Friday, December 25, 2009

A Present From Santa

Santa arrived last night and left you, dear reader, a present in your stocking: two new homework problems for the 4th edition of Intermediate Physics for Medicine and Biology. The problems belong to Chapter 8 on Biomagnetism (one of my favorite chapters), and specifically to Section 8.6 on Electromagnetic Induction. They both explore the idea of skin depth, but from somewhat different perspectives. Please forgive Santa for being a bit long-winded; he got carried away.


Section 8.6

Problem 25.1 The concept of “skin depth” plays a role in some biomagnetic applications.
(a) Write Ampere’s law (Eq. 8.22) for the case when the displacement current is negligible.
(b) Use Ohm’s law (Eq. 6.26) to write the result from (a) in terms of the electric field.
(c) Take the curl of both sides of the equation you found in (b) (Assume the conductivity σ is homogeneous and isotropic).
(d) Use Faraday’s law (Eq. 8.20), ∇·B=0 (Eq. 8.7), and the vector identity ∇×(∇×B)=∇(∇·B)-∇2B to simplify the result from (c).
(e) Your answer to (d) should be the familiar diffusion equation (Eq. 4.24). Express the diffusion constant D in terms of electric and magnetic parameters.
(f) In Chapter 4, we found that diffusion over a distance L takes a time T equal to L2/2D. During transcranial magnetic stimulation, L=0.1 m, σ=0.1 S/m and μo=4π × 10-7 T m/A. How long does the magnetic field take to diffuse into the head? Is this time much longer than or much shorter than the rise time of the magnetic field for the stimulator designed by Barker et al. (1985)?
(g) Solve T= L2/2D for L, using the expression for D found in (e). Calculate L for T=0.1 ms. Is L much larger than or much smaller than the size of your head? L is closely related to the “skin depth” defined in electromagnetic theory.
(h) During magnetic resonance imaging (see Chapter 18), an 85 MHz radio-frequency magnetic field is applied to the body. Calculate L using half a period for T. How does L compare to the size of the head? The frequency of the RF field is proportional to the strength of the static magnetic field in an MRI device, and 85 MHz corresponds to 2 T. If the static field is 7 T (common in modern high-field MRI), calculate L. Is it safe to ignore skin depth during high-field MRI?

Problem 25.2 During magnetic stimulation, a changing magnetic field B induces eddy currents in the body that produce their own magnetic field B’. The goal of this problem is to compare B’ and B. We can estimate B’ using the following approximations. First, ignore the vector nature of all fields and do not distinguish between components. Second, ignore all negative signs. Third, replace all time derivatives with multiplication by 1/T, where T is a characteristic time. Fourth, replace all space derivatives (such as the curl) by multiplication with 1/L, where L is a characteristic length.
(a) Use Faraday’s law (Eq. 8.20) to estimate the induced electric field E from B.
(b) Use Ohm’s law (Eq. 6.26) to estimate the current density J from E.
(c) Use Ampere’s law (Eq. 8.22, but ignore displacement currents) to estimate B’ from J.
(d) Combine parts (a), (b), and (c) to determine an expression for the ratio B’/B in terms of the conductivity σ, the permeability μo, L, and T.
(e) In magnetic stimulation, L=0.1 m, T=0.1 ms, σ=0.1 S/m and μo=4π × 10-7 T m/A. Calculate B’/B. Is it safe to ignore B’ compared to B during magnetic stimulation?

Friday, December 18, 2009

Where's Albert?

Albert Einstein is considered one of the greatest physicists of the 20th century, and perhaps of all time. He certainly is one of the best-known physicists, being selected by TIME magazine as their Person of the Century in 1999. Yet, Einstein is curiously absent in the 4th edition of Intermediate Physics for Medicine and Biology. If you look in the index under Einstein, you find only one entry: on page 393, where Russ Hobbie and I introduce the unit of an einstein (a mole of photons) in a homework problem.

Does Einstein’s work appear anywhere else in Intermediate Physics for Medicine and Biology? Certainly his masterpiece, the general theory of relativity, has little or no direct impact on biology or medicine. I don’t believe we even refer indirectly to this monumental description of gravity. However, Einstein’s earlier theory, special relativity, does appear occasionally in our book. In Chapter 8 on biomagnetism, we write “the appearance of the magnetic force is a consequence of special relativity,” a topic we explore further in Homework Problems 5 and 23. Yet, the relationship between electrodynamics and relativity is mentioned as an aside, and is not a central feature of our analysis of magnetism. We could have left out mention of relativity from Chapter 8 altogether, and the rest of the chapter would be unaffected.

