Polonium-210, The Perfect Poison

Figure 17.27 in the 5th edition of Intermediate Physics for Medicine and Biology shows the decay series arising from the radioactive isotope radon-222, which itself is produced by the decay of the long-lived isotope uranium-238. The last step in this long chain of reactions is the alpha decay of polonium-210 to the stable isotope lead-206. The half-life of this decay is 138 days. This is not the only isotope of polonium in radon’s decay series. A heavier isotope polonium-214 has a half-life of 160 microseconds, and polonium-218 has a half-life of 3 minutes.

Polonium was discovered by Marie and Pierre Curie in 1898 when analyzing pitchblende, a uranium containing ore. It was named after Marie’s homeland, Poland. Now ²¹⁰Po is produced by bombarding bismuth-209 with neutrons, forming bismuth-210, which undergoes beta decay to ²¹⁰Po.

²¹⁰Po is infamous for being a deadly poison. For a given mass, ²¹⁰Po is 250,000 times more toxic than hydrogen cyanide. Its toxicity comes from the 5.3-MeV alpha particle it emits. Because alpha particles are easily stopped by clothing and even skin, ²¹⁰Po is dangerous primarily when breathed or ingested, so that the alpha particles are emitted inside the body. A nearly pure alpha emitter, ²¹⁰Po rarely emits a gamma ray, making it difficult to detect this poison unless one measures the alpha particles directly. A lethal dose comes from ingesting about a microgram.

²¹⁰Po was used in the 2006 assassination of Alexander Litvinenko, a former Russian spy who was apparently given some polonium-laced tea by Russian agents (the investigation into this complicated murder continues--see here and here--and the details are still debated). Death by ²¹⁰Po is slow; the 44-year old Litvinenko needed 22 days for the radiation to eventually take his life.

Polonium was also suspected to play a role in the 2004 death of Palestinian leader Yasser Arafat. Just this month, a French investigation has concluded that there is not enough evidence for pressing charges. The issue is complicated because ²¹⁰Po is found in cigarette smoke, and Arafat was a heavy smoker. The National Council on Radiation Protection and Measurements reports that the annual effective dose equivalent to a smoker from radiation in tobacco is about 13 mSv, which is over four times the average annual dose of 3 mSv we are all exposed to (see Section 16.12 in IPMB), but is still a tiny dose.

The Environmental Protection Agency has published a report titled “Occurrence of ²¹⁰Po and Biological Effects of Low-Level Exposure: The Need for Research.” As with all studies of low-level radiation exposure, the results are difficult to assess, and depend on our assumptions about radiation risks at small doses. But Alexander Litvinenko’s death proves that at high doses ²¹⁰Po is very dangerous indeed; it’s perhaps the perfect poison.

Boltzmann’s Tomb

Asimov’s Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

In Chapter 3 of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Boltzmann factor and Boltzmann’s constant. Later in the book, we analyze the Poisson-Boltzmann equation and the Stefan-Boltzmann law. Who was Boltzmann? In Asimov’s Biographical Encyclopedia of Science and Technology (2nd revised edition), Isaac Asimov writes

BOLTZMANN, Ludwig Edward (bohlts’mahn)
Austrian physicist
Born: Vienna, February 20, 1844
Died: Duino, near Trieste (then in Austria, now in Italy), September 5, 1906

Boltzmann, the son of a civil servant, received his Ph.D. from the University of Vienna in 1866. His work on the kinetic theory of gases was done independently of Maxwell and they share the credit.

Beginning in 1871, Boltzmann increased the rigor of the mathematical treatment and emphasized the statistical interpretation of the second law of thermodynamics thus founding “statistical mechanics.” He showed that Clausius’ concept of increasing entropy of disorder [could be based on statistical ideas], laying the groundwork for the later achievements of Gibbs.

