Div, Grad, Curl, and All That

Russ Hobbie and I assume that readers of the 4th edition of Intermediate Physics for Medicine and Biology know the basics of calculus (our preface states that “calculus is used without apology”). We even introduce some concepts from vector calculus, such as the divergence, gradient, and curl. Although these vector derivatives are crucial for understanding topics such as diffusion and electricity, many readers may be unfamiliar with them. These functions are even more complicated in curvilinear coordinate systems, and in Appendix L we summarize how to write the divergence, gradient, curl, and Laplacian in rectangular, cylindrical, and spherical coordinates.

Div, Grad, Curl, and All That,
by H. M. Schey.

When I was a young physics student at the University of Kansas, Dr. Jack Culvahouse gave me a book that helped explain vector calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, by H. M. Schey. For me, this book made clear and intuitive what had been confusing and complicated. By defining the divergence and curl in terms of surface and line integrals, I suddenly could understand what these seemingly random collections of partial derivatives meant. One can hardly make sense of Maxwell’s equations of electromagnetism without vector calculus (try reading a textbook from Maxwell’s era before vector calculus was invented if you don't believe me). In fact, Schey introduces vector calculus using electromagnetism as his primary example:

In this text the subject of vector calculus is presented in the context of simple electrostatics. We follow this procedure for two reasons. First, much of vector calculus was invented for use in electromagnetic theory and is ideally suited to it. This presentation will therefore show what vector calculus is and at the same time give you an idea of what it's for. Second, we have a deep-seated conviction that mathematics—in any case some mathematics—is best discussed in a context that is not exclusively mathematical. Thus, we will soft-pedal mathematical rigor, which we think is an obstacle to learning this subject on a first exposure to it, and appeal as much as possible to physical and geometric intuition.

For readers of Intermediate Physics for Medicine and Biology who get stuck when we delve into vector calculus, I suggest setting our book aside for a few days (but only a few!) to read Div, Grad, Curl, and All That. Not only will you be able to understand our book better, but you’ll find this background useful in many other fields of physics, math, and engineering.

Sigma Xi

Here at Oakland University, this Tuesday, March 31, is our annual Sigma Xi lecture (4 P.M. in 201 Dodge Hall of Engineering). Each year, we invite a leading scientist to OU to give a lecture for a general audience. This year Dr. Vicki Chandler, Chief Program Director of the Gordon and Betty Moore Foundation, will give a talk about “Epigenetic Silencing Across Generations.” (The term “epigenetic gene silencing” describes the switching off of a gene by a mechanism other than genetic modification. That is, a gene that would be expressed, or turned on, under normal circumstances is switched off by machinery in the cell.)

For six years, I served as the president of the Oakland University chapter of Sigma Xi, the Scientific Research Society. As readers of the 4th edition of Intermediate Physics for Medicine and Biology become biomedical researchers, they should consider joining Sigma Xi. I joined as a graduate student at Vanderbilt University.

Sigma Xi is an international, multidisciplinary research society whose programs and activities promote the health of the scientific enterprise and honor scientific achievement. There are nearly 60,000 Sigma Xi members in more than 100 countries around the world. Sigma Xi chapters, more than 500 in all, can be found at colleges and universities, industrial research centers and government laboratories. The Society endeavors to encourage support of original work across the spectrum of science and technology and to promote an appreciation within society at large for the role research has played in human progress.

The mission of Sigma Xi is “to enhance the health of the research enterprise, foster integrity in science and engineering, and promote the public's understanding of science for the purpose of improving the human condition.” As a member of Sigma Xi, you automatically receive a subscription to American Scientist, the award-winning illustrated magazine of science and technology. I particularly enjoy Henry Petroski’s monthly essay on topics in engineering, and the book reviews are outstanding. The magazine alone is worth the cost of membership. Another benefit that I look forward to each day is Science in the News, a free e-mail bulletin featuring top science and technology stories. Sigma Xi also has an annual meeting, including a student research conference. Last year, the meeting was November 20–23 in Washington, DC. The society is a strong advocate of scientific research, and is worthy of support.

