My Journey into the Hyperbubble

In 2012 I finished writing a pretty weird paper which I published a year later in the journal Physics of Fluids, about the behaviour of bubbles in spatial dimensions greater than three. This is the story of how I started writing it, what lead to me to finish it, and how it got published.

The journey is a metaphor.

After my undergrad, I got a job in a lab that developed ultrasound technology for treating diseases, for example, applying intense sound waves to a tumour until it heats up and coagulates and melts (I want to link a video here but they're all terrible). My job seemed unrelated to this: figure out how bubbles behave in a tube*. Bubbles are used as ultrasound contrast agents: a gas bubble in a liquid will oscillate under an external ultrasound field, getting bigger and smaller. This in turn releases more sound waves as the bubble displaces the fluid around it, which can then be detected. A lot of people ask if bubbles in the bloodstream cause heart attacks: not these ones, they are microscopic.

There are several ways to model how bubbles behave. The first attempt was by Lord Rayleigh, of scattering fame, who in 1917 wanted to understand the damage caused to propellers by the collapse of the bubbles they produce. He considered an infinite fluid with a spherical cavity in it, and calculated how long it would take for the fluid it fill up the cavity. A bubble with gas in it will not get totally filled by fluid: the pressure of the gas will increase until it can reverse the collapse and expand the bubble again, and this oscillation produces sound waves. By treating this as harmonic motion, Michael Minnaert derived the eponymous frequency of a bubble's oscillation in 1933, explaining that "musical air-bubbles" are responsible for the sound of babbling brooks.  These concepts were combined by Milton Plesset to describe gas-containing bubbles, leading to the Rayleigh-Plesset equation, a second-order ordinary differential equation describing how a bubble's radius evolves over time due to internal and external pressures. It is still used today to model bubble dynamics.

My task was to figure out how putting the bubbles in a tube (a proxy for a capillary blood vessel) modifies their behaviour, particularly their resonance frequency and response to ultrasound. My supervisor had grander plans for me, but I wasn't yet aware of them. This was my first independent delve into scientific literature. I found a few papers using fairly advanced modelling techniques to calculate bubble response in vessels, but I wanted something I could reproduce. Ultimately I found an equation that a Russian scientist had derived in 1992 but not solved, and I "solved" it numerically and showed that it gave the same results as some of the more complex models. I wrote a paper explaining all this, and couldn't get it published in most journals so it ended up in an obscure Russian publication. I learned an important lesson about academic publishing from this: just because you did something new, doesn't mean people will care.

The main project my supervisor wanted me to work on involved using this bubble knowledge to figure out how much heat is transferred to the brain during focused ultrasound. I obtained a 3D microscopy map of part of a rat brain from another group in the division, developed a way to populate the blood vessels with bubbles, simulate their oscillations based on the width of the vessels, calculate the heat transfer according to the "thermal damping" of the bubble, and use somebody else's solutions to the Heat Equation to figure out the effect on the tissue. This eventually lead to my first paper, in Physics in Medicine in Biology. If I were to tackle that again, I would probably not try to hack together the different methods the way I did. Unfortunately, after I left the person who was supposed to take over the modelling kind of dropped the ball, and the person who was supposed to test it experimentally quit, so the project kind of died.

Temperature in the mouse brain.

When I started graduate school, it took a while for my project to get going. In fact, it took over a year before we had a lab and I could really get into my experiments. I was still thinking about some of the stuff  from my bubble days, and one of the things that was bugging me was that the Rayleigh-Plesset equation was unsolvable. By unsolvable, I mean that there isn't an expression one can write down that describes a bubble's oscillation over time. In my first semester at McGill I was taking a course on string theory for some reason, and one thing that stuck with me was how they took something that was unsolvable in three dimensions, re-derived it in an arbitrary number of dimensions, and found the one dimensionality that it actually worked in (26, in this case). I figured I could try the same for bubble equations.

I could not. That was probably a ridiculous thing to think. I tried anyway though.

