What’s Wrong With These Equations?

The 4^th edition of Intermediate Physics for Medicine and Biology is full of equations: thousands of them. Each one must fit into the text in a way to make the book easy to read. How?

N. David Mermin wrote a fascinating essay that appeared in the October 1989 issue of Physics Today titled “What’s Wrong With These Equations?” You can find it online at www.cvpr.org/doc/mermin.pdf. It begins

A major impediment to writing physics gracefully comes from the need to embed in the prose many large pieces of raw mathematics. Nothing in freshman composition courses prepares us for the literary problems raised by the use of displayed equations.

Mermin then presents three rules “that ought to govern the marriage of equations to readable prose”

• Rule 1 (Fisher’s rule): Number all displayed equations.


• Rule 2 (Good Samaritan rule): When referring to an equation identify it by a phrase as well as a number.


• Rule 3 (Math is Prose rule): End a displayed equation with a punctuation mark.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I violate Fisher’s rule: some of our displayed equations are not numbered. All I can say is, there are lots of equations in our book, and revising it to obey Fisher’s rule would require more effort than we are willing to expend.

I know you are wondering how an essay about punctuating and numbering equations could possibly be interesting, but Mermin makes the subject entertaining. And if you ever find yourself writing an article that contains equations, obeying his three rules will make the article easier to read.

Boojums All the Way Through,
by N. David Mermin.

Many physicists know Mermin for his renowned textbook Solid State Physics with Neil Ashcroft. His series of “Reference Frame” essays in Physics Today are all delightful, particularly the ones with Professor Mozart. Several Reference Frame essays are reprinted in his book Boojums All the Way Through: Communicating Science in a Prosaic Age. The title essay describes Mermin’s quest to establish the whimsical word “Boojum” as a scientific term for a phenomenon in superfluidity. If you want to learn to write physics well, read Mermin.

The Feynman Lectures

The Feynman Lectures on Physics,
by Richard Feynman.

On page 318 of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite The Feynman Lectures on Physics. Reading The Feynman Lectures, written by Nobel Prize winner Richard Feynman, is a rite of passage for future physicists. Feynman describes how he came to present the lectures in his preface:

These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory section once a week.

Although written for freshman and sophomores, most physics students read The Feynman Lectures a bit later in their education. I recall reading them the summer between graduation from the University of Kansas and starting graduate school at Vanderbilt University. There is some biological and medical physics in the lectures. For instance, Chapters 35 and 36 of Volume 1 are about vision and the eye. In Chapter 3 (The Relation of Physics to Other Subjects), Feynman describes his reductionist point of view about biology:

Certainly no subject or field is making more progress on so many fronts at the present moment, than biology, and if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of jigglings and wigglings of atoms.

And in Volume 2 (Chapter 1), Feynman had this to say about the impact of electricity and magnetism on life:

Now we realize that the phenomena of chemical interaction and, ultimately, of life itself are to be understood in terms of electromagnetism.

He closed that chapter with a favorite quote of mine:

From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

Anyone wondering what to get an aspiring physicist for a holiday gift might want to consider The Feynman Lectures. If you are looking for lighter reading, I suggest two autobiographical books by Feynman: Surely You’re Joking Mr. Feynman, and What Do You Care What Other People Think? They’re delightful and hilarious.

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles

Quantum Physics of Atoms,
Molecules, Solids, Nuclei, and Particles,
by Eisberg and Resnick.

