The Psychic Probe

Foundation,
by Isaac Asimov.

In the summer my wife and I sometimes take long car trips, and I often listen to audiobooks while I drive to keep me awake and alert. During a recent trip I listened to Isaac Asimov’s Foundation trilogy. Regular readers of this blog know that I’m a huge Asimov fan. I first read the Foundation series about forty years ago, and this was my third or fourth time through these delightful books.

In brief, the Foundation series tells the history of the decaying galactic empire, and describes the work of the psychohistorian Hari Seldon who has calculated mathematically how to reduce the duration of the dark ages following the empire’s fall from 30,000 years to merely 1000. All goes according to plan until the Mule, a mutant who can control other people’s emotions, causes all to go awry.

Foundation and Empire,
by Isaac Asimov.

One of Asimov’s inventions in this future history is a device that can read minds, called the Psychic Probe. He writes in Foundation and Empire,

The general threw away his shredded, never-lit cigarette, lit another, and shrugged. “Well, it is beside the immediate point, this lack of first-class tech-men. Except that I might have made more progress with my prisoner were my Psychic Probe in proper order.”

The secretary’s eyebrows lifted. “You have a Probe?”

“An old one. A superannuated one which fails me the one time I needed it. I set it up during the prisoner’s sleep, and received nothing. So much for the Probe. I have tried it on my own men and the reaction is quite proper, but again there is not one among my staff of tech-men who can tell me why it fails upon the prisoner. Ducem Barr, who is a theoretician of parts, though no mechanic, says the psychic structure of the prisoner may be unaffected by the Probe since from childhood he has been subjected to alien environments and neural stimuli. I don’t know. But he may yet be useful. I save him in that hope.” 

Second Foundation,
by Isaac Asimov.

Russ Hobbie and I don’t mention the Psychic Probe in the 5th edition of Intermediate Physics for Medicine and Biology … or do we? Asimov didn’t explain the physical mechanism behind the Probe, but I can speculate. Four candidates are:

• Magnetoencephalography. The probe could be a form of MEG, as discussed in Section 8.5 of IPMB. The probe would then contain a sensitive magnetic field detector, like modern SQUID magnetometers, that could record brain activity whether we wanted it to or not. 
• Functional Magnetic Resonance Imaging. Section 18.12 of IPMB describes how fMRI can use magnetic field heterogeneities caused by oxygenated hemoglobin to detect brain activity. Essentially, one measures blood flow, which is related to neural firing. 
• Positron Emission Tomography. PET images the distribution of injected positron emitters (Section 17.10 of IPMB). For example, the isotope fluorine-18 can be attached to molecules of glucose, which are then taken up by parts of the brain that are metabolically active and become trapped inside neurons. 
• Transcranial Magnetic Stimulation. In Section 8.7 of IPMB, Russ and I discuss using a rapidly changing magnetic field to excite nerve axons. TMS can disrupt brain function, and has even been suggested as a treatment for depression. 

Asimov's Foundation Trilogy.

PET and fMRI are too slow to accurately follow rapid brain activity. PET detects brain metabolism and fMRI detects blood flow, both of which are only indirectly related to neuron firing. My best guess for the Psychic Probe is some combination of MEG and TMS. Apparently the probe can damage the brain when used aggressively, which suggests TMS. But it can also read minds when used more gently, which points toward MEG. A combo TMS/MEG unit could therefore both detect and alter brain function.

While working at NIH in the 1990s, I studied both magnetoencephalography and transcranial magnetic stimulation. Yikes! I may be partially responsible for the invention of the Psychic Probe!

