You Can Hear About a Nickel’s Worth of Difference

In Chapter 13 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe a logarithmic scale for sound intensity: the decibel. In general, an increase in intensity is perceived as a greater loudness (although this relationship is surprisingly complex). Is there an analogous relationship between frequency and pitch? Yes! Loudness is determined using a logarithmic scale with a base of ten, whereas pitch is measured using a logarithmic scale of base two, because a doubling of the frequency corresponds to raising the pitch by one octave. Those familiar with music are accustomed to associating different pitches with different notes in a musical scale. Most instruments are tuned using the equal tempered scale, dividing an octave into twelve equal logarithmic steps. One step, called a semitone, corresponds to the difference in pitch between, say, F and F-sharp. The fractional change of one semitone is 2^1/12 = 1.0595, or an increase in frequency of roughly 6%. This frequency shift is so important that it is expressed by a special unit: one semitone is equal to 100 cents. The cent is to pitch as the decibel is to loudness. Doubling the intensity corresponds to an increase of 3 dB, whereas doubling the frequency corresponds to an increase of 1200 cents.

How finely can the human ear resolve pitch? In other words, if you play two pure tones one right after the other, by how much must their frequency differ for you to notice that they are indeed different? A typical listener can detect a change of about 5 cents, leading to the rule of thumb that you can hear about a nickel’s worth of difference. Try testing your own pitch perception online here. When I tried, I could always detect a difference of 50 cents (tones of 440 and 453 Hz, presented one after another; I assume this is a control used by the testing software, because the difference is obvious). I could never detect a difference of 4 or 6 cents (440 compared to 441 or 441.5 Hz). I had inconsistent results with 8 and 11 cents (440 compared to 442 or 443 Hz); sometimes I perceived slightly different tones, and sometimes I didn’t. Ten cents is a reasonable approximation to my just noticeable difference, which confirms what I have always suspected: I have poor but not pathological pitch perception. When playing the tuba in my high school band, the director often had us “tune up” relative to a standard note, typically played by the first clarinet. I always had trouble with this task; he had to tell me if I was sharp or flat. Poor pitch perception has its advantages: you don’t need to hire a piano tuner! Pity my poor wife—my only audience, other than my dog Suki—but my wrong notes are probably more bothersome than the out-of-tune piano, so spending for a piano tuner would not help.

Humans can hear pitches from roughly 20 to 20,000 Hz. Because 2¹⁰ is about 1000, humans can hear frequencies that range over roughly ten octaves, or 12,000 cents, and therefore you (but not I) can distinguish about 2400 different tones. A piano keyboard plays notes 28 to 4186 Hz, which is a little more than seven octaves, or roughly 8700 cents (87 semitones between the 88 keys). Sometimes changes in frequency are measured in millioctaves: 1 mO = 1.2 cents. Although the cent is not a metric unit, I still worry that the SI police, who insist that all things centi- are going out of fashion compared to all things milli-, will demand we start using the millioctave. I hope not.

If two tones are played at the same time, rather than one after another, you can perceive small differences in pitch using beats. Consider the triginometric identity

If frequencies A and B are similar, then their sum consists of a carrier frequency equal to the average of the two original frequencies modulated by the difference of the two frequencies. For instance, if you have a tone corresponding to concert A (440.00 Hz) and another tone out of tune with concert A by 3 cents (440.78 Hz), when played together the sound consists of a tone having frequency 440.39 Hz modulated by a sinusoid that gets louder and softer with a frequency of 0.78 Hz (or a period of 1.3 seconds). Your ear can’t tell that the pitch of the carrier tone is different from concert A, but it can detect the variation in loudness caused by the beats. You can hear beats generated online here.

If several tones have widely separated frequencies, your ear (or more properly, your brain) detects distinct notes played together; a chord. If two tones have nearly the same frequency (like in the example above), you hear a single tone with modulated amplitude; beats. For intermediate frequency differences (say, between a note an octave above concert A, 880 Hz, and a severely out-of-tune A at 897 Hz, a difference of 17 Hz or 33 cents) you hear neither a chord nor beats. Instead, your brain perceives dissonance. A London police whistle generates frequencies of 1904 and 2136 Hz, a difference of 232 Hz or more than 200 cents. It sounds annoying. Try it yourself.

The frequency ratio between a C and G (a perfect fifth) should be 3:2, which is 702 cents. However, in equal tempered tuning the shift between C and G is 700 cents. This difference of 2 cents is indistinguishable to all but the best ears. The major third is more of a problem. It should have a ratio of 5:4, or 386 cents. In the equal tempered scale, a third is 400 cents, off by 14 cents, which a good ear can hear. If all these tonal relations are different than what they would be in just intonation, then why do we use an equal tempered scale? It allows us to change keys without retuning the instrument. But we pay a price in that a major chord does not have precisely the desired 4:5:6 frequency ratio. Pythagoras must be turning over in his grave.

Both pitch perception and color vision arise from the physics of frequency detection. But the tones we hear and colors we see are determined by more than just physics. There is a lot of information processing by the brain, which leads to many fascinating and unexpected results including surprising pathologies. To completely appreciate how we perceive frequency, you also need to understand the brain.

Biomagnetism Therapy: Pseudoscientific Twaddle

Voodoo Science,
by Robert Park.

Ever since Robert Park—author of Voodoo Science and the weekly column What’s New—suffered a stroke in 2013, I have been searching for another debunker of pseudoscience. Finally, I’ve found her! Harriet Hall is a retired physician and former Air Force flight surgeon. Every week she battles nonsense in the blog sciencebasedmedicine.org. In her November 20 entry, she addressed “biomagnetic therapy.”

Biomagnetism Therapy: Pseudoscientific Twaddle

In a television interview, a practitioner of biomagnetic therapy claimed she had cured her own breast lump and the metastatic cancer of another person. I wonder how many viewers believed her. On the “official website” of biomagnetism therapy, http://biomagnetism.net/, they claim it is “the answer to ALL your health problems… an all-natural, non-invasive therapy proven to prevent, diagnose and treat countless diseases, chronic illnesses and degenerative health problems.”

