Period Three Implies Chaos

Chaos: Making a New Science,
by James Gleick.

With the coronavirus keeping me home, I have been reading Chaos: Making a New Science, by James Gleick. I was particularly struck by Gleick’s discussion of the logistic map, and how it predicts behavior with period three. Russ Hobbie and I discuss the logistic map in Chapter 10 of Intermediate Physics for Medicine and Biology.

We considered the logistic differential equation as a model for population growth. The differential equation assumes that the population changes continuously. For some species each generation is distinct, and a difference equation is a better model of the population than a differential equation. An example might be an insect population where one generation lays eggs and dies, and the next year a new generation emerges. A model that has been used for this case is the logistic difference equation or logistic map

y_j+1 = a y_j (1 – y_j/y_∞)

with a > 0 and j the generation number. It can again be cast in dimensionless form by defining x_j = y_j/y_∞:

x_j+1 = a x_j (1 – x_j) .

…[Fig. 10.24 in IPMB] shows the remarkable behavior that results when a is increased to 3.1. The values of x_j do not come to equilibrium. Rather, they oscillate about the former equilibrium value, taking on first a larger value and then a smaller value. This is a period-2 cycle. The behavior of the map has undergone period doubling…

The period doubling continues with increasing a. For a > 3.449 there is a cycle of period 4… For a > 3.54409 there is a cycle of period 8. The period doubling continues with periods 2^N occurring at more and more closely spaced values of a. When a > 3.569946, for many values of a the behavior is aperiodic, and the values of x_j never form a repeating sequence. Remarkably, there are ranges of a in this region for which a repeating sequence again occurs, but they are very narrow. The details of this behavior are found in many texts. In the context of ecology they are reviewed in a classic paper by May (1976).

For a < 3.569946, starting from different initial values x₀ leads after a number of iterations to the same set of values for the x_j. For values of a larger than this, starting from slightly different values of x₀ usually leads to very different values of x_j, and the differences become greater and greater for larger values of j… This is an example of chaotic behavior, or deterministic chaos.

So I thought that you could have 1, 2, 4, 8, 16, etc., values of x_j, or you could have chaos. I didn’t realize there were other choices. Then I read Gleick’s analysis of James Yorke’s paper “Period Three Implies Chaos.”

He proved that in any one-dimensional system, if a regular cycle of period three ever appears, then the same system will also display regular cycles of every other length, as well as completely chaotic cycles. This was the discovery that came as an “electric shock” to physicists like Freeman Dyson. It was so contrary to intuition. You would think it would be trivial to set up a system that would repeat itself in a period-three oscillation without every producing chaos. Yorke showed that it was impossible.

This sent me scurrying back to IPMB to see if we saw any hint of period-three behavior in the logistic map. Sure enough, Fig. 10.27 shows a narrow range around a = 3.8 with period three. Not entirely believing my eyes, I wrote a program to do the calculation (it’s an easy program to write) and found a period-three cycle. I made a plot using a format similar to Fig. 10.24 in IPMB.

A plot of x_j vs. j using the logistic map and a = 3.83, showing how the sequence of values converges to three values of x.

Wow! Period three behavior and chaos; who would have thought they go hand-in-hand.

Return to Once-A-Week Blog Posts

Two months ago—starting March 16 when the coronavirus pandemic closed down in-person classes at Oakland University, where I teach—I began posting to this blog five times a week, Monday to Friday. My hope is that someone out there who was stuck at home found these posts useful. It’s been a grueling pace, and I worry that the quality of the posts has been slipping lately (see, for example, The Barium Enema). Now that the country is opening back up, I’m returning to my traditional schedule: once a week, on Friday mornings.

I’ve quoted some excellent authors during these coronavirus posts, such as Robert Rodieck, James Gleick, Michael Goitein, Mark Denny, Howard Berg, and especially Steven Vogel. It’s been an honor to share their writing with you.

Before I go, let me remind you that the website for Intermediate Physics for Medicine and Biology is https://sites.google.com/view/hobbieroth/home. There you can find the errata, a mapping that relates blog posts to sections of IPMB (updated this weekend), instructions and game cards for playing Trivial Pursuit IPMB (the perfect game for someone stuck at home because of Covid-19; play it with other students via zoom), and other useful stuff.