One of the sources that Russ Hobbie and I cite most often in the 4th edition of Intermediate Physics for Medicine and Biology is Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Robert Eisberg and Robert Resnick. I used the first (1974) edition of this textbook when I was an undergraduate studying physics at the University of Kansas. Quantum Physics was the book where I was first introduced to the ideas of quantum mechanics, to the Schrodinger equation, and to nuclear physics. A second edition was published in 1985, but I can find nothing about a third edition in the last 25 years. Despite it being somewhat out-of-date, I still consider this book to be one of the best sources of information about modern physics. Below is the first paragraph of the preface:

The basic purpose of this book is to present clear and valid treatments of the properties of almost all the important quantum systems from the point of view of elementary quantum mechanics. Only as much quantum mechanics is developed as is required to accomplish the purpose. Thus we have chosen to emphasize the applications of the theory more than the theory itself. In so doing we hope that the book will be well adapted to the attitudes of contemporary students in a terminal course on the phenomena of quantum physics. As students obtain an insight into the tremendous explanatory power of quantum mechanics, they should be motivated to learn more about the theory. Hence, we hope that the book will be equally well adapted to a course that is to be followed by a more advanced course in formal quantum mechanics.

I have never taught the modern physics class here at Oakland University, but if I did I would certainly consider using Eisberg and Resnick’s book. When I have taught the undergraduate quantum mechanics class (taken after modern physics) I used another wonderful book, Introduction to Quantum Mechanics by David Griffiths. There are several good quantum mechanics books at the graduate level, but I—a biomedical physicist—have never been asked to teach graduate quantum mechanics. (Are they telling me something?)

Intermediate Physics for Medicine and Biology doesn’t make much use of quantum ideas, except at a very qualitative level. Schrodinger’s equation is only mentioned once (on page 49), and is never written out. The idea of discrete quantum energy levels is introduced in Chapter 3 when we discuss statistical mechanics, and again in Chapter 14 when explaining atomic spectra. However, concepts related to quantization of light are important. For instance, thermal (blackbody) radiation is discussed in Section 14.7 (and is covered elegantly in the first chapter of Eisberg and Resnick) and Compton scattering is analyzed in Sec. 15.4. Quantum Physics should provide all the background you will need to understand these and other modern physics topics.

Clark and Plonsey

Problem 30 in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology is based on a paper by John Clark and Robert Plonsey (“The Extracellular Potential of a Single Active Nerve Fiber in a Volume Conductor,” Biophysical Journal, Volume 8, Pages 842–864, 1968). This paper shows how to calculate the extracellular potential from the transmembrane potential, with results shown in our Fig. 7.13.

The calculation involves some mathematical concepts that are slightly advanced for Intermediate Physics for Medicine and Biology. First, the potentials are written in terms for their Fourier transforms. Russ Hobbie and I don’t cover Fourier analysis until Chapter 11, so the problem just assumes a sinusoidal spatial dependence. We also introduce Bessel functions for the first time in the book (to be precise, modified Bessel functions of the first and second kind). Bessel functions arise naturally when solving Laplace’s equation in cylindrical coordinates.

I have admired Clark and Plonsey’s paper for years, and was glad to see this problem introduced into the 4th edition of our book. Robert Plonsey was a professor at Case Western Reserve University from 1968-1983. He then moved to Duke University, where he was when I came to know his work while I was a graduate student. I am most familiar with his research on the bidomain model of cardiac tissue, often in collaboration with Roger Barr (e.g., “Current Flow Patterns in Two-Dimensional Anisotropic Bisyncytia with Normal and Extreme Conductivities,” Biophysical Journal, Volume 45, Pages 557–571 and “Propagation of Excitation in Idealized Anisotropic Two-Dimensional Tissue,” Biophysical Journal, Volume 45, Pages 1191–1202). Plonsey was elected as a member of the National Academy of Engineering in 1986 for “the application of electromagnetic field theory to biology, and for distinguished leadership in the emerging profession of biomedical engineering.” He retired from Duke in 1996 as the Pfizer Inc./Edmund T. Pratt Jr. University Professor Emeritus of Biomedical Engineering. He has won many awards, such as the 2000 Millennium Medal from the IEEE Engineering in Medicine and Biology Society and the 2004 Ragnar Granit Prize from the Ragnar Granit Foundation. John Clark is currently a Professor of Electrical and Computer Engineering at Rice University. He is a Life Fellow in the Institute of Electrical and Electronics Engineers (IEEE) “for contributions to modeling in electrophysiology and cardiopulmonary systems.”