Finally, you have to love the society’s motto: “Companions in Zealous Research.”

The West-Brown-Enquist Model for Allometric Scaling

Chapter 2 of the 4th edition of Intermediate Physics for Medicine and Biology ends with a section on “Food Consumption, Basal Metabolic Rate, and Scaling.” Here Russ Hobbie and I discuss the famous “3/4-power law” (also known as Kleiber’s law), which relates the metabolic rate R (in Watts) to the body mass M (in kg) by the equation R = 4.1 M^0.751 (Eq. 2.32c in our book). We conclude the section by writing

A number of models have been proposed to explain a 3/4-power dependence [McMahon (1973);  Peters (1983); West et al. (1999); Banavar et al. (1999)]. West et al. argue that the 3/4-power dependence is universal: they derive it from a model that supplies nutrients through a branching network that reaches all parts of the organism, minimizes the energy required for distribution, and ends in capillaries (or terminal xylem in plants) that are all the same size. Whether it is universal is still debated [Kozlowski and Konarzewski (2004)]. West and Brown (2004) review quarter-power scaling in a variety of circumstances.

When we wrote this paragraph, the origin of the 3/4th power law was still being hotly debated in the literature. Readers of Intermediate Physics for Medicine and Biology might like an update.

First, this work is highly cited. West, Brown, and Enquist’s first paper in Science (“A General Model for the Origin of Allometric Scaling Laws in Biology,” Volume 276, Pages 122–126, 1997; not cited in our book) now has over 1000 citations. Their second paper, which we list in the references at the end of Chapter 2, has nearly 400 citations. The paper by Banavar, Maritan and Rinaldo cited in Chapter 2 has over 200 citations. Clearly, these studies have had a major impact on the field.

Second, the work has generated quite a bit of discussion in the press. The December 2008 issue of The Scientist has an article by Bob Grant titled “The Powers That Might Be” about West and his colleagues and how they have coped with criticisms of their work. An interview with Geoffrey West can be found at physicsworld.com, and one with Brian Enquist at www.in-cities.com. In 2004, John Whitfield published a feature in the open access journal PLOS Biology reviewing the field (“open access” means that anyone can access the paper over the internet, without the need for a journal subscription).

Third, several recent papers in scientific journals have addressed this topic. Savage et al. have analyzed what they refer to as the WBE model in an article appearing in the open access journal PLOS Computational Biology (Volume 4, Article e1000171, 2008). The authors’ summary states

The rate at which an organism produces energy to live increases with body mass to the 3/4 power. Ten years ago West, Brown, and Enquist posited that this empirical relationship arises from the structure and dynamics of resource distribution networks such as the cardiovascular system. Using assumptions that capture physical and biological constraints, they defined a vascular network model that predicts a 3/4 scaling exponent. In our paper we clarify that this model generates the 3/4 exponent only in the limit of infinitely large organisms. Our calculations indicate that in the finite-size version of the model metabolic rate and body mass are not related by a pure power law, which we show is consistent with available data. We also show that this causes the model to produce scaling exponents significantly larger than the observed 3/4. We investigate how changes in certain assumptions about network structure affect the scaling exponent, leading us to identify discrepancies between available data and the predictions of the finite-size model. This suggests that the model, the data, or both, need reassessment. The challenge lies in pinpointing the physiological and evolutionary factors that constrain the shape of networks driving metabolic scaling.

In another paper, published in the December 2006 issue of Physics of Life Reviews (Volume 3, Pages 229–261), de Silva et al. write that

One of the most pervasive laws in biology is the allometric scaling, whereby a biological variable Y is related to the mass M of the organism by a power law, Y = Y0M^b, where b is the so-called allometric exponent. The origin of these power laws is still a matter of dispute mainly because biological laws, in general, do not follow from physical ones in a simple manner. In this work, we review the interspecific allometry of metabolic rates, where recent progress in the understanding of the interplay between geometrical, physical and biological constraints has been achieved.