One last thing (and it’s a big one). For those of you at institutions of higher education, my understanding is that Springer currently is providing access to IPMB at no cost! Their Covid-19 website states

With the Coronavirus outbreak having an unprecedented impact on education, Springer Nature is launching a global program to support learning and teaching at higher education institutions worldwide. We want to support lecturers, teachers and students during this challenging period and hope that this initiative will go some way to help. Institutions will be able to access more than 500 key textbooks across Springer Nature’s eBook subject collections for free.

IPMB is on their list of books. The Oakland University library has always had an electronic version of IPMB available to OU students, but now it seems that all universities will have access to it. I’m not sure how you go about taking advantage of the offer, but I suggest you go to https://www.springernature.com/gp/librarians/landing/covid19-library-resources and see if you can figure it out.

Stay safe and healthy. See you Friday.

The Potassium Conductance

Yesterday’s post led me to reflect on how Russ Hobbie and I describe the potassium conductance of a nerve membrane in Chapter 6 of Intermediate Physics for Medicine and Biology. In the Hodgkin and Huxley model, the time dependence of the potassium conductance is proportional to n⁴(t), where n is a dimensionless variable called the potassium gate that takes values from zero (potassium channels are closed) to one (channels open).

6.13.2 Potassium Conductance

Hodgkin and Huxley wanted a way to describe their extensive voltage-clamp data, similar to that in Figs. 6.34 and 6.35,. with a small number of parameters. If we ignore the small nonzero value of the conductance before the clamp is applied, the potassium conductance curve of Fig. 6.34 is reminiscent of exponential behavior, such as g_K(v,t) = g_K(v) (1 – e^-t/τ(v)), with both g_K(v) and τ(v) depending on the value of the voltage. A simple exponential is not a good fit. Figure 6.36 shows why. The curve (1 – e^-t/τ) starts with a linear portion and is then concave downward. The potassium conductance in Figs. 6.34 and 6.35 is initially concave upward. The curve (1 – e^-t/τ)⁴ in Fig. 6.36 more nearly has the shape of the conductance data.

This is all correct, but the story has another part. Hodgkin and Huxley focus on this missing part in their 1952 paper “The Components of Membrane Conductance in the Giant Axon of Loligo,” Journal of Physiology, Volume 116, Pages 473-496. 

The experiment shows that whereas the potassium conductance rises with a marked delay it falls along an exponential type of curve which has no inflexion corresponding to that on the rising phase. [my italics]

Russ and I showed in Fig. 6.36 how our model predicts that the potassium conductance rises with a marked delay, but we didn’t check if it falls with “inflexion.” Will the fall have a sigmoidal shape like the rise does, or will it be abrupt like Hodgkin and Huxley observed?

To check, I derived a simple exponential solution for n(t) during both the rising phase (0 < t < 4τ) and the falling phase (t > 4τ). This is a toy model for an experiment in which we clamp the voltage at a depolarizing value for a duration of 4τ, and then return it to rest.

This may look like a complicated expression, but it’s simply the solution to the differential equation

when n(0) = 0, 

and τ is independent of time.

I made a plot of n(t) and n⁴(t), which is an extension of IPMB’s Fig. 6.36 to longer times. 

A plot of n(t) (blue) and n⁴(t) (red) versus time. This is an extension of Fig. 6.36 in IPMB.

The gate itself, n(t) (blue), rises exponentially and then falls exponentially, with no hint of sigmoidal behavior. However, n⁴(t) (red) rises with a sigmoidal shape but then falls exponentially. This is exactly what Hodgkin and Huxley observed experimentally.

The simple toy model that Russ and I use to illustrate the potassium conductance works better than we realized!

The Five 1952 Hodgkin and Huxley Papers

Alan Hodgkin and Andrew Huxley published a series of five papers in the Journal of Physiology that explained the nerve action potential. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite only the last one, which contains their mathematical model and is the most famous. Today, I’ll analyze all five papers. We’d better get started, because we have much to discuss.

Hodgkin AL, Huxley AF & Katz B (1952) “Measurement of Current‐Voltage Relations in the Membrane of the Giant Axon of Loligo.” J. Physiol., Volume 116, Pages 424—448.

