Hi, in this module we'll talk about how we can take the signal that was acquired using the MR scanner and form an image. So just to recap, the subject is placed into the MR scanner and this causes the nuclei of hydrogen 1 atoms to align with the magnetic field. Now the nuclei precess about the field at similar frequencies but at random phase with respect to each other, and this causes a net longitudinal magnetization in the direction of the field. So within a slice of the brain, a radio frequency pulse is used to align the phase and then tip over the nuclei. This causes the longitudinal magnetization to decrease and establishes a new transversal magnetization. And then once we return the RF pulse, the system seeks to return to equilibrium. So again, everything is pointing in the longitudinal direction. Then we remove the RF pulse and it seeks to go back to the transverse direction. So this causes the transverse magnetization to disappear in a process we call transversal relaxation. And the longitudinal magnetization grows back to its original size in something we call longitudinal relaxation. So longitudinal relaxation is an exponential growth which is described by time constant T1. And in contrast, transverse relaxation is an exponential decay described by a time constant T2. And so during this process of returning to the equilibrium, a signal is created that can be measured using a receiver coil. And so now, in this module, we want to talk about how we can use that signal to create an image. So to start with, most structural MRI and fMRI scans involve the construction of a three dimensional brain volume from a set of two dimensional slices. So we acquire a set of two dimensional slices and we put them together, we sort of glue them together, to make a three-dimensional brain volume as illustrated here. So for now we're just going to concentrate on how do we acquire a single slice? So let's imagine that this brain slice is split into a number of equally sized volume elements, or voxels. So again, voxels are sort of the three-dimensional analog of pixels. And so voxels, we have equally sized boxes that we split up. And so in this little cartoon image, we have a little purple brain here and it's split up into 16 separate voxels. So what we want to do is we want to get a measurement of, let's just say for the sake of argument, the number of protons within each of these voxels. So we're going to call this measurement the rho(x,y) where (x,y) represents the location of the voxel. So basically if we were to kind of measure how many protons we had in each of these voxels, we could make an image of that information and we'd get a grayscale image of the object. So this is a very low spatial resolution because we only have 16 different voxels, but what we see here is that the darker voxels illustrate that we have more hydrogen atoms, while the lighter ones illustrate that we have few, and the white ones that we have very little hydrogen atoms. So this sort of gives a representation of the object that we're interested in. Now to get a better representation we would have to split the slice into more voxels, and that comes with a cost which we'll illustrate in a few slides. So the measured signal that we have, so what we want is we want a measurement over each of these voxels. But alas, the measured signal that we have combines information from the whole brain. So basically the signal that we're measuring is basically the combination of all the hydrogen atoms over the whole slice. So we sort of get a measure of the total number of hydrogen atoms. Unfortunately, this doesn't give us the information we need to figure out how many hydrogen atoms are in each of the individual voxels. So we need to make more measurements to get this information. And also we need to make different types of measurements. And so here is one of the clever things about MR scanners, is here we use a second magnetic field which is called magnetic field gradient. And using these magnetic field gradients we can sort of sequentially control the spatial inhomogeneities of the magnetic field, and so we can change the magnetic field across the brain. And so this allows us to make a new measurement which is now a weighted integral of the hydrogen concentration across the brain. So here you see that now we have for two constants, kx and ky, we can measure S(kx, ky), which is this row of x,y weighted by this exponential term which depends on x and y, and kx and ky. Where, again, kx and ky is controlled by S. So basically what we can do is we can alter values of kx and ky. And we can get new measurements of row xy until we have enough for which we can solve this inverse problem and get a reconstruction of the image, get the individual rho x y's back. Now some of you might be familiar with this equation, this is an example of the Fourier transform. And so what we can see here is that the measurements that we make, S(kx, ky), are actually the Fourier transform of the image that we want to reconstruct. So this is a very useful kind of relationship between the measurements made by the scanner and the image that we want to view and work with. Because of this, we can say that the measurements required in the frequency domain, and in the MR lingo this is usually called k-space. So by making measurements for multiple values of kx and ky we can ultimately gain enough information to solve the inverse problem, and reconstruct rho(x,y). And by doing this, once we have enough measurements of S(kx,ky), we can use the inverse Fourier transform and reconstruct rho(x,y). And that's the image that we want to get. I've shown you the measurements are integrals, but in practice the data measurements are made discretely over a finite region. So instead of using the continuous Fourier transform we use discrete Fourier transforms. And so the number of k-space measurements we make will ultimately influence the spatial resolution of the image. So we need enough measurements to solve the inverse problem. So for example, if I want to reconstruct a 4x4 image with 16 different voxels, I have 16 unknowns. So I have rho(x,y) in 16 different locations. So ultimately I have to make 16 k-space measurements so I have enough information to estimate those guys. So I need 16 equations because I have 16 unknowns. If I had made a lower resolution image, only a 2x2 image with 4 voxels, I would only have 4 unknowns. So in this case I'd only have to make four different k-space measurements in order to reconstruct that. So basically there's a tradeoff between the number of k-space measurements I need to make and the spatial resolution. So if I wanted to make a 64x64 image, I would have to make 4,096 k-space measurements. And this takes a little bit longer time, and so there's a temporal cost in doing this. There's lots of different ways of acquiring data in k-space. Here are two examples. One is a echo planar imaging, which basically samples k-space in sort of a Cartesian grid. And the other is a spiral which starts from the center and measures outward, so it changes the values of kx and ky in these manners. So those are just two popular ways of acquiring the data in k-space. Another thing that we need to know is that the measured k-space data is complex values. So the measurements are complex numbers, they have a real and imaginary part. And hence, because the k-space data is complex valued, the measurement at each voxel is also going to be complex value. So what we typically do here is instead of working with this complex value data, we work with the magnitude images, or we just take the square of the real part plus the square of the imaginary part. We add them up and take the square root, and this is what's called the magnitude of the complex number, and so we work with magnitude images. So we take the magnitude of rho(x,y) at each spatial location. So here's sort of the cartoon showing what happens. So basically in k-space, we change values of kx and ky and we make a number of different measurements here using a kind of an EPI trajectory. And then we make enough measurements of k-space and then we apply the inverse Fourier transform, and then we get this pretty picture of the brain. So this is sort of a cool illustration just showing that the measurements are very indirect. If you just look at the measurements by itself, it looks like some salt on a black table here. But once we do the inverse Fourier transform and we move into the image space, we get this beautiful brain image. So this is sort of just illustrating how we can use the signal from an MR scanner and reconstruct an image of the brain. So this is the end of the module. And so here we've talked about how we can take the signal that's acquired by the MR scanner and we can use it to create an image of the brain that we can use to study brain function and brain structure. Okay, I'll see you next time. Bye.