The lasttwo blog posts have dealt with biomagnetism: the magnetic fields produced by our bodies. Some of you might have noticed hints about how these posts originated in “another publication.” That other publication is now published! This week, my review article “Biomagnetism: The First Sixty Years” appeared in the journal Sensors. The abstract is given below.
Biomagnetism is the measurement of the weak magnetic fields produced by nerves and muscle. The magnetic field of the heart—the magnetocardiogram (MCG)—is the largest biomagnetic signal generated by the body and was the first measured. Magnetic fields have been detected from isolated tissue, such as a peripheral nerve or cardiac muscle, and these studies have provided insights into the fundamental properties of biomagnetism. The magnetic field of the brain—the magnetoencephalogram (MEG)—has generated much interest and has potential clinical applications to epilepsy, migraine, and psychiatric disorders. The biomagnetic inverse problem, calculating the electrical sources inside the brain from magnetic field recordings made outside the head, is difficult, but several techniques have been introduced to solve it. Traditionally biomagnetic fields are recorded using superconducting quantum interference device (SQUID) magnetometers, but recently new sensors have been developed that allow magnetic measurements without the cryogenic technology required for SQUIDs.
The “First Sixty Years” refers to this year (2023) being six decades since the original biomagnetism publication in 1963, when Baule and McFee first measured the magnetocardiogram.
My article completes a series of six reviews I’ve published in the last few years.
When I was on the faculty at Vanderbilt University, my student Marcella Woods and I examined the magnetic field produced by electrical activity in a sheet of cardiac muscle. I really like this analysis, because it provides a different view of the mechanism producing the magnetic field compared to that used by other researchers studying the magnetocardiogram. In another publication, here is how I describe our research. I hope you find it useful.
Roth and Marcella Woods examined an action potential propagating through a two-dimensional sheet of cardiac muscle [58]. In Fig. 6, a wave front is propagating to the right, so the myocardium on the left is fully depolarized and on the right is at rest. Cardiac muscle is anisotropic, meaning it has a different electrical conductivity parallel to the myocardial fibers than perpendicular to them. In Fig. 6, the fibers are oriented at an angle to the direction of propagation. The intracellular voltage gradient is in the propagation direction (horizontal in Fig. 6), but the anisotropy rotates the intracellular current toward the fiber axis. The same thing happens to the extracellular current, except that in cardiac muscle the intracellular conductivity is more anisotropic than the extracellular conductivity, so the extracellular current is not rotated as far. Continuity requires that the components of the intra- and extracellular current densities in the propagation direction are equal and opposite. Their sum therefore points perpendicular to the direction of propagation, creating a magnetic field that comes out of the plane of the tissue on the left and into the plane on the right (Fig. 6) [58–60].
Figure 6. The current and magnetic field produced by a planar wave front
propagating in a two-dimensional sheet of cardiac muscle. The muscle is
anisotropic with a higher conductivity along the myocardial fibers.
This perspective of the current and magnetic field in cardiac muscle is unlike that ordinarily adopted when analyzing the magnetocardiogram, where the impressed current is typically taken as in the same direction as propagation. Nonetheless, experiments by Jenny Holzer in Wikswo’s lab confirmed the behavior shown in Fig. 6 [61].
One of the key limitations of the magnetoencephalogram (MEG) is that it’s not sensitive to a radial dipole. What does this mean? The MEG is the magnetic field outside the head produced by the electrical activity of neurons in the brain. Often the source of this activity can be described by a current dipole, p, representing the intracellular current in the neurons. Because current flows in continuous loops, a dipole is surrounded by extracellular “return currents” flowing throughout the brain. Often the brain can be approximated as a sphere.
One can see from the symmetry argument in the caption
of Fig. 8.19 that in a spherically symmetric conducting
medium the radial component of p and its return currents do
not generate any magnetic field outside the sphere. Therefore
the MEG is most sensitive to detecting activity in the fissures
of the cortex, where the trunk of the postsynaptic dendrite
is perpendicular to the surface of the fissure. A tangential
component of p does produce a magnetic field outside a
spherically symmetric conductor.
While this text and figure do explain why a radial dipole has zero magnetic field, the explanation is a bit cryptic. Here is an alternative explanation that I wrote for another publication, and a better (or at least more colorful) figure.
