The Heating of Metal Electrodes

Twenty years ago, I published a paper titled “The Heating of Metal Electrodes During Rapid-Rate Magnetic Stimulation: A Possible Safety Hazard” (Electroencephalography and Clinical Neurophysiology, Volume 85, Pages 116–123, 1992). My coauthors were Alvaro Pascual-Leone, Leonardo Cohen, and Mark Hallett, all working at the National Institutes of Health at that time. The paper motivated two new homework problems in Chapter 8 of the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 24 Suppose one is measuring the EEG when a time-dependent magnetic field is present (such as during magnetic stimulation). The EEG is measured using a disk electrode of radius a = 5 mm and thickness d = 1 mm, made of silver with conductivity σ = 63 × 10⁶ S m⁻¹. The magnetic field is uniform in space, is in a direction perpendicular to the plane of the electrode, and changes from zero to 1 T in 200 μs.
(a) Calculate the electric field and current density in the electrode due to Faraday induction.
(b) The rate of conversion of electrical energy to thermal energy per unit volume (Joule heating) is the product of the current density times the electric field. Calculate the rate of thermal energy production during the time the magnetic field is changing.
(c) Determine the total thermal energy change caused by the change of magnetic field.
(d) The specific heat of silver is 240 J kg⁻¹ ◦C⁻¹, and the density of silver is 10 500 kg m⁻³. Determine the temperature increase of the electrode due to Joule heating. The heating of metal electrodes can be a safety hazard during rapid (20 Hz) magnetic stimulation [Roth et al. (1992)].

Problem 25 Suppose that during rapid-rate magnetic stimulation, each stimulus pulse causes the temperature of a metal EEG electrode to increase by ΔT (see Prob. 24). The hot electrode then cools exponentially with a time constant τ (typically about 45 s). If N stimulation pulses are delivered starting at t = 0 m with successive pulses separated by a time Δt, then the temperature at the end of the pulse train is T(N,Δt) = ΔT Σ e^−iΔt/τ [the sum goes from 0 to N-1]. Find a closed-form expression for T(N,Δt) using the summation formula for the geometric series: 1 + x + x² + ... + xⁿ⁻¹ = (1 − xⁿ)/(1 − x). Determine the limiting values of T(N,Δt)for NΔt [much less than] τ and NΔt [much greater than] τ . [See Roth et al. (1992).]

Both problems walk you through parts of our paper. I like Problem 24 because it provides a nice example of Faraday’s law of induction, one of the topics discussed in Chapter 8 (Biomagnetism). Problem 25 could easily have been placed in Chapter 3 (Systems of Many Particles) because of its emphasis on thermal heating and Newton’s law of cooling.

Problem 25 also provides a physical example illustrating the mathematical expression for the summation of a geometric series. If you are not familiar with this sum, it is one that you don’t have to memorize because it is so easy to derive. Let S = 1 + x + x² + … + x^N-1. If you multiply this expression by x, you get xS = x + x² + … + x^N. Now (and this is the clever trick), subtract xS from S, which gives (1 – x) S = 1 – x^N. Note that all the terms x + x² + … + x^N-1 cancel out! Solving for S gives you the equation in Problem 25. If x is between -1 and 1, and N goes to infinity, you get for the infinite sum S = 1/(1 – x).

I should say a few words about my coauthors, who are all leaders in the field of transcranial magnetic stimulation. The project started when Alvaro Pascual-Leone, who had just arrived at NIH, mentioned to me that one patient of his had suffered a burn during rapid-rate magnetic stimulation (see: Pascual-Leone, A., Gates, J.R. and Dhuna, A.K. “Induction of Speech Arrest and Counting Errors with Rapid Transcranial Magnetic Stimulation,” Neurology, Volume 41, Pages 697–702, 1991). This motivated Alvaro and I to launch a study using a variety of metal disks made by the NIH machine shop. Alvaro is now the Director of the Berenson-Allen Center for Noninvasive Brain Stimulation. Leo Cohen and Mark Hallett were both studying magnetic stimulation when I arrived at NIH in 1988. I was lucky to start collaborating with them, providing physics expertise to augment their extensive clinical experience. Both continue at the Human Motor Control Section at NIH in Bethesda, Maryland.

Law of Laplace

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I often introduce topics in the homework problems that we don’t have room to discuss fully in the text. For instance, Problem 18 in Chapter 1 asks the reader to derive the Law of Laplace, f = p R, a relationship between the pressure p inside a cylinder, its radius R, and its wall tension f.

Vital Circuits,
by Steven Vogel.

In his book Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems, Steven Vogel explains the physiological significance of the Law of Laplace, particularly for blood vessels.

The wall of the aorta of a dog is about 0.6 millimeters thick, while the wall of the artery leading to the head is only half that. Pressure differences across the wall are the same…, but the aorta has an internal diameter three times as great. The bigger vessel simply needs a thicker wall. An arteriole is 100 times skinnier yet; it wall is fifteen times thinner than that of the artery going to the head. A capillary is eight times still skinnier, with walls another twenty times thinner… A general rule that wall thickness is proportional to vessel diameter is clearly evident, just the relationship expected from Laplace’s law.

