Fundamental Limits of Spatial Resolution in PET

In Chapter 17 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss positron emission tomography.

17.10 Positron Emission Tomography

If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence.

Anyone who has looked at PET images will be struck by their low spatial resolution. They provide valuable functional information, but little anatomical detail. Why?

“Fundamental Limits of
Spatial Resolution in PET,”
by William Moses.

In a 2011 article in Nuclear Instruments and Methods in Physics Research A (“Fundamental Limits of Spatial Resolution in PET,” Volume 648, Supplement 1, Pages S236–S240), William Moses analyses what factors contribute to PET spatial resolution.

Abstract: The fundamental limits of spatial resolution in positron emission tomography (PET) have been understood for many years. The physical size of the detector element usually plays the dominant role in determining resolution, but the combined contributions from acollinearity, positron range, penetration into the detector ring, and decoding errors in the detector modules often combine to be of similar size. In addition, the sampling geometry and statistical noise further degrade the effective resolution. This paper quantitatively describes these effects, discusses potential methods for reducing the magnitude of these effects, and computes the ultimately achievable spatial resolution for clinical and pre-clinical PET cameras.

Detector size

The most obvious limitation of spatial resolution comes from the detector size. Usually detection occurs in a scintillator crystal that converts a gamma ray to visible light, which is detected by a photomultiplier. The width of the scintillator limits the spatial solution of the image. A typical detector size is about 4 mm.

Positron range

A positron is emitted with an energy of about a million electronvolts. It then travels through tissue until it slows enough to capture an electron and and give off two 0.511 MeV photons. The range of the positron sets a limit to the spatial resolution. Different isotopes emit positrons with different energies. One of the most widely used isotopes for functional studies is ¹⁸F, which has a range of about half a millimeter. Most other common isotopes used in PET have longer ranges.

Acollinearity

If a positron and electron are at rest when they annihilate, they emit two 0.511 MeV photons. To conserve momentum, these photons must travel in opposite directions. If, however, the positron-electron pair has some kinetic energy when they annihilate, the photons are not emitted exactly in opposite directions. Usually they deviate from 180° by up to 0.25°. This translates into about one to two millimeters of blurring in typical detector rings.

Decoding

Decoding is complicated. Many PET devices have more scintillators than photomultipliers, so the photomultipliers take turns recording from different scintillators (a process called multiplexing). The PET scanner must then decode all this information, and this decoding process is not perfect. Moses estimates that decoding introduces an uncertainty of about a third of the detector width, or around a couple millimeters in spatial resolution.

Penetration

The 0.511MeV photons can penetrate into the ring of detectors used in a PET device, causing blurring. In the illustration below, if the source (green) contains an isotope that emits two photons, then for some angles those photons (red) are detected by a single detector, but for other angles (blue) they are detected by multiple detectors. 

An illustration based on Fig. 2 of
“Fundamental Limits of Spatial Resolution in PET,”
by William Moses, showing how penetration of a
photon into different detectors causes blurring.

Sampling Error

A detector ring is more sensitive to sources at some positions compared to others (see Moses’s Fig. 3 for an explanation). This effect tends to degrade by spatial resolution by about 25%.

 

If you add all these uncertainties in quadrature, you get a spatial resolution of about 6 mm. This is worse resolution that you would have for magnetic resonance imaging or computed tomography, which is why PET images look so blurry. They are often overlaid on an MRI (see Fig. 17.25 in IPMB).

If you decided to build a PET system with the best possible spatial resolution (regardless of complexity or cost), you could eliminate all of the sources of uncertainty except positron range and acollinearity, implying a spatial resolution of about 2 mm (worse for isotopes other than ¹⁸F). PET is never going to image small-scale anatomical detail.

Qualifying Exams

First page of the 2020 Physics
Qualifying Exam.

I have discussed the Oakland University Medical Physics PhD Qualifying Exam previously in this blog. It’s a series of three written exams over math, physics, and biology, plus an oral exam.

