Functional MRI of dolphins?

Those of you who follow me on Twitter may have noticed that I've been scanning some post mortem brains of Cetacea over the past year or so. That's whales, dolphins and porpoises to you and me. The brains come in all shapes and sizes, from a rather tiny Amazon river dolphin, about the size of a fist, to fin and sei whale brains that are so wide they have to be inserted sideways, hemisphere-first, into the 3 T (human) head coil. The conditions and ages of the brains vary tremendously as well. Some have been fixed in formaldehyde for decades yet yield remarkably decent signal, others have been stored in ethanol and are as hard as rubber with T2 to match. A few months ago we obtained a recently deceased fresh brain of a white-sided dolphin which we were able to scan within about twelve hours of its demise. The image quality was magnificent.

What do we plan to do with all the post mortem data? That is still being formulated. Initial motivation for the project came from some Berkeley anthropologists with an interest in comparative neuroanatomy across higher mammalian species. Coincidentally, Greg Berns' group at Emory has recently produced a nice example of dolphin brain tractography and his recent study is a good example of what might be done in future. There's a commentary on Greg's study here, and an example image from his paper below. We are now determining how we might combine resources, share data and all that good stuff. More on what will be available to whom and when as we progress.

From: Berns et al. DOI 0.1098/rspb.2015.1203

In any case, after I posted the white-sided dolphin MRI to Twitter someone asked, likely facetiously (I suppose that should be flippantly), whether functional MRI was next. FMRI of dolphins was the subject of an April Fool's Day joke a few years ago, and it does seem far-fetched at first blush. So, too, does studying trained dogs with fMRI, but Greg Berns' team is already doing that. Since thought experiments are cheap I figured I'd write a blog post to consider what might be feasible today if one were sufficiently motivated (read sufficiently well funded) to want to do fMRI of cetaceans. If nothing else we might learn something as we're forced to consider the manifold factors.

Potential of ultralow field T1 and high field T1ρ in evaluating brain trauma

Crazy Scientist sent me a link to a paper, "Neuroimaging after mild traumatic brain injury: Review and meta-analysis," (doi:10.1016/j.nicl.2013.12.009) and it prompted me to do something with a brief review I wrote this time last year as a way to plan some research activities on MRI of mild traumatic brain injury (mTBI). By a remarkable coincidence the review paper was made available online four whole days before I completed my own review. I've yet to read the published review so I can't yet tell if I wasted my time. In any event, the document I wrote was for internal consumption (for my collaborators, and to clarify my own thoughts) and was never designed to become a public document. But since our research direction changed mid-year I figured I might as well stick my review out there in case anyone can make use of it.

The title of my document is the same as the title of this post. You will find the contents pasted below, or if you prefer you can download a PDF from this Dropbox link. I have quickly re-read it to check for major bloopers, and I've added a couple of update notes highlighted in yellow. There may well be some direct copy-paste of parts of a few of the papers I reviewed, especially those with heavy neuroradiology content where I am generally a long way out of my depth and would prefer to accept charges of plagiarism than get the medical terminology wrong!

For the record, we are still interested in mTBI but the logistics of studying acute brain injury in a non-hospital setting, using a home-made machine (the ULFMRI) sitting in a second basement lab in a physics department, made it all too hard to pursue right now. We have shifted instead to studying chronic conditions where we have a fighting chance of getting a few people scanned in our unorthodox facilities.

QA for fMRI, Part 3: Facility QA - what to measure, when, and why

As I mentioned in the introductory post to this series, Facility QA is likely what most people think of whenever QA is mentioned in an fMRI context. In short, it's the tests that you expect your facility technical staff to be doing to ensure that the scanner is working properly. Other tests may verify performance - I'll cover some examples in future posts on Study QA - but the idea with Facility QA is to catch and then diagnose any problems.

We can't just focus on stress tests, however. We will often need more than MRI-derived measures if we want to diagnose problems efficiently. We may need information that might be seem tangential to the actual QA testing, but these ancillary measures provide context for interpreting the test data. A simple example? The weather outside your facility. Why should you care? We'll get to that.


