## Friday, February 23, 2018

### NIST’s Digital Library of Mathematical Functions

Intermediate Physics for Medicine and Biology: NIST's Digital Library of Mathematical Functions
 Physics Today article about the Digital Library of Mathematical Functions.
In my December 22 post, I discussed the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Abramowitz and Stegun. That post ended with “NIST (the National Institutes of Standards and Technology) also maintains an updated electronic math handbook at http://dlmf.nist.gov.” Guess what. The February issue of Physics Today contains a fascinating article by Barry Schneider, Bruce Miller, and Bonita Saunders about NIST’s handbook: the Digital Library of Mathematical Functions.
One classic scientific reference that the revolution [in online information] has radically affected is the Handbook of Mathematical Functions, familiarly known as A+S, edited by Milton Abramowitz and Irene Stegun. In this article we discuss how A+S was transformed into an online 21st-century resource known as the Digital Library of Mathematical Functions, or DLMF, and how that new, modern resource makes far more information available to users in ways that are quite different from the past. The DLMF also contains far more material—in many cases updated—than does A+S.
I'm from Missouri (well, at least my dad is) so you have to show me. I decided to test if DLMF is useful for readers of Intermediate Physics for Medicine and Biology. Homework Problem 30 in Chapter 7 introduces the modified Bessel functions In and Kn when discussing Clark and Plonsey’s solution for the extracellular potential produced by a nerve axon. Students may not know how modified Bessel functions behave, so I wondered if DLMF included plots of them. It sure did; In color!

Derivatives of modified Bessel functions are needed too. Are they there? Yes.

Problem 16 in Chapter 8 of IPMB extends Clark and Plonsey’s analysis by calculating the magnetic field produced by a single axon. That calculation needs integrals of modified Bessel functions, and I found them in DLMF too.

Finally, the crucial test. My June 12, 2009 post told the story of how I calculated the magnetic field of an axon in two ways: using the law of Biot and Savart and Ampere’s law. The two results didn’t look equivalent until I found a Wronskian relating modified Bessel functions. Could I find that Wronskian in DLMF? Easy!

DLMF not only passes all my tests, but I give it an A+.

Schneider, Miller and Saunders conclude
We invite readers to explore the library: Hover the mouse over intriguing objects, symbols, and graphics to see behind the scenes. Open the info boxes to see what other possibly useful data may be available. We believe that the library will prove as useful to scientists and engineers of today and tomorrow as A+S has been since 1964.
I’m a scientist of the past, so I’m keeping my paper copy of Abramowitz and Stegun; I love the feel of the pages as I thumb through then, and I even like the smell of it. But readers of IPMB are scientists of the future, and for them I recommend NIST’s Digital Library of Mathematical Functions.

Enjoy!

## Friday, February 16, 2018

### Robley Dunglison Evans, Medical Physicist

Intermediate Physics for Medicine and Biology: Robley Dunglison Evans Medical Physicist
 The Atomic Nucleus, by Robley Dunglison Evans.
One sentence, and sometimes even one word, can hide a story. For instance, a footnote in Chapter 17 of Intermediate Physics for Medicine and Biology cites The Atomic Nucleus (McGraw-Hill, 1955), a book by Robley Dunglison Evans. His story is told in three oral history interviews on the American Institute of Physics website (you can find them here, here, and here).

Evans was born in University Place, Nebraska in 1907. When he was five his family moved to California. He won a scholarship to the California Institute of Technology, where he studied math and science, and earned spending money playing drums in a jazz band. He considered majoring in the history of science, but ultimately decided to focus on physics. He remained at Caltech for graduate school analyzing cosmic rays. Nobel Prize winner Robert Millikan was his thesis advisor.

I enjoyed his reminiscences about 1932, that famous year in nuclear physics when the deuteron, positron, and neutron were all discovered. Carl Anderson, who first detected the positron, was at Caltech photographing cosmic ray particles using a cloud chamber. Evans recalls when
...in one of these pictures, he [Anderson] got a track … [that] looked like an electron, but it was bent the wrong way [by the magnetic field] and had too long a range, too long a path length to possibly be the only positive particle we knew about, the proton. I remember seeing Carl in the morning coming dashing out of the darkroom, and I guess I was the first one he ran into in the hall, and he said, “My God, Bob, look at this. This thing is going the wrong way. And I checked the film in my camera. I didn’t have the emulsion facing the wrong way; I had it the right way. Everything looks all right here, and I can’t imagine what possibly is wrong, but maybe [it’s] that damn Pinky Klein,” who was a practical joker with a well-established reputation … So Carl suggested that maybe Pinky had reversed the magnetic field on him just to play a joke…
Of course it wasn’t Pinky. Anderson won the Nobel Prize for his discovery of the positron, a positive electron.