For many years, it was a universal belief that the basal metabolic rate (BMR) of all organisms is described by Kleiber’s law (allometric exponent b = 3/4). A few years ago, a theoretical basis for this law was proposed, based on a resource distribution network common to all organisms. Nevertheless, the 3/4-law has been questioned recently. First, there is an ongoing debate as to whether the empirical value of b is 3/4 or 2/3, or even nonuniversal. Second, some mathematical and conceptual errors were found [in] these network models, weakening the proposed theoretical arguments. Another pertinent observation is that the maximal aerobically sustained metabolic rate of endotherms scales with an exponent larger than that of BMR. Here we present a critical discussion of the theoretical models proposed to explain the scaling of metabolic rates, and compare the predicted exponents with a review of the experimental literature. Our main conclusion is that although there is not a universal exponent, it should be possible to develop a unified theory for the common origin of the allometric scaling laws of metabolism.

Now, five years after we included the topic in Intermediate Physics for Medicine and Biology, the controversy continues. It makes for a wonderful example of how ideas from fundamental physics can elucidate biological laws, and a warning about how complicated and messy biology can be, limiting the application of simple models. I can't tell you how this debate will ultimately be resolved. But it provides a fascinating case study in the interaction of physics and biology.

The Discovery of Technetium

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the biomedical properties of the element technetium (Tc), which plays an important role in nuclear medicine.

The most widely used isotope is 99m-Tc. As its name suggests, it does not occur naturally on earth, since it has no stable isotopes... [It decays by emitting] a nearly monoenergetic 140-keV gamma ray. Only about 10% of the energy is in the form of nonpenetrating radiation. The isotope is produced in the hospital from the decay of its parent, 99-Mo, which is a fission product of 235-U and can be separated from about 75 other fission products. The 99-Mo decays to 99m-Tc.

The Search For the Elements,
by Isaac Asimov.

Technetium has an interesting history. When Dmitri Mendeleev proposed the periodic table, he predicted that holes in his table were missing elements that had not yet been discovered. In The Search for the Elements, Isaac Asimov writes

The first elements produced [by artificial transmutation] was the missing number 43 [technetium]. A claim to discovery of this element had been made in 1925 by Noddack, Tacke, and Berg, the discoverers of rhenium. They had named element number 43 “masurium” (after a district in East Prussia). But no one else was able to find “masurium” in the same source material, so their supposed discovery had remained a question mark. It was, in fact, just a mistake. In 1937 Emilio Gino Segre of Italy, an ardent hunter for the element, identified the real number 43.

[Ernest O.] Lawrence had bombarded a sample of molybdenum (element number 42) with protons accelerated in his cyclotron. Finally he got some radioactive stuff which he sent to Segre in Italy for analysis. Segre and an assistant, C. Perrier, traced some of the radioactivity to an element which behaved like manganese. Since the missing element 43 belonged in the vacancy in the periodic table next to manganese, they were sure this was it.

It turned out that element number 43 had several isotopes. Oddly, all of them were radioactive. There were no stable isotopes of the element!... Segre named the element number 43 “technetium,” from a Greek work meaning “artificial,” because it was the first element made by man.

Asimov tells the standard history of the discovery of technetium, but recently there has been a new twist to the story. John Armstrong of the National Institute of Standards and Technology (NIST) suggested that maybe masurium really was technetium. In an abstract to a NIST Sigma Xi colloquium in 2000 titled “The Disputed Discovery of Element 43 (Technetium),” Armstrong and P. H. M. Van Assche write

In 1925, Noddack, Tacke and Berg reported discovery of element Z = 43, which they named Masurium, based on line identification of x-ray emission spectra from chemically concentrated residues of various U-rich minerals. Their results were disputed and eventually the discovery of element 43 (Technetium) was generally credited to Perrier and Segre, based on their chemical separation of neutron-irradiated molybdenum in 1937. Using first principles x-ray emission spectral generation algorithms from the N.I.S.T. DTSA spectral processing program, we have simulated the x-ray spectra that would be expected using their likely analytical conditions (from their papers and contemporaneous reports) and the likely residue compositions suggested by Noddack et al. and Van Assche. The resulting spectra are in close agreement with that reported by Noddack et al., place limits on the possible residue compositions, and are supportive of the presence of detectable amounts of element 43 in their sample. Moreover, the calculated mass of element 43 shown in their spectrum is consistent with the amount that would be now expected from the spontaneous fission of U present in the ores they studied. The history of the original masurium/technetium controversy and the means used to reexamine the original record will be presented in this scientific detective story.

