In Vitro Evaluation of a 4-leaf Coil Design for Magnetic Stimulation of Peripheral Nerve

In the comments to last week’s blog entry, Frankie asks if there is a way to “safely, reversibly block nerve conduction (first in the lab, then in the clinic) with an exogenously applied E and M signal?” This is a fascinating question, and I may have an answer.

When working at the National Institutes of Health in the early 1990’s, Peter Basser and I analyzed magnetic stimulation of a peripheral nerve. The mechanism of excitation is similar to the one Frank Rattay developed for stimulating a nerve axon with an extracellular electrode. You can find Rattay’s method described in Problems 38–41 of Chapter 7 in the 4th edition of Intermediate Physics for Medicine and Biology. The bottom line is that excitation occurs where the spatial derivative of the electric field is largest. I have already recounted how Peter and I derived and tested our model, so I won’t repeat it today.

If you accept the hypothesis that excitation occurs where the electric field derivative is large, then the traditional coil design for magnetic stimulation—a figure-of-eight coil—has a problem: the axon is not excited directly under the center of the coil (where the electric field is largest), but a few centimeters from the center (where the electric field gradient is largest). What a nuisance. Doctors want a simple design like a crosshair: excitation should occur under the center. X marks the spot.

As I pondered this problem, I realized that we could build a coil just like the doctor ordered. It wouldn’t have a figure-of-eight design. Rather, it would be two figure-of-eights side by side. I called this the four leaf coil. With this design, excitation occurs directly under the center.

A four-leaf-coil used for
magnetic stimulation of nerves.

John Cadwell of Cadwell Labs built a prototype of this coil; an x ray of it is shown above. We wanted to test the coil in a well-controlled animal experiment, so we sent it to Paul Maccabee at the State University of New York Health Science Center in Brooklyn. Paul did the experiments, and we published the results in the journal Electroencephalography and clinical Neurophysiology (Volume 93, Pages 68–74, 1994). The paper begins

Magnetic stimulation is used extensively for non-invasive activation of human brain, but is not used as widely for exciting limb peripheral nerves because of both the uncertainty about the site of stimulation and the difficulty in obtaining maximal responses. Recently, however, mathematical models have provided insight into one mechanism of peripheral nerve stimulation: peak depolarization occurs where the negative derivative of the component of the induced electric field parallel to nerve fibers is largest (Durand et al. 1989; Roth and Basser 1990). Both in vitro (Maccabee et al. 1993) and in vivo (Nilsson et al. 1992) experiments support this hypothesis for uniform, straight nerves. Based on these results, a 4-leaf magnetic coil (MC) design has been suggested that would provide a well defined site of stimulation directly under the center of the coil (Roth et al. 1990). In this note, we perform in vitro studies which test the performance of this new coil design during magnetic stimulation of a mammalian peripheral nerve.

Maccabee’s experiments showed that the coil worked as advertised. In the discussion of the paper we concluded that “the 4-leaf coil design provides a well defined stimulus site directly below the center of the coil.”

This is a nice story, but it’s all about exciting an action potential. What does it have to do with Frankie’s goal of blocking an action potential? Well, if you flip the polarity of the coil current, instead of depolarizing the nerve under the coil center, you hyperpolarize it. A strong enough hyperpolarization should block propagation. We wrote

In a final type of experiment, performed on 3 nerves, the action potential was elicited electrically, and a hyperpolarizing magnetic stimulus was applied between the stimulus and recording sites at various times. The goal was to determine if a precisely timed stimulus could affect action potential propagation. Using induced hyperpolarizing current at the coil center, with a strength that was approximately 3 times greater than that needed to excite by depolarization at that location, we never observed a block of the action potential. Moreover, no significant effect on the latency of the action potential propagating to the recording site was observed… Our magnetic stimulator was able to deliver stimuli with strengths up to only 2 or 3 times the threshold strength, and therefore the magnetic stimuli were probably too weak to block propagation. It is possible that such phenomena might be observed using a more powerful stimulator.

Frankie, I have good news and bad news. The good news is that you should be able to reversibly block nerve conduction with magnetic stimulation using a four-leaf coil. The bad news is that it didn’t work with Paul’s stimulator; perhaps a stronger stimulator would do the trick. Give it a try.