This is the only one of the five papers with an extra coauthor, Bernard Katz. The first four articles were all submitted on the same day, October 24, 1951, and were published on the same day, in the April 28, 1952 issue of the Journal of Physiology. This first paper had two primary goals: describe the voltage clamp method that was the experimental basis for all the papers, and illustrate the behavior of the membrane current for a constant membrane voltage. For each paper, I’ll select one figure that illustrates the main idea (I present a simplified version of the figure, with the sign convention for voltage and current changed to match modern practice). For this first paper, I chose Fig. 11, which shows a voltage clamp experiment. A feedback circuit holds the membrane voltage at a depolarized value and records the membrane current, which consists of an initial inward current lasting about 2 milliseconds, followed by a longer-lasting outward current.

The membrane current at a constant depolarized membrane voltage during a voltage clamp experiment. Adapted from Fig. 11 of Hodgkin et al. (1952).

Hodgkin AL & Huxley AF (1952) “Currents Carried by Sodium and Potassium Ions Through the Membrane of the Giant Axon of Loligo.” J. Physiol., Volume 116, Pages 449—472.

The goal is the second paper is to separate the membrane current into two parts: one carried by sodium, and the other by potassium. The key experiment is to record the membrane current when the axon is immersed in normal seawater (mostly sodium chloride), then replace the seawater with a fluid consisting mainly of choline chloride, and finally restore normal seawater as a control to ensure the process is reversible. When sodium is replaced by the much larger choline cation the initial inward current disappears, while the outward current is little changed. This experiment, plus others, convinced Hodgkin and Huxley that the initial inward current is carried by sodium, and the long-lasting outward current by potassium.

The membrane current at a constant depolarized membrane voltage, when the axon is immersed in normal seawater (“sodium”) and in an artificial seawater with sodium replaced by choline (“choline”). Adapted from Fig. 1 of Hodgkin & Huxley (1952).

Hodgkin AL & Huxley AF (1952) “The Components of Membrane Conductance in the Giant Axon of Loligo.” J. Physiol., Volume 116, Pages 473—496.

I had a difficult time identifying on the main point of this paper, but finally I realized that the goal was to demonstrate that the behavior is best explained using sodium and potassium conductances, rather than currents or voltages. The experimental method was modified slightly by making the depolarization last only a brief one and a half milliseconds. The membrane current changes discontinuously. To see why, imagine that the extreme case of the depolarization going up to the sodium reversal potential (the membrane voltage is not depolarized quite that far in the figure below). The sodium current would be zero during the depolarization because you’re at the reversal potential. Once the membrane voltage drops back down to rest, the sodium current jumps to a large value; the sodium channels are open and now there is a voltage driving it. The sodium conductance, however, changes continuously. Hodgkin and Huxley observed, moreover, that the conductance turns on with a sigmoidal shape (not as obvious in the figure below as it is in their Fig. 13 showing the potassium conductance) but turns off exponentially, with no sign of sigmoidal behavior.

The membrane current (blue) and conductance (green) during a brief voltage clamp experiment. Adapted from Fig. 8c of Hodgkin & Huxley (1952).

Hodgkin AL & Huxley AF (1952) “The Dual Effect of Membrane Potential on Sodium Conductance in the Giant Axon of Loligo.” J. Physiol., Volume 116, Pages 497—506.

This is my favorite of the four experimental papers. It is the shortest (a mere ten pages). It examines the inactivation of the sodium channel, which is crucial for understanding the axon’s refractory period. The figure below shows their ingenious two-step voltage clamp protocol that reveals inactivation. In all three cases shown the membrane voltage changes from rest on the left to 44 mV depolarized on the right. Between these two values, however, is a tens-of-milliseconds-long holding period at different membrane voltages. When the membrane is hyperpolarized (top) the eventual sodium inward current is larger, whereas when the membrane is weakly depolarized (bottom) the inward current is smaller. Hyperpolarization removes the inactivation of the sodium channel.

The membrane voltage (red) and current (blue) during a two-step voltage clamp protocol. Adapted from Fig. 4 of Hodgkin & Huxley (1952).

Hodgkin AL & Huxley AF (1952) “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve.” J. Physiol., Volume 117, Pages 500—544.

