Hysteresis and Bistability in the Direct Transition from 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells

When preparing the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added two homework problems (Problems 37 and 38) in Chapter 10 (Feedback and Control) about “cardiac restitution.” These problems contain a fascinating and elegantly simple example of restitution that provides insight into nonlinear dynamics and chaos. Problem 37 begins

Problem 37 The onset of ventricular fibrillation in the heart can be understood in part as a property of cardiac “restitution.” The action potential duration (APD) depends on the previous diastolic interval (DI): the time from the end of the last action potential until the start of the next one. The relationship between APD and DI is called the “restitution curve.” In cardiac muscle, a typical restitution curve has the form

APD_i+1 = 300 (1 – exp(–DI_i/100))

where all times are given in ms. Suppose we apply to the heart a series of stimuli, with period (or “cycle length”) CL. Since APD + DI = CL, we have DI_i+1 = CL – APD_i+1.

The problem then goes on to have the reader do some numerical calculations using various cycle lengths and initial diastolic intervals. Depending on the parameters, you can get (a) a simple 1:1 response between stimulation and action potential, (b) a 2:2 response in which every stimulus triggers an action potential but the APD alternates between long and short, a behavior called “alternans,” (c) a 2:1 response where an action potential is triggered by every second stimulus, with the tissue being refractory and not responding to the other stimuli, and (d) chaos. I have found this model is an excellent way to introduce students to chaotic behavior; even students with a weak mathematics background can understand it. When discussing this mathematical model with students, I often hand out a particularly clear paper to serve as background reading: J. N. Weiss, A. Garfinkel, H. S. Karagueuzian, Z. Qu, and P.-S. Chen (1999) “Chaos and the Transition to Ventricular Fibrillation: A New Approach to Antiarrhythmic Drug Evaluation,” Circulation, Volume 99, Pages 2819–2826.

Problem 38 explores how to understand this behavior by analyzing the slope of the restitution curve. If the slope is too steep, the behavior becomes more complex. Part (d) of Problem 38 says

Suppose you apply a drug to the heart that can change the restitution curve to

APD_i+1 = 300 (1 – b exp(–DI_i/100)) .

Plot APD as a function of DI for b = 0, 0.5, and 1. What value of b ensures that the slope of the restitution curve is always less than 1? Garfinkel et al. (2000) have suggested that one way to prevent ventricular fibrillation is to use drugs to “flatten” the restitution curve.

There is yet another type of behavior that is not discussed in Problems 37 or 38: a bistable response. Below is a new homework problem that discusses bistable behavior.

Problem 38 ½ Use the restitution curve from Problem 38, with b = 1/3 and CL = 250, to analyze the response of the system with initial diastolic intervals of 50, 60, 70, 80, and 90. You should find that the qualitative behavior depends on the initial condition. Which values of the initial diastolic interval give a 1:1 response, and which give 2:1? Determine the initial value of the DI, to three significant figures, for which the system makes a transition from one behavior to the other. When two qualitatively different behaviors can both occur, depending on the initial conditions, the system is “bistable.” To learn more about such behavior, see Yehia et al. (1999).

The full citation to the paper mentioned at the end of the problem is

Yehia, A. R., D. Jeandupeux, F. Alonso, and M. R. Guevara (1999) “Hysteresis and Bistability in the Direct Transition From 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells,” Chaos, Volume 9, Pages 916–931.

The senior author on this article is Michael Guevara, of the Centre for Applied Mathematics in Bioscience and Medicine at McGill University. The introductory paragraph of their paper is reproduced below.

The majority of cells in the heart are not spontaneously active. Instead, these cells are excitable, being driven into activity by periodic stimulation originating in a specialized pacemaker region of the heart containing spontaneously active cells. This pacemaker region normally imposes a 1:1 rhythm on the intrinsically quiescent cells. However, the 1:1 response can be lost when the excitability of the paced cells is decreased, when there are problems in the conduction of electrical activity from cell to cell, or when the heart rate is raised. When 1:1 synchronization is lost in the intact heart, one of a variety of abnormal cardiac arrhythmias can arise. In single quiescent cells isolated from ventricular muscle, 1:1 rhythm can be replaced by a N+1:N rhythm (N≥2), a period-doubled 2:2 rhythm, or a 2:1 rhythm. We investigate below the direct transition from 1:1 to 2:1 rhythm in experiments on single cells and in numerical simulations of an ionic model of a single cell formulated as a nonlinear system of differential equations. We show that there is hysteresis associated with this transition in both model and experiment, and develop a theory for the bistability underlying this hysteresis that involves the coexistence of two stable fixed-points on a two-branched one-dimensional map.

For those interested in exploring the application of nonlinear dynamics to biology and medicine in more detail, two books Russ and I cite in Intermediate Physics for Medicine and Biology—and which I recommend highly—are From Clocks to Chaos by Leon Glass and Michael Mackey (both also at McGill) and Nonlinear Dynamics and Chaos by Steven Strogatz.

Max Planck and Blackbody Radiation

Max Planck is one of the founders of quantum mechanics, and the fundamental constant governing all quantum phenomena bears his name. His historic contribution arose from the study of thermal radiation. Section 14.7 in the 4th edition of Intermediate Physics for Medicine and Biology analyzes thermal radiation (also known as blackbody radiation), but does not tell the fascinating history behind this advance. In fact, Russ Hobbie and I write “we will not discuss the history of these developments, but will simply summarize the properties of the blackbody radiation function that are important to us.” What better place than this blog to fill in the missing history.

Quantum Physics of Atoms,
Molecules, Solids, Nuclei, and Particles,
by Eisberg and Resnick.

Eisberg and Resnick’s textbook Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles is a good place to learn more (I will quote from the first edition having the silver cover, which I used as an undergraduate). In fact, the opening section of their first chapter addresses this very issue.