Nowadays physics grad students often complain about their job prospects, and rightly so. But the situation was worse in 1932 during the depths of the Great Depression. Evans says “I remember that some of them [the grad students] like Jack Workman who was there urged several of us to join together to go to the state of Washington and grow apples, and we fresh new PhD’s in physics were to become apple farmers. It was that bad.”

But not that bad for Evans himself. After graduating, he started a post doc at Berkeley. In 1934, he accepted a job on the faculty at MIT, where he taught the first class in the United States about nuclear physics. The Atomic Nucleus grew out of this class. He started writing the book by creating a giant card file, with the abstract of every nuclear physics research article written out, one per card.

He became interested in the medical applications of nuclear physics after hearing about watch dial painters who swallowed radium paint and got cancer. (The recently published book The Radium Girls by Kate Moore tells the story; it's on my list of books to read this summer.) Also, at that time people like Eben Byers were drinking radium water as a tonic. Evans claimed “We know of one physician in Chicago, for example, who injected more than a thousand patients, the normal regime being 10 microcuries intravenously once a week for a year! That’s 500 microcuries or half a millicurie.”

Evans become an expert in the new field of nuclear medicine. Based on his studies, he estimated the largest permissible load of radioactivity for a person was 0.1 microcurie. This value was something of a rough guess, but has held up well over time.

I was surprised that during World War II Evans was not at Los Alamos helping to build the atomic bomb. He did, however, carry out war work. For instance, he was responsible for measuring radioactivity in the Belgian uranium ores brought from the Congo for the Manhattan Project. He also invented a scheme to mark land mines with the radioactive isotope cobalt-60, so if American troops had to retake ground previously mined, they could easily detect and remove the mines. Finally, he created a technique to monitor the preservation of blood using radioactive iron as a tracer.

Evans developed a method to use radioactive iodine to diagnose and treat goiter. He taught at MIT until his retirement in 1972. As a student, IPMB author Russ Hobbie took a class with Evans in statistical nuclear physics, based on Chapters 26-28 in The Atomic Nucleus.

Evans won many awards throughout his career, including the Enrico Fermi Award for pioneering work “in measurements of body burdens of radioactivity and their affects on human health, and in the use of radioactive isotopes for medical purposes.” Robley Evans died in 1996, at age 88. You can read his obituary here and here.

The Atomic Nucleus was a leading nuclear physics textbook of its day, and according to Google Scholar it has been cited nearly 4000 times. If interested in reading The Atomic Nucleus, you can download it free online at https://archive.org/details/TheAtomicNucleus.

You never know what tales lie buried beneath each word of IPMB.

## Friday, February 9, 2018

### Suki Roth (2002-2018)

Intermediate Physics for Medicine and Biology: Suki Roth (2002-2018)
 Suki Roth (2002-2018).
Regular readers of this blog are familiar with my dog Suki, who I’ve mentioned in more than a dozen posts. Suki passed away this week. She was a wonderful dog and I miss her dearly.

Suki and I used to take long walks when I would listen to audio books, such as The Immortal Life of Henrietta Lacks, Musicophilia, Destiny of the Republic, Galileo’s Daughter, and First American: The Life and Times of Benjamin Franklin. This list just scratches the surface. On my Goodreads account, I have a category called “listened-to-while-dog-walking” that includes 84 books, all of which Suki and I enjoyed together.

In my post about the Physics of Phoxhounds, I mentioned that a photo of Suki and me (right) was included in Barb Oakley’s book A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra). Recently I learned that Barb’s book has sold over 250,000 copies, making Suki something of a celebrity.

Suki helped me explain concepts from Intermediate Physics for Medicine and Biology, such as age-related hearing loss and the biomechanics of fleas. Few people knew that she had this secret career in biomedical education!

