The Mystery of the Flawed Homework Problem

When teaching PHY 325 (Biological Physics) this fall, I assigned my students homework from the 5th edition of Intermediate Physics for Medicine and Biology. One problem comes from Section 7.10 about Electrical Stimulation.

Problem 36. If the medium has a constant resistance, find the energy required for stimulation as a function of pulse duration.

The odd thing is, when I looked in the solution manual to review how to solve this problem, it contained answers to parts (a) and (b), and (b) is the most useful part. Where are (a) and (b)? Somehow when preparing the 5th edition, part (b) was left out (it is missing from the 4th edition too). Nevertheless, part (b) ended up in the solution manual (don’t ask me how). This is what Problem 36 should look like:

Problem 36. The longevity of a pacemaker battery is related to the energy required for stimulation.

(a) Find an expression for the energy U expended by a pacemaker to stimulate the heart as a function of the pulse duration t. Use the Lapicque strength-duration curve (Eq. 7.45), and assume the body and electrodes have a constant resistance R. Sketch a plot of energy versus duration.

(b) In general you want to stimulate using the least energy. Determine what duration minimizes the energy expended per pulse.

I don’t usually solve homework problems from the book in this blog, but because the interesting part of this problem was left out of IPMB I don’t think it will hurt in this case. Also, it provides readers with a sneak peak at the solution manual. Remember that Russ Hobbie and I will only send the solution manual to instructors, not students. So if you are teaching from IPMB and want the solution manual, by all means contact us. If you are a student, however, you had better talk to your instructor.

7.36 Issues such as pacemaker battery life are related to the energy required for electrical stimulation. This problem relates the energy to the strength-duration curve, and provides additional insight into the physical significance of the chronaxie.

(a) Let the resistance seen by the electrode due to the medium be R. The power is i²R. Therefore the total energy is

 (b) The duration corresponding to minimum energy is found by setting dU/dt = 0. We get

which reduces to t = t_C. The minimum energy corresponds to a duration equal to the chronaxie.

In the 5th edition’s solution manual, each problem has a brief preamble (in italics) explaining the topic and describing what the student is supposed to learn. We also mark problems that are higher difficulty (*), that complete a derivation from the text (§), and that are new in the fifth edition (¶). Problem 7.36 didn’t fall into any of these categories. We typically outline the solution, but don’t always show all the intermediate steps. I hope we include enough of the solution that the reader or instructor can easily fill in anything missing.

One thing not in the solution manual is the plot of energy, U, versus duration, t. Below I include such a plot. The energy depends on the rheobase current i_R, the chronaxie t_C, and the resistance R.

The energy of a stimulus pulse as a function of pulse duration.

I wonder if this change to Problem 7.36 should go into the IPMB errata? It is not really an error, but more of an omission. After some thought, I have decided to include it, since it was supposed to be there originally. You can find the errata at the book's website: https://sites.google.com/view/hobbieroth. I urge you to download it and mark the corrections in your copy of IPMB.

I hope this blog post has cleared up the mystery behind Problem 7.36. Yet, the curious reader may have one last question: why did I assign a homework problem to my students that is obviously flawed? The truth is, I chose which homework problems to assign by browsing through the solution manual rather than the book (yes, the solution manual is that useful). Problem 7.36 sure looked like a good one based on the solution manual!

Stokes' Flow around a Sphere

When working on the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem about low Reynolds number flow. We ask the reader to analyze the classic example of “Stokes’ flow” or “creeping flow” around a sphere.

Problem 46. Consider a stationary sphere of radius a placed in a fluid of viscosity η moving uniformly with speed V. For low Reynolds number flow, the radial and tangential components of the fluid velocity and the pressure surrounding the sphere are

(a) Show that the no-slip boundary condition is satisfied.

(b) Integrate the shear force and the pressure force over the sphere surface and find an expression for the net drag force on the sphere (Stoke’s law). What fraction of this force arises from pressure drag, and what fraction from viscous drag?

(Everywhere else in our book we correctly write “Stokes’ law” since the law is named after Sir George Stokes, but in this problem we slip up and write “Stoke’s law”. Sorry. I noted this in the errata available on the book website.)

After solving this problem, the reader is probably thinking “this is all well and nice, and I understand now how you get Stokes’ law from the pressure distribution and the viscous drag, but where in the world did you get those weird velocity and pressure distributions?”

First, this example applies to a sphere in water, and water is nearly incompressible. Problem 1.35 shows that incompressibility implies that the velocity u has zero divergence,

The reader should pause now, look up the expression for the divergence in spherical coordinates, and verify that the given velocity really is divergenceless.

