Physics for understanding fMRI artifacts: Part Nine

Conjugate variables redefined

In this post I'm going to provide the first part of a recipe for generating 2D images. It's going to be somewhat algorithmic. I may occasionally mention what a particular step implies, but for the most part I'm going to step through a sequence of events, produce a final recipe for you to follow, then go back and explain what some of the parts mean physically. This isn't the traditional approach to learning about k-space; most text books assume that you need to understand what it all means before you get to learn "the rules of the game." As is my wont, I'm coming at it backwards. My hope is that you will then be able to go back to your text books - I'll tell you where to look for subsequent explanations - and cement a decent understanding of the "why" of k-space, not just the "how."


Conjugate variables revisited

In Part Five of this series I introduced the Fourier transform and conjugate variables. The post focused on the most common pair of conjugate variables: frequency and time. If we have the time domain representation and we want to transform it into its frequency domain equivalent, we apply a (one-dimensional) FT, and vice versa.

But there is another pair of conjugate variables that is more useful and intuitive for imaging applications. (In this case your intuition for one of the variables may not develop until the end of this post, or later! Bear with me.) Whether it's maps, MRIs or architectural plans, the axes of an image are best described in terms of length. If we choose the centimeter as our unit of length, then FTing an axis in cm will yield an axis in 1/cm. You happen to have an intuitive notion of time, frequency and space from everyday life. Don't worry about what the reciprocal of real space means, just accept for now that it exists. We call this reciprocal space k-space because another term for 1/cm is the wavenumber, and the wavenumber is given the symbol k.

Representing pictures in reciprocal space

Let's take a random picture, in this case it's a digital photograph of a Hawker Hurricane plane. It's clearly a 2D picture. We have a digital version of it so we can do mathematical operations on it with a computer. If we do a 2D (digital) FT of the picture we get its representation in 2D k-space:

Physics for understanding fMRI artifacts: Part Eight

I had initially planned to go into 2D imaging next, but after some consideration I've decided instead to tidy up a few loose ends that follow more naturally from the last post: gradient-recalled echoes and slice selection. Then, in Part Nine I promise to introduce the second in-plane dimension. This route should better allow me to bring everything together at the end of the next handful of posts and permit you to see, and understand, the EPI pulse sequence at a glance. That's the plan. Let's see if we can make it work! (See Note 1.)

Gradient-recalled echoes

In the last post I used a frequency encoding gradient, also called a readout gradient (because it's on while the signal is being recorded, or read out), to produce one-dimensional images - profiles - of water-filled objects. This isn't the typical way that the signal is acquired, however. Instead, it is typical to acquire a refocused, or echoed, signal that has a certain symmetry in time in order to obtain some experimental benefits. I'll mention these benefits later. First, let's see how the gradient echo works.

Here is a simple gradient echo pulse sequence that is adapted from the simple readout gradient-only sequence that was considered in Part Seven:

The first thing to note is that the period of data acquisition (analog-to-digital conversion) has been delayed and now occurs in concert with a readout gradient having a negative sign, rather than being coincident with the positive gradient period labeled 1 in the figure. Also, the duration of data acquisition has been doubled. So instead of acquiring a free induction decay (FID) almost immediately after the 90 degree excitation pulse, we are now acquiring an echo signal at a later time. How and why does this echo form?

Physics for understanding fMRI artifacts: Part Seven

Magnetic field gradients and one-dimensional MRI

Now that you have a basic understanding of the Fourier transform and some of the practical matters that arise from digital signals, it's time to look at a basic imaging pulse sequence and even make some simple images. We're going to use frequency encoding only for the time being, and for now we're going to make one-dimensional images (also called profiles) so that we can introduce an alternative form of timing diagram to represent a pulse sequence.


A magnetic field gradient alters the local resonance frequency

When a sample is placed into the magnet, all the protons (1-H nuclei) resonate at a near-identical frequency. At 3 T that resonance frequency is approximately 123 MHz, as given by the Larmor equation. If we then impose a magnetic field gradient across the sample - your subject's head, say - instead of having the same resonance frequency uniformly across the brain, there will now be a linear dependence in space (see Note 1):

In a real image we might consider 64 different positions along x. These would define the voxels in one (in-plane) dimension of the image. But for the time being we'll consider just three points in the x direction: the central point, and one point either side.

At the center of the magnet the gradient has no net effect, so the resonance frequency at that point is still 123 MHz. We call this point the null crossing, because all three linear gradients, X, Y and Z, are engineered to have no effect here. (See Note 2.) And to keep things symmetric, the gradient null crossing is placed in the geometric center of the magnet - the isocenter - because that's where the main magnetic field has been engineered to be most homogeneous, and we want to do all our imaging in that location to get the best scanner performance.

MRI Claymation

It's a long weekend here in the US of A, it's hotter than Hades here in northern California (they said there would be a fog-cooled sea breeze! I want my money back!), and I am only halfway through the next post in the background physics series on account of having spent a very pleasant week in Quebec at the Human Brain Mapping conference. So, in lieu of anything more useful at short notice, I thought I'd share a truly awesome video I just found online, courtesy of Andre van der Kouwe and colleagues at MGH. The first two minutes demonstrate the method - surface renderings from MRIs of clay figures - and then it gets really fun: MRI making an image of itself.

427 views in two years simply doesn't do this work justice. Let's fix that!