The Mathematical Approach to Physiological Problems

In Chapter 2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I analyze decay with multiple half-lives, and practice fitting data to exponentials. Near the end of this discussion, we write (really, Russ wrote these words, since they go back to his solo first edition of IPMB)

Estimating the parameters [governing exponential decay] for the longest-lived term may be difficult because of the potentially large error bars associated with the data for small values of y. For a discussion of this problem, see Riggs (1970, pp. 146–163).

The Mathematical Approach
to Physiological Problems,
by Douglas S. Riggs.

The citation is to the book The Mathematical Approach to Physiological Problems, by Douglas Riggs. I  wanted to take a look, so I went to the Oakland University library and checked out the 1963 first edition.

It’s a gold mine. I particularly like the beginning of the preface. Riggs starts with what seems like an odd digression about hiking in the mountains, but then in the second paragraph he skillfully brings us back to math.

Before settling down in the village of Shepreth, in Cambridgeshire, England, to start working in earnest upon this book, I had the unusual pleasure of taking my family on a month-long youth hosteling trip through Northern England and the Scottish Highlands. For the most part, we bicycled, but occasionally we would make an excursion on foot up the steep hillsides and along the rocky ridges where no bicycle could go. We soon learned that the British discriminate carefully between “hill walking” and “mountain climbing.” To be a mountain climber, you must coil 100 feet of nylon rope around you slantwise from shoulder to waist, and have a few pitons dangling somewhere about. You are then entitled to adopt an ever-so-faintly condescending attitude toward any hill-walkers whom you may encounter along the trail, even if you meet them where the trail is practically level. Hill-walkers, on the other hand, remain hill-walkers even when the “walk” turns into a hands-and-knees job up a 40° slope of talus which is barely anchored to the mountain by a few wisps of grass and a clump or two of scraggly heather.

Mathematically speaking, this is a hill-walking book. It is necessarily so, since I myself have never learned the ropes of higher mathematics. But I do believe that the amount of wandering I have done on the lower slopes, the number of sorry hours I have spent lost in a mathematical fog, and the miles I have stumbled down false trails have made me a kind of backwoods expert on mathematical pitfalls, and have given me some practical knowledge of how to plan a safe mathematical ascent of the more accessible physiological hills.

One theme I stress in this blog is the value of simple models. Riggs agrees.

Precisely because living systems are so very complex, one can never expect to achieve anything like a complete mathematical description of their behavior. Before the mathematical analysis itself is begun, it is therefore invariably necessary to reduce the complexity of the real system by making various simplifying assumptions about how it behaves. In effect, these assumptions allow us to replace the actual biological system by an imaginary model system which is simple enough to be described mathematically. The results of our mathematical analysis will then be rigorously applicable to the model. But they will be applicable to the original biological system only to the extent that our underlying assumptions are reasonable. Hence, the ultimate value of our mathematical labors will be determined in large part by our choice of simplifying assumptions.

He then advocates for an approach that I call “think before you calculation.”

An investigator who publishes an erroneous equation has no place to hide! It is therefore prudent to check each calculation, each algebraic manipulation, and each transcription from a table of figures before going on to the next step. Above all, whenever you are engaged in mathematical work you should keep asking yourself over and over and over again, “Does this make sense?” and “Is this of the correct magnitude?”

Riggs’s introduction ends with some wise advice.

All too frequently, students are willing to accept on faith whatever mathematical formulations they encounter in their reading. And why not? After all, mathematics is the exact science, and presumably an author would not express his theories or his conclusions mathematically without due regard for mathematical rigor and precision. It is only by bitter experience that we learn never to trust a published mathematical statement or equation, particularly in a biological publication, unless we ourselves have checked it to see whether or not it makes sense… Misprints are common. Copying errors are common. Blunders are common. Editors rarely have the time or training to check mathematical derivations. The author may be ignorant of mathematical laws, or he may use ambiguous notation. His basic premises may be fallacious even though he uses impressive mathematical expressions to formulate his conclusions… Caveat lector! Let the reader beware!

What about the problem of fitting multiple exponentials to data, which is why Russ and I cite Riggs in the first place? After analyzing several specific examples, Riggs concludes

Page 157 of The Mathematical
Approach to Physiological Problems,
showing a fit to a double exponential.