Second, the equation describing flow is the Navier-Stokes equation, which is really nothing more than Newton’s second law (F=ma) applied to the fluid. Problem 1.28 provides some insight by deriving a simplified form of the Navier-Stokes equation

 

If we assume a low Reynolds number, we can ignore the two terms on the left-hand side of this equation because they are “inertial” terms arising from the acceleration of the fluid. The two terms on the right-hand side can be generalized to three dimensions, with the pressure term containing the gradient of the pressure and the viscous term containing the Laplacian of the velocity. The resulting Navier-Stokes equation is

To get the expressions given in the new Problem 1.46, solve the Navier-Stokes equation assuming an incompressible fluid. In addition, the boundary conditions are 1) far from the sphere (r much greater than a) the flow is entirely along the z-axis with speed V, and 2) at the sphere surface (r = a) the radial component of the velocity vanishes because the flow is incompressible and the tangential component of the velocity vanishes because of the no-slip boundary condition.

Stokes’ law for the net drag force F, derived in part (b) of Problem 1.46, is F = 6πηaV. Often the drag force is described by a dimensionless coefficient called the drag coefficient, C, equal to F divided by ½ρV²πa². For creeping flow around a sphere, the drag coefficient is

Using the definition of the dimensionless Reynolds number, Re (Eq. 1.62 in IPMB), we find that C = 12/Re. Often the Reynolds number is written in terms of the diameter of the sphere rather than the radius, in which case we get the more commonly quoted relationship C = 24/Re. In many fluid dynamics textbooks you will see C plotted versus Re (usually on log-log graph paper). At low Reynolds number C is inversely proportional to Re as creeping flow predicts. At high Reynolds number the relationship between C and Re is more complex because a turbulent boundary layer forms near the sphere surface. But that’s another story.

The Magnetic Field of a Single Axon (Part 2)

In my last blog entry, I began the story behind “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment” (Biophysical Journal, Volume 48, Pages 93–109, 1985). I wrote this paper as a graduate student working for John Wikswo at Vanderbilt University. (I use the first person “I” in this blog post because I was usually alone in a windowless basement lab when doing the experiment, but of course Wikswo taught me how to do everything including how to write a scientific paper.) Last week I described how I measured the transmembrane potential of a crayfish axon, and this week I explain how I measured its magnetic field.

A toroid used to measure
the magnetic field of a single axon.

The magnetic field was recorded using a wire-wound toroid (I have talked about winding toroids previously in this blog). Wikswo had obtained several ferrite toroidal cores of various sizes, most a few millimeters in diameter. I wound 50 to 100 turns of 40-gauge magnet wire onto the core using a dissecting microscope and a clever device designed by Wikswo to rotate the core around several axes while holding its location fixed. I had to be careful because a kink in a wire having a diameter of less than 0.1 mm would break it. Many times after successfully winding, say, 30 turns the wire would snap and I would have to start over. After finishing the winding, I would carefully solder the ends of the wire to a coaxial cable and “pot” the whole thing in epoxy. Wikswo—who excels at building widgets of all kinds—had designed Teflon molds to guide the epoxy. I would machine the Teflon to the size we needed using a mill in the student shop. (With all the concerns about liability and lawsuits these days student shops are now uncommon, but I found it enjoyable, educational, and essential.) Next I would carefully place the wire-wound core in the mold with a Teflon tube down its center to prevent the epoxy from sealing the hole in the middle. This entire mold/core/wire/cable would then be placed under vacuum (to prevent bubbles), and filled with epoxy. Once the epoxy hardened and I removed the mold, I had a “toroid”: an instrument for detecting action currents in a nerve. In 1984, this “neuromagnetic current probe” earned Wikswo an IR-100 award. The basics of this measurement are described in Chapter 8 of Intermediate Physics for Medicine and Biology.

In Wikswo’s original experiment to measure the magnetic field of a frog sciatic nerve (the entire nerve; not just a single axon), the toroid signal was recorded using a SQUID magnetometer (see Wikswo, Barach, Freeman, “Magnetic Field of a Nerve Impulse: First Measurements,” Science, Volume 208, Pages 53–55, 1980). By the time I arrived at Vanderbilt, Wikswo and his collaborators had developed a low-noise, low-input impedance amplifier—basically a current-to-voltage converter—that was sensitive enough to record the magnetic signal (Wikswo, Samson, Giffard, “A Low-Noise Low Input Impedance Amplifier for Magnetic Measurements of Nerve Action Currents,” IEEE Trans. Biomed. Eng. Volume 30, Pages 215–221, 1983). Pat Henry, then an instrument specialist in the lab, ran a cottage industry building and improving these amplifiers.

To calibrate the instrument, I threaded the toroid with a single turn of wire connected to a current source that output a square pulse of known amplitude and duration (typically 1 μA and 1 ms). The toroid response was not square because we sensed the rate-of-change of the magnetic field (Faraday’s law), and because of the resistor-inductor time constant of the toroid. Therefore, we had to adjust the signal using “frequency compensation”; integrating the signal until it had the correct square shape.