Kramers’ Law

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a homework problem about Kramers’ law. (We spelled it Kramer’s, but his name is Kramers with an s, so we should have written Kramers’.) Kramers’ law is Eq. 16.3a, the photon energy fluence dΨ/d(hν) as a function of frequency ν for bremsstrahlung radiation

where Z is the atomic number, h is Planck’s constant, ν_o is the frequency of a photon having the same energy as the incident electrons, and C is a constant. In his paper “On the Theory of X-ray Absorption and of the Continuous X-ray Spectrum” (Philosophical Magazine, Volume 46, Pages 836–871, 1923), Kramers writes

The continuous x-ray spectrum has in the course of the last years been investigated by a number of physicists. The problem is here to determine how, for a given tension [voltage] on the tube and a given anticathode material [typically tungsten], the energy in the continuous spectrum is distributed among different frequencies…

The object of the present paper is to show how it is possible to account theoretically for the main features of the phenomena of x-ray absorption and continuous x-ray emission discussed above. The explanation of these phenomena may be traced back to the determination of the radiation processes which may occur when a free electron of given velocity approaches a positive nucleus with given charge.

Who was Kramers? According to the Dictionary of Scientific Biography, Hendrik Anthony (Hans) Kramers was born in Rotterdam, the Netherlands in 1894. He joined Niels Bohr’s Institute of Theoretical Physics, and in 1934 he moved to Leiden University, where he remained until his death in 1952. He’s known for many contributions to physics, including the Kramers-Kronig relations. The Dictionary of Scientific Biography article concludes

Kramers’ work, which covers almost the entire field of theoretical physics, is characterized both by outstanding mathematical skill and by careful analysis of physical principles. It also leaves us with the impression that he tackled problems because he found them challenging, not primarily because they afforded chances of easy success. As a consequence his work is somewhat lacking in spectacular results that can be easily explained to a layman; but among fellow theoreticians he was universally recognized as one of the great masters.

A Tale of Two Continents,
by Abraham Pais.

Here is my favorite Kramers story. Jewish physicist Abraham Pais described in his autobiography A Tale of Two Continents how he spent much of World War II in Holland hiding from the Gestapo. Kramers was one of the few people who knew of his hiding place, and would visit him weekly to talk physics. One day when Kramers was there, Gestapo agents knocked at the door and Pais had to hide in a small enclosure behind a panel in the wall. Pais writes

I kept sitting in the tiny space, practically bent over double, holding onto the panel, when I heard the door to my room, which lay at the other side of my hiding spot, open softly. Someone entered, I did not at first know who. Then that person sat down on a small bench that stood right at the wall behind which I was folded up. The person began to read, not loud but quite softly. It was Kramers. Earlier he had lent me a volume of Bradley’s Lectures on Shakespeare. What this good man was doing now was reading to me from that book, in order to calm my nerves.

The Divergence Theorem and Stokes’ Theorem

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem to Chapter 4.

Problem 4. Integrate Eq. 4.8 over a volume and subtract the result from Eq. 4.4. The resulting relationship is called the divergence theorem.

For those of you who don’t keep a copy of IPMB always close at hand (what’s the matter with you?), Eq. 4.8 is

where “div j” is the divergence of the vector j, and Eq. 4.4 is

When you put the two equations together, you get (spoiler alert) ,

also known as the divergence theorem. It is one of the fundamental results of vector calculus.

Div, Grad, Curl, and All That,
by H. M. Schey.

If you want to learn more about the divergence theorem, I recommend H. M. Schey’s book Div, Grad, Curl and All That. He writes

For the remainder of this chapter we digress from the mainstream of our narrative to discuss a famous theorem that asserts a remarkable connection between surface integrals and volume integrals. Although this relation may be suggested by the work we have done in electrostatics, the theorem is a mathematical statement holding under quite general circumstances. It is independent of any physics and is applicable in many different places. It is called the divergence theorem, and sometimes Gauss’ theorem… It says that the flux of a vector function through some closed surface equals the triple integral of the divergence of that function over the volume enclosed by the surface.

Besides the divergence theorem, another basic tenet of vector calculus is Stokes’ theorem. Can we make a similar homework problem demonstrating that? Yes! Here is a new problem for Chapter 8.

Problem 23 ½. Integrate Eq. 8.22 over a surface and subtract the result from Eq. 8.21. The resulting relationship is called Stokes’ theorem.