Special relativity enters in a more profound way in Chapter 15, on the Interaction of Photons and Charged Particles with Matter. There, we analyze Compton Scattering, and need the relationship between photon energy E and momentum p, given by special relativity as E=pc, where c is the speed of light. Moreover, the concept of rest mass m is introduced in this chapter, and we use Einstein’s most famous equation E=mc2, relating energy E and mass. Rest mass appears again in the discussion of pair production, where enough photon energy must be present to produce an electron-positron pair. The equation appears one more time in Chapter 17 on Nuclear Physics and Nuclear Medicine, where mass can be converted into energy in nuclear reactions.

Besides relativity, Einstein also played a leading role in the development of quantum mechanics, especially as related to the quantization of light and the idea of photons. This idea is first presented in Chapter 9, in a section on the Possible Effects of Weak External Electric and Magnetic Fields, where we compare the photon energy (equal to Planck’s constant times the frequency of the radiation) to the thermal energy. The idea is developed in more detail in Chapter 14, in a section about The Nature of Light: Waves versus Photons. The idea of photons is central to Chapter 15, and particularly Sec. 15.2 on Photon Interactions. There, we discuss the photoelectric effect—one mechanism by which x rays interact with tissue—which is the research that won Einstein the Nobel Prize.

One final place where Einstein’s research impacts Intermediate Physics for Medicine and Biology is in the study of diffusion (Chapter 4). Einstein did fundamental work on diffusion in his doctoral thesis, and derived a relationship between the diffusion constant and the viscosity that we give as Eq. 4.23.

In summary, we rarely mention Einstein by name in our book, but his influence is present throughout, and most fundamentally when we discuss the idea of a photon. For readers interested in Einstein’s life and work, I recommend the brilliant biography Subtle is the Lord by Abraham Pais. I have heard good things about Isaacson’s more recent biography, Einstein: His Life and Universe, although I have not read it. You might also enjoy the American Institute of Physics website about Einstein prepared by the AIP Center for the History of Physics. Einstein published most of the ideas I have discussed in one miraculous year, 1905. John Rigden describes these publications and their impact in his book Einstein 1905: The Standard for Greatness (I have not read this book either, but I understand it is good). Finally, the equation E=mc2 has received a lot of press recently, including a NOVA special and Bodanis’s book E=mc2: A Biography of the World’s Most Famous Equation.

Friday, December 11, 2009

Error Function

In the November 6th entry to this blog, I mentioned one special function introduced in the 4th edition of Intermediate Physics for Medicine and Biology: the Bessel function. Another special function Russ Hobbie and I discuss briefly is the error function, which arises naturally when solving the one-dimensional cable equation (Chapters 6 and 7) or the diffusion equation (Chapter 4). The error function is the integral of the familiar Gaussian function, and has a sigmoidal shape, being minus one for large negative values of its argument and one for large positive values.

To learn more about the error function, see the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables by Milton Abramowitz and Irene Stegun (1972). This classic math handbook is available online at Also, Wikipedia has a very thorough article about the error function, including beautiful plots of the error function in the complex plane.

I’m not sure how the error function got its name. Perhaps it has something to do with experimental errors often being Gaussianly distributed. If anyone knows, please let me know.

P.S. Speaking of errors: For any students or instructors preparing to use the 4th edition of Intermediate Physics for Medicine and Biology next semester, I recommend you download the errata, which can be found at In it, Russ Hobbie and I list all known errors in our book. The number of errors has grown, and in particular some are present in homework problems. Generally I frown on writing in my books, but in this case do yourself a favor: download the errata and mark the corrections in your copy of the text. And as always, let us know if you find additional errors. The only thing worse than finding errors in a book you wrote is having errors in a book you wrote that you are not even aware of.

P.P.S. I have written in this blog about Steven Strogatz, a mathematician and author, and about Kleber's law, which relates metabolic rate to body mass. Here is an article by Strogatz about Kleber's law. It can't get much better than that!