A radial dipole produces no magnetic field (Fig. 8). This result is best proved using Ampere’s law: the magnetic field integrated along a closed loop is proportional to the net current threading the loop. The symmetry is sufficient that the integral over the path (dashed circle in Fig. 8) equals the path length times the magnetic field. The current produced by a dipole, including the return current, must be contained within the sphere because the region outside is not conducting. Hence, the net current threading the loop (the dipole plus the return current) is zero, so the magnetic field of a radial dipole vanishes.
Figure 8. The magnetic field of a radial dipole is zero outside a spherical conductor.
For three years I’ve dodged the bullet, but no more; I have covid. I’m doing fine, thank you. For me the symptoms were similar to a moderate cold. My doctor put me on a five-day regimen of the antiviral drug Paxlovid plus some supplements to support my immune system (vitamin C, vitamin D3, and zinc). I’ve been isolating in our spare bedroom, which is boring but otherwise comfortable. I think I’m over the hump.
During the last few days I’ve taken several of those at-home covid rapid antigen tests. There’s some interesting physics at work in them. The figure below illustrates how they’re constructed.
To perform a test, you typically swab your nose, dip the swab in saline, stir, and then place a few drops of the solution onto the sample pad (A). You’re not detecting the virus itself, but instead the SARS-Cov 2antibody. To explain what that means, I need to delve into a bit of immunology.
Our immune system produces a Y-shaped protein called an antibody, or immunoglobulin, that can selectively bind to an antigen, which is typically a protein that’s part of the coronavirus. The beauty of the antibody-antigen reaction is that it’s so specific: it lets the immune system attack a particular virus, bacteria, or other pathogen, ignoring everything else.
When you get covid, your body launches an immune attack by producing SARS-Cov 2 antibodies. In the illustration above, the yellow Y is the antibody you are trying to detect. See David Goodsell’s marvelous painting of a virus being attacked by antibodies at the bottom of this post.
In the above figure, the conjugate pad (B) is where much of the physics lives. The pad contains gold nanoparticles (AuNP) that are coated with anti-human antibodies. An “anti-human antibody” is a molecule that binds selectively to a human antibody. In the figure, a red dot with a blue Y sticking out is a gold nanoparticle with an anti-human antibody bound to it.
A nitrocellulose membrane (NC membrane) is made from a mesh of nitrocellulose fibers (C). The mesh is porus
and acts something like a wick, pulling the fluid from left to right by capillary action. This is why a device like that in the figure above is sometimes called a lateral flow test. The mesh also provides protected space for the nanoparticles and molecules to move around and interact in. The absorbent pad (D) acts like a sponge, soaking up the fluid
as it reaches the right end of the detector, contributing to the
capillary action and preventing any back flow.
As any SARS-Cov 2 antibody passes by a gold nanoparticle/anti-human antibody, it binds and the entire complex flows to the right together (in the figure, a combined red dot/blue Y/yellow Y).
Some additional molecules are bound to two spots on the nitrocellulose membrane. One,
the test strip, has the SARS-cov 2 antigen. If any SARS-Cov 2 antibody
passes by, it will bind to the antigen, immobilizing the gold
nanoparticles. The other strip is goat
anti-mouse antibody. How did a goat and mouse get involved? I don’t know. As I understand it, gold nanoparticles with antibodies that bind to the goat anti-mouse antibody are included in the conjugate pad, so regardless of if you have covid or not it serves as a control. If the nanoparticles
don’t collect at the control strip, something is wrong.
Why bother with the gold nanoparticles? Their role is to transduce the signal so it becomes visible. Nanoparticles have interesting optical properties. When exposed to an electromagnetic field such as light, the electric field causes electrons to accumulate on one side of the particle creating a negative surface charge, leaving the opposite side positive from a lack of electrons. Such a distribution of charge oscillates at its own natural frequency (its plasma frequency), and when this frequency matches the driving frequency of the light there is a resonance. This “localized surface plasmon resonance” is effective at absorbing or scattering light. Scattering is particularly important because Rayleigh scattering (the scattering of light by particles with a radius much smaller than the wavelength of the light) depends on the sixth power of the particle radius. The binding of nanoparticles (which typically have a diameter of tens of nanometers) with large antibodies and antigens, and the aggregation of these complexes, can increase their effective size, accentuating scattering. In addition, the high concentration of the nanoparticles at the test and control strips enhance any optical effect. The end result is that you see a dark line if the nanoparticles are present.
