IPMB100

I’m a regular reader of TIME magazine. Every year they publish an issue devoted to the TIME100: the hundred most influential people of that year. I thought I would do the same, except I’d focus on Intermediate Physics for Medicine and Biology. So, below is a list of the one hundred scientists, physicians, engineers, and mathematicians who most influenced IPMB. I list them by impact, with the most influential first.

Like for the TIME100, selecting the list is not an exact science. It’s based on mentions in IPMB, numbers of citations, and my own personal opinions. I’m sure your list would be different, and that’s okay.

There are many brilliant scientists who didn’t make the list (for example: Newton, Faraday, Maxwell, Rutherford, and Einstein). I tried to focus on people who had a direct impact on IPMB, rather than fundamental but not biomedical contributions to physics, so these and other luminaries were left off. 

The oldest scientists are Brown (born 1773) and Poiseuille (1797). The youngest are Basser, Goodsell, Hämäläinen, LeBihon, MacKinnon, Mattiello, Strogatz, and Xia (all more or less my age). I’m embarrassed to say there are only three women (Curie, Eleanor Adair, and Mielczarek; four if you count Abramowitz’s coauthor Stegun). Thirty three are alive today. Twenty are Nobel Prize winners (marked with an asterisk). I know ten personally (marked with a §). When I wasn’t sure about the year a scientist was born or died, I guessed and marked it with a question mark. There are many more I would like to honor, but I decided to—like TIME—stop at 100.

Enjoy!

