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

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, P_survival, 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, P_survival = e^{-αD – βD2} (Eq. 16.29) and determine the best-fit values of α and β. Plot P_survival 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 P_survival.

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.

The Viscous Torque on a Rotating Sphere

In Section 9.10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

We saw in Chap. 4 (Stokes' law) that the translational viscous drag on a spherical particle is 6πηav. Similarly, the viscous torque on a rotating sphere is 8πηa³(dθ/dt).

Let’s calculate this torque. We always learn something when we see where such a result comes from.

To begin, we will redo Homework Problem 46 from Chapter 1 that asks you to calculate the translational Stokes’ law by considering a stationary sphere in a moving viscous fluid (equivalent to a sphere moving through a stationary viscous fluid). Below is the analogous problem for a sphere rotating in the same fluid.

Problem 46 ½. Consider a sphere of radius a rotating with angular velocity ω in a fluid of viscosity η. For low Reynolds number flow, the fluid velocity and pressure surrounding the sphere are

          v_φ = ω a³ sinθ/r²

          v_r = v_θ = p = 0.

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

(b) Integrate the shear torque over the sphere surface and find an expression for the net viscous torque on the sphere.

When I first tried to solve part (b), I kept getting an answer that was off by a factor of 2/3. I checked my work several times, but I couldn’t find any mistake. After much fussing, I finally figured out my error. For the shear stress at the sphere surface, I was using η dv_φ/dr. This seemed right at first, but it’s not. The shear stress is actually η (dv_φ/dr – v_φ/r). Why? I could just say that I looked up the expression for the shear strain ε_rφ for spherical coordinates and found it had two terms. But that’s no fair (and no fun). We have to understand what we are doing, not just look things up. Why does the expression for the shear stress have two terms?

Let’s start on page 16 of IPMB, where Russ and I note that the shear stress is the viscosity times the rate of change of the shear strain. We need to see how the shear strain changes with time. There are two cases.

1. The first case will give us the familiar dv/dr expression for the shear stress. Consider an element of fluid with thickness dr, as shown below.

The velocity is in the φ direction, and depends on r. In time T, the top surface of the box moves to the right a distance v_φ(r+dr) T, while the bottom surface moves only v_φ(r) T, forming the dashed box in the figure. The shear strain is the angle θ (see Problem 14 in Chapter 1). Consider the shaded triangle having height dr and angle θ. The length of the bottom side of the triangle is v_φ(r+dr) T – v_φ(r) T. The tangent of θ is therefore

          tanθ = (v_φ(r+dr) T – v_φ(r) T) / dr .

In the limit as dr goes to zero, and for small angles such that tanθ is approximately θ, the shear strain becomes dv_φ/dr T. Therefore, the rate of change of the shear strain is dv_φ/dr, and the contribution to the shear stress is η dv_φ/dr.

This is where I got stuck, until I realized there is a second case we must consider.

2. Even if v_φ does not change with r, we can still get a shear strain because of the curvilinear coordinates. Consider the arc-shaped element of fluid shown below.

Suppose the fluid moves with the same speed, v_φ, on both the top and bottom surfaces. After time T, the fluid element moves to the right and forms the dashed element. The problem is, this dashed shape is no longer an arc aligned with the curvilinear coordinates. It has been sheared! Consider the shaded triangle with angle θ. The top side has a length (r+dr) θ – v_φ T, and the right side has length dr. The ratio of these two sides is tanθ, or for small angles just θ. So

          θ = [(r+dr) θ – v_φ T]/dr

Solving for θ gives

          θ = (v_φ/r) T,

so the shear stress is η v_φ/r.

Notice that in the first case the top side is sheared to the right, whereas in the second case it is sheared to the left. We need a minus sign in case two.

In general, both of these effects act together, so the shear stress is η (dv_φ/dr – v_φ/r).

For a velocity that falls as 1/r², the dv_φ/dr term gives -2/r³, while the -v_φ/r term gives -1/r³, with a sum of -3/r³. I was getting a factor of two when I was supposed to get a factor of three. 