So swab your nose, swish it in some saline, add a few drops to the sample pad, and wait. After about 15 minutes look at the results. If there is no control line, you’ve messed up. Throw the test away and try again. If there’s a control line but no test line, you’re negative. Be happy (but not too happy, because these tests are prone to false negatives). If there’s both a control line and a test line, you’ve got covid. The tests don’t give false positives too often, so you can be fairly confident you have the disease. Isolate yourself and talk to you doctor.
Where is the physics in all this? First, in the flow, which results from the surface tension created by the mesh of fibers, leading to capillary action. Second, in the optical properties of the nanoparticles, which provide the color that you see in the test and control strips. Unfortunately, Intermediate Physics for Medicine and Biology doesn’t discuss capillary action or surface plasmons, so you can’t learn about them there. Sorry; no book can cover everything. But there is interesting physics hidden in these tests.
Stay safe, dear reader, and may all your covid tests be negative.
This painting shows a cross section through a coronavirus surrounded by
blood plasma, with neutralizing antibodies in bright yellow. The painting was commissioned for the cover of a special COVID-19 issue of Nature. From: David S. Goodsell, RCSB Protein Data Bank and Springer Nature; doi: 10.2210/rcsb_pdb/goodsell-gallery-025
A chemist explains how at-home covid tests work. From WIRED.
Let’s see what my favorite books about writing say.
The Elements of Style, by Strunk and White.
Strunk and White (The Elements of Style): “Do not attempt to emphasize simple statements by using a mark of exclamation…The exclamation mark is to be reserved for use after true exclamations or commands.”
On Writing Well, by William Zinsser.
Zinsser (On Writing Well): “Don’t use it unless you must to achieve a certain effect. It has a gushy aura…Resist using an exclamation point to notify the reader that you are making a joke or being ironic…Humor is best achieved by understatement, and there’s nothing subtle about an exclamation point.”
Dreyer (Dreyer’s English):
“Go light on the exclamation points. When overused, they’re bossy,
hectoring, and, ultimately, wearying. Some writers recommend that you
should use no more than a dozen exclamation points per book; others
insist that you should use no more than a dozen exclamation points in a
lifetime.”
Plain Words, by Sir Ernest Gowers.
Gowers (Plain Words): His disdain for the exclamation point is so complete he doesn’t even acknowledge that it exists.
Below I list eleven places where exclamation points appear in IPMB.
In each case, I indicate if we should keep the exclamation point or
toss it out. Note: I don’t include any exclamation point that indicates a
factorial, such as 4! = 1 × 2 × 3 × 4 = 24; that’s a mathematical symbol, not punctuation.
Page 29: Homework Problem 43 in Chapter 1 says “Suppose a student asked you, ‘How can blood be moving more slowly in a capillary than in the aorta? … The capillary has a much smaller cross-sectional area than the aorta. Therefore, the blood should move faster in the capillary than in the aorta!” The exclamation point belongs to the hypothetical student, not to Russ and me. If you don’t like it, take it up with the student (and beware, those students tend to be gushy). Keep.
Page 44: In Chapter 2, Russ and I write “Moreover, commercial graphing software does not impose this constraint on log–log plots, so it is becoming less and less likely that you can determine the exponent by glancing at the plot. Be careful!” The exclamation point is a warning. Keep.
Page 51: In the references at the end of Chapter 2, we cite Albert Bartlett’s delightful book “The Essential Exponential!” The exponential point is Bartlett’s, and is part of the title (like Oklahoma!). Keep.
Page 53: In the introduction to Chapter 3, Russ and I explain why statistical methods are so useful in thermodynamics. We estimate how long is required to simulate all the particles in a cubic millimeter of blood. We conclude “If a computer can do 1012 operations/s, then the complete calculation for a single time interval will require 108 s or 3 years!” In other words, a really long time. My feelings on this one are mixed. Toss.