I’ve collected the written qualifying exams over the last ten years (2011-2020) in a single file that you’re welcome to download. These exams are broad but not deep. They cover material at a level similar to, or somewhat higher than, Intermediate Physics for Medicine and Biology. Anyone working at the intersection of physics with biology should master these topics. The exams allow you to hone your problem solving skills. If you’re a student interested in applying physics to physiology, or math to medicine, but are stuck at home and not able to attend classes because of the covid-19 pandemic, you might want to try solving the 300 problems in this collection. (Slightly less than 300, because there wasn’t a math exam in 2014 and because we sometimes repeated questions from previous years.) If you want to try some of the exams from 2010 or earlier, you can find them at https://sites.google.com/view/bradroth/home/medical-physics-graduate-program/qualifying-exams?authuser=0.

Sorry, but I don’t have written solutions for these exams. You can always email me (roth@oakland.edu) if you’re stuck.

Enjoy!

Boron Neutron Capture Therapy

In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss boron neutron capture therapy.

Boron neutron capture therapy (BNCT) is based on a nuclear reaction which occurs when the stable isotope ¹⁰B is irradiated with neutrons, leading to the nuclear reaction (in the notation of Chap. 17)

Both the alpha particle and lithium are heavily ionizing and travel only about one cell diameter. BNCT has been tried since the 1950s; success requires boron-containing drugs that accumulate in the tumor. The field has been reviewed by Barth (2003).

Barth (2003)
J Neurooncol 62:1–5.

The citation is to an article by Rolf Barth of Ohio State University.

Barth, RF (2003) A Critical Assessment of Boron Neutron Capture Therapy: An Overview. Journal of Neuro-Oncology, Volume 62, Pages 1–5.

The abstract of this seventeen-year-old review states

Boron neutron capture therapy (BNCT) is based on the nuclear reaction that occurs when boron-10 is irradiated with neutrons of the appropriate energy to produce high-energy alpha particles and recoiling lithium-7 nuclei. BNCT has been used clinically to treat patients with high-grade gliomas, and a much smaller number with primary and metastatic melanoma. The purpose of this special issue of the Journal of Neuro-Oncology is to provide a critical and realistic assessment of various aspects of basic and clinical BNCT research in order to better understand its present status and future potential. Topics that are covered include neutron sources, tumor-targeted boron delivery agents, brain tumor models to assess therapeutic efficacy, computational dosimetry and treatment planning, results of clinical trails in the United States, Japan and Europe, pharmacokinetic studies of sodium borocaptate and boronophenylalanine (BPA), positron emission tomography imaging of BPA for treatment planning, and finally an overview of the challenges and problems that must be faced if BNCT is to become a useful treatment modality for brain tumors. Clinical studies have demonstrated the safety of BNCT. The next challenge is an unequivocal demonstration of therapeutic efficacy in one or more of the clinical trails that either are in progress or are planned over the next few years.

I was wondering what’s happening in this field lately, so I searched the Physics World website and found a fascinating and recent article by Tami Freeman.

Boron neutron capture therapy (BNCT), a technique that deposits highly targeted radiation into tumour cells, was first investigated as a cancer treatment back in the 1950s. But the field remains small, with only 1700 to 1800 patients treated to date worldwide. This may be about to change.

“The field of BNCT seems to be progressing rapidly at the moment,” said Stuart Green, director of medical physics at University Hospital Birmingham. “The big difference compared with five or ten years ago is that the commercial interest from a variety of companies is strong now and this is driving the field…”

Speaking at the Medical Physics & Engineering Conference (MPEC), Green updated on the status of BNCT programmes worldwide, noting that clinical experience is continually increasing. The US Food and Drug Administration has now approved two boron drugs for clinical use. But by far the majority of treatments, over 1150 to date, have taken place in Japan, initially using the Kyoto University Reactor in the early 2000s, and more recently using three Sumitomo accelerator systems in Kyoto, Fukushima and Osaka.

“Very importantly, earlier this year we had the first ever medical device approval for BNCT, for treatment in Japan of recurrent head-and-neck cancer,” said Green. “This is a significant marker for the entire field....”