In our Homework Problem 18, Russ and I write that “Sometimes a patient will have an aneurysm in which a portion of an artery will balloon out and possibly rupture. Comment on this phenomenon in light of the R dependence of the force per unit length.” The answer (spoiler alert) is explained by Vogel. He first examines what happens when inflating a balloon.

About the same pressure is needed throughout the inflation, except for an extra bit to get started and (if you persist) another extra bit just before the final explosion… Pressure gets more effective in generating tension—stretch—as the balloon gets bigger [p = f/R], automatically providing the extra force needed as the rubber is expanded.

What is the implication for an aneurysm?

Pressure, we noted, is more effective in generating tension in the walls of a big cylinder than in those of a small cylinder. Blow into a cylindrical balloon, and one part of the balloon will inflate almost fully before the remainder expands. Pressure inside at any instant is the same everywhere, but the responsive stretching is curiously irregular…any part partially inflated is easier to inflate further than any part not yet inflated at all.

Thus, the law of Laplace provides insight into aneurysms: just think of a bulge that develops when inflating a cylindrical balloon. Now the question can be turned on its head: why don’t all cylindrical vessels immediately develop aneurysms as soon as pressure is applied? In other words, why are we not all dead, killed by the Law of Laplace? Vogel addresses this point too. “The primary question isn’t why aneurysms sometimes occur, but why they don’t normally happen…Why an arterial wall...behaves in a much friendlier manner.”

Vogel’s answer is that arteries “should surely stretch strangely” (I love the alliteration). The walls of an artery as designed such that

a disproportionate force is needed for each incremental bit of stretch—the thing gets stiffer as it stretches further…As the vessels expand, pressure inside is increasingly effective at generating tension in their walls—that’s the unavoidable consequence of Laplace’s law. But that tension, the stress in the walls, is decreasingly effective in causing the walls to stretch. It all comes down to that curved, J-shaped line on the stress-strain graph, which means no aneurysm.

Thank goodness nature found a way to avoid the aneurysms predicted by the Law of Laplace!

There are many more biomedical applications of the Law of Laplace. In a review article, Jeffrey Basford describes the “Law of Laplace and Its Relevance to Contemporary Medicine and Rehabilitation” (Archives of Physical Medidine and Rehabilitation, Volume 83, Pages 1165–1170, 2002). Basford considers many examples, including bladder function, compressive wraps to treat peripheral edema, the choice of where in the uterus to perform a Cesarean section. That is a lot of insight from one simple law relating pressure, radius and wall tension.

Andrew Huxley (1917-2012)

Andrew Huxley, the greatest mathematical biologist of the 20th century, died on Wednesday, May 30. Huxley won the Nobel Prize for his groundbreaking work with Alan Hodgkin that explained electrical transmission in nerves.

In Chapter 6 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the Hodgkin-Huxley model of membrane current in a nerve axon.

Considerable work was done on nerve conduction in the late 1940s, culminating in a model that relates the propagation of the action potential to the changes in membrane permeability that accompany a change in voltage. The model [Hodgkin and Huxley (1952)] does not explain why the membrane permeability changes; it relates the shape and conduction speed of the impulse to the observed changes in membrane permeability. Nor does it explain all the changes in current…Nonetheless, the work was a triumph that led to the Nobel Prize for Alan Hodgkin and Andrew Huxley.

The paper we cite (“A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve,” Journal of Physiology, Volume 117, Pages 500–544) is one of my favorites. Whenever I teach biological physics, I assign this paper to my students as an example of mathematical modeling in biology at its best. In 1981 Hodgkin and Huxley wrote a “citation classic” article about their paper, which has now been cited over 9300 times. They concluded

Another reason why our paper has been widely read may be that it shows how a wide range of well-known, complicated, and variable phenomena in many excitable tissues can be explained quantitatively by a few fairly simple relations between membrane potential and changes of ion permeability—processes that are several steps away from the phenomena that are usually observed, so that the connections between them are too complex to be appreciated, intuitively. There now seems little doubt that the main outlines of our explanation are correct, but we have always felt that our equations should be regarded only as a first approximation that needs to be refined and extended in many ways in the search for the actual mechanism of the permeability change’s on the molecular scale.

As one who does mathematical modeling of bioelectric phenomena for a living, I can think of no better way to honor Huxley than to show you his equations.

This set of four nonlinear ordinary differential equations, plus six expressions relating how the ion channel rate constants depend on voltage, not only describes the membrane of the squid giant nerve axon, but also is the starting point for models of all electrically active tissue. Russ and I consider this model to be so important that we dedicate six pages to exploring it, and present in our Fig. 6.38 a computer program to solve the equations. For anyone interested in electrophysiology, becoming familiar with the Hodgkin-Huxley model is job one, just as analyzing the Bohr model for hydrogen is the starting point for someone interested in atomic structure. Remarkably, 60 years ago Huxley solved these differential equations numerically using only a hand-crank adding machine.

How can your learn more about this great man? First, the Nobel Prize website contains his biography, a transcript of his Nobel lecture, and a video of an interview. Another recent, more detailed interview is available on Youtube in two parts, part1 and part 2. Huxley wrote a fascinating description of the many false leads during their nerve studies in a commemorative article celebrating the 50th anniversary of his famous paper. Finally, the Guardian published an obituary of Huxley yesterday.

