A Present From Santa

Santa arrived last night and left you, dear reader, a present in your stocking: two new homework problems for the 4^th edition of Intermediate Physics for Medicine and Biology. The problems belong to Chapter 8 on Biomagnetism (one of my favorite chapters), and specifically to Section 8.6 on Electromagnetic Induction. They both explore the idea of skin depth, but from somewhat different perspectives. Please forgive Santa for being a bit long-winded; he got carried away.

Enjoy.

Section 8.6

Problem 25.1 The concept of “skin depth” plays a role in some biomagnetic applications.
(a) Write Ampere’s law (Eq. 8.22) for the case when the displacement current is negligible.
(b) Use Ohm’s law (Eq. 6.26) to write the result from (a) in terms of the electric field.
(c) Take the curl of both sides of the equation you found in (b) (Assume the conductivity σ is homogeneous and isotropic).
(d) Use Faraday’s law (Eq. 8.20), ∇·B=0 (Eq. 8.7), and the vector identity ∇×(∇×B)=∇(∇·B)-∇²B to simplify the result from (c).
(e) Your answer to (d) should be the familiar diffusion equation (Eq. 4.24). Express the diffusion constant D in terms of electric and magnetic parameters.
(f) In Chapter 4, we found that diffusion over a distance L takes a time T equal to L²/2D. During transcranial magnetic stimulation, L=0.1 m, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. How long does the magnetic field take to diffuse into the head? Is this time much longer than or much shorter than the rise time of the magnetic field for the stimulator designed by Barker et al. (1985)?
(g) Solve T= L²/2D for L, using the expression for D found in (e). Calculate L for T=0.1 ms. Is L much larger than or much smaller than the size of your head? L is closely related to the “skin depth” defined in electromagnetic theory.
(h) During magnetic resonance imaging (see Chapter 18), an 85 MHz radio-frequency magnetic field is applied to the body. Calculate L using half a period for T. How does L compare to the size of the head? The frequency of the RF field is proportional to the strength of the static magnetic field in an MRI device, and 85 MHz corresponds to 2 T. If the static field is 7 T (common in modern high-field MRI), calculate L. Is it safe to ignore skin depth during high-field MRI?

Problem 25.2 During magnetic stimulation, a changing magnetic field B induces eddy currents in the body that produce their own magnetic field B'. The goal of this problem is to compare B' and B. We can estimate B' using the following approximations. First, ignore the vector nature of all fields and do not distinguish between components. Second, ignore all negative signs. Third, replace all time derivatives with multiplication by 1/T, where T is a characteristic time. Fourth, replace all space derivatives (such as the curl) by multiplication with 1/L, where L is a characteristic length.
(a) Use Faraday’s law (Eq. 8.20) to estimate the induced electric field E from B.
(b) Use Ohm’s law (Eq. 6.26) to estimate the current density J from E.
(c) Use Ampere’s law (Eq. 8.22, but ignore displacement currents) to estimate B' from J.
(d) Combine parts (a), (b), and (c) to determine an expression for the ratio B'/B in terms of the conductivity σ, the permeability μ_o, L, and T.
(e) In magnetic stimulation, L=0.1 m, T=0.1 ms, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. Calculate B'/B. Is it safe to ignore B' compared to B during magnetic stimulation?

Where's Albert?

Albert Einstein is considered one of the greatest physicists of the 20^th century, and perhaps of all time. He certainly is one of the best-known physicists, being selected by TIME Magazine as their Person of the Century in 1999. Yet, Einstein is curiously absent in the 4^th edition of Intermediate Physics for Medicine and Biology. If you look in the index under Einstein, you find only one entry: on page 393, where Russ Hobbie and I introduce the unit of an einstein (a mole of photons) in a homework problem.

Does Einstein’s work appear anywhere else in Intermediate Physics for Medicine and Biology? Certainly his masterpiece, the general theory of relativity, has little or no direct impact on biology or medicine. I don’t believe we even refer indirectly to this monumental description of gravity. However, Einstein’s earlier theory, special relativity, does appear occasionally in our book. In Chapter 8 on Biomagnetism, we write “the appearance of the magnetic force is a consequence of special relativity,” a topic we explore further in Homework Problems 5 and 23. Yet, the relationship between electrodynamics and relativity is mentioned as an aside, and is not a central feature of our analysis of magnetism. We could have left out mention of relativity from Chapter 8 altogether, and the rest of the chapter would be unaffected.