At a meeting of the German Physical Society on Dec. 14, 1900, Max Planck read his paper “On the Theory of the Energy Distribution Law of the Normal Spectrum.” This paper, which first attracted little attention, was the start of a revolution in physics. The date of its presentation is considered to be the birthday of quantum physics, although it was not until a quarter century later that modern quantum mechanics, the basis of our present understanding, was developed by Schroedinger and others… Quantum physics represents a generalization of classical physics that includes the classical laws as special cases. Just as relativity extends the range of application of physical laws to the region of high velocities, so quantum physics extends the range to the region of small dimensions; and, just as a universal constant of fundamental significance, the velocity of light c, characterizes relativity, so a universal constant of fundamental significance, now called Planck’s constant h, characterizes quantum physics. It was while trying to explain the observed properties of thermal radiation that Planck introduced this constant in his 1900 paper…

Eisberg and Resnick end their first chapter with “A Bit of Quantum History”

At first Planck was unsure whether his introduction of the constant h was only a mathematical device or a matter of deep physical significance. In a letter to R. W. Wood, Planck called his limited postulate “an act of desperation.” “I knew,” he wrote, “that the problem (of the equilibrium of matter and radiation) is of fundamental significance for physics; I knew the formula that reproduces the energy distribution in the normal spectrum; a theoretical interpretation had to be found at any cost, no matter how high.”

To better understand the mathematics underlying blackbody radiation, try the new homework problem below, based on Eisberg and Resnick’s analysis (you may want to review Sec. 3.7 of our book about the Boltzmann factor before you attempt this problem).

Section 14.7

Problem 22.5 Let us derive the blackbody spectrum, Eq. 14.37.
(a) Assume the energy E_n of radiation with frequency ν is discrete, E_n = h ν n, where n=0, 1, 2, … Let the probability P_n of any state be given by the Boltzmann factor, C e^−nhν/kT. Normalize this probability distribution (that is, find C by setting the sum of the probabilities over all states equal to one).
(b) Find the average energy E_ave for frequency ν by performing the sum E_ave = P₀ E₀ + P₁ E₁ + … .
(c) The number of frequencies per unit volume in the frequency range from ν to ν + dν is 8πν²dν/c³. Multiply the result from (b) by this quantity, to get the energy density of the radiation.
(d) The spectrum of power per unit area emitted from a blackbody is equal to c/4 times the energy density. Find the power per unit area per unit frequency, W_ν(ν,T) (Eq. 14.37).
You may need to use the following two infinite series

1 + x + x² + x³ + … = 1/(1−x) ,

x + 2x² + 3x³ + … = x/(1−x)² .

Therapeutic Touch

Therapeutic touch is a “healing technique” in which a therapist places their hands near a patient and detects or manipulates the patient’s “energy field.” Russ Hobbie and I don’t discuss therapeutic touch in the 4th edition of Intermediate Physics for Medicine and Biology, nor will we include it in future editions. However, since this egregious example of “voodoo science” hasn’t gone away (see http://www.therapeutictouch.org), let me address it here in this blog.

Bob Park described therapeutic touch in his delightful April 3, 1998 entry to his What’s New weekly column.

3. HUMAN ENERGY FIELD: SCIENTIST, AGE 9, TESTS TOUCH THERAPY.
More than 40,000 health professionals have been trained in TT and it's offered by 70 hospitals in the US. And yet no one had ever checked to see if practitioners can, as they claim, tactilely sense such a field—until now. The Journal of the American Medical Association this week published the research of a fourth-grade girl. For a science fair project, the little girl persuaded 21 touch therapists to submit to a beautifully simple test. In 280 trials, the 21 scored 44%. According to the editor of JAMA, reviewers found the study to be “solid gold.” The James Randi Educational Foundation has been offering $1M to anyone who can pass a similar test—only one tried (WN 27 Mar 98) , but a 9-year old must have seemed less threatening. The girl, Emily Rosa of Loveland, CO, now 11, plans to take on magnet therapy next.

Recently, Russ called my attention to Eugenie Mielczarek’s insightful commentary “Magnetic Fields, Health Care, Alternative Medicine and Physics” in the April 2010 edition of Physics and Society, the quarterly newsletter of the Forum of Physics and Society, a division of the American Physical Society. Mielczarek writes

In Therapeutic Touch the protocol requires that a therapist moves his or her hands over the patient’s “energy field,” allegedly “tuning” a purported “aura” of biomagnetic energy that extends above the patient’s body. This is thought to somehow help heal the patient. Although this is less than one percent of the strength of Earth’s magnetic field, corresponds to billions of times less energy than the energy your eye receives when viewing even the brightest star in the night sky, and is billions of times smaller than that needed to affect biochemistry, the web sites of prominent clinics nevertheless market the claims.

Iron, Nature's Universal Element:
Why People Need Iron
and Animals Make Magnets,
by Eugenie Mielczarek.

Mielczarek is an emeritus professor at George Mason University. In 2006 she published a Resource Letter in the American Journal of Physics: “Physical Frontiers in Biology: A Resource for Students and Faculty” (Volume 74, Pages 375–381). Russ and I mentioned this publication in our 2009 “Resource Letter on Medical Physics,” where we wrote that Mielczarek’s letter “begins with a fascinating three-page essay on the role of physics in biology.” This week I discovered that the published black-and-white pictures in that 3-page essay are available in color at Mielczarek’s website. Mielczarek is an editor of the 1993 book Biological Physics, a collection of landmark biological physics papers. One of her research interests is the role of iron in biological systems, and in 2000 she coauthored the book Iron, Nature’s Universal Element: Why People Need Iron and Animals Make Magnets, which I just put onto my summer reading list. We cite this “very readable” book in Section 8.8.3 of Intermediate Physics for Medicine and Biology, but it must have been one of those things that Russ added to the 4th edition because I haven’t read it yet. We also cite Mielczarek’s American Journal of Physics paper “Experimental and Theoretical Models of Nonlinear Behavior" in Chapter 10 of our book.

For more information about the physics of therapeutic touch, see the article “Emerita Professor Makes a Case Against Distance Healing” in the Mason Gazette, and the press release “Think Tank Objects to Taxpayer Funding for Therapeutic Touch, other Alternative Medicine Therapies” from the Center of Inquiry.

