Transcription
Hello everybody and welcome back. So, in the previous three talks, we've been looking specifically at signal localization. We've looked at how we can select a specific slice along the z-axis, and then how we can localize specific signal based on their X and Y locations within that slice, and differentiate where the different signals are coming from.
Now, the data we use to calculate where that signal is within that specific slice is stored in a matrix that's known as k-space. And I want to spend a little bit more time today looking at k-space itself. Now, you can see that k-space is made of multiple different rows, and each row within k-space represents a specific data acquisition period that we used in a specific pulse sequence we've chosen. Now, we acquire each one of these data points at the data acquisition time, which occurs when we're applying the frequency encoding gradient, all of which we've looked at in previous talks. Each sequential row that we add into k-space represents a different phase encoding gradient that we've used to introduce phase differences along the y-axis of our slice. Now, the number of phase encoding steps that we use determines the y-axis resolution that we can get within our image. The number of times that we sample the signal within that frequency encoding step, the number of columns within k-space, is going to determine how much x-axis resolution we're going to get in our final image. And we've seen that we can use this data and use what's known as a 2D Fourier transformation to actually create the image for that specific slice on our MRI scan.
Now, when we look at k-space data, many people get confused as to what k-space actually represents. Now, what it doesn't represent is that each point on here is not representing a specific pixel in our final image. Each data point within k-space represents signal coming from the entire slice that we've selected, and we're going to look at that here today. It also doesn't represent any sort of spatial localization when we're looking at a specific data point in itself. We need to combine the values from all the data points in order to perform that 2D Fourier transformation and get accurate localization of where that signal is coming from. You can't look at k-space itself and say, "Oh, this is an MRI of the knee." We can't do that with a naked eye. These aren't representing pixels in our final image. The grayscale values here that are used in k-space are actually representing a numerical value, a magnetization vector that's been plotted into this matrix, and we use this matrix to apply mathematical formulas to ultimately generate our image.
So now, let's have a look at k-space in a little bit more detail, and we'll start by looking at how we acquire one line of k-space. Now, let's use this line where we're not applying any phase encoding gradients along the y-axis direction. We know that we can sample the entire slice and measure that net magnetization vector over time. That change in net magnetization vector from the entire slice that we've selected using our slice selection gradient is going to give us an analog signal here, a continuous signal that's changing over a period of time. Now, we can't store a continuous signal within a computer. We need to store discrete data values. So, what we do is we convert this analog signal into discrete digital data points here. Now, each one of those data points represents a sampling time that's occurring while the frequency encoding gradient is on, and we can use this data then to figure out where the signal is coming from along the x-axis location because of this frequency encoding gradient.
So, let's look at the slice that we've selected. Assume now that this slice is here, and in this period of time, it's experiencing the B naught magnetic field, and all of the net magnetization vectors are processing in phase with one another because we haven't applied a phase encoding gradient along the y-axis. Now, what we're going to do along this period of time here, for this moment, let's ignore this frequency encoding gradient here. During the frequency encoding gradient, where we acquire our data signal, what's going to happen is a frequency encoding gradient is going to be applied along the x-axis of our slice. That is going to introduce frequency differences based on the net magnetization vector's location along the x-axis. These spins are going to process faster than these spins because of the local magnetic field that they're experiencing. Now, as that frequency encoding gradient is applied, you can see that a net magnetization vector signal for the entire slice can be measured over time, and each point here represents the net magnetization vector of the entire slice at a given period of time. But it represents the entire slice. Each point that we've acquired is representing the entire slice, but the net magnetization vector for that specific period of time.
So now, we've calculated how we get this row within k-space. Now, let's look at how we generate a different row within k-space that has some phase encoding data baked within the signal here. You can see we've applied a specific magnitude of phase encoding gradients along the y-axis, and you can see that our signal has decreased slightly. The data points we're getting out represent slightly lower values than our non-phase encoded line that we've just acquired. Let's look at this slice now and see what happens when we apply a phase encoding gradient along the y-axis. I've represented with these waveforms here the signal that is being generated from each point along the y-axis, and you can see the net magnetization vector that is generated by all of these spins that were spinning in phase with one another. Now, if we apply a phase encoding gradient that is going to dephase these spins with one another, they are no longer processing in phase with one another. There has been some phase shift occurring. Depending on where they're located along the y-axis, that phase shift has caused these individual signals to become out of phase, and we've lost some net magnetization vector from the entire slice. Remember, transverse magnetization is a function of how in phase the spins are with one another.
