Comparative Anatomy is Largely the Story of the Struggle to Increase Surface in Proportion to Volume

On Being the Right Size,
by J. B. S. Haldane.

J. B. S. Haldane’s essay “On Being the Right Size” is a classic. In the first chapter of Intermediate Physics for Medicine and Biology, Russ Hobbie and I quote it.

You can drop a mouse down a thousand-yard mine shaft; and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.

Another line from the essay is nearly as famous.

Comparative anatomy is largely the story of the struggle to increase surface in proportion to volume.

We describe the interplay between surface and volume in Chapter 2 of IPMB

Consider the relation of daily food consumption to body mass. This will introduce us to simple scaling arguments. As a first model, we might suppose that each kilogram of tissue has the same metabolic requirement, so that food consumption should be proportional to body mass [or volume]. However, there is a problem with this argument. Most of the food that we consume is converted to heat. The various mechanisms to lose heat—radiation, convection, and perspiration—are all roughly proportional to the surface area of the body rather than its mass.

If ridding our bodies of excess heat is an important issue, then we need to increase surface area without increasing volume. A similar issue arises when getting oxygen to our cells. Our circulatory and respiratory systems are elaborate strategies to increase the area over which oxygen diffuses. This is a key concept where physics and physiology overlap.

You can read Haldane's essay in its entirety here. Below I quote part of it. Enjoy!

Animals of all kinds find difficulties in size for the following reason. A typical small animal, say a microscopic worm or rotifer, has a smooth skin through which all the oxygen it requires can soak in, a straight gut with sufficient surface to absorb its food, and a single kidney. Increase its dimensions tenfold in every direction, and its weight is increased a thousand times, so that if it is to use its muscles as efficiently as its miniature counterpart, it will need a thousand times as much food and oxygen per day and will excrete a thousand times as much of waste products.

Now if its shape is unaltered its surface will be increased only a hundredfold, and ten times as much oxygen must enter per minute through each square millimetre of skin, ten times as much food through each square millimetre of intestine. When a limit is reached to their absorptive powers their surface has to be increased by some special device. For example, a part of the skin may be drawn out into tufts to make gills or pushed in to make lungs, thus increasing the oxygen-absorbing surface in proportion to the animal’s bulk. A man, for example, has a hundred square yards of lung. Similarly, the gut, instead of being smooth and straight, becomes coiled and develops a velvety surface, and other organs increase in complication. The higher animals are not larger than the lower because they are more complicated. They are more complicated because they are larger. Just the same is true of plants. The simplest plants, such as the green algae growing in stagnant water or on the bark of trees, are mere round cells. The higher plants increase their surface by putting out leaves and roots. Comparative anatomy is largely the story of the struggle to increase surface in proportion to volume. Some of the methods of increasing the surface are useful up to a point, but not capable of a very wide adaptation. For example, while vertebrates carry the oxygen from the gills or lungs all over the body in the blood, insects take air directly to every part of their body by tiny blind tubes called tracheae which open to the surface at many different points. Now, although by their breathing movements they can renew the air in the outer part of the tracheal system, the oxygen has to penetrate the finer branches by means of diffusion. Gases can diffuse easily through very small distances, not many times larger than the average length traveled by a gas molecule between collisions with other molecules. But when such vast journeys—from the point of view of a molecule—as a quarter of an inch have to be made, the process becomes slow. So the portions of an insect’s body more than a quarter of an inch from the air would always be short of oxygen. In consequence hardly any insects are much more than half an inch thick. Land crabs are built on the same general plan as insects, but are much clumsier. Yet like ourselves they carry oxygen around in their blood, and are therefore able to grow far larger than any insects. If the insects had hit on a plan for driving air through their tissues instead of letting it soak in, they might well have become as large as lobsters, though other considerations would have prevented them from becoming as large as man.

Spillover: Animal Infections and the Next Human Pandemic

Spillover, by David Quammen

I recently read Spillover: Animal Infections and the Next Human Pandemic, by David Quammen. This book was written eight years ago, but it helped me understand what’s happening today with the coronavirus. Quammen writes:

A person might construe this list [Ebola, HIV, bird flu, West Nile virus, SARS, and now Covid-19] as a sequence of dire but unrelated events—independent misfortunes that have happened to us, to humans, for one unfathomable reason and another. Seen that way, Machupo and the HIVs and SARS and the others are “acts of God” in the figurative (or literal) sense, grievous mishaps of a kind with earthquakes and volcanic eruptions and meteor impacts, which can be lamented and ameliorated but not avoided. That’s a passive, almost stoical way of viewing them. It’s also the wrong way.