He was a firm proponent of atomism at a time when Ostwald was mounting the final campaign against it. Boltzmann also advanced a mathematical treatment that explained the manner in which, according to the experimental observations of Stefan (whom Boltzmann, in this college years, served as assistant), quantity of radiation increased as the fourth power of the temperature. This is therefore sometimes called the Stefan-Boltzmann law.

Boltzmann turned down a chance to succeed Kirchhoff at Berlin but in 1894 succeeded to Stefan’s post in Vienna.

Though Boltzmann lived longer than Maxwell, his life too was cut short. In his case it was suicide, brought on by recurrent episodes of severe mental depression accentuated, perhaps, by opposition to his atomistic notions by Oswald and others.

His equation relating entropy and disorder was engraved on the headstone of his grave.

I particularly am intrigued by the last sentence of Asimov’s entry. Who puts an equation on their tombstone? Boltzmann did!

Boltzmann's tombstone.

This equation is Eq. 3.20 in IPMB.

S = k_B ln Ω ,

where S is the entropy, k_B is Boltzmann’s constant, ln is the natural logarithm, and Ω is the number of microstates. The equation says that the entropy increases as the number of possible microstates increases. If there are only one or a few states available, the entropy is small; if there are many states available, the entropy is large. Thus, from a statistical mechanics point-of-view, the thermodynamic concept of entropy (developed well before Boltzmann’s work) is a measure of the number of states. The logarithm is important, because if system A has 10 states available and system B has 20 states available, the total number of states is the product, 200. If the entropy were proportional to Ω, the total entropy of the two systems would not be the sum of the entropy in each system. However, the logarithm property ln(Ω_AΩ_B)=ln(Ω_A)+ln(Ω_B) ensures that the entropy is indeed additive.

The definition of entropy in terms of the number of states is a fundamental relationship connecting thermodynamics and statistical mechanics. No wonder Boltzmann wanted it on his tombstone.

Meselson, Stahl, and the Most Beautiful Experiment in Biology

In Chapter 17 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem to the 5th edition about the famous Hershey-Chase experiment. We had two goals: to demonstrate how scientists use radioisotopes as tracers in biological experiments, and to describe a key experiment in modern biology.

A series of homework problems in Chapter 1 of IPMB describe the physics of the ultracentrifuge. Perhaps Russ and I should add a new homework problem in Chapter 1, to demonstrate how the centrifuge provides crucial information about biological mechanisms, and to describe another famous biological experiment. Here is my try at this new problem.

Problem 24 1/2. Suppose you grow E. coli bacteria in a growth medium containing the rare, heavy but stable isotope of nitrogen, N¹⁵. At some time t = 0 remove some of the E. coli from this medium and place it into another growth medium containing the normal isotope of nitrogen, N¹⁴. Then, at different times place DNA from the E. coli into a density gradient centrifuge (see Problem 23) and measure where along the gradient the DNA settles.

a) Describe qualitatively what you would expect to see at t = 0, before any of the E. coli reproduce.

b) Assume DNA replicates semiconservatively: replication produces two new DNA molecules, each containing two strands: one a strand from the original DNA molecule and another new strand produced from the medium. Describe what you would expect to see at t = t₁, where t₁ is the time required to produce one new generation of E. coli. Describe what you would expect to see at t = 2 t₁. 

c) Repeat part b) assuming DNA replicates conservatively: each replication produces two DNA molecules, one containing the original two strands and the other containing two new strands. 

d) Repeat part b) assuming DNA replicates dispersively: each replication produces two new DNA molecules, both containing a mix of the original and new DNA. 

This experiment was performed by Meselson and Stahl in 1958, and is one of the central experiments underlying modern biology. It demonstrates the semiconservative replication of DNA.

The Eighth Day of Creation,
by Horace Freeland Judson.

If you want to learn more about the Meselson-Stahl experiment, I suggest reading Chapter 3 of The Eighth Day of Creation: The Makers of the Revolution in Biology, by Horace Freeland Judson. Below I provide an excerpt.