Special relativity enters in a more profound way in Chapter 15, on the Interaction of Photons and Charged Particles with Matter. There, we analyze Compton Scattering, and need the relationship between photon energy E and momentum p, given by special relativity as E = pc, where c is the speed of light. Moreover, the concept of rest mass m is introduced in this chapter, and we use Einstein’s most famous equation E = mc², relating energy and mass. Rest mass appears again in the discussion of pair production, where enough photon energy must be present to produce an electron-positron pair. The equation appears one more time in Chapter 17 on Nuclear Physics and Nuclear Medicine, where mass can be converted into energy in nuclear reactions.

Besides relativity, Einstein also played a leading role in the development of quantum mechanics, especially as related to the quantization of light and the idea of photons. This idea is first presented in Chapter 9, in a section on the Possible Effects of Weak External Electric and Magnetic Fields, where we compare the photon energy (equal to Planck’s constant times the frequency of the radiation) to the thermal energy. The idea is developed in more detail in Chapter 14, in a section about The Nature of Light: Waves versus Photons. The idea of photons is central to Chapter 15, and particularly Sec. 15.2 on Photon Interactions. There, we discuss the photoelectric effect—one mechanism by which x rays interact with tissue—which is the research that won Einstein the Nobel Prize.

One final place where Einstein’s research impacts Intermediate Physics for Medicine and Biology is in the study of diffusion (Chapter 4). Einstein did fundamental work on diffusion in his doctoral thesis, and derived a relationship between the diffusion constant and the viscosity that we give as Eq. 4.23.

Subtle is the Lord,
by Abraham Pais.

In summary, we rarely mention Einstein by name in our book, but his influence is present throughout, and most fundamentally when we discuss the idea of a photon. For readers interested in Einstein’s life and work, I recommend the brilliant biography Subtle is the Lord by Abraham Pais. I have heard good things about Isaacson’s more recent biography, Einstein: His Life and Universe, although I haven’t read it. You might also enjoy the American Institute of Physics website about Einstein prepared by the AIP Center for the History of Physics. Einstein published most of the ideas I have discussed in one miraculous year, 1905. John Rigden describes these publications and their impact in his book Einstein 1905: The Standard for Greatness (I have not read this book either, but I understand it is good). Finally, the equation E = mc² has received a lot of press recently, including a NOVA special and Bodanis’s book E=mc²: A Biography of the World’s Most Famous Equation.

Error Function

In the November 6th entry to this blog, I mentioned one special function introduced in the 4^th edition of Intermediate Physics for Medicine and Biology: the Bessel function. Another special function Russ Hobbie and I discuss briefly is the error function, which arises naturally when solving the one-dimensional cable equation (Chapters 6 and 7) or the diffusion equation (Chapter 4). The error function is the integral of the familiar Gaussian function, and has a sigmoidal shape, being minus one for large negative values of its argument and one for large positive values.

Handbook of Mathematical Functions
with Formulas, Graphs, and
Mathematical Tables,
by Abramowitz and Stegun.

To learn more about the error function, see the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables by Milton Abramowitz and Irene Stegun (1972). This classic math handbook is available online at http://www.math.ucla.edu/~cbm/aands//. Also, Wikipedia has a very thorough article about the error function, including beautiful plots of the error function in the complex plane.
I’m not sure how the error function got its name. Perhaps it has something to do with experimental errors often being Gaussianly distributed. If anyone knows, please let me know.


P.S. Speaking of errors: For any students or instructors preparing to use the 4^th edition of Intermediate Physics for Medicine and Biology next semester, I recommend you download the errata, which can be found at https://sites.google.com/view/hobbieroth. In it, Russ Hobbie and I list all known errors in our book. The number of errors has grown, and in particular some are present in homework problems. Generally I frown on writing in my books, but in this case do yourself a favor: download the errata and mark the corrections in your copy of the text. And as always, let us know if you find additional errors. The only thing worse than finding errors in a book you wrote is having errors in a book you wrote that you are not even aware of.

P.P.S. I have written in this blog about Steven Strogatz, a mathematician and author, and about Kleber’s law, which relates metabolic rate to body mass. Here is an article by Strogatz about Kleber’s law. It doesn’t get much better than that!

Hot Tubs and Heat Stroke

In Chapter 10 (about Feedback and Control) of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss hot tubs and heat stroke.

The body perspires in order to prevent increases in body temperature. At the same time blood flows through vessels near the surface of the skin, giving the flushed appearance of an overheated person. The cooling comes from the evaporation of the perspiration from the skin. If the perspiration cannot evaporate or is wiped off, the feedback loop is broken ad the cooling does not occur. If a subject in a hot tub overheats, the same blood flow pattern and perspiration occur, but now heat flows into the body from the hot water in the tub. The feedback has become positive instead of negative, and heat stroke and possibly death occurs.