1. Alan Hodgkin* (1914–1998) English physiologist who discovered how nerve action potentials work and developed the Hodgkin-Huxley model. 
2. Andrew Huxley* (1917–2012) English physiologist and computational biologist who discovered how nerve action potentials work and developed the Hodgkin-Huxley model. 
3. Godfrey Hounsfield* (1919–2004) British electrical engineer who invented the first clinical computed tomography scanner. 
4. Paul Lauterbur* (1929–2007) American chemist who developed a method to do magnetic resonance imaging using magnetic field gradients. 
5. Edward Purcell* (1912–1997) American physicist who co-discovered nuclear magnetic resonance, was author of the article “Life at Low Reynolds Number,” and wrote volume 2 of the Berkeley Physics Course titled Electricity and Magnetism. 
6. Allan Cormack* (1924–1998) South African physicist who developed much of the mathematical theory behind computed tomography. 
7. Hermann von Helmholtz (1821–1894) German physicist and physician; First to measure the propagation velocity of a nerve action potential. 
8. Adolf Fick (1829–1901) German physician and physiologist who derived the laws of diffusion (Fick’s laws). 
9. Willem Einthoven* (1860–1927) Dutch medical doctor and physiologist who was the first to accurately measure the electrocardiogram. 
10. Marie Curie** (1867-1934) Polish-French physicist and chemist who discovered the elements radium and polonium; The unit of the curie is named after her. 
11. Jean Léonard Marie Poiseuille (1797–1869) French physicist and physiologist who determined the law governing the flow of blood in small vessels. 
12. Max Kleiber (1893–1976) Swiss biologist who established a ¾ power law relating metabolic rate to mass. 
13. Felix Bloch* (1905–1983) Swiss-American physicist who co-discovered nuclear magnetic resonance and derived the Bloch equations. 
14. Peter Mansfield* (1933–2017) English physicist who developed techniques used in magnetic resonance imaging, including echo planar imaging. 
15. Roderick MacKinnon* (1956) American biophysicist who determined the structure of the potassium ion channel. 
16. Erwin Neher* (1944) German biophysicist who co-invented the patch clamp method to record from single ion channels. 
17. Bert Sakmann* (1942) German physiologist who co-invented the patch clamp method to record from single ion channels. 
18. Tony Barker (1950) English engineer who invented transcranial magnetic stimulation. 
19. Robert Plonsey§ (1924–2015) American engineer who contributed to theoretical bioelectricity and wrote Bioelectric Phenomena and other books. 
20. Peter Basser§ (1959?) American engineer who invented the magnetic resonance imaging technique of diffusion tensor imaging. 
21. William Oldendorf (1925-1992) American medical doctor who first designed a computed tomography device. 
22. J. B. S. Haldane (1892–1964) British evolutionary biologist who published “On Being the Right Size,” an essay about scaling. 
23. Geoffrey West (1940) British theoretical physicist who derived a model to explain the ¾ power law of metabolism. 
24. George Ralph Mines (1886–1914) English cardiac electrophysiologist who demonstrated reentry in cardiac tissue. 
25. Bernard Cohen (1924–2012) American physicist who opposed the linear no-threshold model of radiation risk. 
26. John Wikswo§ (1949) American physicist who measured the magnetic field of a nerve. 
27. Arthur Winfree§ (1942–2002) American mathematical biologist who studied cardiac arrhythmias and wrote When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. 
28. Richard Blakemore (1950?) Researcher who discovered magnetotactic bacteria. 
29. John Moulder (1945–2022) Radiation biologist who debunked suggestions that radiofrequency electromagnetic fields are dangerous. 
30. Kenneth Foster§ (1945) American bioengineer and expert on the biological effects of electromagnetic fields. 
31. Paul Callaghan (1947–2012) New Zealand physicist who wrote Principles of Nuclear Magnetic Resonance Microscopy. 
32. Paul Nunez (1950?) Analyzed electroencephalography using mathematics, and author of Electric Fields of the Brain. 
33. Charles Bean (1923–1996) American physicist who studied porous membranes and reverse osmosis. 
34. John Hubbell (1925–2007) American radiation physicist who measured and tabulated x-ray cross sections. 
35. Arthur Compton* (1892–1962) American physicist who analyzed Compton scattering of x-rays. 
36. William Bragg* (1862–1942) English physicist who discovered the Bragg peak of energy deposition from charged particles in tissue. 
37. Mark Hallett§ (1943) National Institutes of Health neurophysiologist who helped develop transcranial magnetic stimulation and wrote, with Leo Cohen, the article “Magnetism: A New Method for Stimulation of Nerve and Brain.” 
38. Selig Hecht (1892–1947) American physiologist who performed a classic experiment on scotopic vision. 
39. Milton Abramowitz (1915–1958) American mathematician who, with Irene Stegun, coauthored the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 
40. Knut Schmidt-Nielsen (1915–2007) Norwegian-American comparative physiologist and author of How Animals Work and Scaling: Why Is Animal Size So Important? 
41. Steven Vogel (1940–2015) American biomechanics researcher and author of Life in Moving Fluids: The Physical Biology of Flow and other books. 
42. Mark Denny (1951) American physiologist and author of Air and Water: The Biology and Physics of Life’s Media. 
43. Howard Berg (1934–2021) American biophysicist who studied the motility of E. coli and wrote Random Walks in Biology. 
44. Frank Herbert Attix (1925) Radiologist who is the author of Introduction to Radiological Physics and Radiation Dosimetry. 
45. Steven Strogatz (1959) American mathematician and author of Nonlinear Dynamics and Chaos and other books. 
46. Frederick Donnan (1870–1956) British chemist who analyzed Donnan equilibrium. 
47. Gustav Bucky (1880–1963) German-American radiologist who invented the Bucky grid used in x-ray imaging. 