These examples warn us not to take too seriously any particular set of coefficients and rate constants which we may get by plotting data on semi-logarithmic paper and ferreting out the exponential terms in the fashion described above. The need for such a warning is all too evident from the preposterously elaborate exponential equations which are sometimes published. The technique of “peeling off” successive terms is so deceptively easy! Fit a straight line, subtract, plot the differences, fit another straight line, subtract, plot the differences. How solid and impressive the resulting sum of exponentials looks! And how remarkably well the curve agrees with the observations. Surely the investigator can be pardoned a certain self-satisfaction for having so clearly identified the individual components which were contributing to the overall change. Yet, the examples discussed above show how groundless his satisfaction may be. It is undoubtedly true that the particular sum of exponentials which he happened to pick, plot, and publish fits the points with gratifying accuracy. But so also may other equations of the same general form but with quite different parameters. It is not great trick to have found one such equation. Even with a single exponential declining from a known value at time zero toward an unknown constant asymptote there are two parameters—the rate constant and the asymptote—to be fitted to the data. The effect of a considerable change in either may be largely offset by making a compensatory change in the other… Add a second exponential term with two more parameters to be estimated from the data, and the number and variety of “closely fitting” equations becomes truly bewildering. Worst of all, there are no simple statistical measures of the precision with which any of the parameters have been estimated. These considerations do not destroy the value of fitting an exponential equation to experimental data when it is suggested by some underlying theory or when it provides a convenient empirical way of summarizing a group of observations mathematically… But they make very clear the danger of using an empirical exponential equation to predict what may happen beyond the period actually covered by the observations. It is equally clear that we must be exceedingly skeptical when attempts are made to match the individual terms of an empirical exponential equation with supposedly corresponding processes or regions of the body.

Riggs’s book examines many of the same topics that appear in the first half of IPMB, such as exponential growth, diffusion, and feedback. He has a wonderful chapter suggesting questions you should ask when checking the validity of an equation. Is it dimensionally correct? How does it behave when variables approach zero or infinity? Does it give reasonable answers after numerical substitution?

I’m impressed that Riggs, a professor and head of a Department of Pharmacology at SUNY Buffalo, could write so insightfully about mathematics. I give the book two thumbs up.

Life at Low Reynolds Number

“Life at Low Reynolds Number,”
by Edward Purcell.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite Edward Purcell’s wonderful article “Life at Low Reynolds Number” (American Journal of Physics, Volume 45, Pages 3–11, 1977). This paper is a transcript of a talk Purcell gave to honor physicist Victor Weisskopf. The transcript captures the casual tone of the talk, and the hand-drawn figures are charming. Below I quote excerpts from the article, including my own versions of a couple of the drawings. Notice how his words emphasize insight. As Purcell says, “some essential hand-waving could not be reproduced.” Enjoy!

I’m going to talk about a world which, as physicists, we almost never think about. The physicist hears about viscosity in high school when he’s repeating Millikan’s oil drop experiment and never hears about it again, as least not in what I teach. And Reynolds’s number, of course, is something for the engineers. And the low Reynolds number regime most engineers aren’t even interested in… But I want to take you into the world of very low Reynolds number—a world which is inhabited by the overwhelming majority of the organisms in this room. This world is quite different from the one that we have developed our intuitions in…

Based on Figure 1 from
“Life at Low Reynolds Number.”

In Fig. 1, you see an object which is moving through a fluid with velocity v. It has dimension a,… η and ρ are the viscosity and density of the fluid. The ratio of the inertial forces to the viscous forces, as Osborne Reynolds pointed out slightly less than a hundred years ago, is given by avρ/η or av/ν, where ν is called the kinematic viscosity. It’s easier to remember its dimensions; for water, ν = 10⁻² cm²/sec. The ratio is called the Reynolds number and when that number is small the viscous forces dominate… Now consider things that move through a liquid... The Reynolds number for a man swimming in water might be 10⁴, if we put in reasonable dimensions. For a goldfish or a tiny guppy it might get down to 10². For the animals that we’re going to be talking about, as we’ll see in a moment, it’s about 10⁻⁴ or 10⁻⁵. For these animals inertia is totally irrelevant. We know F = ma, but they could scarcely care less. I’ll show you a picture of the real animals in a bit but we are going to be talking about objects which are the order of a micron in size… In water where the kinematic viscosity is 10⁻² cm²/sec these things move around with a typical speed of 10 μm/sec. If I have to push that animal to move it, and suddenly I stop pushing, how far will it coast before it slows down? The answer is, about 0.1 Å. And it takes it about 0.6 μsec to slow down. I think this makes it clear what low Reynolds number means. Inertia plays no role whatsoever. If you are at very low Reynolds number, what you are doing at the moment is entirely determined by the forces that are exerted on you at that moment, and by nothing in the past…

Based on Figure 18 from
“Life at Low Reynolds Number.”

