Friday, December 28, 2012

Determining the Site of Stimulation During Magnetic Stimulation of a Peripheral Nerve

As December draws to a close and I reflect on all that’s happened over the last twelve months, I conclude that 2012 has been a good year. For me, it has also marked some important anniversaries. Thirty years ago (1982) I graduated from the University of Kansas with a bachelors degree in physics. Twenty-five years ago (1987) I obtained my PhD from Vanderbilt University. And twenty years ago (1992) I was at the National Institutes of Health in Bethesda, Maryland working on magnetic stimulation of nerves.

Today I want to focus on one particular paper published in 1992 that examined magnetic stimulation of a peripheral nerve: Determining the Site of Stimulation During Magnetic Stimulation of a Peripheral Nerve (Electroencephalography and Clinical Neurophysiology, Volume 85, Pages 253-264). To understand this article, we must first examine Frank Rattay’s analysis of electrical stimulation. Rattay showed that excitation along a nerve axon occurs where the “activating function” –λ2 d2Ve/dx2 is largest, with λ the length constant, Ve the extracellular potential produced by a stimulating electrode, and x the distance along the axon. Homework problem 38 in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology guides you through Rattay’s derivation. In 1990, Peter Basser and I showed that this result also holds during magnetic stimulation. What is magnetic stimulation? In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Faraday’s law of induction, and then write
“Since a changing magnetic field generates an induced electric field, it is possible to stimulate a nerve or muscle cells with out using electrodes…One of the earliest investigations was reported by Barker et al. (1985) who used a solenoid in which the magnetic field changed by 2 T in 110 μs to apply a stimulus to different points on a subject’s arm and skull.”
The main difference between Rattay’s analysis of electrical stimulation and our analysis of magnetic stimulation was that Rattay expressed his activation function in terms of the electric potential produced by the stimulus electrode, whereas Basser and I considered the induced electric field along the axon, Ex, and wrote the activating function as λ2 dEx/dx. The most interesting feature of this result is that stimulation does not occur where the electric field is strongest, but instead where its gradient along the axon is greatest. In the early 1990s, this result was surprising (in retrospect, it seems obvious), so we set out to test it experimentally.

Basser and I both worked in NIH’s Biomedical Engineering and Instrumentation Program, and we had neither the expertise nor facilities to perform the needed experiments, but we knew who did. Since arriving at NIH in 1988, I had been working with Mark Hallett and Leo Cohen to develop clinical applications of magnetic stimulation. Also collaborating with Hallett was a delightful couple visiting from Italy, Jan Nilsson and his wife Marcela Panizza. Under Hallett’s overall leadership, with Nilsson and Panizza making the measurements, and with me occasionally making suggestions and cheerleading, we carried out the key experiments that confirmed Basser’s and my prediction about where excitation occurs. These studies were performed on human volunteers (at that time there were people who made their living as paid normal volunteers in clinical studies at NIH) and were carried out in the NIH clinical center. The abstract of our now 20-year old paper said
“Magnetic stimulation has not been routinely used for studies of peripheral nerve conduction primarily because of uncertainty about the location of the stimulation site. We performed several experiments to locate the site of nerve stimulation. Uniform latency shifts, similar to those that can be obtained during electrical stimulation, were observed when a magnetic coil was moved along the median nerve in the region of the elbow, thereby ensuring that the properties of the nerve and surrounding volume conductor were uniform. By evoking muscle responses both electrically and magnetically and matching their latencies, amplitudes and shapes, the site of stimulation was determined to be 3.0 ± 0.5 cm from the center of an 8-shaped coil toward the coil handle. When the polarity of the current was reversed by rotating the coil, the latency of the evoked response shifted by 0.65 ± 0.05 msec, which implies that the site of stimulation was displaced 4.1 ± 0.5 cm. Additional evidence of cathode- and anode-like behavior during magnetic stimulation comes from observations of preferential activation of motor responses over H-reflexes with stimulation of a distal site, and of preferential activation of H-reflexes over motor responses with stimulation of a proximal site. Analogous behavior is observed with electrical stimulation. These experiments were motivated by, and are qualitatively consistent with, a mathematical model of magnetic stimulation of an axon.”
Rather than describe this experiment in detail, I will let you analyze it yourself in a new homework problem (your three-days-late Christmas present). It is similar to a problem from an exam I gave to my biological physics (PHY 325) students.
Section 8.7

Problem 26 ½ (a) Rederive the cable equation for the transmembrane potential v (Eq. 6.55) using one crucial modification: generalize Eq. 6.48 to account for part of the intracellular electric field that arises from Faraday induction and therefore cannot be written as the gradient of a potential,
 Assume you measure v relative to the resting potential so Eq. 6.53 becomes jm = gm v, and let the extracellular potential be small so vi = v. Identify the new source term in the cable equation (the "activating function" for magnetic stimulation), analogous to vr in Eq. 6.55.
(b) Let
Calculate the activating function and plot both the electric field and the activating function versus x.
(c) Suppose you stimulate a nerve using this activating function, first with one polarity of the current pulse and then the other. What additional delay in the response of the nerve (as measured by the arrival time of the action potential at the far end) will changing polarity cause because of the extra distance the action potential must travel? Assume a = 4 cm and the conduction speed is 60 m/s.
At about the same time as we were doing this study, Paul Maccabee and his colleagues at the SUNY Health Science Center in Brooklyn were carrying out similar experiments using an in-vitro pig nerve model (a nerve in a dish), and came to similar conclusions (Magnetic Coil Stimulation of Straight and Bent Amphibian and Mammalian Peripheral Nerve In Vitro: Locus of Excitation, Journal of Physiology, Volume 460, Pages 201-219, 1993). Our paper was published first (Yes!!!) but their results were cleaner and more elegant, in part because they didn’t have the complication of the nerve being surrounded by irregularly shaped muscles and bones. Our paper has been fairly influential (53 citations to date in the Web of Science), but theirs has had an even greater impact (147 citations). A year later Maccabee and I together published a study of a new magnetic stimulation coil design.

What has happened to this cast of characters in the last 20 years? Hallett and Cohen remain at NIH, still doing great work. Nilsson is a biomedical engineer and Panizza is a neurophysiologist in Italy. Basser is at NIH, but is now with the Eunice Kennedy Shriver National Institute of Child Health and Human Development, where he works on MRI diffusion tensor imaging. Paul Maccabee is a neurologist and the Director of the EMG Laboratory at SUNY Brooklyn. I left NIH in 1995, and am now at Oakland University, where I teach, do research, and write a blog so I can wish readers of Intermediate Physics for Medicine and Biology a Happy New Year!

Friday, December 21, 2012

Royal Institution Christmas Lectures

With Christmas approaching, my attention naturally turns to the Royal Institution Christmas Lectures. The Royal Institution (Ri) website states:
“The Ri is an independent charity dedicated to connecting people with the world of science. We're about discovery, innovation, inspiration and imagination. You can explore over 200 years of history making science in our Faraday Museum as well as engage with the latest research, ideas and debates in our public science events.

We run science programmes for young people at our Young Scientist Centre, present exciting, demonstration-packed events for schools and run mathematics masterclasses across the UK.

We are most famous for our Christmas Lectures which were started by Michael Faraday in 1825. Check out the 2011 Lectures here and don't miss them this Christmas on BBC Four.

