(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.
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.
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?
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.
[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.
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.
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.
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.
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.
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.”
Firstenberg covers a wide range of issues in The Invisible Rainbow and let me begin by admitting that I’m not an expert in all of these subjects. For instance, I don’t know much about infectious diseases, such as influenza, and I’m not particularly knowledgeable about viruses in general. However, the Centers of Disease Control and Prevention gathers input from authorities on these topics and here is what it says about the causes of the flu.
“Most experts believe that flu viruses spread mainly by tiny droplets made when people with flu cough, sneeze, or talk. These droplets can land in the mouths or noses of people who are nearby. Less often, a person might get flu by touching a surface or object that has flu virus on it and then touching their own mouth, nose or possibly their eyes.”
Firstenberg, on the other hand, claims that the flu is an electrical disease not caused by a virus spread from person to person. He writes
In 1889, power line harmonic radiation began. From that year forward the earth’s magnetic field bore the imprint of power line frequencies and their harmonics. In that year, exactly, the natural magnetic activity of the earth began to be suppressed. This has affected all life on earth. The power line age was ushered in by the 1889 pandemic of influenza.
In 1918, the radio era began. It began with the building of hundreds of powerful radio stations at [low] and [very low] frequencies, the frequencies guaranteed to most alter the magnetosphere. The radio era was ushered in by the Spanish influenza pandemic of 1918.
In 1957, the radar era began. It began with the building of hundreds of powerful early warning radar stations that littered the high latitudes of the northern hemisphere, hurling millions of watts of microwave energy skyward. Low-frequency components of these waves rode on magnetic field lines to the southern hemisphere, polluting it as well. The radar era was ushered in by the Asian flu pandemic of 1957.
In 1968, the satellite era began. It began with the launch of dozens of satellites whose broadcast power was relatively weak. But since they were already in the magnetosphere, they had as big an effect on it as the small amount of radiation that managed to enter it from sources on the ground. The satellite era was ushered in by the Hong Kong flu pandemic of 1968.
No mechanism is offered to explain how electromagnetic fields might cause a flu pandemic. No distinction is made between power line frequency (60 Hz) and radio frequency (MHz) radiation, although their physical effects are distinct. No estimation of “dose” (the distribution and magnitude of electric and magnetic field exposure) is provided. No randomized, controlled, double-blind studies are cited. He merely lists anecdotal evidence and coincidences.
Perhaps we could just ignore such dubious claims, except that The Invisible Rainbow is often quoted as evidence supporting the assertion that the Covid pandemic is somehow related to 5G cell phone radiation. Why would anyone get a Covid vaccine if they erroneously believe that the disease is caused by electromagnetic radiation? Such misinformation is dangerous to us all.
Firstenberg describes old studies without critical analysis. For instance, on page 73 he writes
In 1923, Vernon Blackman, an agricultural researcher at Imperial College in England, found in field experiments that electric currents averaging less than one milliampere (one thousandth of an ampere) per acre increased the yields of several types of crops by twenty percent. The current passing through each plant, he calculated, was only about 100 picoamperes.
One hundred picoamperes is 10−10 amperes. We aren’t told what the crops were, but let’s assume they consist of a thin stalk that I’ll estimate has a cross-sectional area of one square centimeter (10−4 m2). That means the current density would be 10−6 A/m2. Furthermore, let’s assume an electrical conductivity on the order of saline, 1 S/m. The resulting electric field is 10−6 V/m, or one microvolt per meter. This is far less than the electric field that always surrounds us and is caused by thermal fluctuations. The proposition that one milliamp per acre has such an effect defies credulity.
It was the Schwann cells, Becker concluded—the myelin-containing glial cells—and not the neurons they surrounded, that carried the currents that determined growth and healing. And in a much earlier study Becker had already shown that the DC currents that flow along salamander legs, and presumably along the limbs and bodies of all higher animals, are of semiconducting type.
Firstenberg believes cell phones cause many health hazards. On page 176, he writes
[Allan Frey] discovered the blood-brain barrier effect, an alarming damage to the protective shield that keeps bacteria, viruses, and toxic chemicals out of the brain—damage that occurs at levels of radiation that are much lower than what is emitted by cell phones today.
