Fourier Series

One of the most important mathematical techniques for a physicist is the Fourier series. I discussed Joseph Fourier, the inventor of this method, previously in this blog. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Fourier series in Sections 11.4 and 11.5.

The classic example of a Fourier series is the representation of a periodic square wave: y(t) = 1 for t between 0 and T/2, and y(t) = -1 for t between T/2 and T, where T is the period. The Fourier series represents this function as a sum of sines and cosines, with frequencies of k/T, where k is an integer, k = 0, 1, 2, …. The square wave function y(t) is odd, so the contributions of the cosine functions vanish. The sine functions contribute for half the frequencies, those with odd values of k. The amplitude of each non-zero frequency is 4/πk (Eq. 11.34 in IPMB), so the very high frequency terms (large k) don’t contribute much.

Being able to calculate the Fourier series is nice, but much more important is being able to visualize it. When I teach my Medical Physics class (PHY 326), based on the last half of IPMB, I stress that students should “think before you calculate.” One ought to be able to predict qualitatively the Fourier coefficients by inspection. Being able to understand a mathematical calculation in pictures and in physical terms is crucially important for a physicist. The Wikipedia article about a square wave has a nice animation of the square wave being built up by adding more and more frequencies to the series. I always insist that students draw figures showing better and better approximations to a function as more terms are added, at least for the first three non-zero Fourier components. You can also find a nice discussion of the square wave at the Wolfram website. However, the best visualization of the Fourier series that I have seen was brought to my attention by one of the PHY 326 students, Melvin Kucway. He found this lovely site, which shows the different Fourier components as little spinning wheels attached to wheels attached to wheels, each with the correct radius and spinning frequency so that their sum traces out the square wave. Watch this animation carefully. Notice how the larger wheels rotate at a lower frequency, while the smaller wheels spin around at higher frequencies. This picture reminds me of the pre-Copernican view of the rotation of planets based on epicycles proposed by Ptolemy.

What is unique about the development of Fourier series in IPMB? Our approach, which I rarely, if ever, see elsewhere, is to derive the Fourier coefficients using a least-squares approach. This may not be the simplest or most elegant route to the coefficients, but in my opinion it is the most intuitive. Also, we emphasize the Fourier series written in terms of sines and cosines, rather than complex exponentials. Why? Understanding Fourier series on an intuitive level is hard enough with trigonometric functions; it becomes harder still when you add in complex numbers. I admit, the math appears in a more compact expression using complex exponentials, but for me it is more difficult to visualize.

If you want a nice introduction to Fourier series, click here or here (in the second site, scroll down to the bottom on the left). If you prefer listening to reading, click here for an MIT Open Courseware lecture about the Fourier series. The two subsequent lectures are also useful: see here and here. The last of these lectures examines the square wave specifically.

One of the fascinating things about the Fourier representation of the square wave is the Gibbs phenomenon. But, I have discussed that in the blog before, so I won’t repeat myself.

What is the Fourier series used for? In IPMB, the main application is in medical imaging. In particular, computed tomography (Chapter 12) and magnetic resonance imaging (Chapter 18) are both difficult to understand quantitatively without using Fourier methods.

As a new year’s resolution, I suggest you master the Fourier series, with a focus on understanding it on a graphical and intuitive level. What is my new year’s resolution for 2014? It is for Russ and I to finish and submit the 5th edition of IPMB to our publishers. With luck, you will be able to purchase a copy before the end of 2015.

The Last Question

Entropy and its role in the Second Law of Thermodynamics is one of the fundamental ideas of all science. One can think of entropy roughly as a measure of the disorder within a system. An interesting feature of entropy is that it is not conserved. Rather, it tends to increase over time (the system becomes more disordered). In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

…under what conditions can the entropy of a system be made to decrease?

The answer is that the entropy of a system can be made to decrease if, and only if, it is in contact with one or more auxiliary systems that experience at least a compensating increase in entropy. Then the total entropy remains the same or increases. This is one form of the second law of thermodynamics. For a fascinating discussion of the second law, see Atkins (1994).

Nine Tomorrows,
A short story collection by
Isaac Asimov containing
“The Last Question.”

The reference is to P. W. Atkins’ 1994 Scientific American book The 2nd Law: Energy, Chaos and Form. It is a great book written for a layman with little or no mathematics that clearly conveys the insights provided by the second law.

The very first course I ever taught, at Vanderbilt University in the fall of 1995, was an undergraduate thermodynamics class. When we talked about entropy, I had all my students read the short story “The Last Question” by Isaac Asimov. This marvelous tale speculates about what will happen to the universe as entropy slowly but steadily increases. Regular readers of this blog know that I am a big Asimov fan (for instance, see here), and The Last Question is one of his best stories. That is not just my opinion. Asimov himself said it was his favorite short story of all he had written. You can read it here, or listen to the Good Doctor himself read it below.

I Robot, by Isaac Asimov.

For those wanting more Asimov short stories, I recommend “Nightfall”and “The Ugly Little Boy.” Also I, Robot is a collection of loosely related short stories based on the three laws of robotics (please, skip the 2004 film by the same name featuring Will Smith). One big Asimov fan posted reviews of all Asimov’s books. Enjoy.

Finally, let me share one more crucially important point. Amazon.com says you still have time to order IPMB and have it delivered by Christmas. If you plan on putting a copy of IPMB in each of your loved one's stockings (and who doesn’t?), you had better order soon!

Electricity and Magnetism

Electricity and Magnetism, 3rd Edition,
by Edward Purcell and David Morin.

I have always enjoyed Edward Purcell’s textbook Electricity and Magnetism, which is Volume 2 of the Berkeley Physics Course. I love the book so much that I included it in My Ideal Bookshelf that I presented in this blog a few weeks ago. I own a copy of the first edition, with its orange cover that is now faded and falling apart. The second edition was published soon after I completed my undergraduate physics degree from the University of Kansas, so I have not used it much.

This year, a 3rd edition of the book was published, with coauthor David Morin (Purcell died in 1997 so he had no input on the 3rd edition). In the preface to the 3rd edition, Morin describes his goals:

For 50 years, physics students have enjoyed learning about electricity and magnetism through the first two editions of this book. The purpose of the present edition is to bring certain things up to date and to add new material, in the hopes that the trend will continue. The main changes from the second edition are (1) the conversion from Gaussian units to SI units, and (2) the addition of many solved problems and examples.