Page 270: In a footnote in Chapter 10, we write “Strictly speaking, (dV/dt)alveoli is not the derivative of a function V. (It always has a positive value, and the lungs are not expanding without limit!) We use the notation to remind ourselves that it is the rate of air exchange in the alveoli.” If I could delete only one exclamation point from IPMB, this would it. The thought of those lungs getting bigger and bigger is just disturbing. Toss.
Page 308: In Chapter 11, Russ and I compare two methods for doing a least squares fit of an exponentially decaying function. Method 1: take the logarithm of the data and then do a linear fit. Method 2: do a nonlinear fit to the original data. We end the analysis with a moral: “Use nonlinear least squares, Method 2!” Keep.
Page 388: A section in Chapter 14 has tricky units. We remind the reader to “Be careful with units!” Exclamation points are legitimate when issuing a command or warning. Perhaps, however, we shouldn’t have used both bold and an exclamation point. Keep.
Page 484: When explaining the linear-quadratic model for radiation damage, Russ and I concoct an illustrative but unrealistic example. We then warn the reader “(This is not realistic!).” I wonder if we should’ve made a more realistic example and avoided the exclamation point. Toss.
Page 497: In Homework Problem 8 of Chapter 16, Russ and I discuss x-ray imaging devices used to fit kids’ shoes, common in the 1940s. “These marvelous units were operated by people who had no concept of radiation safety and aimed a beam of x-rays upward through the feet and right at the reproductive organs of the children!” My vote is to keep this exclamation point at all costs! Let’s even keep the sarcasm. Keep.
Page 587: In Appendix H, we use an exclamation point when trying to make a joke. We had already developed the binomial probability distribution using p for the probability of success and q for the probability of failure. Then we write “Suppose that we do N independent tests, and suppose that in healthy people, the probability that each test is abnormal is p. (In our vocabulary, having an abnormal
test is ‘success’!).” Lame. Toss.
Page 593: Appendix I has a footnote containing my favorite exclamation point in Intermediate Physics for Medicine and Biology. It appears when we introduce Stirling’s formula: ln n! ≈ n ln n - n. “For more about Stirling’s formula, see N. D. Mermin (1994) Stirling’s formula!Am J Phys 52: 362–365.” The exclamation point is a pun. Keep!
Over the last couple years, I’ve been writing lots of review articles. In the last few weeks three have been published. All of them are open access, so you can read them without a subscription.
Can MRI be Used as a Sensor to Record Neural Activity?
Magnetic resonance provides exquisite anatomical images and functional MRI monitors
physiological activity by recording blood oxygenation. This review attempts to answer the following
question: Can MRI be used as a sensor to directly record neural behavior? It considers MRI sensing of
electrical activity in the heart and in peripheral nerves before turning to the central topic: recording
of brain activity. The primary hypothesis is that bioelectric current produced by a nerve or muscle
creates a magnetic field that influences the magnetic resonance signal, although other mechanisms for
detection are also considered. Recent studies have provided evidence that using MRI to sense neural
activity is possible under ideal conditions. Whether it can be used routinely to provide functional
information about brain processes in people remains an open question. The review concludes with
a survey of artificial intelligence techniques that have been applied to functional MRI and may be
appropriate for MRI sensing of neural activity.
Parts of the review may be familiar to readers of this blog. For instance, in June of 2016 I wrote about Yoshio Okada’s experiment to measure neural activation in a brain cerebellum of a turtle, in August 2019 I described Allen Song’s use of spin-lock methods to record brain activity, and in April 2020 I discussed J. H. Nagel’s 1984 abstract that may have been the first to report using MRI to image action currents. All these topics are featured in my review article. In addition, I analyzed my calculation, performed with graduate student Dan Xu, of the magnetic field produced inside the heart, and I reviewed my work with friend and colleague Ranjith Wijesinghe, from Ball State University, on MRI detection of bioelectrical activity in the brain and peripheral nerves. At the end of the review, I examined the use of artificial intelligence to interpret this type of MRI data. I don’t really know much about artificial intelligence, but the journal wanted me to address this topic so I did. With AI making so much news these days (ChatGPT was recently on the cover of TIME magazine!), I’m glad I included it.