If you don’t have IPMB handy, Eq. 8.21 is

and Eq. 8.22 is

where “curl E” is the curl of the vector E. When you put the two equations together, you get

 Schey writes

We [now] discuss another famous theorem, one strongly reminiscent of the divergence theorem and yet, as we’ll see, quite different from it. This theorem, named for the mathematician Stokes, relates a line integral around a closed path to a surface integral over what is called a capping surface of the path…In words, Stokes’ theorem says that the line integral of the tangential component of a vector function over some closed path equals the surface integral of the normal component of the curl of that function integrated over any capping surface of the path.

The divergence theorem and Stokes’ theorem are a bit too mathematical to develop in IPMB, with its emphasis on biological and medical applications. Yet there they are, implicit in our discussions of diffusion and of transcranial magnetic stimulation. If you want to learn more, start with Schey’s wonderful (and relatively inexpensive) book Div, Grad, Curl.

So You Don’t Like Error Functions?

In Chapter 4 of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the diffusion equation (Equation 4.26)

where C is the concentration and D is the diffusion constant. We then study one-dimensional diffusion where initially (t = 0) the region to the left of x = 0 has a concentration C_o, and the region to the right has a concentration of zero (Section 4.13). We show that the solution to the diffusion equation is (Equation 4.75)

where erf is the error function.

Some students don’t like error functions (really?). Moreover, often we can gain insight by solving a problem in several ways, obtaining solutions that are seemingly different yet equivalent. Let’s see if we can solve this problem in another way and avoid those pesky error functions. We will use a standard method for solving partial differential equations: separation of variables. Before I start, let’s agree to solve a slightly different problem: initially C(x,0) is C_o/2 for x less than zero, and −C_o/2 for x greater than zero. I do this so the solution will be odd in x: C(x,t) = −C(−x,t). At the end we can add the constant C_o/2 and get back to our original problem. Now, let’s begin.

Assume the solution can be written as the product of a function of only t and a function of only x: C(x,t) = T(t) X(x). Plug this into the diffusion equation, simplify, and divide both sides by TX 

The only way that the left and right hand sides can be equal at all values of x and t is if both are equal to a constant, which I will call −k². This gives two ordinary differential equations

 and

The solution to the first equation is an exponential

and the solution to the second is a sine

There is no cosine term because of the odd symmetry. Unfortunately, we don’t know the value of k. In fact, our solution can be a superposition of infinitely many values of k

where A(k) specifies the weighting.

To determine A(k), use the Fourier techniques developed in Chapter 11. The result is

How did I get that? Let me outline the process, leaving you to fill in the missing steps. I should warn you that a mathematician would worry about the convergence of the integrals we evaluate, but you and I’ll brush those concerns under the rug.

At t = 0, our solution becomes

Except for a missing factor of 2π, this looks just like the Fourier transform from Section 11.9 of IPMB. Next, multiply each side of the equation by sin(ωx), and integrate over all x. Then, use Equation 11.66b to express the integral of the product of sines as a delta function. You get

Both C(x,0) and sin(kx) are odd, so their product is even, and for x greater than zero C(x,0) is −C_o/2. Therefore,

You know how to integrate sine (I hope you do!), so

Here is where things get dicey. We don’t know what cosine equals at infinity, but if we say it averages to zero the first term goes away and we get our result

Plugging in this expression for A(k) gives our solution for C(x,t). If we want to go back to our original problem with an initial condition of C_o on the left and zero on the right, we must add C_o/2. Thus

Let’s compare this solution with the one in Equation 4.75 (given above). Our new solution does not contain the error function! Those of you who dislike that function can celebrate. Unfortunately, we traded the error function for an integral that we can’t evaluate in closed form. So, you can have a function that you may be unfamiliar with and that has a funny name, or you can have an expression with common functions like the sine and the exponential inside an integral. Pick your poison.

Boston

Boston, by Boston.

I did something unusual for me last evening: I went to a rock concert. As a birthday present, my daughter Stephanie took me to Freedom Hill Amphitheater to listen to the band Boston.