Friday, December 4, 2009

Hot Tubs and Heat Stroke

In Chapter 10 (about Feedback and Control) of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Hot Tubs and Heat Stroke.
“The body perspires in order to prevent increases in body temperature. At the same time blood flows through vessels near the surface of the skin, giving the flushed appearance of an overheated person. The cooling comes from the evaporation of the perspiration from the skin. If the perspiration cannot evaporate or is wiped off, the feedback loop is broken ad the cooling does not occur. If a subject in a hot tub overheats, the same blood flow pattern and perspiration occur, but now heat flows into the body from the hot water in the tub. The feedback has become positive instead of negative, and heat stroke and possibly death occurs.”
Were we overly alarming about hot tubes? Not according to an article by Nicholas Bakalar in the November 23rd issue of the New York Times, which indicates hot tub accidents are a growing problem.
“A hot tub might not seem an especially dangerous place, but over a period of 18 years, 1990-2007, more than 80,000 people were injured in hot tubs or whirlpools seriously enough to wind up in an emergency room. Almost 74 percent of the injuries occurred at home…. About half the injuries were caused by slipping or falling, but heat overexposure was the problem in 10 percent of the accidents, and near-drowning in about 2.5 percent. Almost 7 percent of the injuries were serious enough to require hospitalization… The Consumer Products Safety Commission reported more than 800 deaths associated with hot tubs since 1990, nearly 90 percent of them in children under age 3”
This means that about 8000 people suffered from heat stroke accidents in hot tubs over 18 years, or over one per day. Perhaps a better understanding of biological thermodynamics and feedback loops has more than merely academic value.

To learn more, see Death in a Hot Tub: The Physics of Heat Stroke, by Albert Bartlett and Thomas Braun (American Journal of Physics, Volume 51, Pages 127-132, 1983).

Friday, November 27, 2009

What’s Wrong With These Equations?

The 4th 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 It begins
“A major impediment to writing physics gracefully comes from the need to imbed 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.

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.

Friday, November 20, 2009

The Feynman Lectures

On page 318 of the 4th 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 are delightful and hilarious.

Friday, November 13, 2009

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

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. It 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 of Atoms, Molecules, Solids, Nuclei, and Particles should provide all the background you will need to understand these and other modern physics topics.

Friday, November 6, 2009

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 45: 557-571 and "Propagation of Excitation in Idealized Anisotropic Two-Dimensional Tissue". Biophysical Journal 45: 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.

Friday, October 30, 2009

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.

Friday, October 23, 2009

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."
At the moment, you can download a reprint of this historic article at

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

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, 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.

Friday, October 16, 2009

The Klein-Nishina Formula

In Chapter 15 of the 4th 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 & 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. (I have to admit, 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 ( 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.”

Friday, October 9, 2009

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

Friday, October 2, 2009

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 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.”
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.

Friday, September 25, 2009

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!

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.

Friday, September 18, 2009

More on "Is Computed Tomography Safe?"

The December 7, 2007 entry to this blog was titled “Is Computed Tomography Safe?” As is often the case with such a difficult question, the answer is yes and no. No—there are risks associated with any exposure to ionizing radiation, so no procedure is entirely safe. Yes—in most cases the risks are small enough that the benefits outweigh the risks. In order to answer this question more precisely, a large scale study with nearly one million patients was conducted over three years. The conclusions were reported in the August 27 issue of the New England Journal of Medicine. The abstract of Fazel et al.’s paper “Exposure to Low-Dose Ionizing Radiation from Medical Imaging Procedures” (NEJM, Volume 361, Pages 849-857, 2009) is reproduced below:
"Background: The growing use of imaging procedures in the United States has raised concerns about exposure to low-dose ionizing radiation in the general population.

Methods: We identified 952,420 nonelderly adults (between 18 and 64 years of age) in five health care markets across the United States between January 1, 2005, and December 31, 2007. Utilization data were used to estimate cumulative effective doses of radiation from imaging procedures and to calculate population-based rates of exposure, with annual effective doses defined as low (less than 3 mSv), high (greater than 20 to 50 mSv), or very high (greater than 50 mSv).