Let us hope that hope that Bob Park and Eugenie Mielczarek continue to debunk the techniques of “alternative medicine” when they violate the laws of physics.

PHY 530, Bioelectric Phenomena

This week I finished up my PHY 530 class (Bioelectric Phenomena), which I discussed once before in this blog. Rather than adopting a textbook, I based this graduate class on a collection of scientific papers. Below I list the three dozen papers we studied. It should not be regarded as a “greatest hits” list. Some are Nobel Prize winning papers, but oftentimes I selected a lesser-known article that happened to cover a specific topic I wanted to teach. Many are cited in the 4th edition of Intermediate Physics for Medicine and Biology (indicated by a *). Students were assigned the 16 papers marked in bold: they had to take a quiz on each of these before we discussed them in class, and the exams often contained questions drawn directly from these papers. The other 20 articles are supplementary: consider them recommended reading, rather than required.

I had two goals in the class: to teach the basic elements of bioelectricity, and to lead a workshop on how to write a scientific paper. The students were given two projects (one was to simulate a squid nerve axon using the Hodgkin-Huxley model, and the other was to determine a dipole source from simulated EEG data) and had to write up their results in a brief (4 page maximum) paper having the classic structure: Abstract, Introduction, Methods, Results, Discussion, References. We read essays related to writing scientific papers, such as "What's Wrong With These Equations?" and "Writing Physics," both by N. David Mermin, and learned to use the Science Citation Index. I am pleased with how the class went, and I hope the students were too.

1. A. L. Hodgkin and A. F. Huxley (1939) “Action Potentials Recorded from Inside a Nerve Fiber,” Nature, Volume 144, Pages 710–711. *

2. A. L. Hodgkin and B. Katz (1949) “The Effect of Sodium Ions on the Electrical Activity of the Giant Axon of the Squid,” Journal of Physiology, Volume 108, Pages 37–77.

3. A. L. Hodgkin and A. F. Huxley (1952) “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve,” Journal of Physiology, Volume 117, Pages 500–544. *

4. D. A. Doyle, J. M. Cabral, R. A. Pfuetzner, A. Kuo, J. M. Gulbis, S. L. Cohen, B. T. Chait, and R. MacKinnon (1998) “The Structure of the Potassium Channel: Molecular Basis of K+ Conduction and Selectivity,” Science, Volume 280, Pages 69–77. *

5. O. P. Hamill, A. Marty, E. Neher, B. Sakmann, and F. J. Sigworth (1981) “Improved Patch-Clamp Techniques for High-Resolution Current Recording From Cells and Cell-Free Membrane Patches,” Pflugers Archive, Volume 391, Pages 85–100. *

6. A. L. Hodgkin and W. A. H. Rushton (1946) “The Electrical Constants of a Crustacean Nerve Fibre,” Proceedings of the Royal Society of London, B, Volume 133, Pages 444–479. *

7. W. A. H. Rushton (1951) “A Theory of the Effects of Fibre Size in Medullated Nerve,” Journal of Physiology, Volume 115, Pages 101–122. *

8. R. FitzHugh (1961) “Impulses and Physiological States in Theoretical Models of Nerve Membrane,” Biophysical Journal, Volume 1, Pages 445–466.

9. W. Rall (1962) “Theory of Physiological Properties of Dendrites,” Annals of the New York Academy of Sciences, Volume 96, Pages 1071–1092.

10. F. Rattay (1989) “Analysis of Models for Extracellular Fiber Stimulation,” IEEE Transactions on Biomedical Engineering, Volume 36, Pages 676–682.

11. A. T. Barker, R. Jalimous, and I. L. Freeston (1985) “Non-Invasive Magnetic Stimulation of Human Motor Cortex,” Lancet, Volume 8437, Pages 1106–1107. *

12. M. Hallett and L. G. Cohen (1989) “Magnetism: A New Method for Stimulation of Nerve and Brain,” Journal of the American Medical Association, Volume 262, Pages 538–541. *

13. B. J. Roth, L. G. Cohen and M. Hallett (1991) “The Electric Field Induced During Magnetic Stimulation,” Electroencephalography and Clinical Neurophysiology, Supplement 43, Pages 268–278.

14. R. Plonsey (1974) “The Active Fiber in a Volume Conductor,” IEEE Transactions on Biomedical Engineering, Volume 21, Pages 371–381.

15. B. J. Roth, D. Ko, I. R. von Albertini-Carletti, D. Scaffidi and S. Sato (1997) “Dipole Localization in Patients with Epilepsy Using the Realistically Shaped Head Model,” Electroencephalography and Clinical Neurophysiology, Volume 102, Pages 159–166.

16. M. Schneider (1974) “Effect of Inhomogeneities on Surface Signals Coming From a Cerebral Current-Dipole Source,” IEEE Transactions on Biomedical Engineering, Volume 21, Pages 52–54.

17. B. J. Roth and J. P. Wikswo (1985) “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment,” Biophysical Journal, Volume 48, Pages 93–109. *

18. M. Hamalainen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa (1993) “Magnetoencephalography: Theory, Instrumentation, and Application to Noninvasive Studies of the Working Human Brain,” Reviews of Modern Physics, Volume 65, Pages 413–497. *

19. T.-K. Truong and A. W. Song (2006) “Finding Neuroelectric Activity Under Magnetic-Field Oscillations (NAMO) with Magnetic Resonance Imaging In Vivo,” Proceedings of the National Academy of Sciences, Volume 103, Pages 12598–12601.

20. B. J. Roth and P. J. Basser (2009) “Mechanical Model of Neural Tissue Displacement During Lorentz Effect Imaging,” Magnetic Resonance in Medicine, Volume 61, Pages 59–64.