The fifth paper was submitted March 10, 1952, more than four months after the first four, and wasn’t published until August 28, 1952, in the next volume of the Journal of Physiology. It was worth the wait. The last paper is Hodgkin and Huxley’s masterpiece, and is the most cited of the five. They introduce their mathematical model based on the voltage clamp experiments described in paper 1. They divide the membrane current up into a part carried by sodium and a part carried by potassium (plus a leak current that plays little role except in setting the resting potential), as they describe in paper 2. The model focuses on the sodium and potassium conductances, controlled by three gates: m, h, and n. By raising m to the third power and n to the fourth, they ensure the conductances turn on slowly with a sigmoidal shape, but turn off abruptly, just as they found in paper 3. The h-gate describes sodium inactivation, as reported in paper 4. The model not only reproduces their voltage clamp data, but also it predicts the action potential, all-or-none behavior, the refractory period, and even anode break excitation.

The calculated (top) and measured (bottom) action potential. Adapted from Fig. 12 of Hodgkin & Huxley (1952).

Wow! That’s an amazing set of papers.

The Barium Enema

From Page 472 of Intermediate
Physics for Medicine and Biology.

Barium (Ba, element 56) is a contrast agent for x-ray imaging of the gastrointestinal tract. In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe such contrast agents.

Abdominal structures are more difficult to visualize because except for gas in the intestine, everything has about the same density and atomic number. Contrast agents are introduced through the mouth, rectum, urethra, or bloodstream. One might think that the highest-Z [highest-atomic number] materials would be best. However the energy of the K edge rises with increasing Z. If the K edge is above the energy of the x-rays in the beam, then only L absorption with a much lower cross-section takes place. The K edge for iodine is at 33 keV, while that for lead is at 88 keV. Between these two limits (and therefore in the range of x-ray energies usually used for diagnostic purposes), the mass attenuation coefficient of iodine is about twice that of lead. The two most popular contrast agents are barium (Z = 56, K edge at 37.4 keV) and iodine (Z = 53). Barium is swallowed or introduced into the colon. Iodine forms the basis for contrast agents used to study the cardiovascular system (angiography), gall bladder, brain, kidney, and urinary tract.

In Table 16.7, we mention barium again.

TABLE 16.7. Typical radiation equivalent doses for the population of the United States. (From AAPM Report 96 2008, Table 2)

       Procedure                               Equivalent dose (mSv)
Chest X-ray (Anterior Posterior)          0.1-0.2
Mammogram                                        0.3–0.6
Barium enema                                          3-6
Nuclear medicine–cardiac                      13-40
Head CT                                                   1-2
Chest CT                                                   5-7
Abdomen CT                                            5-7
Coronary CT angiography                        5-15

Third on this list is the unpleasant-sounding “barium enema.” What’s that?

From Page 492 of Intermediate
Physics for Medicine and Biology.

In a barium enema, the doctor inserts a lubricated enema tube into the patient’s rectum, fills the colon with the contrast agent barium sulfate, and takes an x-ray. Patients prepare for a barium enema like they prepare for a colonoscopy: the night before they are restricted to a clear liquid diet, take laxatives, and use warm water enemas to clear out stool particles.

I like to learn by experiencing something rather than merely reading about it. I prefer to get my nose out of the book and try things, have new experiences, and live life. However, I DON’T FEEL THAT WAY ABOUT A BARIUM ENEMA. I hope you, dear reader, don’t ever gain first-hand experience with this procedure either.

Enjoy?

My opinion of a barium enema.

Beethoven

The first movement of the Moonlight Sonata,
 Sonata No. 14 in C♯ minor, Opus 27 No. 2,
by Ludwig van Beethoven. Jonathan Biss
despises the name “Moonlight Sonata,”
a title not given by Beethoven.

This year we celebrate an important anniversary: 250 years since the birth of Ludwig van Beethoven, one of the world’s greatest composers. Beethoven (1770–1827) was a bridge between the classical era of Haydn and Mozart, and the romantic era of Schubert and Brahms. I know you’ve all heard the opening motif from his Fifth Symphony.

Recently, while stuck at home because of the coronavirus, I enrolled in “Exploring Beethoven’s Piano Sonatas” through Coursera. This class is taught by pianist Jonathan Biss. You can enroll free of charge and, as the old joke goes, it’s worth every penny. No, seriously, the course is outstanding; Biss gives a masterclass on how to appreciate Beethoven’s music and how to teach online (something many of us had to learn quickly when covid-19 shut down in-person classes in March). Biss’s analysis of the Appassionata (Piano Sonata No. 23 in F minor, Opus 57) is particularly memorable.