The derivation for all the equations I mentioned involved writing down the kinetic energy of the bubble by integrating the kinetic energy of a spherical shell of fluid at some distance from the bubble moving at some velocity. The velocity at the wall of the bubble decays according to the continuity equation, 1/r$^2$ in three dimensions; if that were not the case then more bubbles would form, or something. I rewrote this for some dimension N**, and now the surface element was proportional to r$^{N-1}$ and so was the velocity. With the help of Maple, a computer algebra software, I obtained expressions for the arbitrary dimension resonance frequency, the collapse time, and the Rayleigh-Plesset equation. The main result was that the resonance frequency and the collapse are both faster in higher dimensions.

Two dimensional image of a three dimensional calculation of a bubble's kinetic energy, based on the motion of fluid around the bubble. The velocity of the fluid is proportional to the rate of change of the bubble's radius (the wall velocity) and falls off as an inverse square.

As I said in a previous post, I am not a great mathematician. I hoped that I could use the differential equation solving function on Maple to spit out the answer. It didn't spit out the answer for three dimensions, and it didn't spit out the answer for N-dimensions. I could have it generate Taylor series solutions for certain initial conditions which I could try to generalize in terms of messy and boring matrices and that didn't really go anywhere.

Anyway, I figured since I had derived these things I might as well write a paper on them, even though I didn't have much in the way of implications. So I started doing that...I don't even remember when. Looking through my email, it appears I had a draft of sorts in July of 2010, when I was about halfway through my masters. In the process, I looked up a paper to find a source for something in my introduction. It was the paper "Bubbles" by Andrea Prosperetti, one of big names in bubble world, which is a thing. This paper has one of the best introductions of any I've read, and I recommend reading it. I saw that in the theory section of this paper, Prosperetti starts talking about the oscillations of a bubble in N dimensions, and derives the Minnaert frequency and the Rayleigh-Plesset equation. I was pretty bummed, because what I thought was a unique idea had already been done. However, Prosperetti's equation was not the same as mine, and I ran some numerical solutions and showed that my equation was internally consistent and his was not.

No disrespect to Prof. Prosperetti.

 So, I could write a paper about how his paper was slightly wrong and improve upon it, but that's not much of a paper. This, combined with the fact that by then my masters was probably picking up, I basically dropped the whole project and focused on my actual work.

Fast forward to 2012. I was reading through Physical Review E, which is basically a journal for "other" areas of physics. I saw a paper by Danail Obreschkow, about an analytical approximation for bubble collapse. I had encountered Danail's research during my hospital days, and somehow landed on his website. He was a few years older than me and in charge of a science mission to observe bubbles in a zero-gravity aircraft, which is inherently cool. He was also into extragalactic astronomy, oddly shaped dice, quantum dots, and was in a rock band. Coolest physicist ever basically. His new paper looked at an approximation to the solution of a re-arranged version of the Rayleigh-Plesset equation, matching it to numerical solutions and to their experiments. Describing how the radius evolves over time, they guessed that $r=\left(1-t^{2}\right)^{\alpha}$ and found the values of $\alpha$ that best describe the solution. If you care about the start of the collapse, $\alpha$ is 0.4, and if you care more about the end, $\alpha$ is about 0.42. Seeing this paper re-kindled my interest in the project, and I set myself to working out this version of the equation in my formalism. I figured it out (there may have been some guess-and-check involved), and found that the two versions of the exponent are most similar in three dimensions, and in higher dimensions they diverged to one being 23% greater than the other one***. So, I proved that three dimensional bubbles are special.

So I went back to my draft of the paper from 2010, added this section, and was about to call it a day and put it on arXiv, when I had the idea to look at some of the non-linear effects. What I mean by this is that when a bubble is strained it oscillates at its resonance frequency, but when the strain amplitude increases, the frequency starts to change and the oscillations cease to be sinusoidal...they start to resemble a shape called a cycloid****. So I looked at how the frequency changes with initial conditions, and that was a pretty good idea because I found something interesting. In three dimensions, a bigger strain just decreases the frequency. But in higher dimensions, first the frequency increases, then decreases. So three dimensional bubbles are once again proven to be special: it is the only dimension with a monotone strain response.