The amplifier output was recorded by a digital oscilloscope that saved the data to a tape drive. Another of my first jobs at Vanderbilt was to write a computer program that would read the data from the tape and convert it to a format that we could use for signal analysis. We wrote our own signal processing program—called OSCOPE, somewhat analogous to MATLAB—that we used to analyze and plot the data. I spent many hours writing subroutines (in FORTRAN) for OSCOPE so we could calculate the magnetic field from the transmembrane potential, and vice versa.

An experiment to measure the
transmembrane potential, the extracellular potential,
and the magnetic field of a single axon.

Once all the instrumentation was ready, the experiment itself was straightforward. I would dissect the ventral nerve cord from a crayfish and place it in a plexiglass bath (again, machined in the student shop) filled with saline (or more correctly, a version of saline for the crayfish called van Harreveld’s solution). The nerve was gently threaded through the toroid, a microelectrode was poked into the axon, and an electrode to record the extracellular potential was placed nearby. I would then stimulate the end of the nerve. It was easy to excite just a single axon; the nerve cord split to go around the esophagus, so I could place the stimulating electrode there and stimulate either the left or right half. In addition, the threshold of the giant axon was lower than that of the many small axons, so I could adjust the stimulator strength to get just one giant axon.

The magnetic field of a single axon.

When I first started doing these experiments, I had a horrible time stimulating the nerve. I assumed I was either crushing or stretching it during the dissection, or there was something wrong with the saline solution, or the epoxy was toxic. But after weeks of checking every possible problem, I discovered that the coaxial cable leading to the stimulating electrode was broken! The experiment had been ready to go all along; I just wasn’t stimulating the nerve. Frankly, I now believe it was a blessing to have a stupid little problem early in the experiment that forced me to check every step of the process, eliminating many potential sources of trouble and giving me a deeper understanding of all the details. 

As you can tell, a lot of effort went into this experiment. Many things could, and did, go wrong. But the work was successful in the end, and the paper describing it remains one of my favorites. I learned much doing this experiment, but probably the most important thing I learned was perseverance.

The Magnetic Field of a Single Axon (Part 1)

“The Magnetic Field of a Single Axon.”

Thirty years ago, John Wikswo and I published “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment” (Biophysical Journal, Volume 48, Pages 93–109, 1985). This was my second journal article (and my first as first author). Russ Hobbie and I cite it in Chapter 8 of the 5th edition of Intermediate Physics for Medicine and Biology. I reproduce the introduction below.

An active nerve axon can be modeled with sufficient accuracy to allow a detailed calculation of the associated magnetic field. Therefore the single axon provides a simple, yet fundamentally important system from which we can test our understanding of the relation between biomagnetic and bioelectric fields. The magnetic field produced by a propagating action potential has been calculated from the transmembrane action potential using the volume conductor model (1). The purpose of this paper is to verify that calculation experimentally. To make an accurate comparison between theory and experiment, we must be careful to correct for all systematic errors present in the data.

To test the volume conductor model it is necessary to measure the transmembrane potential and magnetic field simultaneously. An experiment performed by Wikswo et al. (2) provided preliminary data from a lobster axon, however the electric and magnetic signals were recorded at different positions along the axon and no quantitative comparisons were made between theory and experiment. In the experiment reported here, these limitations were overcome and improved instrumentation was used (3–5).

As the introduction notes, the volume conductor model was described in reference (1), which is an article by Jim Woosley, Wikswo and myself (“The Magnetic Field of a Single Axon: A Volume Conductor Model,” Mathematical Biosciences, Volume 76, Pages 1–36, 1985). I have discussed the calculation of the magnetic field previously in this blog, so today I’ll restrict myself to the experiment.

I was not the first to measure the magnetic field of a single axon. Wikswo’s student, J. C. Palmer, had made preliminary measurements using a lobster axon; reference (2) is to their earlier paper. One of the first tasks Wikswo gave me as a new graduate student was to reproduce and improve Palmer’s experiment, which meant I had to learn how to dissect and isolate a nerve. Lobsters were too expensive for me to practice with so I first dissected cheaper crayfish nerves; our plan was that once I had gotten good at crayfish we would switch to the larger lobster. I eventually became skilled enough in working with the crayfish nerve, and the data we obtained was good enough, that we never bothered with the lobsters.

I had to learn several techniques before I could perform the experiment. I recorded the transmembrane potential using a glass microelectrode. The electrode is made starting with a glass tube, about 1 mm in diameter. We had a commercial microelectrode puller, but it was an old design and had poor control over timing. So, one of my jobs was to design the timing circuitry (see here for more details). The glass would be warmed by a small wire heating element (much like you have in a toaster, but smaller), and once the glass was soft the machine would pull the two ends of the tube apart. The hot glass stretched and eventually broke, providing two glass tubes with long, tapering tips with a hole at the narrow end of about 1 micron diameter. I would then backfill these tubes with 2 Molar potassium citrate. The concentration was so high that when I occasionally forgot to clean up after an experiment I would comeback the next day and find the water had evaporated leaving impressive, large crystals. The back end of the glass tube would be put into a plexiglass holder that connected the conducting fluid to a silver-chloride electrode, and then to an amplifier.

