Can we separate real and apparent motion in QC of fMRI data?

 

A few years ago, Jo Etzel and I got into a brief but useful investigation of the effects of apparent head motion in fMRI data collected with SMS-EPI. The shorter TR (and smaller voxels) afforded by SMS-EPI generated a spiky appearance in the six motion parameters (three translations, three rotations) produced by a rigid body realignment algorithm for motion correction, such as MCFLIRT in FSL. The apparent head motion is caused by magnetic susceptibility variations of the subject's chest as he/she breathes, leading to a change in the magnetic field across the head which, in turn, adds a varying phase to the phase-encoded axis of the EPI. This varying phase then manifests as a translation in the phase-encoded axis. It's not a real motion, it's pseudo-motion, but unfortunately it is a real image translation that adds to any real head motion. I should emphasize here that this additive apparent head motion arises in conventional multi-slice EPI, too, but it's generally only when the TR gets short, as is often the case with SMS-EPI, that the apparent head motion can be visualized easily (as a spiky, relatively high frequency fluctuation in the six motion parameter traces). In EPI sampled at a conventional TR of 2-3 sec, there are only a small handful of data points (volumes) per breath for an average breathing rate of 12-16 breaths/minute and this leads to aliasing of most of the apparent head motion frequency. It may still be possible to see the spiky respiration frequency riding on the six motion parameters, but it's not always there as it is for TR much less than 2 seconds.

Once we'd satisfied ourselves we'd understood the problem fully, I confess I let the matter drop. After all, we have tools like MCFLIRT that try to apply a correction to all sources of head motion simultaneously, whether real or apparent. But now I'm wondering if we might be able to evaluate the real and apparent motion contributions separately, with a view to devising improved QC measures that can emphasize real head motion over the apparent head motion when it comes to making decisions on things like data scrubbing. Jo has been dealing with the appropriate framewise displacement (FD) threshold to use when including or excluding individual volumes. (See also this paper.)

Let's review one of the motion traces from my second 2016 blog post on this issue:

These traces come from axial SMS-EPI with SMS factor (aka MB factor ) of 6. The x axes are in seconds, corresponding to TR = 1 sec. (The phase-encoded axis is anterior-posterior, which is the magnet Y direction.) On the left is a subject restrained with only foam, on the right the same subject's head is restrained with a printed head case. During each run the subject was asked to take a deep breath and sigh on exhale every 30 seconds or so. We clearly see the deep breath-then-sigh episodes in both traces, regardless of the type of head restraint used. Yet it is also clear the apparent head motion, which is the high frequency ripple, dominates the Y, Z and roll traces on the left plot. On the right plot, the dominant effect of apparent head motion manifests in the Y trace, with a much reduced effect in the roll axis. Already we are seeing a slight distinction between the translations and rotations for apparent head motion. It looks like apparent head motion contributes more to translations than rotations, which makes sense given the physical origin of the problem. In which case, can we assume that by extension real head motion will dominate the rotations?

For now, let's assume that the deep breath-then-exhale episodes are producing considerable real head motion, in addition to the large apparent head motion spike from exaggerated chest movement. The left plot above shows that pitch, yaw and roll all characterize the six deep breaths readily. They are also visible in Z and X, but with considerably reduced magnitude. There's no clear effect in the Y trace which is dominated by the aforementioned apparent head motion. So far so good! When the head can actually move in the foam restraint, we have clear biases towards rotations for real head motion and translations for apparent head motion. 

What about the right plots? Real head motion is far harder to achieve because of the printed head case restraint. But we assume the apparent head motion is basically the same magnitude because it's chest motion, not head motion. So we might think of this condition as being a low (or lowest) real motion condition. As with the foam restraint on the left, we again see Y translations dominated by apparent head motion. The roll axis also displays considerable apparent head motion. And as for the foam restraint, the roll and pitch axes display something that may be real or apparent head motion for each of the deep breath-then-exhale periods. We can't be sure if the head (or the entire head case, or even the entire RF coil!) was really moving during each breath, but let's assume it was. If so, then for good mechanical head restraint we have the same rough biases as for foam restraint in our motion traces: real motion dominates rotations, apparent motion manifests mostly as translations.

Jo sees a similar distinction between real and apparent head motion in the motion parameter plots of her 2023 blog post. In her top plot, which she suggests is a low real motion condition, the apparent motion dominates Y and Z translations and the roll traces, exactly as my example above. Her second plot exhibits considerable real head motion. The apparent head motion is still visible as ripples on the Y and Z translation traces, but now it's clear the biggest changes arise in the three rotations and these changes are probably real head motion. Again, we have real motion dominating rotations while apparent motion manifests more in the translations.