I was in high school in 1976 when I bought Boston’s famous debut album. That was a big year: it was the bicentennial of the United States, Jimmy Carter was elected president, Nadia Comaneci was earning 10s and then-Bruce Jenner won the decathlon in the Olympic games, much to my chagrin the Cincinnati Reds won the world series (but it wasn’t quite as exciting a series as the year before, which was the best world series ever), and the Apple Computer Company was formed by Steve Jobs and Steve Wozniak. My family moved from Fort Wayne, Indiana to Ashland, Ohio, and I spent the year playing tuba in the high school band, managing the high school baseball team, reading my first Isaac Asimov book, wondering if I should study physics in college, and listening to Chicago, the Eagles, Peter Frampton, the Wings, and Boston. The severe winter of 1976-1977 in northern Ohio and the simultaneous energy crisis resulted in my high school missing several weeks of classes, so some of my friends and I had time to establish our own garage band: “Hades.” We didn’t have a singer, and my role was to pick out the melody on an electric keyboard while the guitars and drums banged out behind me. Only a few years later disco music and the Bee Gees drove me from rock to country music, which I have listened to ever since.

My ears are still tingling a bit from the concert. How loud was it? Chapter 13 in the 5th edition of Intermediate Physics for Medicine and Biology discusses the decibel scale for measuring sound intensity, a logarithmic scale defined as log₁₀(I/I_o), where the sound intensity is
-->I and the reference I_o is the minimum perceptible sound (10⁻¹² W m⁻²). Table 13.1 in IPMB says 120 dB is the threshold for pain, and 130 dB is typical for the peak sound at a rock concert. Stephanie and I were sitting in the back of the amphitheater, so I doubt we ever experienced 130 dB, but we were up there pretty high on the decibel scale. I probably didn’t lose any hair cells in my cochlea (see Section 13.5), but I wonder how the band plays concerts night after night without suffering hearing loss. As people age, they tend to lose the ability to hear high tones: presbycusis. I may not have heard the music last night in the same way I heard it in 1976; some of those frequencies may be lost to me forever.
The leader of Boston is Tom Scholz, their 68-year-old guitar player and keyboardist. Scholz was educated as a mechanical engineer, and is one of the few rock musicians who might enjoy reading IPMB. I found that I could identify with Scholz in some respects: he is past his prime, no longer topping the charts or breaking new ground in rock and roll. But, after decades in the business, he’s still out there performing, playing his music, and even sometimes writing new songs. It makes me want to go write another paper!

My daughter Stephanie and I at a Boston concert
at Freedom Hill Amphitheater in Sterling Heights, Michigan.

The Machinery of Life

The Machinery of Life,
by David Goodsell.

In the very first section of the 5th edition of Intermediate Physics for Medicine and Biology (Sec. 1.1), Russ Hobbie and I discuss “Distances and Sizes.”

In biology and medicine, we study objects that span a wide range of sizes: from giant redwood trees to individual molecules. Therefore, we begin with a brief discussion of length scales.

At the end of this section, we conclude

To examine the relative sizes of objects in more detail, see Morrison et al. (1994) or Goodsell (2009).

I have talked about the book Powers of Ten by Morrison et al. previously in this blog. I have also mentioned David Goodsell’s book The Machinery of Life several times, but until today I have never devoted an entire blog entry to it.

In the 4th edition of IPMB, Russ and I cited the first edition of The Machinery of Life (1998), and that is the edition that sits on my bookshelf. When preparing the 5th edition, we updated our references, so we now cite the second edition of Goodsell's book (2009). Is there much difference between the two? Yes! Like when Dorothy left Kansas to enter Oz, the first edition is all black and white but the second edition is in glorious color. And what a difference color makes in a book that is first and foremost visual. The second edition of The Machinery of Life is stunningly beautiful. It is not just a colorized version of the first edition; it is a whole new book. Goodsell writes in the preface

I created the illustrations in this book to help bridge this gulf and allow us to see the molecular structure of cells, if not directly, then in an artistic rendition. I have included two types of illustrations with this goal in mind: watercolor paintings which magnify a small portion of a living cell by one million times, showing the arrangement of molecules inside, and computer-generated pictures, which show the atomic details of individual molecules. In this second edition of The Machinery of Life, these illustrations are presented in full color, and they incorporate many of the exciting scientific advances of the 15 years since the first edition.