48. Gopalasamudram Narayanan Ramachandran (1922–2001) Indian physicist who, with A. V. Lakshminarayanan, developed mathematical methods used in computed tomography. 
49. Peter Agre* (1949) American molecular biologist who discovered membrane water channels called aquaporins. 
50. Eleanor Adair (1926–2013) American physiologist who studied the health risks of microwave radiation. 
51. Robert Adair (1924–2020) American physicist who studied the biological effects of weak, extremely-low-frequency electromagnetic fields. 
52. Yuan-Cheng Fung (1919–2019) Chinese-American biomedical engineer, and author of Biomechanics. 
53. Herman Carr (1924–2008) American physicist and pioneer in magnetic resonance imaging. 
54. Matti Hämäläinen (1958) Finnish physicist who was lead author on the article “Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain”. 
55. Saul Meiboom (1916–1998) Israeli researcher who, with David Gill, co-invented of the Carr-Purcell-Meiboom-Gill pulse sequence used in magnetic resonance imaging. 
56. Oskar Klein (1894–1977) Swedish physicist who, with Japanese physicist Yoshio Nishina, developed the Klein-Nishina formula for Compton scattering of x-rays. 
57. Chad Calland (1934–1972) Medical doctor, kidney transplant patient, and author of the paper “Iatrogenic Problems in End-Stage Renal Failure.” 
58. Walter Blount (1900–1992) American orthopedic surgeon who advocated for the use of a cane. 
59. Albert Bartlett (1923–2013) American physicist and author of The Essential Exponential! For the Future of Our Planet. 
60. Pierre Auger (1899–1993) French physicist who studied the emission of Auger electrons. 
61. Rolf Sievert (1896–1966) Swedish medical physicist who studied the biological effects of ionizing radiation; The unit of the sievert is named after him. 
62. Louis Gray (1905–1965) English physicist who worked on the effects of radiation on biological systems. The unit of the gray is named after him.
63. Richard Frankel (1943?) American researcher who studied magnetotactic bacteria. 
64. Frederick Reif (1927–2019) Austrian-American physicist who wrote volume 5 of the Berkeley Physics Course, titled Statistical Physics. 
65. Richard FitzHugh (1922–2007) Co-inventor, with Jinichi Nagumo, of the FitzHugh-Nagumo model of a neuron. 
66. Arthur Guyton (1919–2003) American physiologist and author of the Textbook of Medical Physiology. 
67. Leon Glass (1943) American researcher and co-author, with Michael Mackey, of From Clocks to Chaos: The Rhythms of Life. 
68. Ken Kwong (1948) Chinese-American nuclear physicist who studied functional magnetic resonance imaging. 
69. Seiji Ogawa (1934) Japanese biophysicist who studied functional magnetic resonance imaging. 
70. Jay Lubin (1947) National Cancer Institute epidemiologist who battled with Bernard Cohen over the linear no-threshold model and the risk of radon. 
71. Eugenie Mielczarek (1931-2017) American physicist and author of Iron: Nature’s Universal Element: Why People Need Iron and Animals Make Magnets. 
72. David Goodsell (1960?) American structural biologist and science illustrator who wrote the book The Machinery of Life. 
73. Philip Morrison (1915–2005) American physicist who was lead author on Powers of Ten. 
74. Henri Becquerel* (1852–1908) French physicist who discovered radioactivity; the unit of the becquerel is named after him. 
75. Wilhem Roentgen* (1845–1923) German physicist who discovered x rays; The unit of the roentgen is named after him. 
76. Bertil Hille (1940) American biologist and author of Ion Channels of Excitable Membranes. 
77. George Benedek (1928) American physicist who co-authored, with Felix Villars, the three-volume Physics with Illustrative Examples from Medicine and Biology. 
78. William Hendee (1938) Coauthor, with E. Russell Ritenour, of Medical Imaging Physics. 
79. John Cameron (1922–2005) Medical physicist and coauthor of Physics of the Body. 
80. Lawrence Stark (1926–2004) American neurologist and expert on the feedback system controlling the size of the pupil in the eye. 
81. Ernst Ruska* (1906–1988) German physicist who invented the electron microscope. 
82. Britton Chance (1913–2010) American physicist who developed biomedical photonics. 
83. Johann Radon (1887–1956) Austrian mathematician who developed the radon transform used in computed tomography. 
84. Alan Garfinkel (1944?) American researcher who analyzed cardiac restitution for controlling heart arrhythmias. 
85. Eric Hall (1950?) Author of Radiobiology for the Radiologist. 
86. Osborne Reynolds (1842–1912) British engineer who studied fluid mechanics; the Reynolds number is named after him. 
87. Bernard Katz* (1911–2003) German-British biophysicist and author of Nerve, Muscle, and Synapse. 
88. William Rushton (1901–1980) British physiologist who worked with Alan Hodgkin studying nerve conduction. 
89. Robert Eisberg (1928) Coauthor with Robert Resnick of Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 
90. John Clark (1936–2017) American bioengineer who worked with Robert Plonsey. 
91. Raymond Ideker§ (1942) American physiologist and medical doctor who studied the electrical activity of the heart. 
92. Denis LeBihan§ (1957) French physicist and medical doctor who developed diffusion magnetic resonance imaging, and worked with Peter Basser on diffusion tensor imaging. 
93. Ronald Bracewell (1921–2007) Author of Fourier Transforms and Their Applications. 
94. Robert Brown (1773–1858) Scottish botanist who discovered Brownian motion. 
95. Louis DeFelice (1940?–2016) Wrote Introduction to Membrane Noise. 
96. H. M. Schey (1930?) Author of Div, Grad, Curl, and All That. 
97. Warren Weaver (1894–1978) American mathematician and science administrator who wrote Lady Luck: The Theory of Probability. 
98. Peter Atkins (1940) English chemist and author of The Second Law. 
99. Yang Xia§ (1955) Oakland University physicist who studied the magic angle in magnetic resonance imaging. 
100. James Mattiello§ (1958-2017) Oakland University alumnus and American physicist who worked with Peter Basser and Denis LeBihan to developed diffusion tensor imaging.