The Bubble Experiment

When I was a graduate student, my mentor John Wikswo assigned to me the job of measuring the magnetic field of a nerve axon. This experiment required me to dissect the ventral nerve cord out of a crayfish, thread it through a wire-wound ferrite-core toroid, immerse the nerve and toroid in saline, stimulate one end of the nerve, and record the magnetic field produced by the propagating action currents. One day as I was lowering the instrument into the saline bath, a bubble got stuck in the gap between the nerve and the inner surface of the toroid. “Drat” I thought as I searched for a needle to remove it. But before I could poke it out I wondered “how will the bubble affect the magnetic signal?”

A wire-wound, ferrite-core toroid,
used to measure the magnetic field of a nerve.

To answer this question, we need to review some magnetism. Ampere’s law states that the line integral of the magnetic field around a closed path is proportional to the net current passing through a surface bounded by that path. For my experiment, that meant the magnetic signal depended on the net current passing through the toroid. The net current is the sum of the current inside the nerve axon and that fraction of the current in the saline bath that threads the toroid—the return current. In general, these currents flow in opposite directions and partially cancel. One of the difficulties I faced when interpreting my data was determining how much of the signal was from intracellular current and how much was from return current.

I struggled with this question for months. I calculated the return current with a mathematical model involving Fourier transforms and Bessel functions, but the calculation was based on many assumptions and required values for several parameters. Could I trust it? I wanted a simpler way to find the return current.

Then along came the bubble, plugging the toroid like Pooh stuck in Rabbit’s front door. The bubble blocked the return current, so the magnetic signal arose from only the intracellular current. I recorded the magnetic signal with the bubble, and then—as gently as possible—I removed the bubble and recorded the signal again. This was not easy, because surface tension makes a small bubble in water sticky, so it stuck to the toroid as if glued in place. But I eventually got rid of it without stabbing the nerve and ending the experiment.

To my delight, the magnetic field with the bubble was much larger than when it was absent. The problem of estimating the return current was solved; it’s the difference of the signal with and without the bubble. I reported this result in one of my first publications (Roth, B. J., J. K. Woosley and J. P. Wikswo, Jr., 1985, “An Experimental and Theoretical Analysis of the Magnetic Field of a Single Axon,” In: Biomagnetism: Applications and Theory, Weinberg, Stroink and Katila, Eds., Pergamon Press, New York, pp. 78–82.).

When taking data from a crayfish nerve, the toroid and axon were lifted out of the bath for a short time. […] When again placed in the bath an air bubble was trapped in the center of the toroid, filling the space between the axon and the toroid inner surface. […] Taking advantage of this fortunate occurrence, data were taken with and without the bubble present. […] The magnetic field with the bubble present […] is narrower and larger than the field with the toroid filled with saline.

The magnetic field of a nerve axon
with and without a bubble trapped
between the nerve and toroid.

On the day of the bubble experiment I was lucky. I didn’t plan the experiment. I wasn’t wise enough or thoughtful enough to realize in advance that a bubble was the ideal way to eliminate the return current. But when I looked through the dissecting microscope and saw the bubble stuck there, I was bright enough to appreciate my opportunity. “Chance favors the prepared mind.”

I have a habit of turning all my stories into homework problems. You will find the bubble story in the 4th edition of Intermediate Physics for Medicine and Biology, Problem 39 of Chapter 8. Focus on part (b).

Problem 39 A coil on a magnetic toroid as in Problem 38 is being used to measure the magnetic field of a nerve axon.

(a) If the axon is suspended in air, with only a thin layer of extracellular fluid clinging to its surface, use Ampere’s law to determine the magnetic field, B, recorded by the toroid.

(b) If the axon is immersed in a large conductor such as a saline bath, B is proportional to the sum of the intracellular current plus that fraction of the extracellular current that passes through the toroid (see Problem 13). Suppose that during an experiment an air bubble is trapped between the axon and the inner radius of the toroid? How is the magnetic signal affected by the bubble? See Roth et al. (1985).

The Bowling Ball and the Feather

Dropping a feather and a ball in a vacuum to show that they fall at the same rate is a classic physics demonstration. We have a version of this demo at Oakland University, but it is not very effective. A small ball and a feather are in a tube about 1 meter long and a few centimeters in diameter. We have vacuum pump to remove the air, but it is difficult to see the objects from the back of the room, and often they bump into the wall of the tube, slowing them down. I have never found it useful. Yet, the physical principle being demonstrated is fundamental. The gravitational mass in Newton’s universal law of gravity and the inertial mass in Newton’s second law of motion cancel out, so that all objects fall downward with acceleration g = 9.8 m/s².

