Radiobiology for the Radiologist

Radiobiology for the Radiologist,
by Eric Hall and Amato Giaccia.

In Section 16.9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the biological effects of radiation. We write

This section provides a brief introduction to radiobiology, but it ignores many important details. For these details see Hall and Giaccia (2012).

Radiobiology for the Radiologist, by Eric Hall and Amato Giaccia, is a leading graduate textbook in radiology and medical physics. It analyzes the “four Rs” of radiobiology:

1. Repair: “If only one strand [of DNA] is broken, there are efficient mechanisms that repair it over the course of a few hours using the other strand as a template” (p. 481, IPMB).
2. Reassortment: “Even though radiation damage can occur at any time in the cell cycle (albeit with different sensitivity), one looks for chromosome damage during the next M [cell division, or mitosis] phase” (p. 481-482, IPMB).
3. Reoxygenation: “A number of chemicals enhance or inhibit the radiation damage…One of the most important chemicals is oxygen, which promotes the formation of free radicals and hence cell damage. Cells with a poor oxygen supply are more resistant to radiation than those with a normal supply” (p. 482, IPMB).
4. Repopulation: IPMB doesn’t address this last “R” specifically, but it is the most obvious of the four: After a dose of radiation, surviving cells grow and divide, repopulating the tumor.

Hall and Giaccia’s Figure 6.13 summarizes the reoxygenation process (below I show a similar, open-access figure by Padhani et al., European Radiology, 17:861-872, 2007).

Reoxygenation is one reason why radiation is divided into small daily fractions rather than one large dose. Hall and Giaccia write

A modest dose of x-rays to a mixed population of aerated and hypoxic cells results in significant killing of aerated cells but little killing of hypoxic cells. Consequently, the viable cell population immediately after irradiation is dominated by hypoxic cells. If sufficient time is allowed before the next radiation dose, the process of reoxygenation restores the proportion of hypoxic cells to about 15%. If this process is repeated many times, the tumor cell population is depleted, despite the intransigence to killing by x-rays of the cells deficient in oxygen.

They later use the four Rs to summarize why fractions are important during radiotherapy.

The basis of fractionation in radiotherapy can be understood in simple terms. Dividing a dose into several fractions spares normal tissues because of repair of sublethal damage between dose fractions and repopulation of cells if the overall time is sufficiently long. At the same time, dividing a dose into several fractions increases damage to the tumor because of reoxygenation and reassortment of cells into radiosensitive phases of the cycle between dose fractions.

The advantages of prolongation of treatment are to spare early reactions and to allow adequate reoxygenation in tumors. Excessive prolongation, however, allows surviving tumor cells to proliferate during treatment.

I like the colorful figures in Hall and Giaccia's book. For instance, is that French ram wearing a beret?

For those planning to buy a copy of the 7^th edition of Radiobiology for the Radiologist, I have news. The 8^th edition will be published later this year! Hang on for a few more months, and then purchase the new edition.

The Radiation Dose from Radon: A Back-of-the-Envelope Estimation

I like Fermi problems: those back-of-the-envelope order-of-magnitude estimates that don’t aim for accuracy, but highlight underlying principles. I also enjoy devising new homework exercises for the readers of this blog. Finally, I am fascinated by radon, that radioactive gas that contributes so much to the natural background radiation. Ergo, I decided to write a new homework problem about estimating the radiation dose from breathing radon.

What a mistake. The behavior of radon is complex, and the literature is complicated and confusing. Part of me regrets starting down this path. But rather than give up, I plan to forge ahead and to drag you—dear reader—along with me.

Section 17.12

Problem 57 1/2. Estimate the annual effective dose (in Sv yr^-1) if the air contains a trace of radon. Use the data in Fig. 17.27, and assume the concentration of radon corresponds to an activity of 150 Bq m^-3, which is the action level at which the Environmental Protection Agency suggests you start to take precautions. Make reasonable guesses for any parameters your need.

Here is my solution (stop reading now if you first want to solve the problem yourself). In order to be accessible to a wide audience, I avoid jargon and unfamiliar units.

One bequerel is a decay per second, and a cubic meter is 1000 liters, so we start with 0.15 decays per second per liter. The volume of air in your lungs is about 6 liters, implying that approximately one atom of radon decays in your lungs every second.