Sound too good to be true? Of course it does! You are already skeptical. If you read further, you will become even more skeptical….”

She concludes

…It pains me to see misinformation such as this fed to gullible patients. Using biomagnetic therapy isn’t likely to harm patients physically, but it’s likely to harm their comprehension of science. It’s likely to waste their money, and it could delay getting treatments that do work. Perhaps the worst thing is that people who practice this therapy are deceiving themselves. They don’t understand science, and they mistake testimonials for evidence of efficacy. They don’t understand the need for controlled studies. They don’t understand placebo effects, suggestion, expectation, regression to the mean, the natural course of illness, and all the other things that can lead people to believe a bogus treatment works. It is particularly tragic that anyone trained as an MD could have such poor critical thinking skills and be misled by such egregious pseudoscience.

Russ Hobbie and I have an entire chapter about biomagnetism in Intermediate Physics for Medicine and Biology. We discuss the measurement of the very small magnetic fields produced by the brain (magnetoencephalography) and the use of rapidly changing magnetic fields to stimulate neurons (transcranial magnetic stimulation). We also devote a chapter to magnetic resonance imaging. These are important topics, but they often get mixed up with phony claims about “biomagnetic therapy.”

If you doubt this is a real problem, go to Google and search for “biomagnetism” (the title of Chapter 8 in IPMB). The first site you get starts “One of the most peculiar therapy systems that FAIM [Foundation for Alternative and Integrative Medicine] is investigating is one that uses ordinary magnets to heal. Although magnets have been used in therapies for a long time, this particular method uses pairs of magnets to neutralize disease-causing pathogens in the body...” The second site begins “Yes! It’s the answer to ALL your health problems…” The third describes “The Revolutionary Therapy based on the Biomagnetic Pairs discovered by Dr. Isaac Goiz Durán, MD in 1988...” The fourth is the “Official website for Biomagnetism classes in the USA with Dr. Isaac Goiz Durán...” Finally, the fifth site in the list is Wikipedia’s entry on biomagnetism (the measurement of weak magnetic fields produced by the body). The first four are twaddle; the fifth is reputable.

Women Aren’t Supposed to Fly,
by Harriet Hall.

If you want to learn more about Harriet Hall, read her autobiography Women Aren’t Supposed to Fly, in which she describes her experiences in medical school and the Air Force. From the Preface:

There’s an old curse “may you live in interesting times.” I lived in an era when society was starting to allow women to enter male-dominated fields, but didn’t yet entirely approve. Someone said, “Whatever women do they must do twice as well as men to be thought half as good. Luckily this is not difficult.” Actually, it was difficult. It was frequently frustrating, sometimes painful, often ridiculously funny, and always interesting. Come with me on a ramble through my education and career and let me tell you what it was like.

What Women Aren’t Supposed to Fly does not explain is how Hall ended up a lampooner of baloney and poppycock. She needs to write a second book, telling that story. I’m sure it would be equally fascinating and amusing.

I’d have preferred another physicist pick up Bob Park’s banner, but I’ll take what I can get. Harriet Hall, keep up the good work and let’s end this “biomagnetic therapy” rubbish.

Charles Bean, Biological Physicist

I’ve always been fascinated by physicists who move into biology, and I collect stories about scientists who have made this transition successfully. Today let me share one example: Charles Bean (1923–1996). Bean spent much of his career studying solid state physics, especially magnetism and superconductivity. He worked for more than 30 years at the General Electric Research and Development Center in Schenectady, New York. You can read about his research for GE in a biographical memoir published by the National Academy of Sciences. I want to focus on his work in biological physics.

Relatively early in his career, Bean expanded his scientific interests beyond magnetism and superconductivity. He also studied biophysics, and he encouraged colleagues to consider the field as well. In an invited talk to the American Physical Society on how to change from physics to biology, one of his recommendations was straightforward: “Start to eat lunch with biologists.” In 1960 Bean managed to convince the now-renowned biophysicist Carl Woese to join the General Electric Research and Development Center in Schenectady, where he stayed for three years before joining the University of Illinois in 1970. And Charlie took his own advice and ate with Carl every day.

Bean was elected the first Coolidge Fellow at the GE laboratory. The Coolidge Fellowship program was a way to recognize the company’s most valuable scientists, and its main advantage was that the recipient could go on sabbatical leave to any another institution with full pay. Charlie decided to go to Rockefeller University, where he stayed (not full time) from 1973 to 1978. There he was exposed to neurophysiology, and he eventually wrote a theory of stimulation of myelinated fibers that was published in the British Journal of Physiology (1974).

At about this time Bean had started to spend his summers in Woods Hole, MA, which he enjoyed enormously, both for its outdoor activities and its laboratories. When asked why, he said, “I like to be at a place where the library is open 24 hours a day.” Here he became interested in sea urchin sperm, and he invented a clever method to determine the average velocity and length they can swim. He did this by having a dilute concentration of sperm in a solution above a clean gold surface, and each time a sperm hit the gold it stuck.

Very soon Bean’s research papers began to be published in biophysics journals rather than physics journals. He first became interested in membranes, for example, and wrote a long treatise for the U.S. Department of the Interior on reverse osmosis (1969). Taking advantage of GE’s Nuclepore membranes, he developed, together with Ralph DeBlois, a virus counter—a variant of the famous Coulter counter (1970). Later he developed a seminal theory of neutral porous membranes (1972). On the basis of this paper he was offered a professorship, which tempted him, though eventually he turned it down.

… Bean also continued more serious research while at RPI [Rensselaer Polytechnic Institute]. For example, being a good friend of French biophysicist P. G. de Gennes, through him Bean became fascinated by electrophoresis—in particular the way a strand of DNA twists through a gel, like a snake through grass. He analyzed and modeled the process, and in 1987 wrote a paper on the subject with H. Hervet.

Bean spent much of his later time looking for the elusive magnetic bacteria, which were rediscovered by R. P. Blakemore at Woods Hole in 1975 (they had been seen in 1963 in Italy by Salvatore Bellini but thereafter largely forgotten). The reason why these bacteria are guided by a magnetic field is that they have small internal magnets corresponding to the chain-of-spheres model. Bean developed simple equipment that enabled him to seek such magnetotactic bacteria virtually everywhere. He thought he had found some in a pothole right outside RPI, but before verifying and publishing his findings Bean died of heart failure.

