The Framingham Heart Study

The Framingham Heart Study is one of the oldest and most widely cited research studies in the history of medicine. Russ Hobbie and I mention the study briefly In Section 2.4 of the 4th edition of Intermediate Physics for Medicine and Biology, when discussing exponential decay.

Figure 2.8 shows the survival of patients with congestive heart failure for a period of nine years. The data are taken from the Framingham study [McKee et al. (1971)]; the death rate is constant during this period.

The data in Fig. 2.8 is from a paper with over 1400 citations in the scientific and medical literature: P. A. McKee, W. P. Castelli, P. M. McNamara, and W. B. Kannel (1971) “The Natural History of Congestive Heart Failure: The Framingham Study,” New England Journal of Medicine, Volume 285, Pages 1441–1446. The abstract to the paper states

The natural history of congestive heart failure was studied over a 16-year period in 5192 persons initially free of the disease. Over this period, overt evidence of congestive heart failure developed in 142 persons. In almost every five-year age group, from 30 to 62 years, the incidence rate was greater for men than for women. Although the usual etiologic precursors were found, the dominant one was clearly hypertension, which preceded failure in 75 per cent of the cases. Coronary heart disease was noted at an earlier examination in 39 per cent, but in 29 per cent of the cases it was accompanied by hypertension. Precursive rheumatic heart disease, noted in 21 per cent of cases of congestive heart failure, was accompanied by hypertension in 11 per cent. Despite modern management, congestive heart failure proved to be extremely lethal. The probability of dying within five years from onset of congestive heart failure was 62 per cent for men and 42 per cent for women.

A Change of Heart:
How the Framingham Study
Helped Unravel the Mysteries
of Cardiovascular Disease,
by Levy and Brink.

In 2005, Daniel Levy and Susan Brink published A Change of Heart: How the Framingham Heart Study Helped Unravel the Mysteries of Cardiovascular Disease. The book is a fascinating history of this landmark study. Levy (the study’s current director) and Brink (formerly a writer for U.S. News and World Report) write

A turning point in our evolving understanding of heart disease was the establishment of the Framingham Heart Study in 1948. It was a large and ambitious community-based research project unlike anything that had been conducted before. It came at a time of growing awareness that cardiovascular disease was sweeping the country, even slowing down what should have been a steady rise in life expectancy. It was also a time, three years after the end of World War II, when resources from the national treasury, no longer needed for military purposes, could be used for research into the nation’s leading killer….

In light of this ignorance [of how to treat coronary disease], the U.S. government in 1948 made a twenty-year commitment to uncovering the root causes of heart disease. That scientific resolve was sponsored by the U.S. Public Health Service with half a million dollars of start-up funding from Congress. A cadre of physicians, scientists, government officials, and academics—many of whom knew each other from having served together at military hospitals during the war—selected a New England town in which to carry out this national scientific experiment. The Framingham Heart Study turned out to be instrumental in changing the attitudes, if not the behavior, of virtually every American, and it put the otherwise ordinary town of Farmingham, Massachusetts, on the map….

They [the Heart Study researchers] needed the 5209 men and women from Framingham at first, followed by 5124 of their sons and daughters, and now 3500 of their grandchildren who have donated their medical histories to science. It is ironic, perhaps, that this most respected—even beloved—piece of epidemiology centers on the heart, the organ that symbolically aches, breaks, longs, and loves like no other. It took a commitment from thousands of volunteers to make the study a success.

I found Chapter 5 (The People Who Changed America’s Heart: Voices from Framingham) to be particularly inspiring. For instance, they quote Evelyn Langley—housewife, mother, and PTA president—who played an early role in promoting the study among potential participants, and was a participant herself.

Langley’s heart still lies with the Study. “When they call me up and tell me it’s time to come in for an exam, I know I have that ritual to do,” she says. She has made the trip to the clinic twenty-seven times so far. “I am trying to give back to the Heart Clinic [Study] what they have given me. I always feel as if I am part of something bigger than myself. It’s not just for the people who live in this town. Many lives have been saved because of the Heart Study.”

You can learn more about the Framingham Heart Study at the study’s website: http://www.framinghamheartstudy.org. The study is currently funded by the National Heart, Lung, and Blood Institute (part of the National Institutes of Health) and Boston University. Let me finish with a fitting quote from the acknowledgments of A Change of Heart.

This book would not have been possible without the more than fifty years of dedication and commitment from three generations of Framingham Heart Study volunteers. We would like to thank them all for providing a gift to the world that has changed untold millions of lives.