Finally, let's consider Frew et al., who looked at head motion in pediatrics. Here's Figure 3 from their paper:

Using framewise displacement (FD), they show a transition from FD dominated by translations to FD dominated by rotations when considering low, medium and high (real) head motion subjects. Rotations and translations are both affected significantly in the medium movement group. Still, the trend here suggests that we might consider rotations alone as an index of real head motion if, as suggested above, apparent head motion contributes mostly to translations.

So, what might we do to separately evaluate real and apparent head motion? This is where you come in. I only have one starting idea, and that's to shift to considering FD using only rotations, rather than rotations and translations, when setting thresholds for the purposes of QC and scrubbing. Based on what I've presented here, we might be able to set a threshold for FD(rotations only) that will capture most of the real head motion and have a much reduced dependency on apparent head motion. This measure could help avoid mischaracterizing large apparent head motions as events to reject when they are inherently fixable with MCFLIRT and similar. (Real head motion produces a big spin history effect and likely introduces non-linear distortions in the images.) Whether the reverse is true - that is, whether FD(translations only) captures most of the apparent head motion and a reduced contribution from real head motion - I leave as an exercise for another day, but my suspicion is that it is not. Put another way, I think the focus should be on using the rotations to capture and evaluate real head motion. Pooling translations and rotations in measures like FD may be complicating the picture for us.

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Core curriculum - Cell biology: taxonomy

 

Most of the biology we need to learn can be treated orthogonal to the mathematics, whereas the mathematics underlies all the physics and engineering to come. As a change of pace, then, I'm going to start covering some of the biology so I can jump back and forth between two separate tracks. One track will involve Mathematics, then Physics, then Engineering, the other will be Cell Biology, Anatomy, Physiology and then Biochemistry.

 

Let's begin with a simple overview of cell structure:

 https://www.youtube.com/watch?v=0xe1s65IH0w

The owner prohibits embedding this video in other media so you'll have to click through the link to watch.

Next, a little more detail on what's in a typical mammalian cell:

All well and good, but we are primarily interested in the types of cells found in neural tissue, whether central nervous system (CNS) or peripheral nervous system (PNS):

A little more taxonomy before we get into the details of neurons and astrocytes. In this video, we start to encounter the chemical and electrical signaling properties in cells, something we will get into in more detail in a later post. Still, it's timely to introduce the concepts.

As we move towards the neural underpinnings of fMRI signals, we need to know a lot more about neurons and astrocytes. Let's do neurons first.

While this next video repeats a lot of what you've already seen, there is enough unique information to make it worth watching.

Finally, a little more taxonomy that relates types of neurons to parts of the body, something that could be very important for fMRI when we are considering an entire organism.

To conclude this introduction to cell biology and types of neural cells, let's look at glial cells in more detail.

 Another simple introduction, to reinforce the main points:

And a nice review to wrap up.

We will look far more closely at astrocytes in a later video, once we've learned more about blood flow and control. For now, just remember that those astrocyte end feet are going to be extremely important for the neurovascular origin of fMRI signals.

 

That will do for this primer. The next post in this series will concern the resting and action potentials, signaling and neurotransmission.

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Coffee Break with practiCal fMRI

 A new podcast on YouTube

We all know the best science at a conference happens either during the coffee breaks or in the pub afterwards. This being the case, practiCal fMRI and a guest sit down for coffee (or something stronger) to discuss some aspect of functional neuroimaging in what we hope is an illuminating, honest fashion. It's not a formal presentation. It's not even vaguely polished. It’s simply a frank, open discussion like you might overhear during a conference coffee break.

In the inaugural Coffee Break, I sit down with Ravi Menon to discuss two recent papers refuting the existence of a fast neuronal response named DIANA that was proposed in 2022. Ravi was a co-author on one of the two refutations. (The other comes from the lab of Alan Jasanoff at MIT.) We then digress into a brief discussion about the glymphatic system and sleep, and finally some other bits and pieces of shared interest. I've known Ravi for three decades and it's been a couple of years since we had a good natter, so we actually chatted on for another hour after I stopped recording. Sorry you don't get to eavesdrop on that conversation. It was all science, zero gossip and the subject of expensive Japanese whisky versus Scotch and bourbon did not feature, honest guv.

 

All the links to the papers and some items mentioned in our discussion can be found in the description under the video on YouTube. 

