Friday, April 30, 2010

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.

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 En of radiation with frequency ν is discrete, En = h ν n, where n=0, 1, 2, … Let the probability Pn 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 Eave for frequency ν by performing the sum Eave = 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, Wν(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 .

Friday, April 23, 2010

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 do not 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.”
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. Actually 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.

Friday, April 16, 2010

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 144:710-711. *

2. A. L. Hodgkin and B. Katz (1949) The effect of sodium ions on the electrical activity of the giant axon of the squid. J. Physiol. 108:37-77.

3. A. L. Hodgkin and A. F. Huxley (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500-544. *

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 280:69-77. *

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 Arch 391:85-100. *

6. A. L. Hodgkin and W. A. H. Rushton (1946) The electrical constants of a crustacean nerve fibre. Proc. Roy. S. Lond B 133:444-479. *

7. W. A. H. Rushton (1951) A theory of the effects of fibre size in medullated nerve. J. Physiol. 115:101-122. *

8. R. FitzHugh (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1:445-466.

9. W. Rall (1962) Theory of physiological properties of dendrites. Ann. N. Y. Acad. Sci. 96:1071-1092.

10. F. Rattay (1989) Analysis of models for extracellular fiber stimulation. IEEE Trans. Biomed. Eng. 36:676-682.

11. A. T. Barker, R. Jalimous, and I. L. Freeston (1985) Non-invasive magnetic stimulation of human motor cortex. Lancet 8437:1106-1107. *

12. M. Hallett and L. G. Cohen (1989) Magnetism: A new method for stimulation of nerve and brain. JAMA 262:538-541. *

13. B. J. Roth, L. G. Cohen and M. Hallett (1991) The electric field induced during magnetic stimulation. Electroenceph. Clin. Neurophysiol. (Suppl 43):268-278.

14. R. Plonsey (1974) The active fiber in a volume conductor. IEEE Trans. Biomed. Eng. 21:371-381.

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. Electroenceph. Clin. Neurophysiol. 102:159-166.

16. M. Schneider (1974) Effect of inhomogeneities on surface signals coming from a cerebral current-dipole source. IEEE Trans. Biomed. Eng. 21:52-54.

17. B. J. Roth and J. P. Wikswo (1985) The magnetic field of a single axon: A comparison of theory and experiment. Biophys. J. 48:93-109. *

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. Rev. Mod. Phys. 65:413-497. *

19. T.-K. Truong and A. W. Song (2006) Finding neuroelectric activity under magnetic-field oscillations (NAMO) with magnetic resonance imaging in vivo. PNAS 103:12598-12601.

20. B. J. Roth and P. J. Basser (2009) Mechanical model of neural tissue displacement during Lorentz effect imaging. Magn. Reson. Med. 61:59-64.

21. A. T. Winfree (1987) When Time Breaks Down. Princeton Univ Press, Princeton, NJ, 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. J. Clin. Invest. 83:1039-1052.

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. Am. J. Physiol. 255: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 99:2819-2826.

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 firillation by flattening cardiac restitution. PNAS 97:6061-6066. *

27. N. G. Sepulveda, B. J. Roth and J. P. Wikswo, Jr. (1989) Current injection into a two-dimensional anisotropic bidomain. Biophys. J. 55:987-999. *

28. B. J. Roth (1992) How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J. Math. Biol. 30:633-646. *

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. Circ. Res. 82:918-925.

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. J. Cardiovasc. Electrophysiol. 8:1031-1045.

31. Roth, B. J. (1995) A mathematical model of make and break electrical stimulation of cardiac tissue using a unipolar anode or cathode. IEEE Trans. Biomed. Eng. 42:1174-1184.

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. J. Cardiovasc. Electrophysiol. 11:998-1007.

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 Trans. Biomed. Eng. 40:899-908.

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. Am. J. Physiol. 260: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 Trans. Biomed. Eng., 34:519-531.

36. Rodriquez, B., J. C. Eason, and N. Trayanova (2006) Differences between left and right ventricular anatomy determine the types of reentrant circuits induced ay an external electric shock. A rabbit heart simulation study. Prog. Biophys. Mol. Biol. 90:399-413.

Friday, April 9, 2010

Galileo's Daughter


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.”
(You can fine 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 quite 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 the 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.

Friday, April 2, 2010

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.

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.' ”