Thanks to Dr. Kelly Totin, and before her Dr. Ann Callahan, and all the folks at Rochester Veterinary Hospital for taking such good care of Suki. In particular I appreciate Dr. Totin’s help during Suki’s last, difficult days. As she said near the end, her focus was on the quality of Suki’s time left rather than the quantity; an important life lesson for us all.

I’ll close with a quote from one of my favorite authors, James Herriot. In his story “The Card Over The Bed,” the dying Miss Stubbs asks Herriot, a Yorkshire vet, if she will see her pets in heaven. She was worried because she had heard claims that animals have no soul. Herriot responded “If having a soul means being able to feel love and loyalty and gratitude, then animals are better off than a lot of humans. You’ve nothing to worry about there.”

 Suki resting.

 Suki with her nephew Auggie.

 Suki with all five editions of IPMB.

 Suki (right), her niece Smokie (the Greyhound, center), and her nephew Auggie (the Foxhound, left), about to get treats from my wife Shirley.

 Suki and me, 15 years ago.
 Young Suki

## Friday, February 2, 2018

### Gauss and the Method of Least Squares

Intermediate Physics for Medicine and Biology: Gauss and the Method of Least Squares
 Asimov’s Biographical Encyclopedia of Science and Technology, by Isaac Asimov.
In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss fitting data using the method of least squares. This technique was invented by mathematicians Adrien Marie Legendre and Johann Karl Friedrich Gauss. Isaac Asimov describes Gauss’s contributions in Asimov’s Biographical Encyclopedia of Science and Technology.
While still in his teens he [Gauss] made a number of remarkable discoveries, including the method of least squares, advancing the work of Legendre in this area. By this [technique] the best equation for a curve fitting a group of observations can be made. Personal error is minimized. It was work such as this that enabled Gauss, while still in his early twenties, to calculate the orbit for [the asteroid] Ceres.
 Of Time and Space and Other Things, by Isaac Asimov.
Asimov tells the story of Ceres in more detail in Of Time and Space and Other Things
Giuseppe Piazzi, an Italian astronomer … discovered, on the night of January 1, 1801, a point of light which had shifted its position against the background of stars. He followed it for a period of time and found it was continuing to move steadily. It moved less rapidly than Mars and more rapidly than Jupiter, so it was very likely a planet in an intermediate orbit …

Piazzi didn't have enough observations to calculate an orbit and this was bad. It would take months for the slow-moving planet to get to the other side of the Sun and into observable position, and without a calculated orbit it might easily take years to rediscover it.

Fortunately, a young German mathematician, Karl Friedrich Gauss, was just blazing his way upward into the mathematical firmament. He had worked out something called the “method of least squares,” which made it possible to calculate a reasonably good orbit from no more than three good observations of a planetary position.

Gauss calculated the orbit of Piazzi's new planet, and when it was in observable range once more there was [Heinrich] Olbers [a German astronomer] and his telescope watching the place where Gauss's calculations said it would be. Gauss was right and, on January 1, 1802, Olbers found it.
What is the role for least-squares fitting in medicine and biology? In many cases you want to fit experimental data to a mathematical model, in order to determine some unknown parameters. One example is the linear-quadratic model of radiation damage, presented in Chapter 16 of IPMB. Below is a new homework problem, designed to provide practice using the method of least squares to analyze data on cell survival during radiation exposure.
Section 16.9

Problem 29 ½. The fraction of cell survival, Psurvival, as a function of radiation dose, D (in Gy), is
D Psurvival
0   1.000
2   0.660
4   0.306
6   0.100
8   0.0229
10   0.0037
Fit this data to the linear-quadratic model, Psurvival = e-Î±D – Î²D2 (Eq. 16.29) and determine the best-fit values of Î± and Î². Plot Psurvival versus D on semilog graph paper, indicating the data points and a curve corresponding to the model. Hint: use the least-squares method outlined in Sec. 11.1, and make this into a linear least squares problem by taking the natural logarithm of Psurvival.
Gauss is mentioned often in IPMB. Section 6.3 discusses Gauss’s law relating the electric field to charge, and Appendix I discusses the Gaussian probability distribution (the normal, or bell-shaped curve). Asimov writes
Gauss…was an infant prodigy in mathematics who remained a prodigy all his life. He was capable of great feats of memory and of mental calculation…. Some people consider him to have been one of the three great mathematicians of all time, the others being Achimedes and Newton.