The use of SI units is interesting, because apparently Purcell resisted this change when preparing the 2nd edition. When I was an undergraduate around 1980, almost all introductory textbooks had switched to SI units, so I “grew up” with them. Therefore, I agree with Morin about the usefulness of this change. In the preface he writes

The first of these changes [to SI units] is due to the fact that the vast majority of courses on electricity and magnetism are now taught in SI units. The second edition fell out of print at one point, and it was hard to watch such a wonderful book fade away because it wasn’t compatible with the way the subject is presently taught. Of course, there are differing opinions as to which system of units is “better” for an introductory course. But this issue is moot, given the reality of these courses.

The other big change is a lot of new homework problems and worked examples. This resonates with me, because one big change that Russ Hobbie and I made to the 4th edition of Intermediate Physics for Medicine and Biology is many new homework problems. Morin offers some advice to the reader in his preface, which applies equally well to readers of IPMB. One reason we only distribute homework solutions to instructors rather than students is to encourage students to struggle with these problems on their own.

Some advice on using the solutions to the problems…If you are having trouble solving a problem, it is critical that you don’t look at the solution too soon. Brood over it for a while. If you do finally look at the solution, don’t just read it through. Instead, cover it up with a piece of paper and read one line at a time until you reach a hint to get you started. Then set the book aside and work things out for real. That’s the only way it will sink in. It’s quite astonishing how unhelpful it is simply to read a solution. You’d think it would do some good, but in fact it is completely ineffective in raising your understanding to the next level.

One unique feature of Electricity and Magnetism is that magnetism is introduced as a consequence of electricity and special relativity. In almost all other books, this relationship is omitted or presented as an advanced topic. It is an interesting approach, about which I have mixed feelings. Morin writes

The intertwined nature of electricity, magnetism, and relativity is discussed in detail in Chapter 5. Many students find this material highly illuminating, although some find it a bit difficult. (However, these two groups are by no means mutually exclusive!)

If I was teaching our undergraduate electricity and magnetism class next semester, would I use the 3rd edition of Electricity and Magnetism? I would certainly consider it. In my opinion, its main competition would be David Griffiths’ textbook Introduction of Electrodynamics. I used that book last time I taught electricity and magnetism, and it is also outstanding. It is not cited in IPMB, which makes me rather sad, as it is another one of those much-treasured books. Just the thought of reading first Purcell and then Griffiths, trying to decide which to use, sounds so fun that I am tempted to volunteer to teach electricity and magnetism again.

For those wanting to learn more about Morin’s new edition of Electricity and Magnetism, read the interview with him on the Physics Today website. You can find a review of the book here. Additional information is on the book's website.

A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times

I am one of the coauthors on a paper published recently that analyzes how to determine properties of a cardiac wave front from measurements of wave front arrival times (Mazeh, Haines, Kay, and Roth, “A Simplified Approach for Simultaneous Measurement of Wavefront Velocity and Curvature in the Heart Using Activation Times,” Cardiovascular Engineering and Technology, Volume 4, Pages 520–534, 2013). The lead author, Nachaat Mazeh, is a former grad student of mine who obtained his PhD from Oakland University, and now works in the Beaumont Health System. David Haines is the Director of the Heart Rhythm Center at Beaumont, and is well known for his work on radiofrequency ablation of cardiac tissue. Matthew Kay is a professor of engineering at The George Washington University. In this paper, we obtain the wave front properties from measurement of four arrival times. The result is just simple enough to make into a new homework problem, typical in difficulty to those in the 4th edition of Intermediate Physics for Medicine and Biology.

Section 10.11

Problem 43 Suppose you measure the arrival time of an action potential wave front at four points (1-4) in a diamond pattern, each a distance b from the central point (red). Calculate the wave front speed, direction, and curvature from these four measurements.

a) Assume the wave front is circular and propagates outward from the origin. Use the law of cosines to write r₁, r₂, r₃, and r₄ (the distance of each electrode to the origin) in terms of r₀, b, and the angle θ.

b) Pull a factor of r₀ outside of the square root in each of your four expressions from part a).

c) Assume r₀ is much greater than b, and perform a Taylor expansion of each of the four expressions in terms of the small parameter ε = b/r₀. Include terms that are constant, linear, and quadratic in ε.

d) Write the arrival time at each electrode (n=1, 2, 3, and 4) as t_n=r_n/v, where v is the wave speed.

e) Let Δt_ij=t_i – t_j. Find expressions for Δt₁₃ and Δt₂₄ in terms of b, θ, and v. Solve these expressions to determine v and θ in terms of Δt₁₃, Δt₂₄, and b.

f) Find expressions for Δt₁₄ and Δt₂₃ in terms of b, θ, and v. Now (and this is the most difficult step), find an expression for the radius of curvature, r₀, in terms of b, Δt₁₃, Δt₂₄, Δt₁₄, and Δt₂₃.

There are several advantages and several disadvantages to the expressions you will derive. The advantages are that the calculations require only four measurements of arrival time, and they provide not only the speed and direction but also (somewhat unexpectedly--at least to me) the radius of curvature, r₀. The radius of curvature is important for propagation, because highly curved wave fronts propagate more slowly than nearly flat wave fronts. The radius of curvature at the core of a spiral wave is highly curved, and this curvature influences properties of the spiral wave such as how fast it rotates. There are some important limitations. First, a close examination of your expression for the radius of curvature will reveal that the method gives an indeterminate expression for propagation at angles of θ = 45, 135, 225, and 315°. Second, the expressions contain the differences of activation times. In fact, the radius of curvature depends on the difference of a difference of activation times. If these activation times are all similar, then they need to be known precisely for the calculation of their differences to be accurate. The calculation assumes the wave front is circular, although really the wave front only needs to be circular locally, so this should not be too bad an approximation. The method also is based on the assumption that b is much less than r₀.

Despite these limitations, I think the expressions should be useful for characterizing properties of wave fronts in the heart. It may be particularly useful for obtaining wave front speed, direction, and curvature in computer simulations, where the calculation is computed over a regular two-dimensional Cartesian grid and where noise in the activation times may not be a big concern.

From Vision to Change: Educational Initiatives and Research at the Intersection of Physics and Biology

A few months ago, the journal CBE—Life Sciences Education published a special issue about education at the intersection of physics and biology. This topic is of great interest to readers of the 4th edition of Intermediate Physics for Medicine and Biology. The special issue was motivated by the American Association for the Advancement of Science report Vision and Change in Undergraduate Biology Education. While there were many interesting articles in this issue, my favorite was the essay “Learning Each Other’s Ropes: Negotiating Interdisciplinary Authenticity” (Volume 12, Pages 175–186, 2013), by Edward Redish and Todd Cooke, both from the University of Maryland. They describe their goals in the paper’s abstract.