An interview with Andrew Huxley, Part 1.
https://www.youtube.com/watch?v=WdL-81i3Qg4

An interview with Andrew Huxley, Part 2.
https://www.youtube.com/watch?v=qL3aTfljBXE

I will conclude by quoting the summary at the end of Hodgkin and Huxley’s 1952 paper, which was the last of a series of five articles describing their voltage clamp experiments on a squid axon.

SUMMARY

1. The voltage clamp data obtained previously are used to find equations which describe the changes in sodium and potassium conductance associated with an alteration of membrane potential. The parameters in these equations were determined by fitting solutions to the experimental curves relating sodium or potassium conductance to time at various membrane potentials.

2. The equations, given on pp. 518–19, were used to predict the quantitative behaviour of a model nerve under a variety of conditions which corresponded to those in actual experiments. Good agreement was obtained in the cases:

(a) The form, amplitude and threshold of an action potential under zero membrane current at two temperatures.

(b) The form, amplitude and velocity of a propagated action potential.

(c) The form and amplitude of the impedance changes associated with an action potential.

(d) The total inward movement of sodium ions and the total outward movement of potassium ions associated with an impulse.

(e) The threshold and response during the refractory period.

(f) The existence and form of subthreshold responses.

(g) The existence and form of an anode break response.

(h) The properties of the subthreshold oscillations seen in cephalopod axons.

3. The theory also predicts that a direct current will not excite if it rises sufficiently slowly.

4. Of the minor defects the only one for which there is no fairly simple explanation is that the calculated exchange of potassium ions is higher than that found in Sepia axons.

5. It is concluded that the responses of an isolated giant axon of Loligo to electrical stimuli are due to reversible alterations in sodium and potassium permeability arising from changes in membrane potential.

The Semiempirical Mass Formula

When revising Chapter 17 about Nuclear Medicine for the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I were tempted to include a discussion of the semiempirical mass formula, one of the fundamental concepts in nuclear physics. We finally decided that you can’t discuss everything in one book, but we did include the following footnote.

This parabola and the general behavior of the binding energy with Z and A can be explained remarkably well by the semiempirical mass formula [Evans (1955, Chapter 11); Eisberg and Resnick, (1985, p. 528)].

The semiempirical mass formula consists of five terms, which together predict the binding energy of a nucleus having atomic number Z and mass number A.

1. The first term is negative, and arises from the binding caused by the short range nuclear force. It is proportional to A, which implies that it increases with the volume of the nucleus (this term assumes that the nuclear density is constant; the “liquid drop model”).
2. The second term represents a positive correction caused by surface tension, arising because nucleons at the surface of the nucleus feel an attractive force from only one side (the nuclear interior). It is proportional to surface area, or A^2/3.
3. All the positively charged protons repel each other, and this effect is accounted for by a positive term for the Coulomb energy, proportional to Z²/A^1/3.
4. Everything else being equal, nuclei tend to be more stable if they have the same number of protons and neutrons. This behavior is reflected in an asymmetry term containing (Z - A/2)²/A. It is zero if A = 2Z (an equal number of protons and neutrons) and is positive otherwise.
5. Finally, a pairing term is negative if both the number of protons and neutrons is even, positive if both are odd, and zero if one is even and the other odd.

The sum of these five terms is the semiempirical mass formula, with the terms weighted by parameters determined by fitting the model to data.

What can this formula explain? One example is the plot of average binding energy per nucleon as a function of A given in Fig. 17.3. At low A, this function predicts a very low binding energy because of the surface term (very small nuclei have a large surface-to-volume ratio). As A increases, the surface term becomes less important, but the Coulomb term increases as the nucleus is packed with more and more positive charge. For nuclei above about A = 60, the Coulomb term causes the binding energy to decrease as A increases. Therefore, the binding energy per nucleon reaches a peak for isotopes of elements such as iron and nickel, the most stable of nuclei, because of a competition between the surface and Coulomb terms. Although Russ and I did not mention it in our book, the smooth curve that most of the data cluster about in Fig. 17.3 is the prediction of the semiempirical mass formula.

If you hold A constant, you can examine the binding energy as a function of Z. This case is important for beta decay (in which a neutron is converted to a proton and an electron) and positron decay (in which a proton is converted to a neutron and a positron). The two terms in the semiempirical mass formula containing Z—the Coulomb term and the asymmetry term—combine to give a quadratic shape for the binding energy, as shown in Fig. 17.6. For odd A, the resulting parabola predicts the stable isotope (Z) for that A. For even A, the pairing term results in two parabolas, one for even Z and one for odd Z (Fig. 17.7).

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

In their textbook, Eisberg and Resnick conclude that

The liquid drop model is the oldest, and most classical nuclear model. At the time the semiempirical mass formula was first developed, mass data was available, but not much else was known about nuclei. The parameters were purely empirical, and there was not even a qualitative understanding of the asymmetry and pairing terms. Nevertheless, the formula was significant because it described fairly accurately the masses of hundreds of nuclei in terms of only five parameters.

Spin Echo

The spin-echo of nuclear magnetic resonance is one of those concepts that anyone interested in medical physics should know. Russ Hobbie and I discuss the spin-echo’s role in magnetic resonance imaging in Chapter 18 of the 4th edition of Intermediate Physics for Medicine and Biology.