What's next for Coffee Break? I have a fairly long list of subject matter and potential guests. I'm hoping to follow some sort of slightly meandering theme, but no promises. I'm also hoping to get new episodes out about once every couple of weeks. But again, no promises.

(PS The series of posts on the core fMRI syllabus will resume shortly with a new branch on biology, starting with basic cell biology.)

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Core curriculum - Mathematics: Linear algebra VI

 

A13. Eigenvectors and Eigenvalues

Let's end this section on linear algebra with a brief exploration of eigenvectors and their eigenvalues. An eigenvector is simply one which is unchanged by a linear transformation except to be scaled by some constant. The constant factor (scalar) by which the eigenvector is scaled is called its eigenvalue. If the eigenvalue is negative then the direction of the vector is reversed as well as scaled.

 Curious about the terminology? Eigen means "proper" or "characteristic" in German. So if you're struggling to understand or remember what eigenvectors are all about, perhaps it helps to rename them "characteristic vectors" instead.

Here's a nice introduction to the concepts. Pay close attention to the symmetry arguments. It turns out eigenvectors represent things like axes of rotational symmetry and the like:

 

 

And with some of the insights under your belt, here's a tutorial on the mechanics of finding eigenvalues and eigenvectors:

 

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Core curriculum - Mathematics: Linear algebra V

 

With some understanding of basic matrix manipulations, we're ready to begin using matrices to solve systems of linear equations. In this post, you'll learn a few standard tools for solving small systems - system defined by a small number of equations - by hand. Naturally, larger systems as found in fMRI will use computers to solve the equations, but you should understand what's going on when you push the buttons.

A11. Elementary row operations and elimination

 

This is just your standard algebraic manipulation to solve multiple simultaneous equations, e.g. dividing both sides of an equation by some constant to be able to simplify, but where the equations are represented as matrices:

 

A12. Cramer's Rule for solving small linear systems

According to Wikipedia:

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations.

 

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Core curriculum - Mathematics: Linear algebra IV

 

Before getting back to the lectures from 3Blue1Brown, try this part review, part preview:

Now let's get back into the meaning with a little more detail.

 

A9. The dot (or scalar) product 

The dot product is a way to estimate how much two vectors interact in a common dimension. If the vectors are orthogonal to each other, they don't interact in a common dimension so their dot product is zero. This is like asking how much north-south movement is involved in an east-west heading: none. But if two vectors are perfectly parallel then this is equivalent to the two vectors lying on the number line and we can use our standard (scalar) multiplication rules. In between, we use a little trigonometry to determine their (dot) product.

 

Still lacking an intuition? This excellent summary from Better Explained (slogan: "Learn Right, Not Rote") should do the trick.

A10. The cross (or vector) product

Both the dot and cross products affect dimensionality. With the dot product, we find how much two vectors interact in one dimension. The cross product of two vectors is perpendicular to them both, telling us how much rotation arises in a third dimension.

A useful real world example use of the cross product is to compute the torque vector. Torque is the rotating force generated by pulling or pushing on a lever, such as a wrench or a bicycle crank. The lever moves in one plane but produces a rotation orthogonal to that plane. 

 

 

Torque is also fundamental to the origins of the MRI signal. We will encounter it later in the physics section. Can you take a guess how torque might be relevant to the MRI signal? Hint: it has to do with the interaction of a nuclear magnet (the protons in H atoms) with an applied magnetic field.

This article from Cuemath covers the rules for computing dot and cross products. And here are a couple of useful visualizations:

 

 

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Core curriculum - Mathematics: Linear algebra III

 

Now we start to think about transformations between dimensions, e.g. taking a 2D vector into a 3D space. Non-square matrices come up frequently in engineering and research applications, including fMRI analysis, so you'll want a good understanding of their meaning. 

 

 A8. Non-square matrices

Let's look at a simple physical interpretation of changing the number of dimensions.

We previously saw how to invert a square matrix. But how do we invert a non-square matrix?

 

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Core curriculum - Mathematics: Linear algebra II

Continuing the series on linear algebra using the lectures from 3Blue1Brown, we are getting into some of the operations that will become mainstays of fMRI processing later on. It's entirely possible to do the processing steps in rote fashion as an fMRI practitioner, but understanding the foundations should help you recognize the limits of different approaches.

A4. Matrix multiplication as composition

In this video we see how to treat more than one transformation on a space, and how the order of transformations is important.

 

Q: While brains come in all shapes and sizes, we often seek to interpret neuroimaging results in some sort of "average brain" space, or template. We need to account for the variable position and size of anatomical structures. However, we also have the variability of where that brain was located in the scanner, e.g. because of different amounts and types of padding, operator error, and so on. When do you think it makes the most sense to correct for translations and rotations in the scanner: before or after trying to put individual brain anatomy into an "average brain" space? Or does it not matter?