One limitation of these measurements was the capacitance between the microelectrode and the perfusing bath. Because the magnetic measurements required that the nerve be completely immersed in saline, I could not reduce the stray capacitance by lowering the height of the bath. This capacitance severely reduced the rate of rise of the action potential, and to correct for it we used “negative capacitance.” We applied a square voltage pulse to the bath, and measured the microelectrode signal. We then adjusted the frequency compensation knob on the amplifier (basically, a differentiator) until the resulting microelectrode signal was a square pulse. That was the setting we used for measuring the action potential. Whenever I changed the position of the electrode or the depth of the bath, I had to recalibrate the negative capacitance.

To record the transmembrane potential, I would poke the axon (easy to see under a dissecting microscope) with a microelectrode. Often the tip of the electrode would not enter the axon, so I would tap on the lab bench creating a vibration that was just sufficient to drive the electrode through the membrane. Usually I had the output of the microelectrode amplifier go to a device that output current with a frequency that varied with the microelectrode voltage. I’d put this current through a speaker, so I could listen for when the microelectrode tip was successfully inside the axon because the DC potential would drop by about 70 mV (the axon’s resting potential) and therefore the pitch of the speaker would suddenly drop.

Next week I will continue this story, describing how we measured the magnetic field.

The measured transmembrane potential.

Clearance and Semilog Plots

I occasionally like to write a new homework problem as a gift to the readers of Intermediate Physics for Medicine and Biology. Here is the latest, for Section 2.5 about clearance.

Problem 11 ½. A patient has been taking the drug digoxin for her atrial fibrillation. Her distribution volume is V = 400 l. At time t = 0 she stops taking the drug and her doctor measures her blood digoxin concentration, C, every 24 hours.

t (hr)	C (ng/ml)
0	0.85
24	0.53
48	0.33
72	0.20
96	0.125
120	0.077

Calculate the clearance, K, in ml/min.

I like this problem because it reinforces two concepts at once: 1) clearance, and 2) using semilog plots. First, let’s analyze clearance. Equation 2.21 in IPMB gives the blood concentration as a function of time

C(t) = C_o exp(−(K/V)t) .

If we can measure the rate of decay of the concentration, b, where the exponential factor is written as exp(−bt), we can calculate the clearance from K = bV. So, this problem really is about estimating b from the given data.

To calculate the rate b, we should plot the data on semilog graph paper. You can download this graph paper (for free!) at http://www.printablepaper.net/category/log, http://customgraph.com, or http://www.intmath.com/downloads/graph-paper.php. The figure below shows the data points (dots).

The concentration as a function of time.

Now, draw a line through the dots. Sophisticated mathematical techniques could be used to fit the best line through this data, but for this homework problem I suggest merely fitting a line by eye. For data with no noise, such as used here, you should be able to calculate b to within a few percent using a ruler, pen, and some care.

Next, our goal is to determine the decay constant b from the equation C = C_o exp(−bt) using the method discussed in Section 2.3. Select two points on the line. They could be any two, but I suggest two widely spaced points. I’ll use the initial data point (t = 0, C = 0.85) and then estimate the time when the line has fallen by a factor of ten (C = 0.085). The vertical dashed line in the above figure indicates this time, which I estimate to be t = 115 ± 2 hr. I include the uncertainty, which reflects my opinion that I can estimate the time when the dashed line hits the time axis to slightly better than plus or minus one half of one of the small divisions shown on the paper, each of which is 24/5 = 4.8 hr wide. So I have two equations: 0.85 = C_o and 0.085 = C_o exp(−115b). When I divide the two equations, C_o cancels out and I find 10 = exp(115b). Therefore, b = ln(10)/115 = 0.0200 ± 0.0003 hr⁻¹. If I write the exponential as exp(−t/τ), then the time constant is τ = 1/b = 50.0 ± 0.9 hr. Often we say instead that the half-life is t_1/2 = ln(2) τ = 34.7 ± 0.6 hr.

At this point, I suggest you inspect the plot and see if your result makes sense. Does the line appear to drop by half in one half-life? Using the plot, at t = 35 hr I estimate that C is about 0.42, which is just about half of 0.85, so it looks like I’m pretty close.