Results: During the study period, 655,613 enrollees (68.8%) underwent at least one imaging procedure associated with radiation exposure. The mean (±SD) cumulative effective dose from imaging procedures was 2.4±6.0 mSv per enrollee per year; however, a wide distribution was noted, with a median effective dose of 0.1 mSv per enrollee per year (interquartile range, 0.0 to 1.7). Overall, moderate effective doses of radiation were incurred in 193.8 enrollees per 1000 per year, whereas high and very high doses were incurred in 18.6 and 1.9 enrollees per 1000 per year, respectively. In general, cumulative effective doses of radiation from imaging procedures increased with advancing age and were higher in women than in men. Computed tomographic and nuclear imaging accounted for 75.4% of the cumulative effective dose, with 81.8% of the total administered in outpatient settings.

Conclusions: Imaging procedures are an important source of exposure to ionizing radiation in the United States and can result in high cumulative effective doses of radiation."
To help put this study in context, the NEJM published an accompanying editorial by Michael Lauer (“Elements of Danger — The Case of Medical Imaging” Volume 361, Pages 841-843). Lauer writes that
“Because the use of ionizing radiation carries 'an element of danger in every . . . procedure,' we need to adopt a new paradigm for our approach to imaging. Instead of investing so many resources in performing so many procedures, we should take a step back and design and execute desperately needed large-scale, randomized trials to figure out which procedures yield net benefits. This approach would require leadership and courage on the part of our profession, our opinion leaders, and the research enterprise, but were we to insist that all, nearly all, procedures be studied in well-designed trials, we could answer many critical clinical questions within a short time. Because we will continue to be uncertain of the magnitude of harm, an accurate understanding of the magnitude of benefit is a moral imperative.“
In Chapter 16 of the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of radiation. While we do not provide a final answer regarding the safety of CT, we do outline many of the important issues one must examine in order to make an informed decision. The safety of computed tomography and other diagnostic imaging procedures will continue to be a crucial question of interest to readers of Intermediate Physics for Medicine and Biology. I will try to keep you posted as new information becomes available.

P.S. Thanks to Russ Hobbie for calling my attention to this paper. He reads the New England Journal of Medicine more than I do.

Friday, September 11, 2009

A New Homework Problem

While in Minneapolis last week, attending the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society, I had the pleasure of co-chairing a session with Professor Michael Joy of the University of Toronto. Joy has done some fascinating work on measuring current and conductivity in biological tissue using magnetic resonance imaging. His research inspired me to write a new homework problem for Chapter 8 of the 4th Edition of Intermediate Physics for Medicine and Biology.

Problem 21.5 The differential form of Ampere’s law, derived in Problem 21, provides a relationship between the current density J and the magnetic field B that allows you to measure biological current with magnetic resonance imaging [see, for example, Scott, G. C., M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman (1991). Measurement of nonuniform current density by magnetic resonance. IEEE Trans. Med. Imag. 10:362-374]. Suppose you use MRI and find the distribution of magnetic field to be

Bx = C (y z2 – y x2)
By = C (x z2 – x y2)
Bz = C 4 x y z

where C is a constant with the units of T/m3. Determine the current density. Assume the current varies slowly enough that the displacement current can be neglected.
To solve this problem, you need the result of Problem 21 in Chapter 8, which asks the reader to derive the differential form of Ampere’s law from the integral form given in the book by Eq. 8.11. If I were teaching a class from the book, I would assign both Problems 21 and 21.5, and expect the student to solve them both. But for readers of this blog, I will tell you the answer to Problem 21 (ignoring displacement current), so you will have the relationship needed to solve the new Problem 21.5: curl B = μ0 J. The curl is introduced in Section 8.6. If you don’t have the 4th Edition of Intermediate Physics for Medicine and Biology handy, take a look in a math handbook for information about how to calculate the curl (or see Schey’s book Div, Grad, Curl, and All That).

The story of how you measure B using MRI is interesting, but a bit too complicated to describe in detail here. In brief, a magnetic resonance imaging device has a strong static magnetic field about which nuclear spins (such as those of hydrogen) precess. The magnetic field produced by the current density modifies the static magnetic field, causing a phase shift in this precession. This phase shift is detected, and the magnetic field can be deduced from it. Technically, this method allows one to determine the component of the magnetic field that is parallel to the static field. Obtaining the other components requires rotating the object and repeating the procedure. See Chapter 18 for more about MRI.

Send me an email ( if you would like the answer to the new Problem 21.5.