Was masurium really technitium? You will have to look at the evidence and decide for yourself. The story certainly is fascinating, and will interest readers of Intermediate Physics for Medicine and Biology.

NCRP Report No. 160

In past entries to this blog, I have reported on a growing controversy over radiation exposure from medical procedures. On December 7, 2007 I described a study by David Brenner and Eric Hall warning that the increased popularity of CT scans, particularly in children, can lead to an increased incidence of cancer. Then, just three weeks ago, I discussed the “Image Gently” website, created to raise awareness in the imaging community of the need to adjust radiation dose when imaging children.

This week the debate intensified, with three simultaneous press releases. On Wednesday, the National Council on Radiation Protection and Measurement (NCRP) issued a new study titled Medical Radiation Exposure of the U.S. Population Greatly Increased Since the Early 1980s. This report, also known as NCRP Report Number 160, updates NCRP Report Number 93, Ionizing Radiation Exposure of the Population of the United States, published in 1987. Readers of the 4th edition of Intermediate Physics for Medicine and Biology may recall that Russ Hobbie and I based much of our discussion in Chapter 16 about the risk of ionizing radiation on Report No. 93. The press release announcing Report No. 160 states that

In 2006, Americans were exposed to more than seven times as much ionizing radiation from medical procedures as was the case in the early 1980s, according to a new report on population exposure released March 3rd by the National Council on Radiation Protection and Measurements (NCRP) at its annual meeting in Bethesda, Maryland. In 2006, medical exposure constituted nearly half of the total radiation exposure of the U.S. population from all sources.

The report triggered an immediate response from the American Association of Physicists in Medicine. Their press release, titled NCRP Report No. 160 on Increased Average Radiation Exposure of the U.S. Population Requires Perspective and Caution, begins

Scientists at the American Association of Physicists in Medicine (AAPM) are offering additional background information to help the public avoid misinterpreting the findings contained in a report issued today by the National Council on Radiation Protection and Measurements (NCRP), a non-profit body chartered by the U.S. Congress to make recommendations on radiation protection and measurements. The report is not without scientific controversy and requires careful interpretation.

Not to be outdone, the American College of Radiology also issued its own press release Wednesday.

A recent National Council on Radiation Protection and Measurements (NCRP) Report (NCRP Report No. 160, Ionizing Radiation Exposure of the Population of the United States) stated that the U.S. population is now exposed to seven times more radiation each year from medical imaging exams than in 1980. The American College of Radiology (ACR), Society for Pediatric Radiology (SPR), Society of Breast Imaging (SBI), and the Society of Computed Body Tomography and Magnetic Resonance (SCBT-MR) urge Americans, including elected officials and medical providers, to understand why this increase occurred, consider the Report’s information in its proper context, and support appropriate actions to help lower the radiation dose experienced each year from these exams.

“It is essential that this Report not be interpreted solely as an increase in risk to the U.S. population without also carefully considering the tremendous and undeniable benefits of medical imaging. Patients must make these risk/benefit decisions regarding their imaging care based on all the facts available and in consultation with their doctors,” said James H. Thrall, MD, FACR, chair of the ACR Board of Chancellors.

Who says medical physics isn’t exciting? Seriously, this is an important topic, and deserves the careful scrutiny of anyone interested in medical physics. As always, I recommend the 4th edition of Intermediate Physics for Medicine and Biology as a good starting point to learn the basic physics that underlies this controversy. And keep coming back to this blog for updates as the debate unfolds.

Hello to the Medical Physics 2 Class at Ball State University

Russ Hobbie and I would like to thank those instructors and students who use the 4th edition of  Intermediate Physics for Medicine and Biology as the textbook for their class. Also, we greatly appreciate those careful readers who find errors in our book and inform us about them. Without our dear readers, all the work preparing the 4th edition would be pointless.