Britton Chance (1913-2010)

Britton Chance died late last year. The website www.brittonchance.org states that

Britton Chance, M.D., Ph.D., D.Sc., for more than 50 years one of the giants of biochemistry and biophysics and a world leader in transforming theoretical science into useful biomedical and clinical applications, died on November 16, 2010, at age 97 in Philadelphia, PA. Dr. Chance had the rare distinction of being the recipient of a National Medal of Science (1974), a Gold Medal in the Olympics (1952, Sailing, Men’s 5.5 Meter Class), and a Certificate of Merit for his sensitive work during World War II.

His obituary in the New York Times describes his early work.

Over a lifetime of research, Dr. Chance focused on the observation and measurement of chemical reactions within cells, tissue and the body. But unlike most researchers, he also had expertise in mechanics, electronics and optics, and a great facility in instrument-building. His innovations helped transform theoretical science into biochemical and biophysical principles, the stuff of textbooks, and useful biomedical and clinical applications.

Early in his career he invented a tool, known as a stopped-flow apparatus, for measuring chemical reactions involving enzymes; it led to the establishment of a fundamental principle of enzyme kinetics, known as the enzyme-substrate complex.

Another obituary, in the December 17 issue of Science magazine, observed that

In his mid-70s, Chance (then emeritus) launched a new field of optical diagnostics that rests on the physics of light diffusion through scattering materials such as living tissue. He showed that scattered near-infrared light pulses could not only measure the dynamics of oxy- and deoxyhemoglobin levels in performing muscles, but also reveal and locate tumors and cancerous tissue in muscles and breast as well as injury in the brain. Because changing patterns of oxy- and deoxyhemoglobin in the brain reflect cognitive activity, the applications of this diagnostic approach widened to include assessing neuronal connectivity in premature babies.

Chance appears in the 4th edition of Intermediate Physics for Medicine and Biology because of his research on light diffusion. In Section 14.4 (Scattering and Absorption of Radiation), Russ Hobbie and I analyze the absorption and scattering coefficients of infrared light, and then give typical values that “are eyeballed from data from various tissues reported in the article by Yodh and Chance (1995),” with the reference being to Yodh, A. and B. Chance (1995) “Spectroscopy and Imaging with Diffusing Light,” Physics Today, March, Pages 34–40.

Then in Sec. 14.5 (The Diffusion Approximation to Photon Transport), we analyze pulsed measurements of infrared light.

A technique made possible by ultrashort light pulses from a laser is time-dependent diffusion. It allows determination of both [the scattering coefficient] and [the absorption coefficient]. A very short (150-ps) pulse of light strikes a small region on the surface of the tissue. A detector placed on the surface about 4 cm away records the multiply-scattered photons. A typical plot of the detected photon fluence rate is shown in Fig. 14.13.

Figure 14.13 is a figure from Patterson, M. S., B. Chance, and B. C. Wilson (1989) “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical Properties,” Applied Optics, Volume 28, Pages 2331–2336, which has been cited over 1000 times in the scientific literature.

Finally, in Sec. 14.6 (Biological Applications of Infrared Scattering), we reproduce a figure from the Physics Today article by Yodh and Chance, which shows the absorption coefficient for water, oxyhemoglobin and deoxyhemoglobin.

The greater absorption of blue light in oxygenated hemoglobin makes oxygenated blood red…The wavelength 800 nm at which both forms of hemoglobin have the same absorption is called the isosbestic point. Measurements of oxygenation are made by comparing the absorption at two wavelengths on either side of this point.

This property of infrared absorption of light is the basis for pulse oximeters that measure oxygenation. Not all measurements of blood oxygen use pulsed light. Russ and I cite one of Chance’s papers that uses a continuous source: Liu, H., D. A. Boas, Y. Zhang, A. G. Yodh, and B. Chance (1995) “Determination of Optical Properties and Blood Oxygenation in Tissue Using Continuous NIR Light,” Physics in Medicine and Biology, Volume 40, Pages 1983–1993. A fourth of Chance’s paper that we include in our references is Sevick, E. M., B. Chance, J. Leigh, S. Nioka, and M. Maris (1991) “Quantitation of Time- and Frequency-Resolved Optical Spectra for the Determination of Tissue Oxygenation,” Analytical Biochemistry, Volume 195, Pages 330–351.

In 1987, Chance won the Biological Physics Prize (now known as the Max Delbruck Prize in Biological Physics) from the American Physical Society

for pioneering application of physical tools to the understanding of Biological phenomena. The early applications ranged from novel spectrometry that elucidated electron transfer processes in living systems to analog computation of nonlinear processes. Later contributions have been equally at the forefront.

The Quantum Ten

The Quantum Ten:
A Story of Passion, Tragedy,
Ambition, and Science,
by Sheilla Jones.