Anyone can join the Ri. If you're interested in how the world works, or how to make it work better through science, the Ri is the place for you.”
The 2012 Christmas Lectures, 'The Modern Alchemist' will be broadcast on BBC Four on December 26, 27, and 28 at 8pm. Don’t get BBC Four? Neither do I. But that is OK, because you can watch the Christmas Lectures at the Ri website. In fact, you can watch the Christmas Lectures from past years too. You will have to open an account, which means you will need to give them your email address and other information, but you don’t need to pay anything; it’s free. Kind of like a Christmas present.

My favorite lecture is from 2010. Mark Miodownik stars in “Why Elephants Can’t Dance but Hamsters Can Skydive”. He talks about an issue discussed in Homework Problem 28 in Chapter 2 of the Fourth edition of Intermediate Physics for Medicine and Biology. Russ Hobbie and I ask the reader to analyze how fast animals of different sizes fall. In Why Elephants Can’t Dance, Miodownik performs a brilliant demonstration using two spherical animals--one about the size of a hamster, and the other about the size of a dog-- made of some sort of jello-like gel. Suffice to say, the hamster-sized blob of gel does just fine when it hits the ground after a fall, but the dog-sized blob has some problems. The audience for the lecture is mostly children, but as Dickens wrote “it is good to be children sometimes, and never better than at Christmas”. The entire lecture is about why size matters in the animal kingdom.

Miodownik then talks about another topic in animal scaling that Russ and I don’t mention in our book, although I often bring it up when I teach Biological Physics at Oakland University. In two animals with the same shape but different sizes, the weight increases as the cube of the linear size, but the cross-sectional area of its legs increases as the square of the size. Therefore, a large animal has a harder time supporting its weight than a small animal does. Miodownik demonstrates this with two rubber pig-like spheres with rubber legs attached. The small sphere easily stands on its legs, while the large sphere just collapses. As the video says, size really does matter. Of course, elephants solve this problem by making their legs thick, which is why they can’t dance.

I recommend watching Why Elephants Can’t Dance while reading Chapter 2 of Intermediate Physics for Medicine and Biology. It will help you understand animal scaling.

Enjoy the Royal Institution Christmas Lectures, and have a Merry Christmas.

Friday, December 14, 2012

Dynamics: The Geometry of Behavior

When I was working at the National Institutes of Health in the 1990s, I ran across a wonderful series of books from the Visual Mathematics Library that had a big impact on the way I thought about math. Dynamics: The Geometry of Behavior, by Ralph Abraham and Christopher Shaw, was published in four volumes: 1 Periodic Behavior, 2 Chaotic Behavior, 3 Global Behavior, and 4 Bifurcation Behavior. The fascinating feature of these books was that they contained almost no equations; everything was explained in pictures. At first glance, they look like comic books, but on closer inspection you realize that the math is presented in a very accurate and rigorous way. There are lots of plots of phase planes, and drawings of experimental apparatus that are being modeled by the math. There is hardly a page without pictures, and 90% of many pages are filled with illustrations. I highly recommend these books for anyone interested in developing an intuitive feeling for nonlinear dynamics (which should be everyone).

Their forward begins
“During the Renaissance, algebra was resumed from Near Eastern sources, and geometry from the Greek. Scholars of the time became familiar with classical mathematics. When calculus was born in 1665, the new ideas spread quickly through the intellectual circles of Europe. Our history shows the importance of the diffusion of these mathematical ideas, and their effects upon the subsequent development of the sciences and technology.

Today, there is a cultural resistance to mathematical ideas. Due to the widespread impression that mathematics is difficult to understand, or to a structural flaw in our educational system, or perhaps to other mechanisms, mathematics has become an esoteric subject. Intellectuals of all sorts now carry on their discourse in nearly total ignorance of mathematical ideas. We cannot help thinking that this is a critical situation, as we hold the view that mathematical ideas are essential for the future evolution of our society.

The absence of visual representations in the curriculum may be part of the problem, contributing to mathematical illiteracy, and to the math-avoidance reflex. This series is based on the idea that mathematical concepts may be communicated easily in a format which combines visual, verbal, and symbolic representations in tight coordination. It aims to attack math ignorance with an abundance of visual representations.

In sum, the purpose of this series is to encourage the diffusion of mathematical ideas, by presenting them visually.”
In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I do not suppress mathematical expressions. In fact, I suspect many of our readers would claim we have too many, rather than too few, equations. Nevertheless, we try to convey our subject in figures as well as math, visually as well as symbolically. We discuss nonlinear dynamics in Chapter 10, and we have some state space figures that are similar to those found in Abraham and Shaw (although they use 4-color figures—green, red, blue, and black—while we use the less attractive black and white). I believe all the illustrations in Abraham and Shaw are hand-drawn, giving them a charm that often is lacking in this age of computer-generated drawings. Unfortunately, Russ and I never cited Abraham and Shaw. One reason I write this blog is to alert our readers to books and articles that don’t appear in the pages of Intermediate Physics for Medicine and Biology.

Dynamics: The Geometry of Behavior is one of those rare gems that you should become familiar with, both for what it can teach and also for its beauty. To learn more about The Visual Mathematics Library, see Ralph Abraham’s webpage.

Friday, December 7, 2012

Lord Rayleigh, Biological Physicist

I am a big fan of Victorian physicists. Among my heroes are Faraday, Maxwell, and Kelvin. Another leading Victorian was John William Strutt, also known as Lord Rayleigh (1842-1919). Russ Hobbie and I mention Rayleigh in the 4th edition of Intermediate Physics for Medicine and Biology, in the context of Rayleigh Scattering. In Chapter 15 on the interaction of x-rays with matter, we write
“A photon can also scatter elastically from an atom, with none of the electrons leaving their energy levels. This (γ, γ) process is called coherent scattering (sometimes called Rayleigh scattering), and its cross section is σcoh. The entire atom recoils; if one substitutes the atomic mass in Eqs. 15.15 and 15.16, one finds that the atomic recoil kinetic energy is negligible.
In Rayleigh scattering, the oscillating electric field in an electromagnetic wave exerts a force on electrons. These electrons are displaced by this force, and therefore oscillate at the same frequency as the wave. An oscillating charge emits electromagnetic radiation. The net result is scattering of the incident wave. If the electrons are free, this is known as Thomson scattering. If the electrons are bound to an atom, and the frequency of the light is less than the natural frequency of oscillation of the bound electrons, then it is known as Rayleigh scattering. Light scattering is complicated when the wavelength is similar to or smaller than the size of the scatterer, because light scattered from different regions within the particle interfere. However, Rayleigh scattering assumes that the wavelength is large compared to the size of the scatterer, so interference is not important.

Rayleigh scattering not only plays a role in the scattering of x-rays, but also is responsible for the scattering of visible light. The Rayleigh scattering cross section varies as the 4th power of the frequency, or inversely with the 4th power of the wavelength. When we look at the sky, we see the scattered light. Since the short wavelength blue light is scattered much more than the long wavelength red light, the sky appears blue.

Lord Rayleigh made other important contributions to physics. For example, he wrote an influential book on the Theory of Sound, and he won the Nobel Prize in 1904 for his discovery of the element argon. He succeeded Maxwell as the Cavendish Professor of Physics (see this video: to learn more).