Now we can get the clearance from K = bV = (0.02 hr⁻¹)(400 l) = 8 l/hr. The problem asks for units of ml/min (a common unit used in the medical literature), so (8 l/hr)(1 hr/60 min)(1000 ml/l) = 133 ml/min. The uncertainty in the clearance is probably determined by the uncertainty in the distribution volume, which we are not given but I’d guess is known to an accuracy of no better than 10%.

Astute readers might be thinking “This is a great homework problem, but I don’t understand why the distribution volume is so big; 400 l is much more than the volume of a person!” The distribution volume does not represent an actual volume of blood or of body fluid. Rather, it takes into account that most of the digoxin is stored in the tissue, with relatively little circulating in the blood. But since the blood concentration is what we measure, the distribution volume must be “inflated” to account for all the drug stored in muscle and other tissues. This is a common trick in pharmacokinetics.

I believe that analyzing data with semilog or log-log graph paper is one of those crucial skills that must be mastered by all science students. As an instructor, you cannot stress it enough. Hopefully this homework problem, and ones like it that you can invent, will reinforce this technique.

The Lewis Number

Last week in this blog I discussed why dolphins don’t breathe through gills like fish do. The take-home message was that using gills would cool the blood to the temperature of the surrounding water, and reheating the blood to the temperature of the dolphin’s body would require a prohibitive amount of energy.

You might be wondering: is there some tricky way that we can adjust things so that oxygen can diffuse without a significant heat transfer? Perhaps alter how long the blood is in thermal and diffusive contact with the seawater so there is time for oxygen diffusion but not time for thermal diffusion. Might that save the day?

You can compare the mechanisms of molecular and thermal heat transfer using the Lewis number, which is a ratio of the molecular diffusion constant and the thermal diffusion constant. Russ Hobbie and I discuss the Lewis number in Problem 20 of Chapter 4 of Intermediate Physics for Medicine and Biology. For oxygen diffusing in water, the diffusion constant is about 2 x 10⁻⁹ m²/s. The diffusion constant for heat is equal to the thermal conductivity divided by the product of the specific heat capacity and the density, which for water is about 1.5 x 10⁻⁷ m²/s. Thus, the diffusion constant for oxygen is about one hundred times less than the diffusion constant for heat. In other words, heat diffuses one hundred times more readily than oxygen, so it’s difficult to imagine how you could ever devise a situation where you could transfer oxygen without transferring heat too. As we concluded last week, physics constrains biology.

If you are exchanging heat and oxygen in air the situation is a bit better: for air the diffusion constant of oxygen and of heat are roughly the same. You can’t have oxygen diffusion without heat diffusion, but at least you aren’t down by a factor of one hundred.

The Lewis number is one of those useful dimensionless numbers—like the Reynolds number and the Peclet number—that summarizes the relative importance of two physical mechanisms. Because these numbers are dimensionless, their values does not depend on the system of units you use.

This all sounds fine and good, so imagine my surprise when one of my Biological Physics students working on this week’s homework assignment told me that the definition of the Lewis number in IPMB differs from the definition used by other sources. Yikes! The question comes down to this: is the Lewis number defined as the molecular diffusion constant over the thermal diffusion constant, or as the thermal diffusion constant over the molecular diffusion constant? In one sense it does not matter which definition you use. Either definition will tell you that in water oxygen has a harder time diffusing than heat. The only difference is that in one case the Lewis number is 1/100, and in the other case it is 100. The definition is arbitrary, like which direction you call right and which you call left. Had the first person to talk about those two directions called right left and left right, it would make no difference; they are just labels. However, I concede that if everyone uses different labels, confusion results. If half the people call left left and the other half call left right, then giving directions will be difficult—you would have to verify that you used the same definition of left and right before you could tell someone how to get across town.

I decided to check that fount of all knowledge: Wikipedia (how did I grow up without it?). There the definition is the opposite of that in IPMB—“Lewis number (Le) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity.” The free dictionary says “A dimensionless number used in studies of combined heat and mass transfer, equal to the thermal diffusivity divided by the diffusion coefficient” and thermopedia says the same, as does this publication. In the book Air and Water, Mark Denny uses our definition, molecules over heat (perhaps I should say we use Denny’s definition, because I am pretty sure we used Air and Water as our source). Interestingly, the CRC Handbook of Chemistry and Physics (I looked at the 59th edition, which is the one sitting in my office) says heat over molecules, but then adds “N.B.: Lewis number is sometimes defined as reciprocal of this quantity”). My conclusion is that the definition is a bit uncertain, but Russ and I (and Denny) appear to have adopted the minority view. What should I do? I’ve added to the IPMB errata the following entry:

Page 109: At the end of Problem 20, add the sentence “Warning: the Lewis number is sometimes defined as the reciprocal of the definition used here.”

Dolphins are not Sharks

Flipper.

I grew up watching the TV show Flipper, about a dophin. These curious creatures are mammals so they are warm blooded, but they have adapted in many ways to living in the sea. They have not, however, completely evolved into fish. For instance, they breathe air like we do rather than extracting oxygen from seawater using gills.