“For the first time, there’s a substantial and sustained effort in the commercial sector to drive this field forward,” Green concluded. “We should keep an eye on BNCT over the next few years, there’s a lot happening, and hopefully our community can play a key role.”

Why the renewed interest in this technique? First, the original clinical applications of BNCT used neutrons from a nuclear reactor. Now accelerator-based neutron sources are available that can be installed in a hospital. Second, researchers are working hard on boron-containing drugs. Currently, boronophenylalanine and sodium borocaptate are the most common drugs used clinically. Improving the delivery of these drugs, or designing entirely new drugs, could increase the usefulness of BNCT. 

We live in exciting times.

Boron Neutron Capture Therapy Animation. 

https://www.youtube.com/watch?v=cQrbfBiQpfk

Generation of Unidirectionally Propagating Action Potentials Using a Monopolar Electrode Cuff

Ungar et al. (1986) Ann.
Biomed. Engin., 14:437–450.

In Chapter 7 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss electrical stimulation of nerves. One of the end-of-the-chapter homework problems asks the students to

Design a stimulator that will result in one-way propagation… For an application of such [a device] during functional electrical stimulation, see Ungar et al. (1986).

The reference is

Ungar IJ, Mortimer JT, Sweeney JD (1986) “Generation of Unidirectionally Propagating Action Potentials Using a Monopolar Electrode Cuff,” Annals of Biomedical Engineering, Volume 14, Pages 437–450.

The abstract to their article states

Unidirectionally propagating action potentials, which can be used to implement transmission failure on peripheral nerve through “collision block,” have been generated electrically on cat myelinated peripheral nerve using a monopolar electrode cuff with the conductor positioned closest to the “arrest” end of the cuff. A single cathode located at least 5 mm from the arrest end resulted in unidirectional propagation with minimal current and charge injection. The range of stimulus current values that produced unidirectional propagation increased with increases in longitudinal asymmetry of cathode placement over the range of asymmetries tested (1.7:1 to 7:1). The stimulus current pulse that minimized charge injection was quasitrapezoidal in shape with a plateau pulse width of approximately 350 μsec and an exponential trailing phase having a fall time (90%–10%) of approximately 600 μsec. These parameters were found to be independent of cuff geometry. Arrest efficiency was not degraded using a cuff of sufficient internal diameter to prevent nerve compression in chronic implantation. The critical current density within the extracellular space of the electrode cuff required to produce conduction failure at the arrest end was estimated to be 0.47 ± 0.08 mA/mm².

I’ll explain how to make a one-way stimulator using the illustration below, adapted from Ungar et al.’s Figure 1.

The design of a one-way neural stimulator.
Adapted from Fig. 1 of Ungar et al. (1986).

The nerve (blue) is threaded through a cylindrical insulating cuff (red), which resembles a short segment of a plastic drinking straw. The cathode (green) is inside the cuff; it stimulates the action potential. The anode (not shown) is far away. Current (purple curves) comes out of the cathode and enters the nerve axons, depolarizing them. Once it reaches the end of the cuff the current spreads out as it returns to the distance anode. Current there leaves the axons, hyperpolarizing them, and lowering their transmembrane potential. The locations where the axons are hyperpolarized are labeled the virtual anodes.

The key to making a one-way stimulator is to place the cathode off-center in the cuff. Most of the current leaves the cuff through the end nearest the cathode (right end), so the current density is stronger there (the purple current lines are crowded together). Only a small fraction of the current leaves the cuff through the end farther from the cathode (left end), so the virtual anode is weaker there.

Depolarization under a cathode excites an action potential, which then propagates outward in both directions (left and right). If, however, the stimulus strength is strong enough, the hyperpolarization at a virtual anode can block propagation. If the current has the correct strength, the stronger virtual anode on the right will block propagation, while the weaker virtual anode on the left won’t. In that case, an action potential will propagate to the left (the escape end of the stimulator) but will not propagate to the right (the arrest end).