I first heard of semiconservative replication on New Year’s Day, 1958, in Chicago—and a bright, windy, iron-cold morning it was. Seven of us who had been undergraduates together at the University of Chicago (we had all overlapped Watson’s last year there) were sitting scratchy-eyed over bacon and eggs and coffee when Matthew Meselson, by then a doctoral candidate with Pauling at Cal Tech, took a photograph from his wallet and passed it around the table. The picture showed a stack of gray stripes, with narrow, dark-gray bands across them—some stripes with one band, some with two or three together near the middle. The photo was the main result of an experiment that Menelson had devised with a post doctoral fellow at Cal Tech, Franklin Stahl.

Their paper was not yet published—not yet written. The work it describes is now recognized as displaying the most rare technical skill, while conceptually its confirmation of the way DNA reproduces itself has become, simply, part of the mainstream. In its place towards the end of the history of the elucidation of the structure and function of DNA, Meselson’s and Stahl’s paper possessed an importance and authority like Oswald Avery’s announcement, fourteen years earlier, of the isolation of the transforming principle and its identification as DNA. “Classic” was Watson’s epithet for Meselson’s and Stahl’s paper. Watson’s predecessor as director of the Cold Spring Harbor Laboratory, John Cairns, startled me in conversation when he described Meselson’s central demonstration without qualification as “the most beautiful experiment in biology.”

Learning Biology

Suppose you are a physicist, mathematician, or engineer who wants to change your research direction toward biology and medicine. How do you learn biology? Let’s assume you don’t quit your day job, so you have limited time. Here are my suggestions.

1. 

   Machinery of Life,
   by David Goodsell.

   Read The Machinery of Life (2nd edition), by David Goodsell. I discussed this book a few weeks ago in this blog. It’s visual, easy to read, not too long, cheap, and doesn’t get bogged down in details. It’s a great introduction; this is where I would start.
2. If you haven’t had an introductory biology class, you might consider taking this online biology class from MIT. It’s free, it has homework assignments and quizzes so you can assess your learning, and you can work at your own schedule. For those who prefer an online class to reading a book, this is the thing to do.
3. If you would prefer reading an introductory biology textbook, a popular one is Campbell Biology by Reece et al., now in its 10th edition. The MIT online course mentioned above and the introductory biology classes here at Oakland University use this book. Its advantages are that it covers all of biology and it is written for introductory students. Its disadvantages are that it is expensive and long. I am not an expert on the different intro biology textbooks; there may be others just as good.

4. 

   The Eighth Day of Creation,
   by Horace Freeland Judson.

   I like to learn a subject by studying its history. If you want to try this, I suggest: The Double Helix by James Watson (of Watson and Crick) and The Eighth Day of Creation by Horace Freeland Judson. Watson’s book is a classic: a first-person account the discovery of the structure of DNA. It is well written, controversial, and should be read by anyone interested in science. Judson’s book is longer and more comprehensive; a fantastic book.
5. The textbook Physical Biology of the Cell by Phillips et al. was written by physicists trying to learn biology. Also from a physicist’s point of view are Biological Physics and Physical Models of Living Systems, both by Philip Nelson. These books don’t cover all of biology, but a physicist may like them.
6. I learned a lot of biology in high school reading Isaac Asimov books. They often take a historical approach, and are qualitative, interesting, clearly written, fairly short, and cheap. I worry about recommending them because biology has progressed so much over the last few decades that these books from the 1960s are out-of-date. However, I suspect they are still useful introductions, and I suggest The Wellsprings of Life, The Genetic Code, The Human Body, The Human Brain, and A Short History of Biology.
7. Some books from my ideal bookshelf cover parts of biology from the point of view of a physicist: Air and Water by Mark Denny, Scaling: Why is Animal Size so Important? by Knut Schmidt-Nielsen, and Random Walks in Biology by Howard Berg. Steven Vogel has many books you might like, including Life in Moving Fluids, Vital Circuits, and Life’s Devices. 
8. Nothing in biology makes sense except in light of evolution. To learn about evolution, read the books of Stephen Jay Gould. I enjoyed his collections of essays from the magazine Natural History. Start with Ever Since Darwin.



