Physics for understanding fMRI artifacts: Part Fourteen

Partial Fourier EPI

(The full contents for the PFUFA series of posts is here.)

In PFUFA Part Twelve you saw how 2D k-space for EPI is achieved in a single shot, i.e. using a repetitive gradient echo series following a single excitation RF pulse. The back and forth gradient echo trajectory permits the acquisition of a 2D plane of k-space in tens of milliseconds. That's fast to be sure, but when one wants to achieve a lot of three-dimensional brain coverage then every millisecond counts.

In the EPI method as presented in PFUFA Part Twelve it was (apparently) necessary to cover - that is, to sample - the entire k-space plane in order to then perform a 2D Fourier transform (FT) and recover the desired image. Indeed, this "complete" sampling requirement was developed earlier, in PFUFA Part Nine, when we looked at 2D k-space and its relationship to image space.

One aspect of the FT that I glossed over in previous posts has to do with symmetry. Perhaps the eagle-eyed among you spotted the symmetry in the 2D k-space of the first couple of pictures in PFUFA Part Nine. If you didn't, don't worry about it because I'm about to show it to you in detail. It turns out that there's actually no need to acquire the entire 2D k-space plane; it suffices to acquire some of it - at least half - and then use post-processing methods to fill in the missing part. At that point one can apply the 2D FT and recover the desired image.

Now, as you would expect, there's no free lunch on offer. There are practical consequences from not acquiring the full k-space plane. In this post we will look briefly at the physical principles of partial Fourier EPI, then in the next post we'll take a look at some example data that will provide a basis for evaluating partial versus full k-space coverage for fMRI.


What's in a name?

Before getting into the crux of the method I want to spend a moment considering the moniker, "partial Fourier." You may have come across alternative terms such as "partial NEX," "half NEX," "half Fourier" or "half scan." Furthermore, those of you familiar with clinical anatomical scans may have come across sequences such as HASTE, which stands for "Half Fourier Acquisition Single shot Turbo spin Echo." If it doesn't have a contrived acronym, it doesn't belong in MRI!

All of these various descriptors refer to the general principle of omitting from the acquisition some fraction of the k-space data, then synthesizing the omitted part by virtue of the property of "Hermitian symmetry," something we'll see in more detail in a moment. Whether exactly half of the k-space plane is acquired, or whether 5/8ths, 6/8ths, 7/8ths or some other fraction, I think it is minimally confusing to use the term "partial Fourier" for the acquisition. (In some cases the moniker is downright misleading when applied to EPI, as for "half NEX." See Note 1.) Thus, from now on I shall use partial Fourier (pF) to encompass all variants where only a portion of the final k-space plane is acquired, and where the k-space step size is unchanged. (See Note 2 for the difference between partial Fourier and parallel imaging when it comes to omitting k-space lines.) But you should note that in order to apply what you see here on your scanner you may need to translate into your scanner's vernacular first.


Conjugate symmetry in the k-space of real objects

PFUFA Part Nine covered the equivalency of information contained in image space and k-space (or reciprocal space), two domains known as conjugate domains and represented by the conjugate variables of cm and 1/cm. Here's a figure reproduced from PFUFA Part Nine:

Take a closer look at the k-space plane on the right. Notice how symmetric it is, left to right and top to bottom? The symmetry is diagonal: what's in the bottom left quadrant is repeated in the top right quadrant. That's because this k-space plane was obtained from the 2D FT of a real object; in this instance the (digital) picture of a Spitfire on the left. The same symmetry property exists for brains as for pictures of planes.

