This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

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From the course by Johns Hopkins University

Principles of fMRI 1

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This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

From the lesson

Week 1

This week we will introduce fMRI, and talk about data acquisition and reconstruction.

- Martin Lindquist, PhD, MScProfessor, Biostatistics

Bloomberg School of Public Health | Johns Hopkins University - Tor WagerPhD

Department of Psychology and Neuroscience, The Institute of Cognitive Science | University of Colorado at Boulder

Hi, in this module we'll talk about how we can

Â take the signal that was acquired using the MR scanner and form an image.

Â So just to recap, the subject is placed into the MR scanner and

Â this causes the nuclei of hydrogen 1 atoms to align with the magnetic field.

Â Now the nuclei precess about the field at similar frequencies but

Â at random phase with respect to each other, and this causes a net

Â longitudinal magnetization in the direction of the field.

Â So within a slice of the brain,

Â a radio frequency pulse is used to align the phase and then tip over the nuclei.

Â This causes the longitudinal magnetization to decrease and

Â establishes a new transversal magnetization.

Â And then once we return the RF pulse,

Â the system seeks to return to equilibrium.

Â So again, everything is pointing in the longitudinal direction.

Â Then we remove the RF pulse and it seeks to go back to the transverse direction.

Â So this causes the transverse magnetization to disappear

Â in a process we call transversal relaxation.

Â And the longitudinal magnetization grows back to its original size

Â in something we call longitudinal relaxation.

Â So longitudinal relaxation is an exponential growth

Â which is described by time constant T1.

Â And in contrast,

Â transverse relaxation is an exponential decay described by a time constant T2.

Â And so during this process of returning to the equilibrium,

Â a signal is created that can be measured using a receiver coil.

Â And so now,

Â in this module, we want to talk about how we can use that signal to create an image.

Â So to start with, most structural MRI and fMRI scans involve the construction

Â of a three dimensional brain volume from a set of two dimensional slices.

Â So we acquire a set of two dimensional slices and we put them together,

Â we sort of glue them together,

Â to make a three-dimensional brain volume as illustrated here.

Â So for now we're just going to concentrate on how do we acquire a single slice?

Â So let's imagine that this brain slice is split into a number of

Â equally sized volume elements, or voxels.

Â So again, voxels are sort of the three-dimensional analog of pixels.

Â And so voxels, we have equally sized boxes that we split up.

Â And so in this little cartoon image, we have a little purple brain here and

Â it's split up into 16 separate voxels.

Â So what we want to do is we want to get a measurement of, let's just say for

Â the sake of argument, the number of protons within each of these voxels.

Â So we're going to call this measurement the rho(x,y) where (x,y)

Â represents the location of the voxel.

Â So basically if we were to kind of measure how many protons we had

Â in each of these voxels, we could make an image of that information and

Â we'd get a grayscale image of the object.

Â So this is a very low spatial resolution because we only have 16 different voxels,

Â but what we see here is that the darker voxels illustrate that we have more

Â hydrogen atoms, while the lighter ones illustrate that we have few,

Â and the white ones that we have very little hydrogen atoms.

Â So this sort of gives a representation of the object that we're interested in.

Â Now to get a better representation we would have to split the slice into

Â more voxels, and that comes with a cost which we'll illustrate in a few slides.

Â So the measured signal that we have, so

Â what we want is we want a measurement over each of these voxels.

Â But alas, the measured signal that we have combines information from the whole brain.

Â So basically the signal that we're measuring is basically the combination

Â of all the hydrogen atoms over the whole slice.

Â So we sort of get a measure of the total number of hydrogen atoms.

Â Unfortunately, this doesn't give us the information we need to figure out

Â how many hydrogen atoms are in each of the individual voxels.

Â So we need to make more measurements to get this information.

Â And also we need to make different types of measurements.

Â And so here is one of the clever things about MR scanners,

Â is here we use a second magnetic field which is called magnetic field gradient.

Â And using these magnetic field gradients we can sort of sequentially control

Â the spatial inhomogeneities of the magnetic field, and so

Â we can change the magnetic field across the brain.

Â And so this allows us to make a new measurement which is now a weighted

Â integral of the hydrogen concentration across the brain.