Textbook of Medical Physiology,
by Guyton and Hall.

9. Once you have a general biology background, what comes next? When I was in graduate school, I sat in on the Vanderbilt Medical School’s Physiology class and their Biochemistry class. These are the two courses that I encourage Oakland University Medical Physics graduate students to take. Typical textbooks are Guyton and Hall’s Textbook of Medical Physiology, now in its 13th edition, and Nelson and Cox's Lehninger Principles of Biochemistry, now in its 6th edition. Both books are long, expensive, and detailed. If interested in cell and molecular biology, a leading text is Molecular Biology of the Cell by Bruce Alberts and Alexander Johnson. 
10. If you have the time, you can do what Russ Hobbie did: between 1971 and 1973 he audited all the courses medical students take in their first two years at the University of Minnesota. Finally, you can always purchase a copy of the 5th edition of Intermediate Physics for Medicine and Biology!

If readers of the blog have their own recommendations, please add them in the comments.

Art Winfree and the Bidomain Model of Cardiac Tissue

Art Winfree was a pioneer in applying physics and mathematics to cardiac electrophysiology. Russ Hobbie and I cite him often in the 5th edition of Intermediate Physics for Medicine and Biology. After his untimely death in 2002, I was asked to write an article for a special issue of the Journal of Theoretical Biology published in his honor. My paper, “Art Winfree and the Bidomain Model of Cardiac Tissue,” appeared in 2004.

My original submission for the special issue was a personal tribute to Art. It began

“Spiral waves have become so popular in Tucson they are even sold in hair styling salons (Figure 1)”

Figure 1.

I had to laugh as I read the above quote in a preprint Art Winfree sent me. It was to be the opening sentence of a chapter appearing in a prestigious textbook on cardiac electrophysiology. Unfortunately, the sentence and the picture were deleted before the book's publication, although the picture (Fig. 1) did appear eventually in the second edition of Art’s The Geometry of Biological Time. For me, the quote captures the essence of Art: his humor, his irreverence, and his uncanny ability to find science in the world around him. I only met Art in person once, but we corresponded often by email, exchanging ideas, frustrations, and gossip. Of all the scientists who have influenced my research career, only my PhD advisor John Wikswo had a greater impact than Art Winfree did. In this paper, I describe several instances where my path crossed Art’s as we each attacked related problems in cardiac electrophysiology. In addition, I hope to show that Art made important contributions to what is known as the “bidomain model” of cardiac tissue.

Later in the article is one of my favorite passages.

I recall vividly a sunny day in April, soon after my second daughter Katherine was born. I was sitting on a rocking chair in the living room of our house in Kensington, Maryland, holding the sleeping infant in one arm and Art’s book When Time Breaks Down in the other. Outside I could see our dogwood tree in full blossom. As I read page after page, I remember thinking “life doesn’t get any better than this.” The book (and the daughter) changed my life.

Unfortunately, the editors of the special issue didn’t like my paper, saying they wanted a more traditional review article. In particular, they objected to my quoting Art’s emails he had sent me. So, I gave the paper a lobotomy and published a harmless but lifeless review. When the issue came out, I found a wonderful article by George Oster about Winfree, full of personal insights and even the text of one of Art’s emails. I wish now I had pushed harder to get my article published in its original form. The best article in the special issue was “Art Winfree, Artist of Science” by his daughter Rachael Winfree.