Diffusion is important because of [a] very peculiar feature of the world at low Reynolds number, and that is, stirring isn’t any good… At low Reynolds number you can’t shake off your environment. If you move, you take it along; it only gradually falls behind. We can use elementary physics to look at this in a very simple way. The time for transporting anything a distance ℓ by stirring is about ℓ divided by the stirring speed v. Whereas, for transport by diffusion, it’s ℓ² divided by D, the diffusion constant. The ratio of those two times is a measure of the effectiveness of stirring versus that of diffusion for any given distance and diffusion constant. I’m sure this ratio has someone’s name but I don’t know the literature and I don’t know whose number that’s called. Call it S for stirring number. It’s just ℓv/D. You’ll notice by the way that the Reynolds number was ℓv/ν. ν is the kinematic viscosity in cm²/sec, and D is the diffusion constant in cm²/sec, for whatever it is that we are interested in following—let’s say a nutrient molecule in water. Now, in water the diffusion constant is pretty much the same for every reasonably sized molecule, something like 10⁻⁵ cm²/sec. In the size domain that we’re interested in, of micron distances, we find that the stirring number S is 10⁻², for the velocities that we are talking about (Fig. 18). In other words, this bug can’t do anything by stirring its local surroundings. It might as well wait for things to diffuse, either in or out. The transport of wastes away from the animal or food to the animal is entirely controlled locally by diffusion. You can thrash around a lot, but the fellow who just sits there quietly waiting for stuff to diffuse will collect just as much.

 

The Physics of Life: Life at Low Reynolds Number
https://www.youtube.com/watch?v=gZk2bMaqs1E

 

 Edward Purcell
https://www.youtube.com/watch?v=0uATCx7WMMs

Selig Hecht (1892-1947)

Selig Hecht,
History of the Marine Biological Laboratory,
http://hpsrepository.asu.edu/handle/10776/3269.

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I analyze the “classic experiment on scotopic vision” by Hecht, Shlaer, and Pirenne. George Wald wrote an obituary about Selig Hecht in 1948 (Journal of General Physiology, Volume 32, Pages 1–16). He writes that Hecht was

Intensely interested in the relation of light quanta (photons) to vision. Reexamining earlier measurements of the minimum threshold for human rod vision, he and his colleagues confirmed that vision requires only fifty to 150 photons. When all allowances had been made for surface reflections, the absorption of light by ocular tissues, and the absorption by rhodopsin (which alone is an effective stimulant), it emerged that the minimum visual sensation corresponds to the absorption in the rods of, at most, five to fourteen photons. An entirely independent statistical analysis suggested that an absolute threshold involves about five to seven photons. Both procedures, then, confirmed the estimation of the minimum visual stimulus at five to fourteen photons. Since the test field in which these measurements were performed contained about 500 rods, it was difficult to escape the conclusion that one rod is stimulated by a single photon.

Wald also describes the coauthor on the study, Shlaer.

Among Hecht’s first students was Simon Shlaer, who became Hecht’s assistant in his first year at Columbia and continued as his associate for twenty years thereafter. A man infinitely patient with things and impatient with people, Shlaer gave Hecht his entire devotion. He was a master of instrumentation, and though he also had a keen grasp of theory, he devoted himself by choice to the development of new technical devices. Hecht and Shlaer built a succession of precise instruments for visual measurement, among them an adaptometer and an anomaloscope that have since gone into general use. The entire laboratory came to rely on Shlaer’s ingenuity and skill. “I am like a man who has lost his right arm,” remarked Hecht on leaving Columbia—and Shlaer—in 1947, “and his right leg.”

In his Columbia laboratory, Hecht instituted investigations of human dark adaptation, brightness discrimination, visual acuity, the visual response to flickered light, the mechanism of the visual threshold, and normal and anomalous color vision. His lab also made important contributions regarding the biochemistry of visual pigments, the relation of night blindness to vitamin A deficiency in humans, the spectral sensitivities of man and other animals, and the light reactions of plants—phototropism, photosynthesis, and chlorophyll formation.

Hecht and Shlaer both contributed to the war effort during the Second World War.