As with the first edition, I have used several themes to tie the pictures together. One is that of scale. Most of us do not have a good concept of the relative sizes of water molecules, proteins, ribosomes, bacteria, and people. To assist with this understanding, I have drawn the illustrations at a few consistent magnifications. The views showing the interiors of living cells, as in the Frontispiece and scattered through the last half of the book, are all drawn at one million times magnification. Because of this consistent scale, you can flip between pages in these chapters and compare the sizes of DNA, lipid membranes, nuclear pores, and all of the other molecular machinery of living cells. The computer-generated figures of individual molecules are also drawn at a few consistent scales to allow easy comparison.

I have also drawn the illustrations using a consistent style, again to allow easy comparison. A space-filling representation that shows each atom as a sphere is used for all the illustrations of molecules. The shapes of the molecules in the cellular pictures are simplified versions of these space-filling pictures, capturing the overall form of the molecule without showing the location of every atom. The colors, of course, are completely arbitrary since most of these molecules are colorless. I have chosen them to highlight the functional features of the molecules and cellular environments.

I have often wondered how much molecular biology a biological or medical physicist needs to know. I suppose it depends on their research specialty, but in general I believe a physicist who has read and understood The Machinery of Life has most of what you need to begin working at the interface of physics and biology: An understanding of the relative scale of biological objects, an overview of the different types of biological molecules and their structures and functions, and a visual sense of how these molecules fit together to form a cell. To the physicist wanting an introduction to biology on the molecular scale, I recommend starting with The Machinery of Life. That’s why it was included in my ideal bookshelf.

Goodsell fans might enjoy visiting his website: http://mgl.scripps.edu/people/goodsell. There you can download a beautiful poster of different proteins, all drawn to scale. There are many other illustrations and publications. Enjoy!

Fermi Problems and the Annual Background Radiation Dose from Potassium-40

In the very first section of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I emphasize the importance of being able to estimate.

One valuable skill in physics is the ability to make order of magnitude estimates, meaning to calculate something approximately right.

Our first four homework problems at the end of Chapter 1 ask the student to estimate. These exercises are examples of Fermi Problems, after physicist Enrico Fermi who was a master of the skill.

Let us try another of these problems, based on some of the concepts from Chapter 15 about the natural background radiation dose. To be specific, let’s estimate the annual background dose from the radioactive isotope potassium-40 inside our bodies.

Problem 4 ½ Estimate the annual background dose (in mSv/year) from potassium-40 in our bodies. Look up or guess any information you need for this calculation, and clearly explain any assumptions you make.

Begin by considering a single cell. Cells are about 10 microns in size, so their volume is about (10⁻⁵ m)³ = 10⁻¹⁵ m³ (the volume we use is not important; it will cancel out in the end). In nerves, the intracellular potassium ion concentration is about 100 mM, but this may overestimate the amount of potassium inside all types of cells. Moreover, the concentration of potassium ions in the extracellular space is much less that in the intracellular space. Let’s guess 50 mM for the average concentration of potassium in our body, meaning 50 millimoles/liter, or 50 moles/m³. If we multiply by Avogadro’s number (6 x 10²³ molecules/mole), we get about 3 x 10²⁵ molecules/m³. So, the number of potassium atoms per cell is 3 x 10¹⁰. This intermediate result is already interesting; there are over ten billion potassium ions in just one cell.

The ⁴⁰K isotope is radioactive. Its abundance is about 0.01% (abundance data can be found in any table of the isotopes, or even by looking at wikipedia; I don’t know how you could guess that value from first principles). So, this means there are 3 x 10^{6 40}K atoms/cell (a little over a million). How rapidly do these decay? ⁴⁰K has a half life of 1.25 x 10⁹ years (again, see wikipedia), implying a decay rate of 0.693/1.25 x 10⁹ = 5.5 x 10⁻¹⁰ decays per atom per year. Multiplying by the number of atoms/cell, we get 0.0017 decays per cell per year. This is another interesting result: an average cell has less than a one-percent chance of experiencing a ⁴⁰K decay in a year. But we have a lot of cells (Russ and I estimate 2 x 10¹⁴ in IPMB), so your body suffers from about 3.4 x 10¹¹ decays per year, or about ten thousand per second. (According to Wikipedia, this estimate is a factor of two too high, but we are not much worried about factors of two in such order-of-magnitude Fermi problems.)

