A Dozen of My Favorite New Homework Problems

The 4th edition of Intermediate Physics for Medicine and Biology contains 44% more homework problems than did the 3rd Edition. What are some of these new problems about? Here are a dozen of my favorites:

Chapter 1, Problem 25: Poisson’s ratio

Chapter 4, Problem 22: MRI Diffusion Tensor Imaging

Chapter 4, Problem 23: The effect of buffers on the intracellular diffusion of calcium

Chapter 5, Problem 6: Osmotic pressure in articular cartilage

Chapter 5, Problem 17: Countercurrent heat exchangers

Chapter 7, Problem 30: Clark and Plonsey’s calculation of the intracellular and extracellular potential of a nerve axon

Chapter 8, Problem 17: The magnetic field produced by a planar action potential wave front in anisotropic cardiac tissue

Chapter 9, Problem 9: An analytical solution to the nonlinear Poisson-Boltzmann equation

Chapter 10, Problem 37: Ventricular fibrillation of the heart, chaos, and action potential restitution

Chapter 10, Problem 39: A cellular automata model for cardiac arrhythmias

Chapter 12, Problem 23: Analytical example of how to calculate an image from its projection using the method of reconstruction by Fourier transform

Chapter 18, Problem 18: The “magic angle” in MRI

Many of these twelve problems are more difficult than average for our book, but undergraduate physics majors should be able to handle them all. Often we introduce new concepts in the problems. For instance, Poisson’s ratio is never discussed in the text, but other biomechanics topics are, and Problem 25 of Chapter 1 introduces Poisson’s ratio by relating it to concepts we introduced previously.

If you want to get the most out of the 4th edition of Intermediate Physics for Medicine and Biology, work the problems. Otherwise, you may miss some new and fascinating applications of physics to the biomedical sciences.

PhysicsCentral

The American Institute of Physics has a website for the public called PhysicsCentral (http://www.physicscentral.com). The purpose of the site is to help you “learn how your world works.” According to the July 2008 issue of the APS News, the site was redesigned recently to include podcasts and vodcasts, blogs and RSS feeds, and other interactive new features. The website describes its mission.

The American Physical Society represents some 45,000 physicists, and most of our work centers on scientific meetings and publications—the primary ways that physicists communicate with each other. With PhysicsCentral, we communicate the excitement and importance of physics to everyone. We invite you to visit our site every week to find out how physics is part of your world. We’ll answer your questions on how things work and keep you informed with daily updates on physics in the news. We'll describe the latest research and the people who are doing it and, if you want more, where to go on the web. So stick with us. It’s a big, interesting world out there, and we look forward to showing you around.

Of particular interest to readers of the 4th edition of Intermediate Physics for Medicine and Biology is the Biology and Medicine Section of the site. Here you can find fascinating stories about medical and biological physicists, concise descriptions about the frontiers of research, and beautiful pictures. For instance, the current featured story is “The Theory of Everything...Everything Alive!” that describes the work of physicist Geoffrey West on scaling, a topic discussed in Chapter 2 of Intermediate Physics for Medicine and Biology. Students at all levels will find much to inspire and interest them. Take a look at what all the excitement is about, and have fun.

Max Delbruck, Biological Physicist

Niels Bohr's Times:
In Physics, Philosophy, and Polity,
by Abraham Pais.

Last week, I discussed the book Niels Bohr's Times: In Physics, Philosophy, and Polity by Abraham Pais. In particular, I summarized Pais’s view that Bohr played a key role in the development of nuclear medicine through his collaboration with Georg Charles von Hevesy. This week, I address another contribution of Bohr to biological physics through his influence on Max Delbruck.

Bohr had some unorthodox views about biology that were motivated by his idea of “complementarity” in quantum mechanics. Even Pais admits Bohr’s “thoughts on biology have not borne fruit,” which is a polite way to put it. But Delbruck, then a young physicist, wrote “I am perhaps the only one of his associates of those days who took [Bohr] so seriously that it determined [my] career, changing over into biology to find out whether indeed there was anything to this point of view.” Pais writes

Delbruck’s professional switch was Bohr’s greatest contribution to biology. In an obituary of Delbruck it has been written: “Odd though these views [expressed in Bohr’s lecture ‘Light and Life’] may seem to us now, in retrospect, this lecture confirmed Max’s decision to turn to biology... It is fair to say that with Max, Bohr found his most influential philosophical disciple outside the domain of physics, in that through Max, Bohr provided one of the intellectual fountainheads for the development of 20th century biology...”

