Figure 13.5

In Chapter 13 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss imaging using ultrasound. One of the key concepts in this chapter is the reflection of an ultrasonic wave at the boundary between tissues having different acoustic impedances. The acoustic impedance is equal to the square root of ρ/κ, where ρ is the density and κ is the compressibility. An ultrasonic image is formed by recording echoes of an ultrasonic wave and using the timing of those echoes to determine the distance from tissue boundaries.

Figure 13.5 from Intermediate Physics for Medicine and Biology.

Figure 13.5 shows schematically how part of a pressure wave approaching a surface (the incident wave) passes through the surface (the transmitted wave) and part travels back toward the wave’s source (the reflected wave, or echo). When you look closely, however, there is something odd about this figure: the transmitted wave has a larger amplitude than the incident wave. How can this be? Doesn’t this violate some conservation law? If we consider particles rather than waves, we would expect that if 100 particles were incident on a surface, perhaps 75 of them would be transmitted and 25 reflected. Figure 13.5 seems to imply that 100 incident particles result in 133 transmitted and 33 reflected! How?

The figure is not wrong; the problem is with our intuition. Pressure is not a conserved quantity. There is no reason to expect the sum of the pressures of the transmitted and reflected waves to equal the pressure of the incident wave. The amplitudes are consistent with equations 13.26 and 13.27 in IPMB relating the three waves. Yet there is a conserved quantity, one we all know: energy.

The intensity of a wave is the energy per unit area per unit time. The intensity I, pressure p, and acoustic impedance Z are related by equation 13.29: I = p²/2Z. The transmitted wave in Fig. 13.5 has a pressure amplitude that is 1.33 times the amplitude of the incident wave, but it is moving through a tissue that has twice the acoustic impedance (the caption says that for this figure, Z₂ = 2 Z₁). For simplicity, take the acoustic impedance on the left (the incident side of the boundary, region 1) to be Z₁ = 0.5 and the amplitude of the incident wave to be p_i = 1 (let’s not worry about units for now, because our goal is to compare the relative intensities of the three waves). In this case, the intensity of the incident wave is equal to one. If the transmitted pressure is 1.33 and the acoustic impedance on the right (region 2) is Z₂ = 1 (twice Z₁), then the transmitted intensity is (1.33)²/2 = 0.89. The reflected wave has amplitude 0.33, and is propagating through the tissue on the left, so its intensity is (0.33)²/1 = 0.11. The sum of the intensities of the transmitted and reflected waves, 0.89 + 0.11, is equal to the intensity of the incident wave. Energy is conserved! The figure is correct after all.

Here is another way to think about intensity: it’s one half of the product of the pressure times the tissue speed. When I say “tissue speed” I don’t mean the propagation speed of the wave, but the speed of the tissue itself as it oscillates back and forth. The acoustic impedance relates the pressure and tissue speed. The large pressure of the transmitted wave is associated with a small tissue speed. The transmitted wave in Fig. 13.5 looks “big” only because we plot the pressure. Had we plotted tissue speed instead, we would not be wondering why the transmitted wave has such a large amplitude. We would, however, be scratching our head about a funny phase shift in the reflected wave, which the version of the figure showing the pressure hides.

So, Fig. 13.5 is correct. Does that mean it looks exactly the same in the 5th edition of IPMB (due out this summer)? No, we did change the figure; not to correct an error, but to emphasize another point. Figure 13.5, as presently drawn, shows the wavelength of the transmitted wave to be the same as the wavelength of the incident wave. The wavelength does not depend on the acoustic impedance, but it does depend on the wave speed (the propagation speed of the wave itself, usually given the symbol c, which is not the same as the speed of the oscillating tissue). The wave speed is equal to the square root of the reciprocal of the product of the density and compressibility. One can cook up examples where two tissues have the same wave speed but different acoustic impedances. For instance (again, not worrying about units and only comparing relative sizes), if the tissue on the left had twice the compressibility and half the density of the tissue on the right, then the left would have half the acoustic impedance and the same wave speed as the right, just as shown in Fig. 13.5. But tissues usually differ in compressibility by a greater factor than they differ in density. If we assume the two regions have the same density but the right has one-fourth the compressibility, then Z₂ = 2 Z₁ as before but also c₂ = 2 c₁, so the wavelength is longer on the right. In the 5th edition, the figure now shows the wavelength longer on the right.

What is the moral to this story? Readers (and authors) need to think carefully about illustrations such as Fig. 13.5. They tell a physical story that is often richer and more complicated than we may initially realize.

Physical Models of Living Systems

Physical Models of
Living Systems,
by Philip Nelson.

Philip Nelson has a new textbook that came out earlier this year: Physical Models of Living Systems. It’s an excellent book, well written and beautifully illustrated. The target audience is similar to that for the 4th edition of Intermediate Physics for Medicine and Biology: upper-level undergraduates who have studied physics and math at the introductory level. Like IPMB, it stresses the construction of physical and mathematical models of living systems.