Were we overly alarming about hot tubes? Not according to an article by Nicholas Bakalar in the November 23rd issue of the New York Times, which indicates hot tub accidents are a growing problem.

A hot tub might not seem an especially dangerous place, but over a period of 18 years, 1990–2007, more than 80,000 people were injured in hot tubs or whirlpools seriously enough to wind up in an emergency room. Almost 74 percent of the injuries occurred at home… About half the injuries were caused by slipping or falling, but heat overexposure was the problem in 10 percent of the accidents, and near-drowning in about 2.5 percent. Almost 7 percent of the injuries were serious enough to require hospitalization… The Consumer Products Safety Commission reported more than 800 deaths associated with hot tubs since 1990, nearly 90 percent of them in children under age 3.

This means that about 8000 people suffered from heat stroke accidents in hot tubs over 18 years, or over one per day. Perhaps a better understanding of biological thermodynamics and feedback loops has more than merely academic value.

To learn more, see “Death in a Hot Tub: The Physics of Heat Stroke,” by Albert Bartlett and Thomas Braun (American Journal of Physics, Volume 51, Pages 127–132, 1983).

What’s Wrong With These Equations?

The 4^th edition of Intermediate Physics for Medicine and Biology is full of equations: thousands of them. Each one must fit into the text in a way to make the book easy to read. How?

N. David Mermin wrote a fascinating essay that appeared in the October 1989 issue of Physics Today titled “What’s Wrong With These Equations?” You can find it online at www.cvpr.org/doc/mermin.pdf. It begins

A major impediment to writing physics gracefully comes from the need to embed in the prose many large pieces of raw mathematics. Nothing in freshman composition courses prepares us for the literary problems raised by the use of displayed equations.

Mermin then presents three rules “that ought to govern the marriage of equations to readable prose”

• Rule 1 (Fisher’s rule): Number all displayed equations.


• Rule 2 (Good Samaritan rule): When referring to an equation identify it by a phrase as well as a number.


• Rule 3 (Math is Prose rule): End a displayed equation with a punctuation mark.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I violate Fisher’s rule: some of our displayed equations are not numbered. All I can say is, there are lots of equations in our book, and revising it to obey Fisher’s rule would require more effort than we are willing to expend.

I know you are wondering how an essay about punctuating and numbering equations could possibly be interesting, but Mermin makes the subject entertaining. And if you ever find yourself writing an article that contains equations, obeying his three rules will make the article easier to read.

Boojums All the Way Through,
by N. David Mermin.

Many physicists know Mermin for his renowned textbook Solid State Physics with Neil Ashcroft. His series of “Reference Frame” essays in Physics Today are all delightful, particularly the ones with Professor Mozart. Several Reference Frame essays are reprinted in his book Boojums All the Way Through: Communicating Science in a Prosaic Age. The title essay describes Mermin’s quest to establish the whimsical word “Boojum” as a scientific term for a phenomenon in superfluidity. If you want to learn to write physics well, read Mermin.

The Feynman Lectures

The Feynman Lectures on Physics,
by Richard Feynman.

On page 318 of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite The Feynman Lectures on Physics. Reading The Feynman Lectures, written by Nobel Prize winner Richard Feynman, is a rite of passage for future physicists. Feynman describes how he came to present the lectures in his preface:

These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory section once a week.

Although written for freshman and sophomores, most physics students read The Feynman Lectures a bit later in their education. I recall reading them the summer between graduation from the University of Kansas and starting graduate school at Vanderbilt University. There is some biological and medical physics in the lectures. For instance, Chapters 35 and 36 of Volume 1 are about vision and the eye. In Chapter 3 (The Relation of Physics to Other Subjects), Feynman describes his reductionist point of view about biology:

Certainly no subject or field is making more progress on so many fronts at the present moment, than biology, and if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of jigglings and wigglings of atoms.

And in Volume 2 (Chapter 1), Feynman had this to say about the impact of electricity and magnetism on life:

Now we realize that the phenomena of chemical interaction and, ultimately, of life itself are to be understood in terms of electromagnetism.

He closed that chapter with a favorite quote of mine:

From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

Anyone wondering what to get an aspiring physicist for a holiday gift might want to consider The Feynman Lectures. If you are looking for lighter reading, I suggest two autobiographical books by Feynman: Surely You’re Joking Mr. Feynman, and What Do You Care What Other People Think? They’re delightful and hilarious.