I could go on. Firstenberg believes electromagnetic fields are responsible for diabetes, heart disease, and cancer. His views on the mechanism of hearing are at odds with what most researchers believe. He thinks the “qi” that supposedly underlies acupuncture is electric in nature (similar to Becker’s view).
I do have some sympathy for Firstenberg. He’s been plagued by a variety of symptoms that he associates with electromagnetic hypersensitivity. I have no doubt his suffering is real. Yet, the evidence from controlled, double-blind experiments does not support his claim that electromagnetic radiation causes his illness. Rubin et al. reviewed many experiments and concluded that “at present, there is no reliable evidence to suggest that people with [idiopathic environmental intolerance attributed to electromagnetic fields] experience
unusual physiological reactions as a result of exposure to [electromagnetic fields]. This supports suggestions that
[electromagnetic fields are] not the main cause of their ill health” (Bioelectromagnetics, Volume 32, Pages 593–609, 2011). The World Health Organizationconcludes
EHS [electromagnetic hypersensitivity] is characterized by a variety of non-specific symptoms that differ from individual to individual. The symptoms are certainly real and can vary widely in their severity. Whatever its cause, EHS can be a disabling problem for the affected individual. EHS has no clear diagnostic criteria and there is no scientific basis to link EHS symptoms to EMF [electromagnetic field] exposure. Further, EHS is not a medical diagnosis, nor is it clear that it represents a single medical problem.
I put Arthur Firstenberg in the same category as Martin Pall, Robert Becker, Paul Brodeur, and Devra Davis: well-meaning scientific mavericks whose hypotheses have not been confirmed. The Invisible Rainbow is an interesting read, but beware: as science it is flawed.
Listen to Arthur Firstenberg, author of The Invisible Rainbow, answer questions about the hidden dangers of wireless and cellular phone radiation (I post this video so you can hear his side of the story, not because I agree with him).
Robert Resnick is professor emeritus at Rensselaer and the former Edward P. Hamilton Distinguished Professor of Science Education, 1974–93. Together with his co-author David Halliday, he revolutionized physics education with their now famous textbook on general physics, still one of the most highly regarded texts in the field today.
He is author or co-author of seven physics textbooks, which appear in 15 editions and more than 47 languages.
Resnick introduced Rensselaer’s interdisciplinary science curriculum in 1973 and was its chair for 15 years. He was awarded the American Association of Physics Teachers’ highest honor, the Oersted Medal, in 1975, and served as its president, 1986–90. A Distinguished Service Citation issued in 1967 by the association said, “Few physicists have had greater or more direct influence on undergraduate physics students than has Robert Resnick.”
Rensselaer named its Robert Resnick Center for Physics Education in his honor.
Today is Edith Anne Stoney’s birthday; she was born on January 6, 1869. In an article that appeared in the December, 2013 issue of Scope (the quarterly magazine of the Institute for Physics and Engineering in Medicine), Francis Duck describes Stoney as “the first woman medical physicist.” This week’s blog post includes excerpts from Duck’s fascinating article.
Stoney began her education in math and physics, then later switched to medicine.
As a young woman, Edith demonstrated
considerable mathematical talent, gaining a scholarship
at Newnham College, Cambridge, where she achieved
a First in the Part I Tripos examination in 1893.
Extraordinarily, she was never awarded her Cambridge
degree: women were excluded from graduation, a
situation that would not change for another 50 years.
She was later awarded [bachelor’s and master’s] degrees from Trinity College Dublin, after they accepted women in 1904.
Career possibilities for university women were limited.
She carried out some difficult calculations on gas turbines and searchlight design for Sir Charles Parsons,
and then took a mathematics teaching post at
Cheltenham Ladies’ College.
The 1876 Medical Act had made it illegal for academic
institutions to prevent access to medical education on
the basis of gender. Anticipating this change in the law,
the London School of Medicine for Women was
established in 1874 as the first medical school for
women in Britain. It soon became part of the University of London, with clinical teaching at the Royal Free Hospital. Edith’s sister Florence studied there,
obtaining her [medical degree] in 1898. By this time, changed
regulations had embedded physics firmly into medical
training, and Edith gained an appointment as a physics
lecturer there in 1899.