Now, when we stop that phase encoding gradient, the net magnetization vectors are going to process at the same frequency, no matter where they are located within the slice. They are only experiencing the B naught magnetic field. We have not got any frequency changes along the x-axis or any frequency changes along the y-axis. What we've got here is phase changes along the y-axis. Now, this phase encoding gradient that we have applied here has now got some memory. As we are heading towards our frequency encoding gradient, we then apply a frequency encoding gradient along the x-axis here, as we've looked at here. That frequency encoding gradient is going to cause the frequency of the spins to change depending on their x-axis location. We've still got that phase encoding memory that is going to be baked into the signal that we acquire here. So, we apply that frequency encoding gradient and we measure the net magnetization vector of the entire slice. We get a separate signal that has the same frequency encoding gradient baked into that signal here. We can still figure out where those signals are coming from based on their x-axis location, but now we've introduced some phase shift in the y-axis direction. We can take this separate line of data that we generated here and compare it to that first line of data, and the only thing that's changed is the degree of phase shift that we've introduced into the y-axis. So, intuitively, you should be able to see that based on the amount of phasing that we've applied to our slice and based on the amount of dephasing a specific spin experiences depending on where it is on the y-axis location, we can use some form of mathematical formula to figure out where exactly those spins are based on their y-axis location within the slice.
Now, I've touched on this before, and you may be wondering, why is the signal getting stronger and then weaker if we are dephasing these spins along this frequency encoding gradient? Because those spins are now processing at different rates, and the different rates of procession means they're going to dephase, shouldn't that signal get smaller and smaller? There's actually two reasons why the signal gets stronger and then gets weaker again, as you can see that's represented throughout this k-space data. This is happening from here all the way down to the middle of k-space data. Now, the first reason that happens has to do with how we go about selecting our slice, and this specific pulse sequence we're using is what's known as a spin echo sequence. When we acquire signal, we first flip our spins to 90 degrees based on the slice that we've selected. Now, the signal that we're measuring is the transverse magnetization of those spins that we flip to 90 degrees. Those spins will lose transverse magnetization because they will start to dephase with one another, and the rate at which they dephase with one another is dependent on the type of tissue that they're in. And we've seen that rate of dephasing can be represented by what is known as the free induction decay or T2 star. Now, spins lose their transverse magnetization predominantly due to spin-spin interaction that we've looked at before. They also lose their transverse magnetization because of local magnetic field inhomogeneities. In an ideal world, we would have a perfect magnetic field, and spins would lose their transverse magnetization at a rate that's known as T2. Now, we want to create a signal that is going to allow us to generate a signal that's similar to T2 and not to T2 star, and this is what's known as a spin echo sequence that we've looked at in our T2 relaxation talk. Spin echo means that spins that were dephasing based on local magnetic field inhomogeneities are now flipped 180 degrees and will begin to rephase at a time point known as TE. We will get an increase in transverse magnetization and then again a loss of transverse magnetization at the free induction decay. Now, we can see that signal will increase until TE and then decrease because of that rephasing, and the signal that we get at TE is representative of the T2 signal from those tissues. Now, remember, the T2 signal or the transverse magnetization signal will differ in different tissues over time. You can see that in CSF, it maintains its transverse magnetization for a much longer period than fat does, and it's those differences that allow us to get contrast within our image.