This result is unexpected because in everyday life we experience air friction. When you include air friction, objects do not all fall at the same rate. Russ Hobbie and I illustrate this point in Problem 28 of Chapter 2 in the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 28 When an animal of mass m falls in air, two forces act on it: gravity, mg, and a force due to air friction. Assume that the frictional force is proportional to the speed v.

(a) Write a differential equation for v based on Newton’s second law, F = m(dv/dt).

(b) Solve this differential equation. (Hint: Compare your equation with Eq. 2.24.)

(c) Assume that the animal is spherical, with radius a and density ρ. Also, assume that the frictional force is proportional to the surface area of the animal. Determine the terminal speed (speed of descent in steady state) as a function of a.

(d) Use your result in part (c) to interpret the following quote by J. B. S. Haldane [1985]: “You can drop a mouse down a thousand-yard mine shaft; and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.”

If we ignore air fraction, v = gt; the acceleration is g and does not depend on mass. With air friction, objects reach a terminal velocity that depends on their mass. We are all so used to seeing a feather float downward with its motion dominated by air friction that it is difficult to believe it could ever fall as fast as a ball. To persuade students that this behavior does indeed happen, to convince them that in a vacuum a feather drops like a rock, we need a powerful demonstration. The result is so significant, and so nonintuitive, that the demo must be dramatic and memorable.

Now we have it. Watch this amazing video with British Physics Professor Brian Cox. He found the biggest vacuum chamber in the world—a large room used by NASA to test space vehicles—and inside it he dropped a bowling ball and a feather simultaneously from the same height. When the room was filled with air, the feather slowly fluttered to the ground. When the room was evacuated, the feather stayed right beside the bowling ball all the way down. The visual effect is stunning. Cox has a fine sense of drama, building the tension until the final sensational experiment. The video is less than five minutes long. You’ve gotta see it.

The MCAT and IPMB

The Medical College Admission Test, famously known as the MCAT, is an exam taken by students applying to medical school. The Association of American Medical Colleges will introduce a new version of the MCAT next year, focusing on competencies rather than on prerequisite classes. How well does the 4th edition of Intermediate Physics for Medicine and Biology prepare premed students for the MCAT?

The new MCAT will be divided into four sections, and the one most closely related to IPMB deals with the chemical and physical foundations of biological systems. Within that section are two foundational concepts, of which one is about how “complex living organisms transport materials, sense their environment, process signals, and respond to changes that can be understood in terms of physical principles.” This concept is further subdivided into five categories. Below, I review the topics included in these categories and indicate what chapter in IPMB addresses each.

MCAT: Translational motion, forces, work, energy, and equilibrium in living systems

IPMB: Chapter 1 discusses mechanics, including forces and torques, with applications to biomechanics. Work and energy are introduced in Chapter 1, and analyzed in more detail in Chapter 3 on statistical mechanics and thermodynamics (parts of thermodynamics are included under another foundational concept dealing mostly with chemistry). Periodic motion is covered in Chapter 11, which discusses the amplitude, frequency and phase of an oscillator. Waves are analyzed in Chapter 13 about sound and ultrasound.

MCAT: Importance of fluids for the circulation of blood, gas movement, and gas exchange

IPMB: Chapter 1 analyzes fluids, including buoyancy, hydrostatic pressure, viscosity, Poiseuille flow, turbulence, and the circulatory system. Much of this material is not covered in a typical introductory physics class. Chapter 3 introduces absolute temperature, the ideal gas law, heat capacity, and Boltzmann’s constant.

MCAT: Electrochemistry and electrical circuits and their elements

IPMB: Chapters 6 and 7 cover electrostatics, including charge, the electric field, current, voltage, Ohm’s law, resistors, capacitors, and nerve conduction. Chapter 8 discusses the magnetic field and magnetic forces.

MCAT: How light and sound interact with matter

IPMB: Sound is analyzed in Chapter 13, including the speed of sound, the decibel, attenuation, reflection, the Doppler effect, ultrasound, and the ear. Chapter 14 covers light, photon energy, color, interference, and the eye. This chapter also describes absorption of light in the infrared, visible, and ultraviolet. Chapter 18 analyzes nuclear magnetic resonance.

MCAT: Atoms, nuclear decay, electronic structure, and atomic chemical behavior

IPMB: Chapter 17 is about nuclear physics and nuclear medicine, covering isotopes, radioactive decay, and half life. Atoms and atomic energy levels are explained in Chapter 14.

