Transcription
Hello everybody and welcome back. So, in the previous talk, we looked at how we select a specific slice along the z-axis in the Cartesian plane that's going to contribute to the signal that generates our MRI image. And we've seen that we use a slice selection gradient along that z-axis and we match the precessional frequencies on that gradient with a 90-degree RF pulse. When the radio frequency pulse frequency matches that precessional frequency, we get flipping of those spins in the transverse plane, only along that specific slice, because of that slice selection gradient.
Now, if we take that specific slice and look at it end-on, where we've got our x-axis and y-axis to the slice, and we look at what the net magnetization vectors from the various regions in the slice are doing, we see that they're precessing in phase with one another. We've applied that 90-degree RF pulse and the slice selection gradient. We've then applied a 180-degree radio frequency pulse to account for those T2 star differences, and we've created an echo where we're going to be sampling that tissue at TE time.
Now, at this moment, at TE, the only magnetic field that these spins are experiencing is that B naught magnetic field. We've turned off our radio frequency pulse, and we've turned off the slice selection gradient. Those vectors that have gained transverse magnetization are now losing transverse magnetization over time, and they're precessing at a frequency that's proportional to the main magnetic field, the B naught, that is always on during these pulse sequences. That precessional frequency is a function of the B naught, as well as the gyromagnetic ratio of the specific elements, in this case, hydrogen.
Now, when we look at this slice, we see that the entire tissue is resonating in phase, and it's going to generate a signal that is a sinusoidal wave because all of those net magnetization vectors are processing in phase. We're going to get accumulation of those net magnetization vectors, giving us a very regular sinusoidal wave at the same frequency as the procession of those spins.
Now, the way that I've drawn this slice here, I've shown the net magnetization vectors at different locations in this slice as having the same vector value. Now, that's incorrect. The vector value, or the amplitude of the signal that's going to be generated, is dependent on the type of tissue from which those spins are spinning. And it's the TE and the TR that we select that's going to determine what signal is being generated. Are we getting contrast and signal from T2 differences, or from T1 differences, or are we specifically looking at the proton density from that specific region? The amplitude is going to change depending on the type of tissue within that slice. What's not changing is the frequency of the procession. These spins are processing at the same frequency.
Now, because all the spins are processing at the same frequency, we can't take a specific region along the signal that's being generated and figure out what contribution this specific net magnetization vector is giving to that overall net magnetization vector from that slice. And when we think of TE, we talk about TE as being a certain time period away from that first 90-degree RF pulse. We say a TE is 14 microseconds, 40 microseconds after we apply the 90-degree RF pulse, we then sample the tissue at TE.
Now, when we say 40 microseconds, we're giving a specific period in time. And if we were to look at the signal that's being generated here, and we were to only take one specific period of time from that signal, we would get one data point back. We would get one value for the net magnetization vector of that entire slice. Now, from one value, there's no way that we are going to be able to generate an entire image with multiple pixels in it. We've only got one data point. What we need is multiple data points over time to see how that signal is changing over time. Only when we see how the signal is changing over time can we then infer what's happening to that net magnetization vector over time.
So, in fact, instead of sampling at one point at TE, we sample at multiple points surrounding TE. So we actually sample the signal at multiple discrete points. Now, what's receiving that signal is what's known as our receiver coil, something that we're going to look at in a future talk. And the ability of that receiver coil to sample the signal over a period of time is related to what's known as the bandwidth of the receiver coil, again, another topic that we're going to look at in the future. But you can see now, if we sample this signal over a period of time, we can start to see what changes are happening in the signal over that period of time.
Now, if you were to imagine a compass sitting next to this slice of tissue that we've selected, and the signal changing, that compass would move as the signal changed. That's what's known as an analog signal. We're getting a continuous movement of that compass as the net magnetization vector changes within that slice of tissue. Now, when we talk about digital values, we're talking about discrete values over a period of time. We need to convert that analog signal into a digital signal. And each point that we sample that analog signal, we are creating a digital signal, a discrete value that we can enter or store in the computer. And we'll be getting multiple discrete values across a period of time with this signal.
Now, you see that I've drawn this signal as getting slightly bigger before then decreasing over time. Why have I drawn it like this? Well, if you cast your mind back to how we go about acquiring the signal, we first flip the specific slice into the 90-degree plane. We get a transverse magnetization vector here. We then switch off that 90-degree RF pulse, and we switch off the slice selection gradients. And what happens is we lose transverse magnetization at a rate known as the free induction decay, or T2 star decay. We then apply a 180-degree RF pulse that allows those spins that were dephasing at the free induction decay rate to then rephase with one another. And that rephasing causes gain again of that transverse magnetization vector. Remember, transverse magnetization vector is predominantly a measure of how in phase those spins are in the transverse plane. Again, then that transverse magnetization vector will reach its maximum transverse magnetization once those spins have rephased, and then they will continue to dephase again at the rate of T2 star.