Bean appears often in Intermediate Physics for Medicine and Biology. In Section 5.9, Russ Hobbie and I describe a continuum model for volume and solute transport in a pore. We cite his Department of the Interior report and his model of neutral pores repeatedly.

Bean CP (1969) Characterization of cellulose acetate membranes andultrathin films for reverse osmosis. Research and Development Progress Report No. 465 to the U.S. Department of Interior, Office of Saline Water. Contract No. 14-01-001-1480. Washington, Superintendent of Documents, October 1969.

Bean CP (1972) The physics of porous membranes—neutral pores. In: Eisenman G (ed) Membranes, vol 1. Dekker, New York, pp 1–55.

The description of his work in this field is even more extensive in earlier editions of IPMB (see, for example, the third edition which Russ authored before I came along and ruined it). In the fifth edition, we added a homework problem about the Coulter counter, in which we cite another publication by Bean.

DeBlois RW, Bean CP (1970) Counting and sizing of submicron particlesby the resistive pulse technique. Rev Sci Inst 41:909–916.

Given my interest in neural stimulation, I was particularly curious about Bean’s theory of microstimulation of myelinated nerve axons. I had trouble finding his paper on this topic, until I realized that it’s an appendix of a paper by Abzug, Maeda, Peterson and Wilson (“Cervical branching of lumbar vestibulospinal axons”). Bean develops a theory of neural stimulation very similar to that of the activating function introduced in Homework Problem 38 of Chapter 7 in IPMB. Bean assumed a myelinated axon with discrete nodes of Ranvier rather than a continuous cable. This assumption makes little difference if the stimulating electrode is far from the nerve, but if it is closer to the nerve than one internodal space, the discrete model is more appropriate.

Figure 8.25 in IPMB shows a magnetotactic bacterium containing a chain of small particles of magnetite. Bean developed a theory for the magnetic properties of such a “chain-of-spheres” before they were known to occur in bacteria.

In summary, Charles Bean is a fine example of a physicist who moved from physics to biology, and was able to contribute a unique perspective on important biological problems. In my experience, such physicists can contribute to a wide variety of biological topics. Often their insights ignore much of biology’s complexity, but they are based on universal physical principles that may be unfamiliar to many biologists.

Now, I need to go find a biologist to have lunch with.

The Royal Institution's 2015 Christmas Lecutres: How to Survive in Space

Michael Faraday lecturing
at the Royal Institution.

190 years ago at the Royal Institution in London, Michael Faraday presented the first Christmas Lectures. These lectures about science have continued annually except during the worst years of World War II, and are now broadcast on television and online. Each year is a different topic, and this year the topic is How to Survive in Space.

In December 2015, Tim Peake will become the first Briton in space for more than 20 years and a new member of the European Astronaut Corps. As Tim adjusts to life on board the International Space Station (ISS), Kevin Fong’s CHRISTMAS LECTURES will take us on a journey from planet Earth into Low Earth Orbit and beyond. This is the story of human survival against all the odds; the story of how science, medicine and engineering come together to help answer our biggest questions about Life, the Earth, the Universe and our place in it.

From artificial gravity and greenhouses in space to plasma drives and zero-G surgical suits, the Lectures will reveal how what once was the stuff of science fiction is fast becoming today’s science fact.

Throughout the three-part series, Kevin will be accompanied by special guest appearances from ISS astronauts who will reveal what daily life is like 400 kilometres above the Earth, demonstrate the technology and techniques that help them stay safe and healthy, and explain the scientific experiments they are part of that are helping to stretch the limits of our understanding of human physiology and survival in a way that no experiment back on Earth could.

Intermediate Physics for Medicine and Biology does not address space medicine specifically, but many of the topics examined in these Christmas Lectures require a deep understanding of how physics and medicine interact. I plan to watch. The lectures will be first broadcast by the BBC on December 28, 29, and 30. I won’t be able to view them until they are posted afterward (hopefully, soon afterward) on the Ri Channel.

A Christmas Carol,
by Charles Dickens.

Because this year I’m posting on Christmas Day, I’ll conclude with a Christmas present—the final lines of my all-time favorite book (yes, even better that IPMB): A Christmas Carol, by Charles Dickens. I reread it every December.

Scrooge was better than his word. He did it all, and infinitely more; and to Tiny Tim, who did not die, he was a second father. He became as good a friend, as good a master, and as good a man, as the good old city knew, or any other good old city, town, or borough, in the good old world. Some people laughed to see the alteration in him, but he let them laugh, and little heeded them; for he was wise enough to know that nothing ever happened on this globe, for good, at which some people did not have their fill of laughter in the outset; and knowing that such as these would be blind anyway, he thought it quite as well that they should wrinkle up their eyes in grins, as have the malady in less attractive forms. His own heart laughed: and that was quite enough for him.

He had no further intercourse with Spirits, but lived upon the Total Abstinence Principle, ever afterwards; and it was always said of him, that he knew how to keep Christmas well, if any man alive possessed the knowledge. May that be truly said of us, and all of us! And so, as Tiny Tim observed, God bless Us, Every One!

Star Wars

With The Force Awakens opening in theaters, now is the perfect time to answer your questions about how Intermediate Physics for Medicine and Biology relates to Star Wars. (Warning: This post is spoiler laden for Episodes I - VI, but not for Episode VII which I haven’t seen yet.)

1. With so many light saber duels, why is there so little blood? Even blasters kill without gore or carnage. In Section 14.11 of IPMB, Russ Hobbie and I discuss tissue ablation. According to Wookieepedia, ablation cauterizes wounds, preventing bleeding. This is the same reason we use lasers in surgery.

 

2. Why do blaster shots not propagate at the speed of light? Chapter 14 of IPMB gives the speed of light as 3 × 10⁸ m/s. In Star Wars, shots travel not much faster than a hard-thrown fastball, maybe 100 m/s. Apparently this far-away galaxy has a large permeability of free space, μ_o. Next time you view these films, watch for exaggerated magnetic effects.