One of my earliest papers was an extension of Clark and Plonsey’s model to a strand of cardiac tissue, using the bidomain model (“A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue,” IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467–469, 1986.) The mathematics is almost the same as in their paper—Fourier transforms and Bessel functions—but the difference is that I modeled a multicellular strand of tissue, like a papillary muscle in the heart, that contains of both intracellular and interstitial spaces (the two domains of the “bidomain” model). A comparison of my paper to Clark and Plonsey’s earlier work indicates how influential their research was on my early development as a scientist. They were cited in the first sentence of my paper.

Hobbie and Roth, Back in the Saddle Again

In the November, 2009 issue of the American Journal of Physics, Russ Hobbie and I published “Resource Letter MP-2: Medical Physics.” Our resource letter “provides a guide to the literature on the uses of physics for the diagnosis and treatment of disease.” Think of it (along with Ratliff’s “Resource Letter MPRT-1: Medical Physics in Radiation Therapy” discussed in the August 28, 2009 entry to this blog) as an updated bibliography to the 4th edition of Intermediate Physics for Medicine and Biology. Together, these two publications provide over 300 citations to the best and most recent books, articles, and websites about medical physics. We even slipped a mention of this blog into the list of references.

Felix Bloch

One hundred and four years ago today, Felix Bloch (1905–1983) was born in Zurich, Switzerland. Bloch received his PhD in physics in 1928 from the University of Leipzig working under Werner Heisenberg, and then immigrated to the United States after Hitler came to power in Germany. He worked for a time at Los Alamos on the Manhattan Project, and had a long career in the Physics Department at Stanford University.

Bloch is most familiar to readers of the 4th edition of Intermediate Physics in Medicine and Biology because of his contributions to our understanding of nuclear magnetic resonance. He shared the 1952 Nobel Prize with Edward Purcell for “their development of new ways and methods for nuclear magnetic precision measurements.” In Chapter 18 on Magnetic Resonance Imaging, Russ Hobbie and I present the Bloch Equations (Eq. 18.15), which govern the magnetization of a collection of spins in a static magnetic field. Essentially all of MRI begins with the Bloch equations, so they are part of the essential toolkit for any medical physicist. Bloch’s most cited paper is “Nuclear Induction” (Physical Review, Volume 70, Pages 460–474, 1946). The abstract is reproduced below.

The magnetic moments of nuclei in normal matter will result in a nuclear paramagnetic polarization upon establishment of equilibrium in a constant magnetic field. It is shown that a radiofrequency field at right angles to the constant field causes a forced precession of the total polarization around the constant field with decreasing latitude as the Larmor frequency approaches adiabatically the frequency of the r-f field. Thus there results a component of the nuclear polarization at right angles to both the constant and the r-f field and it is shown that under normal laboratory conditions this component can induce observable voltages. In Section 3 we discuss this nuclear induction, considering the effect of external fields only, while in Section 4 those modifications are described which originate from internal fields and finite relaxation times.

Bloch also appears in Chapter 15 of Intermediate Physics for Medicine and Biology, because of his contribution to the development of the Bethe-Bloch formula (Eq. 15.58) governing the stopping power of a charged particle by interaction with a bound electron. He is also known for his fundamental contributions to solid state physics, including his seminal calculation of the electron wave function in a periodic potential, derived when he was only 23. You can download a Biographical Memoir about Bloch by Robert Hofstadter at books.nap.edu/html/biomems/fbloch.pdf.