A common feature of the recent calls for reform of the undergraduate biology curriculum has been for better coordination between biology and the courses from the allied disciplines of mathematics, chemistry, and physics. Physics has lagged behind math and chemistry in creating new, biologically oriented curricula, although much activity is now taking place, and significant progress is being made. In this essay, we consider a case study: a multiyear conversation between a physicist interested in adapting his physics course for biologists (E.F.R.) and a biologist interested in including more physics in his biology course (T.J.C.). These extended discussions have led us both to a deeper understanding of each other’s discipline and to significant changes in the way we each think about and present our classes. We discuss two examples in detail: the creation of a physics problem on fluid flow for a biology class and the creation of a biologically authentic physics problem on scaling and dimensional analysis. In each case, we see differences in how the two disciplines frame and see value in the tasks. We conclude with some generalizations about how biology and physics look at the world differently that help us navigate the minefield of counterproductive stereotypical responses.

I found this paper to be fascinating, and it will be helpful as Russ Hobbie and I prepare the 5th edition of IPMB. It is interesting that the authors use the word “negotiating” in the title, because I felt that Redish and Cooke were involved in an extended negotiation about how much physics to include in an introductory biology class. This process is not restricted to instruction; I go through an often painful negotiation regarding the emphasis of biology versus physics with the reviewers of almost every research article I’ve ever published. I like the conversational tone of Redish and Cooke’s paper, and how it describes the growth of a close collaboration between a biologist and a physicist, each with a different worldview. Probably the most important contribution of the article is the story of how they uncovered and dealt with their hidden biases (they use the term epistemologies, which is one of those words from the science education literature that I dislike). Readers of this blog may remember Redish; one of my blog entries earlier this year discussed his article in Physics Today about “Reinventing Physics for Life Science Majors.” At first, I was annoyed by Redish and Cooke’s habit of referring to themselves collectively in the first person and individually in the third person (as in, “our physicist” and “our biologist”), but as I read on this technique began to grow on me and in the end I found it endearing. I particularly enjoyed their discussion about the role of problem solving in physics and biology, and what makes a good homework problem.

In our interdisciplinary discussions, we also learned that biologists and physicists had dramatically different views of what makes a good biological example in physics… We came to understand that what would be of value in a physics class is biological authenticity—examples in which solving a physics problem in a biological context gives the student a deeper understanding of why the biological system behaves the way it does.

Russ and I strive to achieve authenticity in our end-of-chapter homework problems. Our book is aimed at an intermediate level--we assume the student is comfortable with calculus--so we may have an easier time constructing nontrivial physics exercises applied to biology than an introductory instructor would, but I sometimes wonder if the biologists and medical doctors find them as useful as we think they are.

Redish and Cooke present a list of “cultural components” of both physics and biology that are illuminating.

Physics: Common Cultural Components

• Introductory physics classes often stress reasoning from a few fundamental (usually mathematically formulated) principles. 

• Physicists often stress building a complete understanding of the simplest possible (often highly abstract) examples— “toy models”—and often do not go beyond them at the introductory level.
 
• Physicists quantify their view of the physical world, model with math, and think with equations, qualitatively as well as quantitatively.
 
• Physicists concern themselves with constraints that hold no matter what the internal details (conservation laws, center of mass, etc.).

Biology: Common Cultural Components

• Biology is often incredibly complex. Many biological processes involve the interactions of component parts leading to emergent phenomena, which include the property of life itself. 

• Most introductory biology does not emphasize quantitative reasoning and problem solving to the extent these are emphasized in introductory physics.
 
• Biology contains a critical historical constraint in that natural selection can only act on pre-existing molecules, cells, and organisms for generating new solutions.
 
• Much of introductory biology is descriptive (and introduces a large vocabulary).
 
• However, biology—even at the introductory level—looks for mechanism and often considers micro–macro connections between the molecules involved and the larger phenomenon.
 
• Biologists (both professionals and students) focus on and value real examples and structure–function relationships.

As I read these lists, it is clear to me that I am definitely in the physics camp. It would not be an exaggeration to say that a primary goal of IPMB is to force introduce the physicist’s culture, as described above, to students interested in biology and medicine.

I do have one minor criticism. Initially I was impressed by Redish’s analysis of the relationship of flow to pressure in a blood vessel (Hagen-Poiseuille flow), with the appearance of the fourth power of the radius, using simple arguments involving no calculus. Upon further reflection, however, I’m not totally comfortable with the derivation. Here it is in brief:

F_pressure = ΔP A

F_drag = b L v

Q = A v

where ΔP is the pressure drop, A is the cross-sectional area of the vessel, L is the vessel length, v is the speed (assumed uniform, or plug flow), b is a frictional proportionality constant, and Q is the volume flow. If you set the two forces equal, and eliminate v in favor of Q, you get

ΔP = (b L/A²) Q

The 1/A² dependence implies that the flow increases as the fourth power of the radius. My concern is this: suppose a student approaches Redish and says “I follow your derivation, but shouldn’t the drag force be proportional to the surface area where the flow contacts the vessel wall? In other words, shouldn’t the drag force be given by F_drag = c (2πrL) v?” (I use c for the proportionality constant because it now has different units that b.) Of course, if you do the calculation using this expression for the drag force, you get the wrong answer (a 1/r³ dependence)! I wonder if any of his students ever brought this up, and how he responded? The complete derivation is given in Chapter 1 of IPMB, and the central point is that the frictional force depends on dv/dr rather than v, but the analysis uses some rather advanced calculus that would be inappropriate in the introductory biology class that Redish and Cooke consider. The trade-off between simplifying a concept so it is accessible versus being as accurate as possible is always difficult. I don’t know what the best approach would be in this case. (I can always make one recommendation: buy a copy of IPMB!)

Despite this one reservation, I enjoyed Redish and Cooke’s paper very much. Let me give them the last word.

We conclude that the process [of interdisciplinary collaboration aimed at revising the biology introductory course] is significantly more complex than many reformers working largely within their discipline often assume. But the process of learning each other’s ropes—at least to the extent that we can understand each other’s goals and ask each other challenging questions—can be both enlightening and enjoyable. And much to our surprise, we each feel that we have developed a deeper understanding of our own discipline as a result of our discussions.

My Ideal Bookshelf

My Ideal Bookshelf,
by Thessaly La Force, with
artwork by Jane Mount.

While browsing at the library recently, I ran across the book My Ideal Bookshelf, edited by Thessaly La Force, with artwork by Jane Mount. Their preface describes the book well.