 3. How did Han Solo freeze so quickly (and reversibly!) in carbonite? The bioheat equation  developed in Chapter 14 of IPMB implies that heat diffuses into tissue, and over long distances diffusion is slow. My guess is that carbonite freezing makes use of the blood flow term in the bioheat equation, perhaps by rapidly injecting cold carbonite intravenously. However, I don’t see any IV tubes coming out of Han. Do you? Carbonite freezing has to be dangerous. But no matter; had Han died during freezing, Chewbacca would have saved Princess Leia.

4. How does Luke Skywalker’s artificial hand work? Chapter 7 of IPMB discusses electrical stimulation of nerves, which is crucial for developing neural prostheses. Luke had such a prosthesis after he lost his hand during his epic duel with Darth Vader (Luke’s FATHER!). Functional neural stimulation is becoming so sophisticated that artificial hands may arrive sooner than you think.

5. How does Darth Vader control his breathing? Chapter 10 in IPMB analyzes feedback loops, and our main example describes how we control the carbon dioxide in our blood by adjusting our ventilation rate. I wonder, does Vader’s mask break that feedback loop? Perhaps a carbon dioxide sensor in the mask adjusts the rate of his slow, heavy breathing. I suspect something is horribly wrong with his physiological control mechanism, because once the mask comes off Vader dies.

6. What determines the anatomy of all those alien life forms? In Chapter 2 of IPMB, Russ and I describe how size impacts the structure of animals: scaling. One famous result involves the strength of bones. Body mass scales as length cubed, but the strength of a bone scales as its radius squared, meaning that bones must get thicker relative to their length in larger animals. This is why Jabba the Hutt has no legs. Yoda lied when he said “size matters not.”

7. How did Luke avoid freezing to death on Hoth? When an animal dies, it loses heat according to Newton’s law of cooling (Chapter 3 in IPMB). After Han killed the tauntaun and shoved Luke into its belly, its temperature began to fall exponentially. Luke must have gotten really cold. What saved him was immersion into a bacta tank, which speeds rewarming by taking advantage of convection as well as conduction. I suspect bacta may also contain suspended stem cells, but who knows? Whatever the mechanism, Luke survived. During his recuperation is the most disturbing event in the entire Star Wars saga: the incestuous kiss. Ewwwwww!

8. What did leaders of the Rebel Alliance say when they realized that Vader knew about their plans to attack the Death Star? One way to measure radiation dose is to use thermoluminescent phosphors, which Russ and I describe in Chapter 16 of IPMB as a “dielectric material that has been doped with impurities or has missing atoms in the crystal lattice to form metastable energy levels or traps.” So, Admiral Ackbar must have just discovered the mechanism of thermoluminescence when he exclaimed “It’s a trap!” 

If you want answers to all of your Star Wars questions, keep a copy of Intermediate Physics for Medicine and Biology handy. Before watching the new film, find time for a Star Wars marathon (with episodes in machete order, thereby avoiding Jar Jar Binks). Finally, don’t miss my favorite version of the Star Wars theme song, performed in the video below by lounge singer Nick Winters. Now, off to the movies. And, May The Force Be With You.

The Art of Insight in Science and Engineering

The Art of Insight
in Science and Engineering,
by Sanjoy Mahajan.

I recently read The Art of Insight in Science and Engineering, by Sanjoy Mahajan (MIT Press, 2014). The Preface begins:

Science and engineering, our modern ways of understanding and altering the world, are said to be about accuracy and precision. Yet we best master the complexity of our world by cultivating insight rather than precision.

We need insight because our minds are but a small part of the world. An insight unifies fragments of knowledge into a compact picture that fits in our minds. But precision can overflow our mental registers, washing away the understanding brought by insight. This book shows you how to build insight and understanding first, so that you do not drown in complexity.

I think that Mahajan makes a good point. I’ve noticed that many of my students can use complicated algorithms to calculate numbers correctly, but often lack the ability to estimate a solution. This leads students to make mistakes in their homework that result in ridiculously wrong results. When this happens, I often print “THINK BEFORE YOU CALCULATE” in big red letters on their paper, and take off more than my normal number of points. On the other hand, when a homework question requires an intricate calculation that the student skips but instead writes down a reasonable estimate for the solution, I take off a few points but give them most of the credit because they actually thought about what they were doing.

Russ Hobbie and I stress estimation in the very first section of Intermediate Physics for Medicine and Biology: “One valuable skill in physics is the ability to make order of-magnitude estimates, meaning to calculate something approximately right.”. We provide several “back-of-the-envelope” homework problems in our book, such as “Estimate the number of hemoglobin molecules in a red blood cell,” “Estimate the size of a box containing one air molecule”, and “Estimate the density of water”. In fact, if you search through our book for the word “estimate” you will find many such questions. Students often object to these problems. When I explore why they are having difficulty, I find that many are reluctant to guess some number. For instance, a homework problem in Chapter 6 says “Suppose that an action potential in a 1-μm diameter unmyelinated fiber has a speed of 1.3 m s⁻¹. Estimate how long it takes a signal to propagate from the brain to a finger.” Some students grind to a halt because the length of the arm is not given (or they use 1 μm for the length, getting a ridiculous result). When I tell them that the word “estimate” means “make a reasonable guess,” they’re not any happier; they think that guessing the length of the arm is cheating. But I’m just asking them to estimate the time to send a signal from the brain to the hand, and then THINK about the result. That’s why the last sentence in the problem says “Speculate on the significance of these results for playing the piano.”

In The Art of Insight, Mahajan also talks about units. In homework assignments, I often ask my students to algebraically solve for some expression; for example a relationship involving a distance x, time t, and speed v. A student will sometimes hand in a result like “v + x t”. Yikes! The units don’t work! (I use a lot of exclamation points when grading assignments.) I am stingy with partial credit when the derived equation is dimensionally incorrect, not because I want to punish a minor error, but because the student didn’t take the time to look at their equation and ask “do the units make sense?”