I have an indirect connection to Felix Bloch. When in graduate school at Vanderbilt University in the 1980s, I had several classes from Ingram Bloch, who—if I recall correctly—was Felix’s cousin. At that time, Ingram Bloch was teaching many of the graduate classes, so I took classical mechanics, two semesters of quantum mechanics, and general relativity from him. I remember spending days working on his infamous “take-home” exams. They weren’t easy. With two physicists in the family, the Blochs made quite an impact on 20th century physics.


P.S. Right now, amazon.com has the 4th edition of Intermediate Physics for Medicine and Biology on sale at 40% off. I have no control over if and when amazon reduces prices on books, so the price may go back up anytime.

P.P.S. Last night I finished Steven Strogatz’s book The Calculus of Friendship: What a Teacher and a Student Learned About Life While Corresponding About Math, mentioned in the July 3rd entry to this blog. In a word, the book is charming.

The Klein-Nishina Formula

In Chapter 15 of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I present the Klein-Nishina formula (Eq. 15.17).

The inclusion of dynamics, which allows us to determine the relative number of photons scattered at each angle, is fairly complicated. The quantum-mechanical result is known as the Klein-Nishina formula.

At first glance, Eq. 15.17 doesn’t look quantum-mechanical, because it does not appear to contain Planck’s constant, h. However, closer inspection reveals that the variable x in the equation, defined on the previous page (Eq. 15.15), does indeed contain h. Russ and I don’t derive the Klein-Nishina formula, nor do we give much background about it. Yet, this equation played an important role in the development of quantum mechanics, and specifically of quantum electrodynamics.

In the book Nishina Memorial Lectures: Creators of Modern Physics, the Nobel Prize winning physicist Chen Ning Yang wrote a chapter about “The Klein-Nishina Formula and Quantum Electrodynamics.”

One of the greatest scientific revolutions in the history of mankind was the development of Quantum Mechanics. Its birth was a very difficult process, extending from Planck’s paper of 1900 to the papers of Einstein, Bohr, Heisenberg, Schrodinger, Dirac, and many others. After 1925–1927, a successful theory was in place, explaining many complicated phenomena in atomic spectra. Then attention moved to higher energy phenomena. It was in this period, 1928–1932, full of great ideas and equally great confusions, that the Klein-Nishina formula played a crucial role. The formula was published in 1929, in the journals Nature and Z. Physik. It dealt with the famous classical problem of the scattering of light rays by a charged particle…

Oskar Klein and Yoshio Nishina derived their formula starting from the Dirac equation, which is a relativistic version of Schrodinger’s equation for an electron, including the effect of spin. During the summer of 1928, Klein and Nishina performed the lengthy calculations necessary to derive their formula. They would work independently during the day, and then compare results each evening (as Russ and I say, the calculation is “fairly complicated”). The final result was published in the German journal Zeit. f. Phys. (“Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac,” Volume 52, Pages 853–868, 1929). I don’t read German, so I can’t enjoy the original.

Later, the theory of Quantum Electrodynamics (QED) was developed to more completely describe the quantum mechanical interactions of electrons and photons. For an elementary introduction to this subject, see Richard Feynmann’s book QED. (Although I took several semesters of quantum mechanics in graduate school, I never really mastered quantum electrodynamics.) When the problem of the scattering of light by electrons was reexamined using QED, the result was identical to the Klein-Nishina formula derived earlier. To learn more about how these results were obtained, see “The Road to Stueckelberg's Covariant Perturbation Theory as Illustrated by Successive Treatments of Compton Scattering,” by J. Lacki, H. Ruegg, and V. Telegdi (http://arxiv.org/abs/physics/9903023). But beware, the paper is quite mathematical and not for the faint of heart.

Who were the two men who derived this formula? Oskar Klein (1894–1977) was a Swedish theoretical physicist. He is known for the Kaluza-Klein theory, the Klein-Gordon equation, and the Klein paradox. Yoshio Nishina (1890–1951) was a Japanese physicist. He was a friend of Niels Bohr, and a close associate of Albert Einstein. The crater Nishina on the Moon is named in his honor. During World War II he was the head of the Japanese atomic program.