At the start of the book, Nelson provides a section labeled “To the Student.” I hope students read this, as it provides much wisdom and advice. In fact, most of this advice applies as well to IPMB. I found his discussion of “skills” to be so valuable that I reproduce it here.

Science is not just a pile of facts for you to memorize. Certainly you need to know many facts, and this book will supply some as background to the case studies. But you also need skills. Skills cannot be gained just by reading through this (or any) book. Instead you’ll need to work through at least some of the exercises, both those at the ends of chapters and others sprinkled throughout the text.

Specifically, this book emphasizes

• Model construction skills: It's important to find an appropriate level of description and then write formulas that make sense at that level. (Is randomness likely to be an essential feature of this system? Does the proposed model check out at the level of dimensional analysis?) When reading others’ work, too, it’s important to be able to grasp what assumptions their model embodies, what approximations are being made, and so on.

• Interconnection skills: Physical models can bridge topics that are not normally discussed together, by uncovering a hidden similarity. Many big advances in science came about when someone found an analogy of this sort.

• Critical skills: Sometimes a beloved physical model turns out to be . . . wrong. Aristotle taught that the main function of the brain was to cool the blood. To evaluate more modern hypotheses, you generally need to understand how raw data can give us information, and then understanding.

• Computer skills: Especially when studying biological systems, it’s usually necessary to run many trials, each of which will give slightly different results. The experimental data very quickly outstrip our abilities to handle them by using the analytical tools taught in math classes. Not very long ago, a book like this one would have to content itself with telling you things that faraway people had done; you couldn’t do the actual analysis yourself, because it was too difficult to make computers do anything. Today you can do industrial-strength analysis on any personal computer.

• Communication skills: The biggest discovery is of little use until it makes it all the way into another person’s brain. For this to happen reliably, you need to sharpen some communication skills. So when writing up your answers to the problems in this book, imagine that you are preparing a report for peer review by a skeptical reader. Can you take another few minutes to make it easier to figure out what you did and why? Can you label graph axes better, add comments to your code for readability, or justify a step? Can you anticipate objections?

You'll need skills like these for reading primary research literature, for interpreting your own data when you do experiments, and even for evaluating the many statistical and pseudostatistical claims you read in the newspapers.

One more skill deserves separate mention. Some of the book’s problems may sound suspiciously vague, for example, “Comment on . . .” They are intentionally written to make you ask, “What is interesting and worthy of comment here?” There are multiple “right” answers, because there may be more than one interesting thing to say. In your own scientific research, nobody will tell you the questions. So it’s good to get the habit of asking yourself such things.

Nelson begins the book by discussing virus dynamics, and specifically analyzes the work of Alan Perelson, who constructs mathematical model of how the human immunodeficinecy virus interacts with our immune system. Here at Oakland University, Libin Rong (a former postdoc of Perelson’s) does similar research, and their work is an excellent case study in using mathematics to model a biological process.

A large fraction of the book examines the role of randomness in biology, leading to a detailed analysis of probability. Nelson provides an elegant discussion of the experiments of Max Delbruck and Salvador Luria.

S. Luria and M. Delbruck set out to explore inheritance in bacteria in 1943. Besides addressing a basic biological problem, this work developed a key mode of scientific thought. The authors laid out two competing hypotheses, and sought to generate testable quantitative predictions from them. But unusually for the time, the predictions were probabilistic in character. No conclusion can be drawn from any single bacterium—sometimes it gains resistance; usually it doesn’t. But the pattern of large numbers of bacteria has bearing on mechanism. We will see how randomness, often dismissed as an unwelcome inadequacy of an experiment, turned out to be the most interesting feature of the data.

Perhaps one way to appreciate the differences between Physical Models of Living Systems and IPMB is to compare how each handles diffusion. Russ Hobbie and I consider diffusion in Chapter 4 of IPMB; we describe diffusion as arising from a concentration gradient (Fick’s first law), and use the continuity equation to derive the diffusion equation (Fick’s second law). These relationships are macroscopic descriptions of diffusion. Then, at the end of the chapter—almost as an afterthought—we show that diffusion can be thought of as a random walk. Nelson, on the other hand, starts by analyzing the random nature of diffusion using probabilistic ideas, and then—almost as an afterthought—derives the diffusion equation (or at least a discrete approximation of it). I think this example reflects the different approaches of the two books: IPMB generally takes a macroscopic approach but sometimes reaches down with an example at the microscopic level, whereas Physical Models in Living Systems typically starts with a microscopic description and then sometimes works its way up to the macroscopic level.

