The Number and Distribution of Capillaries in Muscles with Calculations of the Oxygen Pressure Head Necessary for Supplying the Tissue

Oxygen diffuses from capillaries into tissue where it is used for metabolism. Russ Hobbie and I discuss diffusion in Chapter 4 of Intermediate Physics for Medicine and Biology. Below is a new homework problem on this topic.

Section 4.11

Problem 37 ½. Consider a cylindrical capillary of radius a, containing blood having an oxygen concentration C_o (molecules/m³). The capillary is surrounded by a cylinder of tissue of radius b that has an oxygen concentration C(r), consumes oxygen at a rate per unit volume Q (molecules/m³s), and has a diffusion constant D (m²/s). At r = a, C = C_o and at r = b, dC/dr = 0. Within the tissue, C(r) obeys the steady-state diffusion equation

(a) Calculate C(r). Hint: guess a solution of the form C(r) = A + B r² + E ln(r), and determine values for the constants A, B, and E.

(b) Plot C(r) versus r assuming b = 10a and Qb²/(C_oD) = 1

(c) Determine the minimum value of C_o as a function of a, b, D, and Q, assuming the oxygen concentration is nowhere negative.

(d) Describe what assumptions underlie this model.

This problem plays an important role in the history of physiology. August Krogh used the model to infer that when Q increased during exercise, b must decrease (by additional vessels opening that were closed when the muscle was at rest) in order to supply the tissue with sufficient oxygen. For “his discovery of the capillary motor regulating mechanism,” he was awarded the Nobel Prize. Krogh’s model also represents an early contribution of mathematical modeling to medicine and biology. He presented his model in the paper:

Krogh, A. (1919) The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol., 52: 409–415.

He acknowledges mathematician K. Erlang for deriving the mathematical formula for C(r).

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

Isaac Asimov includes an entry for Krogh in Asimov’s Biographical Encyclopedia of Science and Technology (Second Revised Edition).

KROGH, Schack August Steenberg (krawg)

Danish physiologist

Born: Grena, Jutland, November 15, 1874

Died: Copenhagen, September 13, 1949

Krogh, the son of a brewer, was educated at the University of Copenhagen, where he intended to study medicine but shifted his interest to physiology. He obtained his master’s degree in 1899.

He was particularly involved in respiration, following the path of oxygen, nitrogen, and carbon dioxide in and out of the body. In 1908 he gained a professorial position at the University of Copenhagen and there his studies of respiration led him to suggest that the capillaries (the tiniest blood vessels) of the muscles were open during muscular work and partially closed during rest. He went on to demonstrate this and to show the importance of such capillary control to the economy of the body.

For this work, he was awarded the Nobel Prize in Physiology and Medicine in 1920. He went on thereafter to show that this capillary control was brought about by the action of both muscles and hormones.

After Denmark was occupied by Nazi Germany in 1940, Krogh was forced to go underground and then to escape to Sweden. He remained there till the end of the war, then returned to liberated Denmark.

A Brief History of Human Functional Brain Mapping

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe functional magnetic resonance imaging.

The term functional magnetic resonance imaging (fMRI) usually refers to a technique developed in the 1990s that allows one to study structure and function simultaneously. The basis for fMRI is inhomogeneities in the magnetic field caused by the differences in the magnetic properties of oxygenated and deoxygenated hemoglobin. No external contrast agent is required. Oxygenated hemoglobin is less paramagnetic than deoxyhemoglobin. If we make images before and after a change in the blood flow to a small region of tissue (perhaps caused by a change in its metabolic activity), the difference between the two images is due mainly to changes in the blood oxygenation. One usually sees an increase in blood flow to a region of the brain when that region is active. This BOLD contrast in the two images provides information about the metabolic state of the tissue, and therefore about the tissue function.

Brain Mapping: The Systems.

The amazing story of how, 25 years ago, two competing teams developed fMRI simultaneously is told by Marcus Raichle in his chapter “A Brief History of Human Functional Brain Mapping,” published in the book Brain Mapping: The Systems. Below I provide excerpts.

A race was on to produce the first human functional images with MRI even though the participants were unaware of the activities of each other! Who were the participants? [Seiji] Ogawa and his colleagues working with Kamil Ugurbil and friends at the University of Minnesota and a group at the Massachusetts General Hospital led by Ken Kwong...