Now, that free induction decay, or T2 star decay, is occurring because of both spin-spin interactions and because of the local magnetic field inhomogeneities. In an ideal world, we would have a perfectly homogeneous magnetic field, and the decay would happen at a rate that's known as T2 decay, not T2 star decay. Now, what this rephasing, this 180-degree RF pulse, allows us to do is measure signal that is similar to what the T2 signal would have been. We've negated some of those local inhomogeneities in the magnetic field, and we're getting a stronger signal than we would have got if we had just looked at T2 star decay by itself.
So, when we are sampling, remember, we're not sampling at one point in TE. We're sampling at multiple points surrounding TE. TE is the time period at the middle of those sampling points. We are going to get signal that increases slightly and decreases slightly. Now, for this talk, this change in signal doesn't make much difference, but when we look at k-space later on, we want to remember that this signal intensity changes over time surrounding that TE. So, what I'm going to do is show that signal here on our pulse sequence so that we can remember that this is where we're measuring the signal.
Now, we've taken our slice of tissue, and all of those spins are processing in phase, and they're processing at a rate that's proportional to the main magnetic field. If we think of our patient as lying within the bore of the MRI scanner, the head to feet going along the z-axis here, the longitudinal plane, we have selected a specific slice here, and there's no way for us to tell at the moment where the signal is coming from that slice. We've got no X or Y coordinates along that slice.
Now, what we want to do is figure out how we can differentiate the signal based on its x-axis location. Now, the problem previously was all those spins were spinning in phase with one another, and their magnetization vectors were a sum of all those spins because they were exactly in phase. And there's no way for us to tease out where those spins are coming from in space because of those vectors being in phase.
To get around this, what we do is apply what's known as a frequency encoding gradient. We take this gradient coil, this purple coil that we've looked at before, and you see that this coil is separated into two halves. We apply a different current to this half than we do to this half, and what we get is a magnetic field differential along the x-axis into our screen here. That magnetic field differential is going to apply a magnetic field strength difference along the x-axis of the slice that we've selected. This is what's known as the frequency encoding gradient.
Now, we apply this frequency encoding gradient only when we are reading out the signal. We only apply the gradient here. So, up until that frequency encoding gradient, all of those spins are spinning in phase with one another, applying a net magnetization vector that we can measure. As we're about to measure that net magnetization vector, we apply this magnetic field gradient. Now, look what happens when we apply a magnetic field gradient to this slice. The frequency of those spins differs depending on the x-axis location. And because the frequency differs, we can now figure out where signal is coming from based on the frequency of that signal.
Now, you can see that that magnetic field strength in this example is getting stronger from left to right. The right-hand side of our image is now processing at a faster rate, or a faster frequency, than the left-hand side of our image. And we can see that frequency of the spins now correlates to a specific x-axis location within the slice that we've selected.
Now, you might be thinking, if I take this slice here and we cause these spins now to spin out of phase, surely we're going to get a very low net magnetization vector. And that's true, because these are now losing phase, and we've said that transverse magnetization is a function of how in phase these spins are. So, actually, what happens is before we apply this frequency encoding gradient, we apply an equal and opposite frequency encoding gradient prior to readout. So, the left-hand side of our image will have dephased faster because it would have experienced a higher magnetic field, and the right-hand side of this slice would have dephased slower. Then, when we apply this frequency encoding gradient, they will start to rephase with one another. And the period of time that we're sampling those spins will be much more in phase. Although there will be more in phase for that data acquisition time, they will be at different frequencies. And it's the different frequencies that's going to give us a non-sinusoidal wave when we measure the net transverse magnetization vector of this slice.
So, let's have a look at what the net magnetization vector would look like for this specific slice with differing frequencies along the x-axis. You can see now that our net magnetization vector is much more irregular. We're not getting that sinusoidal wave that we saw when all the spins were in phase with one another. We can't just sample one point along this wave and infer what's happening within the entire slice. We need to sample multiple periods of time, that's our data acquisition time. Again, we sample multiple times over a period of time, and we get discrete values depending on the amplitude of that net magnetization vector. When all of those frequencies are generally pointing towards our receiver, we're going to get a high value. When they're canceling each other out, we're going to get a much lower value. When they're all pointing away from the receiver coil, we're going to get a much lower value. And these discrete values we can represent as a grayscale here.