Over the holiday break, I read The Quantum Ten: A Story of Passion, Tragedy, Ambition and Science, by Sheilla Jones. The book is about the development of quantum mechanics in the 1920s.

The seeds of the shift currently taking place in science were sown eighty years ago, from 1925 to 1927. That’s when a dramatic two-year revolution in physics reached a climax, the denouement set the course for what was to follow. It’s the story of a rush to formalize quantum physics, the work of just a handful of men fired by ambition, philosophical conflicts, and personal agendas…

Remarkably, this dramatic shift in science was primarily the work of ten men, and they were ten fallible men, some famous and some not so famous, although they also had a large supporting cast. The triumphs and tragedies, loves and betrayals, dreams realized and ambitions thwarted, shaped the competition over who would get to define truth and reality. There never was a consensus. By the time of the pivotal Fifth Solvay Conference in Brussels in 1927, there was so much ill will and disappointment among the creators of quantum physics over their various competing theories and over who deserved credit that most were barely on speaking terms.

The Brussels conference was the first time so many of them had come together: Albert Einstein, the lone wolf; Niels Bohr, the obsessive but gentlemanly father figure; Max Born, the anxious hypochondriac; Werner Heisenberg, the intensely ambitious one; Wolfgang Pauli, the sharp-tongued critic with a dark side; Paul Dirac, the quiet one; Erwin Schrodinger, the enthusiastic womanizer; Prince Louis de Broglie, the French aristocrat; and Paul Ehrenfest, who was witness to it all. Their coming together, however, lasted only for the duration of the conference.

I enjoyed the book, but couldn’t help wishing that it would focus less on the personal problems of the scientists and more on their science. I prefer my scientific biographies to be a bit more rigorous with an emphasis on the science, like Pais’s Subtle is the Lord. Nevertheless, the story was fascinating in a gossipy sort of way. The book is full of tidbits like this:

From time to time [Schrodinger] did consult on the mathematics with his Zurich colleague Hermann Weyl, who was at that point embroiled in a passionate love affair with Schrodinger’s wife, Anny. Wince the Weyls were part of the same sexually permissive crowd as the Schrodingers, the affair was no cause for tension between the two colleagues.

I found myself oddly attracted to Paul Ehrenfest, “an intense physicist with a debilitating streak of self-doubt who could rarely see the valuable gift he offered to physics and a passionate friend to both Einstein and Bohr.” Then, near the end of the book, I discovered—to my horror—that not only did Ehrenfest take his own life (I had heard that before), but that just before he committed suicide he shot and killed his son. My admiration vanished.

There was no biological physics in The Quantum Ten, but I couldn’t help wonder how these great scientists fared in the 4th edition of Intermediate Physics for Medicine and Biology. A quick survey gave the following results:

• Albert Einstein. I discussed Einstein’s presence in our textbook about a year ago in this blog, and concluded that “we rarely mention Einstein by name in our book, but his influence is present throughout, and most fundamentally when we discuss the idea of a photon.”
• Niels Bohr. His model for the hydrogen atom is referred to, but not derived. His contributions to calculating the stopping power of a charged particle in tissue are discussed in Chapter 15 (Interaction of Photons and Charged Particles with Matter).
• Paul Ehrenfest. His name never appears in our book.
• Max Born. The Born charging energy is discussed in Chapter 6 (Impulses in Nerve and Muscle Cells).
• Erwin Schrodinger. The Schrodinger equation is mentioned in Chapter 3 (Systems of Many Particles), but never written down.
• Wolfgang Pauli. The Pauli exclusion principle is stated in Chapters 14 (Atoms and Light) and 15 (Interaction of Photons and Charged Particles with Matter).
• Louis de Broglie. His name is not in the book, although I have mentioned him in this blog before.
• Werner Heisenberg. He and his uncertainty principle are not in the book.
• Paul Dirac. I discussed Dirac in the blog before. His delta function shows up in Chapter 11 (The Method of Least Squares and Signal Analysis).
• Pascual Jordan. His name never appears in our book.

I am not overly concerned that the quantum ten don’t figure prominently in Intermediate Physics for Medicine and Biology. Russ Hobbie and I do not focus on microscopic phenomena, where quantum mechanics is essential. Probably the greatest contribution to biological physics from any of the quantum ten is Schrodinger’s book What is Life?, which had a major impact on the early development of molecular biology (see The Eighth Day of Creation).

P.S. We had a significant revision of the errata this week. It is available at the book’s website: https://sites.google.com/view/hobbieroth. A big thank you to Gabriela Castellano for finding many mistakes and pointing them out to us. If you, dear reader, find additional mistakes, please let us know.