Russ Hobbie and I mention dolphins in the 5th edition of Intermediate Physics for Medicine and Biology, in a homework problem in Chapter 3.

Problem 50. Fish are cold blooded, and “breathe” water (in other words, they extract dissolved oxygen from the water around them using gills). Could a fish be warm blooded and still breathe water? Assume a warm-blooded fish maintains a body temperature that is 20 °C higher than the surrounding water. Furthermore, assume that the blood in the gills is cooled to the temperature of the surrounding water as the fish breathes water. Calculate the energy required to reheat 1 l of blood to the fish’s body temperature. One liter of blood carries sufficient oxygen to produce about 4000 J of metabolic energy. Is the energy needed to reheat 1 l of blood to body temperature greater than or less than the metabolic energy produced by 1 l of blood? What does this imply about warmblooded fish? Why must a warm-blooded aquatic mammal such as a dolphin breathe air, not water? Use c = 4200 J K⁻¹ kg⁻¹ and ρ = 10³ kg m⁻³ for both the body and the surrounding water. For more on this topic, see Denny (1993).

The basic idea is that the gills would need to “process” a lot of seawater to raise the oxygen concentration in a small amount of blood. The seawater and blood have similar specific heats (that of water), so the heat capacity of the blood is much less than the heat capacity of the processed water. In other words, the surrounding seawater cools the blood to the temperature of the water, rather than the dolphin warming the seawater to its body temperature. This cold blood in the gills must then be warmed to the dolphin body temperature, which takes a lot of energy—much more than you would get by using the extracted oxygen for metabolism. You can’t win.

Air and Water:
The Biology and Physics of Life's Media,
by Mark Denny.

The reference at the end of the homework problem is to the wonderful book Air and Water: The Biology and Physics of Life’s Media, by Mark Denny (Princeton University Press). Denny writes

Consider a hypothetical example. It could conceivably be advantageous for a warm-blooded animal such as a dolphin to breathe water instead of air. Such an adaption would remove the necessity for the animal to return periodically to the water’s surface, thereby increasing the time available in which to hunt food. However, if a 100 kg dolphin swimming in 7 °C water were to breathe water and still maintain a body temperature of 37 °C, it would expend energy at a rate of 3361 W just to heat its respiratory water. This is more than thirty times greater than its resting metabolic rate of 107 W! It suddenly becomes clear why marine mammals and birds continue to breathe air, and why water-breathing organisms (such as fish) are seldom much warmer than their watery surroundings.

A dolphin (a warm-blooded, air-breathing mammal) is very different from a shark (a cold-blooded, gill-breathing fish), even if they look similar.

Physics constrains biology. Evolution can do marvelous things, but it can’t violate the laws of physics.

Herman Carr, MRI pioneer

Nuclear Magnetic Resonance, Magnetic Resonance Spectroscopy, and Magnetic Resonance Imaging have resulted in Nobel Prizes to eight famous scientists.

• Otto Stern, 1943, Physics, “for his contribution to the development of molecular ray method and his discovery of the magnetic moment of the proton.” 
• Isidor Rabi, 1944, Physics, “for his resonance method for recording the magnetic properties of atomic nuclei.” 
• Felix Bloch and Edward Purcell, 1952, Physics, “for their discovery of new methods for nuclear magnetic precision measurements and discoveries in connection therewith.” 
• Richard Ernst, 1991, Chemistry, “for his contributions to the development of the methodology of high resolution nuclear magnetic resonance (NMR) spectroscopy.”
• Kurt Wuthrich, 2002, Chemistry, “for his development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution.” 
• Paul Lauterbur and Peter Mansfield, 2003, Physiology or Medicine, “for their discoveries concerning magnetic resonance imaging.”

Other pioneers are also well-known, such as Raymond Damadian and Erwin Hahn. Yet one crucial scientist who helped establish nuclear magnetic resonance is less know: Herman Carr.

Russ Hobbie and I mention Carr in Intermeidate Physics for Medicine and Biology in the context of a well-known MRI technique: the Carr-Purcell sequence (see Section 18.8). This sequence consists of a 90-degree radio-frequency magnetic pulse that tips the proton spins into the transverse plane, followed by a series of 180-degree RF pulses that form spin echos (I discussed spin echoes in this blog previously).

Herman Carr (1924-2008) grew up in Alliance, Ohio (about 60 miles east of the town where I attended my junior year of high school, Ashland, Ohio). He was a sergeant in the US army air corps during World War II, serving in Italy. He earned his physics PhD in 1953 from Harvard under the direction of Purcell, and spent most of his career at Rutgers University.