Both books also have an extensive analysis of feedback. The canonical example in IPMB comes from physiology at the organism level: breathing rate controls and is controlled by blood carbon dioxide concentration. In Physical Models in Living Systems, a central example is how bacteria use feedback to regulate the synthesis of the amino acid tryptophan. Both case studies are excellent examples of negative feedback, but at different spatial scales. One example is not better than the other; they’re merely different illustrations of the same idea.

One strength of Physical Models of Living Systems is its emphasis on using computer simulations to describe a system’s behavior. IPMB has a few computer programs (for example, a program is provided to simulate the Hodgkin-Huxley model of a nerve axon), but Physical Models of Living Systems has a much heavier reliance on numerical simulation. Again, one approach isn’t better than the other, just different. One can learn a lot about biology using toy models and analytical analysis, but many more-complicated (often nonlinear) processes need the numerical approach. Anyone who uses, or plans to use, MATLAB for simulations may benefit from the Student’s Guide to Physical Models of Living Systems (available free to all at www.macmillanhighered.com/physicalmodels1e).

In conclusion, Nelson states that Physical Models of Living Systems is about how “physical science and life science illuminate each other,” and I can’t think of a better description of the goal of Intermediate Physics for Medicine and Biology. Students are lucky they have both to choose from. Finally, what is the very best thing about Physical Models of Living Systems? At the end of the “To the Student” section, Nelson lists several other books that complement his, and cites…you guessed it…Intermediate Physics for Medicine and Biology.

The Steradian

Angles are measured in radians, but solid angles are measured in steradians. Russ Hobbie and I discuss solid angles in Appendix A of the 4th edition of Intermediate Physics for Medicine and Biology.

A plane angle measures the diverging of two lines in two dimensions. Solid angles measure the diverging of a cone of lines in three dimensions. Figure A.3 shows a series of rays diverging from a point and forming a cone. The solid angle Ω is measured by constructing a sphere of radius r centered at the vertex and taking the ratio of the surface area S on the sphere enclosed by the cone to r²:

Ω = S/r².

…The unit of solid angle is the steradian (sr). A complete sphere subtends a solid angle of 4π steradians, since the surface area of a sphere is 4πr².

It is useful to have an intuitive idea of how big a steradian is. Viewed from the center of the earth, Asia subtends about one steradian, and Switzerland subtends about one millisteradian. From its center a sphere subtends 4π steradians, so one steradian is 1/4π = 0.08, or 8% of the sphere area. Suppose we use spherical coordinates to determine the area, centered at the north pole (θ = 0), that subtends one steradian. It is the area subtended by the “cap” of the sphere having an angle θ = cos^-1(1-1/2π) = cos^-1(0.84) = 32.8 degrees, or 0.57 radians.

One square degree is (π/180)² = 0.000305 sr = 305 μsr. In other words, there are 3283 square degrees per steradian. Put in yet another way, a steradian is one square radian. The moon has a radius of 1737 km, and the distance between the earth and the moon is 384,400 km. The solid angle subtended by the moon in the night sky is therefore π 1737²/384400² = 0.000064 sr, or 64 μsr. Interestingly, the sun, with a radius of 696,000 km and an earth-sun distance of 149,600,000 km, subtends almost the same solid angle, which makes solar eclipses so interesting. Viewed from earth, Mars at its closest approach subtends about 12 nanosteradians.

At the Battle of Bunker Hill, the order went out to “don't fire until you see the whites of their eyes.” This may be a figure of speech, but let’s take it literally. You can see the whites of a person’s eyes at a distance of about 10 meters (I would definitely be shooting before the enemy got that close). The area of the “whites of the eye” is difficult to estimate accurately, but let’s approximate it as one square centimeter. What solid angle is subtended by the whites of the eye at a distance of ten meters? It would be about (0.01 m)²/(10 m)², or one microsteradian. This is not bad for an estimate of our visual acuity. We may sometimes do a little better than this, but probably not during battle.

On Writing Well

Oakland University, where I work, has an ADVANCE grant from the National Science Foundation, with the goal of increasing the representation and advancement of women in academic science and engineering careers. I am part of the leadership team for the Women in Science and Engineering at Oakland University (WISE@OU) Program, and one of my roles is to help mentor young faculty. Last Tuesday I led a WISE@OU workshop on “Best Practices in Scientific Writing.” The event was videotaped, and you can watch it here. I’m the bald guy who is standing and wearing the red shirt.

On Writing Well,
by William Zinsser.

A list of writing resources was provided to all workshop participants (see below). It begins “For the most benefit in the least time with no cost, work through the Duke online Scientific Writing Resource, then read Part 1 (about 50 pages) of Zinsser’s book On Writing Well (available free online), and finally go through the online material for the Stanford Writing in the Sciences class.” If you don’t have enough time for even these three steps, then just read Zinsser, which is a delight.
Russ Hobbie and I try to write well in the 4th edition of Intermediate Physics for Medicine and Biology. You can decide if we succeed. Many readers of this blog are from outside the United States (I can tell from the “likes” on the book’s Facebook page). As I noted in the workshop, it is not fair that scientists from other countries must write science in a language other than their native tongue. Yet, most science is published in English, and scientists need to be able to write it well. So, my advice is to do whatever it takes to become a decent writer.