Sine and Cosine Integrals and the Delta Function

Trigger warning: This post is for mature audiences only; it may contain Fourier transforms and Dirac delta functions. 

In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I examine some properties of Fourier transforms. In particular, we consider three integrals of sines and cosines. After some analysis, we conclude that these integrals are related to the Dirac delta function, δ(ω – ω’), equal to infinity at ω = ω’ and zero everywhere else (it’s a strange function consisting of one really tall, thin spike).

Are these equations correct? I now believe that they’re almost right, but not entirely. I propose that instead they should be 

You’re probably thinking “what a pity, the second three equations looks more complicated than the first three.” I agree. But let me explain why I think they’re better. Hang on, it’s a long story.

Let’s go back to our definition of the Fourier transform in Eq. 11.57 of IPMB

The first thing to note is that y(t) consists of two parts. The first depends on cos(ωt), which is an even function, meaning cos(–ωt) = cos(ωt). There’s an integral over ω, implying that many different frequencies contribute to y(t), weighted by the function C(ω). But one thing we know for sure is that when you add up the contributions from all these many frequencies, the result must be an even function (the sum of even functions is an even function). The second part depends on sin(ωt), which is an odd function, sin(–ωt) = – sin(ωt). Again, when you add up all the contributions from these many frequencies weighted by S(ω), you must get an odd function. So, we can say that we’re writing y(t) as the sum of an even part, y_even(t), and an odd part, y_odd(t). In that case, we can rewrite our Fourier transform expressions as

We should be able to take our expression for y_even(t), put our expression for C(ω) into it, and then—if all works as it should—get back y_even(t). Let’s try it and see if it works. To start I’ll just rewrite the first of the four equations listed above

Now for C(ω) I’ll use the third of the four equations listed above. In that expression, there is an integral over t, but t is a dummy variable (it’s an “internal” variable; after you do the integral, the result does not depend on t), so to keep things from getting confusing we’ll call the dummy variable by another name, t'

Next switch the order of the integrals, so the integral over t' is on the outside and the integral over ω is on the inside

Ha! There, inside the bracket, is one of those integrals were’re talking about. Okay, the variables ω and t are swapped, but otherwise it’s the same. So, let’s put in our new expression for the integral

The 2π’s cancel, and a factor of one half comes out. An integral containing a delta function just picks out the value where the argument of the delta function is zero. We get

But, we know that y_even(t) is an even function, meaning y_even(–t) equals y_even(t). So finally

It works! We go “around the loop” and get back our original function.

You could perform another calculation just like this one but for y_odd(t). Stop reading and do it, to convince yourself that again you get back to where you started from, y_odd(t) = y_odd(t).

Now, you folks who are really on the ball might realize that if you had used the old delta function relationships given in IPMB (the first three equations in this post), they would also work. (Again, try it and see.) So why use my fancy new formulas? Well, if you have an integral that adds up a bunch of cos(ωt), you know you’re gonna get an even function. There’s no way it can be equal to δ(ω – ω’), because that function is neither even nor odd. So, it just doesn’t make sense to say that summing up a bunch of even functions gives you something that isn’t even. In my new formula, that sum of two delta functions is an even function. The same argument holds for the integral with sin(ωt), which must be odd.

Finally (and this is what got me started down this rabbit hole), you often see the delta function written as

Jackson even gives this equation, so it MUST be correct. (For those of who aren’t physicists, John David Jackson wrote the highly regarded graduate textbook Classical Electrodynamics, known by physics graduate students simply as “Jackson.”)