MCAT: General mathematical concepts and techniques

IPMB: Chapter 1 and many other chapters require students to estimate numerically. Chapter 2 covers linear, semilog, and log-log plots, and exponential growth. Metric units and dimensional analysis are used everywhere. Probability concepts are discussed in Chapter 3 and other chapters. Basic math skills such as exponentials, logarithms, scientific notation, trigonometry, and vectors are reinforced throughout the book and in the homework problems, and are reviewed in the Appendices.

The MCAT section about biological and biochemical foundations of living systems includes diffusion and osmosis (discussed in Chapters 4 and 5 of IPMB), membrane ion channels (covered in Chapter 9), and feedback regulation (analyzed in Chapter 10).

Overall, Intermediate Physics for Medicine and Biology covers many of the topics tested on the MCAT. A biological or medical physics class based on IPMB would prepare a student for the exam, and would reinforce problem solving skills and teach the physical principles underlying medicine, resulting in better physicians.

I’m a realist, however. I know premed students take lots of classes, and they don’t want to take more physics beyond a two-semester introduction, especially if the class might lower their grade point average. I have tried to recruit premed students into my Biological Physics (PHY 325) and Medical Physics (PHY 326) classes here at Oakland University, with little success. Perhaps if they realized how closely the topics and skills required for the MCAT correspond to those covered by Intermediate Physics for Medicine and Biology they would reconsider.

To learn more about how to prepare for the physics competencies on the MCAT, see Robert Hilborn’s article “Physics and the Revised Medical College Admission Test,” published in the American Journal of Physics last summer (Volume 82, Pages 428–433, 2014).

Faraday, Maxwell, and the Electromagnetic Field

Faraday, Maxwell, and the
Electromagnetic Field,
by Nancy Forbes and Basil Mahon.

Michael Faraday and James Clerk Maxwell are two of my scientific heroes. So, when I saw the book Faraday, Maxwell, and the Electromagnetic Field displayed in the new book section of the Rochester Hills Public Library, I had to check it out. In their introduction, Nancy Forbes and Basil Mahon write

It is almost impossible to overstate the scale of Faraday and Maxwell’s achievement in bringing the concept of the electromagnetic field into human thought. It united electricity, magnetism, and light into a single, compact theory; changed our way of life by bringing us radio, television, radar, satellite navigation, and mobile phones; inspired Einstein’s special theory of relativity; and introduced the idea of field equations, which became the standard form used by today’s physicists to model what goes on in the vastness of space and inside atoms.

I have read previous biographies of both Faraday and Maxwell, so their story was familiar to me. But one anecdote about Faraday I had never heard before.

The Royal Institution’s Friday Evening Discourses had by now become an institution in their own right. The lecture on April 3, 1846, turned out to be a historic occasion, although none of the audience recognized it as such and the whole thing happened by chance in a rather bizarre fashion. Charles Wheatstone was to have been the latest in a long line of distinguished speakers, but he panicked and ran away just as he was due to make his entrance[!]. Although amply confident in his professional dealings as a scientist, inventor, and businessman, Wheatstone was notoriously shy of speaking in public, and Faraday had taken a gamble when engaging him to talk about his latest invention, the electromagnetic chronoscope—a device for measuring small time intervals, like the duration of a spark. The gamble had failed, and Faraday was left with the choice of sending disappointed customers home or giving the talk himself. He chose to talk, but he ran out of things to say on the advertised topic well before the allotted hour was up.

Caught off-guard, he did what he had never done before and gave the audience a glimpse into his private meditations on matter, lines of force, and light. In doing so, he draw an extraordinary prescient outline of the electromagnetic theory of light, as it would be developed over the next sixty years….

Writing about Maxwell’s electromagnetic theory, Forbes and Mahon say

The theory’s construction had been an immense creative effort, sustained over a decade and inspired, from first to last, by the work of Michael Faraday. Thanks to Faraday’s meticulous recording of his findings and thoughts in his Experimental Researches in Electricity, Maxwell had been able to see the world as Faraday did, and, by bringing together Faraday’s vision with the power of Newtonian mathematics, to give us a new concept of physical reality, using the power of mathematics. But mathematics would not have been enough without Maxwell’s own near-miraculous intuition; witness the displacement current, which gave the theory its wonderful completeness. The theory belongs to both Maxwell and Faraday.