Provided the object we are trying to image is real - and with the possible exception of a few of the extraordinary people I've seen around downtown Berkeley, human brains are real objects - then the k-space representation of that object will exhibit what is called Hermitian symmetry. Signals, S, in one half of k-space are the complex conjugate of the other half of k-space:

S(kx,ky) = S*(-kx,-ky)

where * denotes the complex conjugate. Here is what that relationship looks like:

where the magnitude of the signal at coordinate (kx, ky) is given by either (a + ib) or its complex conjugate, (a - ib). (Recall that the magnitude of the real part of the signal is 'a' while the magnitude of the imaginary part of the signal is 'b.') If you're interested in learning more about the complex conjugate symmetry of k-space and related phenomena then this wiki page provides a nice summary, while this excellent tutorial from SCOPE Online reviews many issues pertaining to k-space, including issues of partial k-space. Or, you can just accept the general idea and move on because it's all you really need to know to comprehend partial Fourier EPI.


Going faster by leaving stuff out

The above symmetry implies that, in principle, we don't actually need to acquire the entire 2D k-space plane in order to reconstruct an image from a 2D FT. I'll get to the practical issues associated with that statement below, but for now let's assume that practice and theory are in agreement. Furthermore, let's put aside the different spatial encoding mechanisms - phase encoding and frequency encoding - that are typically used for 2D MRI sequences such as EPI. What are the implications of conjugate symmetry of k-space for MRI?

Recognizing the symmetry in k-space we could choose to omit the top or bottom, or the left or right half of k-space, and use the mathematics of complex symmetry to "fill in" the omitted half. Or we could compromise somewhere between full and half k-space acquisition, and omit just a portion of the top or bottom, or the left or right half of the acquired k-space, then reconstruct the missing part. Here's an illustration of the process applied to a 2D k-space for which the bottom quarter has been omitted, requiring some sort of mathematical "transfer" of parts of the diagonal quadrants into the gaps:

An illustration of the utility of conjugate symmetry. Some of the bottom half of k-space has been omitted (solid black region), requiring that the  k-space portions from the diagonal quadrants (regions above the blue dashed line) be copied (dashed white arrows) into the missing spaces prior to 2D FT.

I trust it's clear that it doesn't have to be the bottom half of k-space that's omitted. There will be experimental implications for which portion of k-space we omit, but the mathematics is the same whether we omit an upper, lower, left or right portion. And, while I have omitted just one quarter of k-space - half of the bottom half - we could, in principle, omit an entire half of k-space and reconstruct the missing half from the half we acquire.


How do we reconstruct the missing part of k-space?

Before we look at the reconstruction options we first need to consider the experimental limitations that can hamper any method we select. As already mentioned, the imaged object must be real; that is, its mathematical description must not require any imaginary component. This requirement restricts us to obtaining the magnitude of the resulting image; we won't be able to reconstruct properly any phase variations across the image, assuming that we might be interested in such information. (Phase information isn't often used in fMRI, nearly all experiments use magnitude EPIs as the starting point.)

But there's another limitation concerning phase that might affect us, even in the world of magnitude EPI. Since the raw data - the 2D k-space data - is intrinsically complex i.e. it is composed of real and imaginary parts, even when the object being imaged is purely real, we have to take care that any phase variations that do arise across the image - even though we're ignoring them by looking only at the magnitude of the image - are relatively small. If there are large phase variations across the image then there may be insufficient information in the truncated k-space to properly reconstruct the missing part. We would violate a basic premise of the Nyquist sampling frequency for some of the information in what would otherwise be the fully sampled k-space, and that would produce N/2-like ghost artifacts.

The final restriction seems ridiculous at first glance, but when we come to look at pF EPI in future posts you'll see that it can actually be the most pernicious of all: the signal we're imaging has to be somewhere in the sampled k-space. "What?" you say? "It's possible to miss the signal?" Yup. Magnetic susceptibility gradients, in particular, can cause the actual (partial) k-space trajectory to differ significantly from the theoretical one that's imposed by the imaging gradients alone. (See Note 3.)

To illustrate the last restriction, take a look at this example of 2D k-space for a T2-weighted anatomical scan:

Example k-space and resulting image, taken from the SCOPE online tutorial on k-space.