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles

Quantum Physics of Atoms,
Molecules, Solids, Nuclei, and Particles,
by Eisberg and Resnick.

One of the sources that Russ Hobbie and I cite most often in the 4th edition of Intermediate Physics for Medicine and Biology is Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Robert Eisberg and Robert Resnick. I used the first (1974) edition of this textbook when I was an undergraduate studying physics at the University of Kansas. Quantum Physics was the book where I was first introduced to the ideas of quantum mechanics, to the Schrodinger equation, and to nuclear physics. A second edition was published in 1985, but I can find nothing about a third edition in the last 25 years. Despite it being somewhat out-of-date, I still consider this book to be one of the best sources of information about modern physics. Below is the first paragraph of the preface:

The basic purpose of this book is to present clear and valid treatments of the properties of almost all the important quantum systems from the point of view of elementary quantum mechanics. Only as much quantum mechanics is developed as is required to accomplish the purpose. Thus we have chosen to emphasize the applications of the theory more than the theory itself. In so doing we hope that the book will be well adapted to the attitudes of contemporary students in a terminal course on the phenomena of quantum physics. As students obtain an insight into the tremendous explanatory power of quantum mechanics, they should be motivated to learn more about the theory. Hence, we hope that the book will be equally well adapted to a course that is to be followed by a more advanced course in formal quantum mechanics.

I have never taught the modern physics class here at Oakland University, but if I did I would certainly consider using Eisberg and Resnick’s book. When I have taught the undergraduate quantum mechanics class (taken after modern physics) I used another wonderful book, Introduction to Quantum Mechanics by David Griffiths. There are several good quantum mechanics books at the graduate level, but I—a biomedical physicist—have never been asked to teach graduate quantum mechanics. (Are they telling me something?)

Intermediate Physics for Medicine and Biology doesn’t make much use of quantum ideas, except at a very qualitative level. Schrodinger’s equation is only mentioned once (on page 49), and is never written out. The idea of discrete quantum energy levels is introduced in Chapter 3 when we discuss statistical mechanics, and again in Chapter 14 when explaining atomic spectra. However, concepts related to quantization of light are important. For instance, thermal (blackbody) radiation is discussed in Section 14.7 (and is covered elegantly in the first chapter of Eisberg and Resnick) and Compton scattering is analyzed in Sec. 15.4. Quantum Physics should provide all the background you will need to understand these and other modern physics topics.

Clark and Plonsey

Problem 30 in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology is based on a paper by John Clark and Robert Plonsey (“The Extracellular Potential of a Single Active Nerve Fiber in a Volume Conductor,” Biophysical Journal, Volume 8, Pages 842–864, 1968). This paper shows how to calculate the extracellular potential from the transmembrane potential, with results shown in our Fig. 7.13.

The calculation involves some mathematical concepts that are slightly advanced for Intermediate Physics for Medicine and Biology. First, the potentials are written in terms for their Fourier transforms. Russ Hobbie and I don’t cover Fourier analysis until Chapter 11, so the problem just assumes a sinusoidal spatial dependence. We also introduce Bessel functions for the first time in the book (to be precise, modified Bessel functions of the first and second kind). Bessel functions arise naturally when solving Laplace’s equation in cylindrical coordinates.

I have admired Clark and Plonsey’s paper for years, and was glad to see this problem introduced into the 4th edition of our book. Robert Plonsey was a professor at Case Western Reserve University from 1968-1983. He then moved to Duke University, where he was when I came to know his work while I was a graduate student. I am most familiar with his research on the bidomain model of cardiac tissue, often in collaboration with Roger Barr (e.g., “Current Flow Patterns in Two-Dimensional Anisotropic Bisyncytia with Normal and Extreme Conductivities,” Biophysical Journal, Volume 45, Pages 557–571 and “Propagation of Excitation in Idealized Anisotropic Two-Dimensional Tissue,” Biophysical Journal, Volume 45, Pages 1191–1202). Plonsey was elected as a member of the National Academy of Engineering in 1986 for “the application of electromagnetic field theory to biology, and for distinguished leadership in the emerging profession of biomedical engineering.” He retired from Duke in 1996 as the Pfizer Inc./Edmund T. Pratt Jr. University Professor Emeritus of Biomedical Engineering. He has won many awards, such as the 2000 Millennium Medal from the IEEE Engineering in Medicine and Biology Society and the 2004 Ragnar Granit Prize from the Ragnar Granit Foundation. John Clark is currently a Professor of Electrical and Computer Engineering at Rice University. He is a Life Fellow in the Institute of Electrical and Electronics Engineers (IEEE) “for contributions to modeling in electrophysiology and cardiopulmonary systems.”