Russ Hobbie and I discuss electricity and magnetism in the 4th edition of Intermediate Physics for Medicine and Biology. Chapters 6 and 7 show how electrostatics can be used to describe how nerves and muscles behave. Chapter 8 discusses magnetism and electromagnetism. Chapter 9 examines in more detail how electromagnetic fields interact with the body, and Chapter 18 describes how magnetism leads to magnetic resonance imaging. So, it’s safe to say that IPMB has Maxwell and Faraday’s influence throughout.

If you want to learn more about Maxwell’s work, I suggest Maxwell on the Electromagnetic Field: A Guided Study by Thomas K. Simpson. He reproduces Maxwell’s three landmark papers, and provides the necessary context and background to understand them. Forbes and Mahon talk briefly at the end of their book about the scientists who came after Maxwell and firmly established his theory. For more on this topic, read The Maxwellians, one of the best histories of science I know. I enjoyed Faraday, Maxwell, and the Electromagnetic Field. It provides a great introduction to a fascinating story in the history of science.

Low Reynolds Number Flows

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Reynolds number.

The importance of turbulence (nonlaminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by N_R = LVρ/η where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs….

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V , and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces. Since viscosity is a form of internal friction in the fluid, work done on the system by the external forces is transformed into thermal energy. The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive. They are nicely described by Purcell (1977).

“Life at Low Reynolds Number,”
by Edward Purcell.

Edward Purcell’s 1977 paper in the American Journal of Physics provides much insight into low Reynolds number flow, and is a classic. But to learn from this paper you have to read it. Nowadays, students often want to learn from videos rather than reading text (don’t get me started...). Fortunately, a good video exists to explain low-Reynolds-number flow, and it has been around for many years. Click here to watch G. I. Taylor illustrate low Reynolds flow. Sir Geoffrey Ingram Taylor (1886–1975) was an English physicist and an expert in fluid dynamics. He contributed to the Manhattan Project by analyzing the hydrodynamics of implosion needed to develop a plutonium bomb. Among his many contributions is the description of Taylor-Couette flow between two rotating cylinders.

The video shows a beautiful example of reversibility of low Reynolds number flow. A blob of dye is placed into the fluid between the cylinders, one of the cylinders is rotated, and the dye spreads throughout the fluid. They rotation is then reversed, and the dye eventually returns to its original localized blob. This demonstration always reminds me of the formation of a spin echo during magnetic resonance imaging (see Chapter 18 of IPMB), where all spins begin in phase after a 90 degree radio-frequency magnetic field pulse. Then, because of slight heterogeneities in the static magnetic field, the spins dephase as they all rotate at slightly different Larmor frequencies. If you reverse their positions using a 180 degree RF pulse, the spins eventually return to their original configuration, all in phase (the echo). When you think about it, the formation of spin echoes during MRI is nearly as “magical” as the reformation of the dye blob in Taylor’s cylinder.

The video also analyzes how small machines can “swim” at low Reynolds numbers. He even has built small devices, one a machine that zooms through water but just sits there in a viscous fluid, and another that has a helix for a tail and that swims—slowly but steadily—through the viscous fluid. This example reminds me of both Purcell’s article and the research of Howard Berg, who studies how E coli bacteria swim.

To learn more about Taylor’s life and work, watch Katepalli Sreenivasan’s lecture, or read The Life and Legacy of G. I. Taylor by George Batchelor.

The Biological Risk of Ultraviolet Light From the Sun

In Section 14.9 (Blue and Ultraviolet Radiation) of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the biological impact of ultraviolet radiation from the sun. Figure 14.28 in IPMB illustrates a remarkable fact about UV light: only a very narrow slice of the spectrum presents a risk for damaging our DNA. Why? For shorter wavelengths, the UV light incident upon the earth’s atmosphere is almost entirely absorbed by ozone and never reaches the earth’s surface. For longer wavelengths, the UV photons do not have enough energy to damage DNA. Only what is known as UVB radiation (wavelengths of 280–315 nm) poses a major risk.

This does not mean that other wavelengths of UV light are always harmless. If the source of UV radiation is, say, a tanning booth rather than the sun, then you are not protected by miles of ozone-containing atmosphere, and the amount of dangerous short-wavelength UV light depends on the details of the light source and the light’s ability to penetrate our body. Also, UVA radiation (315–400 nm) is not entirely harmless. UVA can penetrate into the dermis, and it can cause skin cancer by other mechanisms besides direct DNA damage, such as production of free radicals or suppression of the body’s immune system. Nevertheless, Fig. 14.28 shows that UVB light from the sun is particularly effective at harming our genes.