The assignment sounds straightforward enough. Select a small shelf of books that represent you—the books that have changed your life, that have made you who you are today, you favorite favorites. You begin, perhaps, by walking over to your bookshelf and skimming the spines on the top shelf. You pull down a handful that you remember loving; you grab a couple that you read over and over again. Some you know just by the color of their dust jackets. One is in tatters—it was passed down by your mother—and it’s dog-eared and carefully held together by tape and tenderness. The closer you look, the trickier the task turns out to be….

We asked more than one hundred creative people in a variety of disciplines from around the world to do exactly this. Chefs and architects, writers and fashion designers, filmmakers and ballet dancers all agreed to share their ideal bookshelves.

It’s not the words that make this book is so interesting; it is the pictures. Each contributor had their bookshelf painted. The dust jackets says “Jane Mount’s original paintings of the colorful and delightful book spines and occasional objects d’art from the contributors’ personal bookshelves showcase the selections.” You can probably guess where this is going. I just had to collect the ideal bookshelf for the 4th edition of Intermediate Physics for Medicine and Biology. (Someone had too much time on his hands this week.) Here it is:

My Ideal Bookshelf, for
Intermediate Physics for Medicine and Biology.

My illustration does not have the charm of Mount’s quirky paintings (see her create one of these illustrations below). I put mine together in powerpoint, and every book spine is a rectangle. If you like this sort of thing, browse through My Ideal Bookshelf and enjoy her enchanting artwork. If you REALLY like My Ideal Bookshelf, you can pay big bucks and get prints of various book collections (see http://www.idealbookshelf.com).

How did I choose which books to include in my illustration? The book had to be cited in IPMB, it had to deal with either physics or the application of physics to biology, and I had to have access to a copy (either from my own office bookshelf or from the Oakland University library) so I could recall what the spine looked like. The towering blue volume of IPMB dominates the skyline that is my bookshelf. Some great authors are represented, including Edward Purcell, Philip Morrison, Art Winfree, Knut Schmidt-Nielsen, Leon Glass, and Howard Berg. Two volumes of the wonderful Berkeley Physics Course appear. Powers of Ten and The Machinery of Life are both beautifully illustrated, and would be the best choices from my shelf for mathophobes. I wanted to include Strogatz’s Nonlinear Dynamics and Chaos, but I couldn’t reproduce the marbled coloration of the spine using powerpoint.

Do you want to learn more about books related to IPMB? I listed my top ten books in an earlier blog entry, and a catalog of 40 books can be found in an Amazon.com Listmania! list for IPMB that I put together a while back. Click one button on that webpage and you can buy them all (I hope your credit card has a high limit).

Whose collection of books in My Ideal Bookshelf is closest to my taste? I resonated with the bookshelves of Atul Gawande (doctor and writer), John Maeda (graphic designer and computer scientist), and most of all Jonathan Zittrain (legal scholar and professor). For you voyeurs who want to peak further into my personal bookshelf, see another of my Listmania! lists containing my favorite books.

Upcoming Events

I like to use this blog to remind readers of the 4th edition of Intermediate Physics for Medicine and Biology about upcoming events they might enjoy. Here are a few:


1. Each December, the Howard Hughes Medical Institute webcasts its Holiday Lectures on Science. This year, the topic is “Medicine in the Genomic Era.”

Sixty years after James Watson and Francis Crick revealed the structure of the DNA double helix and only a decade after scientists published the first complete read-through of all three billion DNA bases in the human genome, the ability to routinely sequence and analyze individual genomes is revolutionizing the practice of medicine—from how diseases are first diagnosed to how they are treated and managed. In the 2013 Holiday Lectures on Science, Charles L. Sawyers of Memorial Sloan-Kettering Cancer Center and Christopher A. Walsh of Boston Children’s Hospital will reveal the breathtaking pace of discoveries into the genetic causes of various types of cancers and diseases of the nervous system, and discuss the impact of those discoveries on our understanding of normal human development and disease.

Although there may not be a lot of physics in these lectures, the analysis of genomic data is a fine example of how mathematics and medicine are intertwined. The webcast schedule is

December 5th, 2013
Live webcast
10:00 a.m. ET “Sizing up the Brain, Gene by Gene,” by Christopher A. Walsh
11:30 a.m. ET “Cancer as a Genetic Disease,” by Charles L. Sawyers
Re-webcast
10:00 a.m. PT “Sizing up the Brain, Gene by Gene,” by Christopher A. Walsh
11:30 a.m. PT “Cancer as a Genetic Disease,” by Charles L. Sawyers 

December 6th, 2013
Live webcast
10:00 a.m. ET “Decoding the Autism Puzzle,” by Christopher A. Walsh
11:30 a.m. ET “From Cancer Genomics to Cancer Drugs,” by Charles L. Sawyers
Re-webcast
10:00 a.m. PT “Decoding the Autism Puzzle,” by Christopher A. Walsh
11:30 a.m. PT “From Cancer Genomics to Cancer Drugs,” by Charles L. Sawyers

I have seen these HHMI lectures in the past, and they are excellent. Aimed at the layman, the audience consists primarily of young scientists (high school students interested in science, if I recall correctly). They say you should register for the lectures, but I don’t think it costs anything. Enjoy.


2. During the Christmas season, the Royal Institution (home to one of my heroes, Michael Faraday) presents its Christmas Lectures. This year, they are about developmental biology.

The 2013 CHRISTMAS LECTURES® presented by Dr Alison Woollard from the University of Oxford, will explore the frontiers of developmental biology and uncover the remarkable transformation of a single cell into a complex organism. What do these mechanisms tell us about the relationships between all creatures on Earth? And can we harness this knowledge to improve or even extend our own lives? - See more at: http://richannel.org/christmas-lectures/2013/life-fantastic#sthash.eGBmtufx.dpuf

These lectures will take place December 14, 17, and 19, in front of a live audience (again, mainly of students) at the Royal Institution in London. We Yanks who can’t afford to cross the pond will have to wait until January to view the recordings at the Royal Institution website.

If you think there is no physics in developmental biology, then you haven’t read Biological Physics of the Developing Embryo. (Honestly, I haven’t read the whole thing either, but I have skimmed through it and there is a lot of physics there.)


3. This time of year is when students should start thinking about research opportunities for summer 2014. One way to find such opportunities is by looking at the Pathways-to-Science website. I recommend the National Institutes of Health Internship Program in Biomedical Research. Having worked at NIH, I know that it is the best place in the world to do biomedical research. For an undergraduate student (or even high school student), working at NIH is the chance of a lifetime. Apply!

Oakland University, where I work, has a website listing many other summer research programs, such as the many Research Experiences for Undergraduates programs funded by the National Science Foundation. Some of them are for OU students, but we also list many programs that anyone can apply to. Set some time aside over the holiday break to review these programs.


