyme or reason for choosing 30,000 Hertz, and it turns out the bandwidth that we choose will influence the gradient that happens along the x-axis of our slice, and that's going to influence the type of image, especially the signal to noise ratio that we're going to represent in this image. After that talk, we're going to look at two artifacts that occur in MRI imaging, both aliasing and the chemical shift artifact, and you're going to see how understanding these concepts become crucially important when going about trying to explain how those artifacts occur.
Now, these types of questions, these topics come up over and over again in exams. And if you are studying for a physics exam or for specifically an MRI physics exam, I've linked below in the description a question bank that I've curated, multiple different questions from multiple different past papers, taking the ones that come up most frequently and answered them there in that question bank in video form, showing you why particular answers are correct and why other answers are incorrect. I also show you how examiners can ask questions in multiple different ways and take you through those questions depending on how the examiners are asking those questions. So that's all for this video. Well done if you've made it all the way through this video, you really are committed to learning MRI physics. I'll see you all in the next talk where we take a deeper dive looking at specifically how we pick the bandwidth for our slice. So until then, goodbye everybody.