In Jackson’s equation, i is the square root of minus one. So, this representation of the delta function uses complex numbers. You won’t see it in IPMB because Russ and I avoid complex numbers (I hate them).

Let’s use the Euler formula e^iθ = cosθ + i sinθ to change the integral in Jackson’s delta function expression to

Now use a couple trig identities, cos(A–B) = cosA cosB + sinA sinB and sin(A–B) = sinA cosB –cosA sinB, to get

This is really four integrals,

Then, using the relations between these integrals and the delta function given in IPMB (the first three equations at the top of this post), you get that the sum of these integrals is equal to

which is obviously wrong; we started with 2πδ and ended up with 4πδ. Even worse, do the same calculation for δ(ω + ω') with a plus instead of a minus in front of the ω'. I’ll leave it to you to work out the details, but you’ll get zero! Again, nonsense. However, if you use the integral relations I propose above (second set of three integrals at the top of this blog), everything works just fine (try it).

Gene Surdutovich, my new coauthor for the sixth edition of IPMB, and I are still deciding how to discuss this issue in the new edition (which we are hard at work on but I doubt will be out within the next year). I don’t want to get bogged down in mathematical minutia that isn’t essential to our book’s goals, but I want our discussion to be correct. Once the sixth edition is published, you can see how we handle it.

I haven’t seen my new delta function/Fourier integral relationships in any other textbook or math handbook. This makes me nervous. Are they correct? Moreover, Intermediate Physics for Medicine and Biology does not typically contain new mathematical results. Maybe I haven’t looked hard enough to see if someone else published these equations (if you’ve seen them before, let me know where…please!). Maybe I’ll find these results in Morse and Feshbach (another one of those textbooks known to all physics graduate students) or some other mathematical tome. I need to make a trip to the Oakland University library to look through their book collection, but right now its too cold and snowy (we got about four to five inches of the white stuff in the last 48 hours).

The Physics of Birds

In this second installment of my series on the physics of native gardening, I’ll talk about the physics of birds. We get a lot of birds in our yard. Robins visit the lawn and our crabapple tree. Too many house sparrows come to our bird feeders; they’re invasive pests. We see lots of blue jays, those big bullies, as well as goldfinches, downy woodpeckers, and black-capped chickadees. Every fall we know that winter is approaching when the dark-eyed juncos come down to Michigan from Canada. Canadian geese fly overhead, but they never stop at our house.

Flight

I often see birds high in the sky, soaring through the air without flapping their wings. I suspect many are red-tailed hawks, but I’ve never gotten close enough to one to say for sure. How does soaring work? First, it requires a thermal updraft. The sun heats the earth and the earth heats the air next to it, resulting in a temperature gradient: the air near the ground is hotter than the cooler air high above. However, hot air is lighter and therefore tends to rise. This unstable situation results in thermal updrafts. Hot air at one location will rise, and then as it cools will sink at some nearby location. The hawk can glide in the uprising air, so it slowly sinks with respect to the air but rises with respect to the ground. Once high up, it can then glide anywhere while searching for food, until it is low enough that it must seek another updraft.

Life in Moving Fluids,
by Steven Vogel.

Most birds don’t soar but instead flap their wings to fly. This flapping is complicated enough that I’ll let Steven Vogel—my favorite expert on biological fluid dynamics—explain it. The following excerpt is from his wonderful book Life in Moving Fluids.

In birds, bats, and insects, flapping wings combine the functions that airplanes divide between fixed wings and propellers—in a sense, they’re closer to helicopters than to airplanes, and it’s all too easy to be misled by our habit of calling the propulsive appendages “wings” rather than “propeller blades.” But they aren’t quite like ordinary propellers either, since flapping wings produce both thrust and lift directly, rather than producing thrust directly and getting lift by diverting some of the thrust to pay for the drag of fixed, lift-producing wings. The composite function, as well as their reciprocating rather than rotational motion, mean that the motion of flapping wings is inevitably complex… The downstroke moves a wing forward as well as downward and produces mainly upward force but usually some rearward force as well. The upstroke goes backward as well as upward, producing mainly rearward force but often some upward force.

Scaling

Scaling,
by Knut Schmidt-Nielsen.