Carr is best known for his early work on magnetic resonance imaging. In this PhD thesis, he applied a magnetic field that varied with position and produced a one-dimensional image, thus introducing the use of magnetic field gradients for MRI. This idea was later developed by Paul Lauterbur. The gist of the method is that the magnetic field varies in space, and therefore the Larmor frequency of the proton spins varies in space. If you measure the magnetic resonance signal and separate it into different frequencies (Fourier analysis), each frequency component corresponds to the signal from a different location (see Section 18.9 of IPMB).

A controversy arose about the MRI Nobel prize to Lauterbur and Mansfield. Some claim that either Damadian (who did medical imaging without gradients) or Carr (who used gradients but did not do medical imaging), or both, should have shared in the prize. This is primarily a historical debate, about which I am not an expert. My impression is that while Lauterbur, Mansfield, Damadian and Carr all deserve credit for their work, the Nobel committee was not wrong in singling out the two winners.

Carr also made important contributions to using nuclear magnetic resonance to measure diffusion. Below is the abstract to Carr and Purcell’s article “Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments” (Physical Review, Volume 94, Pages 630–638, 1954)

Nuclear resonance techniques involving free precession are examined, and, in particular, a convenient variation of Hahn's spin-echo method is described. This variation employs a combination of pulses of different intensity or duration (“90-degree” and “180-degree” pulses). Measurements of the transverse relaxation time T2 in fluids are often severely compromised by molecular diffusion. Hahn's analysis of the effect of diffusion is reformulated and extended, and a new scheme for measuring T2 is described which, as predicted by the extended theory, largely circumvents the diffusion effect. On the other hand, the free precession technique, applied in a different way, permits a direct measurement of the molecular self-diffusion constant in suitable fluids. A measurement of the self-diffusion constant of water at 25°C is described which yields D=2.5(±0.3)×10⁻⁵ cm² /sec, in good agreement with previous determinations. An analysis of the effect of convection on free precession is also given. A null method for measuring the longitudinal relaxation time T1, based on the unequal-pulse technique, is described.

This paper was named a citation classic, and excerpts from Carr’s reminisces (written in 1983) of the paper are reproduced below.

In the fall of 1949 at Harvard University, I began reading about nuclear magnetic resonance (NMR) under the guidance of E. M. Purcell. In early November, Purcell read E. L. Hahn’s historic abstract about the fascinating phenomenon of ‘spin echoes.’ Purcell suggested that I try to understand this effect.

During Christmas recess I traveled to a student conference at the University of Illinois where ... I made a visit to the physics building where Hahn showed me his laboratory—a cramped hallway at the top of a high stairwell. There for the first time I saw spin echoes and learned about their discovery.

Hahn had explained his echoes using a model involving only equatorial components. Purcell suggested using a three-dimensional model, and this greatly simplified the understanding of the relatively complicated echoes associated with Hahn’s equal pulses. It was during lunch one day in the spring of 1950 that I realized the explanation could be simplified even more by using two unequal 90°and 180°pulses, and indeed a sequence consisting of a 90° pulse followed by a series of 180° pulses ... By the end of the summer of 1950, we had seen our own echoes at Harvard.

The 1954 paper—drafts of which were written in a cabin on a Cache Lake-island in Ontario’s Algonquin Park — included work done both at Harvard and using, in 1952- 1953, Henry Torrey’s excellent new magnet at Rutgers University. In addition ... the 1954 paper included an explanation of the effect of a 180°pulse in partially eliminating the artificial decay caused by diffusion in an inhomogeneous magnetic field ... The absolute value of the water self-diffusion coefficient D reported in the paper was measured at Rutgers using “anti- Helmholtz” coils to obtain the nearly uniform gradient ... To the best of my knowledge, this was the first use of intentionally applied gradients to obtain spatial information.

The extensive citation of this 1954 paper is undoubtedly due both to its very simple explanation of important basic phenomena, and to the exceedingly extensive—indeed, beyond all our expectations—applications of free precession techniques, especially when coupled with fast computer technology...

An obituary of Carr is given here.

Polonium-210, The Perfect Poison

Figure 17.27 in the 5th edition of Intermediate Physics for Medicine and Biology shows the decay series arising from the radioactive isotope radon-222, which itself is produced by the decay of the long-lived isotope uranium-238. The last step in this long chain of reactions is the alpha decay of polonium-210 to the stable isotope lead-206. The half-life of this decay is 138 days. This is not the only isotope of polonium in radon’s decay series. A heavier isotope polonium-214 has a half-life of 160 microseconds, and polonium-218 has a half-life of 3 minutes.

Polonium was discovered by Marie and Pierre Curie in 1898 when analyzing pitchblende, a uranium containing ore. It was named after Marie’s homeland, Poland. Now ²¹⁰Po is produced by bombarding bismuth-209 with neutrons, forming bismuth-210, which undergoes beta decay to ²¹⁰Po.

