A Toy Model For Tomography

In Chapter 12 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss tomography. The algorithms for solving the tomography problem involve calculus, Fourier analysis, and convolutions. I love all that mathematics, but some people don’t (😮). Is there a way to introduce tomography to students who don’t have a mathematical background?

Let’s start with a simple object divided into six pixels. Our goal is to determine the value of some property for each pixel. If this were positron emission tomography, this property would be the concentration of a radioactive substance. If this were a CT scan, this property would be the x-ray absorption. How we interpret the property doesn’t matter; we’re just going to assign a number to each pixel.

The Froward Problem

Suppose this is your object.

We assume what you can measure is the sum of the pixels along one direction: a projection.

You can take projections from different orientations; sum the pixels in that direction.

A tomography machine in the hospital measures projections.

The Inverse Problem

So far we have examined the forward problem: determine the projections from the object. Next, consider the inverse problem: determine the image from the projections. Here’s another example.

How do you obtain an image of the object from multiple projections? That is the fundamental problem of tomography. In other words, how do you figure out what number to put into each pixel so that it gives the projections shown above? Stop reading and try to guess the image. When you’re done, continue reading.

Perhaps you found this image.

Good job, but your friends working from the same projections may have found different images.

Check for yourself; they all have the same three projections. If you permit negative values, you can find even more images.

The values don’t have to be integers.

The number of solutions is infinite. What can we do to pick the correct image? Use more projections!

You’ll find only one image consistent with all six projections: the first one I listed (try it yourself). If you have enough projections, you can find a unique image. You’ve solved the problem of tomography.

If you add noise to the projections, you may have no solution. In that case, you would need to use something like the least squares method to estimate the image. But that’s another story.

If you have only six pixels (and no noise) you can compute the image using trial and error, and some logic, sort of like Sudoku. A medical image, however, might have tens of thousands of pixels! What do you do then? That’s exactly what Russ and I discuss in Chapter 12 of Intermediate Physics for Medicine and Biology.

I will leave you with a final example to solve on your own. Enjoy!

IPMB Scavenger Hunt!

Here in Michigan Big Gretch has us all stuck at home cause of rona, and some of you might be quarantined with a younger brother. What can you do with him that is both educational and fun? Try the Intermediate Physics for Medicine and Biology Scavenger Hunt!

In the first section of IPMB, Russ Hobbie and I talk about distances and sizes. We show a couple illustrations (Figures 1.1 and 1.2) containing biological objects drawn to scale. Our goal is to help you build intuition about the relative size of things. This is an important skill, both for a sophomore premed student and for your third-grade niece. So, grab your little rug-rat and let’s play. Start with a picture of severe acute respiratory syndrome coronavirus, the virus that causes covid-19.

The starting point for playing IPMB Scavenger Hunt!

This game is best played in Powerpoint, so I suggest downloading the .pptx version of the figure. Now, you and your little sister search the internet for pictures of biological objects, copy and paste them into the figure, and label them. Make sure you expand or contract the image so its size matches the scale bars in the figure. You should be able to fit eight to ten objects in the picture before it gets crowded. When done, you will have created your own version of Fig. 1.1 or 1.2. But yours will be cooler that ours, because it will be in color and will probably contain real images as opposed to poorly drawn cartoons. An example that I created is shown below.

An example resulting from playing IPMB Scavenger Hunt! A couple good sources of pictures are the CDC Image Library, and the Molecule of the Month Website.

What are the goals of this activity?

• Learn the relative sizes of objects.
• Become familiar with macromolecules, viruses, bacteria, organelles, and cells.
• Understand metric units such as microns and nanometers.
• Practice using Powerpoint to make slides.
• Above all else, have fun!

If your young cousin doesn’t want to stop, you can create your own version of the game using a larger or smaller initial object (perhaps a nerve axon and a 10 micron scale bar).

Your tenderfoot friend may select objects that don’t fit into the figure. For instance, she may want to include a red blood cell, which is too large for the slide. Don’t say no. Find an object around the house (perhaps a pillow) that is about the size of a red blood cell on this scale, thereby reinforcing the relative size of things. Don’t worry if you don’t have Powerpoint. Instead, pick your own size scale (for example, one foot equals one micron) and go around the house collecting everyday objects to represent “viruses,” “bacteria,” and “cells.” The only requirement is that they are all to scale.

If the little fella can’t get enough biology, have him look at some of David Goodsell’s drawings (be sure to see his latest coronavirus painting, below). If you have plenty of time and money, buy him a plush covid-19 toy.

Enjoy!

The coronavirus.
Illustration by David S. Goodsell, RCSB Protein Data Bank;
doi: 10.2210/rcsb_pdb/goodsell-gallery-019

Faust in Copenhagen

Faust in Copenhagen,
by Gino Segrè.

My friend and fellow biological physicist Gene Surdutovich loaned me his copy of Faust in Copenhagen: A Struggle for the Soul of Physics, by Gino Segrè. It’s about seven physicists—Niels Bohr, Max Delbrück, Paul Dirac, Paul Ehrenfest, Werner Heisenberg, Lise Meitner, and Wolfgang Pauli—who played a key role in the development of quantum mechanics in the 1920s. Segrè uses a 1932 meeting in Copenhagen as the hinge about which his story revolves.

This is a book about seven physicists, six men and one woman, who attended a small annual gathering in Copenhagen in April 1932. To be honest, only six of them were actually there. The seventh, Wolfgang Pauli, had originally intended to go, as he had in earlier years and would do so again, but he decided that spring instead to take a vacation. He was there in spirit, as you will see.