Was Rayleigh a biological physicist? Yes! Rayleigh was one of the first to explain how we localize sound. His Duplex Theory suggests that we can determine the direction a sound came by sensing the arrival time difference at each of our two ears for low frequencies, and sensing the intensity difference between the ears for high frequencies.

Lord Rayleigh was born 170 years ago this fall (November 12, 1842). J. J. Thomson studied under Rayleigh, and Ernest Rutherford studied under Thomson. Previously in this blog, I described how I am descended, academically speaking, from Rutherford. This means Lord Rayleigh is, again academically speaking, my great-great-great-great-great-great grandfather.

Friday, November 30, 2012

A Dangerous Error in the Dilution of 25 Percent Albumin

In Problem 5 in Chapter 5 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I consider what happens if a drug is administered without carefully considering osmotic effects. The problem refers to a letter to the editor in the New England Journal of Medicine from Donald Steinmuller.
"A Dangerous Error in the Dilution of 25 Percent Albumin
To the Editor: Physicians and pharmacists should be alert to a serious error that can occur in the preparation of replacement albumin solutions for plasmapheresis.

Plasmapheresis was performed in an elderly man who had myeloma with renal insufficiency. One plasma volume exchange was ordered, with 5 percent albumin as the replacement solution, with calcium, potassium, and magnesium supplements. Because of the lack of availability of 5 percent albumin, the hospital pharmacy used 25 percent albumin and diluted this solution 1:4 with sterile water to achieve a 5 percent solution. Reference was made to Trissel’s Handbook on Injectable Drugs, the 1994 edition of which states, 'A 5% solution may be prepared from the 25% product by adding 1 volume of the 25% albumin to 4 volumes of sterile water or an infusion solution such as dextrose 5% in water or sodium chloride 0.9%.' The pharmacist used sterile water as described by the handbook, resulting in a hypo-osmolar solution that caused severe hemolysis in the patient. The hematocrit dropped 7.3 points, and renal failure developed.

This flagrant error in instructing dilution with water was only partially corrected in the 1996 edition of Trissel's handbook. This edition states, 'If sterile water for injection is the diluent, the tonicity of the diluted solution must be considered. Substantial reduction in tonicity creates the potential for hemolysis.' In view of the osmolarity of the 25 percent albumin solution diluted with sterile water (approximately 36 mOsm per liter), one should never use water to dilute 25 percent albumin.

The problem is aggravated by the label on the 25 percent albumin solution. The label states that 100 ml of 25 percent albumin is 'osmotically equivalent to 500 ml of plasma.' This statement is not true. It confuses the osmotic and oncotic effects. The oncotic effect of 100 ml of the 25 percent solution is equivalent to 500 ml of plasma, but since the concentration of saline in the 25 percent solution is isotonic with plasma, the osmotic effect of 100 ml of the 25 percent is equivalent to only 100 ml of plasma.

The handbook and product label need to be corrected as soon as possible to prevent this error in the future. Pharmacies and the medical community should be alert to this potentially life-threatening error.”
This letter makes the distinction between osmotic pressure and oncotic pressure, that part of the osmotic pressure caused by large colloidal particles such as albumin.

Before publication, the editor sent the letter to experts at the FDA, who responded with a separate letter that immediately followed Steinmuller’s
“…Including the case reported by Dr. Steinmuller, the FDA is aware of four cases of hemolysis that have occurred since 1994 during or after plasmapheresis when albumin (human) 25 percent was diluted to a 5 percent solution with the use of sterile water for injection…. The FDA has taken the following steps to advise the medical community of this potentially serious problem. First, the FDA is recommending to manufacturers of albumin (human) 25 percent that the package inserts for their products be revised to include … information on acceptable diluents, such as 0.9 percent sodium chloride or 5 percent dextrose in water…”
The FDA letter triggered a third letter from Richard Kravath, of Kings County Hospital Center in Brooklyn
“…Although the response by the representatives of the Food and Drug Administration (FDA) is correct with regard to the recommendation that dilution with 5 percent glucose, instead of water alone, would prevent hemolysis, use of this solution would not prevent the possible development of hyponatremia and brain swelling if the solution were used rapidly in large volumes, as in plasma exchange. Owing to its oncotic pressure, the albumin would tend to remain in the plasma compartment, whereas the glucose would rapidly leave the circulation, enter the interstitial fluid and then enter cells, become metabolized, and no longer exert an osmotic effect….

Glucose solutions should not be used to replace plasma or other extracellular fluids. Sodium chloride 0.9 percent (154 mmol per liter) is a reasonable alternative, but a more physiologic solution, one that more closely resembles plasma, would be even better.”
The authors of the handbook quoted by Steinmuller also wrote a letter, in which they quoted the corrected 9th edition of their book (Steinmuller had cited the 8th edition) and claimed (somewhat lamely, in my opinion) that “…It is common knowledge that large volumes of very hypotonic solutions should not be administered intravenously….” and “…This situation points to the need for health care practitioners to use up-to-date references...”.

Finally, the FDA experts responded to these last two letters.
“…Plasma exchange or plasmapheresis represents a unique circumstance because, in formulating the replacement solution, one must take into account not only the loss of endogenous plasma proteins (principally, but not exclusively, albumin) but also the fact that significant quantities of electrolytes such as sodium and chloride are being removed by the procedure...”
This exchange of letters is fascinating, and emphasizes the importance of the osmotic effects that Russ and I discuss in Chapter 5. The original letter by Steinmuller, which highlights an important medical issue, also makes for an instructive homework problem.

Let me conclude by noting an error (now listed in the book errata, downloadable at the book website) in the 4th edition of Intermediate Physics for Medicine and Biology. Russ and I have the title of Steimnuller’s letter incorrect. On page 133 we write “15 percent” when the title of the letter actually says “25 percent”. The homework problem correctly uses 25 %, and needs no change. Below is the corrected citation:
Steinmuller, D. R. (1998) A dangerous error in the dilution of 25 percent albumin. N. Engl. J. Med. 338:1226.

Friday, November 23, 2012

Marie Curie

Marie Curie (1867-1934) is one of the few scientists who received two Nobel Prizes: for Physics in 1903, and for Chemistry in 1911. Russ Hobbie and I don’t discuss Curie extensively in the 4th edition of Intermediate Physics for Medicine and Biology, but in Chapter 17 on Nuclear Medicine we do introduce the unit of radioactive activity named for her.
“The activity A(t) is the number of radioactive transitions (or transformations or disintegrations) per second. The SI unit of activity is the becquerel (Bq):

1 Bq = 1 transition s−1.

The earlier unit of activity, which is still used occasionally, is the curie (Ci):