Radon decays by emitting an alpha particle. You don’t, however, get just one. Radon-222 (the most common isotope of radon) alpha-decays to polonium-218, which alpha-decays to lead-214, which beta-decays twice to polonium-214, which alpha-decays to lead-210 (see Fig 17.27 in Intermediate Physics for Medicine and Biology). The half-life of lead-210 is so long (22 years) that we can treat it as stable. Each decay of radon therefore results in three alpha particles. An alpha particle is ejected with an energy of about 6 MeV. Therefore, roughly 18 MeV is deposited into your lungs each second. If we convert to SI units (1 MeV = 1.6 × 10^-13 joule), we get about 3 × 10^-12 joules per second.

Absorbed dose is expressed in grays, and one gray is a joule per kilogram. The mass of the lungs is about 1 kilogram. So, the dose rate for the lungs is 3 × 10^-12 grays per second. To find the annual dose, multiply this dose rate by one year, or 3.2 × 10⁷ seconds. The result is about 10^-4 gray, or a tenth of a milligray per year.

If you want the equivalent dose in sieverts, multiply the absorbed dose in grays by 20, which is the radiation weighting factor for alpha particles. To get the effective dose, multiply by the tissue weighting factor for the lungs, 0.12. The final result is 0.24 mSv per year.

This all seems nice and logical, except the result is a factor of ten too low! It is probably even worse than that, because my initial radon concentration was higher than average and in Table 16.6 of IPMB Russ Hobbie and I report a value of 2.28 mSv for the average annual effective dose. My calculation here is an estimate, so I don’t expect the answer to be exact. But when I saw such a low value I was worried and started to read some of the literature about radon dose calculations. Here is what I learned:

1. The distribution of radon progeny (such as ²¹⁴Po) is complicated. These short-lived isotopes are charged and behave differently than an unreactive noble gas like radon. They stick to particles in the air. Your dose depends on how dusty the air is.
2. How these particles interact with our lungs is even more difficult to understand. Some large particles are filtered out by the upper respiratory track. 
3. The range of a 6-MeV alpha particle is only about 50 microns, so some of the energy is deposited harmlessly into the gooey mucus layer lining the airways (see https://www.ncbi.nlm.nih.gov/books/NBK234233). Ironically, if you get bronchitis your mucus layer thickens, protecting you from radon-induced lung cancer.  
4. The progeny and their dust particles stick to the bronchi walls like flies to flypaper, increasing their concentration.
5. Filtering out dust and secreting a mucus layer reduces the dose to the lungs, while attaching the progeny to the airway lining increases it. My impression from the literature is that the flypaper effect dominants, and explains why my estimate is so low.
6. The uranium-238 decay chain shown in Fig. 17.27 is the source of radon-222, but other isotopes arise from other decay chains. The thorium-232 decay chain leads to radon-220, called thoron, which also contributes to the dose.
7. I am not confident about my value for the mass. The lungs are a bloody organ; about half of their mass is blood. I don’t know whether or not the blood is included in the reported 1 kg mass. The radon literature is oddly silent about the lung mass, and I don’t know how these authors calculate the dose without it. 
8. I ignored the energy released when progeny beta-decay, which would cause a significant error if my aim was to calculate the absorbed dose in grays. But if I want the effective dose in sieverts I should be alright, because the radiation weighting factor for electrons is 1 compared to 20 for alpha particles. 
9. The radon literature is difficult to follow in part because of strange units, such as picocuries per liter and working level months (see https://www.ncbi.nlm.nih.gov/books/NBK234224).
10. Radon can get into the water as well as the air. If you drink the water, your stomach gets a dose. With a half-life of days, the radon in this elixir has time to enter your blood and irradiate your entire body.
11. Does the dose from radon lead to lung cancer? That depends on the accuracy of the linear no-threshold model. If there is a threshold, then such a small dose may not represent a risk.
12. If you want to learn more about radon, read NCRP Report 160, ICRP Publication 103, or BEIR VI. Of course, you should start by reading Section 17.12 in IPMB.

What do I take away from this estimation exercise? First, radon dosimetry is complicated. Second, biology problems are messy, and while order-of-magnitude estimates are still valuable, your results need large error bars.