Of the seven, the one who plays the least significant role in the story is Delbrück. Nevertheless, he is the one most closely related to Intermediate Physics for Medicine and Biology. One topic I have examined in this blog is how scientists make the transition from physics to biology. Below I provide brief excerpts from Faust in Copenhagen, explaining how Delbrück did it. It appears in the book’s epilogue; I will add links and fill in background that you would have known had you read the rest of the book.

How Delbrück Became a Biologist

While working as the “family-theorist” in Meitner’s laboratory [in Berlin, she was the only experimental physicist of the seven], Delbrück often returned to see Bohr [his former mentor] and the continually changing group at Blegdamsvej [a street in Copenhagen where Bohr’s institute was located]. After the April 1932 meeting he had gone back to Berlin but in August was once again in Copenhagen. Sleepily stepping out of the carriage of the overnight train from Berlin, Delbrück was surprised to see Bohr’s trusted collaborator Leon Rosenfeld waiting for him with a message from Bohr. Delbrück was to go straightaway to the great meeting hall of the Rigsdag, the Danish parliament, where Bohr was about to deliver the opening lecture at the International Congress of Light Therapists… Knowing Delbrück’s interest in biology, Bohr wanted him to attend. This turned out to be the talk that changed Delbrück’s life [my italics].

In 1932 some biologists still clung to the notion that special forces not described by the ordinary laws of physics and chemistry were responsible for the existence of life… [In his talk, Bohr] asked: Could complementarity apply to life itself? [Complementarity was the principle that two opposing properties could both be true, but could not be observed simultaneously.] Perhaps the distinction between living and nonliving was not so easy to understand. Might the act of measurement be a critical step in the assessment [as it is in the Copenhagen interpretation of quantum mechanics]?...

Bohr thought there might be some intangible quality that differentiates living from nonlinving, a quality that could not be precisely quantified. He was not alluding to a divine spark, but rather to something analogous to the impossibility of saying with certainty whether light is a photon or a wave. Bohr’s conjecture was provocative, as it was meant to be, but in the end it turned out to be wrong. DNA and RNA are the answer to life, not complementarity.

But the potential connection between physics and biology had surfaced. Would Delbrück have become a biologist if he had not thought he might find something like complementarity at the root of life’s existence? Would molecular biology have developed as it did without the structure imposed on it by the discipline Delbrück had learned from Bohr? Many who know the field think the answer to both questions is no. When James Watson wrote Delbrück in 1953 that he and Francis Crick had found their double helix model of DNA, Delbrück was struck by the elegance of the structure, but disappointed by its simplicity. Hearing that life did not require any basic new principles, he remarked that if felt to him as if the hydrogen atom had been fully explained in the 1920s without the need for quantum mechanics. [The Bohr model explained the spectrum of hydrogen in 1913, but the more modern theory of quantum mechanics refined and improved that model in the 1920s.]

By then Delbrück had become a Bohr-life figure in the new field of molecular biology [Bohr, one of the senior members of the group of seven along with Ehrenfest and Meitner, was much loved as a mentor and father-figure as well as a scientist]. He had created at Cold Spring Harbor Laboratory on Long Island and at the California Institute of Technology Copenhagen-like atmospheres for young biologists…

[Delbrück], together with Salvador Luria and Alfred Hershey, with whom he shared the 1969 Nobel Prize in Physiology or Medicine, had shown how bacteriophages, the viruses that attack bacteria, were the simplest and most efficient tool for studying genetics… In many ways the importance of these findings were comparable to Bohr’s 1913 discovery of the rules for the hydrogen atom’s radiation...

Perhaps these qualities [of personality that are similar to Bohr’s, of integrity and intellectual openness] are the basis of the deep lifelong link between Pauli and Debrück. Luria, in his memoirs, remembered his first meeting with Delbrück. At the time both were recent refugees from Europe. It was New York City in late December 1941, only a few weeks after Pearl Harbor:

Max took me to dinner with two other scientists, one of them the great physicist Wolfgang Pauli. I was properly intimidated, but Pauli simply asked me “Sprechen Sie Deutsch?” and without waiting for a reply proceeded to eat and speak German so prodigiously fast that I understood not a word [of the seven, only Bohr (Danish) and Dirac (English) did not speak German as their first language]. I would have been even more scared had I known of Pauli’s classic remark “So young and he has already contributed so little.” [Pauli was known and loved for his quick and biting wit.]

Fortunately young Luria soon contributed a great deal.

As for Pauli, his last significant letter, written two months before he died, was a twelve-page one to Max. Beginning with a discussion of the festivities for Lise Meitner’s eightieth birthday, it ends on a more personal note: “I cannot forget the dear manner, which goes back to old times, when you took leave of me. I have the impression that here something has been renewed between us that is important.”

Delbrück would hike and camp with young biologists in the Southern California desert rather than walk with then along the North Sea shore of Denmark [where Bohr had walked with young Heisenberg, discussing quantum mechanics], but the spirit was the same as Bohr’s had been. On the other hand, rather than Bohr’s polite, “I don’t mean to criticize, only to understand,” Max remained famous for his rudeness... Though the spirit was Bohr’s, the style was Pauli’s...

Delbrück died in 1981 in California.

Looking for a good read while cooped up in your house because of the coronavirus? I recommend Faust in Copenhagen. It’s wunderbar.

Radiation Oncology: A Physicist's Eye View

I miss Oakland University’s library. It’s been locked up now for about six weeks because of the coronavirus pandemic, so I can’t checkout books. As the physics department’s liaison to the library, I’ve used our meager annual allocation to purchase many of the books cited in Intermediate Physics for Medicine and Biology. I enjoy browsing through the stacks more than I realized.

