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.