Physics for understanding fMRI artifacts: Part Eleven

Resolution and the field-of-view as seen in k-space

Understanding how distances in k-space manifest as distances in image space is quite straightforward. All you really need to remember is that the relationships are reciprocal. The discrete steps in k-space define the image field-of-view (FOV), whereas the maximum extents of k-space define the image resolution. In other words, small in k-space determines big in image space, and vice versa. In this post we will look first at the implications of the reciprocal relationship as it affects image appearance. Then we'll look at the simple mathematical relationships between lengths in k-space and their reciprocal lengths in image space.


Spatial frequencies in k-space: what lives where?

I mentioned in the previous post that there's no direct correspondence between any single point in k-space and any single point in real space. Instead, in k-space the spatial properties of the object are "turned inside out and sorted according to type" (kinda) in a symmetric and predictable fashion that leads to some intuitive relationships between particular regions of k-space and certain features of the image.

Here is what happens if you have just the inner (left column) or just the outer (right column) portions of k-space, compared to the full k-space matrix arising from 2D FT of a digital photograph (central column):

An illustration of the effect of nulling different regions of k-space from a full k-space matrix, applied to a digital picture of a Hawker Hurricane aircraft. The full k-space matrix and corresponding image are shown in the central column.

Inner k-space only:

The inner portion of k-space (top-left) possesses most of the signal but little detail, leading to a bright but blurry image (bottom-left). (See Note 1.) Most features remain readily apparent in the blurry image, however, because most contrast is preserved; image contrast is due primarily to signal intensity differences, not edges. If this weren't true we would always go for the highest signal-to-noise MRIs we could get, when in practice what we want is the highest contrast-to-noise images we can get! Imagine an MRI that had a million-to-one SNR but no contrast. How would you tell where the gray matter ends and the white matter begins? Without contrast no amount of signal or spatial resolution would help. So much for SNR alone!

Outer k-space only:

If we instead remove the central portion of k-space (top-right) then we remove most of the signal and the signal-based contrast to leave only the fine detail of the image (bottom-right). Strangely, though, it's still possible for us to make out the main image features because our brains are able to interpret entire objects from just edges. In actuality, however, there is very little contrast between the dark fuselage of the Hurricane, the dark shadow underneath it and the dark sky. Our brain infers contrast because we know what we should be seeing! If we were to try doing fMRI, say, on a series of edges-only images we would run into difficulties because we process the time series pixelwise. With a relatively low and homogeneous signal level you can bet good money the statistics would be grim.

Physics for understanding fMRI artifacts: Part Ten

(For the answer to the homework k-space diagram given at the end of Part Nine, see Note 1.)

K-space in two dimensions

As anyone knows who has encountered MRI professionally, whether in research or medicine, there seems to be an endless array of pulse sequences to choose between. The variety can be overwhelming at first. Nor is the situation helped by different vendors using different acronyms - we always use acronyms in MRI! - for what are essentially the same sequence.

It's little wonder, then, that most neophytes' eyes glaze over when it comes to comparing and contrasting any two pulse sequences if the taxonomy appears to be ad hoc. Where on earth to start? But it turns out that most pulse sequences can be categorized fairly easily, and their heritage traced, by separating the part(s) of the sequence that is responsible for spatial encoding, from the part(s) of the sequence that will provide the tissue or functional contrast. Occasionally there is overlap within the sequence of these two missions, but even then it's usually straightforward to understand the spatial encoding and interpret its genesis.

A useful pictorial representation of imaging pulse sequences

It turns out that there are only a handful of spatial encoding methods in common use these days, almost all with roots in the late 1970s or early 1980s. While new pulse sequences appear in the literature all the time, when you look at their k-space representations you'll be able to see how each new method has developed from a small number of key ideas from those early years. It's possible to categorize the encoding methods without k-space, but the k-space formalism makes comparisons trivial (in MR terms).

Spatial encoding methods can be separated into families derived from a central idea. For instance, following Lauterbur's original imaging paper in 1973 (which led to the family of projection reconstruction methods), in 1975 Richard Ernst's group came up with a sequence that utilized a 2D Fourier transform to yield the final image. (See Note 2.) It was a remarkable breakthrough and is the grandparent of nearly all medical/biological sequences still in common use today.