Gaussian integration

Chapter 8 in the 4th edition of Intermediate Physics for Medicine and Biology covers Biomagnetism: the measurement of the magnetic field produced by electrical currents in nerve and muscle. One issue that arises during biomagnetic recordings is that the magnetic field is not measured at a point, but is averaged over a pickup coil. Therefore, when comparing theoretical calculations to experimental data, you need to integrate the calculated magnetic field over the coil.

One way to do this is Gaussian quadrature, which approximates the integral by a weighted sum. Homework problem 40 in Chapter 8 shows a three-point Gaussian quadrature formula for integrating over a circular coil. At the end of the problem Russ Hobbie and I write

Higher-order formulas for averaging the magnetic field can be found in Roth and Sato (1992).

The reference is to Roth, B. J. and S. Sato (1992) “Accurate and Efficient Formulas for Averaging the Magnetic Field over a Circular Coil,” In M. Hoke, S. N. Erne, T. C. Okada, and G. L. Romani, eds. Biomagnetism: Clinical Aspects. Amsterdam, Elsevier. This book is the proceedings of the 8th International Conference on Biomagnetism, held in Munster, Germany on August 19–24, 1991. I didn’t attend that meeting, but my colleague Susumu Sato did. Sato is a senior scientist in the Epilepsy Research Branch of the National Institute of Neurological Disorders and Stroke, part of the National Institutes of Health in Bethesda, Maryland. When I worked with him he had an active research program in magnetoencephalography (MEG), including a large and expensive shielded room and a multi-channel SQUID magnetometer.

The introduction of our paper states

The MEG is measured by detecting the magnetic flux through a pickup coil, usually circular, that is coupled to a SQUID magnetometer. Often the source of the MEG is modeled as a current dipole, whose position, orientation and strength are determined iteratively by fitting the MEG data to a dipolar magnetic field pattern. To obtain an accurate result, this dipole field must be integrated over the pickup coil area to obtain the magnetic flux. Since this integration is repeated for each dipole considered in the iteration, the numerical algorithm used to estimate this integral should be efficient. In this note, several integration formulas are presented that allow the magnetic field to be integrated over the coil area quickly with little error. These formulas are examples of a general technique of approximating multiple integrals described by Stroud [1].

Reference [1] is to: Stroud AH (1971) Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, New Jersey, Pages 277–289.

I remember deriving several of these formulas independently before discovering Stroud’s textbook (it is always deflating to find you’ve been scooped). The derivation requires solving a system of nonlinear equations (which I rather enjoyed). Each formula requires evaluating the magnetic field at N points, and the integral is accurate to m^th order. We presented a 1-point formula accurate to first order, a 3-point formula accurate to second order (this was the formula examined in the homework problem), a 4-point formula accurate to third order, a 6-point formula accurate to fourth order, a 7-point formula accurate to fifth order, and a 12-point formula accurate to seventh order.

The general formulation of Gaussian quadrature was developed by Carl Friedrich Gauss (1777–1855), one of the greatest mathematicians of all time. Gauss’s name appears often in the 4th edition of Intermediate Physics for Medicine and Biology, including the Gaussian function (Chapter 4), Gauss’s law (Chapter 6), the cgs unit for the magnetic field of a gauss (Chapter 8), the fast Fourier transform (FFT, Chapter 11) about which we write “the grouping used in the FFT dates back to Gauss in the early nineteenth century,” and the Gaussian Probability Distribution (Appendix I).

DNA animation by Drew Berry

I know that the 4th edition of Intermediate Physics for Medicine and Biology doesn’t discuss much about the physics of life at the molecular level. In the preface, Russ Hobbie and I wrote that “molecular biophysics has been almost completely ignored.” Nevertheless, I recently ran across an animation of DNA that is so good I have to tell you about it.

My story starts with the January-February issue of American Scientist, the science and technology magazine published by Sigma Xi, The Scientific Research Society. The cover of this issue shows DNA, packed “tightly in some chromosomal territories and loosely in others, forming sheer walls and intergenic fissures, as seen in the cover image from a 3D animation by renowned molecular animator Drew Barry.” When I read this, I asked myself: Who is Drew Berry, and where can I find his animations?

It turns out you can find Berry’s wonderful animation “Molecular Visualizations of DNA” on Youtube. Trust me, you really want to watch this video. It explains DNA packing into chromosomes, transcription, and translation in a visual way that is unforgettable. Other Berry animations can be found at http://www.wehi.edu.au/education/wehitv.