Russ and I obtained Fig. 14.28 from a book chapter by Sasha Madronich (“The Atmosphere and UV-B Radiation at Ground Level,” In Environmental UV Photobiology, Plenum, New York, 1993). Madronich begins

Ultraviolet (UV) radiation emanating from the sun travels unaltered until it enters the earth’s atmosphere. Here, absorption and scattering by various gases and particles modify the radiation profoundly, so that by the time it reaches the terrestrial and oceanic biospheres, the wavelengths which are most harmful to organisms have been largely filtered out. Human activities are now changing the composition of the atmosphere, raising serious concerns about how this will affect the wavelength distribution and quantity of ground-level UV radiation.

Madronich wrote his article in the early 1990s, when scientists were concerned about the development of an “ozone hole” in the atmosphere over Antarctica. Laws limiting the release of chlorofluorocarbons, which catalyze the break down of ozone, have resulted in a slow improvement in the ozone situation. Yet, the risk of skin cancer continues to be quite sensitive to ozone concentration in the atmosphere.

Our exposure to ozone is also sensitive to the angle of the sun overhead. Figure 14.28 suggests that at noon in lower latitudes, when the sun is directly overhead, the ozone exposure is about three times greater than when the sun is at an angle of 45 degrees (here in Michigan, this would be late afternoon in June, or noon in September; we never make it to 45 degrees in December). The moral of this story: Beware of exposure to UV light when frolicking on the Copacabana beach at noon on a sunny day in January!

A Log-Log Plot of the Blackbody Spectrum

Section 14.7 (Thermal Radiation) of the 4th edition of Intermediate Physics for Medicine and Biology contains one of my favorite illustrations: Figure 14.24, which compares the blackbody spectrum as a function of wavelength λ and as a function of frequency ν. One interesting feature of the blackbody spectrum is that its peak (the wavelength or frequency for which the most thermal radiation is emitted) is different depending on if you plot it as a function of wavelength (W_λ(λ,T) in units of W m⁻³) or frequency (W_ν(ν,T) in units of W s m⁻²). The units make more sense if we express the units of W_λ as W m⁻² per m, and the units of W_ν as W m⁻² per Hz.

A few weeks ago I discussed the book The First Steps in Seeing, in which the blackbody spectrum was plotted using a log-log scale. This got me to thinking, “I wonder how Fig. 14.24 would look if all axes were logarithmic?” The answer is shown below.

Figure 14.24 from Intermediate Physics for Medicine and Biology,
but plotted using a log-log scale.

The caption for Fig. 14.24 is “The transformation from W_λ(λ,T) to W_ν(ν,T) is such that the same amount of power per unit area is emitted in wavelength interval (λ, dλ) and the corresponding frequency interval (ν, dν). The spectrum shown is for a blackbody at 3200 K.” I corrected the wrong temperature T in the caption as printed in the 4th edition.

The bottom right panel of the above figure is a plot of W_λ versus λ. For this temperature the spectrum peaks just a bit below λ = 1 μm. At longer wavelengths, it falls off approximately as λ⁻⁴ (shown as the dashed line, known as the Rayleigh-Jeans approximation). At short wavelengths, the spectrum rises abruptly and is exponential.

The top left panel contains a plot of W_ν versus ν. The spectrum peaks at a frequency just below about 0.3 THz. At low frequencies it increases approximately as ν² (again, the Rayleigh-Jeans approximation). At high frequencies the spectrum falls dramatically and exponentially.

The connection between these two plots is illustrated in the upper right panel, which plots the relationship ν = c/λ. This equation has nothing to do with blackbody radiation, but merely shows a general relationship between frequency, wavelength, and the speed of light for electromagnetic radiation.