When I was in graduate school, my dissertation advisor John Wikswo gave me a copy of The Complete Plain Words, a wonderful book about writing originally published by Sir Ernest Gowers. Read it for free online. The version Wikswo loaned me was a later edition coauthored by Bruce Fraser. (You always should be concerned when a perfectly good book picks up a coauthor in later editions). This spring, Gowers’ great-granddaughter Rebecca Gowers is publishing a new edition of Plain Words. I can’t wait. Another oldie but goodie is Strunk and White’s The Elements of Style. The original, by William Strunk, is available online. (The second author of “Strunk and White” is E. B. White who wrote Charlotte’s Web; I vividly remember Mrs. Sheets reading Charlotte’s Web aloud to my third grade class at Northside Elementary School.) If you have time for only three words about writing, let them be Strunk’s admonition “omit needless words.”

I’ve come up with my own Three Laws of Writing Science, patterned after Isaac Asimov’s Three Laws of Robotics (regular readers of this blog know that Asimov influenced me greatly when I was in high school).

• First Law: What you write must be scientifically correct. 

• Second Law: Write clearly, except when clarity would put you in conflict with the First Law.

• Third Law: Write concisely, except when conciseness would put you in conflict with the First or Second Laws.

Writing is easier when you enjoy doing it, and I always have. I once became secretary of the Parent-Teacher Association at my daughters’ elementary school because that job allowed me to write the minutes of the PTA meetings. If you don’t enjoy writing, take heart. You don’t need to be a great writer to succeed in science. Slipping into NSF-speak, if you can improve from “poor” or “fair” to “good” you will get almost the full benefit. Go from “good” to “very good” or “excellent” only if you like to write.

Best Practices in Scientific Writing

Below is a list of resources about scientific writing. For the most benefit in the least time with no cost, work through the Duke online Scientific Writing Resource, then read Part 1 (about 50 pages) of Zinsser’s book On Writing Well (available free online), and finally go through the online material for the Stanford Writing in the Sciences class.

Books about writing:

• Gowers R, Gowers E. 2014. Plain Words
• Gray-Grant D. 2008. 8 ½ Steps to Writing Better, Faster
• Pinker, S. 2014. The Sense of Style
• Silvia PJ. 2007. How to Write a Lot 
• Strunk W, White EB. 1979. The Elements of Style
• Zinsser W. 1976. On Writing Well (free online: archive.org/details/OnWritingWell)

American Scientist article “The Science of Scientific Writing”

Video of Steven Pinker discussing good writing

A free online course from Stanford about Writing in the Sciences

Kamat, Buriak, Schatz, Weiss. 2014. “Mastering the art of scientific publicaition: Twenty papers with 20/20 vision on publishing,” J. Phys. Chem. Lett., 5:3519–3521.

Kotz, Cals, Tugwell, Knottnerus. 2013. “Introducing a new series on effective writing and publishing of scientific papers,” J. Clinical Epidemiology, 66:359–360.

How to Get Published. A discussion with Mike Sevilla and myself, moderated by George Corser, about writing and publishing scientific papers, hosted the OU graduate student group GradConnection.

A free online webinar debating the use of the active or passive voice

Duke University’s online Scientific Writing Resource, open to all.

Nonnative English speakers (and the rest of us too) should see the website Scientific English as a Foreign Language.

Projections and Back Projections

Tomography is one of the most important contributions of mathematics to medicine. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe two methods to solve the problem of tomographic reconstruction.

The reconstruction problem can be stated as follows. A function f(x,y) exists in two dimensions. Measurements are made that give projections: the integrals of f(x,y) along various lines as a function of displacement perpendicular to each line…F(θ,x'), where x' is the distance along the axis at angle θ with the x axis. The problem is to reconstruct f(x,y) from the set of functions F(θ,x'). Several different techniques can be used… We will consider two of these techniques: reconstruction by Fourier transform […] and filtered back projection…. The projection at angle θ is integrated along the line y':

[where x = x' cosθ – y' sinθ and y = x' sinθ + y' cosθ]… The definition of the back projection is

where x' is determined for each projection using Eq. 12.27 [x' = x cosθ + y sinθ].

In IPMB, Homework Problem 32 asks you can take the function (the “object”)

and calculate the projection using Eq. 12.29, and then calculate the back projection using Eq. 12.30. The object and the back projection are different. The moral of the story is that you cannot solve the tomography problem by back-projection alone. Before you back-project, you must filter.*

I like having two homework problems that illustrate the same point, one that I can do in class and another that I can assign to the students. IPMB contains only one example of projecting and then back-projecting, but recently I have found another. So, dear reader, here is a new homework problem; do this one in class, and then assign Problem 32 as homework.