1 Ci=3.7 × 1010 Bq,
1 μCi = 3.7 ×104 Bq.”

Several excellent articles were published about Marie Curie and her husband/collaborator Pierre Curie for the centennial of her 1898 discovery of radium. Saenger and Adamek’s article in the journal Medical Physics states
“Marie Curie’s activities and research left her imprint on nuclear medicine, which continues to this day. Much of her impact is related to the role of women in science, biology, and medicine. She successfully overcame struggles for recognition in the first decades of this century. One of her major achievements was the development of field-radiography for wounded soldiers in World War I. Her continued endeavors to provide radium therapy for cancer was a giant step for humanity. She worked unceasingly in the laboratory to separate and identify radioactive elements of the periodic table. The standardization of these elements resulted in the 1931 report of the International Radium-Standards Commission and the posthumous two-volume Radio-aktivite´.”
The abstract of Mould’s article in the British Journal of Radiology begins
“This review celebrates the events of 100 years ago to the month of publication of this December 1998 issue of the British Journal of Radiology, when radium was discovered by the Curies. This followed the earlier discovery in November 1895 of X-rays by Röntgen, which has already been reviewed in the British Journal of Radiology [1] and the discovery in March 1896, by Becquerel, of the phenomenon of radioactivity, which introduces this review. This is particularly relevant as Marie Curie was in 1897 a research student in Becquerel's laboratory. Marie Curie's life in Poland prior to her 1891 departure for Paris is included in this review as are other aspects of her life and work such as her work in World War I with radiological ambulances (known as "Little Curies") on the battlefields of France and Belgium, early experiments with radium and the founding of the Institut du Radium in Paris and of the Radium Institute in Warsaw. Wherever possible I have included appropriate quotations in Marie Curie's own words [2-4] and each section is related in some way to the life and work of Maria or Pierre. This review is completed with details of the re-interment of the bodies of Pierre and Marie on 20 April 1995 in The Panthéon, Paris.”
Excellent overviews of Curie’s life and work are provided by the AIP Center for the History of Physics and the Official Website of the Nobel Prize. You can read about the discovery of radium in Maria’s own words here. And for all you dear readers who prefer Saturday morning cartoons to learned articles, watch this; it doesn’t include any complex or controversial stuff, like the Langevin affair, but it is enjoyable in its own simple way.

Recently, the visual artist/filmmaker/writer Quintan Ana Wikswo was granted access to Marie Curie’s laboratory in Paris for “creating performance films and photographs for ... LUMINOSITY: THE PASSIONS OF MARIE CURIE, a multimedia opera by composer Pamela Madsen". Wikswo describes her ongoing work and previews some of her photographs in her blog Bumblemoth.
“To see her books, her equipment, to stand at her desk, to see her beakers and centrifuges and shelves of chemicals…it’s a kind of searing existential therapy, and anyone visiting Paris should make the effort to spend a few moments at her lab. Why? It’s an antidote, at the very least. I work half-days at her lab, and then explore art museums of Paris in the off hours. The contrast is shocking and disturbing. Inspiring and sorrowful.”
At Oakland University, I work in the College of “Arts and Sciences”. Wikswo has found her own niche at the intersection of these two rarely-overlapping endeavors. I look forward to seeing the completed project.

Friday, November 16, 2012

The Sinogram

I love sinograms. They are rare and fascinating mixtures of science and art, and often are quite beautiful. One should be able to look at a sinogram and intuitively picture the two-dimensional image. Unfortunately, I rarely can do this, except for the most simple examples.

Russ Hobbie and I define the sinogram in the 4th edition of Intermediate Physics for Medicine and Biology. We explain how to calculate the projection, F(θ, x’), from the image, f(x,y). This transformation and its inverse--determining f(x,y) from F(θ,x’)--is at the heart of many imaging algorithms, such as those used in computed tomography.
“The process of calculating F(θ, x’) from f(x, y) is sometimes called the Radon transformation. When F(θ, x’) is plotted with x’ on the horizontal axis, θ on the vertical axis, and F as the brightness or height on a third perpendicular axis, the resulting picture is called a sinogram. For example, the projection of f(x, y) = δ(x − x0)δ(y − y0) is F(θ, x’) = δ(x’ − (x0 cos θ + y0 sin θ)). A plot of this object and its sinogram is shown in Fig. 12.17.”
Figure 12.17 does indeed contain a sinogram, but a very simple one: the sinogram of a point is just a sine wave. The reader is asked to produce a somewhat more complicated sinogram in homework Problem 29.
Problem 29 An object consists of three δ functions at (0, 2), (√3,−1), and (−√3,−1). Draw the sinogram of the object.
This sinogram consists of three braided sine waves. I like this example, because it is simple enough that you the reader should be able to reason out the structure of the sinogram by imagining the projection in your head, but it is complicated enough that it is not trivial.

When preparing the 4th edition of Intermediate Physics for Medicine and Biology, I derived a couple new homework problems (Chapter 12, Problems 23 and 24) for which the inverse transformation can be solved analytically. I think these are useful exercises that build intuition with the Fourier Transform method of reconstructing an image (see Fig. 12.20, top path). It occurs to me now, however, that while these problems do provide insight and practice for the mathematically inclined reader, they also offer the opportunity to further illustrate the sinogram. So this week I made the figures below, showing the image f(x,y) on the left and the corresponding sinogram F(θ,x’) on the right, for the functions in Problems 23 and 24.
Problem 23

Problem 24

Let us try to interpret these pictures qualitatively. The vertical axis in the sinogram (right panel) indicates the angle, specifying the direction of the projection (the direction that the x-rays come from, to use CT terminology). The bottom of the θ axis is an angle of zero indicating x-rays are incident on the image from the bottom, the middle of the θ axis is x-rays incident from the left, and the top of the θ axis is x-rays incident from the top (see Fig. 12.12). Some authors extend the θ axis so it ranges from 0 to 360°, but to me that seems unnecessary since having the x-rays come from one side or the opposite side does not matter; it provides no new information. It is best if you, dear reader, pause now and stare at these sinograms until you understand how they relate to the image. If you really want to build your intuition, cover the left panel, and try to predict what the hidden image looks like from just the right panel. Or, solve homework Problems 30 and 31 in Chapter 12, and then plot both the image and its sinogram like I do above.

This website has some nice examples of sinograms. For instance, a sinogram of a line is just a point. Think about it and sketch some projections to convince yourself this is correct. Also this website shows a sinogram of a square located away from the center of the image (it looks like the sinogram above for Fig. 23, but with interesting bright curves tenuously weaving throughout the sinogram arising from the corners of the square). Finally, the website shows the sinogram of an image known as a Shepp-Logan head phantom. (Warning, the website displays its sinograms rotated by 90° compared to the way Russ and I plot them; it plots the angle along the horizontal axis.) This video provides additional insight into the construction of the sinogram for the Shepp-Logan head phantom.

One of my favorite images is shown here; a detailed picture of a cross section of a torso, and its lovely sinogram. If you can do the inverse transformation of this complicated sinogram in your head, you’re a better medical physicist than I am.

Friday, November 9, 2012

The Hydrogen Spectrum

One of the greatest accomplishments of atomic physics is Neils Bohr’s model for the structure of the hydrogen atom, and his prediction of the hydrogen spectrum. While Bohr gets the credit for deriving the formula for the wavelengths, λ, of light emitted by hydrogen—one of the early triumphs of quantum mechanics—it was first discovered empirically from the spectroscopic analysis of Johannes Rydberg. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce Rydberg’s formula in homework Problem 4 of Chapter 14.
Problem 4 (a) Starting with Eq. 14.7, derive a formula for the hydrogen atom spectrum in the form

where n and m are integers. R is called the Rydberg constant. Find an expression for R in terms of fundamental constants. 
(b) Verify that the wavelengths of the spectral lines a-d at the top of Fig. 14.3 are consistent with the energy transitions shown at the bottom of the figure.
Our Fig. 14.3 is in black and white. It is often useful to see the visible hydrogen spectrum (just four lines, b-e in Fig 14.3) in color, so you can appreciate better the position of the emission lines in the spectrum.