Still, even geniuses miss opportunities every now and then. And in 1980 a group at Aberdeen came up with a far more practical implementation of Fourier imaging, using amplitude-modulated gradients in a "constant time" pulse sequence, rather than the fixed amplitude, variable time scheme of Kumar, Welti and Ernst. It is this constant time scheme, which the Aberdeen group termed "spin warp" phase encoding, that provides the basis for most clinical (anatomical) scanning used today. It's also a good scheme to look at when first encountering 2D k-space, so we'll consider it in detail in this post.

The goal revisited

In the first part of the last post (see Part Nine) I used two examples of digital images to illustrate how the information content in a 2D plane of image pixels can be equivalently represented in reciprocal 2D space, or k-space. I mentioned that both the images and the k-space comprised 512x512 points, but later on when I started to draw (one-dimensional) k-space trajectories I did so on a k-space plane that was represented by just a set of axes, not discrete points. In case you think that image space and k-space in MRI are continuous, I'm going to spend a moment considering the digital k-space plane explicitly. (Like real space, k-space can also be continuous rather than digital, but that's not how MRI works.)

Here is a 16x16 plane of k-space points (see Note 3) overlaid on some actual signals to reinforce the point that we're digitizing a continuous process:

Courtesy: Karla Miller, FMRIB, University of Oxford.

The goal is to traverse the entire k-space plane, i.e. to use our gradients to follow a trajectory that crosses every single point (as defined by the white grid itself), acquiring data (with our receiver coil), one point for each grid coordinate, as we go. Once we have traversed the entire 2D plane (and assuming a suitable data acquisition scheme) we will have 16x16 k-space data points and will then be in a position to apply a 2D FT and get a 16x16 image out. (See Note 4.)

Lessons from epidemiology

Ben Goldacre, psychiatrist, occasional fMRIer and critic of rubbish medical research over at BadScience.net, has produced a radio documentary that covers many of the pitfalls of modern medical science:

Science: From Cradle to Grave

It's aimed at a general audience but there are important reminders for us in fMRI-land.

Confounds abound

Epidemiology is a lot like fMRI when it comes to discriminating correlation from causation. As with many areas of research using human subjects, there are usually limits to the factors that can be controlled between groups, or even across time for an individual subject.

But there are often some simple things that we can measure - like heart and respiration rates during fMRI - and thus control for. Surely we should be measuring (and ideally controlling for) as many parameters as we can get our hands on, especially when the time and expense are comparatively minor. Get as much data as you can!

Resting state fMRI: a motion confound in connectivity studies?

Neuroskeptic has done us a favor and covered a recently accepted paper from Randy Buckner's lab concerning the role of motion when determining connectivity from resting state fMRI. Not only was the amount of motion found to differ systematically between male and female subjects, but this systematic difference was preserved across sessions, suggesting that it is a stable trait. The implications for group studies are discussed in the paper, and Neuroskeptic adds further perspective. It's a warning that all resting state fMRIers should heed.

Non-neural physiology.... again

There are some important limitations to consider, however. While ventricular and white matter regions were used as ways to remove some effects of heart rate and motion, the study did not acquire breathing or heart rate data and so the authors were unable to perform the more advanced BOLD-based model corrections developed by Rasmus Birn and Catie Chang (references below). Instead, they followed what might be considered the "typical" post-processing steps, including global mean signal removal. The methods are fine, my point is to highlight the limitations of the "typical" processing stream in the absence of independent physiological data.

So, could the gender differences be explained with improved physiological corrections? What about the motion correction methods in current use: might they not be up to the job we give them? We'll have to wait for further studies to find out. In the mean time, surely it only makes sense to acquire physiological data with resting state fMRI - heart rate and respiration at the very least, although there are suggestions that time course blood pressure might also be useful - and to try to explain as many confounds as possible before concluding there's a group difference due to brain activity.





References for physiological corrections:

Birn et al., Neuroimage 31: 1536 –1548, 2006.
Birn et al., Neuroimage 40:  644-654, 2008.
Chang & Glover, Neuroimage 47: 1381–1393, 2009. Also 1448 –1459 in the same issue.