Beethoven slowly lost his hearing as he grew older, and composed many of his later works (including his masterpiece the Ninth Symphony) when he was deaf. I wonder if he would have benefited from a cochlear implant? Russ Hobbie and I mention such auditory prostheses briefly in Chapter 13 of Intermediate Physics for Medicine and Biology.

The cochlear implant… [is] a way to use functional electrical stimulation to partially restore hearing. A row of electrodes is inserted along the cochlea to stimulate the nerves that are usually excited by the hair cells. Some pitch perception can be restored by performing a Fourier analysis of a sound and stimulating neurons at different places along the cochlea.

The adagio from Sonata Pathétique,
Sonata No. 8 in C minor, Opus 13,
by Ludwig van Beethoven.
The date of Feb. 10, 1975 written at the top
was probably when my sister studied it,
as I don't remember playing it in high school.

Whether or not Beethoven would have been helped by such a device depends on why he went deaf. If he lost hair cells in the organ of Conti but had a healthy auditory nerve, then a cochlear implant would have been beneficial. If the auditory nerve itself was the problem, an implant would have been of no use. I don’t know what caused Beethoven’s deafness, and I’m not sure anyone does.

Beethoven’s later years were lonely, and an auditory prosthesis might have let him interact more with people. However—and with all due respect to the heroic scientists and engineers who design and build cochlear implants—he probably would have been disappointed (no, horrified) when listening to music. For a virtuoso like Beethoven, I suspect he would rather hear the music in the privacy of his own thoughts than listen through an imperfect device. If only the cochlear implant had been invented 200 years earlier, Beethoven could have decided for himself.

To learn more about auditory transduction watch this excellent video,
which includes music from Beethoven’s the Ninth Symphony.

https://www.youtube.com/watch?v=46aNGGNPm7s

Jonathan Biss discusses playing Beethoven's sonatas.
He’s recording all 32.

https://www.youtube.com/watch?v=qmsSxH6ay3A

What does Jonathan Biss do when quarantined because of the coronavirus?
He gives us a message of hope inspired by Beethoven’s struggles.

https://www.youtube.com/watch?v=JSKIT2--9c8

 Jonathan Biss playing Sonata No. 12 in A♭ major, Opus 26.
The third movement is a funeral march played at Beethoven’s own funeral.
https://www.youtube.com/watch?v=8LF0TPAWXV4

Are Tens of Thousands Dying from Radon Each Year or Not?

Radon Action Month Proclamation
by Michigan Governor Gretchen Whitmer.

Governor Whitmer declared last January to be Radon Action Month in Michigan. Just how serious of a health hazard is radon?

Russ Hobbie and I discuss radon in Chapter 16 of Intermediate Physics for Medicine and Biology.

Radon is produced naturally in many types of rock. It is a noble gas, but its decay products can become lodged in the lung. An excess of lung cancer has been well documented in uranium miners, who have been exposed to fairly high radon concentrations as well as high dust levels and tobacco smoke. Radon at lower concentrations seeps from soil into buildings and contributes a large fraction of the exposure to the general population.

The Environmental Protection Agency published A Citizen’s Guide to Radon: The Guide to Protecting Yourself and Your Family from Radon. It recommends that you fix your home if your radon level is greater than 4 pCi/L (a picocurie per liter is equal to 37 decays per second per cubic meter of air).

Let’s put that into perspective. According to EPA’s 2003 Assessment of Risks from Radon in Homes, radon causes about 21,000 lung cancer deaths per year.

Based on its analysis, EPA estimates that out of a total of 157,400 lung cancer deaths nationally in 1995, 21,100 (13.4%) were radon related. Among NS [non-smokers], an estimated 26% were radon related... The estimated risks from lifetime exposure at the 4 pCi/L action level are: 2.3% for the entire population, 4.1% for ES [smokers], and 0.73% for NS. A Monte Carlo uncertainty analysis that accounts for only those factors that can be quantified without relying too heavily on expert opinion indicates that estimates for the U.S. population and ES may be accurate to within factors of about 2 or 3.

The data listed above are 25 years old. The American Cancer Society estimates that in 2020 about 136,000 deaths in the US will be from lung cancer. The reduction compared to 1995 is probably because fewer people smoke. I don’t see any way to accurately estimate the current number of radon-related deaths in the USA, but 20,000 per year is a reasonable guess based on the EPA’s 2003 Assessment.