 A5. Three-dimensional linear transformations

 Now we're going to move on from 2D to 3D spaces. Same basic rationale, just more numbers to track!

 

A6. The determinant 

 

A7. Inverse matrices, column space and null space

 

 

Perhaps it's not fully clear why we might need the inverse matrix. It turns out to be the way to achieve the equivalent of division using matrices. To galvanize this insight, let's look at the concept of an inverse matrix for solving an equation without division. Leaving aside the slightly goofy intro, it's a useful tutorial on the mechanics of determining an inverse matrix. 

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Core corriculum - Mathematics: Linear algebra I

 

What is linear algebra? To get us going, I'm going to use the excellent lecture series by 3Blue1Brown and do my best to add some MRI-related questions after each video. Hopefully the connections won't be too cryptic. Don't worry if you can't answer my questions. It's more important that you understand the lectures. No doubt you'll find other material on YouTube and web pages to clarify things.

Let's start with a couple of definitions. While you'll find many examples online, for our purposes we can assume that a linear system is one where the size of the output or outputs scales in proportion to the input or inputs. The take-home pay of a worker paid an hourly rate is linear. They might receive their base amount, say 40 hours per week, plus some amount of overtime at twice their hourly rate. The total is still the linear combination of the base plus overtime amounts.

Non-linear systems don't have this simple proportionality. Gravity is the classic physics example. The strength of the interaction between two massive objects changes as the reciprocal of the squared distance (r^2) between them, that is, as 1/r^2. Finding yourself dangling ten meters in the air above the earth is very different from finding yourself ten more meters away from the earth at a height of 1000 km. In the first case you are about a second away from impacting the ground. In the second case you are in orbit and your more immediate health concerns are lack of oxygen and your temperature.

And what about the term algebra? It's just fancy speak for using symbols to represent the relationships between things that vary. We're going to be interested in changes at different positions in space - points in an image - and so we shall eventually use matrices to perform linear algebra. But we have to build up to a matrix from its skinnier cousin, the vector.

A1. Vectors: Essence of linear algebra

 

Q: We will use both a physicist's and a computer scientist's view of vectors at different points in the fMRI process. Given what you know today, can you guess where these different viewpoints might come up? Hint: fMRI is based on MRI, which is a physical measurement technique, while fMRI is typically the analysis of a time series of a certain type of dynamic MRI scans.

 

A2. Linear combinations, span, and basis vectors

 

 

We might as well look at changes of basis at this point:

 

 

Q: Changes of basis are quite common in MRI. Even the way we usually label image axes involves a change of basis. The magnet bore direction is labeled the z-axis, while left-to-right is the x-axis and up-down is the y-axis. We refer to this assignment as the lab (or magnet) frame of reference. Now consider an axial MR image of a person's brain. An axial slice lies in the x-y plane in the magnet basis (or lab frame if you prefer). Yet we don't generally label the image with (x,y) dimensions. Instead we use (L-R, A-P) where L-R is left-to-right and A-P means anterior-to-posterior. This is an anatomical basis. How might an anatomical basis be more useful than using a magnet basis in MRI?

A3. Linear transformations and matrices:

 

Q: We usually label images using a basis (or reference frame) related to the subject's anatomy, i.e. with the (orthogonal) axes labeled head-to-foot (HF), left-to-right (LR) and anterior-posterior (AP). This means if a subject's head isn't perfectly straight in the magnet - let's say, the head is rotated 20 degrees to the left - the brain still appears straight in the 2D image. But here's the thing. The MRI hardware is controlled using the (x,y,z) "lab" reference frame. The anatomical and lab bases can be related to each other through a rotation matrix. Can you write down what a rotation matrix might look like that relates the subject's anatomical reference frame to the scanner's lab (x,y,z) reference frame?

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Core curriculum: How to learn from videos

 

Make coffee, fire up YouTube, click, watch, go about your day. Not so fast! To actually learn the material you'll see, you will need a minimum of the lecture itself, some sort of reading around the lecture (which could be reviewing a transcript or supporting documents), and then answer some questions on that material. So, as this excellent didactic lecture from an anesthesiologist makes clear, questioning is key:

 

  https://www.youtube.com/watch?v=d7IPiNE4_QE

 

I don't have banks of questions ready to pepper you with at the end of each video, I'm afraid, although I will try to come up with a few questions as homework.