Each summer my wife puts out a feeder filled with sugar water, and near it we plant red cardinal flowers, to attract hummingbirds. The hummingbirds are tiny and are constantly eating. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I explain how an animal’s metabolic rate scales with its size and mass. The heat produced from metabolism increases with the volume of the animal, but heat is lost by an animal at its surface. As you compare larger animals to smaller ones, the volume increases with size faster than the surface area does. This means that large animals have trouble getting rid of excess heat, while small animals find it difficult to stay warm. The tiny hummingbird is smaller than other birds, so it tends to cool down quickly (it has a large surface-to-volume ratio). To keep warm, it has to boost its metabolism, which means it must eat a lot. A high metabolic rate requires not only much food but also oxygen, which implies that the hummingbird’s heart must pump a lot of blood. The heart rate of a hummingbird can be upwards of 1000 beats per minute (a normal heart rate for a human is 60 to 100 bpm).

Scaling relationships like we just saw for the hummingbird are common in biology. If you want to learn more about this topic, I suggest Knut Schmidt-Nielsen’s fascinating book Scaling: Why is Animal Size so Important?

Drinking

My favorite bird is the mourning dove. We sometimes will have eight or more of these sweet, gentle birds around our bird feeder. I love their low-pitched coo… coo… coooooooooo song. They mate for life.

Doves are unique among birds in the way they drink. Most birds fill their bill with water and then gravity pulls it down to their stomach. Sometimes they tilt their head back to help the water flow. Mourning doves, on the other hand, suck water into their bill, like we suck water through a straw. Professor Gart Zweers, from the University of Leiden, took high-speed x-ray photos, and concluded that doves draw a partial vacuum which pulls the water up.

Singing

Bird songs are analyzed using plots of time and frequency. As discussed in Chapter 11 of Intermediate Physics for Medicine and Biology, you can resolve any function of time into its component frequencies: Fourier analysis. If you plot the instantaneous frequency versus time, you get a sonogram. The higher the frequency, the higher the pitch that we hear. The northern cardinal’s song starts on a high pitch (around 4 kilohertz, which is about the frequency of highest pitched note on a piano) and then slurs downward an octave (to 2 kilohertz) in about half a second.

Trevisan and Mindlin (Philosophical Transactions A, Volume 367, Pages 3239–3254, 2009) have modeled the bird’s vocal organ, called the syrinx. Their model might be familiar to physics students: it is Newton’s second law, force equals mass times acceleration. The important parameters that enter the model are the mass, stiffness, and a constant characterizing the dissipation or attenuation of the motion. The dissipation can be nonlinear, leading to all sorts of complex dynamics. The model predicts an oscillatory behavior (like that for a mass on a spring). Furthermore, the beak acts as a resonance tube (somewhat like an organ pipe).

We get majestic red cardinals visiting our birdfeeders all the time. Next time you hear a cardinal singing, think of all the physics going on.

Magnetoreception

Are Electromagnetic Fields
Making Me Ill?

Many birds make long migrations, and one wonders how they find their way. One method is magnetoreception: the sensing of magnetic fields. Most organisms cannot detect magnetic fields, but some birds can. Magnetoreception is possible because the birds have small particles of magnetite, called magnetosomes, which provide a magnetic moment that can interact with a magnetic field. I discussed magnetoreception in my book Are Electromagnetic Fields Making Me Ill?

In 1963, German zoologist Wolfgang Wiltschko placed European robins inside a chamber and turned on a magnetic field comparable in strength to the earth’s field. He did not expect a response, but to his surprise the birds oriented with the field… The robins proved adept at sensing magnetic signals during their annual migration.

Some researchers believe there are other mechanisms for magnetoreception besides magnetite particles. I wrote

A few animals, including the European robin, may take advantage of free radical reactions instead of magnetite for magnetoreception. Sonke Johnsen and Kenneth Lohmann [Physics Today, Volume 61, Pages 29–35, 2008], after reviewing the data, conclude that “the current evidence for the radical-pair hypothesis is maddeningly circumstantial…” The jury is still out on this issue.

To tell you the truth, I’m skeptical that free radical reactions are important.

Another animal that may detect the earth’s magnetic field and use it to navigate is the bee. Next week we will continue this series on the physics of native gardening by examining the physics of bees.