Make no mistake, they are connected, these disease outbreaks coming one after another. And they are not simply happening to us; they represent the unintended results of the things we are doing. They reflect the convergence of two forms of crisis on our planet. The first crisis is ecological, the second medical. As the two intersect, their joint consequences appear as a pattern of weird and terrible new diseases, emerging from unexpected sources and raising deep concern, deep foreboding, among the scientists who study them. How do such diseases leap from nonhuman animals into people, and why do they seem to be leaping more frequently in recent years? To put the matter in its starkest form: Human-caused ecological pressures and disruptions are bringing animal pathogens ever more into contact with human populations, while human technology and behavior are spreading those pathogens ever more widely and quickly.

Spillover doesn’t contain much physics, but it does allude to the math describing epidemics, making it relevant to Intermediate Physics for Medicine and Biology. Chapter 3 of Spillover discusses the 1927 model of William Kermack and Anderson McKendrick. I admire Quammen for including the mathematical biology of epidemics in his book, but he seems uncomfortable talking about math, and hesitant about subjecting his readers to it. I’m glad the readers of IPMB don’t place me under the same constraint.

Amid a dense flurry of mathematical manipulations, [Kermack and McKendrick] derived a set of three differential equations describing the three classes of living individuals: the susceptible, the infected, and the recovered. During an epidemic, one class flows into another in a simple schema, S → I → R, with mortalities falling out of the picture because they no longer belong to the population dynamic. As susceptible individuals become exposed to the disease and infected, as infected individuals either recover (now with immunity) or disappear, the numerical size of each class changes at each moment in time. That’s why Kermack and McKendrick used differential calculus. Although I should have paid better attention to the stuff in high school [I didn’t take calculus until college!], even I can understand (and so can you) that dR/dt = γI merely means that the number of recovered individuals in the population [I would have said “the rate of increase of the number of recovered individuals…”], at a given moment, reflects the number of infected individuals times the average recovery rate.  So much for R, the “recovered” class. The equations for S (“susceptibles”) and I (“infected”) are likewise opaque [not the word I would choose] but sensible. All this became known as an SIR model. It was a handy tool for thinking about infectious outbreaks, still widely used by disease theorists.

Covid-19 is caused by a zoonotic virus: a pathogen that leaps from an animal “reservoir” to infect humans. Quammen focuses on zoonotic viruses in Spillover, but he points out that not all viruses originate in animals. For instance, polio and smallpox are viruses that infect only humans. Once we remove those viruses from the human population, they are gone forever. A zoonotic virus “hides” in some wide animal (bats are a common reservoir) until it makes the jump to humans, so they are extraordinarily difficult to eradicate. Spillover is at its best when it describes these jumps, and the scientists who study them. Moreover, Quammen’s book is an extended case study of the scientific method. Everyone should read it.

Does Quammen predict the Covid-19 pandemic? Sort of. He predicts future pandemics arising from virulent and transmissible viruses that spill over from animal reservoirs. He predicts that our growing population and technology will make such spillovers more common. He even pinpointed coronaviruses as one of the likely suspects that could cause a future plague. What scares me is that Covid-19—as disruptive as it’s been for society—is not virulent enough to be the “Next Big One.” I fear it may be only a hint of things to come.

Worried? Me too. I’ll let Quammen have the final—somewhat hopeful—word [my italics].

I don’t say these things about the ineradicability of zoonoses to render you hopeless and depressed. Nor am I trying to be scary for the sake of scariness. The purpose of this book is not to make you more worried. The purpose of this book is to make you more smart.

 

David Quammen talking about Spillover.

https://www.youtube.com/embed/pFIqVsYa4vA

Charlotte's Web

Charlotte's Web,
by E. B. White

“Fern was up at daylight, trying to rid the world of injustice. As a result, she now has a pig.”

        From Charlotte’s Web, by E. B. White

Intermediate Physics for Medicine and Biology never mentions spiders like Charlotte, does it? It does! Chapter 1 has a homework problem about the strength of a spider’s thread. Steven Vogel discusses this in his terrific book Life’s Devices.

Anything with a strength at or above 100 MPa has to be considered a good tensile material—wood with the grain and collagen have about this value. Nylon [1000 MPa] is outstanding… and spider silk [2000 MPa] is superb—one can only wonder why, if one kind of creature can make a protein this good, the others, with the same synthetic machinery, don’t do as well.