Would you look at that nonlinearity!

So, with the combinations of my derivations, pointing out that Prosperetti's equation was wrong, generalizing Obreschkow's approximation, and the investigation of nonlinear effects, I wrote up a paper and submitted it to arXiv in 2012. And there it sat, I was happy to have "gotten it out there."  By this time I was in the swing of my PhD and had a lot of actual stuff going on. A week later, Danail Obreschkow emailed me and said he liked my paper, and I thought that was pretty cool, it is literally the same as Alexander Friedmann getting an email from Albert Einstein in 1922.

Fast forward again to March 2013. I got another email from Danail Obreschkow, again saying he liked my paper. This time, he gave me some more encouragement and suggested that I try to get it published. I had sort of assumed that it wasn't realistic enough to be accepted in a physics journal, and not rigorous to be accepted in a math journal, but I took this encouragement to heart. That day I submitted it to Physics of Fluids, which has a self-explanatory scope.

Six weeks later I heard back from them. In typical peer-review fashion, Reviewer #1 was all positive and Reviewer #2 was a total jerk. His main points were that a) pointing out a typo in an old paper doesn't make a novel contribution and b) I should focus on something physically realistic. Personally I didn't think b) was a fair point given that we all agree that we only live in three dimensions, but the reviewers must be satisfied.

I was a bit lucky, that after I had submitted my paper, another paper came out analyzing Obreschkow's approximation. In his paper, Obreschkow improved upon his approximation by multiplying it by a truncated Taylor series, and claimed that even without truncation this approximation would never be fully accurate. Amore and Fernandez argued that it would be fully accurate, citing numerical evidence. I re-derived their analysis in arbitrary dimensions, and showed that not only is there numerical evidence in three dimensions, there is numerical evidence in every dimension, which makes a much stronger case than just a single numerical coincidence. In effect, I had used my higher-dimensional equation to settle a question in three dimensions, about whether this approximation can be exact or not. I also got to add some bigger equations, which I liked.

We're talking tertiary parentheticals here, man.

I rewrote the paper, removing some of the higher-dimensional simulation and focusing more on this issue, while toning down my criticism of Prosperetti. What once said "Obviously, experiments cannot be performed in higher dimensions to resolve this discrepancy, but the author finds fault in Prosperetti’s derivation" now reads "There is disagreement between the equations presented by Prosperetti1 for the unbounded case and those of this paper because a term was dropped from the second derivative of radius between his Eqs. (1) and (3)."  Eventually it was accepted and published in August of 2013, the same month that I finished by tunnel-through-the-Earth paper.

When I was younger I had made several naive attempts to publish something mathematical, and now I had finally succeeded. I remember at my old job thinking that Physics of Fluids was too mathy for me to understand, and now I was in it. I also learned an update to the lesson from four years earlier: even if nobody cares, work on it anyway.


There is one more addendum to the story. One of the reasons I started down this path was to try to solve the Rayleigh-Plesset equation. In 2015 two Russian mathematicians did manage to solve my equation in 4 and 6 dimensions, in terms of really complex functions. I emailed them to congratulate them, and one of the authors told me, almost word for word, the same thing I told Danail: that he was inspired to write his paper after seeing my paper.

Oh and if anyone cares but doesn't want to read the paper, here are the main results:

*My main research projects have been bubbles in tubes, DNA in tubes, and falling through tubes.

**I think the convention for an arbitrary dimension is actually D. I stuck with N.

***Coincidentally, this ratio of 1.23 is the square of the ratio of the uniform-density fall through the Earth and the constant-gravity fall through the Earth.

****Which is also the shape of the brachistochrone!