Friday, September 4, 2009

31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society

I’m posting this blog from Minneapolis, Minnesota, where I am attending the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society. The theme of the conference is “Engineering the Future of Biomedicine,” and there are many fascinating talks and posters that would interest readers of the 4th Edition of Intermediate Physics for Medicine and Biology. Conference chair Bin He and his colleagues have put together a great meeting.

My friend Ranjith Wijesinghe and I have a poster later today about the “Effect of Peripheral Nerve Action Currents on Magnetic Resonance Imaging.” We analyzed if the magnetic field of action currents can be used to generate an artifact in an MRI, allowing direct imaging of biocurrents in the brain. There has been a lot of interest, and many publications, on this topic recently, but we conclude that the magnetic fields are just too small to have a measureable effect.

Last night, I got to hear Earl Bakken give a talk on “The History of Short-Term and Long-Term Pacing.” Bakken is a giant in the history of artificial pacemakers, and is the founder of Medtronic Corportion based in Minneapolis. He talked about the early years when Medtronic was a small electronics laboratory in a garage. He recommended a 10-minute video on YouTube, which he said told his story well. He also quoted one of my favorite books, Machines in our Hearts, a wonderful history of pacemakers and defibrillators. Tonight a social is being held at the Bakken Museum, “the only museum of its kind in the country, [where you can] learn about the history of electricity and magnetism and how it relates to medicine.” For a guy like me, this is great stuff.

So far, the presentations are fascinating and inspirational. I must admit, the students who attend these conferences always stay the same age as I grow older. I don’t think these meetings used to be this exhausting for me. As that old Garth Brooks song says, “the competition's getting younger.” They are also getting more diverse. The speaker who welcomed us to the Bakken talk said that in just a few years, Americans will be a minority within the IEEE Engineering in Medicine and Biology Society. This is not difficult to believe, after seeing researchers from so many countries attending this year.

As I survey all the research presented at this meeting, I am proud that so much of the underlying science is described in Intermediate Physics for Medicine and Biology. I am more convinced than ever that Russ Hobbie and I have written a book that will be of great value to future biomedical engineers.

Friday, August 28, 2009

Resource Letter MPRT-1: Medical Physics in Radiation Therapy

When Russ Hobbie and I were preparing the 4th Edition of Intermediate Physics for Medicine and Biology, we tried to update our book with the most recent references. But, inevitably, as time passes the book becomes increasingly out-of-date. How does one keep up with the literature? This blog is meant to help our readers stay current, but sometimes more drastic measures are required. Fortunately, the American Journal of Physics publishes Resource Letters, in which the author reviews important sources (mainly textbooks and research articles) on a particular topic. In the September 2009 issue of AJP, Steven Ratliff of Saint Cloud State University published Resource Letter MPRT-1: Medical Physics in Radiation Therapy (volume 77, pages 774-782, 2009). The abstract is reproduced below.
“This resource letter provides a guide to the literature on medical physics in the field of radiation therapy. Journal articles, books, and websites are cited for the following topics: radiological physics, particle accelerators, radiation dose measurements, protocols for radiation dose measurements, radiation shielding and radiation protection, neutron, proton, and heavy-ion therapies, imaging for radiation therapy, brachytherapy, quality assurance, treatment planning, dose calculations, and intensity-modulated and image-guided therapy."
I highly recommend this Resource Letter for anyone interested in radiation therapy. Particularly useful is Ratliff’s concluding section "Recommended Path Through the Literature".
“The best single reference for a newcomer to the field is Goitein (Ref. 14). It is clear, up to date, readable, complete, and gives a good explanation of what medical physicists do. For a person who does not want to enter the field but is just curious or needs to get some information and does not want to spend any money, a good place to start is the free on-line book by Podgorsak (Ref. 153). Van Dyk (Ref. 17) is a good place to start for those who want a clinical emphasis. The book by Turner (Ref. 91) has good problems (some with answers) and covers many aspects of the subject.

For those wanting to make a career of Medical Physics, a small but good starting library would consist of Goitein (Ref. 14), Hendee et al. (Ref. 30), Johns and Cunningham (Ref. 15), Khan (Ref. 16), Podgorsak (Ref. 153), Turner (Ref. 91), and van Dyk (Ref. 17). Khan is more useful once you have learned the material. If you have more money, you could add Attix (Ref. 19) and Podgorsak's book on radiation physics (Ref. 26). Cember and Johnson (Ref. 92) is a good addition if you are interested in the health-physics aspects of radiotherapy.