Another skill Mahajan stresses is proportional reasoning. Russ and I emphasize this in Chapter 2, when we discuss scaling. For instance, here is a question from an exam I gave earlier this semester in my Biological Physics class:

Assume the specific metabolic rate R* scales with mass M as R* = C M^−1/4 , where C is a constant. If an 81 kg person has a specific metabolic rate of 1.5 W/kg, what is the specific metabolic rate of a 1 kg guinea pig?

Some students get stuck because I didn’t give them a value for C. The point is, R* times M^1/4 is a constant, so R*_person M_person^1/4 = R*_{guinea pig} M_{guinea pig}^1/4. You don’t need C. (You don't even need your calculator if you recall that 81 = 3⁴.)

Mahajan talks about examining “easy cases,” which is similar to our emphasis on taking limits of equations. Consider the analysis of Eq. 2.25 in IPMB: dy/dt = a – by. The solution (Eq. 2.26) is y = a/b (1 – e^−bt). When I derive this equation in class, I always stop and ask the students what is the limit when t goes to inifinity? The answer is clearly y = (a/b), which they usually get right. Then I ask what is the limit when t goes to zero? This case requires more thought. A student will often say “zero,” but then I respond “yes, but HOW does it go to zero?” Using the Taylor series e^x = 1 + x gives a limit of y = at. This skill is even more important for more complicated expressions. For instance, Section 4.12 in IPMB compares drift and diffusion. Equation 4.63 gives the concentration of a material that drifts with fluid speed v and diffuses with diffusion constant D through a tube of length x₁. The student gains much insight by taking the limits when x₁v/D is much greater than one and much less than one. If the student doesn’t take these limits, then this example is just a mathematical exercise that provides no insight.

Finally, Mahajan discusses briefly the value of log-log plots when analyzing scaling laws. This is a topic Russ and I emphasize in Chapter 2 of IPMB. It’s amazing how much information you can get from a log-log plot. One little fact I never realized until I read Mahajan’s book is that the geometric mean, √ab, of two numbers a and b is just the half-way point between the two numbers on a logarithmic scale.

A Mind for Numbers,
by Barb Oakley.

I had a chance to offer my views on insight and learning when my Oakland University colleague and friend Barb Oakley published her book A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra). Barb asked several people, including me, for short “sidebars” to include in her book. Here is mine:

Insights on Learning from Physics Professor Brad Roth, A Fellow of the American Physical Society and Co-Author of Intermediate Physics for Medicine and Biology

One thing I stress in my classes is to think before you calculate. I really hate the ‘plug and chug’ approach that many students use. Also, I find myself constantly reminding students that equations are NOT merely expressions you plug numbers into to get other numbers. Equations tell a story about how the physical world works. For me, the key to understanding an equation in physics is to see the underlying story. A qualitative understanding of an equation is more important than getting quantitatively correct numbers out of it.

Here are a few more tips:

1. Often, it takes way less time to check your work than to solve a problem. It is a pity to spend twenty minutes solving a problem and then get it wrong because you did not spend two minutes checking it.

2. Units of measurement are your friend. If the units don’t match on each side of an equation, your equation is not correct. You can’t add something with units of seconds to something with units of meters. It’s like adding apples and rocks—nothing edible comes of it. You can look back at your work, and if you find the place where the units stop matching, you probably will find your mistake. I have been asked to review research papers that are submitted to professional journals that contain similar errors.

3. You need to think about what the equations means, so that your math result and your intuition match. If they don’t match, then you have either a mistake in your math or a mistake in your intuition. Either way, you win by figuring out why the two don’t match.

4. (Somewhat more advanced) For a complicated expression, take limiting cases where one variable or another goes to zero or infinity, and see if that helps you understand what the equation is saying.

Books like The Art of Insight and A Mind for Numbers reinforce fundamental yet critical skills. I hope IPMB reinforces these skills too.

A Mathematical Model of Make and Break Electrical Stimulation of Cardiac Tissue by a Unipolar Anode or Cathode

“A Mathematical Model of Make and Break
Electrical Stimulation of Cardiac Tissue
by a Unipolar Anode or Cathode.”

Suppose I was going to die tomorrow and I could choose only one paper to cite on my tombstone. Which would I pick? I’d select

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

Below is the introduction, with references removed. I like the way it starts with a question.

What is the mechanism by which an electrical current, passed through a unipolar electrode, excites cardiac tissue? This simple question appears to have a straightforward answer: The stimulus current depolarizes the tissue under the electrode until the transmembrane potential reaches threshold, triggering an action potential wave front. Excitation of cardiac tissue, however, is more complicated than one might initially expect. Stimulation with a cathode might be explained by depolarization of the tissue under the electrode, but how does one explain stimulation with an anode? Even more intriguing, excitation is elicited by turning a stimulus off (break) as well as by turning it on (make). Why should turning off the stimulus excite the tissue? Indeed, four distinct mechanisms are responsible for stimulation of cardiac tissue—cathode make, anode make, cathode break, and anode break—and only cathode-make stimulation can be explained by depolarization under the electrode. To understand the other three mechanisms, we make detailed calculations of the transmembrane potential distribution induced by current through a unipolar electrode. We have three goals: to explain the mechanisms of excitation qualitatively; to predict stimulation thresholds quantitatively; and to determine how the threshold varies with electrode size and with stimulus pulse duration and frequency.

Our calculations are based on the bidomain model of cardiac tissue, which is useful for predicting the transmembrane potential induced by an extracellularly applied electric field. The bidomain model is a two- or three-dimensional cable model that accounts for the resistance of both the intracellular and the extracellular spaces. Many of the most interesting and nonintuitive predictions of the bidomain model occur when the ratios of the electrical conductivities parallel to and perpendicular to the myocardial fibers in the intracellular and extracellular spaces differ. For instance, current that is passed through a point extracellular electrode into a two-dimensional bidomain with unequal anisotropy ratios induces adjacent areas of depolarization and hyperpolarization. Such a region of hyperpolarization near a cathode is called a virtual anode; a region of depolarization near an anode is called a virtual cathode. The existence of virtual anodes and cathodes is predicted by the bidomain model and is essential for three of the four mechanisms of stimulation. Recently, virtual anodes and cathodes were observed experimentally in cardiac tissue.

