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of going through a detailed illustration of how that occurs, which is sort of beyond my level of PowerPoint capability, I'm just gonna go to to the mathematics of it because the concept very easily extends from the one dimensional case. So here I showed you, in

the case of one dimension, the signal as a function of time was given by this. In two dimensions, it's relatively easy to extend that. So now the distribution of water instead of just varying with the x position, now varies as a function of both x and y, and the frequency of the signal now

is dependent both a sinusoidal variation on the x position and the x gradient, times a sinusoidal variation of the y position and the y gradient. And the overall composite signal is gonna be a sum over all the

x and y positions. [BLANK_AUDIO] In one dimension, we applied this k substitution where we had the k variable is equal to gradient times time. And from that, we were able to produce this signal as a function of k-space position, which was just sum of, again, the distribution of water, which is what we're ultimately interested in, times the sine of this k variable times time. So two dimensions

is exactly the same manner, except in this case, we need two k variables. We need an x k variable, which is equal to the x gradient times time, and a y variable, which is equal to the y gradient times time. And the overall signal, as a function of kx and ky, will be given by this formula here. So in this case, you can see again, we have the distribution of water here and we have multiplied by the x k

position and the y k-space position. So now, with these kx and ky variables, we can now describe two dimensional

just see how that works. So before you remember that we have signal evolving as a function of the both kx and ky variables. So if I just turn on an x gradient then we know that we're just moving

along the x direction of k-space. So I haven't done anything with the y gradient, so my y gradient is zero, so there's no movement in ky. So as I turn my x gradient on as a function time, I'm just simply evolving in k-space as a function of the kx position. On the other hand, if I put my y gradient on, if my x gradient is now zero, that means I'm gonna be moving purely in the y direction

of k-space. So if I put these two together, what you'll see is that I'm essentially moving in orthogonal directions in k-space. So here I have my full k-space data matrix, and the kx variable just simply corresponds to the horizontal direction, and the ky line corresponds to the vertical direction. So by turning on these X and Y gradients, I can move in orthogonal directions

of k-space. And as I've showed you earlier where we had 1D, you can produce many different k-space trajectory, but now we have more full flexibility of two dimensional variability. So let's say, initially, I turn on my x gradient as a function of time. So I just simply, in k-space, I'm evolving in the kx direction. If I then turn on my y gradient, that moves me in the in the horizontal, sorry, in the vertical direction, so I

move up a line of k-space. And if I didn't repeat that procedure by turning on the x gradient again, I then sweep out another line of k-space. Again, because I've already turned on my y gradient, I'm up at a higher ky position, so I'm not acquiring the same k-space data anymore. I'm acquiring it at a different

ky line. And I then repeat that procedure, but in this case, I acquire with a larger ky variable, so that's gonna move me up to a larger line in k-space. And I can then repeat that. So simply by repeating this sort of pattern, acquiring x, turning on my y gradient a little bit, acquiring another x, and turning my y gradient to a larger amount, I'm gonna be acquiring different lines of k-space each time. And

they're gonna be adjacent to each other rather than in the same one. Because each time I'm putting on a different y gradient, I'm going up to a different position along the y direction of k-space. Now, in general, as I mentioned before, there's an infinite number of possibilities, so you could also get to the equivalent position in y k-space by having a gradient that's half as large but twice

as long. And that's another alternative way of doing it. That's typically not the way it's done. Typically, you acquire a series of gradients that have the same duration but different amplitudes. So now with what we've described, we can now have a fairly complete description of a basic pulse sequence. So that's this diagram here which shows

you a basic gradient echo pulse sequence. And we all know what all the different elements of this pulse sequence do. So the RF pulse tips down the magnetization so we can get a signal, the Y gradient moves us to different lines in k-space, and the X Gradient is where we read out a line of k-space. Typically, the Y Gradient is depicted by this ladder-like appearance to indicate that on each iteration

of this, we increase the amplitude of the Y Gradient. Now one additional piece I have to mention is that, as I said, the data acquisition, when we turn these gradients on, that's always moving us to k-space. We're always going to different positions in k-space. But we don't necessarily have to take a measurement at every position in k-space, and as we'll see in later lectures, there's a good