Â So here you see that now we have for two constants, kx and

Â ky, we can measure S(kx, ky), which is this row of x,y

Â weighted by this exponential term which depends on x and y, and kx and ky.

Â Where, again, kx and ky is controlled by S.

Â So basically what we can do is we can alter values of kx and ky.

Â And we can get new measurements of row xy until we have enough for

Â which we can solve this inverse problem and get a reconstruction of the image,

Â get the individual rho x y's back.

Â Now some of you might be familiar with this equation,

Â this is an example of the Fourier transform.

Â And so what we can see here is that the measurements that we make, S(kx,

Â ky), are actually the Fourier transform of the image that we want to reconstruct.

Â So this is a very useful kind of relationship between the measurements

Â made by the scanner and the image that we want to view and work with.

Â Because of this, we can say that the measurements required in the frequency

Â domain, and in the MR lingo this is usually called k-space.

Â So by making measurements for multiple values of kx and

Â ky we can ultimately gain enough information to solve the inverse problem,

Â and reconstruct rho(x,y).

Â And by doing this, once we have enough measurements of S(kx,ky),

Â we can use the inverse Fourier transform and reconstruct rho(x,y).

Â And that's the image that we want to get.

Â I've shown you the measurements are integrals, but

Â in practice the data measurements are made discretely over a finite region.

Â So instead of using the continuous Fourier

Â transform we use discrete Fourier transforms.

Â And so the number of k-space measurements we make

Â will ultimately influence the spatial resolution of the image.

Â So we need enough measurements to solve the inverse problem.

Â So for example, if I want to reconstruct a 4x4 image with 16 different voxels,

Â I have 16 unknowns.

Â So I have rho(x,y) in 16 different locations.

Â So ultimately I have to make 16 k-space measurements so

Â I have enough information to estimate those guys.

Â So I need 16 equations because I have 16 unknowns.

Â If I had made a lower resolution image, only a 2x2 image with 4 voxels,

Â I would only have 4 unknowns.

Â So in this case I'd only have to make four different k-space measurements

Â in order to reconstruct that.

Â So basically there's a tradeoff between the number of k-space

Â measurements I need to make and the spatial resolution.

Â So if I wanted to make a 64x64 image,

Â I would have to make 4,096 k-space measurements.

Â And this takes a little bit longer time, and so

Â there's a temporal cost in doing this.

Â There's lots of different ways of acquiring data in k-space.

Â Here are two examples.

Â One is a echo planar imaging,

Â which basically samples k-space in sort of a Cartesian grid.

Â And the other is a spiral which starts from the center and measures outward, so

Â it changes the values of kx and ky in these manners.

Â So those are just two popular ways of acquiring the data in k-space.

Â Another thing that we need to know is that the measured k-space data is

Â complex values.

Â So the measurements are complex numbers, they have a real and imaginary part.

Â And hence, because the k-space data is complex valued,

Â the measurement at each voxel is also going to be complex value.

Â So what we typically do here is instead of working with this complex value data,

Â we work with the magnitude images, or

Â we just take the square of the real part plus the square of the imaginary part.

Â We add them up and take the square root, and this is what's called the magnitude

Â of the complex number, and so we work with magnitude images.

Â So we take the magnitude of rho(x,y) at each spatial location.

Â So here's sort of the cartoon showing what happens.

Â So basically in k-space, we change values of kx and ky and

Â we make a number of different measurements here using a kind of an EPI trajectory.

Â And then we make enough measurements of k-space and then we apply the inverse

Â Fourier transform, and then we get this pretty picture of the brain.

Â So this is sort of a cool illustration just showing that the measurements

Â are very indirect.

Â If you just look at the measurements by itself,

Â it looks like some salt on a black table here.

Â But once we do the inverse Fourier transform and

Â we move into the image space, we get this beautiful brain image.

Â So this is sort of just illustrating how

Â we can use the signal from an MR scanner and reconstruct an image of the brain.

Â So this is the end of the module.

Â And so here we've talked about how we can take the signal that's acquired by the MR

Â scanner and we can use it to create an image of the brain that we can use

Â to study brain function and brain structure.

Â Okay, I'll see you next time.

Â Bye.

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