I now better appreciate online access to textbooks. IPMB can be downloaded through the OU library catalog at no charge for anyone logging on as an OU student, faculty,or staff. I have access to other books online, and the number seems to be increasing every year. 

Radiation Oncology:
A Physicist's Eye View,
by Michael Goitein.

For example, I can download a textbook cited in Chapter 16 (Medical Uses of X-Rays) of IPMB.

Goitein M (2008) Radiation Oncology: A Physicist’s Eye View. Springer, New York.

It’s an excellent book, and you may hear more about it from me in the coming weeks. The preface begins:

This book describes how radiation is used in the treatment of cancer. It is written from a physicist’s perspective, describing the physical basis for radiation therapy, and does not address the medical rationale or clinical aspects of such treatments. Although the physics of radiation therapy is a technical subject, I have used, to the extent possible, non-technical language. My intention is to give my readers an overview of the broad issues and to whet their appetite for more detailed information, such as is available in textbooks.

“Whetting their appetite” is also a goal of IPMB, and of this blog. In fact, education in general is a process of whetting the appetite of readers so they will go learn more on their own.

I particularly like Michael Goitein’s chapter on uncertainty. The lessons it provides apply far beyond radiation oncology. In fact, it’s a lesson that is never more relevant than during a pandemic. Enjoy!

UNCERTAINTY MUST BE MADE EXPLICIT

ISO [International Organization for Standardization] (1995) states that “the result of a measurement [or calculation] … is complete only when accompanied by a statement of uncertainty.” Put more strongly, a measured or computed value which is not accompanied by an uncertainty estimate is meaningless. One simply does not know what to make of it. For reasons which I do not understand, and vehemently disapprove of, the statement of uncertainty in the clinical setting is very often absent. And, when one is given, it is usually unaccompanied by the qualifying information as to the confidence associated with the stated uncertainty interval—which largely invalidates the statement of uncertainty.

The importance of first estimating and then providing an estimate of uncertainty has led me to promulgate the following law:

[Goitein's First Law of Uncertainty.]

There is simply no excuse for violating either part of Law number 1. The uncertainty estimate may be generic, based on past experience with similar problems; it may be a rough “back-of-the-envelope” calculation; or it may be the result of a detailed analysis of the particular measurement. Sometimes it will be sufficient to provide an umbrella statement such as “all doses have an associated confidence interval of ±2% (SD) unless otherwise noted.” In any event, the uncertainty estimate should never be implicit; it should be stated.

In graphical displays such as that of a dose distribution in a two- dimensional plane, the display of uncertainty can be quite challenging. This is for two reasons. First, it imposes an additional dimension of information which must somehow be graphically presented. And second because, in the case, for example, of the value of the dose at a point, the uncertainty may be expressed as either a numerical uncertainty in the dose value, or as a positional uncertainty in terms of the distance of closest approach. One approach to the display of dose uncertainty is shown in Figure 6.4 of Chapter 6.

HOW TO DEAL WITH UNCERTAINTY

To act in the face of uncertainty is to accept risk. Of course, deciding not to act is also an action, and equally involves risk. One’s decision as to what action to take, or not to take, should be based on the probability of a given consequence of the action and the importance of that consequence. In medical practice, it is particularly important that the importance assigned to a particular consequence is that of the patient, and not his or her physician. I know a clinician who makes major changes in his therapeutic strategy because of what I consider to be a trivial cosmetic problem. Of course, some patients might not find it trivial at all. So, since he assumes that all patients share his concern, I judge that he does not reflect the individual patient’s opinion very well. Parenthetically, it is impressive how illogically most of us perform our risk analyses, accepting substantial risks such as driving to the airport while refusing other, much smaller ones, such as flying to Paris (Wilson and Crouch, 2001). (I hasten to add that I speak here of the risk of flying, not that of being in Paris.)

People are often puzzled as to how to proceed once they have analyzed and appreciated the full range of factors which make a given value uncertain. How should one act in the face of the uncertainty? Luckily there is a simple answer to this conundrum, which is tantamount to a tautology. Even though it may be uncertain, the value that you should use for some quantity as a basis for action is your best estimate of that quantity. It’s as simple as that. You should plunge ahead, using the measured or estimated value as though it were the “truth”. There is no more correct approach; one has to act in accordance with the probabilities. To reinforce this point, here is my second law:

[Goitein's Second Law of Uncertainty.]

It may seem irresponsible to promote gambling when there are life-or-death matters for a patient at stake; the word has bad connotations. But in life, since almost everything is uncertain, we in fact gamble all the time. We assess probabilities, take into account the risks, and then act. We have no choice. We could not walk through a doorway if it were otherwise. And that is what we must do in the clinic, too. We cannot be immobilized by uncertainty. We must accept its inevitability and make the best judgment we can, given the state [of] our knowledge.

The Toroid Illustration (Fig. 8.26)

In Chapter 8 (Biomagnetism) of Intermediate Physics for Medicine and Biology, Russ Hobbie and I show an illustration of a nerve axon threaded through a magnetic toroid to measure its magnetic field (Fig. 8.26).

Fig. 8.26. A nerve cell preparation is threaded through the magnetic
toroid to measure the magnetic field. The changing magnetic flux in
the toroid induces an electromotive force in the winding. Any external
current that flows through the hole in the toroid diminishes the magnetic field.

While this figure is clear and correct, I wonder if we could do better? I started with a figure of a toroidal coil from a paper I published with my PhD advisor John Wikswo and his postdoc Frans Gielen.

Gielen FLH, Roth BJ, Wikswo JP Jr (1986) Capabilities of a Toroid-Amplifier System for Magnetic Measurements of Current in Biological Tissue. IEEE Trans. Biomed. Eng. 33:910-921.