Does this number make sense? Assume a 5% excess risk of dying from cancer per 1 Sievert of radiation dose. If we use the background dose from radon given in IPMB of about 2 mSv/year, then the excess risk is 0.0001/year. The current population of the US is about 330 million. Multiplying 330,000,000 times 0.0001 gives 33,000. This is the same order of magnitude as our 20,000 ballpark guess. Both of these estimates are uncertain, but they suggest that a few tens of thousands of deaths in the US each year are caused by radon. While this is not as bad as the coronavirus (80,000 deaths in a couple months), it’s still worrisome.

Tens of thousands dead. Really? Such estimates are based on the controversial linear-no-threshold model. A 2016 open-access article “Rectifying Radon’s Record: An Open Challenge to the EPA” by Jeffry Siegel, Charles Pennington, Bill Sacks, and James Welsh (International Journal of Radiology and Imaging Technology, Volume 2, Article Number 014) states

The American Lung Association has recently led a national workgroup to develop The National Radon Action Plan: A Strategy for Saving Lives. The U.S. Environmental Protection Agency (EPA) is the lead governmental organization projected to implement this plan. The stated intent of the plan is to address the “radon problem” in the United States, with the aim of saving 3,200 lives by the year 2020 through preventing at least a portion of the lung cancer mortality that is assumed to arise from inhaling modest doses of radon in homes, offices, and buildings. The plan identifies a number of actions that government can take in the spirit of saving lives by avoiding the inhalation of radon and its progeny. We are among a growing number of investigators who recognize the substantial body of evidence demonstrating that the radiation doses associated with indoor radon inhalation are not harmful. Radon, at these doses, is unlikely to be a cause of lung cancer, and, on the contrary, may be beneficial in various ways, including its paradoxical tendency to protect against lung cancer. In the present paper, we review and critique the past policies of the EPA with respect to indoor radon and the very impetus for the plan. We indicate that the plan should not be implemented because a preponderance of the evidence indicates an unintended consequence: implementation of the plan is likely to increase, rather than decrease, the risk of lung cancer.

What are we to believe? I don’t know the real risk of radon exposure. One thing I do know is that we need to figure out whether or not the linear-no-threshold model is correct. Are tens of thousands of our citizens dying each year from radon exposure? It seems to me that with so many lives at stake, our nation needs to invest the time and money necessary to answer this question. Either the EPA overestimates the risk, in which case we can focus on other more pressing issues, or it accurately estimates the risk, in which case we have an epidemic on our hands.

Ping Pong

When I was in a teenager, I played a lot of ping pong. It was the sport that I played best (which isn’t saying much) and loved most (except for baseball). Dad bought a ping-pong table for the family when I was young, and he was my first opponent. As I grew up, I spent hours practicing with friends, perfecting our serves, slices, and slams. In high school, my friend Terry Fife and I played hundreds of games, and we were evenly matched. The psychological warfare during those battles was fierce.

I continued playing ping pong in college. I bought one of those paddles that has foam under the rubber so the ball stays in contact with the surface for a long time, letting you impart more spin. During my first two years at the University of Kansas, I studied but also spent a lot of time playing ping pong at the dorm. I was better than most of the guys, although my friend Son Do could beat me, much to my chagrin. Too many evenings were squandered playing cards or ping pong. Only during my last couple years at KU did I get serious about physics.

Life in Moving Fluids,
by Steven Vogel.

Understanding ping pong requires us to discuss the Magnus effect, which Steven Vogel illustrates by considering a spinning cylinder in a flowing fluid. In Life in Moving Fluids—a book often cited in Intermediate Physics for Medicine and Biology—Vogel writes

A look at the resulting streamlines (Figure 10.12 [a modified version of which is shown below]) clarifies what’s happening in this superposition of rotation and translation of a cylinder. On one side of the cylinder the two motions in the fluid oppose one another, so the velocities are lower and the streamlines are farther apart. On the other side, the motions are additive, velocities are increased, and the streamlines are closer together. By Bernoulli’s principle pressure will be elevated on the side where flow speeds are lower and will be reduced on the side where the speeds are higher. Thus a net pressure or force will act in a direction normal to the free-speed flow—in short, lift…

Figure 10.12. If a solid body such as a cylinder rotates as it translates through a fluid, the resulting asymmetry of flow generates a force normal to the free-stream flow. We call this force lift. Adapted from Life in Moving Fluids by Steven Vogel.

