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 31^st 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 4^th 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.

Earl Bakken; Ready, Fire, Aim!

http://www.youtube.com/watch?v=XgEssNsMXoA

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.

Resource Letter MPRT-1: Medical Physics in Radiation Therapy

When Russ Hobbie and I were preparing the 4^th 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, 4^th ed. (Charles C. Thomas, Springfield, Illinois, 1983).

16. The Physics of Radiation Therapy, Faiz M. Khan, 3^rd 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, 3^rd ed. (Wiley, Hoboken, NJ, 2005).

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

92. Introduction to Health Physics, Herman Cember and Thomas E. Johnson, 4^th 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). (www-naweb.iaea.org/nahu/dmrp/pdf_files/ToC.pdf)

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.

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.

 Living Arrhythmias with John Grammer.
https://www.youtube.com/embed/deigcmtDV74

The Bell Curve

I was browsing through the 4^th 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.

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.

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,” Physics in Medicine and Biology, Volume 53, Pages 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 http://medicalphysicsweb.org. 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.

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 (k_x, k_y), 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 http://personalwebs.oakland.edu/~roth/hobbie.htm. I hope to have a more permanent home for the website soon.

Random Walks in Biology

Random Walks in Biology,
by Howard Berg.

In Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the role of diffusion in biology. One source we cite in this chapter is Random Walks in Biology, by Howard Berg. Below is the introduction to this fascinating book, which I recommend highly. In particular, I love Berg’s first sentence.

Biology is wet and dynamic. Molecules, subcellular organelles, and cells, immersed in an aqueous environment, are in continuous riotous motion. Alive or not, everything is subject to thermal fluctuations. What is this microscopic world like? How does one describe the motile behavior of such particles? How much do they move on the average? Questions of this kind can be answered only with an intuition about statistics that very few biologists have. This book is intended to sharpen that intuition. It is meant to illuminate both the dynamics of living systems and the methods used for their study. It is not a rigorous treatment intended for the expert but rather an introduction for students who have little experience with statistical concepts.

The emphasis is on physics, not mathematics, using the kinds of calculations that one can do on the back of an envelope. Whenever practical, results are derived from first principles. No reference is made to the equations of thermodynamics. The focus is on individual particles, not moles of particles. The units are centimeters (cm), grams (g), and seconds (sec).

Topics range from the one-dimensional random walk to the motile behavior of bacteria. There are discussions of Boltzmann’s law, the importance of kT, diffusion to multiple receptors, sedimentation, electrophoresis, and chromatography. One appendix provides an introduction to the theory of probability. Another is a primer on differential equations. A third lists some constants and formulas worth committing to memory. Appendix A should be consulted while reading Chapter 1 and Appendix B while reading Chapter 2. A detailed understanding of differential equations or the methods used for their solution is not required for an appreciation of the main theme of this book.

Buoyancy

Air and Water,
by Mark Denny.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new section on Buoyancy (Sec. 1.12). After a fairly standard derivation of the buoyant force, we discuss the biological applications.

The buoyant force on terrestrial animals is very small compared to their weight. Aquatic animals live in water, and their density is almost the same as the surrounding fluid. The buoyant force almost cancels the weight, so the animal is essentially “weightless.” Gravity plays a major role in the life of terrestrial animals, but only a minor role for aquatic animals. Denny (1993) explores the differences between terrestrial and aquatic animals in more detail.

The reference to Mark Denny is for his excellent book Air and Water. It’s the best book I know of to gain insights into how physics impacts physiology, and it influenced many of the revisions to the 4th edition of Intermediate Physics for Medicine and Biology. Below is a sampler from Denny’s Chapter 4, Density: Weight, Pressure, and Fluid Dynamics

What are the effective densities of plants and animals? Because the density of air is so small, it has little effect on the effective density of terrestrial organisms. For example, a typical density for an animal is 1080 kg m⁻³ and in air its effective density is 1079 kg m⁻³, a negligible difference. The effective weight in air of a 5000 N cow is 4995 N, for instance. For the same animal immersed in fresh water, however, its effective density is 80 kg m⁻³ and its effective weight is 370 N, only 7% of its actual weight. Water obviously has a profound effect on effective density.

Furthermore, because the density of water is so close to the body density of animals, the effective density (and therefore the effective weight) of aqueous organisms is very sensitive to small changes in density of either the body or the surrounding fluid. For example, seawater is only about 2.5% more dense that fresh water, but the effective density of a typical animal (…1075 kg m⁻³) is only 50 kg m⁻³ in the ocean compared to 75 kg m⁻³ in a lake. In this case, a 2.5% increase in water density results in a 33% decrease in effective weight. The same holds true if the density change is in the animal. For instance, if an animal reduces its density from 1075 to 1065 kg [m⁻³], its effective weight in air changes by only about 1%. In seawater, the same change in body density incurs a 20% change in effective density (from 50 to 40 kg m⁻³) and a concomitant change in effective weight.