If you want to take it to the pro level, try to explain what you've just learned to a novice. Nothing makes you learn something like having to teach it:

  https://www.youtube.com/watch?v=_f-qkGJBPts

 

Nobody to discuss it with or teach it to? Try preparing a one or two slide summary as if you are about to give a presentation on it. Practice presenting the summary just like you would any other presentation. Dry runs are almost as good as the real thing.

A major benefit of using videos on YouTube is that you can stop and rewind as much as you like! (If you didn't know, your back and forward arrow keys take you back or forward 5 seconds on YouTube videos. Saves from the imprecision of the progress bar.) You can watch the video then listen to the words, then watch again, as you like.

Also consider taking notes as you go. No need to worry about missing something. Just pause and/or rewind as needed. One of my better high school teachers used to berate us if we claimed to be studying without a pen in hand. He claimed reading alone was almost useless. We had to read and write to learn. Perhaps you have some tips to share in the comments. Maybe even a link to a good video on how to learn from videos:

 

https://www.youtube.com/watch?v=fRo26gpgvV4

 

Whatever you do, set up a system for yourself and don't just be a passive viewer.

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HOMEWORK: Some people are of the opinion that taking notes during a lecture is a bad idea. I reviewed at least one video telling me as much. And yet 99% of my undergraduate classes were exactly that: someone droned on at the front, writing on a chalk board (no dry erase back in them days!), while we scribbled as fast as we could. For a technical subject like neuroimaging (or chemistry), what is a major benefit of writing notes during a lecture? What is a potential cost of writing notes instead of just listening and perhaps trying to summarize afterwards in a debrief?

Core curriculum: An introduction

After much delay, I am finally going to start developing the core curriculum I suggested in December 2021. At that time, I imagined recruiting a group of 10-15 domain experts to provide the bulk of material under each separate discipline. That might have worked. Indeed, it could still work if an appropriate group such as the OHBM education committee decides to have a go. But I'm going to try something different. To borrow a phrase from blockchain folks, I want to be permissionless. I'm going to try to collate publicly available material myself, with occasional assistance from others if and when I get proper stuck. Trying to do it all myself should provide me with an interesting set of learning experiences, I hope, and it should also help guarantee that anyone, anywhere with access to YouTube can participate.

So, how's this gonna go? Not sure, it's an experiment. I have the following main disciplines listed and as of now I plan on tackling them in this order (although I may well start on some of the later ones before finishing the earlier ones). I'm just gonna start and see what happens. I will aim for one post a week, equivalent to 1-2 hours of learning. As I go, I will do my best to organize the collection - for example, all will have Core curriculum somewhere in the title, plus appropriate labels - and once there are enough of them I'll create a main page with links; a virtual contents table.

Likely major themes, in likely order:

• How to learn from videos
• Mathematics
• Physics
• Engineering
• Biology
• Physiology
• Biochemistry
  
• Biophysics
• Image processing & analysis
• Statistics
• Psychology
• Experimental design
• Practical issues

Why this order? The logic is to try to build concepts on concepts. It's hard to understand most important engineering concepts without a decent understanding of some physics, which itself requires some decent understanding of certain mathematics, and so on. And, as noted in my Dec 2021 post, the goal here is to cover material that is non-volatile over decades. It's about the fundamental concepts, not the state-of-the-art. 

Right, enough preamble. Time to get going! 

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Infrequently asked questions:

Q: Where's your Twitter?  A: Gawn, all gawn. Got X'd out.

Q: Can we comment or make suggestions?  A: Yup. I'll do my best to answer comments to the posts, and my email still works.

Q: What do you mean by "non-volatile over decades?"  A: I'm taking my inspiration from the established sciences. Consider chemistry. Any chemist trained in a university anywhere in the world understands the Periodic Table and why the first row transition elements are different from the noble gases. They also understand carbon valence, pH, catalysis and hopefully some thermodynamics. These subjects are all fundamental to the field of chemistry and are unchanged whether they are learned in England, Sri Lanka or Venezuela. They also haven't changed fundamentally since I learned about them in the 1980s. 

Q: Why Blogger and not Substack or some newer platform?  A: Inertia. There's a dozen years of history on this site and a lot of it still applies. Indeed, I hope some of it will be getting re-used in the core curriculum! 

Q: Are you going to go back to more topical tips?  A: I don't have plans to, but if there's something important to cover then I may. However, I won't be going back to writing the series on fMRI artifacts or physiological confounds, at least not at this time. I'm focused on the fundamentals right now. Seeing way too many un(der)prepared folk still coming into neuroimaging.