95gTc and 96gTc as Alternatives to Medical Radioisotope 99mTc

In Chapter 17 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the most widely used radioisotope in nuclear medicine: ^99mTc (technetium-99m). Previously in this blog (here, here, and here) I described the looming shortage of ^99mTc. In a recent paper in the open access journal Heliyon (Volume 4, Article Number e00497, 2018), Hayakawa et al. review “^95gTc and ^96gTc as alternatives to medical radioisotope ^99mTc.” I don’t know enough nuclear medicine to judge if ^95gTc and ^96gTc are realistic alternatives to ^99mTc, but the idea is intriguing. Below I reproduce an abridged and annotated version of the introduction to this interesting paper (my comments are in italics and enclosed in brackets []). Enjoy!

Various radioisotopes, such as ^99mTc (half-life 6.02 h [hours]), ²⁰¹Tl [thallium-201] (half-life 3.04 d [days]), and ¹³³Xe [xenon-133] (half-life 5.27 d), are used for single-photon emission computed tomography (SPECT) in medical diagnostic scans. In particular, ^99mTc has become the most important medical radioisotope at present… Over 30 commonly used radiopharmaceuticals are based on ^99mTc [for example, ^99mTc–sestamibi, ^99mTc–tetrofosmin, and ^99mTc-exametazime]... The ^99mTc radioisotopes are supplied by ⁹⁹Mo/^99mTc generators, which continuously generate ^99mTc through the β-decay of the parent nucleus ⁹⁹Mo [molybdenum-99 is trapped in alumina (Al₂O₃) where it decays to pertechnetate (TcO₄^-); eluting solution flowing through the alumina collects the ^99mTc]... This supply method provides two excellent advantages. First, it is possible to transport ⁹⁹Mo/^99mTc generators from a production facility to any place in the world because the half-life of ⁹⁹Mo is... 2.75 d. Second, when a ⁹⁹Mo/^99mTc generator is transported to a hospital, ^99mTc can be produced fresh for up to 2 weeks by daily milking/elution from this ⁹⁹Mo/^99mTc generator. At present, the parent nucleus ⁹⁹Mo is produced in nuclear reactors by the neutron-induced fission of ²³⁵U [uranium-235] in highly enriched uranium (HEU) targets, in which the fraction of ²³⁵U is approximately 90%. However, some nuclear reactors that have supplied ⁹⁹Mo require major repairs or shutdown [for example, the Chalk River reactor in Ontario, Canada], which may lead to a ^99mTc shortage. Thus, many alternative methods to produce ⁹⁹Mo or ^99mTc [such as in a cyclotron] without HEU have been proposed…

The September 11th terrorist attacks in Washington D.C. [these attacks actually took place in New York City, at the Pentagon in Arlington County Virginia, and near Shanksville, Pennsylvania] in 2001 also affected medical radioisotope production from the viewpoint of the safeguards of nuclear materials. The control of fissionable nuclides such as ²³⁵U and ²³⁹Pu [plutonium-239] is important for the safeguards of nuclear materials… The International Atomic Energy Agency (IAEA) hopes to discontinue ^99mTc production using HEU targets, which can be transmuted into nuclear weapons... In the near future, ^99mTc will be supplied by nuclear reactors using LEU [low-enriched uranium] targets in addition to HEU. The Nuclear Energy Agency (NEA) reported the prediction that the ⁹⁹Mo/^99mTc supply will be larger than the world demand when the scheduled nuclear reactors using LEU start ⁹⁹Mo production…

Because the Tc [technitium] chemistry is the same, all the radiopharmaceuticals based on ^99mTc can, in principle, be applied to other Tc isotopes. There are five Tc isotopes with half-lives in the range from hours to days: ^94mTc (half-life 52 m [minutes]), ^94gTc (half-life 4.88 h), ^95mTc (half-life 60 d), ^95gTc (half-life 20 h), and ^96gTc (half-life 4.28 d) [superscript “g” stands for ground state, whereas superscript “m” stands for metastable excited state]… The half-life of ^96gTc (4.28 d) is long enough for worldwide delivery from a production facility and lengthy use of up to 2 weeks in hospitals. ^95gTc (20 h) can also be transported to a wide area and used for 3–5 days in hospitals. Thus, ^95gTc and ^96gTc are candidates for alternative γ-ray emitters. However, the decay rates of ^95gTc and ^96gTc are lower than that of ^99mTc by a factor of 3.3 and 17, respectively, because the decay rate of a radioisotope is inversely proportional to its half-life. This fact leads to the question of whether these isotopes can work as ^99mTc medical radioisotopes.