Now, the second reason we get an increase in signal and a decrease in signal at TE here, or when we are sampling the data, is because of the frequency encoding gradients or sequence that we use. We first dephase the frequency encoding gradients in the opposite direction to what we would be applying our frequency encoding gradient. So, at the start of our frequency encoding gradient, at the start of data acquisition, when we're filling those points in k-space, these spins will start in different phases with one another because they've got different frequencies. But because that frequency encoding gradient has been applied in an equal and opposite direction along this slice, these frequency encoding gradients will start to catch up with one another, and at TE, all of them, for a brief moment, will be in phase with one another. At TE, we will get the strongest net transverse magnetization from this specific slice. So, have a close look at the spins here as we apply our main frequency encoding gradient. You can see that the spins on the right-hand side are spinning faster than those on the left, and as we get nearer and nearer to TE, they sync up with one another, and then they start to dephase because of their differing frequencies again. We get an increase in signal and then a decrease in signal, and that allows us to get maximum signal at TE here, at the middle of k-space, when we are sampling that data over a period of time.
Now, what we can do is actually take this matrix that's filled with data points and put it on its side, and we can plot that signal that we've been measuring in these various different steps. The first step that we did was apply a sequence that had no phase encoding gradient. We didn't have any phase encoding within our signal. We apply our frequency encoding gradient and we measure that signal over time as we are sampling the entire slice. We then repeat this. We reach TE, all those spins have gained their longitudinal magnetization, we flip them again to 90 degrees and repeat the cycle. We then repeat it with a different phase encoding gradient. The signal that we are generating now will be slightly less than that initial signal because we've got dephasing in the y-axis direction, and any dephasing will cause some form of signal loss. We repeat this process over and over again with more and more stronger and stronger phase encoding steps along the y-axis direction, and the stronger that phase encoding, the less the actual signal that we are going to be getting out. Eventually, we will use our strongest phase encoding step along the y-axis direction and measure out some form of data acquisition. Now, you can see the strongest signal we get is at the center of k-space, when everything is most in phase with one another. The more we head out to the peripheries, the more we lose that signal.
Now, what is giving us signal within our image? It's the net magnetization vector, the amount of transverse magnetization that we have. So, these central regions of k-space are going to give us most of our signal. But the center of k-space here has very little phase encoding or frequency encoding occurring. All of those spins are generally in phase with one another. The further we head out to the periphery, the more difference there is based on the phase and the frequency within the specific slice. So, we can take this k-space data here and look at only specific regions of that k-space data and see how that applies to our final image. Now, again, every single point here is measuring a net magnetization vector from the entire slice. So, each point in here contributes to the entire image, and any pixel within the entire image is getting contributions from all of these data points within k-space.
Now, say we were to only read the central region of k-space here. What have we lost? Now, we've lost multiple phase encoding steps and we've lost multiple frequency encoding steps. We are unable here to accurately represent where this signal is coming from. However, the signal that we are getting is very strong. We're getting a high amplitude of signal. We're getting a measurement that gives us good data as to how strong the signal is coming from from those various regions within the image. Now, depending on the tissue types within the image and depending on the sequence that we have chosen, we are going to get signal coming from specific tissues based on their T2 and T1 relaxation times. If we choose a sequence that is T1 weighted, then signal is going to be coming predominantly from fat. Signal from water will be very small. So, any bright signal that is coming from this data here in a T1 weighted image is likely going to represent fat. Lower signals are going to be coming from water. Now, sampling only the central region of k-space means we lose a lot of spatial encoding. We're unable to see where that signal is coming from, but we are able to see differences in signal very easily. So, this region of k-space is actually what's giving us contrast within the image.
Now, although each single data point in k-space contributes to the image, they contribute different features to the image. The central region of k-space, because of that high signal amplitude, contributes contrast to the image. Now, there are certain types of sequences where we predominantly want contrast in the image. We're not too concerned about spatial resolution. And not only that, certain MRI sequences, we want to be able to acquire that data very rapidly. We want to see changes in time, such as in angiography. Now, we can write a pulse sequence that only creates this region of k-space here. We only need a specific number of phase encoding steps, and therefore we can acquire that data much faster, and we can generate certain types of MRI images that have adequate data to give us the diagnostic information that we need. We don't necessarily need all this spatial information in the peripheries.
Now, if we were to look at the inverse of this, we were to isolate only the peripheral signals coming from k-space. What we have now is multiple phase encoding steps and multiple frequency encoding steps, but we are losing a lot of these signal amplitudes that's coming from the center of k-space. So, the signal that we are generating, the image that we regenerated, we did a two-dimensional transformation here, would give us very good spatial resolution. Would give us very good definition between the edges of tissues. However, we won't be getting much contrast in that image. We won't be able to tell different types of tissues apart very well, but we will see where those boundaries are. The peripheries of k-space is giving us what's known as spatial resolution.