Starting with Figure 1 from that paper (you can find a copy of that figure in a previous post), I modified it to resemble Fig. 8.26, but with a three-dimensional appearance. I also added color. The result is shown below.

An axon (purple) is threaded through a toroid to measure the magnetic field.
The toroid has a ferrite core (green) that is wound with insulated copper wire
(blue). It is then sealed in a coating of epoxy (pink). The entire preparation
is submerged in a saline bath. The changing magnetic flux in the ferrite induces
an electromotive force in the winding. Any current in the bath that flows
through the hole in the toroid diminishes the magnetic field.

Do you like it?

To learn more about how I wound the wire and applied the epoxy coating, see my earlier post about The Magnetic Field of a Single Axon. The part about "any current in the bath that flows through the hole in the toroid diminishes the magnetic field" is described in more detail in my post about the Bubble Experiment.

Chernobyl Then and Now: A Global Perspective

Last year I was supposed to give a talk at Oakland University for a symposium about “Chernobyl Then and Now: A Global Perspective.” It was part of an exhibition at the OU Art Gallery titled “McMillan’s Chernobyl: An Intimation of the Way the World Would End.” My role at the symposium was to explain the factors that led to the explosion of the Chernobyl nuclear power plant. I was chosen by the organizer, OU Professor of Art History Claude Baillargeon, because I had taught a class about The Making of the Atomic Bomb in the Oakland’s Honors College.

Readers of Intermediate Physics for Medicine and Biology should become familiar with the Chernobyl disaster because it illustrates how exposure to radiation can affect people over different time scales, from short term acute radiation sickness to long-term radiation-induced cancer.

It turned out I could not attend the symposium. My friend Gene Surdutovich stepped in at the last minute to replace me, and because he is from Ukraine—where the disaster occurred—he provided more insight than I could have. However, I thought the readers of this blog might want to read a transcript of the talk I planned to present. It was supposed to be my “TED Talk,” aimed at a broad audience with limited scientific background. No Powerpoint, no blackboard; just a few balls and a pencil as props.

The nuclear reactor in Chernobyl had an inherently unstable design that led to the worst nuclear accident in history. To understand why the design was so unstable, we need to review some physics.

The nucleus of an atom contains protons and neutrons. The number of protons determines what element you have. For instance, a nucleus with 92 protons is uranium. The number of neutrons determines the isotope. If a nucleus has 92 protons and 146 neutrons it is uranium-238 (because 92 + 146 = 238). Uranium-238 is the most common isotope of uranium (about 99% of natural uranium is uranium-238). If the nucleus has three fewer neutrons, that is only 143 neutrons instead of 146, it’s uranium-235, a rare isotope of uranium (about 1% of natural uranium is uranium-235).

No stable isotopes of uranium exist, but both uranium-235 and uranium-238 have very long half-lives (a half-life is how long it takes for half the nuclei to decay). Their half-lives are several billion years, which is about the same as the age of the earth. So many of the atoms of uranium that originally formed with the earth have not decayed away yet, and still exist in our rocks. We can use them as nuclear fuel.

Although uranium-235 is the rarer of the two long-lived isotopes, it is the one that is the fuel for a nuclear reactor. The uranium-235 nucleus is “fissile” meaning that it is so close to being unstable that a single neutron can trigger it to break in two pieces, releasing energy and two additional neutrons. This is called nuclear fission.

A nuclear chain reaction can start with a lot of uranium-235 and a single neutron. The neutron causes a uranium-235 nucleus to fission, breaking into two pieces plus releasing two additional neutrons and energy. These two neutrons hit two other uranium-235 nuclei, causing each of them to fission, releasing a total of four neutrons plus more energy. These four neutrons hit four other uranium-235 nuclei, releasing eight neutrons….and so on. The atomic bomb dropped on Hiroshima at the end of World War Two was based on just such an uncontrolled uranium-235 chain reaction. Fortunately, there are ways to control the chain reaction, so it can be used for more peaceful purposes, such as a nuclear reactor.

One surprising feature of uranium-235 is that SLOW neutrons are more likely to split the nucleus than FAST neutrons. How this effect was discovered is an interesting story. Enrico Fermi, an Italian physicist, was studying nuclear reactions in the 1930s by bombarding different materials with neutrons. He observed more nuclear reactions if his apparatus sat on a wooden table top than if it sat on a marble table top! What? It turns out wood was better at slowing the neutrons than marble. Think how confusing this must have been for Fermi. He was so confused that he tried submerging the apparatus in a pond behind the physics building and the reactions increased even more!

A uranium-235 chain reaction triggered by neutrons works best with slow neutrons. Therefore, nuclear reactors need a “moderator”: a substance that slows the neutrons down. The moderator is the key to understanding what happened at Chernobyl.

The best moderators are materials whose nuclear mass is about the same as the mass of a neutron. If the nucleus was a lot heavier than the neutron, the neutron would not slow down after the collision. Imagine this tennis ball is the light neutron, and this big basketball is the heavy nucleus. When the neutron hits the nucleus, it just bounces off. It changes direction but doesn’t slow down. Now, imagine this neutron collides with a very light particle, represented by this ping pong ball. When the relatively heavy neutron hits the light particle, it will just push it out of the way like a train hitting a mosquito. The neutron itself won’t slow down much. To be effective at slowing the neutron down, the nucleus needs to be about the same mass as the neutron. What has a similar mass to a neutron? A proton. What nucleus contains a single proton? Hydrogen. Watch what happens when a neutron and a hydrogen nucleus collide? This ball is the neutron, and this ball is the proton: the hydrogen nucleus. Right after the collision, the neutron stops! It is like when a moving billiard ball slams into a stationary billiard ball; the one that was moving stops, and the one that was stationary starts moving. Interacting with hydrogen is a great way to slow down neutrons. Therefore, hydrogen is a great moderator. Where do you find a lot of hydrogen? Water (H₂O). It was the hydrogen in the water of the wooden table top that was so effective at slowing Fermi’s neutrons. The water in the pond behind the physics building was even better; it had even more hydrogen.

