3 K-space
What is k-space and how is it related to gradients?
We’ll start by reviewing what we’ve already covered about gradients and how they’re used for slice selection, and how they’re used to create gradient echos.
Exercises
1.1. If you “play” a z-gradient with an amplitude of 1.2 G/cm for 20 ms and then turn it off and turn on (“play”) a z-gradient of -1.2 G/cm and start recording data, at what time (relative to the onset of the 2nd gradient, when you started recording data) will you record the most signal?
1.2 In that scenario, what is your echo time (pretending that the first gradient started immediately after excitation and there was no time lag between the first and the second)?
Now for the equations. I find them daunting each time I pick them up after being away for a while, because there are a lot of different variables and substitutions of variables. With enough time, however, it all starts coming together and spells out a really elegant understanding of how the details of gradients and timing create an image. On the first time through, it’s OK just to get the gist. The bottom line is that each point in your image has a different history, and that history determines its relative phase … and if you do it right, that pattern of phases lets us read out the Fourier transform of an image.
Exercises
2.1. Describe in your own words what the variable k represents.
2.2. Is k always increasing when a gradient is turned on? Why or why not?
Now, armed with the knowledge that we can navigate k-space by turning on and off gradients on different axes (and leaving them on for different times), we need to figure out how to systematically cover k-space. The gradient echo samples one line of k-space in one direction, but if we turn 2 gradients on at the same time, we just sample a line in a diagonal direction. So how do we systematically cover a Cartesian grid without over-sampling the center of k-space?
Exercises
3.1. If you’re acquiring an axial image, which gradient was on during slice selection?
3.2. If you want an anterior/posterior phase encode direction in your axial image, which gradient do you use for phase encoding?
3.3. That leaves us with only 1 possible answer for the third part of this question: which gradient is on when you’re doing the read-out (actually recording data) for an axial image with an A/P phase-encode direction?
Here’s a movie that illustrates an entire 2D FLASH acquisition. It’s not accurate in the sense that the phase blips and read pre-encoding gradients are stretched out (k is increased by increasing t instead of G … in the real world, you play a strong gradient for a short amount of time). But it shows every moment of phase evolution as a FLASH sequence repeatedly excites, encodes, and reads out one line of k-space at a time. Note that the 2nd half is redundant with the first. We’re only showing magnitude of the k-space matrix here. Since the signal induced in the coil has a phase, as well as a magnitude, the values in k-space are complex. In a perfect world. the k-space matrix has Hermitian symmetry, and each value is the complex conjugate of (the imaginary part has a negative sign compared to) of the value on the opposite side of k-space (mirrored through the center). Code for making this movie is in https://github.umn.edu/caolman/MRIsimulations/demos/FLASH_with_phase_encode.py.
Exercises
4.1. Kind of a trick question (i.e., the answer depends on how you define echo): how many echoes form during the read-out illustrated in the above movie?
We’ll end this section with a brief discussion of how our sampling rate determines our field of view, how our matrix size determines our resolution, and what aliasing is. Code for playing around with aliasing in 1D FFTs is in https://github.umn.edu/caolman/MRIsimulations/FFT/FFT_demo.py
Exercises
5.1. If the only thing you can change is your gradient strength, and you want to decrease your field of view, will you need to increase or decrease your gradient strength during read-out?
5.2. If you want higher resolution without changing your gradient strength, your two options are to sample faster or to sample for longer. Which approach will have the effect of increasing your resolution without changing your field of view?
Answers for exercises