 DNA packing into chromosomes, by Drew Berry.
https://www.youtube.com/watch?v=7wpTJVWra7I

In 2010, Berry was awarded a MacArthur Fellowship from the John D and Catherine T MacArthur Foundation, the so-called “genius award”. The MacArthur website says

Drew Berry is a biomedical animator whose scientifically accurate and aesthetically rich visualizations are elucidating cellular and molecular processes for a wide range of audiences. Trained as a cell biologist as well as in light and electron microscopy, Berry brings a rigorous scientific approach to each project, immersing himself in the relevant research in structural biology, biochemistry, and genetics to ensure that the most current data are represented. In three- and four-dimensional renderings of such key biological concepts as cell death, tumor growth, and the packaging of DNA, Berry captures the details of molecular shape, scale, behavior, and spatio-temporal dynamics in striking form. His groundbreaking series of animations of the intricate biochemistry of DNA replication, translation, and transcription demonstrates these multifaceted processes in ways that enlighten both scientists and the scientifically curious. The sequence and pace of each molecular interaction are precisely coordinated, at the same time as the ceaseless motion of the molecules reveals the complex and seemingly random choreography of the molecular world. Committed to educating the public about critical topics in medical research, Berry created a two-part animation of the malaria life cycle that illustrates the pathogen’s development in the mosquito host and its invasion of and diffusion throughout human cells. In these and many other projects in progress, Berry synthesizes data across a variety of fields and presents them in engaging and lucid animations that both inspire a sense of wonder and enhance understanding of biological systems.

Drew Berry received B.Sc. (1993) and M.Sc. (1995) degrees from the University of Melbourne. Since 1995, he has been a biomedical animator at the Walter and Eliza Hall Institute of Medical Research. His animations have appeared in exhibitions and multimedia programs at such venues as the Museum of Modern Art, the Guggenheim Museum, the Royal Institute of Great Britain, and the University of Geneva.

 Note added in 2019: Watch Berry’s TED talk below.

 Drew Berry: Animations of Unseeable Biology.
https://www.youtube.com/embed/WFCvkkDSfIU

Convergence

This week researchers at the Massachusetts Institute of Technology released a white paper titled “The Third Revolution: The Convergence of the Life Sciences, Physical Sciences, and Engineering.” It begins

There are few challenges more daunting than the future of health care in this country. This paper introduces the dynamic and emerging field of convergence—which brings together engineering and the physical and life sciences—and explains how convergence provides a blueprint for addressing the health care challenges of the 21st century by producing a new knowledge base, as well as a new generation of diagnostics and therapeutics. We discuss how convergence enables the innovation necessary to meet the growing demand for accessible, personalized, affordable health care. We also address the role of government agencies in addressing this challenge and providing funding for innovative research. Finally, we recommend strategies for embedding convergence within agencies like the National Institutes of Health (NIH), which aims to optimize basic research, improve health technology, and foster important medical advances.

If “convergence” is the melding of physics and engineering with the life sciences, then I suggest that a good place to start your search for convergence is the 4th edition of Intermediate Physics for Medicine and Biology. The MIT white paper is singing our song about the integration of physics with biology. But I am a Johnny-come-lately to convergence compared to my coauthor, Russ Hobbie, who pioneered this approach decades ago.

Between 1971 and 1973 I audited all the courses medical students take in their first two years at the University of Minnesota. I was amazed at the amount of physics I found in these courses.

You can find more about the white paper in an article in the Science Insider. The authors talk about three revolutions in biomedicine: the first was molecular and cellular biology, the second was genomics, and the third will be convergence. I must admit that I find the white paper a little self-serving; most of their examples feature MIT researchers (says the guy who writes a weekly blog about physics in medicine and biology with the goal of peddling textbooks!). But I agree with its premise. Indeed, the first sentence of their concluding paragraph sounds as if it could be a promotion for our book.

The merger of the life, engineering, and physical sciences promises to fundamentally alter and speed our scientific trajectory. NIH and other affected agencies, if adequately funded and made ready, can be thought leaders in this next scientific revolution. The time is right for NIH and other agencies to take up convergence as the wave of the future, creating dramatic new opportunities in medicine for new therapies and diagnostics, economic opportunity, as well as promise in many other scientific fields, from energy to climate to agriculture.

Brownian Motion

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Brownian motion. We first address this topic in Chapter 3 when deriving the equipartition of energy: the average thermal kinetic energy of an object at temperature T is 3k_BT/2, where k_B is Boltzmann’s constant.

This result is true for particles of any mass: atoms, molecules, pollen grains, and so forth. Heavier particles will have a smaller velocity but the same average kinetic energy. Even heavy particles are continually moving with this average kinetic energy. The random motion of pollen particles in water was first seen by a botanist, Robert Brown, in 1827. This Brownian motion is an important topic in the next chapter.

We next address this topic in Chapter 4, as we motivate the reader for a discussion of diffusion.

This movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by the English botanist Robert Brown in 1827 and is called Brownian motion. Solute particles are also subject to this random motion. If the concentration of particles is not uniform, there will be more particles wandering from a region of high concentration to one of low concentration than vice versa. This motion is called diffusion.