Ira Ungar and his collaborators were able to test their stimulator for different cathodal current strengths. For a very weak stimulus, the cathode is below threshold and no action potential is excited. For a moderately weak stimulus, the cathode excites an action potential that then propagates to both the left and the right; both virtual anodes are too weak to block propagation. For a moderately strong stimulus, the right virtual anode is strong enough to block the action potential and you have one-way propagation. For a very strong stimulus, both ends block propagation, and no action potential leaves the stimulator.

Why construct a one-way stimulator? Suppose you have a nerve that’s constantly firing action potentials, causing unwanted muscle contraction (spasticity). You could stop the downstream propagation of those action potentials by electrically stimulating action potentials further downstream. The stimulated action potentials propagating upstream will collide with the original action potentials propagating downstream, annihilating them (colliding action potentials can’t pass through each other because they’re each followed by a region of refracotoriness). This works great unless your stimulator sends its own volley of action potentials propagating downstream to excite the muscle. To avoid this problem, you need a one-way stimulator, so you only excite action potentials propagating upstream to block those causing the trouble, but none propagating downstream.

The senior author on this article was J. Thomas Mortimer, now emeritus professor in the Neural Engineering Center at Case Western Reserve University. He earned his PhD from Case and then spent his entire career there. He has developed an online Applied Neural Control Toolkit to teach how nerves work. I heard Mortimer speak during one of the Neural Prosthesis Workshops at the National Institutes of Health; he was inspirational.

Mortimer and his team’s development of a one-way stimulator is a classic example of how physics and engineering can contribute to medicine and biology.

The Death and Life of the Great Lakes

Many readers of this blog live far from me—sometimes in countries on the other side of the world—and may not be interested in local topics germane to Michigan. Yet the Great Lakes are important to everyone; they hold 20% of the earth’s liquid freshwater. The challenges faced by the Great Lakes aren’t unique; they’re relevant to other watersheds worldwide.

For those who aren’t Michiganders, a drawing of the five Great Lakes looks like a map of Michigan. 

The Great Lakes
From Phizzy at Wikipedia.

Between its upper and lower peninsulas, Michigan borders four of the five lakes. I live half way between the southern tip of Lake Huron and the western edge of Lake Erie. I’ve visited all five lakes, and I’m concerned about their welfare.

The Death and Life of the Great Lakes,
by Dan Egan.

 

I recently read The Death and Life of the Great Lakes (2017), by Dan Egan. (This book was recommended to me by Congresswoman Elissa Slotkin, who’s now running for reelection in my home district). I’ve lived in Michigan for 22 years, but much of this story was new to me. In his introduction, Egan writes

Iconic disasters have a history of prompting government action. Three years after the Cuyahoga River fire of 1969, Congress passed the Clean Water Act. Two decades later, when the Exxon Valdez ran aground and dumped 10.8 million gallons of crude oil into Alaska’s Prince William Sound, images of cleanup crews using paper towels to cleanse tarred birds helped press Congress into doing something it should have done years earlier. It mandated double-hulled oil tankers.

But the disaster unfolding today on the Great Lakes didn’t ignite like a polluted river or gush like oil from a cracked hull, and so far there is no galvanizing image of this slow-motion catastrophe, though a few come to my mind. One is the bow of an overseas ship easing its way into the first navigation lock on the St. Lawrence Seaway, the Great Lakes’ “front door” to fresh waves of biological pollution. Another is a satellite photo of a green-as-paint toxic algae slick smothering as much as 2,000 square miles of Lake Erie.

Yet another is the grotesque mug of an Asian carp, a monster-sized carp imported to the United States in the 1960s and used in government experiments to gobble up excrement in Arkansas sewage lagoons. The fish, which can grow to 70 pounds and eat up to 20 percent of their weight in plankton per day, escaped into the Mississippi River basin decades ago and have been migrating north ever since. There are now mustering at the Great Lakes’ “back door”—the Chicago canal system that created a manmade connection between the previously isolated Great Lakes and the Mississippi basin, which covers about 40 percent of the continental United States. The only thing blocking the fish’s swim through downtown Chicago and into Lake Michigan is an electrical barrier in the canal—one that has a history of unexpected shutdowns.