An outline of the process

Let's outline the steps in a comprehensive Facility QA routine and then we can get into the details:

• Select an RF coil to use for the measurements. 
• Select an appropriate phantom.
• Decide what to measure from the phantom.
• Determine what other data to record at the time of the QA testing.
• Establish a baseline.
• Make periodic QA measurements.
• Look for deviations from the baseline, and decide what sort of deviations warrant investigation.
• Establish procedures for whenever deviations from "normal" occur.
• Review the QA procedure's performance whenever events (failures, environment changes, upgrades) occur, and at least annually.

In this post I'll deal with the first six items on the list - setting up and measuring - and I'll cover analysis of the test results in subsequent posts.

Free online fMRI education!

UCLA has their excellent summer Neuroimaging Training Program (NITP) going on as I type. Most talks are streamed live, or you can watch the videos at your leisure. Slides may also be available. Check out the schedule here.

I am grateful to Lauren Atlas for tweeting about the NIH's summer fMRI course. It's put together by Peter Bandettini's FMRI Core Facility (FMRIF). It started in early June and runs to early September, 3-4 lectures a week. The schedule is here. Videos and slides are available a few days after each talk.

Know of others? Feel free to share by commenting!

Using partial Fourier EPI for fMRI

Back in August I did a post on the experimental consequences of using partial Fourier for EPI. (An earlier post, PFUFA Part Fourteen introduces partial Fourier EPI.) The main point of that post was to demonstrate how, with all other parameters fixed, there are two principal effects on an EPI obtained with partial Fourier (pF) compared to using full phase encoding: global image smoothing, and regionally enhanced signal dropout. (See Note 1.)

In this post I want to look a little more closely at how pF-EPI works in practice, on a brain, with fMRI as the intended application, and to consider what other parameter options we have once we select pF over full k-space. I'll do two sets of comparisons. In the first comparison all parameters except the phase encoding k-space fraction will be fixed so that we can again consider the first stage consequences of using pF. In the second comparison each pF-EPI scheme will be optimized in a "maximum performance" test. The former is an apples to apples comparison, with essentially one variable changing at a time, whereas the latter is how you would ordinarily want to consider the pF options available to you.


Why might we want to consider partial Fourier EPI for fMRI anyway?

If we assume a typical in-plane matrix of 64 x 64 pixels, an echo spacing (the time for each phase-encoded gradient echo in the train, as explained in PFUFA Part Twelve) of 0.5 ms and a TE of 30 ms for BOLD contrast then it takes approximately 61 ms to acquire each EPI slice. (See Note 2 for the details.) The immediate consequence should be obvious: at 61 ms per slice we will be limited to 32 slices in a TR of 2000 ms. If the slice thickness is 3 mm then the total brain coverage in the slice dimension will be ~106 mm, assuming a 10% nominal inter-slice gap (i.e. 32 x 3.3 mm slices). With axial slices we aren't going to be able to cover the entire adult brain. We will have to omit either the top of parietal lobes or the bottom of the temporal lobes, midbrain, OFC and cerebellum. Judicious tilting might be able to capture all of the regions of primary interest to you, but we either need to reduce the time taken per slice or increase the TR to cover the entire brain.

Partial Fourier is one way to reduce the time spent acquiring each EPI slice. There are two basic ways to approach it: eliminate either the early echoes or the late echoes in the echo train, as described at the end of PFUFA: Part Fourteen. Eliminating the early echoes doesn't, by itself, save any time at all. Only if the TE is reduced in concert is there any time saving. But omitting the late echoes will mean that we complete the data acquisition for the current slice earlier than we would for full Fourier sampling, hence there is some intrinsic speed benefit. I'll come back to the time savings and their consequences later on. Let's first look at what happens when we enable partial Fourier without changing anything else.