Special thanks go to Dr. Ranjith Wijesinghe, Assistant Professor of Physics and Astronomy at Ball State University in Muncie, Indiana. This semester, Ranjith is teaching APHYS 316 (Medical Physics 2) using Intermediate Physics for Medicine and Biology. As he prepares his class lectures, Ranjith emails me all the mistakes he finds in our book, which I dutifully add to the errata. I can keep track of what the class is covering by the location of the errors Ranjith finds. In mid January the class was studying Fourier series, and he found a missing “sin” in Eq. 11.26d. By early February they were analyzing images, and Ranjith noticed some missing text in the figure associated with Problem 12.7. Then in mid February they began studying ultrasound, and eagle-eyed Ranjith emailed me that the derivative in Eq. 13.2 should be a partial derivative. I’m expecting some newly-discovered typo in Chapter 14 next week.

Electric Fields of the Brain:
The Neurophysics of EEG,
by Paul Nunez.

Ranjith is an old friend of mine. We were graduate students together at Vanderbilt University in the late 1980s, and both worked in the lab of John Wikswo. I took care of the crayfish (which have some giant axons that are useful for studying action currents) and Ranjith looked after the frogs (whose sciatic nerve is an excellent model for analyzing the compound action potential). After leaving Vanderbilt, Ranjith was a postdoc at Tulane University with Paul Nunez, an expert in electroencephalography and author of the acclaimed textbook Electric Fields of the Brain: The Neurophysics of EEG. While a member of Nunez’s group, Ranjith coauthored several papers, including “EEG Coherency.1. Statistics, Reference Electrode, Volume Conduction, Laplacians, Cortical Imaging, and Interpretation at Multiple Scales” in the journal Electroencephalography and Clinical Neurophysiology (Volume 103, Pages 499–515, 1997). According to Google Scholar, this landmark paper has been cited 277 times, which is quite an accomplishment (and is more citations than my most cited paper has).

I hope Ranjith keeps on sending me errors he finds, and I encourage other careful readers to do so too. And a big HELLO! to Ball State students taking Medical Physics 2. The true measure of a textbook is what the students think of it. I hope you all find it useful, and best of luck to you as the end of the semester approaches. Don’t give Dr. Wijesinghe too hard a time in class. If he finishes early one day and you have a few minutes to spare, ask him for some old stories from graduate school. He has a few, if he will tell you!

Allan Cormack

This Monday (February 23) will mark the 85th anniversary of the birth of Allan Cormack (1924–1998), who won the 1979 Nobel Prize in Physiology or Medicine (along with Godfrey Hounsfield) for the “development of computer assisted tomography.”

Imagining the Elephant:
A Biography of Allan MacLeod Cormack,
by Christopher Vaughan.

Last year Christopher Vaughan published a book titled Imagining the Elephant: A Biography of Allan MacLeod Cormack. A book review by Reginald Greene in the December 18 issue of the New England Journal of Medicine (Volume 359, Pages 2735–2736) states that

This brief book is a fascinating biography. The author, Christopher Vaughan, warmly sketches Cormack as a quietly gregarious man, traces his Scottish parentage and antecedents, follows his schooling and family life in South Africa, and mines the origins of his research into CT [Computed Tomography] at the University of Cape Town, latter at Cambridge University, and during his subsequent years in the United States at Tufts University and at the Harvard University Cyclotron Laboratory.

I haven’t read Vaughan's book yet, but it’s high on my list of things to do. You can learn more about Cormack online at the website published by the American Physical Society. For those of you who prefer to go straight to the original source, take a look at Cormack’s two highly cited papers, both in the Journal of Applied Physics: “Representation of a Function by Its Line Integrals, with Some Radiological Applications” (Volume 34, Pages 2722–2727, 1963), and “Representation of a Function by Its Line Integrals, with Some Radiological Applications. II” (Volume 35, Pages 2908–2913, 1964). Warning: these papers are highly mathematical. For those who would rather not wade through the math (and shame on you for that attitude!), I recommend looking at Section 4 (An Experimental Test) of the second paper, to see perhaps the first CT scan ever made, of an aluminum phantom in air. Or, see Chapter 12 of the 4th edition of Intermediate Physics for Medicine and Biology for a discussion of the numerical algorithms underlying tomography.