If you were restricted to one book and wanted to learn as much as possible, then the handbook of Mayles et al. (Ref. 18) is worthy of serious consideration.”
The references Ratliff cites in his conclusion (less than 10% of the 183 publications included in the entire Resource Letter) are listed below.

14. Radiation Oncology—A Physicist's Eye View, Michael Goitein (Springer Science+Business Media, LLC, New York, 2008).

15. The Physics of Radiology, Harold Elford Johns and John Robert Cunningham, 4th ed. (Charles C. Thomas, Springfield, Illinois, 1983).

16. The Physics of Radiation Therapy, Faiz M. Khan, 3rd ed. (Lippincott Williams and Wilkins, Philadelphia, PA, 2003).

17. The Modern Technology of Radiation Oncology—A Compendium for Medical Physicists and Radiation Oncologists, Vols. 1 and 2, edited by Jacob Van Dyk (Medical Physics, Madison, WI, 1999 and 2005).

18. Handbook of Radiotherapy Physics—Theory and Practice, edited by P. Mayles, A. Nahum, and J. C. Rosenwald (Taylor & Francis, New York, 2007).

19. Introduction to Radiological Physics and Radiation Dosimetry, Frank Herbert Attix (Wiley-VCH, Weinheim, Germany, 1986).

26. Radiation Physics for Medical Physicists, E. B. Podgorsak (Springer-Verlag, New York, 2006).

30. Radiation Therapy Physics, William R. Hendee, Geoffrey S. Ibbott, and Eric G. Hendee, 3rd ed. (Wiley, Hoboken, NJ, 2005).

91. Atoms, Radiation, and Radiation Protection, James E. Turner, 2nd ed. (Wiley, New York, 1995).

92. Introduction to Health Physics, Herman Cember and Thomas E. Johnson, 4th ed. (McGraw-Hill Medical, New York, 2009).

153. Radiation Oncology Physics: A Handbook for Teachers and Students, edited by E. B. Podgorsak (International Atomic Energy Agency, Vienna, 2005). (

By the way, if you look in the acknowledgments of Ratliff’s publication you will find the ubiquitous Russ Hobbie among those thanked for their helpful suggestions.

Friday, August 21, 2009

The ECG Dance

In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the electrocardiogram. I always thought that the best way to teach the ECG was an online cardiac rhythm simulator. But now, thanks to a tip from my former student Debbie Janks, I have found an even better way to teach the ECG. Check out this video on youtube. I will have to try this myself next time I teach Biological Physics.

Friday, August 14, 2009

The Bell Curve

I was browsing through the 4th Edition of Intermediate Physics for Medicine and Biology the other day (I do this sometimes; don’t ask why), and I noticed the footnote at the bottom of page 566 in Appendix H: The Binomial Probability Distribution, which states
“See also A. Gawande, The bell curve. The New Yorker, December 6, 2004, pp. 82-91.”
I thought to myself, “that must be one of the changes Russ Hobbie made when we were preparing the 4th edition, because I don’t remember ever reading the article.” Well, if Russ recommends it, then I want to read it, so I found the article on the web. It turns out to be a lovely, well-written piece about cystic fibrosis (CF), modern medicine, self-evaluation, and striving for excellence. The excerpt below is the one Russ probably had in mind when he added the citation of the article to our book. It describes a discussion between a teenage CF patient, Janelle, her physician Dr. Warwick, and the article’s author Atul Gawande, who is a surgeon and was observing Janelle’s interview as part of an effort to improve cystic fibrosis care. In the quote below, Janelle’s doctor is speaking.

"Let’s look at the numbers,” he said to me, ignoring Janelle. He went to a little blackboard he had on the wall. It appeared to be well used. “A person’s daily risk of getting a bad lung illness with CF is 0.5 per cent.” He wrote the number down. Janelle rolled her eyes. She began tapping her foot. “The daily risk of getting a bad lung illness with CF plus treatment is 0.05 per cent,” he went on, and he wrote that number down. “So when you experiment you’re looking at the difference between a 99.95-per-cent chance of staying well and a 99.5-per-cent chance of staying well. Seems hardly any difference, right? On any given day, you have basically a one-hundred-per-cent chance of being well. But”—he paused and took a step toward me—“it is a big difference.” He chalked out the calculations. “Sum it up over a year, and it is the difference between an eighty-three-per-cent chance of making it through 2004 without getting sick and only a sixteen-per-cent chance.