[Delbruck’s] celebrated work on bacteriophages (viruses that infect bacteria) began after his move to Cal Tech in 1937, where he became professor of biology in 1946. It is worthy of note that, in the 1963 meeting at Copenhagen commemorating the 50th anniversary of Bohr’s first papers on atomic constitution, only one paper on complementarity was presented—by Delbruck, on biology. He received the Nobel Prize in 1969.

Delbruck, a German native, spent the years during World War II with the Department of Physics at Vanderbilt University in Nashville, Tennessee (where I later obtained my PhD). While at Vanderbilt, Delbruck collaborated with Salvadore Luria on an experiment using bacteriophages to show that mutations in viruses are random rather than directed events. The American Physical Society has named its prize in biological physics the Max Delbruck Prize for his work on genetics.

Why do I discuss Bohr, Hevesy and Delbruck in a blog about the 4th edition of Intermediate Physics for Medicine and Biology? Because they represent classic examples of how physics and physicists can make fundamental and lasting contributions to medicine and biology. And that is the whole point of the book.

Georg Charles von Hevesy

Niels Bohr’s Times:
In Physics, Philosophy, and Polity,
by Abraham Pais.

Recently I finished reading the biography Niels Bohr’s Times: In Physics, Philosophy, and Polity, by Abraham Pais. It is an excellent book, but not quite as good as Pais’s masterpiece, Subtle Is the Lord: The Science and the Life of Albert Einstein. Although I was familiar with Bohr’s research in atomic and nuclear physics, I was surprised to discover his pioneering role (perhaps more as a fund raiser, administrator, and mentor than researcher) in medical and biological physics.

Much of this research was in collaboration with Georg Charles von Hevesy (1885–1966). Pais writes “The work of Hevesy in the 1930s at Bohr’s institute made isotope tracers methods flourish. And so Bohr became the godfather, and Hevesy the father, of nuclear medicine.” Hevesy was a Hungarian of Jewish descent who was a professor in Germany in the 1930s until “he left Freiburg because of his disgust with the Nazi regime.” In 1934 he joined Bohr in Copenhagen. Hevesy began using heavy radioactive isotopes to study the uptake and loss of elements. In Pais’s words, Hevesy made

the first application anywhere of tracers in the life sciences... This led Hevesy to tracer research... on the resorption, distribution, and excretion of labeled bismuth compounds administered to rats, the first use of tracers in the study of animal metabolism...

In 1948 Hevesy wrote: “During our early work with natural radioisotopes as indicators, we often mentioned what an attractive place that Fairyland must be where radioactive isotopes of all elements are available. This utopia became reality almost in a single stroke, when the Joliot-Curies made their most important discovery of artificial radioactivity [man-made radioactive isotopes]. The path was thus paved for investigation of the fate of the atoms of the common constituents of the animal and plant organisms...”

Hevesy was 50 years old in 1935 when he turned his attention to applications of induced radioactivity. That year marks the beginning of the most important phase in his scientific career. [An inspiring thought to this 47-year-old writer.] From then on he very rarely published anything in physics; nearly all his further oeuvre, over 200 papers, deal with tracers in biology. That work was more influential than anything he had done before...

Hevesy certainly added to the life in Copenhagen. There is the story of the radioactive cat that had jumped out of a window of the institute for theoretical physics and was retrieved only after hours of hunting for it in the nearby park, when saliva tests on about a dozen captured cats showed one of them to be radioactive...

In 1944 Hevesy was awarded the 1943 Nobel Prize in chemistry “for his work on the use of isotopes as tracers in the study of chemical processes...”

In the post-war period the development of nuclear reactors provided for vastly enlarged production of radioisotopes. This in turn made possible much wider application of the tracer method in medicine. In hospitals all over the world one now finds special departments for nuclear medicine, a discipline which unquestionably was founded by Hevesy. Also from that period dates the use of numerous biological tracers with half-lives much longer that that of the popular P-32, notably the carbon isotope C-14 discovered in 1940.

Chapter 17 of the 4th edition of Intermediate Physics for Medicine and Biology discusses nuclear medicine, and the work of those “special departments for nuclear medicine.” Pais’s excellent biography makes clear that these important medical applications arose from the pioneering work of Neils Bohr and Georg Charles von Hevesy. Next week: the story continues, with more on Neils Bohr’s impact on biological physics.