The pulse sequence shown in Fig. 18.17 can be used to determine T₂ [the true or non-recoverable spin-spin relaxation time] and T^*₂ [the experimental spin-spin relaxation time]. Initially a π/2 [90°] pulse nutates M [the magnetization] about the x’ axis so that all spins lie along the rotating y’ axis. Figure 18.17(a) shows two such spins. Spin a continues to precess at the same frequency as the rotating coordinate system; spin b is subject to a slightly smaller magnetic field and precesses at a slightly lower frequency, so that at time T_E/2 it has moved clockwise in the rotating frame by angle θ, as sown in Fig. 18.17(b). At this time a π [180°] pulse is applied that rotates all spins around the x' axis. Spin a then points along the –y' axis; spin b rotates to the angle shown in Fig. 18.17(c). If spin b still experiences the smaller magnetic field, it continues to precess clockwise in the rotating frame. At time T_E both spins are in phase again, pointing along –y' as shown in Fig. 18.17(d). The resulting signal is called an echo, and the process for producing it is called a spin-echo sequence.

When I discuss this concept in class, I use the analogy of a footrace. Suppose all runners line up at the starting line, and at the sound of the starter’s gun they begin to run clockwise around a track. Because they all run at somewhat different speeds, the pack of runners spreads until eventually (after many laps) they are distributed nearly evenly, and seemingly randomly, around the track. At this time another gun is fired, commanding all runners to turn around and run counterclockwise. Now, the fast runners who were ahead of the others are suddenly behind, and the slow runners who were behind the others are miraculously ahead. As time goes on, the fast runners catch up to the slow ones, and eventually they all meet in one tight pack as they run past the starting line. This unexpected regrouping of the runners is the echo. The analogy is not perfect, because the spins always precess in the same direction. Nevertheless, the 180° pulse has the effect of placing the fast spinners behind the slow spinners, which is the essence of both the spin echo effect and the runner analogy.

The spin-echo was first observed by physicist Erwin Hahn. His paper “Spin Echos” (Physical Review, Volume 80, Pages 580–594, 1950) has been cited over 3000 times. Hahn wrote a citation classic article about this paper, in which he describes how he made his great discovery by accident.

One day a strange signal appeared on the oscilloscope, in the absence of a pulse pedestal, so I kicked the apparatus and breathed a sigh of relief when the signal went away. A week later, the signal returned, and this time it checked out to be a real spontaneous spin echo nuclear signal from the test sample of protons in the glycerine being used. In about three weeks, I was able to predict mathematically what I suspected to be a constructive interference of precessing nuclear magnetism components by solving the Bloch nuclear induction equations. Here for the first time, a free precession signal in the absence of driving radiation was observed first, and predicted later. The spin echo began to yield information about the local atomic environment in terms of various amplitude and frequency memory beat effects, certainly not all understood in the beginning.

As I look back at this experience, it was an awesome adventure to be alone with the apparatus showing one new effect after another at a time when there was no one at Illinois experienced in NMR with whom I could talk.

You can learn more about Hahn and his discovery of the spin-echo from the transcript of an oral history interview published by the Niels Bohr Library and Archives, part of the American Institute of Physics.

For those of you who are visual learners, Wikipedia has a nice animation of the formation of a spin-echo. Another animation is at http://mrsrl.stanford.edu/~brian/mri-movies/spinecho.mpg.

You can find an excellent video about spin-echo NMR on Youtube, narrated by Sir Paul Callaghan, a New Zealand physicist (this is part of a series of videos that nicely support the discussion in Chapter 18 of Intermediate Physics for Medicine and Biology). Callaghan was a leader in MRI physics, and wrote Principles of Nuclear Magnetic Resonance Microscopy and, more recently, Translational Dynamics and Magnetic Resonance: Principles of Pulsed Gradient Spin Echo NMR. Tragically, Callaghan lost his battle to colon cancer this March.

Paul Callaghan discusses the spin echo.

https://www.youtube.com/watch?v=B2HMAJQJ7ok

Stopping Power and the Bragg Peak

Proton therapy is becoming a popular treatment for cancer. Russ Hobbie and I discuss proton therapy in Chapter 16 of the 4th edition of Intermediate Physics for Medicine and Biology.

Protons are also used to treat tumors. Their advantage is the increase of stopping power at low energies. It is possible to make them come to rest in the tissue to be destroyed, with an enhanced dose relative to intervening tissue and almost no dose distally (“downstream”) as shown by the Bragg peak in Fig. 16.51. … The edges of proton fields are much sharper than for x rays and electrons. This can provide better tissue sparing, but it also means that alignments must be much more precise [Moyers (2003)]. Sparing tissue reduces side effects immediately after treatment. It also reduces the incidence of radiation-induced second cancers many years later.

Stopping power and range are a key concepts in describing how radiation interacts with matter, and are defined in Chapter 15.

It is convenient to speak of how much energy the charged particle loses per unit path length, the stopping power, and its range—roughly, the total distance it travels before losing all its energy. The stopping power is the expectation value of the amount of kinetic energy T lost by the projectile per unit path length. (The term power is historical. The units of stopping power are J m⁻¹ not J s⁻¹.)

To illustrate these concepts, I have devised a new homework problem. It’s a bit like Problem 31 in Chapter 16, but uses a simpler expression for the energy dependence of the stopping power, and focuses on how this leads to a Bragg peak. This problem occasionally appears on the qualifier exam taken by our Medical Physics graduate students at Oakland University.