In the acknowledgments of my paper is the line “I would like to thank Jesse Malouf for his help editing this paper.” Jesse was a student in my honors college course about Pacemakers and Defibrillators. At Oakland University, Honors College has many of the best students in the university, but they are from all backgrounds and often have weak math skills. Jesse was a mathaphobe, but a wonderful writer. On one of my exams I had a mixture of questions, some requiring mathematical analysis and others needing an essay. Jesse skipped the math questions, but to make up for it he not only answered all the essay questions elegantly but also wrote a “bonus essay”. I never had a student hand in a bonus essay before! The next semester, I hired him to help me write the Winfree article. I fear many of his contributions to the original version were not included in the published one.

In the “olden days” the original draft of my Winfree article would be lost forever, or maybe would sit in some file cabinet unseen for decades. But nowadays, you can find anything on the internet (how did we live without it?). I have posted the original submission on my ResearchGate page. You can find it here.

The Coulter Counter

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I often include applications of important topics in the homework problems. One such problem, new in Chapter 6 of the 5th edition, is an analysis of a Coulter counter.

Problem 23. The Coulter counter or resistive pulse technique is used to count and size particles in a wide variety of applications (Kubitschek 1969; DeBlois and Bean 1970), including the automated counting of blood cells. The cells being counted are assumed to be nonconducting and immersed in a conducting fluid. The fluid is made to flow through a narrow channel. When a suspended particle enters the channel there is a change in resistance. Assume a long channel of radius b with no end effects.

(a) What is the resistance of pure fluid of resistivity ρ = 1/σ in a segment of channel of length 2a?

(b) A cylindrical non-conducting cell of radius a and length 2a is in the channel. Its axis and the axis of the channel coincide. What is the resistance of a segment of channel of length 2a? Ignore end effects.

(c) Show that the resistance change (the difference between these two results) is proportional to the volume of the cell, V=2πa³, and inversely proportional to b⁴.

In the August issue of Physics Today is an article about extending the Coulter counter to sequencing DNA. Murugappan Muthukumar, Calin Plesa, and Cees Dekker write

In the 1940s Wallace Coulter set about finding a way to quickly count blood cells, which at the time was a slow and inefficient process. His approach was to pass cells, one by one, through a small hole connecting to compartments filled with electrolyte solution. Simultaneously, he applied a voltage across the compartment and measured the ionic current passed through the hole. As a cell passed through the hole, it would partially block the flow of electric charges, and the current would drop by an amount proportional to the volume of the cell….Coulter’s technique worked out wonderfully and revolutionized cell counting.

Then, the authors describe how this method can be used to sequence DNA.

The last two decades have seen a renaissance of the Coulter counter concept. The principle remains essentially the same, but nanopores—holes with a diameter of merely a few nanometers—have shrunk the length scale from that of single cells to that of single molecules. When DNA molecules are added to one side of the pore and an electric field is applied, the resulting electrophoretic force on the negatively charged DNA can pull the molecule through the pore in a head-to-tail fashion, leading to an observable blockade in the ionic current…

In the 1990s several research groups … began probing whether the different bases on a DNA strand might block measurably different amounts of ionic current as they pass through a nanopore. If so, the pattern of current generated by a DNA strand threaded through a nanopore might provide a linear readout of the strand’s base sequence… Although significant challenges remain to turn that vision into a practical reality, the goal appears to be within reach.

The authors then describe more details about the technique. Some use transmembrane proteins like the membrane channels described in Chapter 9 of IPMB. Others use tiny holes drilled into sheets of silicon nitride. Still others use a hybrid of these two.

Clearly the method will not work unless the DNA is a single strand. Wanunu (2012) discusses the molecular dynamics involved in unzipping a double strand to obtain two single strands, one of which can then be threaded through the pore to do the sequencing. The nanopores must be very narrow if you are to have any chance of distinguishing different bases attached to the DNA backbone.

Russ and I had no idea about these modern uses of the Coulter counter when we added the homework problem. This new application of the Coulter idea shows how a strong understanding of the fundamentals of physics as applied to medicine and biology can allow one to quickly move to the forefront of cutting-edge new technologies.

The Psychic Probe

Foundation,
by Isaac Asimov.