21. A. T. Winfree (1987) When Time Breaks Down. Princeton Univ Press, Princeton, NJ, Pages 106–107. *

22. B. J. Roth (2002) “Virtual Electrodes Made Simple: A Cellular Excitable Medium Modified for Strong Electrical Stimuli,” The Online Journal of Cardiology, http://sprojects.mmi.mcgill.ca/heart/pages/rot/rothom.html

23. D. W. Frazier, P. D. Wolf, J. M. Wharton, A. S. L. Tang, W. M. Smith and R. E. Ideker (1989) “Stimulus-Induced Critical Point: Mechanism for Electrical Initiation of Reentry in Normal Canine Myocardium,” Journal of Clinical Investigation, Volume 83, Pages 1039–1052.

24. N. Shibata, P.-S. Chen, E. G. Dixon, P. D. Wolf, N. D. Danieley, W. M. Smith, and R. E. Ideker (1988) “Influence of Shock Strength and Timing on Induction of Ventricular Arrhythmias in Dogs,” American Journal of Physiology, Volume 255, Pages H891–H901.

25. J. N. Weiss, A. Garfinkel, H. S. Karagueuzian, Z. Qu and P.-S. Chen (1999) “Chaos and the Transition to Ventricular Fibrillation: A New Approach to Antiarrhythmic Drug Evaluation,” Circulation, Volume 99, Pages 2819–2826.

26. A. Garfinkel, Y.-H. Kim, O. Voroshilovsky, Z. Qu, J. R. Kil, M.-H. Lee, H. S. Karagueuzian, J. N. Weiss, and P.-S. Chen (2000) “Preventing Ventricular Fibrillation by Flattening Cardiac Restitution,” Proceedings of the National Academy of Sciences, Volume 97, Pages 6061–6066. *

27. N. G. Sepulveda, B. J. Roth and J. P. Wikswo, Jr. (1989) “Current Injection into a Two-Dimensional Anisotropic Bidomain,” Biophysical Journal, Volume 55, Pages 987–999. *

28. B. J. Roth (1992) “How the Anisotropy of the Intracellular and Extracellular Conductivities Influences Stimulation of Cardiac Muscle,” Journal of Mathematical Biology, Volume 30, Pages 633–646. *

29. Efimov I. R., Y. Cheng, D. R. Van Wagoner, T. Mazgalev, and P. J. Tchou (1998) “Virtual Electrode-Induced Phase Singularity: A Basic Mechanism of Defibrillation Failure,” Circulation Research, Volume 82, Pages 918–925.

30. Efimov, I. R., Y. N. Cheng, M. Biermann, D. R. Van Wagoner, T. N. Mazgalev, and P. J. Tchou (1997) “Transmembrane Voltage Changes Produced by Real and Virtual Electrodes During Monophasic Defibrillation Shock Delivered by an Implantable Electrode,” Journal of Cardiovascular Electrophysiology, Volume 8, Pages 1031–1045.

31. Roth, B. J. (1995) “A Mathematical Model of Make and Break Electrical Stimulation of Cardiac Tissue Using a Unipolar Anode or Cathode,” IEEE Transactions on Biomedical Engineering, Volume 42, Pages 1174–1184.

32. Cheng, Y., V. Nikolski, and I. R. Efimov (2000) “Reversal of Repolarization Gradient Does Not Reverse the Chirality of the Shock-Induced Reentry in the Rabbit Heart,” Journal of Cardiovascular Electrophysiology, Volume 11, Pages 998–1007.

33. Trayanova, N. A., B. J. Roth, and L. J. Malden (1993) “The Response of a Spherical Heart to a Uniform Electric Field: A Bidomain Analysis of Cardiac Stimulation,” IEEE Transactions on Biomedical Engineering, Volume 40, Pages 899–908.

34. Nielsen, P. M. F., I. J. Le Grice, B. H. Smaill, and P. J. Hunter (1991) “Mathematical Model of Geometry and Fibrous Structure of the Heart,” American Journal of Physiology, Volume 260, Pages H1365–H1378.

35. Krassowska, W., T. C. Pilkington, and R. E. Ideker (1987) “The Closed Form Solution to the Periodic Core-Conductor Model Using Asymptotic Analysis,” IEEE Transactions on Biomedical Engineering, Volume 34, Pages 519–531.

36. Rodriquez, B., J. C. Eason, and N. Trayanova (2006) “Differences Between Left and Right Ventricular Anatomy Determine the Types of Reentrant Circuits Induced by an External Electric Shock: A Rabbit Heart Simulation Study,” Progress in Biophysics and Molecular Biology, Volume 90, Pages 399–413.

Galileo's Daughter

Galileo's Daughter,
by Dava Sobel.

As is my habit, I listen to recorded books when I walk my dog Suki each day. Recently, I listened to the book Galileo’s Daughter, by Dava Sobel. I was surprised how touching I found this story (like Galileo, I have two daughters). It is a biography of Galileo Galilei (1564–1642), the famous Italian scientist, but also tells the parallel story of Sister Maria Celeste (1600–1634), Galileo’s daughter who was a nun at the San Matteo convent near Florence. The book quotes Maria Celeste’s letters to Galileo, which Sobel herself translated from Italian. (Unfortunately, Galileo’s replies are lost.) Maria Celeste comes across as a loving, intelligent and extremely loyal daughter who played a central role in Galileo’s life. “She alone of Galileo’s three children mirrored his own brilliance, industry, and sensibility, and by virtue of these qualities became his confidante.”

I tend to see biological physics everywhere, and I found some in this story. Late in his life, Galileo published his final book, Two New Sciences. One of these sciences was the motion of projectiles, and the other was what we would now call the strength of materials. In the part about materials, Galileo addressed the issue of scaling in animals. I quote Sobel, who quotes Galileo:

I have sketched a bone whose natural length has been increased three times and whose thickness has been multiplied until, for a correspondingly large animal, it would perform the same function which the small bone performs for its small animal. From the figures here shown you can see how out of proportion the enlarged bone appears. Clearly then if one wishes to maintain in a great giant the same proportion of limb as that found in an ordinary man he must either find a harder and stronger material for making the bones, or he must admit a diminution of strength in comparison with men of medium stature.

Scaling: Why is Animal
Size so Important?
by Knut Schmidt-Nielsen.