Section 16.11

Problem 31 ½   Assume the stopping power of a particle, S = − dT/dx, as a function of kinetic energy, T, is S = C/T. 
(a) What are the units of C? 
(b) If the initial kinetic energy at x = 0 is T_o, find T(x) .
(c) Determine the range R of the particle as a function of C and T_o. 
(d) Plot S(x) versus x. Does this plot contain a Bragg peak? 
(e) Discuss the implications of the shape of S(x) for radiation treatment using this particle.

The stopping power often does fall as 1/T for large energies, as assumed in the above problem, but it rises as the square root of T for small energies (See Fig. 15.17 in Intermediate Physics for Medicine and Biology). To find a more accurate expression for S(x), try repeating this problem with

S(T) = C/(T + A/√T) .

Warning: I wasn’t able to find a simple analytical expression for S(x) in this case. Can you?

One can imagine a proton incident with such low energy that it lies entirely on the rising part of the stopping power versus energy curve. In that case, a good approximation for the stopping power would be simply

S(T) = B √T .

I was able to solve for the stopping power in this case, although the expression is cumbersome. Interestingly, for these low energy particles the range is now infinite, because as the particle slows down it loses energy more slowly. I suppose once the particle’s energy is similar to the thermal energy, the entire model breaks down, so I am not too worried about this result.

These considerations illustrate how we gain much insight by examining simple toy models. That tends to be the view Russ and I adopt in our book, which is at odds with the traditional view of biologists and medical doctors, who relish the diversity and complexity of life.

The Optics of Life

The Optics of Life:
A Biologist's Guide to Light in Nature,
by Sonke Johnsen.

As I mentioned two weeks ago, I’ve been reading The Optics of Life: A Biologist’s Guide to Light in Nature, by Sonke Johnsen. The book is delightful, exploring the biological implications of many fascinating phenomena such as scattering, interference, fluorescence, and bioluminescence. To me, The Optics of Life does for light what Life in Moving Fluids does for fluid dynamics; it explains how basic principles of physics apply to the diversity of life. Today, I want to focus on Chapter 8 of Johnsen’s book, about polarization.

The polarization of light is one of those topics Russ Hobbie and I don’t cover in the 4th edition of Intermediate Physics for Medicine and Biology. We only hint at its importance in Chapter 14 about Atoms and Light, when discussing Terahertz radiation.

Classical electromagnetic wave theory is needed to describe the interactions [of Terahertz radiation with the body], and polarization (the orientation of the E vector of the propagating wave) is often important.

Had you asked me two weeks ago why Russ and I skipped polarization, I would have said “because there are so few biological applications.” Johnsen proves me wrong. He writes

As I mentioned earlier, aside from the few people who can see Haidinger’s Brush in the sky, the polarization characteristics of light are invisible to humans. However, a host of animals can detect one or both aspects of linearly polarized light (see Talbot Waterman’s massive review [1981] and Horvath and Varju’s even more massive book [2004] for comprehensive lists of taxa). Arthropods are the big winners here, especially insects, though also most crustaceans and certain spiders and scorpions. In fact, it is unusual to find an insect without polarization sensitivity. Outside the arthropods, the other major polarization-sensitive invertebrates are the cephalopods. Among vertebrates, polarization sensitivity is rarer and more controversial, but has been found in certain fish (primarily trout and Talbot salmon), some migratory birds and a handful of amphibians and reptiles. It is important to realize, though, that there is a serious sampling bias. Testing for polarization sensitivity is difficult and so has usually only been looked for in migratory animals and those known to be especially good at navigation, such as certain desert ants. The true prevalence of polarization sensitivity is unknown.

The ability to sense the polarization of light has been divided into two types. One is known as “polarization sensitivity.” Animals that have polarization sensitivity are not much different from humans wearing Polaroid sunglasses. Polarization affects the intensity of what they see—but without a lot of head-turning and thinking, they cannot reliably determine the angle or degree of polarization or even separate polarization from brightness. The other type is known as “polarization vision.” Animals with polarization vision perceive the angle and degree of polarization as something separate from simple brightness differences. Behaviorally, this means that they can distinguish two lights with different degrees and/or angles of polarization regardless of their relative radiances and colors. This is much like the definition of color vision, which involves the ability to distinguish two lights of differing hue and/or saturation regardless of their relative radiances.

How I’d love to have polarization vision! It would be an entirely new sensory experience. When Dorothy entered the land of Oz, she went from a black and white world to the richness of color. Now imagine a similar experience when going from our drab nonpolarized vision to polarization vision; it would offer a whole new way to view the world; a sixth sense. Alas, not all animals have polarization sensitivity, and even fewer have polarization vision. How these senses work is still unclear.

While polarization sensitivity is certainly rarer among vertebrates [than invertebrates], it does exist… The mechanism of polarization sensitivity in vertebrates remains—along with the basis of animal magnetoreception—one of the two holy grails of sensory biology.