How much energy does each decay deposit in our tissue? Beta decay accounts for 90% of all decays of ⁴⁰K, and each decay releases an energy of about 1.3 MeV (decay energy data is a little harder to find, but appears in any good table of the isotopes; if you had merely guessed that 1 MeV is the order of magnitude of nuclear decay energies, you would not be too far off). Some of that energy goes to a neutrino, which leaves the body. Let’s assume that on average about 30% of the beta decay energy goes to the electron and that no electrons escape the body, so each decay deposits 0.4 MeV into the cell, or 6 x 10⁻¹⁴ joules. A gray (the unit of dose) is a joule per kilogram, so to calculate dose we need the mass of a cell, which to a first approximation is the product of the density of water (10³ kg/m³) times the volume of the cell, or 10⁻¹² kg (I told you the volume of the cell would cancel out). So, the dose is the number of decays per year (0.0017) times the energy per decay (6 x 10⁻¹⁴ J) divided by the mass (10⁻¹² kg), or 0.0001 gray/year. Another unit of dose is the sievert, which accounts for biological damage in addition to energy deposition. A gray and a sievert are the same for electrons, so the annual background dose is 0.0001 Sv, or 0.1 mSv.

In Table 16.6 of IPMB, Russ and I estimate the annual background dose from all internal sources is about 0.3 mSv. Because ⁴⁰K is not the only isotope in our body that is decaying (for example, carbon-14 is another), we seem to have gotten our order-of-magnitude estimate pretty close. One goal of Chapters 16 and 17 in IPMB is to refine such calculations. For medical purposes you need more accuracy; for a Fermi problem we did okay.

The Electric Potential of a Rectangular Sheet of Charge - Resolved

In the January 9 post of this blog, I challenged readers to find the electrical potential V(z) that will give you the electric field E(z) of Eq. 6.10 in the 5th edition of Intermediate Physics for Medicine and Biology

In other words, the goal is to find V(z) such that E = − dV/dz produces Eq. 6.10. In the comments at the bottom of the post, a genius named Adam Taylor made a suggestion for V(z) (I love it when people leave comments in this blog). When I tried his expression for the potential, it almost worked, but not quite (of course, there is always a chance I have made a mistake, so check it yourself). But I was able to fix it up with a slight modification. I now present to you, dear reader, the potential:

How do you interpret this ugly beast? The key is the last term, z times the inverse tangent. When you take the z derivative of V(z), you must use the product rule on this term. One derivative in the product rule eliminates the leading z and gives you exactly the inverse tangent you need in the expression for the electric field. The other gives z times a derivative of the inverse tangent, which is complicated. The two terms containing the logarithms are needed to cancel the mess that arises from differentiating tan⁻¹.

I don’t know what there is to gain from having this expression for the potential, but somehow it comforts me to know that if there is an analytic equation for E there is also an analytic equation for V.

Dr. Euler’s Fabulous Formula Cures Many Mathematical Ills

Dr. Euler's Fabulous Formula,
by Paul Nahin.

I’ve decided that Leonhard Euler (1707-1783) is my favorite mathematician, and I’m mad at myself for not getting his name mentioned in the 5th edition of Intermediate Physics for Medicine and Biology. I’ve discussed Euler before in this blog, when I reviewed William Dunham’s book Euler: The Master of Us All. I’ve just finished another book, this one by Paul Nahin, about Dr. Euler’s Fabulous Formula. Of course, Nahin means Eq. 11.28 in IPMB,
e^iθ = cosθ + i sinθ .

I liked the book, in part because Nahin and I seem to have similar tastes: we both favor the illustrations of Norman Rockwell over the paintings of Jackson Pollock, we both like to quote Winston Churchill, and we both love limericks:

I used to think math was no fun,
‘Cause I couldn’t see how it was done.
    Now Euler’s my hero
    For I now see why zero,
Equals e^πi + 1.