As so often happens when you look deeply into a subject, the story is more complicated than can be described in an introductory (or even an intermediate) textbook. In the December 2010 issue of the American Journal of Physics, Philip Pearle and his colleagues published the fascinating article “What Brown Saw and You Can Too” (Volume 78, Pages 1278–1289).

A discussion of Robert Brown’s original observations of particles ejected by pollen of the plant Clarkia pulchella undergoing what is now called Brownian motion is given. We consider the nature of those particles and how he misinterpreted the Airy disk of the smallest particles to be universal organic building blocks. Relevant qualitative and quantitative investigations with a modern microscope and with a “homemade” single lens microscope similar to Brown’s are presented.

One interesting conclusion of their study is that Brown did not actually see pollen grains move.

We emphasize that Brown did not observe the pollen move. Instead, he observed the motion of much smaller objects that reside within the pollen.¹⁵ Nonetheless, statements that Brown saw the pollen move are common.¹⁶

Fortunately, reference 16 does not cite Intermediate Physics for Medicine and Biology. But it raises the question: Did Russ and I get it wrong? Our discussion in Chapter 4 seems safe. The text in Chapter 3 depends on if you interpret “pollen particles” as the entire “pollen grain” or “particles arising from pollen.” Russ may have been aware of this distinction when he wrote the original text, but I confess I was not. I always thought Brown saw the entire pollen grain move.

Pearle et al. show electron microscope pictures of pollen grains, which are 50–100 microns in diameter. I summarize their analysis about what Brown saw as a new homework problem for Chapter 4.

Problem 5 1/2. This problem looks at the original observations of Robert Brown that established Brownian motion.

(a) Combine Eqs. 4.23 and 4.71 to determine an expression for the average distance a particle of radius a will diffuse through a fluid of viscosity η in time t.

(b) Assume you observe a pollen grain with a radius of 50 microns in water at room temperature, and that your visual perception is particularly sensitive to motions occurring over a time of about one second. What is the average distance you observe the grain to move?

(c) Now assume your eye cannot see movements that occur over angles of less than 1 minute of arc, or 3 × 10⁻⁴ radians (in Chapter 14, we estimate 3 minutes of arc, but use 1 arc min to be conservative). Most eyes cannot focus on objects closer than 25 cm. Determine the smallest displacement you can observe with the naked eye.

(d) Robert Brown had a microscope that could magnify objects by a factor of about 370. What is the smallest displacement he could observe with his microscope? Is this larger or smaller than the displacement of a pollen grain in one second?

In fact, Brown did not observe the motion of entire pollen grains. He observed fat and starch particles about 2 microns in diameter that are released by pollen. For more on Brown’s original observations, see Pearle et al. (2010).

The authors also analyze the microscope that Brown used, and estimate the diffraction effects he had to contend with. Using an analysis similar to that presented in Section 13.7 (Medical Uses of Ultrasound) of Intermediate Physics for Medicine and Biology, they show that Brown probably could not resolve some of the smaller particles, but instead observed their diffraction pattern. As in Eq. 13.40 in our textbook, the diffraction pattern involves a Bessel function, and implies that the apparent size of an object is larger than the real size. The effect is minor for large objects but dominates for small objects.

I find the history and analysis of Brown’s original studies to be fascinating. For me, Pearle et al.’s paper reminds me that 1) the American Journal of Physics is still my favorite journal, and 2) physics has much to offer biology and medicine.

The littlest things can drive you nuts

I hope that our readers (and Russ Hobbie and I do value and appreciate all our dear readers) find the list of references at the end of each chapter in the 4th edition of Intermediate Physics for Medicine and Biology useful. We tried to include books and articles that you would enjoy, and that would help you understand the material in our textbook better. But, you may wonder, what do I see when I look at those lists of references? The first thing I see—the thing that jumps out of the page and screams at me—is that in each list, the first reference is not indented like the rest!!! As I recall, it is some issue in LaTex that is difficult to fix. I think it is related to the policy of not indenting the first paragraph of a section (a practice that I don’t care for).

I suppose what really should worry me are the errors that creep into the book. But at least we can correct those in the erratum, available at the book website. For some reason, I can live with those errors (que sera, sera) but the indentation issue is killing me. You can find a lot of other useful information at the book website, including an interview with Russ Hobbie published in the December 2006 issue of the American Physical Society Division of Biological Physics newsletter, a movie of Russ Hobbie explaining how radiation interacts with tissue based on his Mac Dose computer program, an American Journal of Physics resource letter that Russ and I published last year, and other supplementary material.