The experimental consequences of using partial Fourier for EPI

PFUFA Part Fourteen introduced the idea of acquiring partial k-space and explained how the method, hereafter referred to as partial Fourier (pF), is typically used for EPI acquisitions. At this point it is useful to look at some example data and to begin to assess the options for using pF-EPI for experiments.


Image smoothing

The first consequence of using pF is image smoothing. It arises because we've acquired all of the low spatial frequency information twice - on both halves of k-space - but only half of some of the high spatial frequency information. We've then zero-filled that part of k-space that was omitted. This has the immediate effect of degrading the signal-to-noise ratio (SNR) for the high spatial frequencies that reside in the omitted portion of k-space. (PFUFA Part Eleven dealt with where different spatial frequencies are to be found in k-space.) Thus, the final image has less detail and is smoother than it would have been had we acquired the full k-space matrix, and because of the smoothing the final image SNR tends to be higher for pF-EPI than for the full k-space variant.

It was surprising to me that pF-EPI has higher SNR - due to smoothing - than full Fourier EPI in spite of the reduced data sampling in the acquisition. Conventional wisdom, which is technically correct, states that acquiring less data will degrade SNR. To understand this conundrum, we can think of pF as being like a square filter applied asymmetrically to the phase encoding dimension of an EPI obtained from a complete k-space acquisition. Indeed, as we start to evaluate the costs and benefits of pF for EPI we should probably be thinking about a minimum of a three-way comparison. Firstly, we obviously want to compare our pF-EPI to the full k-space alternative having the same nominal resolution. But we should also consider whether there is any advantage over a lower resolution EPI with full k-space coverage, too. Why? Because this lower resolution version is, in effect, what you get when partial Fourier is applied symmetrically, i.e. when the high spatial frequencies are omitted from both halves of the phase encoding dimension!

Let's do our first assessment of pF on a phantom. There are four images of interest: the full k-space image, two versions of pF - omitting the early or the late echoes from the echo train - and, for the sake of quantifying the amount of smoothing, a lower resolution full k-space image which is tantamount to omitting both the early and late echoes. (See Note 1.) From this point on I'm going to refer to omission of the early and late echo variants as pF(early)-EPI and pF(late)-EPI, respectively.

Images acquired from a structural phantom with a 12-channel head coil on a Siemens Trio. All parameters except the phase encode k-space sampling were fixed. Top left: 64x64 full Fourier EPI. Top right: 64x48 full Fourier EPI. Bottom left: 6/8ths pF(early)-EPI, reconstructed to 64x64. Bottom right: 6/8ths pF(late)-EPI, reconstructed to 64x64. (Click image to enlarge.)

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.

Introduction to MR principles: online resources

I recently came across some extremely informative online resources for learning the basics of (nuclear) magnetic resonance. The first (via Agilent's Spinsights.net blog) is an online simulator that is nicely introduced in a series of four YouTube tutorials (see below). The simulator allows you to demonstrate such concepts as RF excitation, the rotating frame of reference, relaxation and even a 1D gradient for spatial encoding. If you are brand new to MR then you might need some assistance in understanding things for yourself, and I would think this tool (and the supporting tutorials) would be best used by an instructor in a class, but I don't want to dissuade you from taking a stab on your own. Watch the videos first (see below), then check out the simulator. (You can also find technical info and links to the tutorials at www.drcmr.dk/bloch.)

The other resource I found just about blew me away, not so much for the NMR lectures themselves, as good as they are, but because they are part of an extensive biophysics course covering everything from electromagnetic radiation to flow cytometry and sedimentation methods! The lectures are by Yair Meiry, a fellow who is apparently now working as a skydiving instructor in Canada (assuming my Internet sleuthing has improved since yesterday's attempt to divine the Scandinavian country of origin of another YouTube video). Channeling his inner Garrett Lisi, perhaps? I know I'm impressed.