Readers of Intermediate Physics for Medicine and Biology will find this review to be a useful extension of Section 18.12 (“Functional MRI”), especially the last paragraph of that section beginning with “Much recent research has focused on using MRI to image neural activity directly, rather than through changes in blood flow...”
Next is “Magneto-Acoustic Imaging in Biology,” published in the journal Applied Sciences (Volume 13, Article Number 3877). The abstract states
This review examines the use of magneto-acoustic methods to measure electrical conductivity.
It focuses on two techniques developed in the last two decades: Magneto-Acoustic Tomography
with Magnetic Induction (MAT-MI) and Magneto-Acousto-Electrical Tomography (MAET). These
developments have the potential to change the way medical doctors image biological tissue.
The only place in IPMB where Russ Hobbie and I talked about these topics is in Homework Problem 31 in Chapter 8, which analyzes a simple example of MAT-MI.
This article reviews the mechanical bidomain model, a mathematical description of how the extracellular matrix and intracellular cytoskeleton of cardiac tissue are coupled by integrin membrane proteins. The fundamental hypothesis is that the difference between
the intracellular and extracellular displacements drives mechanotransduction. A one-dimensional example illustrates the model, which
is then extended to two or three dimensions. In a few cases, the bidomain equations can be solved analytically, demonstrating how
tissue motion can be divided into two parts: monodomain displacements that are the same in both spaces and therefore do not
contribute to mechanotransduction, and bidomain displacements that cause mechanotransduction. The model contains a length
constant that depends on the intracellular and extracellular shear moduli and the integrin spring constant. Bidomain effects often occur
within a few length constants of the tissue edge. Unequal anisotropy ratios in the intra- and extracellular spaces can modulate
mechanotransduction. Insight into model predictions is supplied by simple analytical examples, such as the shearing of a slab of cardiac
tissue or the contraction of a tissue sheet. Computational methods for solving the model equations are described, and precursors to the
model are reviewed. Potential applications are discussed, such as predicting growth and remodeling in the diseased heart, analyzing
stretch-induced arrhythmias, modeling shear forces in a vessel caused by blood flow, examining the role of mechanical forces in
engineered sheets of tissue, studying differentiation in colonies of stem cells, and characterizing the response to localized forces applied
to nanoparticles.
This review is similar to my article that I discussed in a blog post about a year ago, but better. I originally published it as a manuscript on the bioRxiv, the preprint server for biology, but it received little attention. I hope this version does better. If you want to read this article, download the pdf instead of reading it online. The equations are all messed up on the journal website, but they look fine in the file.
If you put these three reviews together with my previous ones about magnetic stimulation and the bidomain model of cardiac electrophysiology, you have a pretty good summary of the topics I’ve worked on throughout my career. Are there more reviews coming? I’m working feverishly to finish one more. For now, I’ll let you guess the topic. I hope it’ll come out later this year.
Why Should We Abolish Daylight Saving Time? J. Biol. Rhythms, 34:227–230, 2019.
Last Sunday, we all switched from standard time to daylight saving time, losing an hour of sleep in the process. Should we stop this changing of clocks every six months? We have three options: 1) we can continue to switch between standard time in the winter and daylight saving time in the summer (our current practice), 2) we can change to permanent daylight saving time (a change that the Senate has passed, but has not yet been approved by the House of Representatives), or 3) we can change to permanent standard time. A position paper from the Society of Research on Biological Rhythms addresses this issue. The citation and abstract are given below.
Local and national governments around the world are currently considering the elimination of the annual switch to and from Daylight Saving Time (DST). As an international organization of scientists dedicated to studying circadian and other biological rhythms, the Society for Research on Biological Rhythms (SRBR) engaged experts in the field to write a Position Paper on the consequences of choosing to live on DST or Standard Time (ST). The authors take the position that, based on comparisons of large populations living in DST or ST or on western versus eastern edges of time zones, the advantages of permanent ST outweigh switching to DST annually or permanently. Four peer reviewers provided expert critiques of the initial submission, and the SRBR Executive Board approved the revised manuscript as a Position Paper to help educate the public in their evaluation of current legislative actions to end DST.