He turned to Janelle. “How do you stay well all your life? How do you become a geriatric patient?” he asked her. Her foot finally stopped tapping. “I can’t promise you anything. I can only tell you the odds.”

In this short speech was the core of Warwick’s world view. He believed that excellence came from seeing, on a daily basis, the difference between being 99.5-per-cent successful and being 99.95-per-cent successful. Many activities are like that, of course: catching fly balls, manufacturing microchips, delivering overnight packages. Medicine’s only distinction is that lives are lost in those slim margins.

The article describes how one CF center began measuring its own success against the top programs in the country, and their efforts to improve. Gawande concludes

”The hardest question for anyone who takes responsibility for what he or she does is, What if I turn out to be average? If we took all the surgeons at my level of experience, compared our results, and found that I am one of the worst, the answer would be easy: I’d turn in my scalpel. But what if I were a C? Working as I do in a city that’s mobbed with surgeons, how could I justify putting patients under the knife? I could tell myself, Someone’s got to be average. If the bell curve is a fact, then so is the reality that most doctors are going to be average. There is no shame in being one of them, right?

Except, of course, there is. Somehow, what troubles people isn’t so much being average as settling for it. Everyone knows that averageness is, for most of us, our fate. And in certain matters—looks, money, tennis—we would do well to accept this. But in your surgeon, your child’s pediatrician, your police department, your local high school? When the stakes are our lives and the lives of our children, we expect averageness to be resisted. And so I push to make myself the best. If I’m not the best already, I believe wholeheartedly that I will be. And you expect that of me, too. Whatever the next round of numbers may say.”

Friday, August 7, 2009

Technetium Shortage....Again

Readers of this blog (are there any?) may recall two earlier entries on December 14, 2007 and the May 23, 2008, when I discussed a shortage of technetium for medical imaging. It seems that this problem just won’t go away. According to a recent article in the New York Times, we are once again experiencing a global shortage of technetium, caused by the shutdown of nuclear reactors in Canada and the Netherlands. I fear that although the current shortage may be temporary, disruptions of the supply of technetium will reoccur with increasing frequency as nuclear power plants age. A reactor dedicated to technetium production in the United States would go a long way toward solving the problem, but would be expensive.

Russ Hobbie and I discussed technetium in the 4th Edition of Intermediate Physics for Medicine and Biology. Technetium-99m--the key isotope of technetium for medical imaging--is a decay product of Molybdenum-99, which in turn is a nuclear fragment that is produced during the fission of uranium. It is widely used in part because its 140 keV gamma emission and its 6 hour half life are particularly suited to nuclear medicine diagnostic procedures. 99mTc is often combined with other molecules to make radiopharmaceuticals, such as 99mTc-sestamibi and 99mTc-tetrofosmin, that can have very specific effects as tracers. For more about the discovery of technetium, see the March 13, 2009 entry of this blog.

Friday, July 31, 2009

Roberts Prize

One journal that readers of the 4th Edition of Intermediate Physics for Medicine and Biology may enjoy is Physics in Medicine and Biology. Below is part of an editorial that recently appeared in PMB.
“The publishers of Physics in Medicine and Biology (PMB), IOP Publishing, in association with the journal owners, the Institute of Physics and Engineering in Medicine (IPEM), jointly award an annual prize for the 'best' paper published in PMB during the previous year.

The procedure for deciding the winner has been made as thorough as possible, to try to ensure that an outstanding paper wins the prize. We started off with a shortlist of the 10 research papers published in 2008 which were rated the best based on the referees' quality assessments. Following the submission of a short 'case for winning' document by each of the shortlisted authors, an IPEM college of jurors of the status of FIPEM assessed and rated these 10 papers in order to choose a winner, which was then endorsed by the Editorial Board.