Physics for understanding fMRI artifacts: Part Thirteen

A tour through a real EPI pulse sequence

In some posts I've got planned it will be important for you to know something about all of the different functional modules that are included in a real EPI pulse sequence. So far in this PFUFA series I've used schematics of the particular segment of the sequence that I was writing about, e.g. the echo train that covers 2D k-space for single-shot EPI. Except that there comes a time when you need to know about the sequence in its entirety, as it is implemented on a scanner. Why? Because there are various events that I've given short shrift - fat saturation and N/2 ghost correction, for instance - that have significant temporal overheads in the sequence, and these additional delays obviously affect how quickly one can scan a brain.

So, without further ado, here is a pulse sequence for fat-suppressed, single-shot gradient echo EPI, as used for fMRI:

(Click to enlarge.)

Okay, so it's not the entire pulse sequence. The readout gradient echo train in this diagram has been curtailed after just nine of 64 total gradient echoes that will be acquired, for EPI with a matrix of 64x64 pixels. The omitted 55 echoes are simply clones of the nine echoes that you can see. (Note that there are no additional gradient episodes at the end of this particular EPI sequence; all the crusher gradients occur at the start of the sequence and these are visible in the above figure. More on crusher gradients below.) I should also point out that this is the timing diagram for acquisition of a single 64x64 matrix EPI slice. The pulse sequence as shown would be repeated n times for n slices within each TR. (See Note 1.)


Interpreting what you see

Let's first determine what information is being displayed on the figure above. There are five axes, all handily labeled on the far right-hand side of the figure. The top axis is the RF transmit channel; we've got two RF pulses in this sequence. The second axis down is the receiver, or analog-to-digital converter (ADC) channel. The scanner is receiving signals only when there's a rectangle specified on the second axis. Finally, the bottom three axes represent the pulsed field gradients, in the order X, Y, Z.

Just for fun, let's quickly determine what the scanner is doing in the logical frame of reference, before we delve into the nitty-gritty. The slice selection gradient will occur in concert with an RF excitation pulse, and we have two RF pulses to choose from. Slice selection can't be the first RF pulse because that pulse occurs without any concomitant gradients. Thus, the slice excitation pulse must be the second one and we can deduce that slice selection is along the Z axis, which is the magnet bore axis. We're doing axial slices.

Physics for understanding fMRI artifacts: CONTENTS

Figured it might be useful to have some summary/contents pages. I'll do similar admin posts as the other series mature, too. And I will label these pages with "Contents" to make them easier to find via the sidebar.


Part One

An introduction to the series, followed by an introductory video courtesy of Sir Paul Callaghan: What is NMR and how does it work?


Part Two

Further videos explaining the principles of nuclear magnetic resonance - how the intrinsic spin of certain atomic nuclei interacts with applied magnetic fields to yield useful information.


Part Three

Videos showing the anatomy of a miniature scanner, a basic NMR experiment, why shimming is important for NMR (and MRI), how and why a spin echo works, and the relaxation of spins back towards their ground state.


Part Four

Mathematics of oscillations: an introduction to imaginary and complex numbers, and frequency and phase.

Physics for understanding fMRI artifacts: Part Twelve

Apologies for the lengthy delay getting this post out. New academic year, teaching, talks, etc. etc. Anyway, I hope that this opus will be the final post in the background physics series for the time being. I reserve the right to append further posts down the road, but with this post I hope you will be in a position to understand the origins of artifacts in real (EPI-based) fMRI data. So, after today we'll change tacks and start reviewing what "good" data should look like. First things first though. Time to put all your k-space knowledge to good use, and review the pulse sequence that the majority of us use for fMRI.


The Echo Planar Imaging (EPI) pulse sequence

In Part Ten we looked at a pulse sequence and its corresponding k-space representation for a gradient-recalled echo (GRE) imaging method. That sequence used conventional, or spin warp, phase encoding to produce the second spatial dimension of the final image. A single row of the k-space matrix was acquired per RF excitation, with successive rows of (frequency-encoded) k-space being sampled after stepping down (or up) in the 2D k-space plane following each new RF pulse.