In the summer my wife and I sometimes take long car trips, and I often listen to audiobooks while I drive to keep me awake and alert. During a recent trip I listened to Isaac Asimov’s Foundation trilogy. Regular readers of this blog know that I’m a huge Asimov fan. I first read the Foundation series about forty years ago, and this was my third or fourth time through these delightful books.

In brief, the Foundation series tells the history of the decaying galactic empire, and describes the work of the psychohistorian Hari Seldon who has calculated mathematically how to reduce the duration of the dark ages following the empire’s fall from 30,000 years to merely 1000. All goes according to plan until the Mule, a mutant who can control other people’s emotions, causes all to go awry.

Foundation and Empire,
by Isaac Asimov.

One of Asimov’s inventions in this future history is a device that can read minds, called the Psychic Probe. He writes in Foundation and Empire,

The general threw away his shredded, never-lit cigarette, lit another, and shrugged. “Well, it is beside the immediate point, this lack of first-class tech-men. Except that I might have made more progress with my prisoner were my Psychic Probe in proper order.”

The secretary’s eyebrows lifted. “You have a Probe?”

“An old one. A superannuated one which fails me the one time I needed it. I set it up during the prisoner’s sleep, and received nothing. So much for the Probe. I have tried it on my own men and the reaction is quite proper, but again there is not one among my staff of tech-men who can tell me why it fails upon the prisoner. Ducem Barr, who is a theoretician of parts, though no mechanic, says the psychic structure of the prisoner may be unaffected by the Probe since from childhood he has been subjected to alien environments and neural stimuli. I don’t know. But he may yet be useful. I save him in that hope.” 

Second Foundation,
by Isaac Asimov.

Russ Hobbie and I don’t mention the Psychic Probe in the 5th edition of Intermediate Physics for Medicine and Biology … or do we? Asimov didn’t explain the physical mechanism behind the Probe, but I can speculate. Four candidates are:

• Magnetoencephalography. The probe could be a form of MEG, as discussed in Section 8.5 of IPMB. The probe would then contain a sensitive magnetic field detector, like modern SQUID magnetometers, that could record brain activity whether we wanted it to or not. 
• Functional Magnetic Resonance Imaging. Section 18.12 of IPMB describes how fMRI can use magnetic field heterogeneities caused by oxygenated hemoglobin to detect brain activity. Essentially, one measures blood flow, which is related to neural firing. 
• Positron Emission Tomography. PET images the distribution of injected positron emitters (Section 17.10 of IPMB). For example, the isotope fluorine-18 can be attached to molecules of glucose, which are then taken up by parts of the brain that are metabolically active and become trapped inside neurons. 
• Transcranial Magnetic Stimulation. In Section 8.7 of IPMB, Russ and I discuss using a rapidly changing magnetic field to excite nerve axons. TMS can disrupt brain function, and has even been suggested as a treatment for depression. 

Asimov's Foundation Trilogy.

PET and fMRI are too slow to accurately follow rapid brain activity. PET detects brain metabolism and fMRI detects blood flow, both of which are only indirectly related to neuron firing. My best guess for the Psychic Probe is some combination of MEG and TMS. Apparently the probe can damage the brain when used aggressively, which suggests TMS. But it can also read minds when used more gently, which points toward MEG. A combo TMS/MEG unit could therefore both detect and alter brain function.

While working at NIH in the 1990s, I studied both magnetoencephalography and transcranial magnetic stimulation. Yikes! I may be partially responsible for the invention of the Psychic Probe!