Ugurbil turned to a new postdoctoral fellow in the laboratory, Ravi Menon, to help in the effort to obtain the first functional MRI BOLD images in humans...He was joined by Ogawa and [David] Tank from Bell Labs along with members of the Ugurbil lab and a pair of Grass Goggles for visual stimulation!….It was early summer of 1991 that believable results were finally obtained. This was obviously too late to submit an abstract to the upcoming Society of Magnetic Resonance Conference to be held in San Francisco in August. Members of the laboratory, nevertheless, left for the meeting with slides in their pockets hopeful that they would have a chance to show some of their new results.

Meanwhile, a very parallel but completely independent set of events was unfolding in Boston. Ken Kwong, a member of the group at the Massachusetts General Hospital, was anxious to develop a method for measuring blood flow with MRI….Kwong saw a poster by Bob Turner, another MR physicist working at the NIH, which was of related interest. Turner had been studying hypoxia/ischemia in cats produced by brief periods of ventilatory arrest….They choose a visual activation paradigm. A pair of well-known Grass Goggles resided in the lab to support the function activation work using contrast agents….

Buoyed by the results obtained with the goggles and BOLD imaging, the MGH group rushed to submit a “Works in Progress” abstract to the Society of Magnetic Resonance Conference… Much to the MGH group’s dismay and to this day unexplained, this particular abstract failed to reach those putting the program together…. Recognizing by this time the significance of their results, they persuaded Tom Brady to include their results in his plenary lecture. The group from Minneapolis had no such opportunity! Not only did the scientific world get its first glimpse of fMRI, but the two groups working on the concept also realized for the first time who the competition was!

By the early fall of 1991 both the Minneapolis and the Boston groups had publishable results. With great anticipation, papers were submitted to Nature (Minneapolis) and Science (Boston) and summarily rejected. The basic judgment of both journals was that they contained nothing new! It is fitting that the work of the two groups appeared together in the Proceedings of the National Academy of Science (Kwong et al., 1992; Ogawa et al., 1992). A new and very important chapter on functional brain imaging had begun.

Functional MRI has since been used for many studies of how the brain works. I consider it one of the best examples of physics applied to medicine in the last 25 years. Raichle not only tells the story of fMRI’s development well (but perhaps with too many exclamation points for my taste), but also reviews the long history of mapping brain function, dating back into the 19th century. The chapter is well worth a read. Enjoy!

You Can Hear About a Nickel’s Worth of Difference

In Chapter 13 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe a logarithmic scale for sound intensity: the decibel. In general, an increase in intensity is perceived as a greater loudness (although this relationship is surprisingly complex). Is there an analogous relationship between frequency and pitch? Yes! Loudness is determined using a logarithmic scale with a base of ten, whereas pitch is measured using a logarithmic scale of base two, because a doubling of the frequency corresponds to raising the pitch by one octave. Those familiar with music are accustomed to associating different pitches with different notes in a musical scale. Most instruments are tuned using the equal tempered scale, dividing an octave into twelve equal logarithmic steps. One step, called a semitone, corresponds to the difference in pitch between, say, F and F-sharp. The fractional change of one semitone is 2^1/12 = 1.0595, or an increase in frequency of roughly 6%. This frequency shift is so important that it is expressed by a special unit: one semitone is equal to 100 cents. The cent is to pitch as the decibel is to loudness. Doubling the intensity corresponds to an increase of 3 dB, whereas doubling the frequency corresponds to an increase of 1200 cents.

How finely can the human ear resolve pitch? In other words, if you play two pure tones one right after the other, by how much must their frequency differ for you to notice that they are indeed different? A typical listener can detect a change of about 5 cents, leading to the rule of thumb that you can hear about a nickel’s worth of difference. Try testing your own pitch perception online here. When I tried, I could always detect a difference of 50 cents (tones of 440 and 453 Hz, presented one after another; I assume this is a control used by the testing software, because the difference is obvious). I could never detect a difference of 4 or 6 cents (440 compared to 441 or 441.5 Hz). I had inconsistent results with 8 and 11 cents (440 compared to 442 or 443 Hz); sometimes I perceived slightly different tones, and sometimes I didn’t. Ten cents is a reasonable approximation to my just noticeable difference, which confirms what I have always suspected: I have poor but not pathological pitch perception. When playing the tuba in my high school band, the director often had us “tune up” relative to a standard note, typically played by the first clarinet. I always had trouble with this task; he had to tell me if I was sharp or flat. Poor pitch perception has its advantages: you don’t need to hire a piano tuner! Pity my poor wife—my only audience, other than my dog Suki—but my wrong notes are probably more bothersome than the out-of-tune piano, so spending for a piano tuner would not help.

