A Simple Mathematical Function Representing the Intracellular Action Potential

In Problem 14 of Chapter 7 in Intermediate Physics for Medicine and Biology, Russ Hobbie and I chose a strange-looking function to represent the intracellular potential along a nerve axon, v_i(x). It’s zero everywhere except in the range −a < x < a, where it’s

 

 

Why this function? Well, it has several nice properties, which I’ll leave for you to explore in this new homework problem.

Section 7.4

Problem 14 ¼. For the intracellular potential, v_i(x), given in Problem 14

(a) show that v_i(x) is an even function,

(b) evaluate v_i(x) at x = 0,

(c) show that v_i(x) and dv_i(x)/dx are continuous at x = 0, a/2 and a, and

(d) plot v_i(x), dv_i(x)/dx, and d²v_i(x)/dx² as functions of x, over the range −2a < x < 2a.

This representation of v_i(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 v_i(x) would be triangular waveform, like that shown in Figure 7.4 of IPMB. However, that function has a discontinuous derivative 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 v_i(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, v_i(x), having the following properties. 

(a) v_i(x) is zero outside the region −a < x < 2a. 

(b) v_i(x) and its derivative dv_i(x)/dx are continuous. 

(c) v_i(x) is maximum and equal to one at x = 0. 

(d) v_i(x) can be represented by a polynomial b_i + c_i x + d_i x², 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 v_i(x) versus x over the region −a < x < 2a,

(b) Show that v_i(x) and its derivative are each continuous. 

(c) Calculate the maximum value of v_i(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.

Abraham Liboff (1927–2023)

Abe Liboff, in his
office at Oakland University

Oakland University physicist Abe Liboff died recently. A notice from President Ora Hirsch Pescovitz, published on the OU website, stated:

It is with deep sadness that I inform you of the death of Professor Emeritus Abraham Liboff who passed away on January 9, 2023. Dr. Liboff joined the Oakland University community in the Department of Physics on August 15, 1972, where he served until his retirement in August 2000.

During his tenure here at OU, Dr. Liboff was Chair of the Department of Physics. He is credited with 111 research publications, more than two dozen patents and nearly 3,400 scholarly citations during his career.

I arrived at OU in 1998, so his time at OU and mine overlapped by a couple years. I remember having a delightful breakfast with him during my job interview. He was one of the founders of OU’s medical physics PhD program that I directed for 15 years. His office was just a few doors down the hall from mine and he helped me get started at Oakland. I’ll miss him.

Although I loved the man, I didn’t love Abe’s cyclotron resonance theory of how magnetic fields interact with biological tissue. It’s difficult to reconcile admiration for a scientist with rejection of his scientific contributions. Rather than trying to explain Abe’s theory, I’ll quote the abstract from his article “Geomagnetic Cyclotron Resonance in Living Cells,” published in the Journal of Biological Physics (Volume 13, Pages 99–102, 1985).

Although considerable experimental evidence now exists to indicate that low-frequency magnetic fields influence living cells, the mode of coupling remains a mystery. We propose a radical new model for electromagnetic interactions with cells, one resulting from a cyclotron resonance mechanism attached to ions moving through transmembrane channels. It is shown that the cyclotron resonance condition on such ions readily leads to a predicted ELF-coupling at geomagnetic levels. This model quantitatively explains the results reported by Blackman et al. (1984), identifying the focus of magnetic interaction in these experiments as K⁺ charge carriers. The cyclotron resonance concept is consistent with recent indications showing that many membrane channels have helical configurations. This model is quite testable, can probably be applied to other circulating charge components within the cell and, most important, leads to the feasibility of direct resonant electromagnetic energy transfer to selected compartments of the cell.

In my book Are Electromagnetic Fields Making Me Ill? I didn’t have the heart to attack Abe in print. When discussing cyclotron resonance effects, I cited the work of Carl Blackman instead, who proposed a similar theory. What’s the problem with this idea? If you calculate the cyclotron frequency of a calcium ion in the earth’s magnetic field, you get about 23 Hz (see Eq. 8.5 in Intermediate Physics for Medicine and Biology). However, the thermal speed of a calcium ion at body temperature is about 440 m/s (Eq. 4.12 in IPMB). At that speed, the radius of the cyclotron orbit would be 3 meters (roughly ten feet)! The mean free path of a ion in water, however, is about an angstrom, which means the ion will suffer more than a billion collisions in one orbit; these interactions should swamp any cyclotron motion. Moreover, ion channels have a size of about 100 angstroms. In order to have a orbital radius similar to the size of a ion channel, the calcium ion would need to be moving extremely fast, which means it would have a kinetic energy vastly larger than the thermal energy. The theory just doesn’t work.