reason why we don't do that because it makes the interpretation of the image more difficult. So there's often typically a fourth line in these pulse sequence timing diagrams that indicates when we turn our data acquisition on and off. In this case, you can see that the data acquisition is concurrent with the X Gradient, so that's essential when we're reading out each line of k-

space. And, as I had mentioned in previous lectures, the x-gradient is called the "readout gradient" typically, because that's when we read out the k-space data, and the y gradient is called the "phase encode" gradient. So just a bit of MR jargon that's used quite commonly. And as I've also mentioned in previous lectures, for any pulse sequence,

going on with the gradient waveform. So, let's say you see, in some paper somewhere or some journal, these gradient waveforms that are used, so this is quite a complicated ones, and you have to say well what is the k-space trajectory that would be produced by this? Well you just simply have to run them through these formulas and determine what the effect is on the

k-space trajectory. So, for an example, in this k-space trajectory that you follow is this spiral trajectory, and for this particular sets of gradients the k-space trajectory you follow is this one here. Now one thing I should just briefly mention, just for the purists who might be watching, this description of the k-space variables, gradient times time, is a slight simplification

from what really occurs. So it's actually slightly more complicated than this, but I'm not gonna go into that for these lectures, just in the interest of simplicity. Now, up to now we've been talking about two-dimensional imaging. But you can also extend this exact same concept

we have the signal as function of time in two dimension with

two orthogonal gradients x and y. So, obviously, you could probably guess that to make this a three-dimensional image we could just add in a gradient in third orthogonal direction. So a gradient in the z direction as well. And the signal is just a simple extension of the two decay. So now we have, again, a distribution of magnetization that depends on x, y, and z, and now we have three

terms that determine the frequency of the signal Gx, Gy, and Gz. And to define our k-space variables, in the case of 2D, we had two variables kx and ky, and three dimensions we have correspondingly three variables kx, ky, kz. So again, the interpretation the signal is exactly the same. The signal evolves as a function of all three of these k-space variables. So it's slightly more complicated,

but the basic content is exactly the same. So now instead of having a 2D k-space trajectory, we would have a three-dimensional k-space trajectory, so we'd have to essentially fill up a cube rather than just a square. [BLANK_AUDIO] So a 3D pulse sequence looks like what you see here. So in this case, now you can see we have x, y, and z gradients.

And again, the y and z gradients have this ladder-like appearance because, again, we're increasing the y and z gradients on each subsequent iteration of our acquisition scheme. And again, we typically have one read out direction, along the x, and the y and z, in this case, two phase encode directions. Now one thing I should also mention at this stage is that, previously, I was talking about

information by doing what's called a 2D multislice scan. So this is essentially where you acquire a series of two-dimensional images that correspond to a very thin section of the anatomy. So previously, when we did a full three dimensional scan, we had this pulse sequence here. We had phase encoding in the y and z gradient. In the case

of the two-dimensional scan, so we only have to have gradients in the two dimension cause we're only encoding a two-dimensional image, but what we do is we just excite only a thin section of the anatomy. And we do that by turning/g on our z gradient with the RF pulse, and that's called

a slice selection. And then we acquire data acquisition just as before. So those are the two different methods of acquiring three dimensional scans. You either acquire a full 3D scan where you're phase encoding in two dimensions, or you can acquire a 2D multislice scan where you're only imaging a thin slice of the anatomy at any one time, and you're encoding the two dimensional image at different positions

along the anatomy. So just to summarize what we've been through in this talk. So we've shown that gradients create a mapping between

spatial position and frequency. We showed that by using a Fourier transform we can determine the relative signal content at different spatial positions. We've shown that the resolution is inversely proportional

to how far we were in k-space. Shown the field of view is inversely related to the spacing of the k-space. And we've shown that gradient in orthogonal directions can be used to encode multidimensional images. So if we need to encode two dimensions, we need two orthogonal gradients, three dimensions we need three orthogonal gradients. And we've shown that the k-space trajectory is determined

directly by what's going on with the gradients. The x, y, and z gradient correspond to an x, y, and z position in k-space, that's it. [BLANK_AUDIO].

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