Humans can hear pitches from roughly 20 to 20,000 Hz. Because 2¹⁰ is about 1000, humans can hear frequencies that range over roughly ten octaves, or 12,000 cents, and therefore you (but not I) can distinguish about 2400 different tones. A piano keyboard plays notes 28 to 4186 Hz, which is a little more than seven octaves, or roughly 8700 cents (87 semitones between the 88 keys). Sometimes changes in frequency are measured in millioctaves: 1 mO = 1.2 cents. Although the cent is not a metric unit, I still worry that the SI police, who insist that all things centi- are going out of fashion compared to all things milli-, will demand we start using the millioctave. I hope not.

If two tones are played at the same time, rather than one after another, you can perceive small differences in pitch using beats. Consider the triginometric identity

If frequencies A and B are similar, then their sum consists of a carrier frequency equal to the average of the two original frequencies modulated by the difference of the two frequencies. For instance, if you have a tone corresponding to concert A (440.00 Hz) and another tone out of tune with concert A by 3 cents (440.78 Hz), when played together the sound consists of a tone having frequency 440.39 Hz modulated by a sinusoid that gets louder and softer with a frequency of 0.78 Hz (or a period of 1.3 seconds). Your ear can’t tell that the pitch of the carrier tone is different from concert A, but it can detect the variation in loudness caused by the beats. You can hear beats generated online here.

If several tones have widely separated frequencies, your ear (or more properly, your brain) detects distinct notes played together; a chord. If two tones have nearly the same frequency (like in the example above), you hear a single tone with modulated amplitude; beats. For intermediate frequency differences (say, between a note an octave above concert A, 880 Hz, and a severely out-of-tune A at 897 Hz, a difference of 17 Hz or 33 cents) you hear neither a chord nor beats. Instead, your brain perceives dissonance. A London police whistle generates frequencies of 1904 and 2136 Hz, a difference of 232 Hz or more than 200 cents. It sounds annoying. Try it yourself.

The frequency ratio between a C and G (a perfect fifth) should be 3:2, which is 702 cents. However, in equal tempered tuning the shift between C and G is 700 cents. This difference of 2 cents is indistinguishable to all but the best ears. The major third is more of a problem. It should have a ratio of 5:4, or 386 cents. In the equal tempered scale, a third is 400 cents, off by 14 cents, which a good ear can hear. If all these tonal relations are different than what they would be in just intonation, then why do we use an equal tempered scale? It allows us to change keys without retuning the instrument. But we pay a price in that a major chord does not have precisely the desired 4:5:6 frequency ratio. Pythagoras must be turning over in his grave.

Both pitch perception and color vision arise from the physics of frequency detection. But the tones we hear and colors we see are determined by more than just physics. There is a lot of information processing by the brain, which leads to many fascinating and unexpected results including surprising pathologies. To completely appreciate how we perceive frequency, you also need to understand the brain.

Biomagnetism Therapy: Pseudoscientific Twaddle

Voodoo Science,
by Robert Park.

Ever since Robert Park—author of Voodoo Science and the weekly column What’s New—suffered a stroke in 2013, I have been searching for another debunker of pseudoscience. Finally, I’ve found her! Harriet Hall is a retired physician and former Air Force flight surgeon. Every week she battles nonsense in the blog sciencebasedmedicine.org. In her November 20 entry, she addressed “biomagnetic therapy.”

Biomagnetism Therapy: Pseudoscientific Twaddle

In a television interview, a practitioner of biomagnetic therapy claimed she had cured her own breast lump and the metastatic cancer of another person. I wonder how many viewers believed her. On the “official website” of biomagnetism therapy, http://biomagnetism.net/, they claim it is “the answer to ALL your health problems… an all-natural, non-invasive therapy proven to prevent, diagnose and treat countless diseases, chronic illnesses and degenerative health problems.”

Sound too good to be true? Of course it does! You are already skeptical. If you read further, you will become even more skeptical….”