Are you still not convinced about the second term in the stress? Look at it this way. Suppose the velocity were proportional to r. This would imply that the fluid was rotating as if it were a solid body (all the fluid would have the same angular velocity). Such a pure rotation should not result in shear. If we only include the dv_φ/dr term, we would still predict a shear stress. But if we include both terms they cancel, implying no stress.

Let me outline how you do the integral in part (b) of the homework problem above. The torque is the force times distance. The distance from the axis of rotation to the surface where the shear acts (the moment arm) is a sinθ. The force is the shear stress times the area, and the area element is a² sinθ dθ dφ. You end up getting three factors of sinθ: one from the moment arm, one from the shear stress, and one from the area element, so you have to integrate sin³θ.

If you want, you can add a third part to Homework Problem 35 in Chapter 1:

(c) Show that the velocity distribution in Problem 46 ½ is incompressible by verifying that the divergence of the velocity is zero.

Appendix L will help you calculate the divergence in spherical coordinates.

Finally, how did I get the velocity distribution v_φ = ω a³ sinθ/r² that appeared in the homework problem? When the pressure is zero, the velocity during low Reynolds number flow, known as Stokes flow, obeys ∇²v = 0. This is a complicated equation to solve, because v is a vector. In Cartesian coordinates, the Laplacian of a vector is just the Laplacian of its components. In curvilinear coordinates, however, the r, θ, and φ components of the vector mix together in a complicated mess. I will let you try to sort that all out. Don’t say I didn't warn you.

Is Magnetic Resonance Imaging Safe?

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss safety issues in magnetic resonance imaging.

Safety issues in MRI include forces on magnetic objects in and around the patient such as aneurysm clips, hairpins, pacemakers, wheel chairs, and gas cylinders (Kanal et al. 2007), absorbed radio-frequency energy (Problem 21), and induced currents from rapidly-changing magnetic field gradients. The rapid changes of magnetic field can stimulate nerves and muscles, cause heating in electrical leads and certain tattoos, and possibly induce ventricular fibrillation. Induced fields are reviewed by Schaefer et al. (2000). Cardiac pacemakers are being designed to be immune to the strong—and rapidly varying—magnetic and rf fields (Santiniet al. 2013).

Recently, two more safety issues have emerged. The first is the possibility of genetic damage caused by MRI. This question is examined in the article “Will an MRI Examination Damage Your Genes?” by Kenneth Foster, John Moulder, and Thomas Budinger (Radiation Research, Volume 187, Pages 1-6, 2017). Foster and Moulder are cited extensively in Chapter 9 of IPMB, when we discuss the risks of low-frequency electric and magnetic fields. They are two of my heroes in the fight against pseudoscience. Budinger has studied MRI safety for years. They write

We conclude that while a few studies raise the possibility that MRI exams can damage a patient’s DNA, they are not sufficient to establish such effects, let alone any health risk to patients. Based on the failure of decades of research on biological effects of static and RF fields to establish genotoxic effects of such fields at levels comparable to those used in clinical MRI, we consider that genotoxic effects of MRI are highly unlikely. The likely increase in risk, if it were present at all, from a one-off MRI exam would surely be very small and possibly nil, but could not be proven to be zero.

In my opinion, the phrase “highly unlikely” is generous.

A second, more serious, safety issue is risks associated with the MRI contrast agent gadolinium. In IPMB, Russ and I explain

Differences in relaxation time are easily detected in an [MRI] image. Different tissues have different relaxation times. A contrast agent containing gadolinium (Gd³⁺), which is strongly paramagnetic, is often used in magnetic resonance imaging. It is combined with many of the same pharmaceuticals used with ^99mTc [an isotope used extensively in nuclear medicine], and it reduces the relaxation time of nearby nuclei. Gadolinium has been used to assess ischemic myocardium.