Other elements that have relatively light nuclei are also good moderators, such as carbon (carbon’s nucleus has 6 protons and 6 neutrons). It’s somewhat heavier than you want in order to slow neutrons optimally, but it’s not bad, and its abundant, cheap, and dense. During the Manhattan Project, Fermi (who had fled fascist Italy and settled in the United States) built the first nuclear reactor in a squash court under the football stadium at the University of Chicago. His reactor was a “pile” of uranium balls, with each ball surrounded by blocks of graphite (almost pure carbon, like the lead in this pencil). The uranium was the fuel and the graphite was the moderator.

Before we talk more about moderators, you might be wondering why Fermi’s reactor didn’t explode, destroying Chicago? One reason was that his uranium was a mix of uranium-235 and uranium-238, and was in fact 99% uranium-238. The uranium-238 doesn’t contribute to the chain reaction; it’s not fissile. To make matters worse, uranium-238 can absorb a neutron and dampen the chain reaction. When uranium-238 captures a neutron to become uranium-239, it takes a neutron “out of action” so to speak. During the Manhattan Project the United States spent enormous amounts of time and money separating uranium-235 from uranium-238, so it could use almost pure uranium-235 in the atomic bomb. But Fermi didn’t have any such enriched uranium. Also, Fermi controlled his reactor using a super-duper neutron absorber, cadmium. Cadmium sucks up the neutrons, stopping the chain reaction. Fermi could push in or pull out cadmium control rods to keep the speed of the reaction “just right.” As an emergency backup he had one big cadmium control rod suspended over the reactor by a rope. One of Fermi’s assistants stood by with an axe. If things started to go out of control, his job was to cut the rope dropping the cadmium rod and stopping the reaction. Fortunately, Fermi took great pains to operate the reactor carefully, and no such problems occurred. Had things gone wrong, the reactor probably wouldn’t have exploded like a bomb. It would have just gotten very hot and melted, causing a “meltdown” with all sorts of radiation release, like at Chernobyl. It’s a scary thought because it was in the middle of Chicago, but we were at war against the Nazis, so people took some risks.

Now back to the moderator. Let’s consider three different moderators. First, “heavy water.” This is water containing a rare, heavy isotope of hydrogen, hydrogen-2 (its nucleus consists of one proton and one neutron). While it is not quite as good as hydrogen-1 at slowing down neutrons, it’s still very good, and it has one advantage. Hydrogen-1 (a single proton) can sometimes absorb a neutron to become hydrogen-2. It’s as if occasionally these two balls stick together when they hit. This capture of a neutron slows the chain reaction. Hydrogen-2, however, rarely absorbs a neutron to become hydrogen-3, so it’s a great moderator: it slows the neutrons without absorbing them. During World War Two, the Germans tried to construct a nuclear reactor using heavy water as the moderator. The problem was, heavy water is difficult and expensive to make. There was a plant in Norway that produced heavy water, and it was controlled by the Germans. The British sent in a commando raid that sabotaged the plant, causing all that precious heavy water to flow down the drain. Heavy water is so expensive it isn’t used nowadays in reactors, and we won’t discuss it anymore.

The second moderator we’ll consider is regular water made using hydrogen-1 (I’ll call it just “water” as opposed to “heavy water”). Nowadays most nuclear reactors in the United States use water as the moderator. They also use water as the coolant. You need a coolant to keep the reactor from getting too hot and melting. Also, the coolant is how you get the heat out of the reactor so you can use it to run a steam engine and generate power. So in the United States, water in a nuclear reactor has two purposes: it’s the moderator and the coolant. Suppose that the reactor, for some reason, gets too hot and the water starts boiling off. That will cause the moderator to boil away. No more moderator, no more slowing down the neutrons. No more slowing down the neutrons, no more chain reaction. This is a type of a negative feedback loop that makes the reactor inherently safe. It’s like the thermostat in your house: if the house gets too hot, the thermostat turns off the furnace, and the house cools down. Recall that hydrogen-1 can also absorb neutrons, and in theory that could cause the reactor to speed up when the water boils away because there is less neutron absorption. So neutron absorption and moderation are opposite effects. But a reduction of neutron absorption is less important than the disappearance of the moderator, so on the whole when water boils the reaction slows down. We say that the reactor has a “negative void coefficient.” The “void” means the water is boiling, forming bubbles. The “negative” means this negative feedback loop occurs, keeping the reaction from increasing out of control.

Now for the third moderator: carbon. The Russians built something called an RBMK reactor. This is a Russian acronym, so I won’t try to explain what the different letters mean. Suffice to say, an RBMK reactor is a nuclear reactor that uses carbon as the moderator. Chernobyl was an RBMK reactor. Like Fermi’s original reactor, the carbon was in the form of graphite. In addition, an RBMK reactor uses water as the coolant. Graphite is the moderator and water is the coolant. Now, suppose this type of reactor begins to heat up and the water starts to boil away. The hydrogen in the water is not the primary moderator, the carbon in the graphite is. So, the reaction doesn’t slow down when the water boils away; the carbon moderator is still there, slowing the neutrons. But remember, the hydrogen in water sometimes absorbs a neutron, taking it out of action. This neutron capture decreases as the water boils away, so the reaction increases. Increased heat causes water to boil, causing the reaction to speed up, causing increased heat, causing more water to boil, causing the reaction to speed up even more, causing yet more increased heat … This is a positive feedback loop; a vicious cycle. The reactor has a “positive void coefficient.” It’s as if the thermostat in your house was wired wrong, so when the house got hot the furnace turned ON, heating the house more. Normally the reactor is designed with all sorts of controls to prevent this positive feedback loop from taking off. For instance, control rods can be pushed in and out as needed. But, if for some reason these controls are not in place, the reactor will heat up dramatically and quickly, just as it did at Chernobyl.