Numerical Computing

My research involves computer simulations, so I spend a lot of time implementing numerical algorithms. I view a numerical technique as a tool for solving a problem, not as something intrinsically interesting itself. But as a numerical modeler it pays to become familiar with your tools, so I have.

The first question is which programming language to use? I use FORTRAN, and I’m sure that makes me a dinosaur in the eyes of many. But FORTRAN is still common among physicists, and has served me well since I first learned it as a senior in high school. If you look in the solution manual for the 4th edition of Intermediate Physics for Medicine and Biology, you will find a few programs written in FORTRAN. Russ Hobbie humored me by letting me write these in FORTRAN rather than C, although C appears in the textbook itself.

Numerical Recipes:
The Art of Scientific Computing,
by Press et al.

I tend to avoid software like Matlab, Mathematica, and Maple as being to “black-boxy.” I like to tinker with the guts of my program, and often you can’t do that with high-level software packages. Perhaps younger scientists disagree. I do use Matlab for graphics.

When I face a numerical problem that is new to me, the first place I look is Numerical Recipes: The Art of Scientific Computing, by Press, Teukolsky, Vetterling, and Flannery . The copy on my bookshelf is the 2nd edition of Numerical Recipes in FORTRAN 77: The Art of Scientific Computing. In the preface Press et al. write

We call this book “Numerical Recipes” for several reasons. In one sense, the book is indeed a “cookbook” on numerical computation. However there is an important distinction between a cookbook and a restaurant menu. The latter presents choices among complete dishes in each of which the individual flavors are blended and disguised. The former—and this book—reveals the individual ingredients and explains how they are prepared and combined.

Numerical Methods That Work,
by Forman Acton.

For a guide to some of the lore of numerical computing, I recommend two delightful books by Forman Acton: Numerical Methods that Work, and Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations. In Real Computing, Acton writes

This book addresses errors of the third kind. You’ve never heard of them? But you've made them; we all make them every time we write a computer program to solve a physical problem.

Errors of the first kind are grammatical—we write things that aren’t in our programming language. The compiler finds them.

Errors of the second kind are our mistakes in programming the algorithms we sought to use. They include n−1 errors, inversions of logical tests, overwriting array limits (in Fortran) and a lot of other little technical mistakes that just don’t happen to be ungrammatical. We have to find them.

Then, Mirabile visu, the program runs—and even gives the correct answers to the two test problems we happen to have already solved.

Errors of the third kind are the ones we haven’t found yet. They show up only for as-yet-untested input values—often for quite limited ranges of these parameters. They include (but are not limited to) loss of significant digits, iterative instabilities, degenerative inefficiences in algorithms and convergence to extraneous roots or previously docile equations. Since some of these errors occur only for limited combinations of parameters inputs they may never disturb our results; more likely some of them will creep into our answers, but so quietly that we don't notice them—until our bridge has collapsed!

Real Computing Made Real,
by Forman Acton.

In the rest of the book, Acton serves up a feast of tricks, tips, and techniques. Even if you don’t particularly like numerical methods, you will enjoy these books. Readers of Intermediate Physics for Medicine and Biology will find them useful when trying to write a computer program to solve the Hodgkin and Huxley equations or implement the Fourier method for computed tomography.

I end with a quote from Acton’s book that he attributes to Richard Hamming. It is one of my favorite quotes, and one I believe is worth repeating:

The purpose of computing is insight, not numbers.

Physicist Playing Cards

Physicist Playing Cards.

Last Christmas, my daughter Katherine gave me a unique and fascinating present: Physicist Playing Cards. Each card features a picture of an eminent physicist. You can order the cards from the American Institute of Physics at http://store.aip.org. Two decks are available: one of historical physicists, and one of modern physicists. I browsed through the deck of historical physicists and found ten cards that have relevance to biology and medicine. These ten physicists, each a Nobel Prize winner, contributed greatly to the application of physics to the life sciences. Many of these physicists are mentioned in the 4th edition of Intermediate Physics for Medicine and Biology (the appropriate page is given in parentheses).

Queen of Diamonds: Marie Curie, pioneer in the study of radiation (p. 489).

Ace of Diamonds: Wilhelm Roentgen, discoverer of X-rays (p. 440).

Nine of Clubs: William Bragg, analyzed crystal structures using X-ray diffraction (p. 466).

Three of Hearts: Felix Bloch, co-discoverer of nuclear magnetic resonance (p. 519).

Ten of Hearts: Henri Becquerel, discoverer of radioactivity (p. 472).