Nahin’s book contains a lot of math, and I admit I didn’t go through it all in detail. A large chunk of the text talks about the Fourier series, which Russ Hobbie and I develop in Chapter 11 of IPMB. Nahin motivates the study of the Fourier series as a tool to solve the wave equation. We discuss the wave equation in Chapter 13 of IPMB, but never make the connection between the Fourier series and this equation, perhaps because biomedical applications don’t rely on such an analysis as heavily as, say, predicting how a plucked string vibrates.

Nahin delights in showing how interesting mathematical relationships arise from Fourier analysis. I will provide one example, closely related to a calculation in IPMB. In Section 11.5, we show that the Fourier series of the square wave (y(t) = 1 for t from 0 to T/2 and equal to -1 for t from T/2 to T) is

y(t) = Σ b_k cos(k2πt/T)

where the sum is over all odd values of k (k = 1, 3, 5, ....) and b_k = 4/(π k). Evaluate both expressions for y(t) at t = T/4. You get

π/4 = 1 – 1/3 + 1/5 – 1/7 +…

This lovely result is hidden in IPMB’s Eq. 11.36. Warning: this is not a particularly useful algorithm for calculating π, as it converges slowly; including ten terms in the sum gives π = 3.04, which is still over 3% off.

In Figure 11.17, Russ and I discuss the Gibbs phenomenon: spikes that occur in y(t) at discontinuities when the Fourier series includes only a finite number of terms. Nahin makes the same point with the periodic function y(t) = (π – t)/2 for t from 0 to 2π. He describes the history of the Gibbs phenomena, which arises from a series of published letters between Josiah Gibbs, Albert Michelson, A. E. H. Love, and Henri Poincare. Interestingly, the Gibbs phenomenon was discovered long before Gibbs by the English mathematician Henry Wilbraham.

Fourier series did not originate with Joseph Fourier. Euler, for example, was known to write such trigonometric series. Fourier transforms (the extension of Fourier series to nonperiodic functions), on the other hand, were first presented by Fourier. Nahin discusses many of the same topics that Russ and I cover, including the Dirac delta function, Parseval’s theorem, convolutions, and the autocorrelation.

Nahin concludes with a section about Euler the man and mathematical physicist. I found an interesting connection to biology and medicine: when hired in 1727 by the Imperial Russian Academy of Sciences, it was as a professor of physiology. Euler spent several months before he left for Russia studying physiology, so he would not be totally ignorant of the subject when he arrived in Saint Petersburg!

I will end with a funny story of my own. I was working at Vanderbilt University just as Nashville was enticing a professional football team to move there. One franchise that was looking to move was the Houston Oilers. Once the deal was done, folks in Nashville began debating what to call their new team. They wanted a name that would respect the team’s history, but would also be fitting for its new home. Nashville has always prided itself as the home of many colleges and universities, so a name out of academia seemed appropriate. Some professors in Vanderbilt’s Department of Mathematics came up with what I thought was the perfect choice: call the team the Nashville Eulers. Alas, the name didn’t catch on, but at least I never again was uncertain about how to pronounce Euler.

Circularly Polarized Excitation Pulses, Spin Lock, and T1ρ

In Chapter 18 of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic resonance imaging. We describe how spins precess in a magnetic field, B_o = B_o k, at their Larmor frequency ω, and show that this behavior is particularly simple when expressed in a rotating frame of reference. We then examine radio-frequency excitation pulses by adding an oscillating magnetic field. Again, the analysis is simpler in the rotating frame. In our book, we apply the oscillating field in the laboratory frame’s x-direction, B₁ = B₁ cos(ωt) i. The equations of the magnetization are complicated in the rotating frame (Eqs. 18.22-18.24), but become simpler when we average over time (Eq. 18.25). The time-averaged magnetization rotates about the rotating frame’s x' axis with angular frequency γB₁/2, where γ is the spin’s gyromagnetic ratio. This motion is crucial for exciting the spins; it rotates them from their equilibrium position parallel to the static magnetic field B_o into a plane perpendicular to B_o.