Now, if you look at the different phase encoding gradients that we apply along the y-axis of our slice that we've selected here, we can see that these stronger phase encoding gradients have a larger rate of change of dephasing as we head up this y-axis here. The rate of change of the amount of dephasing that will happen in the y-axis is much higher for the stronger phase encoding gradients along the y-axis here. So, these phase encoding gradients are our strongest phase encoding gradients, and they induce the largest change in phase along the y-axis direction. The smaller the phase encoding gradients, the smaller the rate of change of phase as we head along our y-axis. Now, the way I like to think about this is the rate of change of dephasing increases as we head out to the peripheries, and we can measure rate of change over time. You may hear that the peripheries of k-space contain what is known as higher frequency information. Now, the way I like to think about this is that the rate of change of phase or dephasing that occurs with stronger phase encoding gradients is much higher as we head up the y-axis than the rate of change that there is at phase encoding gradients that are weaker. Now, we can represent the rate of change by a waveform, and the stronger the phase encoding gradient, the faster that rate of change of dephasing as we head up our y-axis. The more frequency that rate of change has as we head out into the peripheries.
Now, in actual fact, the way in which we store k-space data is slightly different to how I've represented it here, with phase encoding and frequency encoding steps happening in the X and Y axis directions. Now, this, in my opinion, is a level above what we need for part one radiology exams. The way in which we've explained it here will give us enough intuition to allow us to create specific pulse sequences that will generate the types of images that we are trying to acquire. There are much more detailed mathematical equations where we start talking about imaginary numbers and we start talking about 2D waveforms in multiple different planes. However, the way in which we've represented the data here is sufficient for our understanding of how we go about generating specific pulse sequences that we're going to be looking at in the next talks. What we need to know is that the amplitude or the main signal is coming from the center of k-space, and that gives us contrast. The rate of change of dephasing in both the phase encoding direction and the frequency encoding direction can be plotted as a waveform, and as we head out to the peripheries of k-space, that waveform gets shorter wavelengths and higher frequencies. This data represents higher frequency data and ultimately gives us better spatial resolution within our image, allows us to see edges between tissue boundaries.
Now, with that being said, if we look at k-space itself, and we apply phase encoding gradients that increase in degree as we apply stronger and stronger phase encoding gradients, we can see that there are certain phase encoding gradients that have an equal and opposite phasing coding gradient. So, this line of data, say that we acquire here at a specific phase encoding gradient, should in theory have an equal and opposite phase encoding gradient along the y-axis here, if you take dark red and dark blue here. Now, as you can see, a specific spin at a specific location on this y-axis will experience an equal and opposite phase encoding along that y-axis data point, depending on the direction of that gradient that we have applied. Now, it's that principle that allows us to take k-space and actually see that k-space has what's known as conjugate symmetry. If we took k-space and separated it into two, the top half here represents all of these phase encoding steps, and the bottom half represents all of these phase encoding steps here. What we can actually do is take the top half, flip it over, and we see that it's created two identical regions of k-space here. The two halves of k-space have what's known as conjugate symmetry. Now, this piece of knowledge allows us to see that we could, at least in theory, only acquire half of k-space, only use these phase encoding gradients here, and then we can use a mathematical formula to calculate what the second half of k-space would be. And in an ideal world, with a perfect machine, with no magnetic field inhomogeneities and no noise in our ability to pick up signal, we would be able to create a perfect image from only half of k-space.
Now, the reason I'm showing you all these variations within k-space is when we look at specific pulse sequences in the future, hopefully you'll see how we can selectively sample specific regions of k-space to get the image that we are trying to generate. So, I hope all of that made at least some sense to you. Again, I've linked a question bank in the description below. Go and test yourself on these concepts. Otherwise, join me in the next talk where we're going to look at how we sample the data, we're going to look at receiver coils and a concept known as receiver bandwidth that's going to show us how certain artifacts come up in MRI imaging. Until then, goodbye.