Problem 32 ½ Consider the object f(x,y) = √(a² − x² − y²)/a for √(x² +y²) less than a, and 0 otherwise.

(a) Plot f(x,0) vs. x 

(b) Calculate the projection F(θ,x'). Plot F(0,x') vs. x'.

(c) Use the projection from part (b) to calculate the back projection f_b(x,y). Plot f_b(x,0) vs. x.

(d) Compare the object and the back projection. Explain qualitatively how they differ.

The nice thing about this function

(as well as the function in Problem 32) is that f(x,y) does not depend on direction, so F(θ,x') is independent of θ; you can make you life easier and solve it for θ = 0. Similarly, you can calculate the back projection along any line through the origin, such as y = 0. I won’t solve Problem 32½ here in detail, but let me outline the solution.

Below is a plot of f(x,0) versus x

To take the projection in part (b) use θ = 0, so x' = x and y' = y. If |x| is greater than a, you integrate over a function that is always zero, so F(0,x') = 0. If |x| is less than a, you must do more work. The limits of the integral over y become −√(a² – x²) to +√(a² – x²). The integral is not too difficult (it’s in my mathematical handbook), and you get F(0,x) = π(a² – x²)/2a. Because the projection is independent of θ, you can generalize to

The plot, an inverted parabola, is

In part (c) you need to find the back-projection. I suggest calculating it for the line y = 0. Once you have it, you can find it for any point by replacing x by √(x² +y²). The back-projection for |x| less than a is easy. The integral in Eq. 12.30 gives another inverted parabola. For |x| is greater than a, the calculation is more complicated because some angles give zero and some don’t. A little geometry will convince you that the integral should range from cos^-1(a/x) to π - cos^-1(a/x). Because the function is even around π/2, you can make your life easier by multiplying by two and integrating from cos^-1(a/x) to π/2. The only way I know to show how you get cos^-1(a/x) is to draw a picture of a line through the point x greater than a, y = 0 that is tangent to the circle x² + y² = a², and then do some trigonometry. When you then evaluate the integral you get

A plot of this complicated looking function is

To answer part (d), compare your plots in (a) and (c). The object in (a) is confined entirely inside the circle x² + y² = a², but the back-projection in (c) spreads over all space. One could say the back-projection produced a smeared-out version of the object. Why are they not the same? We didn’t filter before we back projected.

If you really want to have fun, show that the limit of the back-projection goes to zero as x goes to infinity. This is surprisingly difficult—that last term doesn’t look like it’ll go to zero—and you need to keep more than one term in your Taylor series to make it work.

The back projection shown above looks similar to the back projection shown in Fig. 12.23 of IPMB. My original goal was to calculate the back projection in Fig. 12.23 exactly, but I got stuck trying to evaluate the integral

I sometimes ask myself: why do I assign these analytical examples of projections and back-projections? Anyone solving the tomography problem in the clinic will certainly use a computer. Is anything gained by doing analytical calculations? Yes. The goal in doing these problems is to gain insight. A computer program is a black box: you put projections in and out comes an image, but you don’t know what happens in between. Analytical calculations force you to work through each step. Please don’t skip the plots in parts (a) and (c), and the comparison of the plots in part (d), otherwise you defeat the purpose; all you have done is an exercise in calculus and you learn nothing about tomography.

I wish I had come up with this example six months ago, so Russ and I could have included it in the 5th edition of IPMB. But it’s too late, as the page proofs have already been corrected and the book will be printed soon. This new homework problem will have to wait for the 6th edition!


*Incidentally, Prob. 12.32 has a typo in the second line: "|x|" should really be "√(x² +y²)".

Consider an Impervious Plane Containing A Circular Disk

When I teach, one of my goals is to help students analyze a mathematical equation to uncover the underlying physics. Equations are not merely expressions you plug numbers into to get other numbers; equations tell a story. Students often begin the semester knowing the required math and physics, but have difficulty connecting the two. One way students learn this connection is by taking limits of mathematical equations, so they can show that the equation behaves as it should when some key variable is large or small. If the equation is too complicated, taking the limit may be so difficult that the student gets lost in the algebra (see the February 20 blog entry about “the ugliest equation”). If the equation is too simple, taking the limit may be trivial. The best examples are of intermediate difficulty: students must do some work to take the limit, but their result provides physical insight.

One of my favorite examples is Problem 29 of Chapter 4 in the 4th edition of Intermediate Physics for Medicine and Biology. I should be more specific: parts (b)-(d) of that problem are just the sort of practice the students need (part (a) of the problem is for masochists).