The Chicago canal has also turned the Great Lakes’ ballast water problem [ships from the Atlantic Ocean having exotic species in their ballast water and then dumping it into the Great Lakes] into a national one, because there are dozens of invasive species poised to ride its waters out of the lakes and into the rivers and water bodies throughout the heart of the continent. Species like the spiny water flea, the threespine stickleback, the bloody red shrimp and the fishhook water flea. All organisms you probably haven’t heard of. Yet.

Few out West, after all, had ever heard of quagga mussels—until they tumbled down the Chicago canal and metastasized across the Mississippi basin and, eventually, into the arid West, likely as hitchhikers aboard recreational boats towed over the Rocky Mountains. The mussels have since unleashed havoc on hydroelectric dams, drinking water systems and irrigation networks in Utah, Nevada and California and the federal government estimates that if the mussels make their way into the Northwest’s Columbia River hydroelectric dam system they could do a half billion dollars of damage—per year.

Much of Egan’s book is about how invasive species interact with native animals. In Chapter 2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the predator-prey problem modeled by the Lotka-Volterra equations. It is a useful toy model for understanding the interplay of two species, but it doesn’t explain the workings of a large and complex ecosystem.

The Great Lakes (at least the four that border Michigan) were once isolated from the Atlantic Ocean by Niagara Falls, a barrier than even the most determined fish could not surmount. The opening of the Saint Lawrence Seaway allowed a series of nonnative animals to enter the lakes. First came the sickening sea lamprey that nearly caused the native lake trout to go extinct. Next was an explosion of alewife (a river herring). Scientists introduced coho and Chinook salmon to prey upon the alewives. In each case, the ramifications of a new species were difficult to predict. Then zebra mussels and the quagga muscles arrived, and finally the round goby. It’s a horror story, and I suspect an invasion of the Asian carp is inevitable.

Besides these exotic species, the Great Lakes (especially Lake Erie) suffer from algae blooms triggered by fertilizer runoff. Add in climate change, which is causing the water level in the lakes to fluctuate, and you have an unstable and unhealthy ecosystem. 

I’ll give Egan the last word.

If we can close these doors to future invasions [of invasive species], we may give the lakes, and the rest of the country, time to reach a new equilibrium, a balance between what is left of the natural inhabitants and all the newcomers... And if we do this, then we can focus on the major problems that still plague the lakes, which include the overapplication of farm fertilizer that is helping to trigger the massive toxic algae outbreaks, the impact a warming globe is having on the lake’s increasingly unstable water levels and the need to protect lake waters from outsiders seeking to drain them for their own profit.

Dan Egan on The Death and Life of the Great Lakes at the 2018 L.A. Times Festival of Books. https://www.youtube.com/watch?v=jtq_cN1kPkg

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.

An Advanced Undergraduate Laboratory in Living State Physics

One weakness of Intermediate Physics for Medicine and Biology is that it doesn’t have an associated laboratory. Students need to learn how to perform experiments and use instruments.

An Advanced Undergraduate Laboratory
In Living State Physics,
by Wikswo, Vickery, and Venable.

Fortunately, instructors wanting to develop a lab don’t need to start from scratch. My PhD advisor, John Wikswo, and his colleagues Barbara Vickery and John Venable created An Advanced Undergraduate Laboratory in Living State Physics at Vanderbilt University around 1980. I didn’t take this lab class, but my wife Shirley did (she obtained a masters degree in physics from Vanderbilt), and she still has the lab manual. 

Wikswo obtained a grant from the National Science Foundation to support the development of the lab. He collaborated with John Venable, a biologist on the Vanderbilt faculty. When I was a graduate student, Venable was the Associate Dean of the College of Arts and Sciences. Barbara Vickery was a Vanderbilt undergraduate biomedical engineering major.

The lab wasn’t designed for any particular textbook, but Wikswo was an early adopter of Russ Hobbie’s Intermediate Physics for Medicine and Biology, and I think I can see its influence. I don’t have an electronic copy of the 250-page lab manual; you would have to contact Wikswo for that. Below I quote parts of it.