Since Liboff isn’t around to defend himself, I’ll let Louis Slesin—the editor and publisher of Microwave News—tell Abe’s side of the story. Read Slesin’s Reminiscence on the Occasion of Abe Liboff’s 90th Birthday. Although I don’t agree with Slesin on much, we both concur that Abe was a “wonderful and generous man.” If you want to hear about cyclotron resonance straight from the horse’s mouth, you can hear Abe talk about his career and work in a series of videos posted on the Seqex YouTube channel. (Seqex is a company that sells products based on Abe’s theories.) Below I link to the most interesting of these videos, in which Abe tells how he conceived of his cyclotron resonance idea.

Abe Liboff discussing the cyclotron resonance theory.
https://www.youtube.com/watch?v=YL-wqJ-PMAQ&list=PLCO-VktC6wofkMeEeZknT9Y4WhMnP76Ee&index=6

Water, Cavendish, and Lavoisier

To understand biological physics, you must know the properties of water. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss water’s density, compressibility, viscosity, heat capacity, surface tension, thermal conductivity, dielectric constant, and index of refraction. It’s behavior is critical for osmosis, diffusion, absorption of x-rays, propagation of ultrasonic waves, and magnetic resonance imaging.

In the very first section of IPMB, about distances and sizes, we say

At the 1-nm scale and below, we reach the world of small molecules and individual atoms. Water is the most common molecule in our body. It consists of two atoms of hydrogen and one of oxygen. The distance between adjacent atoms in water is about 0.1 nm.

Every schoolchild learns that water is H₂O. But how do we know that water is made from hydrogen and oxygen? In other words, how did we first learn that water is not an element itself, but is a compound of two elements?

A Short History of Chemistry,
by Isaac Asimov.

I’ll let Isaac Asimov tell the story. In A Short Histor of Chemistry, he writes

In 1783 [English scientist Henry] Cavendish was … working with his inflammable gas… He burned some of it and studied the consequences. He found that the vapors produced by the burning condensed to form a liquid that, on investigation, proved to be nothing more or less than water.

This was a crucially important experiment. In the first place, it was another hard blow at the Greek theory of the elements [air, water, earth, fire], for it showed that water was not a simple substance but was the sole product of the combination of two gases.

[French chemist Antoine] Lavoisier, hearing of the experiment, named Cavendish’s gas, hydrogen (“water-producer”) and pointed out that hydrogen burned by combining with oxygen and that therefore water was a hydrogen-oxygen combination.

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

So now you know. But these two brilliant scientists had tragic fates. You can find the stories in Asimov’s Biographical Encyclopedia of Science and Technology.

Cavendish (1731-1810):

[Cavendish] was excessively shy and absent-minded. He almost never spoke and when he did it was with a sort of stammer… He build a separate entrance to his house so he could come and leave alone… he even literally insisted on dying alone.

The eccentric had one and only one love, and that was scientific research. He spent almost sixty years in exclusive preoccupation with it. It was a pure love, too, for he did not care whether his findings were published, whether he got credit, or anything beyond the fact that he was sating his own curiosity. He wrote no books and published only twenty articles altogether. As a result, much of what he did remained unknown until years after his death…

Lavoisier (1743-1794):

In the same year that [Lavoisier’s] textbook [Elementary Treatise on Chemistry] appeared the French Revolution broke out. By 1792 the radical antimonarchists were in control…. Lavoisier… was guillotined on May 8, 1794, and buried in an unmarked grave. Two months later the radicals were overthrown. His was the most deplorable single casualty of the revolution.

50 Prominent Medical Physicists

Fifty Outstanding Medical Physicists.