Now, these grayscales are not representing the pixels within our image. They're representing a numerical value, an actual data value of this wave that we are measuring here. As the wave increases in amplitude, we're getting a higher numerical data value. And these grayscale values that we're representing here at each data acquisition point represent a data value, a numerical value.
Now, importantly, and this is where most people get confused, when we look at this particular point in time here, that dark square here, this numerical value represents the entire net magnetization vector for that entire slice at a given period of time. We then wait a tiny period of time and we measure the net magnetization vector for the entire slice again. Each data acquisition point in this wave represents the entire net magnetization vector for the whole slice. It has no spatial information. This data point here represents the entire slice. It gives us no information on its own as to where the signal is coming from from that slice.
Now, because the frequencies vary along the x-axis, we can now use this specific signal to figure out what frequencies are contributing to this signal. Not only can we see which frequencies are contributing to this complex signal, but we can also delineate what the amplitude of those frequencies are. Now, essentially, what we're doing here is we're creating multiple simultaneous equations where we have the answers to what the net magnetization vector is at a given period of time. Each one of these data values represents the net magnetization vector at a given period of time in our pulse sequence. So, we've got the answers to what the accumulation of all those magnetization vectors will give us at a period of time.
Now, we've got two variables that we're trying to solve for here. The first variable is frequency. We can see that the frequency changes depending on where we are along the slice, along the x-axis of that slice. The frequency will increase as we move from left to right in our image. That's one variable that we're trying to calculate. The second variable that we're trying to calculate is what is the net magnetization vector, or the amplitude of those net magnetization vectors that we are accumulating to give us our total net magnetization vector. And each column along the x-axis here is going to have a different amplitude depending on the type of tissue that is sitting within that column.
Depending on the pulse sequence we use, that signal is either going to be coming from the proton density, or the signal differences is going to come from T1 or T2 differences. What we want to do now is take these values here and tease out which signals are contributing to these values. And what we can do is use this mathematical equation to figure out which unique combination of frequencies and amplitudes will give us this specific signal that we've read out. Only one specific set of frequencies and amplitude will give us this unique signal that we've read out. And that transformation is what's known as a Fourier transformation, which we're going to look at today.
Now, look at this signal that we've read out, that we've digitized into multiple different numerical values, and see how that is a combination of all of those frequencies coming from different x-axis locations on our image. Again, at any given point in time, each one of these numerical values is representing the entire slice. And it's only the change in time, when we compare each of those numerical values over a period of time, that we can use them to calculate which specific combination of frequencies and amplitudes will give us this unique digital signature. And this mathematical equation will allow us to tease out the various different frequencies that are going to contribute to this ultimate net magnetization vector that we've been measuring from that slice.
Now, remember, the frequencies match up to a specific x-axis location within our image. So, we can use these frequency values, which each have a different amplitude, to figure out what amount of signal is coming from each x-axis location on our image. We've now taken this time data and we've converted it into a set of data that now has some spatial information. Because this certain frequency on the left-hand side of our image is lower, we know that this amplitude is coming from this x-axis location along the slice that we're imaging. Again, we've got no y-axis data here, because all of these spins at a particular point on the x-axis along that y-axis have the same frequency. They're all contributing to the amplitude that we've calculated here for this particular frequency. What we have got now is x-axis location data points.
Now, when we've been looking at the specific slices in my example, I only had eight different net magnetization spins along the x-axis. In reality, there's an infinite amount, or infinite variation, of frequencies as we head along that x-axis. And it turns out that the number of times that we sample the signal determines the amount of frequencies that we can delineate in our sample. The more times that we sample this complex signal, the more separate frequencies we can get along our x-axis here. Again, remember, the frequency is the variable we are trying to calculate for. You can't take a simultaneous equation with 30 different variables and only have four simultaneous equations and calculate for all those variables. You need at least 30 equations in order to be able to calculate for all those variables. Again, here you need multiple data points in order to delineate multiple frequency data points along the x-axis.
My example here, what we've got is what's called oversampling, where we've created more data than we need if we're only looking for eight points here. In fact, this data here will give us an image that looks like this. We've delineated into much more frequencies. I just haven't been able to draw all of those frequencies here. And what we've done is converted this time-based data set into a data set that changes over frequency, and we know that frequency is a proxy for x-axis location. We've taken a time-based data set and changed it into a frequency or location-based data set.
Now, again, this is a point that people often get confused with. This particular point here is not coding specifically for the x-axis location on the left. Each data point here encodes for the entire net magnetization vector of the slice. It's only when comparing all of those data sets with one another that we can then create this entire line here with different x-axis values along our slice. This conversion is what's known as an inverse Fourier transformation, and this particular one is a one-dimensional Fourier transformation.