Problem 29 Consider an impervious plane at z = 0 containing a circular disk of radius a having a concentration C₀. The concentration at large z goes to zero. Carslaw and Jaeger (1959) show that the steady-state solution to the diffusion equation is 

(a) (optional) Verify that C(r, z) satisfies ∇ ²C = 0. The calculation is quite involved, and you may wish to use a computer algebra program such as Mathematica or Maple.

(b) Show that for z = 0, C = C₀ if r is less than a. 

(c) Show that for z = 0, dC/dz = 0 if r is greater than a. 

(d) Integrate J_z over the disk (z = 0, r between 0 and a) and show that i = 4DaC₀.

I’m amazed that the convoluted expression for the concentration, C(r, z), containing an inverse sine function with a complicated argument does indeed obey Laplace’s equation. I once proved that this is true, but have no intention of doing so again—it’s a mess. Before analyzing parts (b)-(d) of the problem, let’s examine the claim that “the concentration at large z goes to zero.” Exactly how does it go to zero? Far from the disc (z is much greater than a) the concentration should fall off as the inverse distance (just as the electrical potential of a point charge falls off as 1/r). When z is large, the argument of the inverse sine function is small, and it can be approximated using the first term of its Taylor series: sin^-1(x) ~ x. In that case, the concentration becomes 2aC₀/πz, falling off with the inverse distance as expected.

The behavior at small z is more puzzling and the solution has the astonishing property that it changes its behavior at r = a. For r less than a the concentration is constant, but for r greater than a the derivative of the concentration is zero. How can such a relatively simple analytical expression display such a discontinuous behavior?

To answer that question, let’s analyze the equation for the concentration. The key step in our analysis is when we factor out (r – a)² from the first square root in the denominator of the argument. This factor is always positive because the quantity is squared. But when removed from inside the square root, it becomes |r – a|, with the absolute value needed to ensure that the expression remains positive. There are two cases: r less than a, when we replace |r – a| by (a – r); and r greater than a, when we replace |r – a| by (r – a). The rest is an exercise in Taylor series resulting from the limit as z goes to zero. Use the expansion √(1 + x²) = 1 + x²/2 for the two square roots, simplify the denominator as much as possible, and then use the expansion 1/(1 - x) = 1 + x to find that for r less than a

and for r greater than a

For all the gory details see the solution manual, available free-of-charge to instructors using IPMB in their class; email Russ Hobbie or me (roth@oakland.edu) for a copy.

Now we can do parts (b)-(d) of the homework problem. For part (b), when r is less than a set z = 0 and use that sin⁻¹(1) = π/2 to find that C(r,0) = C₀, independent of r. For part (c), when r is greater than a take the derivative of the inverse sine function with respect to z, and then set z = 0. Because the argument of the inverse sine is even in z, at z = 0 the derivative dC/dz vanishes.

Part (d) is tricky. Using the same argument as above, we might expect that because the argument of the inverse sine is even for r less than a the derivative dC/dz must be zero. But no! The derivative of sin⁻¹(u(x)) is 1/√(1−u²) du/dx. When we take the derivative of the expression for C(r,z) for r less than a and then set z = 0, we get not only zero in the numerator (because du/dx = 0; the function is even) but also zero in the denominator (because √(1−u²) = 0 in this case). The result is dC/dz = − 2C₀/π√(a² − r²).

To find the total number of particles per unit time i coming out of the disk, we need to integrate the particle current density J_z over the plane z = 0. The current density is J_z = – D dC/dz, where D is the diffusion constant. Because dC/dz is zero for r greater than a, we need to integrate the current density over only the circular area r less than a. You can look this integral up and find that i = 4C₀Da.

There are other aspects of the solution that are not emphasized in this problem, but are equally interesting. For instance, J_z goes to infinity at r = a. In other words, the number of particles leaving the disk is largest at the edge of the disk. This result is analogous to the concentration of electrical current near the edge of a disk electrode, discussed in Problem 22 of Chapter 6 in IPMB; J_z changes abruptly from an extremely large value for r just less than a to zero for r just greater than a. Another interesting result is that the total number of particles leaving the disk per unit time is proportional to the disk radius, not the disk area as you might naively expect. This result also has an analogy in electrostatics: the capacitance of a disk electrode relative to a distant ground is proportional to the radius of the disk, not the radius squared, which surprised me when I first encountered it (see Section 6.19 in IPMB).

In retrospect, maybe this problem is slightly too complicated to be the perfect example of taking the limits of a mathematical expression to gain insight into the physical behavior. But for those whose mathematical skills are up to the job, it provides an excellent case study.

Mathematical equations tell a physical story. You must practice extracting that story from the math.