Because the effective density of aquatic organisms is so sensitive to minor changes in body density, it is likely to have important biological consequences, and as a result, the density of aquatic organisms has received much attention. Some of the results are discussed below when we explore balloons and swim bladders…

Sync

Sync, By Steven Strogatz.

In the October 17, 2008 entry to this blog, I discussed Steven Strogatz’s textbook Nonlinear Dynamics and Chaos, which Russ Hobbie and I cite in Intermediate Physics for Medicine and Biology. Toward the end of that entry, I mentioned that reading Strogatz’s other book,  Sync, was “on my list of things to do.” Well, jobs often sit on my to-do list for a long time, but they eventually get done. This week I finished Sync, a fascinating book about “how order emerges from chaos in the universe, nature, and daily life.” It is an unusual mathematics book, because I don’t recall seeing a single equation. Nevertheless, Strogatz tells a charming tale of his contributions, and that of many others, to nonlinear dynamics. Russ added a citation to Sync to our chapter on Feedback and Control when we were preparing the 4th edition of Intermediate Physics for Medicine and Biology:

Strogatz (2003) discusses phase-resetting and other nonlinear phenomena in an engaging and nonmathematical manner.

The Geometry of Biological Time,
by Art Winfree.

Chapter Eight of Strogatz’s book, “Sync in Three Dimensions,” is my favorite. Here he describes how he first discovered the work of Art Winfree, his “mentor, inspiration, friend.”

I walked across the street to Heffer’s Bookstore to browse the books on biomathematics… As I scanned the shelves, with my head tilting sideways, one title popped out at me:  The Geometry of Biological Time. Now that was a weird coincidence. My senior thesis on DNA had been subtitled “An Essay on Geometric Biology.” I thought I had invented that odd juxtaposition, geometry next to biology. But the book’s author, someone named Arthur T. Winfree, from the biology department at Purdue University, had obviously connected them first.

Strogatz then relates how he corresponded with Winfree, and ended up working with him at Purdue in the summer of 1982. He quotes Winfree’s letters, often written in what Strogatz calls “idiosyncratic code.” This characteristic style brought back memories of my own correspondence with Winfree. Although we only met in person once, I recall us exchanging many emails about cardiac dynamics, with his emails all in that same idiosyncratic code. As I read Winfree’s letters to Strogatz, I found myself thinking “yes, that is exactly the way Winfree would have said it.”

When Time Breaks Down,
by Art Winfree.

I had my own encounter with a Winfree book that influenced my early career. For me, it was not The Geometry of Biological Time, but Winfree’s other influential text, When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. I read the book in the spring of 1991. (I remember the time precisely, because I recall turning the pages of the book with one hand, while holding my newborn daughter Kathy with the other.) I was soon performing calculations of reentrant wave propagation in cardiac tissue, similar to the dynamics described in When Time Breaks Down. I worked at the National Institutes of Health at the time, and a summer student, Josh Saypol, and I performed a calculation of reentry induction using the bidomain model to test a prediction that Winfree had made in a chapter of the book Cardiac Electrophysiology: From Cell to Bedside (1990). His help was invaluable in interpreting our results, and for getting a preliminary note (“The Formation of a Re-entrant Action Potential Wave Front in Tissue with Unequal Anisotropy Ratios”) published in the just-started International Journal of Bifurcation and Chaos (Volume 1, Pages 927–928, 1991).

Strogatz describes Winfree’s untimely death in the epilog of Sync.

Tragically, Art Winfree died on November 5, 2002, at age 60, seven months after being diagnosed with brain cancer. He helped me with this book at every stage, even when he was conscious only for a few hours a day. Though he did not live to see it published, he knew that it would be dedicated to him.

For more about Winfree’s career, see his website (still available through the University of Arizona), the obituary Strogatz wrote for the Society for Industrial and Applied Mathematics, another by Leon Glass in Nature, and also one in the New York Times.

I described my own interactions with Winfree, and some of his contributions to cardiac electrophysiology, in my paper “Art Winfree and the Bidomain Model of Cardiac Tissue,” published in a special issue of the Journal of Theoretical Biology dedicated to his memory (Volume 230, Pages 445–449, 2004). Other particularly interesting contributions to that issue, full of delightful Winfree anecdotes, were the article by his daughter Rachael, and the article by George Oster.

I thoroughly enjoyed Sync. It is a fine introduction to the mathematics of synchronization and nonlinear dynamics. (Don’t, however, consult the book to learn how lasers work!) Sync ends with a lovely paragraph that explains what motivates scientists:

For reasons I wish I understood, the spectacle of sync strikes a chord in us, somewhere deep in our souls. It’s a wonderful and terrifying thing. Unlike many other phenomena, the witnessing of it touches people at a primal level. Maybe we instinctively realize that if we ever find the source of spontaneous order, we will have discovered the secret of the universe.

Alas, my to-do list never gets any shorter. Strogatz has a new book coming out next month, The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math, and I plan to read it as soon as I get a bit of spare time.