This phenomenon, the lift of a rotating cylinder moving through a fluid, is called the “Magnus effect,” after H. G. Magnus (1802-1870).

The Magnus effect (at a little lower intensity) works for spheres as well as for cylinders. It’s a really big deal in sports in which spheres are thrown, hit, or otherwise put into motion since (except for a golf slice) a confusingly nonstraight course is distinctly meritorious. Two pleasant books on such contemporary compulsions are Sport Science, by P. J. Brancazio (1984), with good references, and The Physics of Baseball, by R. K. Adair (1990), with more on the Magnus effect specifically.

Me playing ping pong at the
University of Kansas, circa 1980.

Vogel’s description of the Magnus effect is clear and interesting, but I think it’s not very relevant. In ping pong, the goal of spin is not to make the ball follow a curved path through the air, but instead is to make the ball leave your opponent’s paddle traveling in the wrong direction! After putting backspin on the ball, I loved to see Terry Fife’s return dive into the net. And I enjoyed nothing more than giving the ball some sidespin, and then watching Son Do’s next hit miss the table on the right. Those were the days!

Intensity-Modulated Radiation Therapy

Radiation Oncology: A Physicist's-Eye View,
by Michael Goitein.

In an earlier post, I mentioned Michael Goitein textbook Radiation Oncology: A Physicist’s Eye View. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite Goitein’s book when we discuss intensity-modulated radiation therapy.

In classical radiotherapy, the beam was either of uniform fluence across the field, or it was shaped by an attenuating wedge placed in the field. Intensity-modulated radiation therapy (IMRT) is achieved by stepping the collimator leaves during exposure so that the fluence varies from square to square in Fig. 16.45 (Goitein 2008; Khan 2010, Ch. 20)

It was originally hoped that CT reconstruction techniques could be used to determine the collimator settings at different angles. This does not work because it is impossible to make the filtered radiation field negative, as the CT reconstruction would demand. IMRT with conventional treatment planning improves the dose pattern (Goitein (2008); Yu et al. (2008)), providing better sparing of adjacent normal tissue and allowing a boost in dose to the tumor.

Goitein’s analysis of IMRT is full of insight, and I quote an excerpt below. Enjoy.

INTENSITY-MODULATED RADIATION THERAPY (IMRT)

So far, we have implicitly assumed that each radiation field is near-uniform over its cross section; dose uniformity of a field within the target volume has, in fact, historically been an explicit goal of radiotherapy. However, the radical suggestion to allow the use of non-uniform fields was made some two decades ago, independently by Anders Brahme and Alan Cormack, fresh from co-inventing the CT scanner—and, in the context of π-meson therapy... (Cormack 1987; Brahme 1988; Pedroni 1981). Their idea was based on the judgment that, using mathematical techniques, an irradiation scheme using non-uniform beams could be found which would more closely achieve the ideal of delivering the desired dose to the target volume while limiting the dose to the normal tissues outside the target volume to some predefined value.

Brahme’s and Cormack’s approaches were motivated by the observation that, in CT reconstruction, one can deduce from the intensity reduction of X-rays traversing an object along a series of straight paths what the internal structure of the object is. By inverting the mathematics, one can deduce the intensities ( pencil beam weights) of a series of very small beams (pencil beams) that pass through the object and deliver dose within it. This procedure leads to highly nonuniform individual fields which, in combination, deliver the desired (usually, uniform) dose to the target volume.

There are two very substantial flaws to the original idea. The first is that, when the problem is posed to deliver zero dose outside the target volume as was initially proposed, many of the computed intensitiesare negative—a highly unphysical result. The second is that there is no a priori way of specifying a physically possible dose distribution to serve as the goal of the optimization.

However, the basic idea of using non-uniform beams has proven enormously fruitful. A workable computational solution is to use optimization algorithms to iteratively adjust the pencil beam weights such that the resulting dose distribution maximizes some score function. The search is computationally intensive and therefore poses interesting technical challenges. However, the still bigger challenge is to find score functions which give a viable measure of clinical goodness. Increasingly, biophysical models of the dose-response of both tumors and normal tissues are being investigated and are beginning to be used as elements of such score functions. These matters are discussed in Chapters 5 and 9.