King of Hearts: Edward Purcell, co-discoverer of nuclear magnetic resonance (p. 519).

Ace of Hearts: Pierre Curie, discoverer of the piezoelectric effect (p. 216).

Eight of Spades: Frederic Joliot, co-creator of the first artificial radioisotope (none).

Nine of Spades: Max von Laue, discoverer of X-ray diffraction (none).

Queen of Spades: Irene Joliot Curie, co-creator of the first artificial radioisotope (none).

The Electrocardiogram

http://www.skillstat.com/ECG_sim_demo.html

When I teach biological physics, we spend at least one class discussing the electrocardiogram (ECG, sometimes called the EKG for the German term Elektrokardiogramm). The 4th edition of Intermediate Physics for Medicine and Biology covers the ECG in Chapter 7, but students often need additional practice on interpreting the signals from heart arrhythmias. As homework, I require my students to go to the website http://www.skillstat.com/ECG_sim_demo.html and play the ECG simulator game. It tests the students’ knowledge of ECGs in a fun, interactive way. The same website has units on cardiac anatomy, a cardiac dictionary, and cardiac trivia games.

Danny Smith Heart.

I have found that a plastic model of the heart is useful for teaching cardiac anatomy. I own a “Danny Smith Heart,” which clearly shows all the heart valves, vessels, chambers, and other anatomical features. Unfortunately, these plastic models tend to be expensive. Other good websites about the heart are http://www.visibleheart.com/, http://en.ecgpedia.org/wiki/Main_Page, and http://www.texasheartinstitute.org/HIC/Topics/inde x.cfm. There is also a good online video that explains the ECG.

The 1924 Nobel Prize in Physiology or Medicine was awarded to the Dutch physiologist Willem Einthoven (1860–1927) “for his discovery of the mechanism of the electrocardiogram.” The notation of the P, QRS, and T waves (see Fig. 7.17 of our textbook) was developed by Einthoven, as was his interpretation of the ECG using “Einthoven's triangle” (see Fig. 7.19, and page 188). Einthoven’s work is an excellent example of how physics can applied successfully to medicine and biology.

http://medicalphysicsweb.org

Readers of the 4th edition of Intermediate Physics for Medicine and Biology who are particularly interested in Medical Physics will find the website http://medicalphysicsweb.org interesting. This “community website” is maintained by the Institute of Physics, the United Kingdom’s professional organization for physicists. The IOP created several community websites to “promote innovation, growth and networking,... [and to] provide both a valuable information source and an international forum within which community members can share and exchange their views.”

Medicalphysicsweb provides “a mix of in-depth news, analysis, opinion and primary research papers across the key disciplines of medical physics.” The site contains editorials, job postings, a buyer’s guide, featured journal articles, and research and industry news. Students who are studying from Intermediate Physics for Medicine and Biology will find this website an easy and free way to become familiar with medical physics as a profession. You can become a member (no cost, but there is a registration procedure) and receive a weekly “What's New” newsletter. I highly recommend it. 


Note added in 2019: The website has changed to https://physicsworld.com/c/medical-physics. 

The "Big Three" Partial Differential Equations of Physics

Often a mathematical physics class will focus on the “big three” partial differential equations of physics: the diffusion equation, the wave equation, and Laplace’s equation. The 4th edition of Intermediate Physics for Medicine and Biology provides a gentle introduction to each of these equations. You won’t find much mathematical theory; Russ Hobbie and I are concerned primarily with exploring their relevance to biology and medicine.

• The diffusion equation (Eq. 4.24) is the topic of Chapter 4. Diffusion is not covered in many physics classes, but is crucially important to biology. The diffusion equation is also known by another name: the heat equation. Problem 24 of Chapter 4 explores an extension of this topic: a “reaction-diffusion” equation that governs the nonlinear propagation of calcium waves.
• The wave equation (Eq. 13.5) was not discussed much in previous editions of Intermediate Physics for Medicine and Biology, but it is an essential topic in the 4th edition's new Chapter 13 on sound and ultrasound. Here we meet classic wave behavior, such as the relationship between wavelength and frequency, the difference between propagating and standing waves, reflection, and the Doppler effect.
• Laplace’s equation (the displayed equation before Eq. 7.44a) is encountered in Chapter 7 on the electrocardiogram. We don’t talk as much about solutions to Laplace’s equation as we do the diffusion and wave equations, but it plays a fundamental role in electrostatics and appears when discussing steady-state solutions to the diffusion equation.