In the current study, we present the relative γ-ray flux of these isotopes with simple assumptions. We also estimate the patient radiation does [dose] per Tc-labeled tracer using… PHITS [Particle and Heavy Ion Transport code System, a general purpose Monte Carlo particle transport simulation code]... Various nuclear reactions that are production methods of Tc isotopes, such as (p, n) reactions [in which a proton enters the nucleus and a neutron leaves it]…, deuteron [hydrogen-2]-induced reactions…, and ⁹⁶Ru [ruthenium-96] (n, p)^96gTc reactions…, were studied. We consider the production by the (p, n) reaction on an enriched Mo isotope. We also calculate the production rate using a typical PET [positron emission tomography] medical cyclotron [If you must make ^96gTc in a cyclotron, why not make ^99mTc in the cyclotron instead?]. Because the energies of decay γ-rays of these Tc isotopes are typically higher than 200 keV, they are not suitable for the traditional SPECT cameras. Thus, we also discuss the property of possible ETCC [Electron-Tracking Compton Camera] for high energy γ-rays.

Another Analytical Example of Filtered Back Projection

Some people collect stamps, aged bottles of wine, or fine art. I collect analytical examples of tomographic reconstruction: determining a function f(x,y) from its projections F(θ,x'). Today I’ll share my latest acquisition: an example of filtered back projection. I discussed a similar example in a previous post, but you can’t have too many of these. This analysis illustrates the process that Russ Hobbie and I describe in Section 12.5 of Intermediate Physics for Medicine and Biology.

The method has two steps: filtering the projection (that is, taking its Fourier transform, multiplying by a filter function, and doing an inverse Fourier transform) and then back projecting. We start with a projection F(θ,x'), which in the clinic would be the output of your tomography machine.

This projection is independent of the angle θ, implying that the function f(x,y) looks the same in all directions. A plot of F(θ,x') as a function of x' is shown below.

I suggest you pause for a minute and guess f(x,y) (after all, our goal is to build intuition). Once you make your guess, continue reading.

Step 1a: Fourier transform

The Fourier transform of the projection F(θ,x') is

with S_f(θ, k) = 0 (Russ and I divide the Fourier transform into a cosine part C and a sine part S, see Eq. 11.57 in IPMB). C_f(θ, k) is a function of k, the wave number (also known as the spatial frequency). I won’t show all calculations; to gain the most from this post, fill in the missing steps. C_f(θ, k) is plotted below as a function of k.

Step 1b: Filter

To filter, multiply C_f(θ, k) by |k|/2π (Eq. 12.39 in IPMB) to get the Fourier transform of the filtered projection C_g(θ, k)

with S_g(θ, k) = 0. The result is shown below. Notice that filtering removes the dc contribution at k = 0 and causes the function to fall off more slowly at large |k| (it's a high-pass filter).

Step 1c: Inverse Fourier transform

Next use Eq. 11.57 in IPMB to calculate the inverse Fourier transform of C_g(θ, k). Initially I didn’t think the required integral could be solved analytically. I even checked the best integral table in the world, with no luck. However, when I typed the integral into the WolframAlpha website, it gave me the answer

The expression contains the cosine integral, Ci(z). After evaluating it at the integral’s end points, using limiting expressions for Ci(z) at small z, and expending much effort, I derived the filtered projection G(θ,x')

A plot of G(θ,x') is shown below.

Step 2: Back projection

The back projection (Eq. 12.30 in IPMB) of G(θ,x') requires some care. Because f(x,y) does not depend on direction, you can evaluate the back projection along any radial line. I chose y = 0, which means the rotated coordinate x' = x cosθ + y sinθ in the back projection is simply x' = x cosθ. You must examine two cases.

Case 1: |x| less than a

In this case, integrate using only the section of G(θ,x') for |x'| less than a.

Instead of integrating from θ = 0 to π, use symmetry, multiply by two, and integrate from 0 to π/2.

At this point, I was sure I could not integrate such a complicated function analytically, but again WolframAlpha came to the rescue.

(Beware: Wolfram assumes you integrate over x, so in the above screenshot x is my θ and b is my x.) The solution contains inverse hyperbolic tangent functions, but once you evaluate them at the end points you get a simple expression

Case 2: |x| greater than a

When |x| is greater than a, angles around θ = 0 use the section of G(θ,x') for |x'| greater than a and angles around θ = π/2 use the section for |x'| less than a.