When I teach medical physics (PHY 326 here at Oakland University), I go over this derivation in class, but the students still need practice. I have them analyze some related examples as homework. For instance, the oscillating magnetic field can be in the y direction, B₁ = B₁ cos(ωt) j, or can be shifted in time, B₁ = B₁ sin(ωt) i. Sometimes I even ask them to analyze what happens when the oscillating magnetic field is in the z direction, B₁ = B₁ cos(ωt) k, parallel to the static field. This orientation is useless for exciting spins, but is useful as practice.

Yet another way to excite spins is using a circularly polarized magnetic field, B₁ = B₁ cos(ωt) i – B₁ sin(ωt) j. The analysis of this case is similar to the one in IPMB, with one twist: you don’t need to average over time! Below is a new homework problem illustrating this.

Problem 13 1/2. Assume you have a static magnetic field in the z direction and an oscillating, circularly polarized magnetic field in the x-y plane, B =B_o k + B₁ cos(ωt) i – B₁ sin(ωt) j.

a) Use Eq. 18.12 to derive the equations for the magnetization M in the laboratory frame of reference (ignore relaxation).

b) Use Eq. 18.18 to transform to the rotating coordinate system and derive equations for M'.

c) Interpret these results physically.

I get the same equations as derived in IPMB (Eq. 18.25) except for a factor of one half; the angular frequency in the rotating frame is ω₁ = γ B₁. Not having to average over time makes the result easier to visualize. You don’t get a complex motion that—on average—rotates the magnetization. Instead, you get a plain old rotation. You can understand this behavior qualitatively without any math by realizing that in the rotating coordinate system the RF circularly polarized magnetic field is stationary, pointing in the x’ direction. The spins simply precess around the seemingly static B₁'= B₁ i', just like the spins precess around the static B_o = B_o k in the laboratory frame.

Now let’s assume that an RF excitation pulse rotates the magnetization so it aligns with the x' rotating axis. Once the pulse ends, what happens? Well, nothing happens unless we account for relaxation. Without relaxation the magnetization precesses around the static field, which means it just sits there stationary in the rotating frame. But we know that relaxation occurs. Consider the mechanism of dephasing, which underlies the T₂* relaxation time constant. Slight heterogeneities in B_o mean that different spins precess at different Larmor frequencies, causing the spins to lose phase coherence, decreasing the net magnetization.

Next, consider the case of spin lock. Imagine an RF pulse rotates the magnetization so it is parallel to the x' rotating axis. Then, when the excitation pulse is over, immediately apply a circularly polarized RF pulse at the Larmor frequency, called B₂, which is aligned along the x' rotating axis. In the rotating frame the magnetization is parallel to B₂, so nothing happens. Why bother? Consider those slight heterogeneities in B_o that led to T₂* relaxation. They will cause the spins to dephase, picking up a component in the y' direction. But a component along y' will start to precess around B₂. Rather than dephasing, B₂ causes the spins to wobble around in the rotating frame, precessing about x', with no net tendency to dephase. You just killed the mechanism leading to T₂*! Wow!

Will the spins just precess about B₂ forever? No, eventually other mechanisms will cause them to relax toward their equilibrium value. Their time constant will not be T₁ or T₂ or even T₂*, but something else called T_1ρ. Typcially, T_1ρ is much longer than T₂*. To measure T_1ρ, apply a 90 degree excitation pulse, then apply a RF spin lock oscillation and record the free induction decay. Fit the decay to an exponential, and the time constant you obtain is T_1ρ. (I am not a MRI expert: I am not sure how you can measure a free induction decay when a spin lock field is present. I would think the spin lock field would swamp the FID.)

T_1ρ is sometimes measured to learn about the structure of cartilage. It is analogous to T₁ relaxation in the laboratory frame, which explains its name. Because B₂ is typically much weaker than B_o, T_1ρ is sensitive to a different range of correlation times than T₁ or T₂ (see Fig. 18.12 in IPMB).