Let me use this post to update you on a few issues mentioned previously in this blog. In an October post, I talked about tanning and skin cancer. A recent article in the online newspaper MinnPost.com suggests that the problem is not getting any better, especially in the midwest, and that “people are still not recognizing that indoor tanning use is linked to skin cancer.” An article in medicalphysicsweb.com reports that “supply shortages of molybdenum-99 could become commonplace over the next decade unless longer-term actions are taken.” I discussed this issue several times before: here, here, and here. Felix Baumgartner’s attempt to jump out of a balloon at the edge of space and break the sound barrier in free fall has been put on hold, apparently because of a law suit over who owns the rights to this idea. Finally, you can watch online a series of lectures about the physics of hearing and cochlear implants delivered at the University of Michigan.

I wish you all a peaceful and happy Christmas Eve. If you are lucky, you will wake up tomorrow morning to find that Santa has left the 4th edition of Intermediate Physics for Medicine and Biology in your stocking. For those unfortunate few who received something else from Santa, I suggest amazon.com.

Merry Christmas!

Subtracting Large Numbers

One of the most notorious difficulties in numerical computations is the loss of precision when subtracting two similar, large numbers to obtain a smaller one. Russ Hobbie and I illustrate this hazard in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. We begin this chapter with a discussion of the method of least squares, and we derive the formulas (Eqs. 11.5a and 11.5b) for fitting data to a straight line, y = ax + b. We then add “In doing computations where the range of data is small compared to the mean, better numerical accuracy can be obtained from…” and then present alternative formulas (Eqs. 11.5c, 11.5d, and 11.5e). Homework Problem 7 in Chapter 11 (one of the many new problems in the 4th edition) illustrates the advantage of the second set of equations.

Problem 7 Consider the data

   x       y
100   4004
101   4017
102   4039
103   4063

(a) Fit these data with a straight line y=ax+b using Eqs. 11.5a and 11.5b to find a.

(b) Use Eq. 11.5c to determine a. Your result should be the same as in part (a).

(c) Repeat parts (a) and (b) while rounding all the intermediate numbers to 4 significant figures. Do Eqs. 11.5a and 11.5b give the same result as Eq. 11.5c? If not, which is more accurate?

(Spoiler alert: Don’t continue reading if you want to solve the problem yourself first, as you should.) If you solve this problem, you will find that Eqs. 11.5a and 11.5b do not work very well at all for this problem. Their flaw is that they require you to subtract two really big numbers to get a much smaller one.

Numerical Methods That Work,
by Forman Acton.

A good discussion of this issue can be found in Forman Acton’s book Numerical Methods that Work.

The following problem often appears as a puzzle in Sunday Supplements. The difficulties are numerical rather than formulative and hence it is an especially appropriate challenge to the aspiring numerical analyst. We strongly urge that the reader solve it in his own way before turning to the “official” solution.

A railroad rail 1 mile long is firmly fixed at both ends. During the night some prankster cuts the rail and welds in an additional foot, causing the rail to bow up in the arc of a circle. The classical question concerns the maximum height this rail now achieves over its former position. To put it more precisely: We are faced…with the chord of a circle AB that is exactly 1 mile long and the corresponding arc AB that is 1 mile plus 1 foot and our question concerns the distance d between the chord and the arc at their midpoints. [See Acton’s book for the accompanying figure]

The relationships available are the simple ones from trigonometry involving the subtended half angle, θ, and the Pythagorean relationship. The student at this point should attempt to solve the problem before turning to the solution given in Chapter 2. He should attempt to find the distance d to an accuracy of three significant figures. In his effort he will probably be faced with subtracting two large and nearly equal numbers, which will cause a horrendous loss of significant figures. He can live with this process by shear brute force, but it will involve use eight-significant-figure trigonometric tables to preserve three figures in his answer. The point of the problem here is to find another method of calculating d, one that does not require such extreme measures. The three-figure answer can, indeed, be obtained rather easily using nothing more than pencil, paper, and a slide rule. The student should seek such a method.

If you find numerical methods interesting (as I do), you will love Acton’s delightfully written book. Originally published in 1970, it is all the more charming for its now-quaint references to slide rules and trigonometric tables. Yet, the concepts are not out-of-date. Even with powerful computers, errors can arise from subtracting nearly equal numbers. I’ve run into the issue myself when using the finite difference method and relaxation to solve Laplace’s equation with a fine grid and only single precision arithmetic.

Real Computing Made Real,
by Forman Acton.

Unfortunately, Acton’s book is not cited in the 4th edition of Intermediate Physics for Medicine and Biology (we’ll have to fix that in later editions), although I have mentioned it before in this blog. Acton is an emeritus professor in the Department of Computer Science at Princeton University (a department with an illustrious history). Also interesting is his more recent book Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations.