 Northern cardinal song

https://www.youtube.com/watch?v=e_b4PkcpDe0

A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle

Roth and Gielen,
Annals of Biomedical Engineering,
Volume 15, Pages 591–602, 1987

Today I want to tell you how Frans Gielen and I wrote the paper “A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle” (Annals of Biomedical Engineering, Volume 15, Pages 591–602, 1987). It’s not one of my more influential works, but it provides insight into the kind of mathematical modeling I do.

The story begins in 1984 when Frans arrived as a post doc in John Wikswo’s Living State Physics Laboratory at Vanderbilt University in Nashville. Tennessee. I had already been working in Wikswo’s lab since 1982 as a graduate student. Frans was from the Netherlands and I called him “that crazy Dutchman.” My girlfriend (now wife) Shirley and I would often go over to Frans and his wife Tiny’s apartment to play bridge. I remember well when they had their first child, Irene. We all became close friends, and would go camping in the Great Smoky Mountains together.

Frans had received his PhD in biophysics from Twente University. In his dissertation he had developed a mathematical model of the electrical conductivity of skeletal muscle. His model was macroscopic, meaning it represented the electrical behavior of the tissue averaged over many cells. It was also anisotropic, so that the conductivity was different if measured parallel or perpendicular to the muscle fiber direction. His PhD dissertation also reported many experiments he performed to test his model. He used the four-electrode method, where two electrodes pass current into the tissue and two others measure the resulting voltage. When the electrodes are placed along the muscle fiber direction, he found that the resulting conductivity depended on the electrode separation. If the current-passing electrodes where very close together then the current was restricted to the extracellular space, resulting in a low conductivity. If, however, the electrodes were farther apart then the current would distribute between the extracellular and intracellular spaces, resulting in a high conductivity.

When Frans arrived at Vanderbilt, he collaborated with Wikswo and me to revise his model. It seemed odd to have the conductivity (a property of the tissue) depend on the electrode separation (a property of the experiment). So we expressed the conductivity using Fourier analysis (a sum of sines and cosines of different frequencies), and let the conductivity depended on the spatial frequency k. Frans’s model already had the conductivity depend on the temporal frequency, ω, because of the muscle fiber’s membrane capacitance. So our revised model had the conductivity σ be a function of both k and ω: σ = σ(k,ω). Our new model had the same behavior as Fran’s original one: for high spatial frequencies the current remained in the extracellular space, but for low spatial frequencies it redistributed between the extracellular and intracellular spaces. The three of us published this result in an article titled “Spatial and Temporal Frequency-Dependent Conductivities in Volume-Conduction Calculations for Skeletal Muscle” (Mathematical Biosciences, Volume 88, Pages 159–189, 1988; the research was done in January 1986, although the paper wasn’t published until April of 1988).

Meanwhile, I was doing experiments using tissue from the heart. My goal was to calculate the magnetic field produced by a strand of cardiac muscle. Current could flow inside the cardiac cells, in the perfusing bath surrounding the strand, or in the extracellular space between the cells. I was stumped about how to incorporate the extracellular space until I read Les Tung’s PhD dissertation, in which he introduced the “bidomain model.” Using this model and Fourier analysis, I was able to derive equations for the magnetic field and test them in a series of experiments. Wikswo and I published these results in the article “A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue” (IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467–469, 1986).

By the summer of 1986 I had two mathematical models for the electrical conductivity of muscle. One was a “monodomain” model (representing an averaging over both the intracellular and extracellular spaces) and one was a “bidomain” model (in which the intracellular and extracellular spaces were each individually averaged over many cells). It was strange to have two models, and I wondered how they were related. One was for skeletal muscle, in which each muscle cell is long and thin but not coupled to its neighbors. The other was for cardiac muscle, which is a syncytium where all the cells are coupled through intercellular junctions. I can remember going into Frans’s office and grumbling that I didn’t know how these two mathematical representations were connected. As I was writing the equations for each model on his chalkboard, it suddenly dawned on me that the main difference between the two models was that for cardiac tissue current could flow perpendicular to the fiber direction by passing through the intercellular junctions, whereas for skeletal muscle there was no intracellular path transverse to the uncoupled fibers. What if I took the bidomain model for cardiac tissue and set the transverse, intracellular conductivity equal to zero? Wouldn’t that, in some way, be equivalent to the skeletal muscle model?