Why do we have nuclear reactors? Nuclear reactors produce heat to power a steam engine, which in turn generates electricity. The steam needs to be at high pressure, so it can turn the turbine. Therefore, the reactor is in a pressure container. It’s like a pressure cooker. If the water boils too much, the pressure builds up until the container can’t handle it anymore and bursts, releasing steam. It’s a little like your whistling tea pot, except instead of whistling when the water boils, the reactor explodes. And unlike your teapot, the reactor releases radioactive elements along with the steam. You get a cloud of radioactivity.

Another problem with an RBMK reactor is that graphite burns. It’s pure carbon. It’s like coal. Once the pressure container bursts, oxygen can get in igniting the graphite, starting a fire. The graphite then spews radioactive smoke up into the atmosphere. Many of the people killed in the Chernobyl accident were firemen, trying to put out the fire.

Another issue, a little less important but worth mentioning, was the control rods. Chernobyl had control rods made out of boron, which like cadmium is an excellent neutron absorber. It vacuums up the neutrons and stops the chain reaction. The problem was, the control rods were tipped with graphite. As you push in a control rod, initially it would be like adding moderator, quickening the reaction. Only when the rod was completely pushed in would the boron absorb neutrons, slowing the reaction. So, the control rods would eventually suppress the chain reaction, but initially they made things worse. If, like at Chernobyl, a problem developed quickly, the control rods couldn’t keep up.

I won’t go in to all the comedy of errors that were the immediate cause of the accident at Chernobyl. The reactor was undergoing a test, and several of the controls were turned off. Some safeguards were still in place, but mistakes, poor communication, and ignorance prevented them from working. Whatever the immediate cause of the accident, the crucial point is that the reactor design itself was unstable. It’s like trying to balance this pencil on its tip. You can do it if you are careful and have some controls, but it’s inherently unstable. If you are not always vigilant, the pencil will fall over. The unstable design of the Chernobyl reactor made it a disaster waiting to happen.

If you would like to hear me give this talk (slightly modified), you can watch the YouTube video below. This winter I was teaching the second semester of introductory physics, and when the coronavirus pandemic arrived I had to switch to an online format. I recorded a lecture about Chernobyl when we were discussing nuclear energy.

My Chernobyl talk, given to my Introductory Physics class,
online from home because of Covid-19.

Donnan Equilibrium

Russ Hobbie and I analyze Donnan equilibrium in Chapter 9 of Intermediate Physics for Medicine and Biology.

Section 9.1 discusses Donnan equilibrium, in which the presence of an impermeant ion on one side of a membrane, along with other ions that can pass through, causes a potential difference to build up across the membrane. This potential difference exists even though the bulk solution on each side of the membrane is electrically neutral.

Today I present two new homework problems based on one of Donnan’s original papers.

Donnan, F. G. (1924) “The Theory of Membrane Equilibria.” Chemical Reviews, Volume 1, Pages 73-90.

Here’s the first problem.

Section 9.1

Problem 2 ½. Suppose you have two equal volumes of solution separated by a semipermeable membrane that can pass small ions like sodium and potassium but not large anions like A⁻. Initially, on the left is 1 mole of Na⁺ and 1 mole of A⁻, and on the right is 10 moles of K⁺ and 10 moles of A⁻. What is the equilibrium amount of Na⁺, K⁺, and A⁻ on each side of the membrane?

Stop and solve the problem using the methods described in IPMB. Then come back and compare your solution with mine (and Donnan’s).

In equilibrium, x moles of sodium will cross the membrane from left to right. To preserve electroneutrality, x moles of potassium will cross from right to left. So on the left you have 1 – x moles of Na⁺, x moles of K⁺, and 1 mole of A⁻. On the right you have x moles of Na⁺, 10 – x moles of K⁺, and 10 moles of A⁻.

Both sodium and potassium are distributed by the same Boltzmann factor, implying that

           [Na⁺]_left/[Na⁺]_right = [K⁺]_left/[K⁺]_right = exp(−eV/kT)            (Eq. 9.4)

where e is the elementary charge, V is the voltage across the membrane, k is Boltzmann’s constant, and T is the absolute temperature. Therefore

           (1 – x)/x = x/(10 – x)

or x = 10/11 = 0.91. The equilibrium amounts (in moles) are

                          left       right
           Na⁺        0.09      0.91
           K⁺          0.91      9.09
           A⁻          1.00    10.00

The voltage across the membrane is

           V = kT/e ln([Na⁺]_right/[Na⁺]_left) = (26.7 mV) ln(10.1) = 62 mV .

Donnan writes

In other words, 9.1 per cent of the potassium ions originally present [on the right] diffuse to [the left], while 90.9 per cent of the sodium ions originally present [on the left] diffuse to [the right]. Thus the fall of a relatively small percentage of the potassium ions down a concentration gradient is sufficient in this case to pull a very high percentage of the sodium ions up a concentration gradient. The equilibrium state represents the simplest possible case of two electrically interlocked and balanced diffusion-gradients.

Like this problem? Here’s another. Repeat the last problem, but instead of initially having 10 moles of K⁺ on the right, assume you have 10 moles of Ca⁺⁺. Calcium is divalent; how will that change the problem?