The angle where you switch from one to the other is θ = cos^-1(a/x). (To see why, analyze the right triangle in the above figure with side of length a and hypotenuse of length x.) The back projection integral becomes

If you evaluate this integral, you get zero.


Final Result

Putting all this together, and remembering that f(x,y) doesn’t depend on direction so you can replace x by the radial distance, you find

We did it! We solved each step of filtered back projection analytically, and found f(x,y).

I’ll end with a few observations.

1. Most clinical tomography devices use discrete data and computer computation. You rarely see the analysis done analytically. Yet, I think the analytical process builds insight. Plus, it’s fun.
2. Want to check our result? Calculate the projection of f(x,y). You get the function F(θ,x') that we started with. To learn how to project, click here.
3. Regular readers of this blog might remember that I analyzed this function in a previous post, where you can see what you get if you don’t filter before back projecting. 
4. I have a newfound respect for WolframAlpha. It solved integrals analytically that I thought were hopeless. In addition, it’s online, free, and open to all.
5. Most filtered back projection algorithms don’t filter using Fourier transforms. Instead, they use a convolution. I think Fourier analysis provides more insight, but that may be a matter of taste. 
6. My bucket list includes finding an analytical example of filtered back projection when f(x,y) depends on direction. Wouldn’t that be cool! 
7. Remember, there is another method for doing tomography: the Fourier method (see Section 12.4 in IPMB). Homework Problems 26 and 27 in Chapter 12 provide analytical examples of that technique.

Mechanics in Morphogenesis

Two weeks ago, I attended the Mechanics in Morphogenesis workshop sponsored by the Princeton Center for Theoretical Science. What was I doing at a workshop about morphogenesis? In the past few years, I have been dabbling in biomechanics (long story), with the goal of understanding mechanotransduction (how tissues grow and remodel in response to mechanical forces). My work might have applications to development, and I wanted to educate myself.

The meeting attracted a mix of experimental biologists and theoretical physicists. My two favorite talks were late Thursday afternoon (Feb 22). Ellen Kuhl of Stanford University discussed the “Mechanics of the Developing Brain,” focusing on how the brain folds into gyri and sulci. She said it was an appropriate topic for a talk at Princeton University because of Einstein’s brain (stolen during the autopsy by a Princeton Hospital pathologist). Kuhl discussed ideas from her article “A Mechanical Model Predicts Morphological Abnormalities in the Developing Human Brain” (Budday, Raybaud and Kuhl, Sci. Rep., Volume 4, Article Number 5644, 2014). This article appeared in Scientific Reports, an open access journal, so you can read the entire article online. The abstract appears below.

The developing human brain remains one of the few unsolved mysteries of science. Advancements in developmental biology, neuroscience, and medical imaging have brought us closer than ever to understand brain development in health and disease. However, the precise role of mechanics throughout this process remains underestimated and poorly understood. Here we show that mechanical stretch plays a crucial role in brain development. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, we model the human brain as a living system with a morphogenetically growing outer surface and a stretch-driven growing inner core. This approach seamlessly integrates the two popular but competing hypotheses for cortical folding: axonal tension and differential growth. We calibrate our model using magnetic resonance images from very preterm neonates. Our model predicts that deviations in cortical growth and thickness induce morphological abnormalities. Using the gyrification index, the ratio between the total and exposed surface area, we demonstrate that these abnormalities agree with the classical pathologies of lissencephaly and polymicrogyria. Understanding the mechanisms of cortical folding in the developing human brain has direct implications in the diagnostics and treatment of neurological disorders, including epilepsy, schizophrenia, and autism.

I also enjoyed a physics colloquium about “Motifs in Morphogenesis” by Lakshminarayanan Mahadevan of Harvard University. I liked how he presented simple, toy models that illustrate development (my type of physics). Mahadevan is a MacArthur “Genius Grant” Fellow, and I can see why. Great talk.

On Growth and Form,
by D'Arcy Wentworth Thompson.

Mahadevan is also coeditor with Thomas Lecuit of a special issue published by the journal Development commemorating the groundbreaking book On Growth and Form. In their editorial (Volume 144, Pages 4197-4198, 2017) they write

Morphogenesis, the study of how forms arise in biology, has attracted scientists for aeons. A century ago, D'Arcy Wentworth Thompson crystallized this question in his opus On Growth and Form (Thompson, 1917) using a series of biological examples and geometric and physical analogies to ask how biological forms arise during development and across evolution. In light of the advances in molecular and cellular biology since then, a succinct modern view of the question states: how do genes encode geometry?...