If You Want Healthy Cows Feed Them Magnets

I saw an article on the internet claiming “If you want healthy cows feed them magnets” and I thought “oh no, not more biomagnetism nonsense.” First magnets in shoes to relieve foot pain, then magnetic bracelets for arthritis, and finally “biomagnetic therapy” for all sorts of disorders; I thought it couldn’t get worse, but feeding magnets to heifers? Really? Sounds like bull to me.

A drawing of a cow
with a magnet in its stomach.

Well, I can’t vouch for the accuracy of this story or the effectiveness of the treatment, but at least the mechanism underlying the feeding of magnets to cows is plausible. Cattle swallow a lot of junk while eating, including some that is magnetic (for example, wires and nails...yikes!). The article says

That's where magnets come in. A magnet about the size and shape of a finger is placed inside a bolus gun, essentially a long tube that ensures the magnet goes down the cow's throat. Then it settles in the reticulum, collecting any stray pieces of metal. The magnets, which cost a few bucks a pop, can also be placed preventatively. To check if a cow already has a magnet, farmers use a compass.

Apparently the “bolus gun” is inserted through the mouth; I wasn’t so sure. Wikipedia has a page about cow magnets, titled “hardware disease.” Companies make money selling cow magnets (these are big magnets, about four inches long). But even though calves eat magnets, kids should not (note the plural: the problems arise when magnets interact).

Consider a spherical cow.

The 4th edition of Intermediate Physics for Medicine and Biology has an entire chapter about biomagnetism, but no mention of magnets in bovine stomachs. What is wrong with Russ and me? The only place we mention cattle at all is in Homework Problem 30 in Chapter 4, where we analyze the temperature distribution throughout a spherical cow. A small-scale analogy of magnets in steers’ stomachs are rows of magnetosomes in magnetotactic bacteria (see Fig. 8.25 in IPMB), but I doubt the bacteria use them to collect nails before they can puncture their membrane. Yet, could we misunderstand the biological purpose of magnetosomes?

Finally, I have some good news and bad news about the 5th edition of IPMB. The good news: we submitted the page proofs and the book should be published in the next few months. The bad news: no more mention of livestock in the revised edition.

If you want healthy cows feed them magnets.

A Mathematical Model of Agonist-Induced Propagation of Calcium Waves in Astrocytes

When I was working at the National Institutes of Health in the mid-1990s, I spent most of my time studying transcranial magnetic stimulation and theoretical cardiac electrophysiology. But also I collaborated with James Russell to study calcium waves in astrocytes (a type of glial cell in the brain), and we published a paper in the journal Cell Calcium describing “A Mathematical Model of Agonist-Induced Propagation of Calcium Waves in Astrocytes” (Volume 17, Pages 53–64, 1995). The introduction is reproduced below (with citations removed):

Recent experiments have clearly shown that astroglia in brain participate in long distance signaling together with neurons. Such signalling in astrocytes is supported by intracellular calcium oscillations induced by neurotransmitters that are propagated as waves through the cytoplasm of individual cells and through astrocyte networks. These calcium oscillations generally are triggered by activation of metabotropic receptors which are coupled to inositol 1,4,5-trisphosphate (IP₃) generation and intracellular calcium release through IP₃-gated calcium channels on the endoplasmic reticulum (ER) membrane. Yagodin et al. have shown that, in astrocytes, wave propagation is saltatory, with discrete loci of nonlinear wave amplification separated by regions through which passive diffusion of calcium occurs. These wave amplification loci appear to be intracellular specializations that remain invariant and support a qualitatively characteristic response pattern in any given cell. The loci may each have different intrinsic oscillatory frequencies, resulting in complex spatio-temporal dynamics, with wave collisions and annihilations.

Several mathematical models have been presented that describe the temporal characteristics of agonist-induced calcium oscillations in different types of cells. A few of these models address the spatial characteristics of wave propagation, but none have addressed the complex wave dynamics observed in different types of cells including astrocytes. The purpose of this paper is to extend a previously developed model of calcium oscillations so that it includes spatial diffusion of calcium in a cell with discrete active loci of wave amplification. This model is then used to analyze experimental data and to gain insight into the mechanism of wave collisions and annihilations.

As you might expect, my contribution to this paper was developing the mathematical model, while Russell and his team provided the experimental data as well as the biological knowledge and insight. The model was based on a paper by Li and Rinzel (“Equations for InsP3 Receptor-Mediated [Ca2+]i Oscillations Derived from a Detailed Kinetic Model: A Hodgkin-Huxley Like Formalism,” Journal of Theoretical Biology, Volume 166, Pages 461–473, 1994). At that time, John Rinzel was at NIH, heading the Mathematical Research Branch of the National Institute of Diabetes and Digestive and Kidney Diseases. John, now at the Center for Neural Science at New York University, has contributed much to theoretical biology, but I remember him best for his work on bursting of pancreatic beta cells. He is this year’s winner of the Society of Mathematical Biology’s Arthur T. Winfree Prize “for his elegant work on the analysis of dynamical behavior in models of neural activity and the contributions that work has made in the neurobiological community to the understanding of a host of phenomena (including simple excitability as well as bursting) in single neurons, small populations of neurons, and other excitable cells.”