I immediately went back to my own office and began to work out the details. This calculation starts on page 85 of my Vanderbilt research notebook #15, dated June 13, 1986. There were several false starts, work scratched out, and a whole page crossed out with a red pen. But by page 92 I had shown that the frequency-dependent conductivity model for skeletal muscle was equivalent to the bidomain model for cardiac muscle if I set the bidomain transverse intracellular conductivity to zero, except for one strange factor that included the membrane impedance, which represented current traveling transverse to the skeletal muscle fibers by shunting across the cell membrane. But this extra factor was important only at high temporal frequencies (when capacitance shorted out the membrane) and otherwise was negligible. I proudly marked the end of my analysis with “QED” (quod erat demonstrandum; Latin for “that which was to be demonstrated,” which often appears at the end of a mathematical proof).

Two pages (85 and 92) from my Research Notebook #15 (June, 1986).

Frans and I published this result in the Annals of Biomedical Engineering, and it is the paper I cite at the top of this blog post. Wikswo was not listed as an author; I think he was traveling that summer, and when he returned to the lab we already had the manuscript prepared, so he let us publish it just under our names. The abstract is given below:

We compare two models for calculating the extracellular electrical potential in skeletal muscle bundles: one a bidomain model, and the other a model using spatial and temporal frequency-dependent conductivities. Under some conditions the two models are nearly identical, However, under other conditions the model using frequency-dependent conductivities provides a more accurate description of the tissue. The bidomain model, having been developed to describe syncytial tissues like cardiac muscle, fails to provide a general description of skeletal muscle bundles due to the non-syncytial nature of skeletal muscle.

Frans left Vanderbilt in December, 1986 and took a job with the Netherlands section of the company Medtronic, famous for making pacemakers and defibrillators. He was instrumental in developing their deep brain stimulation treatment for Parkinson’s disease. I graduated from Vanderbilt in August 1987, stayed for one more year working as a post doc, and then took a job at the National Institutes of Health in Bethesda, Maryland.

Those were fun times working with Frans Gielen. He was a joy to collaborate with. I’ll always remember than June day when—after brainstorming with Frans—I proved how those two models were related.

Short bios of Frans and me published in an article with Wikswo in the IEEE Trans. Biomed. Eng.,
cited on page 237 of Intermediate Physics for Medicine and Biology.

 

Integral of the Bessel Function

Have you ever been reading a book, making good progress with everything making sense, and then you suddenly stop at say “wait… what?”. That happened to me recently as I was reading Homework Problem 31 in Chapter 12 of Intermediate Physics for Medicine and Biology. (Wait…what? I’m a coauthor of IPMB! How could there be any surprises for me?) The problem is about calculating the two-dimensional Fourier transform of 1/r, and it supplies the following Bessel function identity 

The function J₀ is a Bessel function of the first kind of order zero. What surprised me is that if you let x = kr, you get that the integral of the Bessel function is one,

Really? Here’s a plot of J₀(x).

It oscillates like crazy and the envelope of those oscillations falls off very slowly. In fact, an asymptotic expansion for J₀ at large x is

The leading factor of 1/√x decays so slowly that its integral from zero to infinity does not converge. Yet, when you include the cosine so the function oscillates, the integral does converge. Here’s a plot of

The integral approaches one at large x, but very slowly. So, the expression given in the problem is correct, but I sure wouldn’t want to do any numerical calculations using it, where I had to truncate the endpoint of the integral to something less than infinity. That would be a mess!

Here’s another interesting fact. Bessel functions come in many orders—J₀, J₁, J₂, etc.—and they all integrate to one.

Who’s responsible for these strangely-behaved functions? They’re named after the German astronomer Friedrich Bessel but they were first defined by the Swiss mathematician Daniel Bernoulli (1700–1782), a member of the brilliant Bernoulli family. The Bernoulli equation, mentioned in Chapter 1 of IPMB, is also named for Daniel Bernoulli. 

There was a time when I was in graduate school that I was obsessed with Bessel functions, especially modified Bessel functions that don’t oscillate. I’m not so preoccupied by them now, but they remain my favorite of the many special functions encountered in physics.