4. I am the Oakland University representative for the Barry M. Goldwater Scholarship, which is the most prestigious honor available for undergraduate science, math, and engineering majors. I suspect many readers of IPMB are top students at their university. If you are currently a sophomore or junior at a United States university, you should ask around and find your Goldwater representative. Hurry, because the deadline is approaching fast! (Sorry, my friends from other countries, but only US citizens and permanent residents can apply.)


5. I just learned about the Science News for Students website. It is interesting, and worth a look. How did we survive without sites like these when I was growing up?


6. This week, Oakland University sent five of our top undergraduate researchers to the Sigma Xi 2013 Student Research Conference. This is a great meeting for young scholars, and undergraduates should explore the possibility of attending with their research mentor. This year’s meeting was November 8 and 9 at the Sheraton Imperial Hotel in Research Triangle Park, North Carolina. I realize I was tardy in announcing this meeting…you just missed this year’s conference. But students should keep the meeting in mind for next year. Plan ahead! I am a big fan of Sigma Xi, the Scientific Research Society.

Tobacco Mosaic Virus

The tobacco mosaic virus.

Table 4.2 in the 4th edition of Intermediate Physics for Medicine and Biology contains list of various particles and their root-mean-square thermal velocities. It is meant to illustrate Equation 4.12, which specifies the rms velocity as a function of a particle’s mass and the temperature. The list contains many common molecules important to life—water, oxygen, glucose, hemoglobin—and even small organisms such as Escherichia coli bacteria. Between the protein hemoglobin and the bacterium E. coli lies the romantically-named tobacco mosaic virus. Why was this particular virus included in the table? One could argue that the table needed at least one virus to fill the large gap between molecules and cells. But the table also includes the bacteriophage, a smaller virus that infects bacteria and that played a leading role in the development of molecular biology. Why then also include the Tobacco mosaic virus? I must admit that the table and its entries predate the 4th edition of the textbook. Russ Hobbie included this table in earlier editions of IPMB, so I can only guess. But a look at the history of biology suggests an answer.

A Short History of Biology,
by Isaac Asimov.

The tobacco mosaic virus was the first virus discovered. Isaac Asimov describes this discovery in A Short History of Biology

But twentieth-century serology [the study of plasma serum] reserved its most spectacular successes for the battle with microorganisms of a type unknown to [Louis] Pasteur and [Robert] Koch [founders of bacteriology] in their day. Pasteur had failed to find the infective agent of rabies, a clearly infectious disease undoubtedly caused, according to his germ theory, by a microorganism. Pasteur suggested that the microorganism existed but that it was too small to be detected by the techniques of the time. In this, he turned out to be correct.

The fact that an infectious agent might be much smaller than ordinary bacteria was shown to be true in connection with a disease affecting the tobacco plant (“tobacco mosaic disease”). It was known that juice from diseased plants would infect healthy ones and, in 1892, the Russian botanist, Dmitri Iosifovich Ivanovski (1864–1920), showed that the juice remained infective even after it had passed through filters fine enough to keep any known bacterium from passing through. In 1895, this was discovered independently, by the Dutch botanist, Martinus Willem Beijerinck (1851–1931). Beijerinck named the infective agent a “filtrable virus” where virus simply means “poison.” This marked the beginning of the science of virology.

The tobacco mosaic virus has other claims to fame. In 1935 it became the first virus to be crystalized, by American biochemist Wendell Stanley, a feat for which he received the 1946 Nobel Prize in Chemistry. Two years later, English plant pathologist Sir Frederick Bawden showed that the Tobacco mosaic virus contained ribonucleic acid (RNA), suggesting that nucleic acids are crucial for life. Rosalind Franklin, famous for her roll in Watson and Crick’s discovery of the structure of DNA, later studied the structure of the tobacco mosaic virus. In their paper “Tobacco Mosaic Virus: Pioneering Research for a Century” (Plant Cell, Volume 11, Pages 301–308, 1999), Creager et al. write

Tobacco mosaic virus (TMV), as we now know the agent that Beijerinck and others were studying, was the first virus to be identified. Perhaps because of this, research on TMV and other plant viruses has continued to be of profound significance in addressing fundamental questions about the nature of viruses in general. Indeed, TMV as a model system has been at the forefront of virology research to the present time. For example, TMV was the first virus to be chemically purified (Stanley, 1935; Bawden et al., 1936), to be detected in an analytical ultracentrifuge and in an electrophoresis apparatus (Eriksson-Quensel and Svedberg, 1936), and to be visualized in an electron microscope (Kausche et al., 1939). TMV RNA was used in the first decisive experiments showing that nucleic acids carry hereditary information and that nucleic acid alone is sufficient for viral infectivity (Fraenkel-Conrat, 1956; Gierer and Schramm, 1956). The TMV coat protein (CP) was the first virus protein to be sequenced (Anderer et al., 1960; Tsugita et al., 1960), and TMV’s particle structure was among the first to be elucidated in atomic detail (Bloomer et al., 1978; Namba et al., 1989).

TMV’s preeminence has extended into the recombinant era, when the first transgenic plants were constructed using TMV to demonstrate the concept of CP-mediated cross-protection (Abel et al., 1986). TMV was also the first virus shown to encode a cell-to-cell movement protein (MP; Deom et al., 1987). MP binds to RNA (Citovsky et al., 1990), associates with cytoskeletal elements (Heinlein et al., 1995; McLean et al., 1995), and increases the permeability of plasmodesmata to mediate cell-to-cell movement of the virus (Wolf et al., 1989; Waigmann et al., 1994).

What is the “mosaic” part of the virus name mean? A mosaic virus infects plants and causes the leaves to appear speckled. To learn more about this virus, read The Life of a Virus: Tobacco Mosaic Virus as an Experimental Model, 1930–1965 by Angela Creager (I have not read this, but it looks interesting, as does her more recent book Life Atomic: A History of Radioisotopes in Science and Medicine).

Russ must have known exactly what he was doing when he compiled Table 4.2 (and created Fig. 4.12), as he selected perhaps the most important virus historically, and one that has become a model system for microbiology.

Isotopes of Carbon

One of the fundamental ideas of nuclear physics is the isotope. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe isotopes at the start of Chapter 17.

Each atom contains a nucleus about 100,000 times smaller than the atom. The nuclear charge determines the number of electrons in the neutral atom and hence its chemical properties. The nuclear mass determines the mass of the atom. For a given nuclear charge there can be a number of nuclei with different masses or isotopes. If an isotope is unstable, it transforms into another nucleus through radioactive decay.