(You can find the picture of the two bones here.) This example of how the strength of bones must scale with animal size did not make it into the 4th edition of Intermediate Physics in Medicine and Biology, although I sometimes discuss it when I teach PHY 325 (Biological Physics) at Oakland University. It serves as an excellent example of how physics can constrain the structure of animals. I won’t hold it against Galileo that he didn’t get his drawing of the bones quite right; it was the 17th century after all. According to Knut Schmidt-Nielsen (Scaling: Why is Animal Size so Important)

The need for a disproportionate increase in the size of supporting bones with increasing body size was understood by Galileo Galilei (1637), who probably was the first scientist to publish a discussion of the effects of body size on the size of the skeleton. In his Dialogues [Two New Sciences was written in the form of a dialogue] he mentioned that the skeleton of a large animal must be strong enough to support the weight of the animal as it increases with the third power of the linear dimensions. Galileo used a drawing to show how a large bone is disproportionately thicker than a small bone. (Incidentally, judging from the drawing, Galileo made an arithmetical mistake. The larger bone, which is three times as long as the shorter, shows a 9-fold increase in diameter, which is a greater distortion than required. A three-fold increase in linear dimensions should give a 27-fold increase in mass, and the cross-sectional area of the bone should be increased 27-fold, and its diameter therefore by the square root of 27 (i.e., 5.2 instead of 9)).

Russ Hobbie and I discuss the issue of scaling in Chapter 2 of Intermediate Physics for Medicine and Biology. In Problem 28 of Chapter 2, we ask the reader to calculate the falling speed of animals of different sizes, taking into account air friction. The solution to the problem indicates that large animals, with their smaller surface-to-volume ratio, have a larger terminal speed (the speed of descent in steady state, once the acceleration drops to zero) than smaller animals. We end the problem with one of my favorite quotes, by J. B. S. Haldane

You can drop a mouse down a thousand-yard mine shaft; and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, and man is broken, a horse splashes.

When listening to Galileo’s Daughter, I was surprised to hear Galileo’s own words on this same subject, which are similar and written centuries earlier.

Who does not know that a horse falling from a height of three or four braccia will break his bones, while a dog falling from the same height or a cat from eight or ten, or even more, will suffer no injury? Equally harmless would be the fall of a grasshopper from a tower or the fall of an ant from the distance of the Moon.

Of course, the climax of Galileo’s Daughter is the great scientist’s trial by the Catholic Church for publishing a book supporting the Copernican view that the earth travels around the sun. Although I was familiar with this trial, I had never read the transcript, which Sodal quotes extensively. Listening to the elderly Galileo being forced into a humiliating recantation of his scientific views almost made me nauseous.

Sobel is a fine writer. Years ago I read her most famous book, Longitude, about finding a method to measure longitude at sea. Galileo himself contributed to the solution of this problem by introducing a method based on the orbits of the moons of Jupiter, which he of course discovered. However, the longitude problem was not definitely solved until clocks that could keep time on a rolling ship were invented by John Harrison. I have also listened to Sobel’s book The Planets, which I enjoyed but, in my opinion, isn’t as good as Longitude and Galileo’s Daughter. I hope Sobel continues writing books. As soon as a new one comes out (and arrives at the Rochester Hills Public Library, because I’m too cheap to buy these audio books), Suki and I plan on taking some long walks. I can’t wait.

Kids: Don’t Try This At Home

Russ Hobbie and I included a new chapter on sound and ultrasound in the 4th edition of Intermediate Physics for Medicine and Biology. In that chapter, we discuss how to calculate the speed of sound from the compressibility and the density of the tissue (Eq. 13.11). We then go on to describe, among other things, hearing, ultrasonic imaging, and the Doppler effect. One topic we do not mention is the behavior of objects moving faster than the speed of sound. Such a discussion, often found in physics and engineering books, usually is based on the Mach number, defined as the speed of an object divided by the speed of sound. If the Mach number is greater than one, the speed is supersonic and a shock wave develops. When an airplane travels faster than the speed of sound, people on the ground can hear the shock wave as a “sonic boom.” This is all very interesting, but it has nothing to do with biology and medicine, right?

Guess again. A recent article in the New York Times describes the plans of Felix Baumgartner, who intends to be the first human to break the sound barrier. I know, dear readers, that some of you are now saying “No, Chuck Yeager was the first to break the sound barrier, and that happened over 60 years ago.” Well, Yeager broke the sound barrier when flying in a plane, whereas Baumgartner plans to break the sound barrier while in free fall! The Times article states

But now Fearless Felix, as his fans call him, has something more difficult on the agenda: jumping from a helium balloon in the stratosphere at least 120,000 feet above Earth. Within about half a minute, he figures, he would be going 690 miles per hour and become the first skydiver to break the speed of sound. After a free fall lasting five and a half minutes, his parachute would open and land him about 23 miles below the balloon.

No one is certain what will happen to a human near the sound barrier. The NYT article says that turbulence may set in, causing havoc. Turbulence is a subject Russ and I discuss briefly in Chapter 1 of Intermediate Physics for Medicine and Biology, when introducing the Reynolds number. Most fluid in the body flows at low Reynolds number, with no danger of turbulence, although blood flow in the heart and aorta can get so fast that some turbulence may develop. Of course, any animal that flies in air will experience turbulence, which includes birds, bats, pterodactyls, and, in Baumgartner’s case, humans.

Physics With Illustrative Examples
From Medicine and Biology, Volume 1,
by Benedek and Villars.

These high altitude exploits remind me of a delightful section in the textbook Physics With Illustrative Examples from Medicine and Biology, by George Benedek and Felix Villars. In their Volume 1 on Mechanics, they describe balloon ascensions and the physiological effects of air pressure. After reviewing the medical implications of a lack of oxygen at high altitudes, they present the fascinating tale of a 19th century balloon ascension.