The analysis of spiders in IPMB cites the paper:

Wilbur looking at
Charlotte's web.

Köhler T, Vollrath F (1995) “Thread biomechanics in the two orb-weaving spiders, Araneus diadematus (Araneae, Arcneidae) and Uloborus walckenaerius (Araneae, Ulobordae),” Journal of Experimental Zoology, Volume 271, Pages 1–17.

Their introduction (below, with references removed) explains how biomechanics is critical for spider webs.

Orb-weaving spiders within the Araneoidea are some of the most diverse and abundant predators of flying insects. As such, orb-weaving spiders depend upon their webs to stop the massive kinetic energy of flying insects and retain those insects long enough for the spiders to attack and subdue them. An orb web consists of a framework of stiff and strong radial threads that supports a spiral of sticky capture silk, the primary means by which prey adhere to the web. In addition to being covered with viscous glue, capture silk is also highly extensible, which allows the silk to gradually decelerate intercepted insects, thereby preventing prey from ricocheting out of webs. Thus, the potential for an orb web to retain prey long enough to be captured by the spider depends intimately upon the mechanical properties of these capture threads. Araneoid capture threads are composite structures that consist of two parts: a core pair of axial fibers spun from flagelliform silk and a surrounding coating of aqueous glue spun from aggregate silk glands. The aggregate silk secretions make capture threads sticky and can modulate the mechanics of the flagelliform axial fibers. However, it is the core axial fibers that provide the primary tensile mechanics of araneoid capture threads.

One of Garth Williams’s radiant drawings from Charlotte’s Web (above) makes me suspect that Charlotte was an orb-weaving spider. 

Charlotte's babies say
good-bye to Wilbur.

When Charlotte’s children were babies, Wilbur (some pig) witnessed them engaged in biological physics.

Then came a quiet morning when Mr. Zuckerman opened a door on the north side. A warm draft of rising air blew softly through the barn cellar. The air smelled of the damp earth, of the spruce woods, of the sweet springtime. The baby spiders felt the warm updraft. One spider climbed to the top of the fence. Then it did something that came as a great surprise to Wilbur. The spider stood on its head, pointed its spinnerets in the air, and let loose a cloud of fine silk. The silk formed a balloon. As Wilbur watched, the spider let go of the fence and rose into the air.

"Good-bye!" it said, as it sailed through the doorway.

Mark Denny describes this behavior in Air and Water.

The young of some spiders exhibit a remarkable behavior in which they climb to the apex of a blade of grass, extend their abdomen into the wind, and pull from their spinnerets a skein of very fine silk fibers. The drag on the fibers is sufficient to carry the young aloft, and Darwin reported having these “ballooning” spiders land on the Beagle while still many miles at sea.

I also enjoyed the animated musical of Charlotte’s Web with Paul Lynde as the voice of Templeton the rat.

A Veritable Smorgasbord. https://www.youtube.com/watch?v=kf1bu5sUXaU

When I was in third grade, my teacher Miss Sheets read Charlotte’s Web to my class, one chapter each day. I remember sitting at my desk crying when Charlotte died.

 Charlotte’s last song. https://www.youtube.com/watch?v=rPh94cOW-MI

The Elements of Style,
by Strunk and White.

E. B. White was an excellent writer. In addition to his children’s books—Charlotte's Web, Stuart Little, and The Trumpet of the Swan—he was coauthor with William Strunk on the famous writing manual The Elements of Style (“Omit Needless Words”).

The closing line of Charlotte’s Web reminds me of Barry Bowman, my humble friend who helped me become a better writer.

“It is not often that someone comes along who is a true friend and a good writer. Charlotte was both.”

Xenon-Enhanced Computed Tomography

Homework Problem 28 in Chapter 16 of Intermediate Physics for Medicine and Biology analyzes xenon-enhanced computed tomography.

Section 16.8

Problem 28. An experimental technique to measure cerebral blood perfusion is to have the patient inhale xenon, a noble gas with Z = 54, A = 131 (Suess et al. 1995). The solubility of xenon is different in red cells than in plasma. The equation used is

(arterial enhancement) = 5.15θ_Xe/[(μ/ρ)_w/(μ/ρ)_Xe]C_Xe(t),

where the arterial enhancement is in Hounsfield units, C_Xe is the concentration of xenon in the lungs (end tidal volume), and

θ_Xe = (0.011)(Hct) + 0.10.

Hct is the hematocrit: the fraction of the blood volume occupied by red cells. Discuss why the equation has this form.