1.1 An Introduction to the Living State Physics Laboratory

The undergraduate physics curriculum at a typical university might include an introductory class in biophysics or medical physics in addition to the more traditional curriculum of mechanics, electricity and magnetism, light and sound, thermodynamics, and modern physics. While introductory and advanced laboratory classes cover these standard fields of physics, generally there has been little opportunity for an undergraduate student to gain laboratory experience in biophysics or medical physics. The need for such experience is particularly acute today for preprofessional and scientifically oriented students. Of these students, physics majors are not being exposed to an important area of experimental physics, and pre-medical students and majors in other departments such as Molecular Biology, Chemistry, and Biomedical Engineering are presently receiving only a minimal exposure to modern biophysical techniques and instrumentation. Thus by introducing an advanced undergraduate laboratory in physics applied to living systems, we expect to broaden the experience in experimental physics for physics majors and non-majors alike.

Several options were available to us in designing this laboratory. We could, for example, have structured the laboratory to emphasize applications of physics to certain living systems such as the nervous system, the cardiovascular system, and the special senses. Rather than take this system-oriented approach, we have chosen to organize the course by areas of physics. The course will draw on techniques and ideas from the whole breadth of physics (mechanics, electricity, thermodynamics, optics, etc.) and apply these to topics of biophysical interest [the same approach as IPMB]. Since we will study intact living systems such as people and frogs, as well as isolated living preparations and inanimate molecules and models, this laboratory will use physics to study topics conventionally identified with both biophysics and medical physics, as well as with electrophysiology, physical chemistry, biomedical engineering and molecular biology. Because of the intended breadth of the planned experiments and their organization by area of physics rather than by biological system, we have chosen to title this laboratory “An Advanced Undergraduate Laboratory in Living State Physics”. The generality of the term “Living State Physics” is intended to parallel the generality of the term “solid state physics”, which as an experimental discipline utilizes the complete spectrum of physical concepts and techniques...

1.2 Summary of Experiments

a. Introduction to Bioelectric Phenomena. The first of the three experiments in this section is an exercise with an oscilloscope and an electronic stimulator which will allow the student to obtain a familiarity with the use of these instruments. In the second and third experiments, the Thornton Modular Plug-In System is used to provide familiarity with the basic physics describing the electromyogram and the electroencephalogram…

b. The Heart Experiments. This section should enable the student to gain an understanding of the basic principles of cardiac physiology. In the laboratory, the student will measure the frog and the human electrocardiogram…

c. Nerve Action Potential… [Students perform an] in-depth study of the properties of nerve propagation in the isolated sciatic nerve of a frog. In both experiments, from extracellular recordings of the nerve action potential it will be possible to demonstrate the graded response of the nerve bundle, the strength-duration relationship of stimuli producing a threshold response, bi-directional conduction, and the monophasic response…

d. Nerve Modeling. In the first experiment, the passive cable properties of the nerve are studied by using a resistor-capacitor network that represents a section of a nerve axon… The active properties of the nerve are investigated in the second experiment. An electronic nerve model which has a design based on a system of equations similar to those developed by Hodgkin and Huxley is used…

e. Skeletal Muscle. The first of the two experiments in this section is an introduction to the active and passive mechanical properties of skeletal muscle using the frog gastrocnemius muscle. The experiment includes measurement of the muscle twitch, the ability of the muscle to do work, and the maximum tension developed by the muscle at different lengths, as well as demonstration of the phenomena of temporal summation and the graded response of muscle. The second experiment involves characterization of the mechanical properties of muscle in its resting and contractile states…

f. Diffusion. In this experiment, a Cenco model is used for qualitative demonstration of the transport phenomenon of diffusion, showing the exponential approach to equilibrium and how the relative sizes of molecules and pores affect diffusion rates.

g. Compartmental Modeling. The usefulness of compartmental modeling in analysis of some systems is demonstrated by constructing one- and two-compartment models for several open and closed thermal systems. The theoretical models are analyzed mathematically…

h. The Physical Aspects of Vision. The minimum number of photons that the human eye can detect in a single detectable flash is the minimum number of photons whose absorption by photoreceptor cells in the eye leads to the firing of an impulse in the brain. This threshold value is determined by recording the fraction of detected flashes as a function of relative intensity of the flashes… by utilizing Poisson statistics.

i. Ultrasound… The experiments introduce the physics of mechanical waves by using ultrasound transducers, a two-dimensional ultrasound target, and an existing ultrasound scanner and transient analyzer to demonstrate wave propagation, attenuation, reflection, refraction, pulse-echo principles, piezoelectric crystals and the concepts of cross-section and spatial resolution.