Biological oscillations are complicated. Readers of Chapter 10 in Intermediate Physics for Medicine and Biology know that nonlinear dynamics makes resetting an oscillator’s phase difficult, and that driving a nonlinear oscillator can lead to complex, and sometimes even chaotic, behavior. I’m glad that the Society for Research on Biological Rhythms sought advice from experts about this issue.
Roenneberg et al.’s paper focuses on health issues related to our three clocks: the sun clock, our body clock, and our social clock (the clock set by society). The authors summarize the problem of synchronizing these three clocks in this way:
We live according to the same social clock time within a time zone, but as long as we still can see the natural day (through windows or on our way to or from work or school), our body clocks still follow more or less the time of the sun clock.
Is this a problem? Apparently so.
We know that DST increases the time difference between the social clock and the body clock. More and more studies show that time differences between the social clock and the body clock challenge our health, are associated with decreased life expectancy, shorten sleep, cause mental and cognitive problems, and contribute to the many sleep disturbances.
I removed the references from this quote, but the authors support these claims by citing many research studies. Note that these problems do not arise only from the change back-and-forth between standard and daylight saving time in the spring and fall. Even permanent daylight saving time would cause chronic health problems.
My preference is permanent standard time. I live in Rochester Hills, Michigan (a suburb of Detroit), which is in the western part of the Eastern Time Zone, so I was surprised to read that
the further west people live within a time zone, the more health problems they may experience and the shorter they live on average.
Yikes!
Michigan is also in the northern part of the United States, where differences in the length of day and night throughout the year are exaggerated compared to our southern neighbors. This year in early January the sun rose in Detroit just after 8 am. Change to year-round daylight saving time and we would have sunrise at 9 am. For a morning person like me, that’s a lot of darkness before the sun comes up.
My personal preference, however, isn’t important. I’m sure there are others who feel just as strongly that year-round daylight saving time is better than year-round standard time. What impresses me is that Roenneberg et al. make a strong case favoring permanent standard time as being better for society overall. They conclude that
if we want to improve human health, we should not fight against our body clock, and therefore, we should abandon DST and return to Standard Time (which is when the sun clock time most closely matches the social clock time) throughout the year.
I agree!
And if Congress refuses to move to year-round standard time, I would rather keep things as they are now (changing with the seasons) rather than have year-round daylight saving time.
[The British medical physicist Leonard George] Grimmett was an expert in the use of radium to treat cancer and in the safe handling and measurement of radiation and radioactive materials in clinical situations. He had spent the best part of his career devising better, safer, and more efficient ways to treat cancer with radiation and he remained in England during [World War II]... Then in 1948 while working for UNESCO in Paris he received an offer he could not refuse the, “…post as physicist to a new ‘Cancer Research Institute and Atomic Center’ in The University of Texas”, one of the original universities in the ORINS [Oak Ridge Institute of Nuclear Studies] consortium. Thus was set in motion the events that would lead Grimmett to Houston, Texas and to be the first person to publish, in 1950, the design of a cobalt-60 radiation therapy unit for the treatment of cancer. For the next 25 years cobalt-60 units would be the mainstay of cancer radiation therapy, treating millions of patients worldwide. Grimmett, however, would not live to see the completion of his work. This is his story.
Grimmett is a fascinating guy. As a young boy he learned to play the piano and was quite good. “He had worked his way through college playing for the silent movies, but with the advent of the ‘talkies,’ he had lost his income. He went to work at Westminster Hospital.” At Westminster and other hospitals he helped develop cancer treatment machines using radium, and later he established the medical physics program at the renowned M. D. Anderson Cancer Center. But he had other talents. He was a pilot, a scriptwriter, a gemologist, and jeweler. He’s remembered today primarily for developing a cobalt-60 therapy machine. Almond writes
It is not known for sure who first had the idea of replacing the radium in teletherapy units with a more suitable and less-expensive artificial radioactive substance. Grimmett, however, had been thinking about it for some years before he went to Houston, and a case can be made that he was the first.