These symptoms are shown very clearly in the tragic balloon ascent of the “Zenith” carrying the balloon pioneers Tissandier, Sivel, and Corce-Spinelli on April 15, 1875. During this ascent Sivel and Corce-Spinelli died. The balloons maximum elevation as recorded on their instruments was 8600 m. Though gas bags carrying 70% oxygen were carried by the balloonists, the rapid and insidious effects of hypoxia reduced their judgment and muscular control and prevented their use of the oxygen when it was most needed. Though these balloonists were indeed trying to establish an altitude record, their account shows clearly that their judgment was severely impaired during critical moments during the maximum tolerable altitude.

Then Benedek and Villars present a 3-page extended quote from the account of the surviving member of the trio, Gaston Tissandier. It makes for fascinating reading. However, concerns about oxygen depletion aren’t relevant for Fearless Felix, because he will be wearing a space suit during his jump, with its own oxygen supply.

The New York Times story ends with the following quote. One wonders if we should admire Baumgartner’s pluck, or commit him to an insane asylum.

Private adventurers have more freedom to take their own risks. The Stratos medical director, Dr. Jonathan Clark, who formerly oversaw the health of space shuttle crews at NASA, says that the spirit of this project reminds him of stories from the first days of the space age.

“This is really risky stuff, putting someone up there in that extreme environment and breaking the sound barrier,” Dr. Clark said. “It’s going to be a major technical feat. It’s like early NASA, this heady feeling that we don’t know what we’re up against but we’re going to do everything we can to overcome it.”

Erwin Neher

I subscribe to a monthly magazine, The Scientist, which was founded by Eugene Garfield (who also was a founder of the Science Citation Index). It provides print and online coverage about biomedical research, technology and business. I’m not sure what I did to deserve it, but I get a paper copy delivered to my office for free, and I can tell you for certain that the magazine is worth the price. Seriously, it is a valuable resource, and the articles are general enough that I can follow them without having to consult my physiology and biochemistry textbooks. The online site contains many of the articles for free, and also has career information for young scientists. I recommend it.

The March 2010 issue of The Scientist contains a profile of Nobel Prize winner Erwin Neher, the developer of the patch clamp technique. Russ Hobbie and I discuss patch-clamp recording in Chapter 9 of the 4th edition of Intermediate Physics for Medicine and Biology.

The next big advance was patch-clamp recording [Neher and Sakmann (1976)]. Micropipettes were sealed against a cell membrane that had been cleaned of connective tissue by treatment with enzymes. A very-high-resistance seal resulted [(2-3) × 10⁷ Ohm] that allowed one to see the opening and closing of individual channels. For this work Erwin Neher and Bert Sakmann received the Nobel Prize in Physiology or Medicine in 1991. Around 1980, Neher’s group found a way to make even higher-resistance (10¹⁰-10¹¹ Ohm) seals that reduced the noise even further and allowed patches of membrane to be torn from the cell while adhering to the pipette [Hamill et al. (1981)]…

The profile in The Scientist provides some insight into how this research began.

[Neher’s former postdoc Fred] Sigworth remembers it well. “I came into lab that Monday morning and Erwin said, with a twinkle in his eye, ‘I think I know how you’re going to see sodium channels,’” he says. These channels—essential to neural communication—had proven elusive because they produce such small currents and remain open for such a short time. But thanks to the team’s new “patch-clamp” technique—and in particular, the formation of an incredibly tight seal, or “gigaseal,” between the pipette tip and the cell membrane—“seeing sodium channels suddenly became really easy,” says Sigworth, who, along with Neher, published these observations (and the first description of the tight-seal patch-clamp technique) in Nature in 1980.

I always enjoy reading about the quirks and odd twists of fate that often accompany scientific advance. The profile in The Scientist provides an entertaining anecdote.

You also needed to suck. “You had to apply a little bit of suction in order to pull some membrane into the orifice of the pipette,” says Neher. “If you did it the right way, it worked.” At least for Neher. “There was a weird period where we could no longer get gigaseals,” recalls [Owen] Hamill [a postdoc at the time]. “Then Bert suggested you have to blow before you suck.” Gently blowing a solution through the pipette as it approaches the surface of the cell keeps the tip from picking up debris during the descent. Between the blowing and the sucking, Hamill says, “our effeiciency went up to 99 percent.”

I fond it interesting that Neher’s undergraduate degree was in physics, and it was only after he arrived at the University of Wisconsin on a Fulbright Scholarship that he began studying biophysics. In his Nobel autobiography he describes his early motivation for studying biological physics.

At the age of 10, I entered the “Maristenkolleg” at Mindelheim [...] the local “Gymnasium” is operated by a catholic congregation, the “Maristenschulbrüder.” The big advantage of this school was that our teachers—both those belonging to the congregation and others—were very dedicated and were open not only to the subject matter but also to personal issues. During my years at the Gymnasium (1954 to 1963) I found out that, next to my interest in living things, I also could immerse myself in technical and analytical problems. In fact, pretty soon, physics and mathematics became my favourite subjects. At the same time, however, new concepts unifying these two areas had seeped into the literature, which was accessible to me. I eagerly read about cybernetics, which was a fashionable word at that time, and studied everything in my reach on the “Hodgkin-Huxley theory” of nerve excitation. By the time of my Abitur—the examination providing access to university—it was clear to me that I should become a “biophysicist.” My plan was to study physics, and later on add biology.

Neher provides a classic example of how a strong background in physics can lead to advances in biology and medicine, a major theme underlying Intermediate Physics for Medicine and Biology.

How Should We Teach Physics to Future Life Scientists and Physicians?