The first time I ever saw my wife was when she was in Wikswo's office asking a question about one of the lab exercises. I needed to talk to him about some very important issue related to my research, and she was in the way! Well, one thing led to another and....

I recall how Shirley and my friend Ranjith Wijesinghe were lab partners doing the vision experiment. It required sitting in a small, dark enclosure for about half an hour while their eyes became adapted to the dark. I had only recently met Shirley, and I recall being jealous of Ranjith for getting to spend such a private time with her! 

One of the most memorable parts of the lab was the pithing of the frog. None of the students liked doing that. Wikswo had a fun way of demonstrating the fight-of-flight response during the electrocardiogram lab. He would measure the ECG on one of the students, and then take out a giant syringe and say something like “now watch what happens to her heart rate when I inject her with this adrenaline.” Of course no one ever got injected, but the student was always so startled that her heart rate would jump dramatically.

If you are considering developing you own laboratory for Intermediate Physics for Medicine and Biology, you could start with Wikswo’s lab, and then add some of the experiments discussed in these American Journal of Physics papers. Good luck!

J. D. Prentice and K. G. McNeill (1962) “Measurement of the Beta Spectrum of I¹²⁸ in an Undergraduate Laboratory,” American Journal of Physics, Volume 30, Pages 66–67.  

Peter J. Limon and Robert H. Webb (1964) “A Magnetic Resonance Experiment for the Undergraduate Laboratory,” American Journal of Physics, Volume 32, Pages 361–364.    

L. J. Bruner (1979) “Cardiovascular Simulator for the Undergraduate Physics Laboratory,” American Journal of Physics, Volume 47, Pages 608–611.  

H. W. White, P. E. Chumbley, R. L. Berney, and V. H. Barredo (1982) “Undergraduate Laboratory Experiment to Measure the Threshold of Vision,” American Journal of Physics, Volume 50, Pages 448–450. 

Colin Delaney and Juan Rodriguez (2002) “A Simple Medical Physics Experiment Based on a Laser Pointer,” American Journal of Physics, Volume 70, Pages 1068–1070. 

Danny G. Miles Jr. and David W. Bushman (2005) “Protein Gel Electrophoresis in the Undergraduate Physics Laboratory,” American Journal of Physics, Volume 73, Pages 1186–1189. 

Luis Peralta (2006) “A Simple Electron-Positron Pair Production Experiment,” American Journal of Physics, Volume 74, Pages 457–461.  

Joseph Peidle, Chris Stokes, Robert Hart, Melissa Franklin, Ronald Newburgh, Joon Pahk, Wolfgang Rueckner, and Aravi Samuel (2009) “Inexpensive Microscopy for Introductory Laboratory Courses,” American Journal of Physics, Volume 77, Pages 931–938. 

Timothy A. Stiles (2014) “Ultrasound Imaging as an Undergraduate Physics Laboratory Exercise,” American Journal of Physics, Volume 82, Pages 490–501.  

Elliot Mylotta, Ellynne Kutschera, and Ralf Widenhorn (2014) “Bioelectrical Impedance Analysis as a Laboratory Activity: At the Interface of Physics and the Body,” American Journal of Physics, Volume 82, Pages 521–528.    

Alexander Hydea and Oleg Batishchevb (2015) “Undergraduate Physics Laboratory: Electrophoresis in Chromatography Paper,” American Journal of Physics, Volume 83, Pages 1003–1011.

Owen Paetkau, Zachary Parsons, and Mark Paetkau (2017) “Computerized Tomography Platform Using Beta Rays,” American Journal of Physics, Volume 85, Pages 896–900.