What motivated him to use cobalt? “What Grimmett was looking for was an artificial radioactive isotope with gamma ray energies of 1–5 MeV with as long a half-life as possible that could be made in large quantities at a reasonable price.” He considered using sodium-24 for therapy. After 24Na beta decays it emits two gamma rays with energies of 4.1 and 1.4 MeV (see Fig. 17.9 of Intermediate Physics for Medicine and Biology). However, the half-life of 24Na is only 15 hours.
The idea that cobalt-60 might be a suitable replacement for radium first occurred to Grimmett while he was reading Physical Review in an air-raid shelter during World War II… Later, after the war, he would have read the paper by J. S. Mitchell in the December 1946 issue of the British Journal of Radiology [82]. This is often cited as the paper that initiated the cobalt-60 era. Mitchell specifically mentions cobalt-60 as a replacement for radium beam therapy, and he gave the half-life as 5.3 years and the gamma ray energies as 1.3 and 1.1 MeV. He also reported that it could be produced in “the pile” (nuclear reactor).
Why did Almond title his book Cobalt Blues? Grimmett had trouble obtaining the needed cobalt-60. It is a by-product of nuclear reactors. He first tried the reactor at Oak Ridge, but ended up getting it from a reactor on Chalk River in Canada. Incidentally, the book cover of Cobalt Blues is a lovely cobalt blue.
Grimmett was not the only person trying to use cobalt-60 to treat
cancer. Almond briefly describes the other groups, including one in
Canada by Harold Johns, and tries to sort out the various priority
claims.
Unfortunately, Grimmett died unexpectedly and never saw his unit in use. His obituary in the Houston Chronicle begins
Doctor Grimmett, Cancer Expert, Dies Suddenly
Dr. Leonard G. Grimmett, 49, eminent physicist whose work in cancer research at M.D. Anderson Hospital, opened a whole new field of treatment of cancer, died of a heart attack at 1:10 a.m. Sunday at his home, 3238 Ewing.
I enjoyed Almond’s book. I learned much about the early years of the M. D. Anderson Cancer Center and about the issues that must be considered when building radiation therapy units. Readers of IPMB will find Cobalt Blues fascinating.
(c) show that vi(x) and dvi(x)/dx are continuous at x = 0, a/2 and a, and
(d) plot vi(x), dvi(x)/dx, and d2vi(x)/dx2 as functions of x, over the range −2a < x < 2a.
This representation of vi(x) has a shape like that of an action potential. Other functions also have a similar shape, such as a Gaussian. But our function is nice because it’s non-zero over only a finite region (−a < x < a) and it’s represented by a simple, low-order polynomial rather than a special function. An even simpler function for vi(x) would be triangular waveform, like that shown in Figure 7.4 of IPMB. However, that function has a discontinuousderivative and therefore its second derivative is infinite at discrete points (delta functions), making it tricky (but not too tricky) to deal with when calculating the extracellular potential (Eq. 7.21). Our function in Problem 14 ¼ has a discontinuous but finite second derivative.
The main disadvantage of the function in Problem 14 ¼ is that the depolarization phase of the “action potential” has the same shape as the repolarization phase. In a real nerve, the upstroke is usually briefer than the downstroke. The next new homework problem asks you to design a new function vi(x) that does not suffer from this limitation.
Section 7.4
Problem 14 ½. Design a piecewise continuous mathematical function for the intracellular potential along a nerve axon, vi(x), having the following properties.
(a) vi(x) is zero outside the region −a < x < 2a.
(b) vi(x) and its derivative dvi(x)/dx are continuous.
(c) vi(x) is maximum and equal to one at x = 0.
(d) vi(x) can be represented by a polynomial bi + cix + dix2, where i refers to four regions:
i = 1, −a < x < −a/2
i = 2, −a/2 < x < 0
i = 3, 0 < x < a
i = 4, a < x < 2a.
Finally, here’s another function that I’m particularly fond of.
Section 7.4
Problem 14 ¾. Consider a function that is zero everywhere except in the region −a < x < 2a, where it is
(a) Plot vi(x) versus x over the region −a < x < 2a,
(b) Show that vi(x) and its derivative are each continuous.
(c) Calculate the maximum value of vi(x).
Simple functions like those described in this post rarely capture the full behavior of biological phenomena. Instead, they are “toy models” that build insight. They are valuable tools when describing biological phenomena mathematically.
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