The American Physical Society publishes a monthly newspaper, the APS News, and the back page of each issue contains an editorial that goes under the name—you guessed it—“The Back Page.” Readers of the 4th edition of Intermediate Physics for Medicine and Biology will want to read The Back Page in the March 2010 issue, subtitled “Physics for Future Physicians and Life Scientists: A Moment of Opportunity.” This excellent editorial—written by Catherine Crouch, Robert Hilborn, Suzanne Amador Kane, Timothy McKay, and Mark Reeves—champions many of the ideas that underlie our textbook. The editorial begins

How should we teach physics to future life scientists and physicians? The physics community has an exciting and timely opportunity to reshape introductory physics courses for this audience. A June 2009 report from the American Association of Medical Colleges (AAMC) and the Howard Hughes Medical Institute (HHMI), as well as the National Research Council’s Bio2010 report, clearly acknowledge the critical role physics plays in the contemporary life sciences. They also issue a persuasive call to enhance our courses to serve these students more effectively by demonstrating the foundational role of physics for understanding biological phenomena and by making it an explicit goal to develop in students the sophisticated scientific skills characteristic of our discipline. This call for change provides an opportunity for the physics community to play a major role in educating future physicians and future life science researchers.

A number of physics educators have already reshaped their courses to better address the needs of life science and premedical students, and more are actively doing so. Here we describe what these reports call for, their import for the physics community, and some key features of these reshaped courses. Our commentary is based on the discussions at an October 2009 conference (www.gwu.edu/~ipls), at which physics faculty engaged in teaching introductory physics for the life sciences (IPLS), met with life scientists and representatives of NSF, APS, AAPT, and AAMC, to take stock of these calls for change and possible responses from the physics community. Similar discussion on IPLS also took place at the 2009 APS April Meeting, the 2009 AAPT Summer Meeting, and the February 2010 APS/AAPT Joint Meeting.

One key distinction between our textbook and the work described in The Back Page editorial is that our book is aimed toward an intermediate level, while the IPLS movement is aimed at the introductory level. Like it or not, premedical students have a difficult time fitting additional physics courses into their undergraduate curriculum. I know that here at Oakland University, I’ve been able to entice only a handful of premed students to take my PHY 325 (Biological Physics) and PHY 326 (Medical Physics) classes, despite my best efforts to attract them and despite OU’s large number of students hoping to attend medical school (these classes have our two-semester introductory physics sequence as a prerequisite). So, I think there’s merit in revising the introductory physics class, which premedical students are required to take, if your goal is to influence premedical education. As The Back Page editorial states, “the challenge is to offer courses that cultivate general quantitative and scientific reasoning skills, together with a firm grounding in basic physical principles and the ability to apply those principles to living systems, all without increasing the number of courses needed to prepare for medical school.” The Back Page editorial also cites the “joint AAMC-HHMI committee … report, Scientific Foundations for Future Physicians (SFFP). This report calls for removing specific course requirements for medical school admission and focusing instead on a set of scientific and mathematical ‘competencies.’ Physics plays a significant role…”

How do you fit all the biomedical applications of physics into an already full introductory class? The Back Page editorial gives some suggestions. For instance, “an extended discussion of kinematics and projectile motion could be replaced by more study of fluids and continuum mechanics... [and] topics such as diffusion and open systems could replace the current focus on heat engines and equilibrium thermal situations.” I agree, especially with adding fluid dynamics (Chapter 1 in our book) and diffusion (Chapter 4), which I believe are absolutely essential for understanding biology. I have my own suggestions. Although Newton’s universal law of gravity, Kepler’s laws of planetary motion, and the behavior of orbiting satellites are fascinating and beautiful topics, a premed student may benefit more from the study of osmosis (Chapter 5) and sound (Chapter 13, including ultrasound). Electricity and magnetism remains a cornerstone of introductory physics (usually in a second semester of a two-semester sequence), but the emphasis could be different. For instance, Faraday’s law of induction can be illustrated using magnetic stimulation of the brain, Ampere’s law by the magnetic field around a nerve axon, and the dipole approximation by the electrocardiogram. In a previous post to this blog, I discussed how Intermediate Physics for Medicine and Biology addresses many of these issues. Russ Hobbie will be giving an invited paper about medical physics and premed students at the July 2010 meeting of the American Association of Physics Teachers. When he gives the talk it will be posted on the book website.

One way to shift the focus of an introductory physics class toward the life sciences is to create new homework problems that use elementary physics to illustrate biological applications. In the 4th edition of Introductory Physics for Medicine and Biology, Russ Hobbie and I constructed many interesting homework problems about biomedical topics. While some of these may be too advanced for an introductory class, others may (with some modification) be very useful. Indeed, teaching a traditional introductory physics class but using a well-crafted set of homework problems may go a long ways toward achieving the goals set out by The Back Page editorial.

Let me finish this blog entry by quoting the eloquent final paragraph of The Back Page editorial. Notice that the editorial ends with the same central question that began it. It is the question that motivated Russ Hobbie to write the first edition of Intermediate Physics for Medicine and Biology (published in 1978) and it is the key question that Russ and I struggled with when working on the 4th edition.

The physics community faces a challenging opportunity as it addresses the issues surrounding IPLS courses. A sizable community we serve has articulated a clear set of skills and competencies that students should master as a result of their physics education. We have for a number of decades incorporated engineering examples into our physics classes. The SFFP report asks us to respond to another important constituency. Are we ready to develop courses that will teach our students how to apply basic physical principles to the life sciences? The challenges of making significant changes in IPLS courses are daunting if we each individually try to take on the task. But with a community-wide effort, we should be able to meet this challenge. The physics community is already moving to develop and implement changes in IPLS courses, and the motivations for change are strong. The life science and medical school communities stress that a working knowledge of physical principles is essential to success in all areas of life science including the practice of medicine. Thus we see significant teaching and learning opportunities as we work to answer the question that opened our discussion: how should we teach physics to future physicians and life scientists?

The Strangest Man

The Strangest Man:
The Hidden Life of Paul Dirac,
Mystic of the Atom,
by Graham Farmelo.

I recently read The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom, by Graham Farmelo, a fascinating biography of the Nobel Prize winning physicist Paul Adrien Maurice Dirac. One thing I did not find in the book was biological or medical physics. Nevertheless, Russ Hobbie and I mention Dirac in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology, in connection with the Dirac delta function.

The δ function can be thought of as a rectangle of width a and height 1/a in the limit [as a goes to zero]… The δ function is not like the usual function in mathematics because of its infinite discontinuity at the origin. It is one of a class of “generalized functions” whose properties have been rigorously developed by mathematicians since they were first used by the physicist P. A. M. Dirac.