Russell remains at NIH with the Microscopy and Imaging Core of the Eunice Kennedy Shriver National Institute of Child Health and Development. He leads a multi-user research facility providing training and instrumentation for high resolution microscopy and image processing.

As so often happens, an echo of my work on calcium wave modeling with Russell appears in the 4th edition of Intermediate Physics for Medicine and Biology. Homework Problem 24 in Chapter 4 contains a simplified model of calcium waves. This system is a classic “reaction-diffusion” system: calcium diffuses down the cell, triggering calcium-induced calcium release, which produces more diffusion, triggering more calcium release, resulting in positive feedback and a calcium wave. The process is analogous to action potential propagation along a nerve.

Alan Turing, Biological Physicist

Recently, my wife and I went to the theater to see The Imitation Game, about Alan Turing and the breaking of the enigma code during World War II. It is a fascinating movie. I’m a big fan of Benedict Cumberbatch, who played Turing (I particularly enjoy his portrayal of Sherlock Holmes in the TV series Sherlock), and I always enjoy performances by Keira Knightly.

Turing was primarily a mathematician, but he did publish one paper that straddled the disciplines of mathematical biology and biological physics: A. M. Turing, 1952, “Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society of London. Series B, Volume 237, Pages 37–72. The abstract is reproduced below.

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

Mathematical Biology,
by James Murray.

You can learn more about Turing’s theory in James Murray’s book Mathematical Biology (I am basing my comments on the edition in the Oakland University library: the Second, Corrected Edition, 1993). Murray writes

Turing’s (1952) idea is a simple but profound one. He said that if, in the absence of diffusion….[two chemicals] A and B tend to a linearly stable uniform steady state then, under certain conditions, which we shall derive, spatially inhomogeneous patterns can evolve by diffusion driven instability… Diffusion is usually considered a stabalising process which is why this was such a novel concept. To see intuitively how diffusion can be destabilizing consider the following, albeit unrealistic, but informative analogy.

Consider a field of dry grass in which there is a large number of grasshoppers…

I don’t know about you, but I gotta love someone who explains mathematics using dry grass and grasshoppers.

Diffusion is a key concept underlying Turing’s work. Russ Hobbie and I discussion diffusion in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology, and it is one of the central ideas in all of biological physics. Diffusion-driven instabilities play a role when analyzing the Belousov-Zhabotinsky oscillating chemical reaction, and are relevant to explaining how leopards get their spots (a spotted leopard graces the cover of Murray’s book; whenever I search for his book in the stacks of the OU library, I just look for the leopard).

Murray continues

A reaction diffusion system exhibits diffusion-driven instability or Turing instability if the homogeneous stead state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present. The usual concept of instability in biology is in the context of ecology, where a uniform steady state becomes unstable to small perturbations and the populations typically exhibit some temporal oscillatory behaviour. The instability we are concerned with here is of a quite different kind. The mechanism driving the spatially inhomogeneous instability is diffusion: the mechanism determines the spatial pattern that evolves. How the pattern or mode is selected is an important aspect of the analysis.

Not only did Turing make a monumental contribution to deciphering the enigma code, but also he helped to develop the field of mathematical biology. In my book, that makes him a biological physicist.

The Ugliest Equation

The 4th edition of Intermediate Physics for Medicine and Biology contains thousands of equations. One of them, Equation 15.22, gives the cross section for energy transferred to electrons during Compton scattering, as a function of photon energy (x = hν/mc²). Of all the equations in the book, it is the ugliest.

Eq. 15.22 in IPMB; the ugliest equation.

Russ Hobbie and I write

Equation 15.22 is a rather nasty equation to evaluate, particularly at low energies, because many of the terms nearly cancel.

Examining the behavior of the expression in brackets at small x should be easy: just take its Taylor’s series (to review Taylor’s series, see Appendix D of IPMB). In order to get the correct answer, however, you need to keep not two terms in the expansion, or three, but four! The Taylor’s series you need are

Three Taylor's series needed to analyze Eq. 15.22.

Oddly, the expansion for the fourth term 4x²/3(1+2x)³ doesn’t even need one term in its expansion; it is small and doesn’t contribute to the limiting behavior. Plug these all in, and you find that the terms in x⁻², x⁻¹, and x⁰ all vanish. The first nonzero term is linear: 4x/3.

Out of curiosity, I evaluated each of the five terms in the expression using x = 0.01. I got

20001.9608 - 0.9900 + 9900.0192 - 0.0001 - 29900.9771  =  0.0128

The terms really do “nearly cancel.”