The hydrogen lines in the visible part of the spectrum are often referred to as the Balmer series, in honor of physicist Johann Balmer who discovered this part of the spectrum before Rydberg. Additional Balmer series lines exist in the near ultraviolet part of the spectrum (the thick band of lines just to the left of line e at the top of Fig. 14.3). All the Balmer series lines can be reproduced using the equation in Problem 4 with n = 2.

An entire series of spectral lines exists in the extreme ultraviolet, called the Lyman series, shown at the top of Fig. 14.3 as the line labeled a and the lines to its left. These lines are generated by the formula in Problem 4 using n = 1. The new homework problem below will help the student better understand the hydrogen spectrum.
Section 14.2

Problem 4 ½ The Lyman series, part of the spectrum of hydrogen, is shown at the top of Fig. 14.3 as the line labeled a and the band of lines to the left of that line. Create a figure like Fig. 14.3, but which shows a detailed view of the Lyman series. Let the wavelength scale at the top of your figure range from 0 to 150 nm, as opposed to 0-2 μm in Fig. 14.3. Also include an energy level drawing like at the bottom of Fig. 14.3, in which you indicate which transitions correspond to which lines in the Lyman spectrum. Be sure to indicate the shortest possible wavelength in the Lyman spectrum, show what transition that wavelength corresponds to, and determine how this wavelength is related to the Rydberg constant.
Many spectral lines can be found in the infrared, known as the Paschen series (n = 3), the Brackett series (n = 4) and the Pfund series (n = 5). The Paschen series is shown as lines f, g, h, and i in Fig. 14.3, plus the several unlabeled lines to their left. The Paschen, Brackett, and Pfund series overlap, making the hydrogen infrared spectrum more complicated than its visible and ultraviolet spectra. In fact, the short-wavelength lines of the Brackett series would appear at the top of Fig. 14.3 if all spectral lines were shown.

Rydberg’s formula, given in Problem 4, nicely summarizes the entire hydrogen spectrum. Johannes Rydberg is a Sweedish physicist (I am 3/8th Sweedish myself). His entry in Asimov's Biographical Encyclopedia of Science and Technology reads
“RYDBERG, Johannes Robert (rid’bar-yeh) Swedish physicist Born: Halmstad, November 8, 1854. Died: Lund, Malmohus, December 28, 1919.

Rydberg studied at the University of Lund and received his Ph.D. in mathematics in 1879, and then jointed the faculty, reaching professorial status in 1897.

He was primarily interested in spectroscopy and labored to make sense of the various spectral lines produced by the different elements when incandescent (as Balmer did for hydrogen in 1885). Rydberg worked out a relationship before he learned of Balmer’s equation, and when that was called to his attention, he was able to demonstrate that Balmer’s equation was a special case of the more general relationship he himself had worked out.

Even Rydberg’s equation was purely empirical. He did not manage to work out the reason why the equation existed. That had to await Bohr’s application of quantum notions to atomic structure. Rydberg did, however, suspect the existence or regularities in the list of elements that were simpler and more regular than the atomic weights and this notion was borne out magnificently by Moseley’s elucidation of atomic numbers.”
Yesterday was the 158th anniversary of Rydberg’s birth.

Friday, November 2, 2012

Art Winfree and Cellular Excitable Media

Ten years ago Art Winfree died. I've written about Winfree in this blog before (for example, see here, and here). He shows up often in the 4th edition of Intermediate Physics for Medicine and Biology; Russ Hobbie and I cite Winfree’s research throughout our discussion of nonlinear dynamics and cardiac electrophysiology.

One place where Winfree’s work impacts our book is in Problems 39 and 40 in Chapter 10, discussing cellular automata. Winfree didn’t invent cellular automata, but his discussion of them in his wonderful book When Time Breaks Down is where I first learned about the topic.

“Box 5.A: A Cellular Excitable Medium
Take a pencil and a sheet of tracing paper and play with Figure 5.2 [a large hexagonal array of cells] according to the following game rules … Each little hexagon in this honeycomb is supposed to represent a cell that may be excited for the duration of one step (put a ‘0’ in the cell) or refractory (after the excited moment, replace the ‘0’ with a ‘1’) or quiescent (after that erase the ‘1’) until such time as any adjacent cell becomes excited: then pencil in a ‘0’ in the next step.
If you start with no ‘0’s,’ you’ll never get any, and this simulation will cost you little effort. If you start with a single ‘0’ somewhere, it will next turn to ‘1’ while a ring of 6 neighbors become infected with ‘0’. As the hexagonal ring of ‘0’s’ propagates, it is followed by a concentric ring of ‘1’ refractoriness, right to the edge of the honeycomb, where all vanish.
Now see what happens if you violate the rules just once by erasing a segment of that ring wave when it is about halfway to the edges: you will have created a pair of counter-rotating vortices (alias phase singularities), each of which turns out to be a source of radially propagating waves.
(Stop reading until you have played some.)
You may feel a bit foolish, since this is obviously supposed to mimic action potential propagation, and the caricature is embarrassingly crude. Which aspects of its behavior are realistic and which others are merely telling us ‘honeycombs are not heart muscle’? The way to find out is to undertake successively more refined caricatures until a point of diminishing returns is reached. For most purposes, it is reached surprisingly soon.”
I consider cellular automata—whose three simple rules can be mastered by a child--to be among the best tools for illustrating cardiac reentry . In fact, I like this model so much that I generalized it to account for electrical simulation that produces adjacent regions of depolarization and hyperpolarization (Sepulveda et al., 1989; read more about that paper here). In “Virtual Electrodes Made Simple: A Cellular Excitable Medium Modified for Strong Electrical Stimuli,” published in the Online Journal of Cardiology, I added a fourth rule
“During a cathodal stimulus, the state of the cell directly under the electrode and its four nearest neighbors in the direction perpendicular to the fibers change to the excited state, and the two remaining nearest neighbors in the direction parallel to the fibers change to the quiescent state, regardless of their previous state.”
Using this simple model, I was able to initiate “quatrefoil reentry” (Lin et al., 1999; read more here). I also could reproduce most of the results of a simulation of the “pinwheel experiment” (a point stimulus applied near the end of the refractory period of a previous planar wave front) predicted by Lindblom et al. (2000). I concluded
“This extremely simple cellular excitable medium--which is nothing more than a toy model, stripped down to contain only the essential features--can, with one simple modification for strong stimuli, predict many interesting and important phenomena. Much of what we have learned about virtual electrodes and deexcitation is predicted correctly by the model (Efimov et al., 2000; Trayanova, 2001). I am astounded that this simple model can reproduce the complex results obtained by Lindblom et al. (2000). The model provides valuable insight into the essential mechanisms of electrical stimulation without hiding the important features behind distracting details.”
My online paper came out in 2002, the same year that Winfree died. In an obituary, Steven Strogatz wrote
“When Art Winfree died in Tucson on November 5, 2002, at the age of 60, the world lost one of its most creative scientists. I think he would have liked that simple description: scientist. After all, he made it nearly impossible to categorize him any more precisely than that. He started out as an engineering physics major at Cornell (1965), but then swerved into biology, receiving his PhD from Princeton in 1970. Later, he held faculty positions in theoretical biology (Chicago, 1969-72), in the biological sciences (Purdue, 1972-1986), and in ecology and evolutionary biology (University of Arizona, from 1986 until his death).