My use of the royal “we” seemed reasonable when I wrote the paper, but now it grates on my ear. According to Google Scholar, in the twenty years since I published this article it has been cited 169 times. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I turned the prediction of break excitation of cardiac tissue into a homework problem (Chapter 7, Problem 48).

I did this research while working at the National Institutes of Health in Bethesda, Maryland. Sometimes on a slow afternoon I would sneak away from my desk and browse the stacks of the NIH library. One day I found a fascinating paper by Egbart Dekker, who measured the threshold for each of the four mechanisms of excitation (E. Dekker, 1970, “Direct Current Make and Break Thresholds for Pacemaker Electrodes on the Canine Ventricle,” Circulation Research, Volume 27, Pages 811–823.) Once I read Dekker’s article, I knew I could simulate this behavior using the then-new bidomain model and perhaps gain insight about mechanisms. At the time I was not well versed in mathematical models of the cardiac membrane kinetics with all their different ion currents, so I just used the Hodgkin-Huxley model of a nerve axon. A paper describing that study was unpublishable because who in their right mind would use a squid nerve axon model to represent a cardiac action potential? After the manuscript using the Hodgkin-Huxley model was rejected, I set to work learning about cardiac ion channel dynamics. I chose the Beeler-Reuter model, and the paper using the BR model (no, I did not choose that model because of my initials) was ultimately accepted for publication.

I sent a draft of my article to my PhD advisor, John Wikswo. He and his post doc Marc Lin immediately verified the model predictions experimentally (see their lovely paper: J. P. Wikswo, S. F. Lin, and R. A. Abbas, 1995, “Virtual Electrodes in Cardiac Tissue: A Common Mechanism for Anodal and Cathodal Stimulation,” Biophysical Journal, Volume 69, Pages 2195–2210). I remember the day Wikswo emailed me asking something like “what would you say if I told you the cathode make, cathode break, and anode make mechanisms all behave exactly as you predicted, but your anode break mechanism is totally wrong?” I began to panic, wondering how in the world I messed up, and sent Wikswo a frantic email asking for more details. His response was along the lines of “I asked ‘what would you say?’ I didn’t claim your prediction was actually wrong.” Ha, ha, ha; all four mechanisms were verified. Their paper was published the same month as mine and now has 300 citations. Your can read a layman’s account of this work in an article published in the Vanderbilt Register.

The figures in my original article were all black-and-white contour plots of action potential wave fronts propagating through the tissue. Wikswo had beautiful color figures in his paper. So, a few years later I “colorized” the figures, including them in a review article (B. J. Roth, S.-F. Lin and J. P. Wikswo, Jr., 1998, “Unipolar Stimulation of Cardiac Tissue,” Journal of Electrocardiology, Volume 31, Supplement, Pages 6–12). This always reminds me of how some of the classic old black-and-white movies have been colorized to look modern.

One reason I like publishing in the IEEE TBME is that they provide a short biographical sketch of the author. Below is my bio from 20 years ago. My how time flies.

Bradley J. Roth was raised in Morrison, Illinois. He received the B.S. degree from the University of Kansas in 1982, where he was a Summerfield Scholar and received the Stranathan Award from the Department of Physics and Astronomy. He received the Ph.D. degree in physics from Vanderbilt University.

From 1988-1995, he worked in the Biomedical Engineering and Instrumentation Program at the National Institutes of Health. One of his primary accomplishments while at NIH was the study of the bidomain model and its application to solving fundamental problems solving the interaction of applied electric fields with cardiac muscle. Using the results of numerical simulations, he has formulated mechanisms for stimulation, defibrillation, and the initiation of arrhythmias in the heart In September, 1995, he became the Robert T. Lagemann Assistant Professor of Living State Physics at Vanderbilt University.

Steven Vogel (1940-2015)

Life in Moving Fluids,
by Steven Vogel.

Steven Vogel died on Tuesday. He was the author of several excellent books about the interface between physics and biology. Two that Russ Hobbie and I cite in the first chapter of Intermediate Physics for Medicine and Biology are Vital Circuits (1992) and Life in Moving Fluids (1994), which is one of the books featured in the IPMB Ideal Bookshelf. I posted two blog entries about Vogel’s book Glimpses of Creatures in Their Physical Worlds, here and here. I quote him extensively in a blog entry about the Law of Laplace, in a blog entry about Murray’s law, and in a blog entry about the Reynolds number. His other books I have enjoyed include Life’s Devices, Cats’ Paws and Catapults, and Prime Mover. Reading The Life of a Leaf remains on my to-do list.

I learned the sad news of Vogel’s death from Raghuveer Parthasarathy’s blog The Eighteenth Elephant. There is little I can add to his eloquent tribute. I attended the same conference that Parthasarathy writes about, which is where I met Vogel. He was a delightful and fascinating man. You can listen to him talk about writing scientific papers here, and read his obituary here.

 Steven Vogel talking about writing scientific papers.

https://www.youtube.com/watch?v=eYBMp5sYchA

I leave you with Vogel’s own words, the first two paragraphs of the Preface from the second edition of Life in Moving Fluids. I don’t own the first edition, but I will try to hunt down for you the “first punning sentence” of the first edition Preface that Vogel refers to. I always love a good pun.

About a dozen years ago, calling up a degree of hubris I now find quite inexplicable, I wrote a book about the interface between biology and fluid dynamics. I had never deliberately written a book, and I had never taken a proper course in fluids. But I had learned through teaching—both something about the subject and something about the dearth of material that might provide a useful avenue of approach for biologist and engineer. Each seemed dazzled and dismayed by the complexity of the other’s domain. The book happened in a hurry, in a kind of race against the impending end of a sabbatical semester, and in a kind of mad fit of passion driven by simple realization (and astonishment) that it was actually happening.