The Principles of Quantum Mechanics,
by Paul Dirac.

Dirac won his Nobel Prize for contributions to quantum mechanics. I bought a copy of his famous textbook The Principles of Quantum Mechanics when I was an undergraduate at the University of Kansas. Farmelo describes it as “never out of print, it remains the most insightful and stylish introduction to quantum mechanics and is still a powerful source of inspiration for the most able young theoretical physicists. Of all the textbooks they use, none presents the theory with such elegance and with such relentless logic.”

One of Dirac’s greatest contributions was the prediction of positive electrons, or positrons, a type of antimatter. His prediction arose from the relativistic wave equation for the electron, now called the Dirac equation. An interesting feature of the Dirac equation is that it implies negative energy states. The only time these negative states are observable is when an electron is missing from one of the states: a hole. Farmelo writes

The bizarre upshot of the theory is that the entire universe is pervaded by an infinite number of negative-energy electrons – what might be thought of as a “sea.” Dirac argued that this sea has a constant density everywhere, so that experimenters can observe only departures from this perfect uniformity… Only a disturbance in Dirac’s sea—a bursting bubble, for example—would be observable. He envisaged just this when he foresaw that there would be some vacant states in the sea of negative-energy electrons, causing tiny departures from the otherwise perfect uniformity. Dirac called these unoccupied states “holes”... Each hole has positive energy and positive charge—the properties of the proton, the only other subatomic particle known at that time [1929]. So Dirac made the simplest possible assumption by suggesting that a hole is a proton.

We now know that these holes are not protons but are positrons, discovered experimentally in 1932 by Carl Anderson. Positrons are vital for understanding how x-rays interact with matter, as Russ and I describe in Section 15.6 of Intermediate Physics for Medicine and Biology

A photon with energy above 1.02 MeV can produce a particle-antiparticle pair: a negative electron and a positive electron or positron… Since the rest energy (mc²) of an electron or positron is 0.51 MeV, pair production is energetically impossible for photons below 2mc² = 1.02 MeV.

One can show, using hν_o = pc for the photon, that momentum is not conserved by the positron and electron if Eq. 15.23 [conservation of energy] is satisfied. However, pair production always takes place in the Coulomb field of another particle (usually a nucleus) that recoils to conserve momentum.

In Sec. 17.14, Russ and I describe the crucial role positrons play in medical imaging.

If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence. This simplifies the attenuation correction, because the total attenuation for both photons is the same for all points of emission along each gamma ray through the body (see Problem 54). Positron emitters are short-lived, and it is necessary to have a cyclotron for producing them in or near the hospital. This is proving to be less of a problem than initially imagined. Commercial cyclotron facilities deliver isotopes to a number of nearby hospitals. Patterson and Mosley (2005) found that 97% of the people in the United States live within 75 miles of a clinical PET facility.

Another famous prediction of Dirac’s was magnetic monopoles. Russ and I only mention monopoles in passing in Section 8.8.1: “Since there are no known magnetic charges (monopoles), we must consider the effect of magnetic fields on current loops or magnetic dipoles.” Dirac predicted that magnetic monopoles could exist. Farmelo tells the story.

In Cambridge, during the spring of 1931, Dirac happened upon a rich new seam of ideas that would crystallize into one of his most famous contributions to science… As usual, Dirac appears to have said nothing of this to anyone, even to his close friends. In the early months of 1931, a quiet time for his fellow theoreticians, he was working on the most promising new theory he had conceived for years. The theory broke new ground in magnetism. For centuries, it had been a commonplace of science that magnetic poles come only in pairs, labeled north and south: if one pole is spotted, then the opposite one will be close by. Dirac had found that quantum theory is compatible with the existence of single magnetic poles. During a talk at the Kapitza Club, he dubbed them magnons, but the name never caught on in this context; the particles became known as magnetic monopoles.

Physicists have searched for magnetic monopoles, and once they even thought they found one. In 1982, physicist Blas Cabrera observed a signal consistent with the experimental signature of a monopole (Physical Review Letters, Volume 48, Pages 1378–1381), but it now appears to have been an artifact, as the result has never been reproduced. I have my own remote (indeed, very remote) connection with this experiment (and thus to Dirac). Cabrera’s PhD advisor, William Fairbank, was John Wikswo’s PhD advisor, and Wikswo was in turn my PhD advisor. Thus, academically speaking, I am one of Cabrera’s scientific nephews.

Dirac was known for saying little and behaving rather oddly (the title of the book is, after all, “The Strangest Man”), and Farmelo suggests a possible reason: Dirac may have been autistic.

[Dirac] always attributed his extreme taciturnity and stunted emotions to his father’s disciplinarian regime; but there is another, quite different explanation, namely that he was autistic. Two of Dirac’s younger colleagues confided in me that they had concluded this, each of them making their disclosure in sotto voce, as if they were imparting a shameful secret. Both refused to be quoted… There is not nearly enough detail in her [Dirac’s mother’s] comments or in reports of Dirac’s behaviour in school to justify a diagnosis that he was then autistic. His behavior as an a adult, however, had all the characteristics that almost every autistic person has to some degree—reticence, passivity, aloofness, literal-mindedness, rigid patterns of activity, physical ineptitude, self-centredness and, above all, a narrow range of interests and a marked inability to empathise with other human beings.

Whatever the cause of Dirac’s unusual behavior, he was a great physicist. Farmelo sums up Dirac’s enduring legacy at the end of his book.

There is no doubt that Dirac was a great scientist, one of the few who deserves a place just below Einstein in the pantheon of modern physicists. Along with Heisenberg, Jordan, Pauli, Schrodinger and Born, Dirac was one of the group of theoreticians who discovered quantum mechanics. Yet his contribution was special. In his heyday, between 1925 and 1933, he brought a uniquely clear vision to the development of a new branch of science: the book of nature often seemed to be open in front of him.