Problem 3 ½. Suppose you have two solutions of equal volume separated by a semi-impermeable membrane that can pass small ions like sodium and calcium but not large anions like A⁻ and B⁻⁻.  Initially, on the left is 1 mole of Na⁺ and 1 mole of A⁻, and on the right is 10 moles of Ca⁺⁺ and 10 moles of B⁻⁻. What is the equilibrium amount of Na⁺, Ca⁺⁺, A⁻ and B⁻⁻ on each side of the membrane?

Again, stop, solve the problem, and then come back to compare solutions.

Suppose 2x moles of Na⁺ cross the membrane from left to right. To preserve electroneutrality, x moles of Ca⁺⁺ move from right to left. Both cations are distributed by a Boltzmann factor (Eq. 9.4)

           [Na⁺]_left/[Na⁺]_right = exp(−eV/kT)

           [Ca⁺⁺]_left/[Ca⁺⁺]_right  = exp(−2eV/kT) .

However,

          exp(−2eV/kT) = [ exp(−eV/kT) ]²

so

      { [Na⁺]_left/[Na⁺]_right }² = [Ca⁺⁺]_left/[Ca⁺⁺]_right

or
        [ (1 –2 x)/(2x) ]² = x/(10 – x)

This is a cubic equation that I can’t solve analytically. Some trial-and-error numerical work suggests x = 0.414. The equilibrium amounts are therefore

                          left       right
           Na⁺        0.172    0.828
           Ca⁺⁺      0.414    9.586
           A⁻          1           0
           B⁻⁻        0          10 

The voltage across the membrane is

           V = kT/e ln([Na⁺]_right/[Na⁺]_left) = (26.7 mV) ln(4.814) = 42 mV .

I think this is correct; Donnan didn’t give the answer in this case, so I’m flying solo.

Frederick Donnan.
From an article in the Journal of Chemical Education,
Volume 4(7), page 819.

Who was Donnan? Frederick Donnan (1870 – 1956) was an Irish physical chemist. He obtained his PhD at the University of Leipzig under Wilhelm Ostwald, and then worked for Henry van’t Hoff. Most of his career was spent at the University College London. He was elected a fellow of the Royal Society and won the Davy Medal in 1928 “for his contributions to physical chemistry and particularly for his theory of membrane equilibrium.”

The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart

Readers of Intermediate Physics for Medicine and Biology may enjoy this story about some of my research as a graduate student, working for John Wikswo at Vanderbilt University. My goal was to determine if the biomagnetic field contains new information that cannot be obtained from the electrical potential.

In 1988, Wikswo, fellow grad student Wei-Qiang Guo, and I published an article in Mathematical Biosciences (Volume 88, Pages 191-221) about the magnetic field at the apex of the heart.

The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart.

B. J. Roth, W.-Q. Guo, and J. P. Wikswo, Jr. 

Living State Physics Group, Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235

This paper describes a volume-conductor model of the apex of the heart that accounts for the spiraling tissue geometry. Analytic expressions are derived for the potential and magnetic field produced by a cardiac action potential propagating outward from the apex. The model predicts the existence of new information in the magnetic field that is not present in the electrical potential.

The analysis was motivated by the unique fiber geometry in the heart, as shown in the figure below, from an article by Franklin Mall. It shows how the cardiac fibers spiral outward from a central spot: the apex (or to use Mall’s word, the vortex).

The apex of the heart.
From Mall, F. P. (1911) “On the Muscular Architecture of the Ventricles of the Human Heart.” American Journal of Anatomy, Volume 11, Pages 211-266.

Our model was an idealization of this complicated geometry. We modeled the fibers as making Archimedean spirals throughout a slab of tissue representing the heart wall, perfused by saline on the top and bottom.

The geometry of a slab of cardiac tissue. The thickness of the tissue is l, the conductivity of the saline bath is σ_e, and the conductivity tensors of the intracellular and interstitial volumes are σ̃_i and σ̃_o. The variables ρ, θ, and z are the cylindrical coordinates, and the red curves represent the fiber direction. Based on Fig. 2 of Roth et al. (1988).

Cardiac tissue is anisotropic; the electrical conductivity is higher parallel to the fibers than perpendicular to them. This is taken into account by using conductivity tensors. Because the fibers spiral and make a constant angle with the radial direction, the tensors have off-diagonal terms when expressed in cylindrical coordinates.

Consider a cardiac wavefront propagating outward, as if stimulated at the apex. Two behaviors occur. First, ignore the spiral geometry. A wavefront produces intracellular current propagating radially outward and extracellular current forming closed loops in the bath (blue). This current produces a magnetic field above and below the slab (green).

The current (blue) and magnetic field (green) created by an action potential propagating outward from the apex of the heart if no off-diagonal terms are present in the conductivity tensors. Based on Fig. 5a of Roth et al. (1988).

Second, ignore the bath but include the spiral fiber geometry. Although the wavefront propagates radially outward, the anisotropy and fiber geometry create an intracellular current that has a component in the θ direction (blue). This current produces its own magnetic field (green).

The azimuthal component of the current (blue) and the electrically silent components of the magnetic field (green) produced by off-diagonal terms in the conductivity tensor, with σ_e = 0. Based on Fig. 5b of Roth et al. (1988).

Of course, both of these mechanisms operate simultaneously, so the total magnetic field distribution looks something like that shown below.

The total magnetic field at the apex of the heart. This figure is only qualitatively correct; the field lines may not be quantitatively accurate. Based on Fig. 5e of Roth et al. (1988).