²¹⁰Po is infamous for being a deadly poison. For a given mass, ²¹⁰Po is 250,000 times more toxic than hydrogen cyanide. Its toxicity comes from the 5.3-MeV alpha particle it emits. Because alpha particles are easily stopped by clothing and even skin, ²¹⁰Po is dangerous primarily when breathed or ingested, so that the alpha particles are emitted inside the body. A nearly pure alpha emitter, ²¹⁰Po rarely emits a gamma ray, making it difficult to detect this poison unless one measures the alpha particles directly. A lethal dose comes from ingesting about a microgram.

²¹⁰Po was used in the 2006 assassination of Alexander Litvinenko, a former Russian spy who was apparently given some polonium-laced tea by Russian agents (the investigation into this complicated murder continues--see here and here--and the details are still debated). Death by ²¹⁰Po is slow; the 44-year old Litvinenko needed 22 days for the radiation to eventually take his life.

Polonium was also suspected to play a role in the 2004 death of Palestinian leader Yasser Arafat. Just this month, a French investigation has concluded that there is not enough evidence for pressing charges. The issue is complicated because ²¹⁰Po is found in cigarette smoke, and Arafat was a heavy smoker. The National Council on Radiation Protection and Measurements reports that the annual effective dose equivalent to a smoker from radiation in tobacco is about 13 mSv, which is over four times the average annual dose of 3 mSv we are all exposed to (see Section 16.12 in IPMB), but is still a tiny dose.

The Environmental Protection Agency has published a report titled “Occurrence of ²¹⁰Po and Biological Effects of Low-Level Exposure: The Need for Research.” As with all studies of low-level radiation exposure, the results are difficult to assess, and depend on our assumptions about radiation risks at small doses. But Alexander Litvinenko’s death proves that at high doses ²¹⁰Po is very dangerous indeed; it’s perhaps the perfect poison.

Boltzmann’s Tomb

Asimov’s Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

In Chapter 3 of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Boltzmann factor and Boltzmann’s constant. Later in the book, we analyze the Poisson-Boltzmann equation and the Stefan-Boltzmann law. Who was Boltzmann? In Asimov’s Biographical Encyclopedia of Science and Technology (2nd revised edition), Isaac Asimov writes

BOLTZMANN, Ludwig Edward (bohlts’mahn)
Austrian physicist
Born: Vienna, February 20, 1844
Died: Duino, near Trieste (then in Austria, now in Italy), September 5, 1906

Boltzmann, the son of a civil servant, received his Ph.D. from the University of Vienna in 1866. His work on the kinetic theory of gases was done independently of Maxwell and they share the credit.

Beginning in 1871, Boltzmann increased the rigor of the mathematical treatment and emphasized the statistical interpretation of the second law of thermodynamics thus founding “statistical mechanics.” He showed that Clausius’ concept of increasing entropy of disorder [could be based on statistical ideas], laying the groundwork for the later achievements of Gibbs.

He was a firm proponent of atomism at a time when Ostwald was mounting the final campaign against it. Boltzmann also advanced a mathematical treatment that explained the manner in which, according to the experimental observations of Stefan (whom Boltzmann, in this college years, served as assistant), quantity of radiation increased as the fourth power of the temperature. This is therefore sometimes called the Stefan-Boltzmann law.

Boltzmann turned down a chance to succeed Kirchhoff at Berlin but in 1894 succeeded to Stefan’s post in Vienna.

Though Boltzmann lived longer than Maxwell, his life too was cut short. In his case it was suicide, brought on by recurrent episodes of severe mental depression accentuated, perhaps, by opposition to his atomistic notions by Oswald and others.

His equation relating entropy and disorder was engraved on the headstone of his grave.

I particularly am intrigued by the last sentence of Asimov’s entry. Who puts an equation on their tombstone? Boltzmann did!

Boltzmann's tombstone.

This equation is Eq. 3.20 in IPMB.

S = k_B ln Ω ,

where S is the entropy, k_B is Boltzmann’s constant, ln is the natural logarithm, and Ω is the number of microstates. The equation says that the entropy increases as the number of possible microstates increases. If there are only one or a few states available, the entropy is small; if there are many states available, the entropy is large. Thus, from a statistical mechanics point-of-view, the thermodynamic concept of entropy (developed well before Boltzmann’s work) is a measure of the number of states. The logarithm is important, because if system A has 10 states available and system B has 20 states available, the total number of states is the product, 200. If the entropy were proportional to Ω, the total entropy of the two systems would not be the sum of the entropy in each system. However, the logarithm property ln(Ω_AΩ_B)=ln(Ω_A)+ln(Ω_B) ensures that the entropy is indeed additive.

The definition of entropy in terms of the number of states is a fundamental relationship connecting thermodynamics and statistical mechanics. No wonder Boltzmann wanted it on his tombstone.