She concludes

…It pains me to see misinformation such as this fed to gullible patients. Using biomagnetic therapy isn’t likely to harm patients physically, but it’s likely to harm their comprehension of science. It’s likely to waste their money, and it could delay getting treatments that do work. Perhaps the worst thing is that people who practice this therapy are deceiving themselves. They don’t understand science, and they mistake testimonials for evidence of efficacy. They don’t understand the need for controlled studies. They don’t understand placebo effects, suggestion, expectation, regression to the mean, the natural course of illness, and all the other things that can lead people to believe a bogus treatment works. It is particularly tragic that anyone trained as an MD could have such poor critical thinking skills and be misled by such egregious pseudoscience.

Russ Hobbie and I have an entire chapter about biomagnetism in Intermediate Physics for Medicine and Biology. We discuss the measurement of the very small magnetic fields produced by the brain (magnetoencephalography) and the use of rapidly changing magnetic fields to stimulate neurons (transcranial magnetic stimulation). We also devote a chapter to magnetic resonance imaging. These are important topics, but they often get mixed up with phony claims about “biomagnetic therapy.”

If you doubt this is a real problem, go to Google and search for “biomagnetism” (the title of Chapter 8 in IPMB). The first site you get starts “One of the most peculiar therapy systems that FAIM [Foundation for Alternative and Integrative Medicine] is investigating is one that uses ordinary magnets to heal. Although magnets have been used in therapies for a long time, this particular method uses pairs of magnets to neutralize disease-causing pathogens in the body...” The second site begins “Yes! It’s the answer to ALL your health problems…” The third describes “The Revolutionary Therapy based on the Biomagnetic Pairs discovered by Dr. Isaac Goiz Durán, MD in 1988...” The fourth is the “Official website for Biomagnetism classes in the USA with Dr. Isaac Goiz Durán...” Finally, the fifth site in the list is Wikipedia’s entry on biomagnetism (the measurement of weak magnetic fields produced by the body). The first four are twaddle; the fifth is reputable.

Women Aren’t Supposed to Fly,
by Harriet Hall.

If you want to learn more about Harriet Hall, read her autobiography Women Aren’t Supposed to Fly, in which she describes her experiences in medical school and the Air Force. From the Preface:

There’s an old curse “may you live in interesting times.” I lived in an era when society was starting to allow women to enter male-dominated fields, but didn’t yet entirely approve. Someone said, “Whatever women do they must do twice as well as men to be thought half as good. Luckily this is not difficult.” Actually, it was difficult. It was frequently frustrating, sometimes painful, often ridiculously funny, and always interesting. Come with me on a ramble through my education and career and let me tell you what it was like.

What Women Aren’t Supposed to Fly does not explain is how Hall ended up a lampooner of baloney and poppycock. She needs to write a second book, telling that story. I’m sure it would be equally fascinating and amusing.

I’d have preferred another physicist pick up Bob Park’s banner, but I’ll take what I can get. Harriet Hall, keep up the good work and let’s end this “biomagnetic therapy” rubbish.

Charles Bean, Biological Physicist

I’ve always been fascinated by physicists who move into biology, and I collect stories about scientists who have made this transition successfully. Today let me share one example: Charles Bean (1923–1996). Bean spent much of his career studying solid state physics, especially magnetism and superconductivity. He worked for more than 30 years at the General Electric Research and Development Center in Schenectady, New York. You can read about his research for GE in a biographical memoir published by the National Academy of Sciences. I want to focus on his work in biological physics.

Relatively early in his career, Bean expanded his scientific interests beyond magnetism and superconductivity. He also studied biophysics, and he encouraged colleagues to consider the field as well. In an invited talk to the American Physical Society on how to change from physics to biology, one of his recommendations was straightforward: “Start to eat lunch with biologists.” In 1960 Bean managed to convince the now-renowned biophysicist Carl Woese to join the General Electric Research and Development Center in Schenectady, where he stayed for three years before joining the University of Illinois in 1970. And Charlie took his own advice and ate with Carl every day.

Bean was elected the first Coolidge Fellow at the GE laboratory. The Coolidge Fellowship program was a way to recognize the company’s most valuable scientists, and its main advantage was that the recipient could go on sabbatical leave to any another institution with full pay. Charlie decided to go to Rockefeller University, where he stayed (not full time) from 1973 to 1978. There he was exposed to neurophysiology, and he eventually wrote a theory of stimulation of myelinated fibers that was published in the British Journal of Physiology (1974).