So the eventual consensus was that he was a theoretical biologist. That was how the MacArthur Foundation saw him when it awarded him one of its "genius" grants (1984), in recognition of his work on biological rhythms. But then the cardiologists also claimed him as one of their own, with the Einthoven Prize (1989) for his insights about the causes of ventricular fibrillation. And to further muddy the waters, our own community honored his achievements with the 2000 AMS-SIAM Norbert Wiener Prize in Applied Mathematics, which he shared with Alexandre Chorin.

Aside from his versatility, what made Winfree so special (and in this way he was reminiscent of Wiener himself) was the originality of the problems he tackled; the sparkling creativity of his methods and results; and his knack for uncovering deep connections among previously unrelated parts of science, often guided by geometrical arguments and analogies, and often resulting in new lines of mathematical inquiry.”

Friday, October 26, 2012

The Logistic Map

In Section 10.8 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the logistic map, a difference equation that can describe phenomena such as population dynamics. We are by no means the first to use the logistic map to illustrate deterministic chaos. Indeed, it has become the canonical example of chaos since Robert May published Simple Mathematical Models With Very Complicated Dynamics in 1976 (Nature, Volume 261, Pages 459-467). This paper has been cited nearly 2500 times, implying that it has had a major impact.

Russ and I write the logistic equation as (Eq. 10.36 in our book)

xj+1 = a xj (1 – xj)

where xj is the population in the jth generation. Our first task is to determine the equilibrium value for xj.
“The equilibrium value x* can be obtained by solving Eq. 10.36 with xj+1 = xj = x*:

x* = a x* (1 – x*) = 1 – 1/a.

Point x* can be interpreted graphically as the intersection of Eq. 10.36 with the equation xj+1 = xj as shown in Fig. 10.22. You can see from either the graph or from Eq. 10.37 that there is no solution for positive x if a is less than 1. For a = 1 the solution occurs at x* = 0. For a = 3 the equilibrium solution is x* = 2/3. Figure 10.23 shows how, for a = 2.9 and an initial value x0 = 0.2, the values of xj approach the equilibrium value x* = 0.655. This equilibrium point is called an attractor.

Figure 10.23 also shows the remarkable behavior that results when a is increased to 3.1. The values of xj do not come to equilibrium. Rather, they oscillate about the former equilibrium value, taking on first a larger value then a smaller value. This is called a period-2 cycle. The behavior of the map has undergone period doubling. What is different about this value of a? Nothing looks strange about Fig. 10.22. But it turns out that if we consider the slope of the graph of xj+1 vs xj at x*, we find that for a greater than 3 the slope of the curve at the intersection has a magnitude greater than 1.”
Usually, when Russ and I say something like “it turns out”, we include a homework problem to verify the result. Homework 34 in Chapter 10 does just this; the reader must prove that the magnitude of the slope is greater than 1 for a greater than 3.

One theme of Intermediate Physics for Medicine and Biology is the use of simple, elementary examples to illustrate fundamental ideas. I like to search for such examples to use in homework problems. One example that has great biological and medical relevance is discussed in Problems 37 and 38 (a model for cardiac electrical dynamics based on the idea of action potential restitution). But when reading May’s review in Nature, I found another example that—while it doesn’t have much direct biological relevance—is as simple or even simpler than the logistic map. Below is a homework problem based on May’s example.
Section 10.8

Problem 33 ½ Consider the difference equation
     (a) Plot xn+1 versus xn for the case of a=3/2, producing a figure analogous to Fig. 10.22.
     (b) Find the range of values of a for which the solution for large n does not diverge to infinity or decay to 0. You can do this using either arguments based on plots like in part (a), or using numerical examples.
     (c) Find the equilibrium value x* as a function of a, using a method similar to that in Eq. 10.37.
     (d) Determine if this equilibrium value is stable or unstable, based on the magnitude of the slope of the xn+1 versus xn curve.
     (e) For a = 3/2, calculate the first 20 values of xn using 0.250 and 0.251 as initial conditions. Be sure to carry your calculations out to at least five significant figures. Do the results appear to be chaotic? Are the results sensitive to the initial conditions?
     (f) For one of the data sets generated in part (e), plot xn+1 versus xn for 25 values of n, and create a plot analogous to Fig. 10.27. Explain how you could use this plot to distinguish chaotic data from a random list of numbers between zero and one.

Friday, October 19, 2012

Ernest Rutherford

Who is the greatest physicist never mentioned by name in the 4th edition of Intermediate Physics for Medicine and Biology? Russ Hobbie and I allude to Newton, Maxwell, Faraday, Bohr, Einstein, and many others. But a search for the name “Rutherford” comes up empty. In my opinion, Ernest Rutherford is the greatest physicist absent from our book. Ironically, he is also one of my favorite physicists; a colorful character who rivals Faraday as the greatest experimental scientist of all time.

Rutherford (1871-1937) was born in New Zeeland, and attended Cambridge University in England on a scholarship. His early work was on radioactivity, a subject discussed in Chapter 17 of our textbook. Asimov’s Biographical Encyclopedia of Science and Technology states
“[Rutherford] was one of those who, along with the Curies, had decided that the rays given off by radioactive substances were of several different kinds. He named the positively charged ones alpha rays and the negatively charged ones beta rays….Between 1906 and 1909 Rutherford, together is his assistant, Geiger, studied alpha particles intensively and proved quite conclusively that the individual particle was a helium atom with its electrons removed.

Rutherford’s interest in alpha particles led to something greater still. In 1906, while still at McGill in Montreal, he began to study how alpha particles are scattered by thin sheets of metal….From this experiment Rutherford evolved the theory of the nuclear atoms, a theory he first announced in 1911….

For working out the theory of radioactive disintegration of elements, for determining the nature of alpha particles, [and] for devising the nuclear atom, Rutherford was awarded the 1908 Nobel Prize in chemistry, a classification he rather resented, for he was a physicist and tended to look down his nose at chemists….

Rutherford was … the first man ever to change one element into another as a result of the manipulations of his own hands. He had achieved the dream of the alchemists. He had also demonstrated the first man-made ‘nuclear reaction’…

He was buried in Westminster Abbey near Newton and Kelvin.”
Rutherford also measured the size of the nucleus. To explain his alpha particle scattering experiments, he derived his famous scattering formula (see Chapter 4 of Eisberg and Resnick for details). He found that his formula worked well except when very high energy alpha particles are fired at low atomic-number metal sheets. For instance, results began to deviate from his formula when 3 MeV alpha particles are fired at aluminum. The homework problem below explains how to estimate the size of the nucleus from this observation. This problem is based on data shown in Fig. 4-7 of Eisberg and Resnick’s textbook.
Section 17.1

Problem ½  An alpha particle is fired directly at a stationary aluminum nucleus. Assume the only interaction is the electrostatic repulsion between the alpha particle and the nucleus, and the aluminum nucleus is so heavy that it is stationary. Calculate the distance of their closest approach as a function of the initial kinetic energy of the alpha particle. This calculation is consistent with Ernest Rutherford’s alpha particle scattering experiments for energies lower than 3 MeV, but deviates from his experimental results for energies higher than 3 MeV. If the alpha particle enters the nucleus, the nuclear force dominates and the formula you calculated no longer applies. Estimate the radius of the aluminum nucleus.
To learn more about Ernest Rutherford and his groundbreaking experiments, I recommend the book Rutherford: Simple Genius by David Wilson.