My favorite example discussed by Johnsen is the Mantis shrimp, which can distinguish between left-handed and right-handed circularly polarized light. They do this by passing the light through a biological quarter-wave plate. The quarter-wave plate was one of my favorite topics in my undergraduate optics class. Incoming linearly polarized light is converted into circularly polarizing light by inducing a phase difference of 90 degrees between the two linear components. Similarly, the plate can convert circularly polarized light into linearly polarized light. Circularly polarized light always struck me as somehow magical. You can’t detect it using a sheet of plastic polarizing film, yet it is as fundamental a polarization state for light as is linear polarization. That the Mantis shrimp could make use of a quarter-wave plate to detect circularly polarized light is awesome.

Let me conclude by quoting the first sentence of Johnsen’s introduction to The Optics of Life, which elegantly sums up the book itself.

Of all the remarkable substances of our experience—rain, leaves, baby toes—light is perhaps the most miraculous.

Added note in the evening of May 4: Russ Hobbie reminds me that on the book’s website is text from the first edition of Intermediate Physics for Medicine and Biology about optics, including much about polarization!

Physics and Medicine

Readers of Intermediate Physics for Medicine and Biology already know how important physics is to medicine. Now, subscribers to the famed British medical journal The Lancet are learning this too. The April 21–27 issue (Volume 379, Issue 9825) of The Lancet contains a series of articles under the heading “Physics and Medicine.” In his editorial introducing this series, Peter Knight (president of the Institute of Physics) calls for UK medical schools to reinstate an undergraduate physics requirement for aspiring premed students. The English don’t require their premed students to take physics? Yikes!

Richard Horton, editor-in-chief of The Lancet, seconds this call for better physics education. He concludes that “Young physicists need to be nurtured to ensure a sustainable supply of talented scientists who can take advantage of the opportunities for health-related physics research in the future. Schools, indeed all of us interested in the future of health care, should declare and implement a passion for physics. Our Series is our commitment to do so.” Bravo! Below I reproduce the abstracts to the five articles in the Physics and Medicine series. In brackets I indicate the chapter or section in the 4th edition of Intermediate Physics for Medicine and Biology where a particular topic is discussed.

Physics and Medicine: a Historical Perspective
Stephen F Keevil

Nowadays, the term medical physics usually refers to the work of physicists employed in hospitals, who are concerned mainly with medical applications of radiation, diagnostic imaging, and clinical measurement. This involvement in clinical work began barely 100 years ago, but the relation between physics and medicine has a much longer history. In this report, I have traced this history from the earliest recorded period, when physical agents such as heat and light began to be used to diagnose and treat disease. Later, great polymaths such as Leonardo da Vinci and Alhazen used physical principles to begin the quest to understand the function of the body. After the scientific revolution in the 17th century, early medical physicists developed a purely mechanistic approach to physiology, whereas others applied ideas derived from physics in an effort to comprehend the nature of life itself. These early investigations led directly to the development of specialties such as electrophysiology [Chpts 6, 7], biomechanics [Secs 1.5–1.7] and ophthalmology [Sec 14.12]. Physics-based medical technology developed rapidly during the 19th century, but it was the revolutionary discoveries about radiation and radioactivity [Secs 17.2–17.4] at the end of the century that ushered in a new era of radiation-based medical diagnosis and treatment, thereby giving rise to the modern medical physics profession. Subsequent developments in imaging [Chpt 12] in particular have revolutionised the practice of medicine. We now stand on the brink of a new revolution in post-genomic personalised medicine, with physics-based techniques again at the forefront. As before, these techniques are often the unpredictable fruits of earlier investment in basic physics research.

Diagnostic Imaging
Peter Morris, Alan Perkins

Physical techniques have always had a key role in medicine, and the second half of the 20th century in particular saw a revolution in medical diagnostic techniques with the development of key imaging instruments: x-ray imaging [Chpt 16] and emission tomography [Secs 12.4–12.6] (nuclear imaging [Secs 17.12-17.13] and PET [Sec 17.14]), MRI [Chpt 18], and ultrasound [Chpt 13] These techniques use the full width of the electromagnetic spectrum [Sec 14.1], from gamma rays to radio waves, and sound [Secs 13.1–13.3]. In most cases, the development of a medical imaging device was opportunistic; many scientists in physics laboratories were experimenting with simple x-ray images within the first year of the discovery of such rays, the development of the cyclotron and later nuclear reactors created the opportunity for nuclear medicine, and one of the co-inventors of MRI was initially attempting to develop an alternative to x-ray diffraction for the analysis of crystal structures. What all these techniques have in common is the brilliant insight of a few pioneering physical scientists and engineers who had the tenacity to develop their inventions, followed by a series of technical innovations that enabled the full diagnostic potential of these instruments to be realised. In this report, we focus on the key part played by these scientists and engineers and the new imaging instruments and diagnostic procedures that they developed. By bringing the key developments and applications together we hope to show the true legacy of physics and engineering in diagnostic medicine.

The Importance of Physics to Progress in Medical Treatment
Andreas Melzer, Sandy Cochran, Paul Prentice, Michael P MacDonald, Zhigang Wang, Alfred Cuschieri

Physics in therapy is as diverse as it is substantial. In this review, we highlight the role of physics—occasionally transitioning into engineering—through discussion of several established and emerging treatments. We specifically address minimal access surgery, ultrasound [Sec 13.7], photonics [Chpt 14], and interventional MRI, identifying areas in which complementarity is being exploited. We also discuss some of the fundamental physical principles involved in the application of each treatment to medical practice.