Why is it useful to show these functions in a log-log plot? First, it reinforces the concepts Russ Hobbie and I introduced in Chapter 2 of IPMB (Exponential Growth and Decay). In a log-log plot, power laws appear as straight lines. Thus, in the book’s version of Fig. 14.24 the equation ν = c/λ is a hyperbola, but in the log-log version this is a straight line with a slope of negative one. Furthermore, the Rayleigh-Jeans approximation implies a power-law relationship, which is nicely illustrated on a log-log plot by the dashed line. In the book’s version of the figure, W_λ falls off at both large and small wavelengths, and at first glance the rates of fall off look similar. You don’t really see the difference until you look at very small values of W_λ, which are difficult to see in a linear plot but are apparent in a logarithmic plot. The falloff at short wavelengths is very abrupt while the decay at long wavelengths is gradual. This difference is even more striking in the book’s plot of W_ν. The curve doesn’t even go all the way to zero frequency in Fig. 14.24, making its limiting behavior difficult to judge. The log-log plot clearly shows that at low frequencies W_ν rises as ν².

Both the book’s version and the log-log version illustrate how the two functions peak at different regions of the electromagnetic spectrum, but for this point the book’s linear plot may be clearer. Another advantage of the linear plot is that I have an easier time estimating the area under the curve, which is important for determining the total power emitted by the blackbody and the Stefan-Boltzmann law. Perhaps there is some clever way to estimate areas under a curve on a log-log plot, but it seems to me the log plot exaggerates the area under the small frequency section of the curve and understates the area under the large frequencies (just as on a map the Mercator projection magnifies the area of Greenland and Antarctica). If you want to understand how these functions behave completely, look at both the linear and log plots.

Yet another way to plot these functions would be on a semilog plot. The advantage of semilog is that an exponential falloff shows up as a straight line. I will leave that plot as an exercise for the reader.

For those who want to learn about the derivation and history of the blackbody spectrum, I recommend Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (although any good modern physics book should discuss this topic). A less mathematical but very intuitive description of why W_λ and W_ν peak at different parts of the spectrum is given in The Optics of Life. For a plot of photon number (rather than energy radiated) as a function of λ or ν, see The First Steps in Seeing.

A Theoretical Model of Magneto-Acoustic Imaging of Bioelectric Currents

Twenty years ago, I became interested in magneto-acoustic imaging, primarily influenced by the work of Bruce Towe that was called to my attention by my dissertation advisor and collaborator John Wikswo. (See, for example, Towe and Islam, “A Magneto-Acoustic Method for the Noninvasive Measurement of Bioelectric Currents,” IEEE Trans. Biomed. Eng., Volume 35, Pages 892–894, 1988). The result was a paper by Wikswo, Peter Basser, and myself (“A Theoretical Model of Magneto-Acoustic Imaging of Bioelectric Currents,” IEEE Trans. Biomed. Eng., Volume 41, Pages 723–728, 1994). This was my first foray into biomechanics, a subject that has become increasingly interesting to me, to the point where now it is the primary focus of my research (but that’s another story; for a preview look here).

A Treatise on the Mathematical
Theory of Elasticity,
by A. E. H. Love.

I started learning about biomechanics mainly through my friend Peter Basser. We both worked at the National Institutes of Health in the early 1990s. Peter used continuum models in his research a lot, and owned a number of books on the subject. He also loved to travel, and would often use his leftover use-or-lose vacation days at the end of the year to take trips to exotic places like Kathmandu. When he was out of town on these adventures, he left me access to his personal library, and I spent many hours in his office reading classic references like Schlichting’s Boundary Layer Theory and Love’s A Treatise on the Mathematical Theory of Elasticity. Peter and I also would read each other’s papers, and I learned much continuum mechanics from his work. (NIH had a rule that someone had to sign a form saying they read and approved a paper before it could be submitted for publication, so I would give my papers to Peter to read and he would give his to me.) In this way, I became familiar enough with biomechanics to analyze magneto-acoustic imaging. Interestingly, we published our paper in the same year Basser began publishing his research on MRI diffusion tensor imaging, for which he is now famous (see here).

As with much of my research, our paper on magneto-acoustic imaging addressed a simple “toy model”: an electric dipole in the center of an elastic, conducting sphere exposed to a uniform magnetic field. We were able to calculate the tissue displacement and pressure analytically for the cases of a magnetic field parallel and perpendicular to the dipole. One of my favorite results in the paper was that we found a close relationship between magneto-acoustic imaging and biomagnetism.

“Magneto-acoustic pressure recordings and biomagnetic measurements image action currents in an equivalent way: they both have curl J [the curl of the current density] as their source.”