The reception of Life in Moving Fluids turned out to surpass my most self-indulgent fantasies—it reached the people I hoped to reach, from ecologist and marine biologist to physical and applied scientists of various persuasions, and it seems to have played a catalytic or instigational role in quite a few instances. Quite clearly the book has been the most important thing of a professional sort that I’ve ever done: certainly that’s true if measured by the frequency with which the first punning sentence of its preface is flung back at me (That my writing has been more important than my research in furthering my area of science suggests that doing hands-on science, which I enjoy, is really just a personal indulgence—quite a curious state of affairs!)

Note added a few hours after the post: Russ has the first edition. He says the first line of the preface is “Fluid flow is not currently in the mainstream of biology, but it has its place.”

The Mystery of the Flawed Homework Problem

When teaching PHY 325 (Biological Physics) this fall, I assigned my students homework from the 5th edition of Intermediate Physics for Medicine and Biology. One problem comes from Section 7.10 about Electrical Stimulation.

Problem 36. If the medium has a constant resistance, find the energy required for stimulation as a function of pulse duration.

The odd thing is, when I looked in the solution manual to review how to solve this problem, it contained answers to parts (a) and (b), and (b) is the most useful part. Where are (a) and (b)? Somehow when preparing the 5th edition, part (b) was left out (it is missing from the 4th edition too). Nevertheless, part (b) ended up in the solution manual (don’t ask me how). This is what Problem 36 should look like:

Problem 36. The longevity of a pacemaker battery is related to the energy required for stimulation.

(a) Find an expression for the energy U expended by a pacemaker to stimulate the heart as a function of the pulse duration t. Use the Lapicque strength-duration curve (Eq. 7.45), and assume the body and electrodes have a constant resistance R. Sketch a plot of energy versus duration.

(b) In general you want to stimulate using the least energy. Determine what duration minimizes the energy expended per pulse.

I don’t usually solve homework problems from the book in this blog, but because the interesting part of this problem was left out of IPMB I don’t think it will hurt in this case. Also, it provides readers with a sneak peak at the solution manual. Remember that Russ Hobbie and I will only send the solution manual to instructors, not students. So if you are teaching from IPMB and want the solution manual, by all means contact us. If you are a student, however, you had better talk to your instructor.

7.36 Issues such as pacemaker battery life are related to the energy required for electrical stimulation. This problem relates the energy to the strength-duration curve, and provides additional insight into the physical significance of the chronaxie.

(a) Let the resistance seen by the electrode due to the medium be R. The power is i²R. Therefore the total energy is

 (b) The duration corresponding to minimum energy is found by setting dU/dt = 0. We get

which reduces to t = t_C. The minimum energy corresponds to a duration equal to the chronaxie.

In the 5th edition’s solution manual, each problem has a brief preamble (in italics) explaining the topic and describing what the student is supposed to learn. We also mark problems that are higher difficulty (*), that complete a derivation from the text (§), and that are new in the fifth edition (¶). Problem 7.36 didn’t fall into any of these categories. We typically outline the solution, but don’t always show all the intermediate steps. I hope we include enough of the solution that the reader or instructor can easily fill in anything missing.

One thing not in the solution manual is the plot of energy, U, versus duration, t. Below I include such a plot. The energy depends on the rheobase current i_R, the chronaxie t_C, and the resistance R.

The energy of a stimulus pulse as a function of pulse duration.

I wonder if this change to Problem 7.36 should go into the IPMB errata? It is not really an error, but more of an omission. After some thought, I have decided to include it, since it was supposed to be there originally. You can find the errata at the book's website: https://sites.google.com/view/hobbieroth. I urge you to download it and mark the corrections in your copy of IPMB.

I hope this blog post has cleared up the mystery behind Problem 7.36. Yet, the curious reader may have one last question: why did I assign a homework problem to my students that is obviously flawed? The truth is, I chose which homework problems to assign by browsing through the solution manual rather than the book (yes, the solution manual is that useful). Problem 7.36 sure looked like a good one based on the solution manual!

Stokes' Flow around a Sphere

When working on the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem about low Reynolds number flow. We ask the reader to analyze the classic example of “Stokes’ flow” or “creeping flow” around a sphere.

Problem 46. Consider a stationary sphere of radius a placed in a fluid of viscosity η moving uniformly with speed V. For low Reynolds number flow, the radial and tangential components of the fluid velocity and the pressure surrounding the sphere are

(a) Show that the no-slip boundary condition is satisfied.

(b) Integrate the shear force and the pressure force over the sphere surface and find an expression for the net drag force on the sphere (Stoke’s law). What fraction of this force arises from pressure drag, and what fraction from viscous drag?

(Everywhere else in our book we correctly write “Stokes’ law” since the law is named after Sir George Stokes, but in this problem we slip up and write “Stoke’s law”. Sorry. I noted this in the errata available on the book website.)

After solving this problem, the reader is probably thinking “this is all well and nice, and I understand now how you get Stokes’ law from the pressure distribution and the viscous drag, but where in the world did you get those weird velocity and pressure distributions?”

First, this example applies to a sphere in water, and water is nearly incompressible. Problem 1.35 shows that incompressibility implies that the velocity u has zero divergence,

The reader should pause now, look up the expression for the divergence in spherical coordinates, and verify that the given velocity really is divergenceless.

Second, the equation describing flow is the Navier-Stokes equation, which is really nothing more than Newton’s second law (F=ma) applied to the fluid. Problem 1.28 provides some insight by deriving a simplified form of the Navier-Stokes equation

 

If we assume a low Reynolds number, we can ignore the two terms on the left-hand side of this equation because they are “inertial” terms arising from the acceleration of the fluid. The two terms on the right-hand side can be generalized to three dimensions, with the pressure term containing the gradient of the pressure and the viscous term containing the Laplacian of the velocity. The resulting Navier-Stokes equation is

To get the expressions given in the new Problem 1.46, solve the Navier-Stokes equation assuming an incompressible fluid. In addition, the boundary conditions are 1) far from the sphere (r much greater than a) the flow is entirely along the z-axis with speed V, and 2) at the sphere surface (r = a) the radial component of the velocity vanishes because the flow is incompressible and the tangential component of the velocity vanishes because of the no-slip boundary condition.