Kramers’ Law

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a homework problem about Kramers’ law. (We spelled it Kramer’s, but his name is Kramers with an s, so we should have written Kramers’.) Kramers’ law is Eq. 16.3a, the photon energy fluence dΨ/d(hν) as a function of frequency ν for bremsstrahlung radiation

where Z is the atomic number, h is Planck’s constant, ν_o is the frequency of a photon having the same energy as the incident electrons, and C is a constant. In his paper “On the Theory of X-ray Absorption and of the Continuous X-ray Spectrum” (Philosophical Magazine, Volume 46, Pages 836–871, 1923), Kramers writes

The continuous x-ray spectrum has in the course of the last years been investigated by a number of physicists. The problem is here to determine how, for a given tension [voltage] on the tube and a given anticathode material [typically tungsten], the energy in the continuous spectrum is distributed among different frequencies…

The object of the present paper is to show how it is possible to account theoretically for the main features of the phenomena of x-ray absorption and continuous x-ray emission discussed above. The explanation of these phenomena may be traced back to the determination of the radiation processes which may occur when a free electron of given velocity approaches a positive nucleus with given charge.

Who was Kramers? According to the Dictionary of Scientific Biography, Hendrik Anthony (Hans) Kramers was born in Rotterdam, the Netherlands in 1894. He joined Niels Bohr’s Institute of Theoretical Physics, and in 1934 he moved to Leiden University, where he remained until his death in 1952. He’s known for many contributions to physics, including the Kramers-Kronig relations. The Dictionary of Scientific Biography article concludes

Kramers’ work, which covers almost the entire field of theoretical physics, is characterized both by outstanding mathematical skill and by careful analysis of physical principles. It also leaves us with the impression that he tackled problems because he found them challenging, not primarily because they afforded chances of easy success. As a consequence his work is somewhat lacking in spectacular results that can be easily explained to a layman; but among fellow theoreticians he was universally recognized as one of the great masters.

A Tale of Two Continents,
by Abraham Pais.

Here is my favorite Kramers story. Jewish physicist Abraham Pais described in his autobiography A Tale of Two Continents how he spent much of World War II in Holland hiding from the Gestapo. Kramers was one of the few people who knew of his hiding place, and would visit him weekly to talk physics. One day when Kramers was there, Gestapo agents knocked at the door and Pais had to hide in a small enclosure behind a panel in the wall. Pais writes

I kept sitting in the tiny space, practically bent over double, holding onto the panel, when I heard the door to my room, which lay at the other side of my hiding spot, open softly. Someone entered, I did not at first know who. Then that person sat down on a small bench that stood right at the wall behind which I was folded up. The person began to read, not loud but quite softly. It was Kramers. Earlier he had lent me a volume of Bradley’s Lectures on Shakespeare. What this good man was doing now was reading to me from that book, in order to calm my nerves.

The Divergence Theorem and Stokes’ Theorem

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem to Chapter 4.

Problem 4. Integrate Eq. 4.8 over a volume and subtract the result from Eq. 4.4. The resulting relationship is called the divergence theorem.

For those of you who don’t keep a copy of IPMB always close at hand (what’s the matter with you?), Eq. 4.8 is

where “div j” is the divergence of the vector j, and Eq. 4.4 is

When you put the two equations together, you get (spoiler alert) ,

also known as the divergence theorem. It is one of the fundamental results of vector calculus.

Div, Grad, Curl, and All That,
by H. M. Schey.

If you want to learn more about the divergence theorem, I recommend H. M. Schey’s book Div, Grad, Curl and All That. He writes

For the remainder of this chapter we digress from the mainstream of our narrative to discuss a famous theorem that asserts a remarkable connection between surface integrals and volume integrals. Although this relation may be suggested by the work we have done in electrostatics, the theorem is a mathematical statement holding under quite general circumstances. It is independent of any physics and is applicable in many different places. It is called the divergence theorem, and sometimes Gauss’ theorem… It says that the flux of a vector function through some closed surface equals the triple integral of the divergence of that function over the volume enclosed by the surface.

Besides the divergence theorem, another basic tenet of vector calculus is Stokes’ theorem. Can we make a similar homework problem demonstrating that? Yes! Here is a new problem for Chapter 8.

Problem 23 ½. Integrate Eq. 8.22 over a surface and subtract the result from Eq. 8.21. The resulting relationship is called Stokes’ theorem.