Robert Millikan

One fundamental constant that appears repeatedly in the 4th edition of Intermediate Physics for Medicine and Biology is the charge of the electron (the elementary charge, e), equal to 1.6 × 10⁻¹⁹ C. The first appearance of e that I can find is in Section 3.8 on the Nernst Equation. It appears in another context in Section 8.9, The Detection of Weak Magnetic Fields, when discussing Superconducting Quantum Interference Device (SQUID) magnetometers and the quantum of flux, equal to Planck’s constant divided by two times e. It shows up repeatedly in Chapter 9 on Electricity and Magnetism at the Cellular Level, and then again in Chapter 14 when discussing the energy levels of the hydrogen atom. It appears in Chapter 15 in the Klein-Nishina formula and in the expression for the classical radius of the electron.

Understanding Physics:
The Electron, Proton, and Neutron,
by Isaac Asimov.

How was the charge of the electron first measured? Isaac Asimov tells the story in Understanding Physics: The Electron, Proton, and Neutron.

The experiments that determined the size of the electric charge on the electron were conducted by the American physicist Robert Andrews Millikan (1868–1953) in 1911.

Millikan made use of two horizontal plates, separated by about 1.6 centimeters, in a closed vessel containing air at low pressure, The upper plate had a number of fine holes in it and was connected to a battery that could place a positive charge upon it. Millikan sprayed fine drops of nonvolatile oil into the closed vessel above the plates. Occasionally, one droplet would pass through one of the holes in the upper plate and would appear in the space between the plates. There it could be viewed through a magnifying lens because it was made to gleam like a star through its reflection of a powerful beam of light entering from one side.

Left to itself, the droplet of oil would fall slowly, under the influence of gravity. The rate of this fall in response to gravity, against the resistance of air (which is considerable for so small and light an object as an oil droplet), depends on the mass of the droplet. Making use of an equation first developed by the British physicist George Gabriel Stokes (1819–1903), Millikan could determine the mass of the oil droplets.

Millikan then exposed the container to the action of X rays. This produced ions in the atmosphere within (see page 110). Occasionally, one of these ions attached itself to the droplet. If it were a positive ion, the droplet, with a positive charge suddenly added, would be repelled by the positively-charged plate above, and would rush downward at a rate greater than could be accounted for by the action of gravity alone. If the ion were negative, the droplet would be attracted to the positively-charged plate and might even begin to rise in defiance of gravity.

The change in velocity of the droplet would depend on the intensity of the electric field (which Millikan knew) and the charge on the droplet, which he could now calculate.

Millikan found that the charge on the droplet varied according to the nature of the ion that was adsorbed and on the number of ions that were adsorbed. All the charges were, however, multiples of some minimum unit, and this minimum unit could reasonably be taken as the smallest possible charge on an ion and therefore, equal to the charge on the electron. Millikan's final determination of this minimum charge was quite close to the value now accepted, which is 4.80298 × 10⁻¹⁰ electrostatic units (“esu”), or 0.000000000480298 esu.

We don’t use electrostatic units in Intermediate Physics for Medicine and Biology (although they appear briefly in homework problem 3 in Chapter 6), but this is equivalent to 1.6 × 10⁻¹⁹ Coulombs.

Selected Papers of
Great American Physicists.

I remember doing Millikan’s oil drop experiment as an undergraduate physics major at the University of Kansas. It required several hours in a dark room staring at small oil drops through a microscope. When in graduate school, I read one of Millikan’s papers in the book Selected Papers of Great American Physicists: The Bicentennial Commemorative Volume of The American Physical Society. I was particularly impressed by Millikan’s careful analysis of sources of systematic error in his experiment. In fact, I used that paper as a model for one of my few experimental papers: “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment” (Roth and Wikswo, Biophysical Journal, Volume 48, Pages 93–109, 1985). Some have claimed that Millikan committed scientific fraud by an improper selection of data to use in his analysis, but that claim has been debunked (see “Data Selection and Responsible Conduct: Was Millikan a Fraud?” by Richard Jennings, Science and Engineering Ethics, Volume 10, Pages 639–653, 2004).

I have a personal reason for being interested in the work of Robert Millikan. According to his Nobel Prize biography, he was born in Morrison Illinois, a small town 120 miles west of Chicago, about 15 miles from the Mississippi River. This is the town I grew up in, from an age of just a few months until I was 12 years old. At the time, I didn’t realize who Robert Millikan was, or that Morrison was the home to a Nobel Prize winning physicist. But over the years I have become a big fan of “Millikan from Morrison.” According to the Morrison chamber of commerce, there is now a downtown park named after Millikan. I must go visit.