Allan Cormack is a role model for all physicists (or physics students) who hope to make important contributions to medicine.

Image Gently

The December 7, 2007 entry of this blog addressed a controversy over the safety of computed tomography scans, particularly for children. In response to these concerns, the Society for Pediatric Radiology, the American Association of Physicists in Medicine, the American College of Radiology, and the American Society of Radiologic Technologists have banded together to establish the Alliance for Radiation Safety in Pediatric Imaging. Its “image gently” website states that

The Alliance for Radiation Safety in Pediatric Imaging—the Image Gently Alliance—is a coalition of health care organizations dedicated to providing safe, high quality pediatric imaging nationwide. The primary objective of the Alliance is to raise awareness in the imaging community of the need to adjust radiation dose when imaging children.

The ultimate goal of the Alliance is to change practice.

The Alliance has chosen to focus first on computed tomography (CT) scans. The dramatic increase in the number of pediatric CT scans performed in the United States in the past five years and the rapid evolution, change and availability of CT technology and equipment well justify this Alliance strategy.

Image Gently offers reasonable recommendations to parents, pediatricians, radiologic technologists, and medical physicists about the risks and benefits of CT scans. While asserting that “there’s no question: CT helps us save kids’ lives!,” it nevertheless provides specific suggestions for reducing radiation dose, such as: “child size the kVp and mA”; “one scan (single phase) is often enough”; and “scan only the indicated area”. Image gently offers such calm, science-based advice on a subject often dominated by emotion and a misunderstanding of risk assessment. You won’t find much physics at the image gently website, but you will benefit from a case study in how to use physics to help patients without scaring them (or hurting them) in the process.

After exploring the image gently website, if you want to know more about how computed tomography works, or about the biological effects of radiation, see Chapter 16 in the 4th edition of Intermediate Physics for Medicine and Biology.

Darwin Day

The Origin of Species,
by Charles Darwin.

Thursday, Feb 12, is Darwin Day: the 200th anniversary of Charles Darwin’s birth. This year is special because it also marks the sesquicentennial of Darwin’s masterpiece The Origin Of Species.

Although Darwin Day is primarily a time to celebrate biology, physics plays two important roles in Darwin’s theory of evolution. First, physics constrains evolution. Natural selection has produced an amazing variety and diversity of organisms, but each and every one obeys the laws of physics. You can dream up all sorts of organisms in your imagination, but some just won't work. Readers of the 4th edition of Intermediate Physics for Medicine and Biology will learn about several of these constraints. For instance, in Chapter 2 Russ Hobbie and I discuss scaling (see my blog entry from August 8, 2008 for an earlier discussion of scaling). One can imagine a giant spider a hundred feet high with thin spider legs, but physics won’t allow this: the spider would be crushed under its own weight (weight scales as the volume, but the strength of the legs scale as the cross-sectional area, so the larger the spider the more difficult it would be to support the weight). In Chapter 4 we show that diffusion is an effective way to transport molecules over short distances, but is a poor method over long distances. One can envision a three-story high single cell—a giant amoeba—but if that cell depends on diffusion to obtain oxygen and get rid of carbon dioxide, it will not survive. So, physics limits biological evolution, and these limitations provide important insights into why animals are designed the way they are.