Intensity-modulated radiation therapy (IMRT), as treatments featuring non-uniform beams are called, has been most intensely developed for X-ray therapy. However, it is equally appropriate for other radiation modalities—including protons. With charged particles one has an extra degree of freedom. One can vary the beam intensity as a function of lateral position and as a function of penetration (energy).

Who was Michael Goitein? Below is an excerpt from his 2017 obituary in the International Journal of Radiation Oncology Biology Physics.

Michael Goitein was a visionary thought provocateur and is rightly judged to have been an exceptionally innovative and creative physicist in radiation oncology (Fig. 1). He was a critical player in the development of proton radiation therapy, with many of his advances widely used in current proton and photon therapy. His highly important and numerous contributions to medical science have been well rewarded with many important awards: Fulbright Fellowship, 1961-65; US Research Career Development Award, 1976-81; Fellow, American Association of Physicists in Medicine, 2000; Gold Medal of the American Society for Radiation Oncology, 2003; and Lifetime Achievement Award of the European Society for Radiotherapy and Oncology, 2014. Significantly, he was a cofounder of the Proton Therapy Co-Operative Oncology Group and served as the second president. Furthermore, he participated in many of its subsequent functions. In addition, Michael was an invited lecturer at a long list of national and international conferences and medical centers. He published 163 articles and 3 books that have positively affected the practice of radiation oncology (1). It must also be mentioned that he has authored 4 books of a personal nature, one of which is a book of his poems (2, 3, 4, 5).

To learn more, read his ASTRO interview.

Never at Rest

Never at Rest, by Richard Westfall.

Isaac Newton’s name appears many times in Intermediate Physics for Medicine and Biology. You can learn more about him in Richard Westfall’s wonderful book Never at Rest: A Biography of Isaac Newton. As we all sit in quarantine because of the coronavirus pandemic, I thought you might like to read about Newton’s experience with the plague. Here is an excerpt from Never at Rest.

In the summer of 1665, a disaster descended on many parts of England including Cambridge. It had “pleased Almighty God in his just severity,” as Emmanuel College put it, “to visit this towne of Cambridge with the plague of pestilence.” Although Cambridge could not know it and did little in the following years to appease divine severity, the two-year visitation was the last time God would choose to chastise them in this manner [until 2020]. On 1 September, the city government canceled Sturbridge Fair [one of the largest fairs in Europe] and prohibited all public meetings. On 10 October, the senate of the university discontinued sermons at Great St. Mary’s and exercises in the public schools. In fact, the colleges had packed up and dispersed long before. Trinity [College, Cambridge] recorded a conclusion on 7 August that “all Fellows & Scholars which now go into the Country upon occasion of the Pestilence shall be allowed [the] usual Rates for their Commons for [the] space of [the] month following...” For eight months the university was nearly deserted…

Many of the students attempted to continue organized study by moving with their tutors to some neighboring village. Since Newton was entirely independent in his studies and had had his independence confirmed with a recent B.A. [Newton received his bachelors degree in August 1665],… he returned… to Woolsthorpe [the Newton family home]...

Much has been made of the plague years in Newton’s life. He mentioned them in his account of mathematics. The story of the apple [hitting him on the head, triggering the discovery of the universal law of gravity], set in the country, implies the stay in Woolsthorpe. In another much-quoted statement written in connection with the calculus controversy [a debate between Newton and Leibniz about who first invented calculus] about fifty years later, Newton mentioned the plague years again.

In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series [the binomial theorem]. The same year in May I found the method of Tangents of Gregory & Slusius [a way of finding the slope of a curve], & in November had the direct method of fluxions [diferential calculus] & the next year in January had the Theory of Colours [later published in Opticks] & in May following I had entrance into [the] inverse method of fluxions [integral calculus]. And the same year I began to think of gravity extending to [the] orb of the Moon & (having found out how to estimate the force with [which a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodical times of the Planets being in sesquialterate proportion [to the 3/2 power] of their distances from the center of their Orbs, I deduced that the forces [which] keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about [which] they revolve [the inverse square law]: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth and found them answer pretty nearly. All this was in the two Plauge years of 1665-1666. For in those days I was in the prime of my age for invention and minded Mathematicks & Philosophy [physics] more than at any time since.

 So what are you doing while stuck at home during the coronavirus pandemic?