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 d²V_e/dx² is largest, with λ the length constant, V_e 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, E_x, and wrote the activating function as λ² dE_x/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 j_m = g_m v, and let the extracellular potential be small so v_i = v. Identify the new source term in the cable equation (the “activating function” for magnetic stimulation), analogous to v_r 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!

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’s 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, their weight increases as the cube of their linear size, but the cross-sectional area of their 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.

Dynamics: The Geometry of Behavior

Dynamics: The Geometry of Behavior,
by Ralph Abraham and Christopher Shaw.

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 foreword 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.

Dynamics: The Geometry of Behavior,
by Ralph Abraham and Christopher Shaw.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I do not suppress mathematical expressions. 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.

Lord Rayleigh, Biological Physicist

Theory of Sound,
by Lord Rayleigh.

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 σ_coh. 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: https://www.youtube.com/watch?v=tkwLavjqsBI 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.

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,” New England Journal of Medicine, Volume 338, Page 1226.

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 Curie’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.

Marie Curie Animated Hero Classics
https://www.youtube.com/watch?v=x-ynt1yY43I&t=205s

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.

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’s 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’s 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’s 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.) The video shown below provides additional insight into the construction of the sinogram for the Shepp-Logan head phantom.

Here is one of my favorite images: a detailed image of a brain, 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. 

An image of a brain, and its sinogram,
adapted from Wikipedia.

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.

(Figure from http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/graphics/hydrogen.gif).

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.

Asimov's Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

Rydberg’s formula, given in Problem 4, nicely summarizes the entire hydrogen spectrum. Johannes Rydberg is a Swedish physicist (I am 3/8th Swedish 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.

Art Winfree and Cellular Excitable Media

When Time Breaks Down,
by Art Winfree.

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. 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.

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)

x_j+1 = a x_j (1 – x_j)

where x_j is the population in the j^th generation. Our first task is to determine the equilibrium value for x_j.

The equilibrium value x* can be obtained by solving Eq. 10.36 with x_j+1 = x_j = 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 x_j+1 = x_j 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 x_j 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 x_j 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 x_j+1 vs x_j 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 x_n+1 versus x_n 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 x_n+1 versus x_n curve.

     (e) For a = 3/2, calculate the first 20 values of x_n 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 x_n+1 versus x_n 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.