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.