If you don’t have IPMB handy, Eq. 8.21 is

and Eq. 8.22 is

where “curl E” is the curl of the vector E. When you put the two equations together, you get

 Schey writes

We [now] discuss another famous theorem, one strongly reminiscent of the divergence theorem and yet, as we’ll see, quite different from it. This theorem, named for the mathematician Stokes, relates a line integral around a closed path to a surface integral over what is called a capping surface of the path…In words, Stokes’ theorem says that the line integral of the tangential component of a vector function over some closed path equals the surface integral of the normal component of the curl of that function integrated over any capping surface of the path.

The divergence theorem and Stokes’ theorem are a bit too mathematical to develop in IPMB, with its emphasis on biological and medical applications. Yet there they are, implicit in our discussions of diffusion and of transcranial magnetic stimulation. If you want to learn more, start with Schey’s wonderful (and relatively inexpensive) book Div, Grad, Curl.

So You Don’t Like Error Functions?

In Chapter 4 of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the diffusion equation (Equation 4.26)

where C is the concentration and D is the diffusion constant. We then study one-dimensional diffusion where initially (t = 0) the region to the left of x = 0 has a concentration C_o, and the region to the right has a concentration of zero (Section 4.13). We show that the solution to the diffusion equation is (Equation 4.75)

where erf is the error function.

Some students don’t like error functions (really?). Moreover, often we can gain insight by solving a problem in several ways, obtaining solutions that are seemingly different yet equivalent. Let’s see if we can solve this problem in another way and avoid those pesky error functions. We will use a standard method for solving partial differential equations: separation of variables. Before I start, let’s agree to solve a slightly different problem: initially C(x,0) is C_o/2 for x less than zero, and −C_o/2 for x greater than zero. I do this so the solution will be odd in x: C(x,t) = −C(−x,t). At the end we can add the constant C_o/2 and get back to our original problem. Now, let’s begin.

Assume the solution can be written as the product of a function of only t and a function of only x: C(x,t) = T(t) X(x). Plug this into the diffusion equation, simplify, and divide both sides by TX 

The only way that the left and right hand sides can be equal at all values of x and t is if both are equal to a constant, which I will call −k². This gives two ordinary differential equations

 and

The solution to the first equation is an exponential

and the solution to the second is a sine

There is no cosine term because of the odd symmetry. Unfortunately, we don’t know the value of k. In fact, our solution can be a superposition of infinitely many values of k

where A(k) specifies the weighting.

To determine A(k), use the Fourier techniques developed in Chapter 11. The result is

How did I get that? Let me outline the process, leaving you to fill in the missing steps. I should warn you that a mathematician would worry about the convergence of the integrals we evaluate, but you and I’ll brush those concerns under the rug.

At t = 0, our solution becomes

Except for a missing factor of 2π, this looks just like the Fourier transform from Section 11.9 of IPMB. Next, multiply each side of the equation by sin(ωx), and integrate over all x. Then, use Equation 11.66b to express the integral of the product of sines as a delta function. You get

Both C(x,0) and sin(kx) are odd, so their product is even, and for x greater than zero C(x,0) is −C_o/2. Therefore,

You know how to integrate sine (I hope you do!), so

Here is where things get dicey. We don’t know what cosine equals at infinity, but if we say it averages to zero the first term goes away and we get our result

Plugging in this expression for A(k) gives our solution for C(x,t). If we want to go back to our original problem with an initial condition of C_o on the left and zero on the right, we must add C_o/2. Thus

Let’s compare this solution with the one in Equation 4.75 (given above). Our new solution does not contain the error function! Those of you who dislike that function can celebrate. Unfortunately, we traded the error function for an integral that we can’t evaluate in closed form. So, you can have a function that you may be unfamiliar with and that has a funny name, or you can have an expression with common functions like the sine and the exponential inside an integral. Pick your poison.