In addition to his fundamental contributions to physics, I have a personal reason for liking Rutherford. Academically speaking, he is my great-great-great-great grandfather. My PhD advisor was John Wikswo, who got his PhD working under William Fairbank at Stanford. Fairbank got his PhD under Cecil Lane, who studied under Etienne Bieler, who worked for James Chadwick (discoverer of the neutron), who was a student of Rutherford’s

Ernest Rutherford died (needlessly) on October 19, 1937; 75 years ago today.

Friday, October 12, 2012

The Gaussian integral

One of my favorite “mathematical tricks” is given in Appendix K of the 4th edition of Intermediate Physics for Medicine and Biology. The goal is to calculate the integral of the Gaussian function, e-x2, or bell shaped curve. (This is often called the Gaussian Integral). The indefinite integral cannot be expressed in terms of elementary functions (in fact, “error functions” are defined as the integral of the Gaussian), but the definite integral integrated over the entire x axis (from –∞ to ∞) is amazingly simple: the square root of π. Here is how Russ Hobbie and I describe how to derive this result:
Integrals involving e-ax2 appear in the Gaussian distribution. The integral
can also be written with y as the dummy variable:
There can be multiplied together to get
A point in the xy plane can also be specified by the polar coordinates r and θ (Fig. K.1). The element of area dxdy is replaced by the element rdrdθ:
To continue, make the substitution u = ar2, so that du = 2ardr. Then
The desired integral is, therefore,
Of course, if you let a =1, you get the simple result I mentioned earlier. Isn’t this a cool calculation?

To learn more, click here. For those of you who prefer video, click here.

The integral and function are, of course, named after the German mathematician Johann Karl Friedrich Gauss (1777-1855). Asimov’s Biographical Encyclopedia of Science and Technology (2nd Revised Edition) says
“Gauss, the son of a gardener and a servant girl, had no relative of more than normal intelligence apparently, but he was an infant prodigy in mathematics who remained a prodigy all his life. He was capable of great feats of memory and of mental calculation. There are those with this ability who are only average or below-average mentality, but Gauss was clearly a genius. At the age of three, he was already correcting his father’s sums, and all his life he kept all sorts of numerical records, even useless ones such as the length of lives of famous men, in days. He was virtually mad over numbers.

Some people consider him to have been one of the three great mathematicians of all time, the others being Archimedes and Newton.”

Friday, October 5, 2012

The Truth About Terahertz

In Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Terahertz Radiation.
“For many years, there were no good sources or sensitive detectors for radiation between microwaves and the near infrared (0.1-100 THz; 1 THz = 1012 Hz). Developments in optoelectronics have solved both problems, and many investigators are exploring possible medical uses of THz radiation (“T rays”). Classical electromagnetic wave theory is needed to describe the interactions, and polarization (the orientation of the E vector of the propagating wave) is often important. The high attenuation of water to this frequency range means that studies are restricted to the skin or surface of organs such as the esophagus that can be examined endoscopically. Reviews are provided by Smye et al. (2001), Fitzgerald et al. (2002), and Zhang (2002).”
(By the way, apologies to Dr. N. N. Zinovev, a coauthor on the Fitzgerald et al. paper whose last name is spelled incorrectly in our book.) 

In the September 2012 issue of the magazine IEEE Spectrum, Carter Armstrong (a vice president of engineering at L-3 Communications, in San Francisco) reviews some of the challenges facing the development of Terahertz radiation. His article The Truth About Terahertz begins
“Wirelessly transfer huge files in the blink of an eye! Detect bombs, poison gas clouds, and concealed weapons from afar! Peer through walls with T-ray vision! You can do it all with terahertz technology—or so you might believe after perusing popular accounts of the subject.

The truth is a bit more nuanced. The terahertz regime is that promising yet vexing slice of the electromagnetic spectrum that lies between the microwave and the optical, corresponding to frequencies of about 300 billion hertz to 10 trillion hertz (or if you prefer, wavelengths of 1 millimeter down to 30 micrometers). This radiation does have some uniquely attractive qualities: For example, it can yield extremely high-resolution images and move vast amounts of data quickly. And yet it is nonionizing, meaning its photons are not energetic enough to knock electrons off atoms and molecules in human tissue, which could trigger harmful chemical reactions. The waves also stimulate molecular and electronic motions in many materials—reflecting off some, propagating through others, and being absorbed by the rest. These features have been exploited in laboratory demonstrations to identify explosives, reveal hidden weapons, check for defects in tiles on the space shuttle, and screen for skin cancer and tooth decay.

But the goal of turning such laboratory phenomena into real-world applications has proved elusive. Legions of researchers have struggled with that challenge for decades.”
Armstrong then explores the reasons for these struggles. The large attenuation coefficient of T-rays places severe limitations on imaging applications. He then turns specifically to medical imaging.
“Before leaving the subject of imaging, let me add one last thought on terahertz for medical imaging. Some of the more creative potential uses I’ve heard include brain imaging, tumor detection, and full-body scanning that would yield much more detailed pictures than any existing technology and yet be completely safe. But the reality once again falls short of the dream. Frank De Lucia, a physicist at Ohio State University, in Columbus, has pointed out that a terahertz signal will decrease in power to 0.0000002 percent of its original strength after traveling just 1 mm in saline solution, which is a good approximation for body tissue. (Interestingly, the dielectric properties of water, not its conductive ones, are what causes water to absorb terahertz frequencies; in fact, you exploit dielectric heating, albeit at lower frequencies, whenever you zap food in your microwave oven.) For now at least, terahertz medical devices will be useful only for surface imaging of things like skin cancer and tooth decay and laboratory tests on thin tissue samples.”
Following a detailed review of terahertz sources, Armstrong finishes on a slightly more optimistic note.
“There is still a great deal that we don’t know about working at terahertz frequencies. I do think we should keep vigorously pursuing the basic science and technology. For starters, we need to develop accurate and robust computational models for analyzing device design and operation at terahertz frequencies. Such models will be key to future advances in the field. We also need a better understanding of material properties at terahertz frequencies, as well as general terahertz phenomenology.

Ultimately, we may need to apply out-of-the-box thinking to create designs and approaches that marry new device physics with unconventional techniques. In other areas of electronics, we’ve overcome enormous challenges and beat improbable odds, and countless past predictions have been subsequently shattered by continued technological evolution. Of course, as with any emerging pursuit, Darwinian selection will have its say on the ultimate survivors.”
Terahertz radiation is such a big field that one year ago the IEEE introduced a new journal: IEEE Transactions on Terahertz Science and Technology. In the inaugural issue, Taylor et al. examine THz Medical Imaging.

Friday, September 28, 2012

Benedek and Villars, Volume 3

This is the third and final entry in a series of blog entries about Benedek and Villars’ textbook Physics With Illustrative Examples From Medicine and Biology. Today I discuss Volume 3, about electricity and magnetism. In the preface to the first edition of Volume 3, Benedek and Villars write
“With this volume on Electricity and Magnetism, we complete the third and final volume of our textbooks on Physics, with Illustrative Examples from Medicine and Biology. We believe that this volume is as unique as our previous books on Classical Mechanics (Vol. 1) and Statistical Physics (Vol. 2). Here, we continue our program of interweaving into the rigorous development of classical physics, an analysis and clarification of a wide variety of important phenomena in physical chemistry, biology, physiology, and medicine.”
The topics covered in Volume 3 are similar to those Russ Hobbie and I discuss in Chapters 6-9 in the 4th edition of Intermediate Physics for Medicine and Biology. Because I do research in the fields of bioelectricity and biomagnetism, you might expect that this would be my favorite volume of the three, but it is not. I don’t find that it contains as many rich and interesting biological examples. Yet it is a solid book, and contains much useful electricity and magnetism.