In 2014, to mark its 50th anniversary, the International Organization for Medical Physics published a list of 50 medical physicists who have made an outstanding contribution to the advancement of medical physics over the last 50 years. I thought it would be interesting to see how many Russ Hobbie and I mention in Intermediate Physics for Medicine and Biology.

Hardly any. Only 11 of these 50 prominent medical physicists appear in IPMB.

1. Peter R. Almond, of the M. D. Anderson Cancer Center, helped develop cancer treatments using photons, electrons, and neutrons. IPMB cites his paper “Review of Electron Beam Therapy Physics” (Physics in Medicine & Biology, Volume 51, Pages R455–R489, 2006) coauthored with Kenneth Hogstrom. 
2. F. Herbert Attix worked at the Naval Research Laboratory and then the University of Wisconsin. IPMB cites Attix a dozen times, primarily for his landmark textbook Introduction to Radiological Physics and Radiation Dosimetry. 
3. John Cameron was the father of the medical physics PhD program at the University of Wisconsin. IPMB cites his book Physics of the Body. 
4. Aaron Fenster, of the University of Western Ontario, co-founded a microCT company eventually sold to GE Healthcare. Russ and I cite his paper with Paul Carson about the “Evolution of Ultrasound Physics and the Role of Medical Physicists in the AAPM and its Journal in that Evolution” (Medical Physics, Volume 36, Pages 411–428, 2008). 
5. William Hendee, of the Medical College of Wisconsin, coauthored over thirty books. IPMB cites one of them, written with E. Russell Ritenour: Medical Imaging Physics. 
6. Godfrey Hounsfield won a Nobel Prize for developing X-ray computed tomography, a central topic in Chapters 12 and 16 of IPMB. 
7. Willi Kalender, of the University of Erlangen-Nuremberg, introduced volumetric spiral computer tomography. IPMB cites his book Computed Tomography: Fundamentals, System Technology, Image Quality and Applications. 
8. Paul Lauterbur won a Nobel Prize for the invention of magnetic resonance imaging, the main topic of Chapter 18 in IPMB. 
9. Peter Mansfield shared the Nobel Prize with Lauterbur, primarily for his contributions to MRI, including echo-planar imaging. 
10. Colon Orton, of Wayne State University, studied radiotherapy and was active in the American Association of Physicists in Medicine. In IPMB, Russ and I cite three of his “Point/Counterpoint” articles in Medical Physics, for which he served as moderator. 
11. Peter Wells, of Cardiff University, pioneered techniques using ultrasound imaging, the topic of Chapter 13 in IPMB. Russ and I cite his paper “Ultrasound Imaging” (Physics in Medicine and Biology, Volume 51, Pages R83–R98, 2006).

Another eight don’t appear in IPMB but have been mentioned in this blog: Anders Brahme, John Cunningham, Charles Mistretta, Ervin Podgorsak, Madan Rehani, Jean-Claude Rosenwald, Jacob Van Dyk, and Steve Webb. I’m not familiar with the other 31. I guess I don’t know as much as I thought I did. 😮 You can find the entire list here. 

 

Enjoy! 

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As I was posting this article on my blog, it occurred to me that the list of 50 medical physicists came out about ten years ago, and that I ought to update it to 60 outstanding medical physicists in the last 60 years. Here are my additional ten. I tried to honor the spirit of the list by restricting myself to those who worked in the era from 1963 to 2023, but I couldn’t resist going back just a little further to select a few who worked in the 1950s.

1. Paul Callaghan developed microscopic MRI 
2. Hermann Carr proposed using gradients to image with magnetic resonance
   
3. Allan Cormack won the Nobel Prize for his work on computed tomography
   
4. Raymond Damadian performed early work on developing MRI
   
5. Erwin Hahn discovered the nuclear magnetic resonance spin echo
6. Eric Hall is the author of Radiobiology for the Radiologist
7. John Hubbell compiled data on photon cross sections and attenuation coefficients
8. Harold Johns is a coauthor of The Physics of Radiology
9. Faiz Khan is the author of The Physics of Radiation Therapy
   
10. William Oldendorf was a pioneer in computed tomography
    

Felix Savart, Biological Physicist

Bust of Félix Savart
in the Institut de France.
From Wikipedia.