One of my earliest papers was an extension of Clark and Plonsey’s model to a strand of cardiac tissue, using the bidomain model (“A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue,” IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467–469, 1986.) The mathematics is almost the same as in their paper—Fourier transforms and Bessel functions—but the difference is that I modeled a multicellular strand of tissue, like a papillary muscle in the heart, that contains of both intracellular and interstitial spaces (the two domains of the “bidomain” model). A comparison of my paper to Clark and Plonsey’s earlier work indicates how influential their research was on my early development as a scientist. They were cited in the first sentence of my paper.

Hobbie and Roth, Back in the Saddle Again

In the November, 2009 issue of the American Journal of Physics, Russ Hobbie and I published “Resource Letter MP-2: Medical Physics.” Our resource letter “provides a guide to the literature on the uses of physics for the diagnosis and treatment of disease.” Think of it (along with Ratliff’s “Resource Letter MPRT-1: Medical Physics in Radiation Therapy” discussed in the August 28, 2009 entry to this blog) as an updated bibliography to the 4th edition of Intermediate Physics for Medicine and Biology. Together, these two publications provide over 300 citations to the best and most recent books, articles, and websites about medical physics. We even slipped a mention of this blog into the list of references.

Felix Bloch

One hundred and four years ago today, Felix Bloch (1905–1983) was born in Zurich, Switzerland. Bloch received his PhD in physics in 1928 from the University of Leipzig working under Werner Heisenberg, and then immigrated to the United States after Hitler came to power in Germany. He worked for a time at Los Alamos on the Manhattan Project, and had a long career in the Physics Department at Stanford University.

Bloch is most familiar to readers of the 4th edition of Intermediate Physics in Medicine and Biology because of his contributions to our understanding of nuclear magnetic resonance. He shared the 1952 Nobel Prize with Edward Purcell for “their development of new ways and methods for nuclear magnetic precision measurements.” In Chapter 18 on Magnetic Resonance Imaging, Russ Hobbie and I present the Bloch Equations (Eq. 18.15), which govern the magnetization of a collection of spins in a static magnetic field. Essentially all of MRI begins with the Bloch equations, so they are part of the essential toolkit for any medical physicist. Bloch’s most cited paper is “Nuclear Induction” (Physical Review, Volume 70, Pages 460–474, 1946). The abstract is reproduced below.

The magnetic moments of nuclei in normal matter will result in a nuclear paramagnetic polarization upon establishment of equilibrium in a constant magnetic field. It is shown that a radiofrequency field at right angles to the constant field causes a forced precession of the total polarization around the constant field with decreasing latitude as the Larmor frequency approaches adiabatically the frequency of the r-f field. Thus there results a component of the nuclear polarization at right angles to both the constant and the r-f field and it is shown that under normal laboratory conditions this component can induce observable voltages. In Section 3 we discuss this nuclear induction, considering the effect of external fields only, while in Section 4 those modifications are described which originate from internal fields and finite relaxation times.

Bloch also appears in Chapter 15 of Intermediate Physics for Medicine and Biology, because of his contribution to the development of the Bethe-Bloch formula (Eq. 15.58) governing the stopping power of a charged particle by interaction with a bound electron. He is also known for his fundamental contributions to solid state physics, including his seminal calculation of the electron wave function in a periodic potential, derived when he was only 23. You can download a Biographical Memoir about Bloch by Robert Hofstadter at books.nap.edu/html/biomems/fbloch.pdf.

I have an indirect connection to Felix Bloch. When in graduate school at Vanderbilt University in the 1980s, I had several classes from Ingram Bloch, who—if I recall correctly—was Felix’s cousin. At that time, Ingram Bloch was teaching many of the graduate classes, so I took classical mechanics, two semesters of quantum mechanics, and general relativity from him. I remember spending days working on his infamous “take-home” exams. They weren’t easy. With two physicists in the family, the Blochs made quite an impact on 20th century physics.


P.S. Right now, amazon.com has the 4th edition of Intermediate Physics for Medicine and Biology on sale at 40% off. I have no control over if and when amazon reduces prices on books, so the price may go back up anytime.

P.P.S. Last night I finished Steven Strogatz’s book The Calculus of Friendship: What a Teacher and a Student Learned About Life While Corresponding About Math, mentioned in the July 3rd entry to this blog. In a word, the book is charming.