In order to understand isotopes, consider the isotopes of carbon, one of the most important elements for life.

¹²C

The most abundant isotope of carbon is ¹²C. This isotope is stable represents 99% of all carbon. The nucleus of ¹²C contains six protons (carbon’s atomic number is six) and six neutrons. The nucleus is basically three alpha particles stuck together.

¹³C

A second stable isotope is ¹³C, which contains six protons and seven neutrons. Only about 1% of carbon is ¹³C. The ratio of ¹³C to ¹²C is used to study ancient climates and oceans. Plants preferentially take up ¹²C, so the ¹³C/¹²C ratio provides a way to determine the amount of photosynthesis production, and contains information about the origin of carbon dioxide emission (to learn more, click here)

¹⁴C

All isotopes of carbon except ¹²C and ¹³C are unstable. The longest-lived unstable isotope is ¹⁴C, with a half-life of 5730 years. As with most isotopes containing an overabundance of neutrons, it decays by β^- emission (a neutron turns into a proton, with the emission of an electron). In Chapter 16 of IPMB, we describe how the decay of ¹⁴C contributes to the background radiation.

We are continuously exposed to radiation from natural sources. These include cosmic radiation, which varies with altitude and latitude; rock, sand, brick, and concrete containing varying amounts of radioactive minerals; the naturally occurring radionuclides in our bodies such as ¹⁴C and ⁴⁰K; and radioactive progeny from radon gas from the earth.

A homework problem in Chapter 17 describes how ¹⁴C is used to determine the age of organic remains.

Problem 58 One way to determine the age of biological remains is “carbon-14 dating.” The common isotope of carbon is stable ¹²C. The rare isotope ¹⁴C decays with a half-life of 5,370 yr. ¹⁴C is constantly created in the atmosphere by cosmic rays. The equilibrium between production and decay results in about 1 of every 10¹² atoms of carbon in the atmosphere being ¹⁴C, mostly as part of a CO₂ molecule. As long as the organism is alive, the ratio of ¹²C to ¹⁴C in the body is the same as in the atmosphere. Once the organism dies, it no longer incorporates ¹⁴C from the atmosphere, and the number of ¹⁴C nuclei begins to decrease. Suppose the remains of an organism have one ¹⁴C for every 10^{13 12}C nuclei. How long has it been since the organism died?

¹⁵C

Isotopes of carbon with even more neutrons can be created in the lab, but they have very short half-lives. For instance, ¹⁵C has a half-life of only 2.4 seconds

¹¹C

Isotopes that have fewer neutrons that found in the stable form often decay by  β⁺ emission, also known as positron decay. A proton transforms into a neutron, with the emission of a positron (an anti-electron). These nuclei can also decay by electron capture (an electron is captured by the nucleus, where it combines with a proton to produce a neutron), however most light elements such as carbon preferentially undergo β⁺ decay rather than electron capture. ¹¹C has a half-life of 20 minutes, and decays by positron emission. It is sometimes used for PET imaging, described in Chapter 17 of IPMB.

If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence. This simplifies the attenuation correction, because the total attenuation for both photons is the same for all points of emission along each γ ray through the body (see Problem 54). Positron emitters are short-lived, and it is necessary to have a cyclotron for producing them in or near the hospital.

Many PET applications use ¹¹C-acetate or ¹¹C-choline, which is administered to the patient. Imaging where the ¹¹C decays provides information about where carbon uptake occurs.

¹⁰C

Lighter isotopes of carbon, such as ¹⁰C, decay quickly; the half-life of ¹⁰C is 19 seconds. Such short half-lives make these isotopes difficult to use in PET imaging, even thought they are positron emitters.


By discussing the isotopes of carbon, we survey many of the most important ideas of nuclear physics, particularly those relevant to nuclear medicine.

From Neuron to Brain

From Neuron to Brain,
by Stephen Kuffler and John Nicholls.

In 1982, when I was accepted into graduate school at Vanderbilt University, I already knew that I wanted to study with John Wikswo, who was measuring the magnetic field of nerve axons. There was just one problem: I didn’t know how nerves worked. So I asked John to recommend some books that would get me up-to-speed before I arrived in Nashville. One text he suggested was From Neuron to Brain, by Stephen Kuffler and John Nicholls. The book taught me the basics of nerve electrophysiology, and allowed me to be (more-or-less) ready to go when I showed up at Vanderbilt.

From Neuron to Brain is now in its 5th edition. I obtained a copy through interlibrary loan, and I’m delighted to say that it still looks to be a great neuroscience textbook. One change is that the authors are different. Kuffler died in 1980, but Nicholls has carried on, now with 5 coauthors (don’t you just hate it when a fine textbook adds additional “coauthors” in later editions!). The Preface to the 5th edition begins

When the First Edition of our book appeared in 1976, its preface stated that our aim was “…to describe how nerve cells go about their business of transmitting signals, how the signals are put together, and how out of this integration higher functions emerge. The book is directed to the reader who is curious about the workings of the nervous system but does not necessarily have a specialized background in biological sciences. We illustrate the main points by selected examples…”

This new, Fifth Edition has been written with the same aim in mind but in a very different context. When the First Edition appeared there were hardly any books, and only a few journals devoted to the nervous system. The extraordinary advances in molecular biology, genetics, and immunology had not been applied to the study of nerve cells or the brain, and the internet was not available for searching the literature. The explosion of knowledge since 1976 means that even though we still want to produce a readable narrative, the topics that have to be addressed and the number of pages have increased. Inevitably, descriptions of certain older experiments have had to be jettisoned, even though they still seem beautiful. Nevertheless, our approach continues to be to follow ideas from their conception to the latest developments. To this end, in this edition we have retained descriptions of classical experiments as well as the newest findings. In this way we hope to present key lines of research of interest for practicing research workers and teachers of neurobiology, as well as for readers who are not familiar with the field.

I’ve always believed the From Neuron to Brain did a great job describing the Hodgkin and Huxley model, a topic that Russ Hobbie and I cover in Chapter 6 of the 4th edition of Intermediate Physics for Medicine and Biology. Many of the key figures from the Hodgkin and Huxley papers are redrawn in a uniform, crisp, elegant style. Some pictures I remember from the first edition, such as the photos of Hodgkin and Huxley, but others—like the illustrations of the detailed structures of potassium and sodium ion channels—are obviously new. Someone put a tremendous amount of time and effort into creating an outstanding collection of illustrations and figures, all with an appropriate and effective use of color, clearly labeled axes, with an uncluttered and simple style. Bravo!