Future Medicine Shaped by an Interdisciplinary New Biology
Paul O'Shea

The projected effects of the new biology on future medicine are described. The new biology is essentially the result of shifts in the way biological research has progressed over the past few years, mainly through the involvement of physical scientists and engineers in biological thinking and research with the establishment of new teams and task forces to address the new challenges in biology. Their contributions go well beyond the historical contributions of mathematics, physical sciences, and engineering to medical practice that were largely equipment oriented. Over the next generation, the entire fabric of the biosciences will change as research barriers between disciplines diminish and eventually cease to exist. The resulting effects are starting to be noticed in front-line medicine and the prospects for the future are immense and potentially society changing. The most likely disciplines to have early effects are outlined and form the main thrust of this paper, with speculation about other disciplines and emphasis that although physics-based and engineering-based biology will change future medicine, the physical sciences and engineering will also be changed by these developments. Essentially, physics is being redefined by the need to accommodate these new views of what constitutes biological systems and how they function.

The Importance of Quantitative Systemic Thinking in Medicine
Geoffrey B West

The study and practice of medicine could benefit from an enhanced engagement with the new perspectives provided by the emerging areas of complexity science [Secs 10.7-10.8] and systems biology. A more integrated, systemic approach is needed to fully understand the processes of health, disease, and dysfunction, and the many challenges in medical research and education. Integral to this approach is the search for a quantitative, predictive, multilevel, theoretical conceptual framework that both complements the present approaches and stimulates a more integrated research agenda that will lead to novel questions and experimental programmes. As examples, the importance of network structures and scaling laws [Sec 2.10] are discussed for the development of a broad, quantitative, mathematical understanding of issues that are important in health, including ageing and mortality, sleep, growth, circulatory systems [Sec 1.17], and drug doses [Sec 2.5]. A common theme is the importance of understanding the quantifiable determinants of the baseline scale of life, and developing corresponding parameters that define the average, idealised, healthy individual.

Frequency versus Wavelength

The Optics of Life:
A Biologist's Guide to Light in Nature,
by Sonke Johnsen.

I am currently reading The Optics of Life: A Biologist’s Guide to Light in Nature, by Sonke Johnsen. I hope to have more to say about this fascinating book when I finish it, but today I want to consider a point made in Chapter 2 (Units and Geometry), which addresses the tricky issue of measuring light intensity as a function of either frequency or wavelength. Johnsen favors using wavelength whenever possible.

However, one critical issue must be discussed before we put frequency away for good. It involves the fact that light spectra are histograms. Suppose you measure the spectrum of daylight, and that the value at 500 nm is 15 photons/cm²/s/nm. That doesn’t mean that there are 15 photons/cm²/s with a wavelength of exactly 500 nm. Instead, it means that, over a 1-nm-wide interval centered on a wavelength of 500 nm, you have 15 photons/cm²/s. The bins in a spectrum don’t have to be 1 nm wide, but they all must have the same width.

Let’s suppose all the bins are 1 nm wide and centered on whole numbers (i.e., one at 400 nm, one at 401 nm, etc.). What happens if we convert these wavelength values to their frequency counterparts? Let’s pick the wavelengths of two neighboring bins and call them λ₁ and λ₂. The corresponding frequencies ν₁ and ν₂ are equal to c/λ₁ and c/λ₂, where c is the speed of light. We know that λ₁−λ₂ equals 1 nm, but what does ν₁−ν₂ equal?

ν₁−ν₂ = … = −c/λ₁²

…So the width of the frequency bins depends on the wavelengths they correspond to, which means they won’t be equal! In fact, they are quite unequal. Bins at the red end of the spectrum (700 nm) are only about one-third as wide as bins at the blue end (400 nm). This means that a spectrum generated using bins with equal frequency intervals would look different from one with equal wavelength intervals. So which one is correct? Neither or both. The take-home message is that the shape of a spectrum depends on whether you have equal frequency bins or equal wavelength bins.

Johnsen goes on to note that the wavelength at which the spectrum is maximum depends on if you use equal frequency or equal wavelength bins. It does not make sense to say that the spectrum of, say, sunlight peaks at a particular wavelength, unless you specify the type of spectrum you are using. Furthermore, you cannot unambiguously say light is “white” (a uniform spectrum). White light using equal wavelength bins is not white using equal frequency bins. Fortunately, if you integrate the spectrum, you get the same value regardless of if you express it in terms of wavelength or frequency.

Russ Hobbie and I discuss this issue in Chapter 14 (Atoms and Light) of the 4th edition of Intermediate Physics for Medicine and Biology.

Early measurements of the radiation function were done with equipment that made measurements vs. wavelength. It is also possible to measure vs. frequency. To rewrite the radiation function in terms of frequency, let λ₁ and λ₂ =  λ₁ + dλ be two slightly different wavelengths, with power W_λ(λ, T) dλ emitted per unit surface area at wavelengths between λ₁ and λ₂. The same power must be emitted between frequencies ν₁ = c/λ₁ and ν₂ = c/λ₂:

W_ν(ν,T) dν = W_λ(λ,T) dλ .     (14.35)

Since ν = c/λ, dν/dλ = − c/λ², and

|dν| = + c λ² |dλ| .                 (14.36)

... This transformation is shown in Fig. 14.24. The amount of power per unit area radiated in the 0.5 μm interval between two of the vertical lines in the graph on the lower right is the area under the curve of W_λ between these lines. The graph on the upper right transforms to the corresponding frequency interval. The radiated power, which is the area under the W_ν curve between the corresponding frequency lines on the upper left, is the same. We will see this same transformation again when we deal with x rays. Note that the peaks of the two curves are at different frequencies or wavelengths.