On Growth and Form raised the question of the origin of biological shape in a physical framework. Since then, advances in our understanding of the biochemical basis of the laws of heredity have provided the modern conceptual understanding for how shapes develop anew at each generation, from a single cell – thus surviving the death of an individual through its offspring. As this Special Issue illustrates, we are now beginning to understand how genes encode geometry. As morphology both enables and constrains function, a natural next question is how biology creates functional (and plastic) shape that begins to link morphology to physiology and behaviour. As you mull this question, we would like to thank all the authors and referees of the articles in this Special Issue for their contributions, and we hope you enjoy reading it!

I had never been to Princeton University, so during lunch breaks I snuck out to look around the campus. It is beautiful (even in February), with many historical buildings. I particularly liked the gorgeous chapel, shown below.

 Princeton University Chapel

Pyne Hall

Alexander Hall

Joseph Henry electromagnet displayed in Jadwin Hall.

My poster with Debabrata Auddya.

My poster with graduate student Kharananda Sharma.

The Mechanics in Morphogenesis workshop highlighted how physics can be applied to biology and medicine, a topic central to Intermediate Physics for Medicine and Biology.

Chromatin Packing and Molecular Biology of the Cell

Molecular Biology of the Cell,
by Bruce Alberts et al.

In Section 16.9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Cellular DNA is organized into chromosomes.… Figure 16.30 shows, at different magnifications, a strand of DNA, various intermediate structures that we will not discuss, and a chromosome as seen during the M phase of the cell cycle. The size goes from 2 nm for the DNA double helix to 1400 nm for the chromosome.

Figure 16.30 in IPMB is an illustration from the book Molecular Biology of the Cell by Bruce Alberts and his coauthors. This acclaimed textbook is currently in its 6th edition (2014). The Oakland University library has the 5th edition (2008), which I quote in this blog post.

The black and white figure Russ and I include in IPMB has evolved into the beautiful color figure 4-72 in the 5th edition of the Molecular Biology of the Cell. Their caption reads

Figure 4-72 Chromatin packing. This model shows some of the many levels of chromatin packing postulated to give rise to the highly condensed mitotic chromosome.

Alberts et al. state at the start of their Chapter 4

In this chapter we begin by describing the structure of DNA. We see how, despite its chemical simplicity, the structure and chemical properties of DNA make it ideally suited as the raw material of genes. We then consider how the many proteins in chromosomes arrange and package this DNA. The packing has to be done in an orderly fashion so that the chromosomes can be replicated and apportioned correctly between the two daughter cells at each cell division. It must also allow access to chromosomal DNA for the enzymes that repair it when it is damaged and for the specialized proteins that direct the expression of its many genes. We shall also see how the packaging of DNA differs along the length of each chromosome in eucaryotes, and how it can store a valuable record of the cell’s developmental history.

One of the most interesting feature of this figure is the “beads-on-a-string” appearance of chromatin. Alberts et al. write

The proteins that bind to DNA to form eukaryotic chromosomes are traditionally divided into two general classes: the histones and the nonhistone chromosomal proteins. The complex of both classes of protein with the nuclear DNA of eukaryotic cells is known as chromatin. Histones are present in such enormous quantities in the cell (about 60 million molecules of each type per human cell) that their total mass in chromatin is about equal to that of the DNA.

Histones are responsible for the first and most basic level of chromosome packing, the nucleosome, a protein-DNA complex discovered in 1974. When interphase nuclei are broken open very gently and their contents examined under the electron microscope, most of the chromatin is in the form of a fiber with a diameter of about 30 nm (Figure 4-22A). If this chromatin is subjected to treatments that cause it to unfold partially, it can be seen under the electron microscope as a series of “beads on a string” (Figure 4-22B). The string is DNA, and each bead is a “nucleosome core particle” that consists of DNA wound around a protein core formed from histones.

For those who prefer videos over text and illustrations, below are two movies about how DNA is packaged in a cell.

I hope this post helps readers of IPMB understand better those “various intermediate structures that we will not discuss.” For more details, I recommend Molecular Biology of the Cell.

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 I_n and K_n 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!

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.

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