From Neuron to Brain uses little math, and the few equations that do appear are most often just presented, not derived. Those wanting to understand the mathematical basis of the Hodgkin-Huxley model would be well advised to keep a copy of IPMB nearby as you read From Neuron to Brain. Conversely, readers of IPMB who have a weak background in biology might want to keep a copy of From Neuron to Brain close at hand as the work their way through IPMB (especially Chapters 6-9). The two books are complimentary. You won’t find the cable equation written down, much less analyzed, in From Neuron to Brain. But with those gorgeous figures to look at, you may not notice the lack of math.

I really like two other features of From Neuron to Brain. They have an extensive glossary at the back of the book, defining important terms. When I was first learning nerve electrophysiology to prepare for graduate school, one of my biggest obstacles was the vocabulary. Biologists use strange words. I suppose it was more difficult back in those days because we didn’t have Google and Wikipedia (how did we get anything done?), but even today I still appreciate having the glossary handy. Also present in the first edition, and still there now, is an extensive bibliography. Perhaps the beginning student doesn’t refer to the bibliography much, but when you really start digging deep into a subject you want to consult the original papers. As Russ and I work on updating IPMB for the 5th edition, there is always a tension between citing older classic papers and adding new modern ones. From Neuron to Brain has an interesting mix of the new and the old. They provide over 60 pages of citations in small font; I estimate about 40 references per page, for something like 2400 articles. Now that’s a bibliography!

Readers of IPMB looking for more details about nerve electrophysiology will find the 5th edition of From Neuron to Brain to be a valuable text. I’m not familiar with the competing neuroscience textbooks, but I would be surprised if they’re all of this high quality.

Osmosis and the Kidneys

Physics of the Body,
by Cameron, Skofronick, and Grant.

One textbook that covers much of the same material as the 4th edition of Intermediate Physics for Medicine and Biology—but at a somewhat lower level—is Physics of the Body, by John Cameron, James Skofronick and Roderick Grant. They have chapter titles such as “Physics of the Skeleton,” “Physics of the Ear and Hearing,” and “Physics of the Lungs and Breathing.” They apparently didn’t have the expertise among the three coauthors to write a chapter on the “Physics of the Kidneys,” so they recruited an outside author familiar with both physics and the renal system to write it for them. That author is none other than Russ Hobbie. In their preface they write

Emeritus Professor Russell Hobbie of the University of Minnesota, the author of the more advanced text Intermediate Physics for Medicine and Biology, kindly contributed Chapter 6 [“Osmosis and the Kidneys”] on the physics of osmosis as it relates to fluid transport across membranes in the body. He also contributed to the revision of Chapter 9 [“Electrical Signals from the Body”]. His cooperation is greatly appreciated.

Russ’s chapter covers some of the same topics as in Chapters 4 and 5 of IPMB, such as diffusion and osmotic pressure. However, his Section 6.4 goes into more detail about the anatomy and physiology of the kidney that we do in IPMB. Here’s an excerpt.

The kidneys excrete much of the body’s metabolic waste products—except carbon dioxide and some water which leave through the lungs. They also regulate the concentration of most chemicals in the blood plasma. Each kidney contains over 1 million nephrons. Each nephron is a complete urine-forming unit. Figure 6.5 shows the kidneys and the ureters through which urine flows to the urinary bladder. Figure 6.6 shows a magnified view of a nephron.

Figure 6.7 shows the essential functioning parts of the nephron. Blood from the renal artery passes first by a membrane in the glomerulus, where a large amount of fluid—about 250 ml per minute (~1 cup)—passes through the basement membrane of the glomerulus. This process is called filtration. Careful measurements of dog kidneys using radioactively tagged solute molecules of different radii suggest that the filtration is by pores of 5 nm radius in the basement membrane. The filtration rate is controlled by valves which control the rate of blood flow through the glomerulus and the pressure drop across the glomerular basement membrane. Substances with a molecular weight of 5000 or less pass easily through the membrane with the water. Most proteins, which have a molecular weight of 69,000 or more, do not pass through the pores and remain in the blood. The filtrate then passes through the tubules, where 99% of it is reabsorbed (if it were not reabsorbed, we would void 360 liters of urine per day [!]). The other 1% passes into the collecting system as urine. Unwanted substances are not reabsorbed, so their concentration in the urine increases. Creatinine, a metabolic waste product, and sucrose are not reabsorbed at all. About half of the urea, a nitrogenous product of protein metabolism, is reabsorbed.

One interesting appendix I found when thumbing through Physics of the Body is the “Standard Man.”

In medical physics, where we are concerned with the anatomy and physiology of humans, it is convenient to define the physical characteristics of a “standard man.” While the standard man is nonexistent, the following somewhat arbitrary values are useful for simulation and for computational purposes:

Age: 30 yr

Height: 1.72 m (5 ft 8 in)

Mass: 70 kg

Weight: 690 N (154 lb)

Surface area: 1.85 m²

Body core temperature: 37.0 C

Body skin temperature: 34.0 C

Heat capacity: 3.6 kJ/kg C (0.86 kcal/kg C)

Basal metabolism: 44 W/m² (38 kcal/m² hr, 70 kcal/hr, 1680 kcal/day)

Heart rate: 70 beats/min

Blood volume: 5.2 liters

Cardiac output: 5 liters/min

Blood pressure—systolic: 16 kPa (120 mm Hg)

Blood pressure—diastolic: 10.5 kPa (80 mm Hg)

Breathing rate: 15/min

O₂ consumption: 0.26 liter/min

CO₂ production: 0.21 liter/min

Total lung capacity: 6 liters

Tidal volume: 4.8 liters

Lung dead space: 0.15 liters

John Cameron, who passed away in 2005, was a giant in the field of medical physics. He was one of the founders of the well-known medical physics program at the University of Wisconsin. James Skofronick is emeritus professor in the department of physics at Florida State University. Roderick Grant is an emeritus professor in the department of physics and astronomy at Denison University.

How Well Does a Three-Sphere Model Predict Positions of Dipoles in a Realistically Shaped Head?

When I worked at the National Institutes of Health, I collaborated with Susumu Sato, a neurophysiologist interested in electroencephalography and magnetoencephalography. One of Sato’s goals was to develop methods to localize the source of epileptic seizures in the brain. In a small percentage of patients, these seizures cannot be controlled by drugs and are severe enough to be debilitating. In such cases, the best alternative is surgery: remove the region of the brain where the seizure originates, and you stop the seizures. Obviously, in these patients the surgeon must know what part of the brain to remove, and the more accurately that is known the better. Ideally, you want to localize the source using a noninvasive procedure such as electroencephalography. One way to model the sources of electrical activity in the brain is as a single dipole source. Russ Hobbie and I discuss dipoles and the EEG in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology.