She became interested in medical imaging through her sister, the first female radiologist in the United Kingdom.
In 1901, the Royal Free Hospital appointed Florence
into a new part-time position of medical electrician. The
two sisters set about selecting, purchasing and
installing x-ray equipment and, the following April, a
new x-ray service was opened in the electrical department.
Edith and Florence with their father George Johnstone Stoney.
During the next few years Edith actively supported
the women’s suffrage movement, though opposed the
direct violent action with which it was later associated.
The years from 1910–1915 did not go smoothly for her.
After her father’s death in 1911 she no longer had his
guidance to call on. As student numbers increased so
did her staff, but they often did not stay long,
finding her difficult to work with. Finally, in March
1915, she left [her teaching position at the University of London].
Edith was now free from other commitments and
could make her own contribution to the war. She
contacted the Scottish Women’s Hospitals (SWH), an
organisation formed in 1914 to give medical support in
the field of battle, financed by the women’s suffrage
movement. In May she set off to Europe, and would be
away for most of the next 4 years… She established
stereoscopy to localise bullets and shrapnel and
introduced the use of x-rays in the diagnosis of gas gangrene… [The war resulted in] traumatically
injured soldiers and difficult working conditions. It
could have crushed a weaker character…
It was hard physical work for the women
to pack up the whole tented hospital, weighing three or
four hundred tons.
In March 1918, and for the third time, she had to
supervise a camp closure and retreat, when Villers-Cotterets was overrun by the advancing front. During
the final months of the war the fighting intensified and
there was a huge increase in workload. In the month of
June 1918 alone the x-ray workload peaked at over 1,300,
partly resulting from an increased use of fluoroscopy... However, [fluoroscopy] also resulted in an increased
incidence of radiation burns to Edith’s staff, two of
whom had to take sick leave to recover.
After the war ended, her work supporting the troops was honored by government awards, but not with an appropriate job.
She retired in 1925, but remained active supporting women in science.
After leaving King’s she retired to Bournemouth
where she lived with Florence who was by then
terminally ill with spinal cancer. She supported the
British Federation [of] University Women (BFUW) for
which she had acted as the first treasurer before the war.
She travelled widely, first with her ailing sister, and then
alone after Florence died in 1932.
Stoney passed away just as Europe was hurtling toward another world war.
Edith Stoney died, aged 69 years, on 25th June 1938.
Obituaries were printed in Nature, The Lancet and The
Times…. She was not noted as a
creative scientist: this was not her forte. She was a tough
and single-minded woman with high academic ability.
Her organisational skills established physics laboratories
and courses in two institutes of higher education. She
showed considerable bravery and resourcefulness in the
face of extreme danger, and imagination in contributing
to clinical care under the most difficult conditions of war.
She was a strong advocate of education for women... At a
time when medical physics was still struggling to become
an identified profession, Edith Stoney stands out as one
of its most able pioneers.
Anyone searching for a female role model in medical physics need look no further. What an amazing life.
Since a changing magnetic field generates an induced electric
field, it is possible to stimulate nerve or muscle cells
without using electrodes. The advantage is that for a given
induced current deep within the brain, the currents in the
scalp that are induced by the magnetic field are far less than
the currents that would be required for electrical stimulation.
Therefore transcranial magnetic stimulation (TMS) is
relatively painless.
The method was invented in 1985 and when I arrived at NIH in 1988 the field was new and ripe for analysis. I spent the next seven years calculating electric fields in the brain and determining how the electric field couples to a nerve.
This review describes the development of transcranial magnetic stimulation in 1985 and the research related to this technique over the following 10 years. It not only focuses on work done at the National Institutes of Health but provides a survey of other related research as well. Key topics are the calculation of the electric field produced during magnetic stimulation, the interaction of this electric field with a long nerve axon, coil design, the time course of the magnetic stimulation pulse, and the safety of magnetic stimulation.
I like magnetic stimulation because it's a classic example of how a fundamental concept from physics can have a major impact in biology and medicine. If you combine this review of transcranial magnetic stimulation together with my earlier review of the bidomain model of cardiac tissue, you get a pretty good summary of my most important research.
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.