One feature of the spin warp imaging scheme should have been relatively obvious: it's slow. Frequency encoding along kx is fast but stepping through all the ky (the phase-encoded) values is some two orders of magnitude slower, resulting in an imaging speed from tens of seconds (low resolution) to minutes (high resolution). That's not the sort of speed we need if we are to follow blood dynamics associated with neural events.

Instead of acquiring a single row of k-space per RF excitation - a process that is always going to be limited by the recovery time to allow the spins to relax via T1 processes - we need a way to acquire multiple k-space rows per excitation, in a sort of "magnetization recycling" scheme. Ideally, we would be able to recycle the magnetization so much that we could acquire an entire stack of 2D planes (slices) in just a handful of seconds. That's what echo planar imaging (EPI) achieves.

Gradient echo EPI pulse sequence

The objective with the EPI sequence, as for the GRE (spin warp) imaging sequence we saw in Part Ten, is to completely sample the plane of 2D k-space. That objective is unchanged. All we're going to do differently is sample the k-space plane with improved temporal efficiency. Then, once we have completed the plane we can apply a 2D FT to recover the desired image. Pretty simple, eh?

As before, sampling (data readout) need only happen along the rows of the k-space matrix, i.e. along kx. So we need a way to hop between the rows quickly, spending as much time as possible reading out signals under the frequency encoding gradients, Gx, and as little time as possible getting ready to sample the next row. EPI is the original recycled pulse sequence, so I'll color the readout gradient echoes in green:

The first four (and a half) gradient echoes in a gradient echo EPI pulse sequence.

To keep things simple I've omitted slice selection and indicated a 90 degree RF excitation; this could of course be any flip angle in practice. (See Note 1.) I've also shown just the first four (and a half) gradient echoes in the echo train. The full sequence repeats as many times as there are phase-encoded rows in the k-space matrix. A typical EPI sequence for fMRI might use 64 gradient echoes, corresponding to 63 little blue triangles in the train shown in the figure above. But for the example k-space plane below, the k-space grid is 16x16 so assume for the time being that the full echo train would consist of 15 little blue triangles separating eight positive Gx gradient periods and eight negative Gx gradient periods.

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.)

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.

Physics for understanding fMRI artifacts: Part Six

Practical issues arising from the use of the Fourier transform in MRI

Here's the plan for this post. We will complete our look at functions undergoing Fourier transformation; there are some really useful relationships to see and to commit to memory (even when the figures are hand drawn for expediency!). Then we will look at the effects of the FT on real signals. We have two issues to consider: 1) a finite sampling window, and 2) digitization. Off we go!

Fourier pairs

We saw in the last post (Part Five) how a single frequency - a sinusoid - can be represented by a single line - a delta function - in a frequency domain plot. This is an example of a Fourier pair because the relationship holds both ways, i.e. if you take a delta function in the time domain and FT it you get a sinusoid in the frequency domain, and vice versa:

What about other useful Fourier pairs? Here's a useful relationship. An exponential decay Fourier transforms into what's known as a Lorentzian line:

Again, remember that the exponential can be in either the time domain or the frequency domain, although in MRI we generally deal with exponential decays (signals) in the time domain. It's also worth pointing out here that the faster the exponential decay, the broader the Lorentzian line in the other domain. This inverse relationship has a number of practical consequences for fMRI. I'll come back to this point below.

I found an interactive online tool on the National High Magnetic Field Lab's website that allows you to change the rate of decay as well as the frequency of an oscillation in the time domain, and see the resulting Lorentzian line in the frequency domain. Tinker with it here. (It's Java, it takes a couple of seconds to load.)

Next, let's look at what could be arguably the most useful Fourier pair for MRI. A boxcar (or top hat) function Fourier transforms into a sinc function, where sinc(x) = sin(x)/x:

Physics for understanding fMRI artifacts: Part Five

An introduction to the Fourier transform - what it does and how it works


Moving along from bathroom design and complex numbers, it's time to look at one of the most fundamental mathematical relationships underpinning all of modern MRI: the Fourier transform (FT). Accordingly, I would strongly suggest that you make sure you fully understand everything in this post before continuing into future posts.