Let me share one last anecdote about Klein, Nishina, and Paul Dirac that I find amusing. Gosta Ekspong tells the story in his chapter “The Klein-Nishina Formula,” in the book The Oskar Klein Memorial Lectures.

When Dirac paid a short visit to Copenhagen in 1928, he met Klein and Nishina. The three of them were once conferring in the library of the Bohr Institute. Dirac was a man of few words, so when the remark came from Nishina that he had found an error of sign in the new Dirac paper on the electron, Dirac drily answered: “But the result is correct.” Nishina, in an attempt to be helpful, said: “There must be two mistakes,” only to get Dirac’s reply that “there must be an even number of mistakes.”

Steven Chu, Biological Physicist

Readers of the 4th edition of Intermediate Physics for Medicine and Biology may wish to see examples of physicists who have contributed to biology. One excellent example is Steven Chu, who until recently was professor of physics and professor of molecular and cellular biology at the University of California, Berkeley. Chu describes his biological physics research on his Berkeley website:

We apply single molecule techniques such as fluorescence resonance energy transfer, atomic force microscopy and optical tweezers, we study enzyme activity, and protein and RNA folding at the single bio-molecule level. Systems being studied include how the ribosome reads m-RNA and manufactures proteins, how vesicles fuse into the cell wall at the synapse of neurons, how cells adhere to each other via adhesive molecules, and how RNA molecules fold into active enzymes.

If you want to hear Chu talk about his biological physics research, watch this video on YouTube.

 Steven Chu asks What Can Physics Say About Life?
https://www.youtube.com/watch?v=lUR3WgVh-Hc

Some of his best known biological physics papers, published while on the faculty at Stanford, are:

• Zhuang XW, Bartley LE, Babcock HP, Russell R, Ha TJ, Herschlag D, Chu S (2000) “A Single-Molecule Study of RNA Catalysis and Folding,” Science, Volume 288, Pages 2048–2051.


• Zhuang XW, Kim H, Pereira MJB, Babcock HP, Walter NG, Chu S (2002) “Correlating structural dynamics and function in single ribozyme molecules,” Science, Volume 296, Pages 1473–1476.


• Ha T, Rasnik I, Cheng W, Babcock HP, Gauss GH, Lohman TM, Chu S (2002) “Initiation and re-initiation of DNA unwinding by the Escherichia coli Rep helicase,” Nature, Volume 419, Pages 638–641.


• Blanchard SC, Gonzalez RL, Kim HD, Chu S, Puglisi JD (2004) “tRNA selection and kinetic proofreading in translation,” Nature Structural and Molecular Biology, Volume 11, Pages 1008–1014.

Steven Chu exemplifies how physicists can contribute to our understanding of biology.


....Oh, did I forget to mention something? Chu is best known for his work on the “development of methods to cool and trap atoms with laser light,” for which he shared the Nobel Prize in Physics in 1997. He is currently Secretary of Energy in the Obama administration, and is leading the US effort to move away from fossil fuels and toward alternative energy sources, thereby combating global warming.

Who says we don’t have wonderful role models anymore?

Are Static Magnetic Fields Dangerous?

Are static magnetic fields dangerous? This question has recently taken on added importance because a European directive is limiting a worker’s exposure to the strong static magnetic field in a magnetic resonance imager, thereby impeding research with MRI. A recent article by Denis Le Bihan on medicalphysicsweb.org asserts that

those limits could end up preventing the technique from being used—just when European scientists are starting to lead the world in ultra-high-field (UHF) MRI magnet research. The initially proposed limits will immediately put the brakes on progress and, moreover, be a big blow to companies that make MRI scanners and magnets, such as Siemens, Philips, Bruker and Magnex. These firms could end up being unable to meet the growing global demand for clinical UHF MRI scanners, the high fields from which could boost the potential of MRI for healthcare and biomedical sciences, particularly for neurological applications.