There is only modest symmetry in the k-space on the left, even for this anatomical scan. In particular, notice how there are signals in the bottom half of the k-space - there's a band about one third of the way up, for instance - that don't have counterparts in the upper half. Thus, if a partial Fourier scheme was used such that the bottom third (or more) of k-space were omitted from the acquisition, we can expect that there would be signal voids and perhaps other artifacts somewhere in the resulting image, compared to the full k-space image. A pF version wouldn't look precisely like the full k-space image that appears on the right-hand side of the figure above.

The degree of symmetry of the signals in (full) k-space has immediate implications for the method used to reconstruct the missing k-space from a partial Fourier acquisition. We are reliant on conjugate symmetry and no matter what post-processing magic we might consider, we can't create information that doesn't exist in the acquired partial k-space! This limitation on the post-processing can, a little perversely, be considered an advantage for pF EPI: there may not be a significant benefit to fancy reconstruction schemes because the primary damage has likely already been done in the acquisition!


Nothing added, nothing taken away

Thus we get the simplest approach to computing the missing k-space data: don't bother. On Siemens scanners, at least, the default is to simply "zero fill" the omitted k-space points. A complete k-space matrix is constructed from the acquired k-space by appending as many lines of zero valued data points as needed to make the matrix rectangular. The resulting matrix is then fed into the 2D FT as usual, producing the final image.

Completing the k-space matrix with zeros adds no signal and no noise to the image and is generally a benign way to prepare the matrix for 2D FT, provided you weren't too aggressive with the omitted portion of k-space in the first place. (I'll deal with real examples of pF EPI in the next post.) But zero filling does have one immediate cost: it smooths the resulting images a bit, because we have effectively multiplied the fully acquired k-space matrix by a step function that produces zeros in that part of the 2D matrix we didn't actually acquire. Or, if you want another way to conceptualize the smoothing, refer back to PFUFA Part Eleven where we looked at "what lives where in k-space" and note how edges in the image reside in the high spatial frequencies in k-space. By acquiring only a partial k-space in one dimension, the signal-to-noise of the edge information has been degraded relative to the low spatial frequencies.

What other alternatives are there for reconstruction of the omitted k-space portion? There are various ways to estimate the omitted portion from the acquired portion, including ways to estimate the phase variation. These are nicely summarized in this PDF (which isn't EPI-specific). However, there isn't much literature comparing the different reconstruction methods for EPI, or more specifically pF EPI for fMRI applications. What's more, they all seem to be a large amount of work for a small potential benefit, and they might have costs that would affect time series EPI in a bad way. Since the acquisition already imposes constraints on the final images, and because most people will use whatever reconstruction is the default on their scanner anyway, I will leave the reconstruction issue for the time being and return to the acquisition options. In the next couple of posts I'll be using zero filling for reconstruction - it is the Siemens default, as already mentioned.


Which portion of k-space should we leave out for EPI?

To complete this post let's look at two options for partial Fourier EPI: omitting either the early or the late echoes from the gradient echo train. (We could conceivably omit readout data points instead - left- or right-hand portions of k-space in the figures to come - but this doesn't save us as much time and we generally focus on the phase encoding dimension for fMRI.) So, let's return to the (full) k-space trajectory for EPI that we saw in PFUFA Part Twelve:

I've already indicated that we could either omit some of the early echoes or some of the late echoes. (Leaving out both early and late echoes is tantamount to reducing the maximum k-space extent, which has the effect of reducing the image resolution.)

If we decide to omit some of the early echoes then our k-space trajectory might look like this:

This trajectory means that we can, if we choose, hit the center of k-space - the point which defines the echo time, TE - sooner than we would have had we acquired the full matrix. Thus, one immediate consequence of omitting early echoes is to permit a shorter minimum TE for what is essentially the same resolution image. (Recall, however, the smoothing that I already mentioned. We'll look at smoothing effects in a later post.)