You may not believe it by looking at us, but take it from me that MR physicists do "mental FTs" every day as we compare pulse sequences, try to understand artifacts and so on. Many of us (me) are not mathematically inclined, either. But being able to switch your mind between domains, as they are called, is a really useful skill if you want to be good at artifact recognition. An MR physicist will see something in one domain and will immediately try to imagine how it would appear in the alternate domain. Thus, if an image contains an artifact - an intense stripe, say - the MR physicist tries to imagine what signal-acquisition domain feature is implied by it, then tries to track it down. Alternatively, when comparing two different pulse sequences - GRAPPA on versus GRAPPA off, perhaps - the MR physicist will project the implications of each into the image domain in order to comprehend the likely practical consequences. You don't have to remember the equations or even understand the maths itself, but it is really useful to grasp the concepts!


Fourier analysis: the art of decomposition

As you have seen in your introductory texts and in previous video lectures, MRI signals are actually time-varying voltages that are induced in receiver coils as a result of magnetization oscillating (precessing) about the polarizing magnetic field. The detection of time-varying signals has several implications for obtaining MR images, the very first of which is choosing a representation for the information content in the signals.

Consider the two waveforms on the left-hand side of the figure below, which have the same amplitude but differ in their frequency:

Courtesy: Karla Miller, FMRIB, University of Oxford.

It is possible in principle - and in practice for simple examples such as these - to take a ruler and measure the amplitude and frequency and then draw a graph representing the time varying signals as a plot of amplitude (along y) against frequency (along x), as shown on the right-hand side of the above figure. Piece of cake, right?

Physics for understanding fMRI artifacts: Part F(f)our

(Wondering why the title has F(four) in it? It's so that Blogger can't trash this post for a third time! I'm giving it a new, unique name. Ha!)

It's finally time to get back to the series of posts on the essential physics concepts that will allow you to interpret and differentiate between acquisition artifacts. There are another five or six posts in this background series, so bear with me. After that we will shift gears and look at "good data," taking some time to assess the normal variations that you can expect to see in time series EPI, and then I promise we'll look at artifacts themselves.


When reality meets imagination

Before we go any further there are a few mathematical properties we need to review. These are actually quite simple relationships that, for the most part, can be explained via a handful of pictures. Like this one....

Maths dude, chillin'.

Complex numbers

The name notwithstanding, complex numbers are quite straightforward to understand from a physical perspective, with a tiny bit of explanation. By the time you've finished reading this post you should have a basic idea of what complex numbers mean and where they come from (they arise quite naturally, as it happens), but for now I am simply going to define some relationships. Hang in there.

We start by defining a so-called imaginary number as any number for which the square is negative. The squares of real numbers - the ones you're used to in everyday life, such as 2, 8.73, -7, pi, and so on - are always positive, whether the number being squared is positive or negative. Thus 2x2 = 4, 8.73x8.73 = 76.2129, -7x-7 = 49 and so on. Squaring a negative number results in a positive number. So how could we possibly get an answer of -4 or -25 out of any square?

Go faster MRI at Berkeley!

With apologies for the continued delay to the artifact recognition series of posts - I've been distracted with some scanner problems - I thought I'd do a quick post on a recent methodological development that's generated some buzz in the field as well as in the media. The media buzz:

ABC 7 News video

UC Berkeley news center story

And in case you want to read the actual publication, it was published at PLoS ONE in early January. The work is part of the Human Connectome Project, an NIH-funded consortium involving Washington University (St Louis), Oxford, Minnesota and Berkeley. David Feinberg is the Berkeley representative.

The implications of these methodological developments could be quite substantial, possibly allowing better interpretation of brain dynamics than is currently permitted with the typical fMRI temporal resolution of two seconds or so. Of course, there are caveats. One is that the BOLD response is still low-pass filtered. And another is that the new "go faster" method involves several separate steps, each of which tends to exacerbate head motion sensitivity. Still, it looks good on highly motivated volunteers!