Stokes’ law for the net drag force F, derived in part (b) of Problem 1.46, is F = 6πηaV. Often the drag force is described by a dimensionless coefficient called the drag coefficient, C, equal to F divided by ½ρV²πa². For creeping flow around a sphere, the drag coefficient is

Using the definition of the dimensionless Reynolds number, Re (Eq. 1.62 in IPMB), we find that C = 12/Re. Often the Reynolds number is written in terms of the diameter of the sphere rather than the radius, in which case we get the more commonly quoted relationship C = 24/Re. In many fluid dynamics textbooks you will see C plotted versus Re (usually on log-log graph paper). At low Reynolds number C is inversely proportional to Re as creeping flow predicts. At high Reynolds number the relationship between C and Re is more complex because a turbulent boundary layer forms near the sphere surface. But that’s another story.

The Magnetic Field of a Single Axon (Part 2)

In my last blog entry, I began the story behind “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment” (Biophysical Journal, Volume 48, Pages 93–109, 1985). I wrote this paper as a graduate student working for John Wikswo at Vanderbilt University. (I use the first person “I” in this blog post because I was usually alone in a windowless basement lab when doing the experiment, but of course Wikswo taught me how to do everything including how to write a scientific paper.) Last week I described how I measured the transmembrane potential of a crayfish axon, and this week I explain how I measured its magnetic field.

A toroid used to measure
the magnetic field of a single axon.

The magnetic field was recorded using a wire-wound toroid (I have talked about winding toroids previously in this blog). Wikswo had obtained several ferrite toroidal cores of various sizes, most a few millimeters in diameter. I wound 50 to 100 turns of 40-gauge magnet wire onto the core using a dissecting microscope and a clever device designed by Wikswo to rotate the core around several axes while holding its location fixed. I had to be careful because a kink in a wire having a diameter of less than 0.1 mm would break it. Many times after successfully winding, say, 30 turns the wire would snap and I would have to start over. After finishing the winding, I would carefully solder the ends of the wire to a coaxial cable and “pot” the whole thing in epoxy. Wikswo—who excels at building widgets of all kinds—had designed Teflon molds to guide the epoxy. I would machine the Teflon to the size we needed using a mill in the student shop. (With all the concerns about liability and lawsuits these days student shops are now uncommon, but I found it enjoyable, educational, and essential.) Next I would carefully place the wire-wound core in the mold with a Teflon tube down its center to prevent the epoxy from sealing the hole in the middle. This entire mold/core/wire/cable would then be placed under vacuum (to prevent bubbles), and filled with epoxy. Once the epoxy hardened and I removed the mold, I had a “toroid”: an instrument for detecting action currents in a nerve. In 1984, this “neuromagnetic current probe” earned Wikswo an IR-100 award. The basics of this measurement are described in Chapter 8 of Intermediate Physics for Medicine and Biology.

In Wikswo’s original experiment to measure the magnetic field of a frog sciatic nerve (the entire nerve; not just a single axon), the toroid signal was recorded using a SQUID magnetometer (see Wikswo, Barach, Freeman, “Magnetic Field of a Nerve Impulse: First Measurements,” Science, Volume 208, Pages 53–55, 1980). By the time I arrived at Vanderbilt, Wikswo and his collaborators had developed a low-noise, low-input impedance amplifier—basically a current-to-voltage converter—that was sensitive enough to record the magnetic signal (Wikswo, Samson, Giffard, “A Low-Noise Low Input Impedance Amplifier for Magnetic Measurements of Nerve Action Currents,” IEEE Trans. Biomed. Eng. Volume 30, Pages 215–221, 1983). Pat Henry, then an instrument specialist in the lab, ran a cottage industry building and improving these amplifiers.

To calibrate the instrument, I threaded the toroid with a single turn of wire connected to a current source that output a square pulse of known amplitude and duration (typically 1 μA and 1 ms). The toroid response was not square because we sensed the rate-of-change of the magnetic field (Faraday’s law), and because of the resistor-inductor time constant of the toroid. Therefore, we had to adjust the signal using “frequency compensation”; integrating the signal until it had the correct square shape.

The amplifier output was recorded by a digital oscilloscope that saved the data to a tape drive. Another of my first jobs at Vanderbilt was to write a computer program that would read the data from the tape and convert it to a format that we could use for signal analysis. We wrote our own signal processing program—called OSCOPE, somewhat analogous to MATLAB—that we used to analyze and plot the data. I spent many hours writing subroutines (in FORTRAN) for OSCOPE so we could calculate the magnetic field from the transmembrane potential, and vice versa.

An experiment to measure the
transmembrane potential, the extracellular potential,
and the magnetic field of a single axon.

Once all the instrumentation was ready, the experiment itself was straightforward. I would dissect the ventral nerve cord from a crayfish and place it in a plexiglass bath (again, machined in the student shop) filled with saline (or more correctly, a version of saline for the crayfish called van Harreveld’s solution). The nerve was gently threaded through the toroid, a microelectrode was poked into the axon, and an electrode to record the extracellular potential was placed nearby. I would then stimulate the end of the nerve. It was easy to excite just a single axon; the nerve cord split to go around the esophagus, so I could place the stimulating electrode there and stimulate either the left or right half. In addition, the threshold of the giant axon was lower than that of the many small axons, so I could adjust the stimulator strength to get just one giant axon.

The magnetic field of a single axon.

When I first started doing these experiments, I had a horrible time stimulating the nerve. I assumed I was either crushing or stretching it during the dissection, or there was something wrong with the saline solution, or the epoxy was toxic. But after weeks of checking every possible problem, I discovered that the coaxial cable leading to the stimulating electrode was broken! The experiment had been ready to go all along; I just wasn’t stimulating the nerve. Frankly, I now believe it was a blessing to have a stupid little problem early in the experiment that forced me to check every step of the process, eliminating many potential sources of trouble and giving me a deeper understanding of all the details. 

As you can tell, a lot of effort went into this experiment. Many things could, and did, go wrong. But the work was successful in the end, and the paper describing it remains one of my favorites. I learned much doing this experiment, but probably the most important thing I learned was perseverance.