For about ten years, our paper had little impact. A few people cited it, including Amalric Montalibet and later Han Wen, who each developed a method to use ultrasound and the Lorentz force to generate electrical current in tissue. I’ve described this work before in a review article about the role of magnetic forces in medicine and biology, which I have mentioned before in this blog. Then, in 2005 Bin He began citing our work in a long list of papers about magnetoacoustic tomography with magnetic induction, which again I've written about previously. His work generated so much interest in our paper that in 2010 alone it was cited 19 times according to Google Scholar. Of course, it is gratifying to see your work have an impact.

But the story continues with a more recent study by Pol Grasland-Mongrain of INSERM in France. Building on Montalibet’s work, Grasland-Mongrain uses an ultrasonic pulse and the Lorentz force to induce a voltage that he can detect with electrodes. The resulting electrical data can then be analyzed to determine the distribution of electrical conductivity (see Ammari, Grasland-Mongrain, et al. for one way to do this mathematically). In many ways, their technique is in competition with Bin He’s MAT-MI as a method to image conductivity.

Grasland-Mongrain also publishes his own blog about medical imaging. (Warning: The website is in French, and I have to rely on Google Translate to read it. It is my experience that Google has a hard time translating technical writing). There he discusses his most recent paper about imaging shear waves using the Lorentz force. Interestingly, shear waves in tissue is one of the topics Russ Hobbie and I added to the 5th edition of Intermediate Physics for Medicine and Biology, due out next year. Grasland-Mongrain’s work has been highlighted in Physics World and Focus Physics, and a paper about it appeared this year in Physical Review Letters, the most prestigious of all physics journals (and one I’ve never published in, to my chagrin).

I am amazed by what can happen in twenty years.


As a postscript, let me add a plug for toy models. Russ and I use a lot of toy models in IPMB. Even though such simple models have their limitations, I believe they provide tremendous insight into physical phenomena. I recently reviewed a paper in which the authors had developed a very sophisticated and complex model of a phenomena, but examination of a toy model would have told them that the signal they calculated was far, far to small to be observable. Do the toy model first. Then, once you have the insight, make your model more complex.

John H Hubbell

In the references at the end of Chapter 15 (Interaction of Photons and Charged Particles with Matter) in the 4th edition of Intermediate Physics for Medicine and Biology, you will find a string of publications authored by John H. Hubbell (1925–2007), covering a 27-year period from 1969 until 1996. Data from his publications are plotted in Fig. 15.2 (Total cross section for the interactions of photons with carbon vs. photon energy), Fig. 15.3 (Cross sections for the photoelectric effect and incoherent and coherent scattering from lead), Fig. 15.8 (The coherent and incoherent differential cross sections as a function of angle for 100-keV photons scattering from carbon, calcium, and lead), Fig. 15.14 (Fluorescence yields from K-, L-, and M-shell vacancies as a function of atomic number Z), and Fig. 15.16 (Coherent and incoherent attenuation coefficients and the mass energy absorption coefficient for water).

Hubbell’s 1982 paper “Photon Mass Attenuation and Energy-Absorption Coefficients from 1 keV to 20 MeV” (International Journal of Applied Radiation and Isotopes, Volume 33, Pages 1269–1290) has been cited 976 times according to the Web of Science. It has been cited so many times that it was selected as a citation classic, and Hubbell was invited to write a one-page reminiscence about the paper. It began modestly

Some papers become highly cited due to the creativity, genius, and vision of the authors, presenting seminal work stimulating and opening up new and multiplicative lines of research. Another, more pedestrian class of papers is “house-by-the-side-of-the-road” works, highly cited simply because these papers provide tools required by a substantial number of researchers in a single discipline or perhaps in several diverse disciplines, as is here the case.

At the time of his death, the International Radiation Physics Society Bulletin published the following obituary. 

The International Radiation Physics Society (IRPS) lost one of its major founding members, and the field of radiation physics one of its advocates and contributors of greatest impact, with the death this spring of John Hubbell.

John was born in Michigan in 1925, served in Europe in World War II [he received a bronze star], and graduated from the University of Michigan with a BSE (physics) in 1949 and MS (physics) in 1950. He then joined the National Bureau of Standards (NBS), later NIST, where he worked during his entire career. He married Jean Norford in 1955, and they had three children. He became best known and cited for National Standards Reference Data Series Report 29 (l969), “Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 keV to 100 GeV.” He was one of the two leading founding members of the International Radiation Physics Society in 1985, and he served as its President 1994–97. While he retired from NIST in 1988, he remained active there and in the affairs of IRPS, until the stroke that led to his death this year.

Learn more about John Hubbell here and here.