 

 

I’m fascinated by scientists who make the transition from medicine to physics, which is the opposite of my own transition from physics to medicine. One example is Félix Savart. In this blog post, I provide several excerpts from a 1959 article by Victor McKusick and H. Kenneth Wiskind, titled Félix Savart (1791–1841), Physician-Physicist: Early Studies Pertinent to the Understanding of Murmurs (Journal of the History of Medicine and Allied Sciences, Volume 14, Pages 411–423).

Savart was born in Meziere, France on June 30, 1791. His family had a long history of excelling in engineering, but Savart chose a different path.

Savart decided on a medical career and about 1808 entered the hospital in Metz. From 1810 to 1814 he served as a regimental surgeon in Napoleon’s armies… After discharge from the army, he completed his medical training in Strasbourg, where he received his doctor’s degree in October 1816. The title of his doctorate thesis was "Du cirsocele." The mundane topic of varicocele [enlarged veins in the scrotum] must have had little intrinsic appeal for him, and it is perhaps slight wonder he did not stay in medicine.

I can understand how that topic might drive a person away from the medical profession. For whatever reason, Savart spent little time practicing medicine. Instead, he was interested in physics, and particularly in sound.

In 1817 Savart returned to Metz with the intention of establishing a medical practice… He spent his time “more in fitting out a laboratory and building instruments than in seeing sick people and perusing Hippocrates…” It was during this period that he… began to devote himself specifically to the study of acoustics, a subject which engaged his attention almost exclusively for the remainder of his life.

McKusick and Wiskind compare Savart to three other physicians who made the transition to physics: Hermann von Helmholtz, Thomas Young, and Jean Leonard Marie Poiseuille. When Savart was 28, he made a life-changing trip to Paris.

In 1819 Savart went to Paris… to consult Jean-Baptiste Biot (1774–1862) in connection with his study of the acoustics of musical instruments. This was undoubtedly a turning point in Savart’s career. Biot encouraged and aided Savart in many ways and took him into collaboration in a study of electricity.

Savart’s name appears in Intermediate Physics for Medicine and Biology only when paired with Biot for the Biot-Savart Law. Russ Hobbie and I write

8.2.3 The Biot-Savart Law

In situations where the symmetry of the problem does not allow the [magnetic] field to be calculated from Ampere’s law, it is possible to find the field due to a steady current in a closed circuit using the Biot-Savart law.

Ironically, Savart is remembered among physicists for this one investigation into magnetism rather than a lifetime studying acoustics. 

Savart was an excellent experimentalist and instrument builder. He made careful measurements of the frequencies produced by a trapezoid violin, which a French commission found to be as good as the violins of Stradivarius. McKusick and Wiskind describe one of his more significant inventions: the Savart wheel.

About 1830 Savart invented a toothed wheel for determining the number of vibrations in a given musical tone. He attached tongues of pasteboard to the hoop of the wheel and arranged for these to strike a projecting object as the wheel was turned… [With this invention] Savart [determined] the frequency limits of audibility of sounds for the human ear [see Section 13.4 in IPMB]. He set the low and high values at 8 and 24,000 cycles per second, respectively... The values he determined are of the same order of magnitude as the 16 to 16,000 cycles per second one usually hears quoted now.

Savart also has a unit named for him.

The savart is a unit related to the perceptible change in frequency; 300 savarts are approximately equal to one octave. However, this unit has not enjoyed general acceptance and usage.

Another unit for frequency interval, discussed previously in this blog, is the cent. A savart is about 4 cents.

Savart became of member of the French Academie des Sciences in 1827, a position he held “until his untimely death on 16 March 1841 at the age of fifty years.”

Félix Savart is a biological physicist in the mold of Helmholtz, Young, and Poiseuille. He’s just the sort of interdisciplinary scientist that Russ and I had in mind when writing Intermediate Physics for Medicine and Biology.

Bart Hopkin describes the Savart wheel.
https://www.youtube.com/watch?v=yhen0XGyheY

A Trapezoid violin, designed by Félix Savart.
https://www.youtube.com/watch?v=Q3npNDKkqsc

 

Jennifer Cash describes the Biot-Savart law.
https://www.youtube.com/watch?v=1BoIH6Quhiw