The first page of
“X-ray-Computed Tomography Contrast Agents,”
by Lusic and Grinstaff.

I found an article that reviews using xenon as a contrast agent to monitor blood flow; Hrvoje Lusic and Mark Grinstaff discuss “X-ray-Computed Tomography Contrast Agents” (Chemical Reviews, Volume 113, Pages 1641–1666, 2013). I will quote the section on xenon, with references removed and comments added.

7.0 Xenon gas in CT imaging applications

“High Z” [high atomic number] noble gasses also represent a class of contrast media used in certain applications of X-ray CT [computed tomography] imaging. The most commonly used noble gas for CT imaging is xenon (Z_Xe = 54; absorption edge k_Xe = 34.6 keV) [compare this to other widely used contrast agents: iodine (Z_I = 53, k_I = 33.2 keV) and barium (Z_Ba = 56, k_Ba = 37.4 keV)]. Xenon is a readily diffusible monoatomic gas with low but not insignificant solubility in blood and fairly good solubility in adipose [fat] tissue. Xenon gas can pass across cell membranes, exchange between blood and tissue, and can cross the blood-brain barrier. Drawbacks to xenon gas use are related to its anesthetic properties, and may include respiratory depression, headaches, nausea, and vomiting. [Xenon-enhanced CT uses stable isotopes of xenon, so there is no dose from radioactive decay, although there is a dose from the X-rays used in CT. Other imaging methods use Xe-133, a radionuclide.]… Undesired side-effects can be adequately managed by controlling the xenon gas concentration and the length of time xenon is inhaled for. In several countries the stable xenon gas (non-radioactive 131Xe) is approved for clinical use in X-ray CT imaging. In the U.S., xenon-CT is not FDA [Food and Drug Administration] approved (as of the writing of this document) and is only available under investigational new drug (IND) status [as best I can tell, this remains true today; I’m not sure why].

Xenon-CT has been used for several decades to evaluate cerebral blood flow and perfusion in patients experiencing cerebrovascular disorders (e.g., following a brain injury, brain surgery, or stroke). It is considered a valuable imaging modality used as an alternative or complement to PET [positron emission tomography], SPECT [single photon emission computed tomography], MRI [magnetic resonance imaging], etc. Current standard for the xenon-CT cerebral blood flow evaluation calls for inhalation of 28 ± 1% medical grade xenon gas with at least 25% oxygen, for the duration of ~4.5 minutes. Following the procedure, xenon is rapidly washed out from cerebral tissues due to its short half-life of < 40 s. In the U.S., xenon-CT is often replaced by perfusion X-ray CT technique (PCT), which commonly employs non-ionic iodinated [containing iodine] small molecule contrast agents, frequently in combination with vasodilatory challenge [the widening of blood vessels] (e.g., acetazolamide) to measure brain hemodynamics…

Outlook

Xenon gas has X-ray attenuating properties similar to iodine. Xenon is chemically inert, biocompatible, and non-allergenic and can be safely used in patients with renal dysfunction. The undesired side-effects of xenon inhalation, related to its anesthetic properties, can be minimized by controlling the concentration of xenon gas being inhaled and the duration of the procedure. The rapid rate of xenon clearance from the body can be advantageous and conducive to repeat examinations. Xenon-CT has so far gained clinical approval in a number of countries, where the technique is most frequently used for cerebral blood flow assessment. Overall, xenon-CT is a useful clinical alternative to CT imaging using iodinated imaging media, especially when and where the diagnostic equipment is readily available.

The next noble gas in the rightmost column of the periodic table is radon (Z_Rn = 86, k_Rn = 98.4 keV), which has no stable isotopes. Being a noble gas, it should be diffusible and cross the blood-brain barrier like xenon. Would radon be a more effective contrast agent than xenon? For x-ray energies when the photoelectric effect dominates the interaction of photons with tissue, the cross section increases a Z⁴ (see Eq. 15.8 in IPMB), indicating that radon should be almost seven times more effective that xenon at increasing the x-ray absorption. Its k-edge is significantly higher than xenon’s, so its advantages would be realized only for x-ray energies above 100 keV. The key question is if the disadvantage of exposure to radiation (alpha decay in the lungs, which could cause lung cancer) would outweigh the advantage of its higher atomic number. If the risk from radon could be made much smaller than the risk of ionizing radiation from the CT scan itself, the use of radon might make sense. I suspect the expense of producing and handling radon, and public fears of even slight radioactivity, would tip the balance toward xenon over radon. Still, it’s an interesting idea.