It was a close run thing between the top two papers this year. The Board feel that we have a very worthy winner... We have much pleasure in advising the readers of PMB that the 2008 Roberts Prize is awarded to J P Schlomka et al for their paper on multi-energy CT.”
The abstract of the paper (J P Schlomka, E Roessl, R Dorscheid, S Dill, G Martens, T Istel, C Bäumer, C Herrmann, R~Steadman, G Zeitler, A Livne and R Proksa, Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography, Phys. Med. Biol. 53 4031–4047, 2008) is reproduced below
“Theoretical considerations predicted the feasibility of K-edge x-ray computed tomography (CT) imaging using energy discriminating detectors with more than two energy bins. This technique enables material-specific imaging in CT, which in combination with high-Z element based contrast agents, opens up possibilities for new medical applications. In this paper, we present a CT system with energy detection capabilities, which was used to demonstrate the feasibility of quantitative K-edge CT imaging experimentally. A phantom was imaged containing PMMA, calcium-hydroxyapatite, water and two contrast agents based on iodine and gadolinium, respectively. Separate images of the attenuation by photoelectric absorption and Compton scattering were reconstructed from energy-resolved projection data using maximum-likelihood basis-component decomposition. The data analysis further enabled the display of images of the individual contrast agents and their concentrations, separated from the anatomical background. Measured concentrations of iodine and gadolinium were in good agreement with the actual concentrations. Prior to the tomographic measurements, the detector response functions for monochromatic illumination using synchrotron radiation were determined in the energy range 25 keV–60 keV. These data were used to calibrate the detector and derive a phenomenological model for the detector response and the energy bin sensitivities.”
You can learn more about the Robert’s award and the winning paper at the IOP’s excellent website I signed up for their weekly email, which is where I learned about this year’s winner. It is a great way for readers of Intermediate Physics for Medicine and Biology to keep up-to-date on recent breakthroughs in medical physics.

Friday, July 24, 2009

Two-Dimensional Image Reconstruction

In Section 12.4 of the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Two-Dimensional Image Reconstruction from Projections by Fourier Transform. The method is summarized in our Fig. 12.20: i) perform a 1-D Fourier transform of the projection at each angle θ, ii) convert from polar coordinates (k, θ) to Cartesian coordinates (kx, ky), and iii) perform an inverse 2-D Fourier transform to recover the desired image.

I wanted to include in our book some examples where this procedure could be done analytically, thinking that they would give the reader a better appreciation for what is involved in each step of the process. The result was two new homework problems in Chapter 12: Problems 23 and 24. In both problems, we provide an analytical expression for the projection, and the reader is supposed to perform the necessary steps to find the image. Both problems involve the Gaussian function, because the Gaussian is one of the few functions for which the Fourier transform can be calculated easily. (Well, perhaps “easily” is in the eye of the beholder, but by completing the square of the exponent the process is fairly straight forward).

I recall spending considerable time coming up with examples that are simple enough to assign as a homework problem, yet complicated enough to be interesting. One could easily do the case of a Gaussian centered at the origin, but then the projection has no angular dependence, which is dull. I tried hard to find examples that were based on functions other than the Gaussian, but never had any success. If you, dear reader, can think of any such examples, please let me know. I would love to have a third problem that I could use on an exam next time I teach medical physics.

For anyone who wants to get a mathematical understanding of image reconstruction from projections by Fourier transform, I recommend solving Problems 23 and 24. But you won’t learn everything. For instance, in medical imaging the data is discrete, as compared to the continuous functions in these homework problems. This particularly complicates the middle step: transforming from polar to Cartesian coordinates in frequency space. Such a transformation is almost trivial in the continuous case, but more difficult using discrete data (see Problem 20 in Chapter 12 for more on that process). Nevertheless, I have found that performing the reconstruction in a couple specific cases is useful for understanding the algorithm better.

Problems 23 and 24 are a bit more difficult than the average homework problem in our book. The student needs to be comfortable with Fourier analysis. But there is something fun about these problems, especially if you are fond of treasure hunts. I find it exciting to know that there is a fairly simple function f(x,y) representing an object, and that it can be determined from projections F(θ,x') by a simple three-step procedure. Perhaps it mimics, in a very simplistic way, the thrill that developers of computed tomography must have felt when they were first able to obtain images by measuring projections.

If you get stuck on these two problems, contact Russ or me about obtaining the solution manual. Enjoy!

P.S. The Oakland University website is currently undergoing some changes. For the moment, if you have trouble accessing the book website, try I hope to have a more permanent home for the website soon.