Students who prefer visual explanations should see Fig. 14.24, which Russ drew. It is one of my favorite pictures in our book, and provides an illuminating comparison of the two spectra.

One detail I should mention: why in Eq. 14.36 do we use absolute values to eliminate the minus sign introduced by the derivative dν/dλ? Typically, when you integrate a spectrum, you start from the lower frequency and go to the higher frequency (say, zero to infinity), and you start from the shorter wavelength and go to the longer wavelength (again, zero to infinity). However, zero frequency corresponds to an infinite wavelength, and an infinite frequency corresponds to zero wavelength. So, really one case should be integrated forward (zero to infinity) and the other backwards (infinity to zero). If we keep the convention of always integrating from zero to infinity in both cases, we introduce an extra minus sign, which cancels the minus sign introduced by dν/dλ.

Sometimes it helps to have an elementary example to illustrate these ideas. Therefore, I have developed a new homework problem that introduces an extremely simple spectrum for which you can do the math fairly easily, thereby allowing you to focus on the physical interpretation. Enjoy.

Section 14.7

Problem 23 ½ Let W_ν(ν) = A ν (ν_ο - ν) for ν less than ν_ο, and W_ν(ν) = 0 otherwise.
(a) Plot W_ν(ν) versus ν.
(b) Calculate the frequency corresponding to the maximum of W_ν(ν), called ν_max.
(c) Let λ_ο = c/ν_ο and λ_max = c/ν_max. Write λ_max in terms of λ_ο.
(d) Integrate W_ν(ν) over all ν to find W_tot.
(e) Use Eqs. 14.35 and 14.36 to calculate W_λ(λ).
(f) Plot W_λ(λ) versus λ.
(g) Calculate the wavelength corresponding to the maximum of W_λ(λ), called λ^*_max, in terms of λ_ο.
(h) Compare λ_max and λ^*_max. Are they the same or different? If λ_ο is 400 nm, calculate λ_max and λ^*_max? What part of the electromagnetic spectrum is each of these in?
(i) Integrate W_λ(λ) over all λ to find W^*_tot. Compare W_tot and W^*_tot. Are they the same or different?

Stirling’s Formula!

Factorials are used in many branches of mathematics and physics, and particularly in statistical mechanics. One often needs the natural logarithm of a factorial, ln(n!). In Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I use Stirling’s approximation to compute ln(n!). We analyze this approximation in Appendix I.

There is a very useful approximation to the factorial, called Stirling’s approximation:

ln(n!) = n ln n – n .

To derive it, write ln(n!) as

ln(n!) = ln 1 + ln 2 + … + ln n = ∑ ln m

The sum is the same as the total area of the rectangles in Fig. I.1, where the height of each rectangle is ln m and the width of the base is one. The area of all the rectangles is approximately the area under the smooth curve, which is a plot of ln m. The area is approximately

∫₁ⁿln m dm = [m ln m – m]₁ⁿ = n ln n – n + 1.

This completes the proof.

David Mermin—one of my favorite writers among physicists—has much more to say about Stirling’s approximation in his American Journal of Physics article “Stirling’s Formula!” (leave it to Mermin to work an exclamation point into his title). He writes Stirling’s approximation as n! = √(2 π n) (n/e)ⁿ. Taking the natural logarithm of both sides gives ln(n!) = ln(2 π n)/2 + n ln n – n . For large n, the first term is small, and the result is the same as Russ and I present. I wonder what affect the first term has on the approximation? For small n, it makes a big difference! In Table I.1 of our textbook, we compute the accuracy of n ln n – n for n = 5. In that case, n! = 120, so ln(n!) = ln(120) = 4.7875 and 5 ln 5 – 5 = 3.047, giving a 36% error. But ln(10 π)/2 + 5 ln 5 – 5 = 4.7708, implying an error of 0.35 %, so Mermin’s formula is much better than ours. (I shouldn’t call it Mermin’s formula; I believe Stirling himself derived n! = √(2 π n) (n/e)ⁿ.)

Mermin doesn’t stop there. He analyzes the approximation in more detail, and eventually derives an exact formula for n! that looks like Stirling’s approximation given above, except multiplied by an infinite product. In the process, he looks at the approximation for the base of the natural logarithms, e, presented in Chapter 2 of Intermediate Physics for Medicine and Biology, e = (1 + 1/N)^N, and shows that a “spectacularly better” approximation for e is (1 + 1/N)^N+1/2. He then goes on to derive an improved approximation for n!, which is his expression for Stirling’s formula times e^(1/12n). Perhaps getting carried away, he then derives even better approximations.

All of this matters little in applications to statistical mechanics, where n is on the order of Avogadro’s number, in which case the first term in Stirling’s formula is utterly negligible. Nevertheless, I urge you to read Mermin’s paper, if only to enjoy the elegance of his writing. To learn more about Mermin’s views on writing physics, see his essay “Writing Physics.”