Now, in the coming talks, we're going to look at both two-dimensional and three-dimensional Fourier transformations that will ultimately give us the image that we look at on our screen when we're looking at MRIs.
Now, what are we actually looking at when we get this data? Well, this one line of data represents a single acquisition time that we took while this frequency encoding gradient was going on. We then wait until our TR, and then we repeat the process again. Remember, when we are creating our pulse signal, we flip the spins to 90 degrees, let them dephase at TE, we're going to measure a signal. We then allow the spins to gain longitudinal magnetization to a certain degree before again flipping them at 90 degrees. That second time that we flip them to 90 degrees is our TR, our time of repetition. The cycle then repeats itself again. We will get a second data set, and a third data set, however many times we repeat that pulse sequence, we're getting a data set for this specific slice.
Now, in this example, we're changing nothing each time we repeat this, so the data set that we get is going to be identical. It's going to give us the same outcome. And if we were to repeat this pulse sequence that we've got here with our slice selection gradient and our frequency encoding gradient, the image that we would end up getting would look like this. We've selected a specific slice, and now we've got the signal data from the x-axis locations along that slice. We've got the total signal for that column along the x-axis. What we haven't got is y-axis values here, and that's what we're going to be looking at in the next talk, how we can start delineating those y-axis values in order to get us a readable MRI image.
So, up until now, we've placed the patient in the scanner and we've used a slice selection gradient along with a radio frequency pulse to select a specific slice. In that slice, now, before we apply our frequency encoding gradient, all of those spins are spinning in phase. We then apply a reverse frequency encoding gradient before applying our frequency encoding gradient to allow those spins to dephase based on their x-axis location. The dephasing happens because the frequency changes. Remember, the frequency of those spins is dependent on the gyromagnetic ratio as well as the magnetic field strength, and we are creating a gradient magnetic field strength along the x-axis. That gradient magnetic field strength changes the frequency of those spins depending on the x-axis location. That's why it's called a frequency encoding gradient. Because we have done that, we can then use an inverse Fourier transformation, or one-dimensional Fourier transformation, to create a data set that represents the entire signal coming from the columns along the x-axis of the slice that we've selected.
Now, we want to create y-axis values. Now, if you think about how would we go about creating a y-axis value? Well, you might think, can't we apply a gradient along the y-axis once we've waited for TR on our next cycle? Can't we apply a frequency encoding gradient, but in this time oriented along the y-axis, and get data values along that point? Now, it turns out you can't do that. If you think about a Sudoku, if you've ever done a Sudoku, all the rows and all the columns in a Sudoku add up to the same value. Now, here we have got the values of the rows and columns, and if we did a frequency encoding gradient along the y-axis, we would get values along all the rows of our image. And you would think that if we combine those values, we'd be able to figure out each of the pixel values.
Now, if you think about a Sudoku, every Sudoku in the world has the same column values and the same row values, but they're all completely different. There are multiple ways that you can put the numbers into a Sudoku. The same thing happens if we were to get values of all our rows here. Now, it's the same as taking a frontal chest x-ray and a lateral chest x-ray and saying, from just those two data points, we are going to create a single slice on a CT image. There's simply not enough data to get all of those pixels that we require for an entire slice.
What we end up needing to do is apply what's known as a phase encoding gradient to the y-axis of this particular slice. Now, the phase encoding gradient is unlike the frequency encoding gradient. The frequency encoding gradient can be done on a single TE. We can get all of that data in one TE. We sample that data in one cycle of our pulse sequence. Here, the phase encoding gradient is going to require multiple repetitions of this particular pulse sequence, and we're going to see why that's the case in the next talk.
Now, I'd highly encourage you, if at any point during this talk you felt lost and you weren't following along with what I was saying, go back in the video and make sure you understand these concepts. This talk and the next talk is the most common point where people feel overwhelmed and they give up on MRI physics. If you can understand these talks, how we go about localizing signal, the rest of MRI physics becomes incredibly easy. If you don't understand these, the next talks are going to become very difficult to follow along with. So, spend time here making sure you understand these concepts. And again, if you wanted to test yourself with these concepts, I've linked question banks in the description below. You can go through the questions, and then I go through the answers with you in video format, showing you why certain answers are correct and why certain answers are not correct. So, go and check that out to identify where you're doing well and where you need more work in your Radiology Physics Exam studying. Otherwise, I'll see you all in the next talk, where we're going to examine the phase encoding gradient along that y-axis and how we can ultimately go about creating an image where each pixel represents the signal coming from the voxel in the slice that we are imaging. So, until then, goodbye everybody.