The original versions of these beautiful figures were prepared by a draftsman in Wikswo’s laboratory. I can’t remember who, but it might have been undergraduate David Barach, who prepared many of our illustrations by hand at the drafting desk. I added color for this blog post.

The main conclusion of this study is that there exists new information about the tissue in the magnetic field that is not present from measuring the electrical potential. The ρ and z components of the magnetic field are electrically silent; the spiraling fiber geometry has no influence on the electrical potential.

Is this mathematical model real, or just the musings of a crazy physics grad student? Two decades after we published our model, Krista McBride—another of Wikswo’s grad students, making her my academic sister—performed an experiment to test our prediction, and obtained results consistent with our calculations.

I’m always amazed when one of my predictions turns out to be correct.

Consequences of the Inverse Viscosity-Temperature Relationship

In Homework Problem 50 of Chapter 1 in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Section 1.19

Problem 50. The viscosity of water (and therefore of blood) is a rapidly decreasing function of temperature. Water at 5° C is twice as viscous as water at 35° C. Speculate on the implications of this extreme temperature dependence for the circulatory system of cold-blooded animals. (For a further discussion, see Vogel 1994, pp. 27–31.)

Life in Moving Fluids,
by Steven Vogel.

Let’s see what Steven Vogel discusses. The citation is to Life in Moving Fluids: The Physical Biology of Flow.

CONSEQUENCES OF THE INVERSE VISCOSITY-TEMPERATURE RELATIONSHIP

At 5° C water is about twice as viscous (dynamically or kinematically) as at 35° C; organisms live at both temperatures and, indeed, at ones still higher and lower. Some experience an extreme range within their lifetimes—seasonally, diurnally, or even in different parts of the body simultaneously. Does the consequent variation in viscosity ever have biological implications?...

Consider the body temperature of animals. At elevated temperatures less power ought to be required to keep blood circulating if the viscosity of blood follows the normal behavior of liquids. And, in our case, it does behave in the ordinary way—human blood viscosity (ignoring blood’s minor non-Newtonianism) is 50% higher at 20° than at 37° C… Is this a fringe benefit of having a high body temperature? Probably the saving in power is not especially significant—circulation costs only about 6% of basal metabolic rate. More interesting is the possibility of compensatory adjustments in the bloods and circulatory systems of animals that tolerate a wide range of internal temperatures. The red blood cells of cold-blooded vertebrates, and therefore presumably their capillary diameters, are typically larger than either the nucleated cells of birds or the nonnucleated ones of mammals… The shear rate of blood is greatest in the capillaries; must these be larger in order to permit circulation at adequate rates without excessive cost in a cold body?...

Is the severe temperature dependence of viscosity perhaps a serendipitous advantage on occasion? A marine iguana of the Galapagos basks on warm rocks, heating rapidly, and then jumps into the cold Humboldt current to graze on algae, cooling only slowly. Circulatory adjustments as the animal takes the plunge have been postulated…, but no one seems to have looked at whether part of the circulatory reduction in cold water is just a passive consequence of an increase in viscosity. A variety of large, rapid, pelagic fish have circulatory arrangements that permit locomotory muscles to get quite hot when they’re in use…; blood flow ought to increase automatically at just the appropriate time.

A less speculative case is that of Antarctic mammals and birds… [They] must commonly contend with cold appendages, since full insulation of feet and flippers would be quite incompatible with their normal functions. The circulation of such an appendage often includes a [countercurrent] heat exchanger at the base of the limb so that, in effect, a cold-blooded appendage and a warm-blooded body can be run on the same circulatory system without huge loses of heat… Changes in blood viscosity will reduce flow to appendages when they get cold quite without active adjustments within the circulatory systems.

Vogel goes on for another couple pages. I love the way he uses comparative physiology to illustrate physics. He also covers a lot of ground, ranging from the Galapagos islands to Antarctica. Such discussions are typical of Life in Moving Fluids.

The Rayleigh-Einstein-Jeans law

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss blackbody radiation. Max Planck’s blackbody radiation formula is given in Eq. 14.33

where λ is the wavelength and T is the absolute temperature. This equation, derived in December 1900, is the first formula that contained Planck’s constant, h.

Often you can recover a classical (non-quantum) result by taking the limit as Planck’s constant goes to zero. Here’s a new homework problem to find the classical limit of the blackbody radiation formula.

Section 14.8

Problem 26 ½. Take the limit of Planck’s blackbody radiation formula, Eq. 14.33, when Planck’s constant goes to zero. Your result should should be the classical Rayleigh-Jeans formula. Discuss how it behaves as λ goes to zero. Small wavelengths correspond to the ultraviolet and x-ray part of the electromagnetic spectrum. Why do you think this behavior is known as the “ultraviolet catastrophe”?

Subtle is the Lord,
by Abraham Pais.

I always thought that the Rayleigh-Jeans formula was a Victorian result that Planck knew about when he derived Eq. 14.33. However, when thumbing through Subtle is the Lord: The Science and the Life of Albert Einstein, by Abraham Pais, I learned that the Rayleigh-Jeans formula is not much older than Planck’s formula. Lord Rayleigh derived a preliminary version of it in June 1900, just months before Planck derived Eq. 14.33. He published a more complete version in 1905, except he was off by a factor of eight. James Jeans caught Rayleigh’s mistake, corrected it, and thereby got his name attached to the Rayleigh-Jeans formula. I am amazed that Planck’s blackbody formula predates the definitive version of the Rayleigh-Jeans formula.

Einstein rederived Rayleigh’s formula from basic thermodynamics principles in, you guessed it, his annus mirabilis, 1905. Pais concludes “it follows from this chronology (not that it matters much) that the Rayleigh-Jeans law ought properly to be called the Rayleigh-Einstein-Jeans law.”