Whether a shorter TE is beneficial for fMRI applications will depend on many factors. We might expect less signal dropout, but we might also expect lower BOLD functional contrast if the TE we use departs significantly from the T₂* of the gray matter. (If these statements are baffling then you might want to read the pertinent sections in my user training guide/FAQ.) But there is one thing that omitting the early echoes doesn't do: it doesn't allow more slices within the TR. Put another way, it doesn't save us any time per slice unless we also shorten TE. I'll deal with the issue of speed - slices/TR - in a subsequent post.

What if we acquire all the early echoes and omit some of the later ones instead? That k-space trajectory looks like this:

The minimum TE is unchanged compared to acquiring the full k-space matrix. Now, however, we reach the end of data acquisition for this slice sooner (relative to TE) than we would for a complete k-space plane. Thus, omitting the later echoes permits us to increase the number of slices per TR without changing TE (or any other timing parameter).


That'll do for this post. In the next post I will show some examples of partial Fourier EPI from phantoms and brains. We will investigate the practical consequences of using pF with zero-filled reconstruction of the omitted k-space portion, in particular the effects on signal dropout and image smoothing. And in the post after that we will look at imaging speed issues, with a view to generating some guidelines for selecting, and using, partial Fourier EPI for fMRI.

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Notes:

1.  The term, half NEX seems to be an historical reference to the application of partial Fourier methodology to a standard spin warp-style phase encoded scan, where just one line of phase-encoded information (one line of k-space) is acquired following each RF pulse. (See PFUFA Part Ten for a description of gradient echo imaging using conventional, spin warp-style phase encoding.) In that case there is one line of k-space detected per excitation RF pulse; there will be N pulses for N phase-encoded lines of k-space in the final image. Hence, omitting some of the N EXcitations makes sense for spin warp phase encoding. But when applied to EPI, which is most commonly acquired as a single shot (single RF) acquisition, the historical term just gets confusing. We only use a single excitation! N=1. I understand there are practical reasons for not changing terminology on a scanner - not least keeping the installed base of users happy - so I'm not pointing fingers, just trying to clarify what might otherwise lead to some bewilderment amongst those not fluent in multiple scanner languages. This stuff's complicated enough as it is!


2.  With partial Fourier encoding the k-space step size is unchanged from that used for the full k-space coverage. In other words, the distance between successive frequency-encoded lines of k-space is exactly as was demonstrated in earlier posts on EPI. All we are going to do is (intentionally) fail to acquire either the first few or the last few frequency-encoded lines, thereby failing to fill the entire 2D k-space plane with real data. But there are other ways to save time by skipping over some fraction of k-space. Parallel imaging (PI) methods, such as SENSE and GRAPPA, make use of spatial information encoded in the receive field of the RF coil, and then save acquisition time by skipping some k-space lines, e.g. all the odd-numbered frequency-encoded k-space lines might be skipped for an acceleration factor (R) of two. Note, however, that parallel imaging changes the k-space step size in the phase encoding direction - it is doubled for R=2, for instance - whereas partial Fourier omits a continuous patch of k-space while leaving the k-space step size unchanged for the acquired portion, as shown here:

The total number of lines of k-space omitted is generally larger for PI than for pF. Thus, the time savings can be larger for PI than for pF, but at the expense, usually, of increased motion sensitivity. The issue of whether (and when) to use PI, pF or other schemes for saving time will be considered in future posts. But there is a brief comparison of PI (GRAPPA) versus pF in my user training guide/FAQ to be going on with.


3.  The restriction that the signals actually get sampled in the (partial) k-space plane is not fundamentally different from the case when we are sampling the entire plane: magnetic susceptibility gradients may interact with the imaging gradients and cause the signals to be moved entirely out of the (theoretical) k-space plane then, too. This is one of the sources of the infamous EPI dropout! So, what we're really doing is imposing an even stricter requirement that the magnetic susceptibility gradients be minimized through processes like shimming, or we can expect to pay a penalty in additional regions of signal loss.