Much can be learned about the brain by measuring the electric potential on the scalp surface. Such data are called the electroencephalogram (EEG). Nunez and Srinivasan have written an excellent book about the physics of the EEG. We briefly examine the topic here. The EEG is used to diagnose brain disorders, to localize the source of electrical activity in the brain in patients who have epilepsy, and as a research tool to learn more about how the brain responds to stimuli (“evoked responses”) and how it changes with time (“plasticity”). Typically, the EEG is measured from 21 electrodes attached to the scalp according to the “10–20 system” (Fig. 7.34). A typical signal from an electroencephalographic electrode is shown in the top panel of Fig. 11.38. One difficulty in interpreting the EEG is the lack of a suitable reference electrode. None of the 21 electrodes in Fig. 7.34 qualifies as a distant ground against which all other potential recordings can be measured. One way around this difficulty is to subtract from each measured potential the average of all the measured potentials. In the problems, you are asked to prove that this “average reference recording” does not depend on the choice of reference electrode; it is a reference independent method.

The reference is to Paul Nunez’s book Electric Fields of the Brain (Oxford University Press, 2005), which is a great starting point to learn about the physics of the EEG.

Sato wanted to localize the dipole as accurately as possible, even if that meant moving beyond the three-sphere model. Therefore, I was recruited to write a computer program to solve the EEG problem for a realistically-shaped head. This was not easy, because no software existed at that time for numerically solving the electric potential produced by a dipole in the brain when it is not spherical (at least, Sato and I didn’t have access to such software). I used a boundary element method to perform the calculation. I needed information about the shape of the skull, scalp, and brain surfaces, and I remember painstakingly digitizing those surfaces by hand from magnetic resonance images, and then tessellating the surfaces with triangles. Our resulting image of the brain graced the cover of the journal Electroencephalography and clinical Neurophysiology for several years.

The cover of Electroencephalography
and Clinical Neurophysiology.

This research culminated in a paper published almost exactly 20 years ago: Roth, B. J., M. Balish, A. Gorbach and S. Sato, 1993, “How well does the three-spheres model predict dipoles in a realistically-shaped head?” Electroenceph. clin. Neurophysiol., Volume 87, Pages 175–184. The introduction of the paper is presented below, with references removed.

Electroencephalographic data, such as interictal spikes and evoked responses, are increasingly analyzed using the moving dipole method. The source of the EEG activity is represented as one or more dipoles within the brain; their location, orientation and strength are determined using an iterative least-squares algorithm to fit the calculated potential to the measured EEG data. Although the dipole approximation is an oversimplification, it is a convenient representation of the complex cortical sources. Most often, the potential produced by a dipole is calculated using the 3-sphere model. In this model the brain, skull and scalp are represented as concentric, spherical shells that differ in conductivity. More computationally demanding models use a realistically shaped head; the electrical potential produced by a dipole source is computed either by solving a system of integral equations governing the potential on the brain, skull, and scalp surfaces or by using a finite element model of the head.

In this paper, we compare the 3-sphere model to a realistically shaped head model, in which the brain, skull and scalp surfaces are obtained from magnetic resonance images. We consider a dipole in the temporal or frontal lobe of the brain, and perform a forward calculation using the realistically shaped head model to determine the potential at the 10-20 electrode positions. We then use these data to predict the dipole position by performing an inverse calculation with the 3-sphere model. The average difference between the original and predicted dipole positions is about 2 cm, though differences as large as 4 cm are seen under certain circumstances. Our results are particularly significant for localization of EEG sources of epileptic spikes, which commonly lie in the temporal and frontal lobes.

Medical Physics Qualifying Exams

We have a Medical Physics PhD program here at Oakland University, and this was the week we administered the oral qualifying exam to our current crop of graduate students. Happily, they all passed. In August they also took a battery of written exams about theoretical physics, mathematical methods, and biophysical sciences (Physics, Math, and Biology for short). I have mentioned these exams before in this blog. We consider them to be a common core that our graduate students are expected to master.

These exams do not require knowing extremely advance material, but they do cover a broad range of topics. I take them to be a minimum that our students must know, rather than a target they should aim for. A student who has a strong undergraduate background in physics, math and biology should be able to survive. Some of the more advanced homework problems and examples from the 4th edition of Intermediate Physics for Medicine and Biology sometimes make their way onto these exams.

Let me add a few words about our PhD program. It is aimed at producing research students who can apply physics to medical and biological problems, rather than preparing students for traditional medical physics positions in a hospital. We are not CAMPEP accredited, because that accreditation is mainly for programs aimed narrowly at producing clinical medical physicists. Our students get broad training in both mathematics and medicine, and in both physics and physiology. Their depth comes from doing their research dissertation. After graduating, they go on to a variety of positions in academia, industry, and research laboratories.

Readers of IPMB who want to see how well they would do on our qualifying exam can find over ten years of the written exams at https://files.oakland.edu/users/roth/web/qualifierexams.htm. I have four reasons for posting these exams on the web. First, I assume the exams, or at least some of the questions from them, would make the rounds among our graduate students, or at least among some subset of the graduate students, and I would rather they all have equal access. Second, I am often asked to provide guidance and suggestions as to what specific topics might be on these exams (I admit, all of physics, math, and biology is a lot to master), and my answer is have them look at the previous exams. Third, it can be a useful recruiting tool; if a potential applicant wants to know what they are expected to master to succeed in our program, I can send them to the old exams and be confident that they realize what they are getting into. Fourth, failing our qualifying exam is a serious issue. The students only get two tries, and then they must leave the program. I prefer to give a student some direction and help rather wondering if the exam was unfair as I tell them that they failed. The downside to posting these exams is that I need to keep coming up with new problems each year. While some identical problems from old exams appear on later exams, I try to minimize this. So, writing the exams (which I do largely myself, although with input from the other faculty in the program) is a little harder than it otherwise would be. One useful side effect of posting the exams is that they are all “out there” available to anyone, including our dear readers of IPMB. So, feel free to use them as you wish. Sorry, but I don’t have solutions I can send you.

In addition to these written exams, each student must stand in front of a group of (intimidating?) faculty and answer questions about “everything”: all the topics from the written exams, plus questions related to their research, and to any other part of physics, mathematics, or biology that might strike the questioner’s fancy. This grilling is what our students went through on Wednesday, successfully. I think the students fear this part of the exam most of all, but I believe they grow from the experience.

Congratulations to this year’s students. I hope readers of IPMB find these exams useful.