What other important partial differential equations are introduced in our book? The Navier-Stokes equation, governing fluid flow, is analyzed in Problem 28 of Chapter 1. The Helmholtz equation is presented under its pseudonyms: the cable equation (Chapter 6) and the linearized Poisson-Boltzmann equation (Chapter 9). Schrödinger's equation, the basic equation of quantum mechanics, is mentioned in the introduction to Chapter 3, but is never written down.

Peter Basser wins ISMRM Gold Medal for Diffusion Tensor Imaging

Earlier this month, at the 16th Scientific Meeting and Exhibition of the International Society for Magnetic Resonance in Medicine (ISMRM) in Toronto, Peter Basser was awarded an ISMRM Gold Medal for “his pioneering and innovative scientific contributions in the development of Diffusion Tensor Imaging (DTI).”

Peter is an old friend of mine from the days when we were both staff fellows in the now-defunct Biomedical Engineering and Instrumentation Program at the National Institutes of Health in Bethesda, Maryland. We collaborated on many projects, including a study of magnetic stimulation of nerves (for example, see: Roth BJ, Basser PJ. “A Model of the Stimulation of a Nerve Fiber by Electromagnetic Induction,” IEEE Transactions on Biomedical Engineering, Volume 37, Pages 588–597, 1990.)

Peter is now the head of the Section on Tissue Biophysics and Biomimetics, which is part of the Eunice Kennedy Shriver National Institute of Child Health and Human Development. The goal of his section is “to understand fundamental physical mechanisms governing tissue-level physiological processes that are essential for life, or necessary to achieve a high quality of life. Examples include understanding the physical basis of nerve excitability and of effective load bearing in cartilage. This entails discovering relationships between physiological function and a tissue's structure, organization, and physical properties. This is done by studying the behavior of biological model systems using novel quantitative approaches (e.g., experimental methods, mathematical models, physical models). Another aim of ours is to transfer these new methodologies to the biomedical research and healthcare communities. An example includes the invention and successful dissemination of diffusion tensor magnetic resonance imaging from the 'bench' to the ‘bedside.’”

Diffusion tensor imaging is one of the topics that Russ Hobbie and I added to the 4th edition of Intermediate Physics for Medicine and Biology (see Chapter 18, Section 13). We also wrote a new homework problem that asked the student to show that the trace of the diffusion tensor is independent of fiber direction. We had trouble deciding if this problem belonged in Chapter 4 (on diffusion) or Chapter 18 (on magnetic resonance imaging), and we ended up putting the problem in both chapters (see Problems 4.22 and 18.40). Another homework problem featuring Peter’s work on cartilage appears in Chapter 5 (Problem 5.6).

The Office of NIH History has published an interview with Peter, in which he explains how he developed diffusion tensor imaging. Below is a brief excerpt of this interview, describing the moment Peter conceived the idea of DTI (I make a cameo):

Actually, the first exposure I had to diffusion imaging was a talk that Denis Le Bihan had given. He had recently come to the NIH from France and talked about how diffusion could be used—I think it was in stroke—and I thought it was very interesting, but I didn’t really initially make a connection to it. But in the early 1990s, Denis Le Bihan and, I believe it was Philippe Douek had a poster presentation at one of the NIH research festivals off in a corner in one of the white tents that they had constructed over here in the parking lot East of Building 30. They had done something very novel. They had shown that they could color code different parts of the brain according to what they thought was the orientation of diffusion. That was a poster that resulted in a paper, I think early in the next year, by Denis and Philippe. But I visited that poster and I was there with my friend and colleague, Brad Roth, the guy I was doing the magnetic stimulation with, and I realized that there was something really fundamentally wrong with the approach that Denis and Philippe were using.

The rest, as they say, is history. One of Peter's first papers on DTI (Basser PJ, Mattiello J, LeBihan D. “MR Diffusion Tensor Spectroscopy and Imaging,” Biophysical Journal, Volume 66, Pages 259–267, 1994) has been cited over 700 times according to the ISI Web of Knowledge. His coauthors were Denis Le Bihan (a previous ISMRM Gold Medal Winner) and James Mattiello (the first graduate of the Oakland University Medical Physics PhD Program). The technique is now widely used to map fiber orientation in the brain and the heart.

Congratulations Peter! 

 Listen to Peter Basser describe the invention and development of Diffusion Tensor Imaging.
https://www.youtube.com/watch?v=1_BeCeDak3w