At about this time Bean had started to spend his summers in Woods Hole, MA, which he enjoyed enormously, both for its outdoor activities and its laboratories. When asked why, he said, “I like to be at a place where the library is open 24 hours a day.” Here he became interested in sea urchin sperm, and he invented a clever method to determine the average velocity and length they can swim. He did this by having a dilute concentration of sperm in a solution above a clean gold surface, and each time a sperm hit the gold it stuck.

Very soon Bean’s research papers began to be published in biophysics journals rather than physics journals. He first became interested in membranes, for example, and wrote a long treatise for the U.S. Department of the Interior on reverse osmosis (1969). Taking advantage of GE’s Nuclepore membranes, he developed, together with Ralph DeBlois, a virus counter—a variant of the famous Coulter counter (1970). Later he developed a seminal theory of neutral porous membranes (1972). On the basis of this paper he was offered a professorship, which tempted him, though eventually he turned it down.

… Bean also continued more serious research while at RPI [Rensselaer Polytechnic Institute]. For example, being a good friend of French biophysicist P. G. de Gennes, through him Bean became fascinated by electrophoresis—in particular the way a strand of DNA twists through a gel, like a snake through grass. He analyzed and modeled the process, and in 1987 wrote a paper on the subject with H. Hervet.

Bean spent much of his later time looking for the elusive magnetic bacteria, which were rediscovered by R. P. Blakemore at Woods Hole in 1975 (they had been seen in 1963 in Italy by Salvatore Bellini but thereafter largely forgotten). The reason why these bacteria are guided by a magnetic field is that they have small internal magnets corresponding to the chain-of-spheres model. Bean developed simple equipment that enabled him to seek such magnetotactic bacteria virtually everywhere. He thought he had found some in a pothole right outside RPI, but before verifying and publishing his findings Bean died of heart failure.

Bean appears often in Intermediate Physics for Medicine and Biology. In Section 5.9, Russ Hobbie and I describe a continuum model for volume and solute transport in a pore. We cite his Department of the Interior report and his model of neutral pores repeatedly.

Bean CP (1969) Characterization of cellulose acetate membranes andultrathin films for reverse osmosis. Research and Development Progress Report No. 465 to the U.S. Department of Interior, Office of Saline Water. Contract No. 14-01-001-1480. Washington, Superintendent of Documents, October 1969.

Bean CP (1972) The physics of porous membranes—neutral pores. In: Eisenman G (ed) Membranes, vol 1. Dekker, New York, pp 1–55.

The description of his work in this field is even more extensive in earlier editions of IPMB (see, for example, the third edition which Russ authored before I came along and ruined it). In the fifth edition, we added a homework problem about the Coulter counter, in which we cite another publication by Bean.

DeBlois RW, Bean CP (1970) Counting and sizing of submicron particlesby the resistive pulse technique. Rev Sci Inst 41:909–916.

Given my interest in neural stimulation, I was particularly curious about Bean’s theory of microstimulation of myelinated nerve axons. I had trouble finding his paper on this topic, until I realized that it’s an appendix of a paper by Abzug, Maeda, Peterson and Wilson (“Cervical branching of lumbar vestibulospinal axons”). Bean develops a theory of neural stimulation very similar to that of the activating function introduced in Homework Problem 38 of Chapter 7 in IPMB. Bean assumed a myelinated axon with discrete nodes of Ranvier rather than a continuous cable. This assumption makes little difference if the stimulating electrode is far from the nerve, but if it is closer to the nerve than one internodal space, the discrete model is more appropriate.

Figure 8.25 in IPMB shows a magnetotactic bacterium containing a chain of small particles of magnetite. Bean developed a theory for the magnetic properties of such a “chain-of-spheres” before they were known to occur in bacteria.

In summary, Charles Bean is a fine example of a physicist who moved from physics to biology, and was able to contribute a unique perspective on important biological problems. In my experience, such physicists can contribute to a wide variety of biological topics. Often their insights ignore much of biology’s complexity, but they are based on universal physical principles that may be unfamiliar to many biologists.

Now, I need to go find a biologist to have lunch with.