The second role of physics in the study of evolution comes from the interplay between evolution and astronomy. A famous example is the idea proposed by physicist Luis Alverez that an asteroid slammed into the earth 65 million years ago, leading to the death of the dinosaurs and many other species. I’d like to highlight a different example, in part because it’s new and less familiar, and in part because it’s been developed by researchers in the Department of Physics at the University of Kansas, my undergraduate alma mater (go jayhawks!). Professor Adrian Melott and his colleagues have proposed that gamma-ray bursts may have caused other mass extinctions. In the January 2004 issue of the International Journal of Astrobiology (Volume 3, Pages 55–61), Melott et al. write

Gamma-ray bursts (GRBs) produce a flux of radiation detectable across the observable universe. A GRB within our own galaxy could do considerable damage to the Earth’s biosphere; rate estimates suggest that a dangerously near GRB should occur on average two or more times per billion years. At least five times in the history of life, the Earth has experienced mass extinctions that eliminated a large percentage of the biota. Many possible causes have been documented, and GRBs may also have contributed. The late Ordovician mass extinction approximately 440 million years ago may be at least partly the result of a GRB. A special feature of GRBs in terms of terrestrial effects is a nearly impulsive energy input of the order of 10 s. Due to expected severe depletion of the ozone layer, intense solar ultraviolet radiation would result from a nearby GRB, and some of the patterns of extinction and survivorship at this time may be attributable to elevated levels of UV radiation reaching the Earth. In addition, a GRB could trigger the global cooling which occurs at the end of the Ordovician period that follows an interval of relatively warm climate. Intense rapid cooling and glaciation at that time, previously identified as the probable cause of this mass extinction, may have resulted from a GRB.

On Darwin Day, as you celebrate Charles Darwin and his theory of evolution by natural selection, remember that a knowledge of physics as well as biology is crucial to understanding this important idea. The 4th edition of Intermediate Physics in Medicine and Biology is a good place to obtain the necessary physics background.

Lady Luck

Lady Luck, by Warren Weaver.

At the bottom of page 50 in the 4th edition of Intermediate Physics for Medicine and Biology is a short footnote: “A good book on probability is Weaver (1963).” The reference given at the end of the chapter is to Lady Luck, by Warren Weaver. This book is one of my favorites, and reflects my interest in probability. I particularly enjoyed Weaver’s description (in Chapter 6) of a game that at first glance is counter-intuitive:

Take three identical cards. Make a red mark on both sides of one, a black mark on both side of the second, and mark the third black on one side and red on the other. Mix them up in a hat, pick out a card at random, and put it down on the table without disclosing to yourself or anyone else what color is marked on the concealed side.

Suppose the upper side of the card is marked red. You say to your opponent, “Obviously we are not dealing with the black-black card. That one is clearly eliminated. We definitely have either the red-black card or the red-red card. We shuffled fairly and drew at random, so it is just as likely to be one of these as the other. I will therefore bet you even money that the other side is red.”

It isn’t too hard to find takers, although ... the odds in favor of your bet are not even, but are actually two to one! The catch, of course, is the clause “so it is just as likely to be one as the other.” It is twice as likely that it is the red-red card! Forty years ago, when graduate students had to work for their living, the author used to teach this particular problem, at reasonable rates and using the experimental method, to his college friends.

Weaver is an interesting figure in 20th century mathematics and science, and fits in well with our theme of applying physics to medicine and biology. He was director of the Division of Natural Sciences at the Rockefeller Foundation from 1932 to 1955. According to Wikipedia,

Weaver early understood how greatly the tools and techniques of physics and chemistry could advance knowledge of biological processes, and used his position in the Rockefeller Foundation to identify, support, and encourage the young scientists who years later earned Nobel Prizes and other honors for their contributions to genetics or molecular biology.

When I teach probability (often in the first week of a class on statistical mechanics or quantum mechanics), I like to use the example of playing craps. Weaver analyzes craps in chapter 15 of Lady Luck.

The game of craps furnishes a good example of probability calculations in a gambling game; for craps is sufficiently more complicated than “heads and tails” to raise some nice little problems, but not so complicated (as is bridge, for example) that the calculations are tedious.

It is also interesting for the students, who have visited the casinos and played craps more than I have. (Anyone who understands probability will find gambling at casinos to be financially unwise).

In Intermediate Physics for Medicine and Biology, probability is particularly important in Chapter 3, when Hobbie and I discuss statistical mechanics. Probabilistic ideas also appear throughout the book in the form of the Poisson probability distribution, which we analyze in detail in our Appendix J.