Voodoo Science,
by Robert Park.

Physicist Bob Park, the author of Voodoo Science and the weekly newsletter What’s New, writes

MAGNETIC FIELDS: THE PRECAUTIONARY PRINCIPLE IN ACTION. According to Denis Le Bihan at the CEA-Saclay Centre, a European directive to prevent workers from being exposed to high magnetic fields could severely impact research into Ultrahigh-Field MRI which shows great promise particularly in neurological applications. It is particularly frustrating that limits on static magnetic fields resulted from the paranoia surrounding EMF, which was associated with everything from power lines to cell phones, Wi-Fi, Bluetooth, and other wireless devices. As I pointed out in an editorial in the Journal of the National Cancer Institute eight years ago, “there will always be some who will argue that the issue has not been completely settled. In science, few things ever are.”

Are these limits justified? Based on my knowledge of biomagnetism, I think not. There are few known mechanisms by which a static magnetic field can have a significant biological impact, except in unusual cases such as a person with a ferromagnetic medical implant, or in some animals (such as magnetotactic bacteria) that are believed to sense magnetic fields, presumably by the presence of ferromagnetic or superparamagnetic nanoparticles (magnetosomes). Russ Hobbie and I discuss the possible effects of weak magnetic fields in Chapter 9 of the 4th edition of Intermediate Physics for Medicine and Biology.

Denis Le Bihan is a leader in the field of MRI, known for his development of diffusion weighted imaging. He is director of NeuroSpin, a French institute aimed at developing and using ultra high field Magnetic Resonance to understand the brain. I knew Denis when we were both working at the National Institutes of Health in the 1990s. He was a close collaborator with my friend Peter Basser, and together they developed diffusion tensor imaging. (Incidentally, one of their early papers on this topic just received its 1000th citation in the citation index!) Let us hope that Le Bihan’s important research is not interrupted unnecessarily by misguided government regulations.

Cochlear Implants

Russ Hobbie and I discuss the ear and hearing in Chapter 13 of the 4th edition of Intermediate Physics for Medicine and Biology. Last Wednesday, Russ attended a colloquium at the University of Minnesota titled “Bionic Hearing: The Science and the Experience,” presented by Ian Shipsey of Purdue University. The talk was about cochlear implants, at topic we mention briefly in Section 7.10 on Electrical Stimulation. You can download the entire powerpoint presentation from the colloquium. Shipsey’s story is itself inspirational. On his website he writes “I had cochlea implant surgery in November 2002 at the Riley Hospital for Children in Indianapolis, IN. The surgeon was Professor Richard Miyamoto. The device was activated in late December. I am now able to hear my daughter for the first time and my wife for the first in 12 years.”

In order to understand cochlear implants, you need to understand how the ear works. For a short lesson, you should watch an absolutely incredible video on YouTube. This animation is a wonderful example of what can happen when science education meets modern technology, and it was selected by Science Magazine as the first place winner in its 2003 Science and Engineering Visualization Challenge. Don’t miss it; especially you Beethoven fans!

Auditory Transduction.

https://www.youtube.com/watch?v=46aNGGNPm7s

You can learn more about cochlear implants by viewing a video of a lecture by Richard Miyamoto titled “Cochear Implants: Past, Present, and Future,” which you can download from a website about cochlear implants maintained by the National Institute of Deafness and Other Communication Disorders, one of the National Institutes of Health. More information is available on a Food and Drug Administration webpage. Also interesting is a National Public Radio report about cochlear implants.

When I worked at NIH in the 1990s, I used to attend the Neural Prosthesis Program workshops held in Bethesda every fall. I recall listening to the researchers each year report on how they were developing these incredible devices to restore hearing. From those workshops, I gained a great appreciation for cochlear implants, and I have come to view them as a prototypical example—along with the cardiac pacemaker—of how physics and engineering can contribute to medicine.