MRI using gadolinium was recently discussed in a point/counterpoint article (“The Use of Gadolinium-Based Contrast Agents Should be Discontinued Until Proven Safe,” Medical Physics, Volume 44, Pages 3371–3374, 2017). Moderator Colin Orton writes

Gadolinium-based contrast agents (GBCAs) are widely used in MRI to increase the visibility of tissues. Some believe, however, that due to their documented toxicity, clinical use of these agents should be discontinued until proven safe. This is the premise debated in this month’s Point/Counterpoint. Arguing for the Proposition is Stacy Matthews Branch, Ph.D. Dr. Branch is a biomedical consultant, medical writer, and veterinary medical doctor…. Arguing against the Proposition is Michael F. Tweedle, Ph.D. Dr. Tweedle is the Stefanie Spielman Professor of Cancer Imaging and Professor of Radiology at The Ohio State University.

I am a big fan of point/counterpoint articles, and we discuss one every Friday in my Medical Physics class. This debate has more substance than the genetic damage controversy, but I tend to agree with Tweedle when he concludes

But is dissociated Gd a risk factor beyond NSF [Nephrogenic Systemic Fibrosis, a disease shown to be associated with some Gadolinium-based contrast agents]? At what level and for what? Research to better understand the risks of GBCAs should certainly continue. But discontinuation of all GBCAs would result in complete loss of their benefit, probably in loss of human life due to inaccurate or imprecise diagnosis, while we search for an hypothesized chronic toxicity of unknown seriousness that we, at this point, have no reason in evidence to anticipate. The reasonable response to the new findings is further research into chronic tolerance and more discriminating use of the available GBCAs.

A recent article featured in medicalphysicsweb highlighted new MRI contrast agents based on manganese instead of gadolinium, that may be safer.

Manganese-based contrast could allow safer MRI. A team at Massachusetts General Hospital has developed a potential alternative to gadolinium-based MRI contrast agents, which carry significant health risks for some patients and cannot be used in patients with poor renal function. In tests on baboons, the researchers demonstrated that the manganese-based agent Mn-PyC3A produced equivalent contrast enhancement of blood vessels to that of a gadolinium-based agent (Radiology doi:10.1148/radiol.2017170977).

A discussion of significant MRI safety issues can be found here.

So is magnetic resonance imaging safe? For the vast majority of MRIs that do not use any contrast agent, I would say overwhelmingly yes. When gadolinium is used, there is a small risk that in most cases will be far less significant than the benefit of obtaining the image.

#MieToo

Many readers of Intermediate Physics for Medicine and Biology are undergraduate students who may be looking for their first research experience. Now is the time to be applying for summer undergraduate research opportunities; often they have application deadlines in early February. I’ve collected a list of many biomedical research programs on the Oakland University Center for Biomedical Research website. Another place to learn about research programs is Pathways to Science. The National Science Foundation funds a large number of Research Experiences for Undergraduates; find one here. If you are on Twitter, search for tweets with the hashtag #undergraduateresearch.

Be sure to apply to the Summer Internship Program at the National Institutes of Health #NIH. I worked at the NIH intramural program in Bethesda, Maryland during the 1990s, and can think of no better place to do a summer internship. They have a special program aimed at biomedical engineers #Bioengineering, which might be of particular interest to readers of IPMB. The deadline for the general program is March 1, but for the biomedical engineering program it is February 9. Sorry international students, but students eligible for the NIH program must be US citizens or permanent residents.

I had my own career-defining undergraduate research experience at the University of Kansas #KansasU. I worked with physics professor Wes Unruh and his graduate student Robert Bunch studying the scattering of light. We analyzed our data using a theory of light scattering developed by German physicist Gustav Mie (#MieToo...that's supposed to be a joke). My first publication was a 1983 abstract to the March Meeting of the American Physical Society titled “Size distributions of Ni and Co colloids within MgO”. Although in graduate school I switched from condensed matter physics to biological physics, this first exposure to scientific research set me on the path that led ultimately to coauthoring Intermediate Physics for Medicine and Biology.

Most applications for summer research programs require an essay, college transcripts, and letters of recommendation. Gathering all this stuff takes time. So, if you plan to apply for summer research programs, get to work now, now, now!