Before leaving this topic, I should say a few words about George Benedek and Felix Villars. Benedek is currently the Alfred H. Caspary Professor of Physics and Biological Physics in the Department of Physics in the Harvard-MIT Division of Health Sciences and Technology. His group' research program “centers on phase transitions, self-assembly and aggregation of biological molecules. These phenomena are of biological and medical interest because phase separation, self-assembly and aggregation of biological molecules are known to play a central role in several human diseases such as cataract, Alzheimer's disease, and cholesterol gallstone formation.” Villars was born in Switzerland. In the late 1940s, he collaborated with Wolfgang Pauli, and developed Pauli-Villars regularization. He began work at the MIT in 1950, where he collaborated with Herman Feshbach and Victor Weisskopf. He became interested in the applications of physics to biology and medicine, and helped establish the Harvard-MIT Division of Health Sciences and Technology. He died in 2002 at the age of 81.

Friday, September 21, 2012

Benedek and Villars, Volume 2

Last week I discussed volume 1 of Benedek and Villars’ three-volume textbook Physics With Illustrative Examples From Medicine and Biology, which dealt with mechanics. The second volume discusses statistical physics. The preface to their first edition of Volume 2 states
“In the present volume we develop and present the ideas of statistical physics, of which statistical mechanics and thermodynamics are but one part. We seek to demonstrate to students, early in their career, the power, the broad range, and the astonishing usefulness of a probabilistic, non-deterministic view of the origin of a wide range of physical phenomena. By applying this approach analytically and quantitatively to problems such as: the size of random coil polymers; the diffusive flow of solutes across permeable membranes; the survival of bacteria after viral attack; the attachment of oxygen to the binding sites on the hemoglobin molecule; and the effect of solutes on the boiling point and vapor pressure of volatile solvents; we demonstrate that the probabilistic analysis of statistical physics provides a satisfying understanding of important phenomena in fields as diverse as physics, biology, medicine, physiology, and physical chemistry.”
Many of the topics in Volume 2 of Benedek and Villars are similar to Chapters 3-5 in the 4th edition of Intermediate Physics for Medicine and Biology: the Boltzmann factor, diffusion, and osmotic pressure. As I said last week, I am most interested in those topics Benedek and Villars discuss that are not covered in Intermediate Physics for Medicine and Biology, such as their fascinating description of the use of Poisson statistics by Luria and Delbruck.
“If a bacterial culture is brought into contact with bacteriophage virus particles, the viruses will attack the bacteria and kill them in a matter of hours. However, a small number of bacteria do survive the attack. These survivors will reproduce and pass on to their descendants their resistance to the virus. The form of resistance of the offspring of the surviving bacteria is that their surface does not adsorb the attacking virus. Bacterial strains can also be resistant to metabolic inhibitors, such as streptomycin, penicillin, and sulphonamide. If a bacterial culture is subjected to attack by these antibiotics, the resistant strain will emerge just as in the case of the phage resistant bacteria.

In the early 1940s, Luria and Delbruck were working on ‘mixed infection’ experiments in which the bacteriophage resistant strain of E. coli bacteria were used as indicators in studies they were making on T1 and T2 virus particles. Starting in the Fall of 1942, they began to put aside the mixed infection experiment and asked themselves: What is the origin of those resistant bacterial strains that they were using as indicators?”
They go on to give a detailed description of how Poisson statistics were used by Luria and Delbruck to study mutations.

Russ Hobbie and I discuss the Poisson distribution in our Appendix J. The Poisson distribution is an approximation of the more familiar binomial distribution, applicable for large numbers and small probabilities. One can see how this distribution would be appropriate for Luria and Delbruick, who had large numbers of viruses and a small probability of a mutation.

Russ and I cite Volume 1 of Benedek and Villars’ text in our Chapter 1 on biomechanics. We draw the data for our Fig. 4.12 from Benedek and Villars’ Volume 2. We never cite their Volume 3, about electricity and magnetism, which I will discuss next week.

Friday, September 14, 2012

Benedek and Villars, Volume 1

One early textbook that served as a precursor to Introductory Physics for Medicine and Biology is the 3-volume Physics With Illustrative Examples From Medicine and Biology, by George Benedek and Felix Villars. The first edition, published in 1973, was just bound photocopies of a typewritten manuscript, but a nicely printed second edition appeared in 2000. The preface to Volume 1 of the first edition states
“This is a unique book. It is an introductory textbook of physics in which the development of the principles of physics is interwoven with the quantitative analysis of a wide range of biological and medical phenomena. Conversely, the biological and medical examples serve to vitalize and motivate the learning of physics. By its very nature, this book not only teaches physics, but also exposes the student to topics in fields such as anatomy, orthodedic medicine, physiology, and the principles of hemostatic control.

This book, and its follow-up, Volumes II and III, grew out of an introductory physics course which we have offered to freshmen and sophomores at MIT since 1970. The stimulus for this course came from Professor Irving M. London, MD, Director of the Harvard-MIT Program in Health Sciences and Technology. He convinced us that continued advances in the biological and medical sciences demand that students, researchers, and physicians should be capable of applying the quantitative methods of the physical sciences to problems in the life sciences. We have written this book in the hope that students of the life sciences will come to appreciate the value of training in physics in helping them to formulate, analyze, and solve problems in their own fields.”
This quote applies almost without change (except for replacing MIT with Russ Hobbie’s University of Minnesota) to Intermediate Physics for Medicine and Biology. Clearly the goals and objectives of the two works are the same.

Many of the topics in Volume 1 of Benedek and Villars are similar to those found in Chapters 1 and 10 of Intermediate Physics for Medicine and Biology: biomechanics, fluid dynamics, and feedback. Particularly interesting to me are the topics that Russ Hobbie and I don’t discuss, such as the physiological effects of underwater diving.
“On ascent and descent the diver must arrange to have the pressure of gas in his lungs be the same as that of the surrounding water. He can do this either by breathing out on ascent or by adjusting the output pressure of his compressed air tanks. Second to drowning, the most serious underwater diving accident is produced by taking a full breath of air at depth, and holding this breath as the diver rises to the surface quickly. For example, if the diver did this at 99 ft he would have gas at 4 atm in his lungs. This if fine at 99 ft, but if he holds this total volume of gas on ascending, then at the surface the surrounding water is at 1 atm, and his lungs are holding air at 3 atm. This can do two things. His lungs can rupture, thereby allowing gas to flow into the space between lungs and ribs. This is called pneumothorax. Also the great pressure of air in the lungs can force air bubbles into the blood stream. These air embolisms can then occlude blood vessels in the brain or the coronary circulation, and this can lead to death. Of course, the obvious necessity of balancing pressure in the ears, sinuses, and intestines must be realized.”
Benedek and Villars also have a delightful description of the physiological effects of low air pressure experienced by balloonists. It is too long to reproduce here, but well worth reading.

In the coming weeks, I will discuss Benedek and Villars’ second and third volumes.