Throughout the late years of World War II, Hecht devoted his energies and the resources of his laboratory to military problems. He and Shlaer developed a special adaptometer for night-vision testing that was adopted as standard equipment by several Allied military services. Hecht also directed a number of visual projects for the Army and Navy and was consultant and advisor on many others. He was a member of the National Research Council Committee on Visual Problems and of the executive board of the Army-Navy Office of Scientific Research and Development Vision Committee.

Explaining the Atom,
by Selig Hecht.

Hecht straddled the fields of physics and physiology, and was comfortable with both math and medicine. He entered college studying mathematics. After World War II ended, he wrote the book Explaining the Atom, which Wald described as “a lay approach to atomic theory and its recent developments that the New York Times (in a September 20, 1947, editorial) called ‘by far the best so far written for the multitude.’”

An obituary in Nature by Maurice Henri Pirenne concludes

The death of Prof. Selig Hecht in New York on September 18, 1947, at the age of fifty-five, deprives the physiology of vision of one of its most outstanding workers. Hecht was born in Austria and was brought to the United States as a child. He studied and worked in the United States, in England, Germany and Italy. After a broad biological training, he devoted his life to the study of the mechanisms of vision, considered as a branch of general physiology. He became professor of biophysics at Columbia University and made his laboratory an international centre of visual research.

Role of Virtual Electrodes in Arrhythmogenesis: Pinwheel Experiment Revisited

The Journal of Cardiovascular Electrophysiology,
with a figure from Lindblom et al. on the cover.

Twenty years ago, I published an article with Natalia Trayanova and her student Annette Lindblom about initiating an arrhythmia in cardiac muscle (“Role of Virtual Electrodes in Arrhythmogenesis: Pinwheel Experiment Revisited,” Journal of Cardiovascular Electrophysiology, Volume 11, Pages 274-285, 2000). We performed computer simulations based on the bidomain model, which Russ Hobbie and I discuss in Section 7.9 of Intermediate Physics for Medicine and Biology. A key feature of a bidomain is anisotropy: the electrical conductivity varies with direction relative to the axis of the myocardial fibers.

Our results are summarized in the figure below (Fig. 14 of our article). An initial stimulus (S1) launched a planar wavefront through the tissue, either parallel to (longitudinal, L) or perpendicular to (transverse, T) the fibers (horizontal). As the tissue recovered from the first wave front, we applied a second stimulus (S2) to a point cathodal electrode (C), inducing a complicated pattern of depolarization under the cathode and two regions of hyperpolarization (virtual anodes) adjacent to the cathode along the fiber axis (see my previous blog post for more about how cardiac tissue responds to a point stimulus). In some simulations, we reversed the polarity of S2 so the electrode was an anode (A). This pair of stimuli (S1-S2) underlies the “pinwheel experiment” that has been studied by many investigators, but never before using the anisotropic bidomain model. 

Fig. 14 from Lindblom et al. (2000).

We found a variety of behaviors, depending on the direction of the S1 wave front, the polarity of the S2 stimulus, and the time between S1 and S2, known as the coupling interval (CI). In some cases, we induced a figure-of-eight reentrant circuit: an arrhythmia consisting of two spiral waves, one rotating clockwise and the other counterclockwise. In other cases, we induced quatrefoil reentry: an arrhythmia consisting of four spiral waves (see my previous post for more about the difference between these two behaviors).

I began working on these calculations in the winter of 1999, shortly after I arrived at Oakland University as an Assistant Professor. The photograph below is of a page from my research notebook on March 5 showing initial results, including my first observation of quatrefoil reentry in the pinwheel experiment (look for “Quatrefoil!”).

The March 5, 1999 entry from my research notebook,
showing my first observation of quatrefoil reentry
induced during the pinwheel experiment.

A few weeks later I got a call from my friend Natalia (see my previous post about an earlier collaboration with her). She was organizing a session for the IEEE Engineering in Medicine and Biology Society conference, to be held in Atlanta that October, and asked me to give a talk. We got to chatting and she started to describe simulations she and Lindblom were doing. They were the same calculations I was analyzing! I told her about my results, and we decided to collaborate on the project, which ultimately led to our Journal of Cardiovascular Electrophysiology paper.

Our article was full of beautiful color figures showing the different types of arrhythmias. Below is a photo of two pages of the article. Those familiar with my previous publications will notice that the color scheme representing the transmembrane potential is different than what I usually used. Lindblom and Trayanova had their own color scale, and we decided to adopt it rather than mine. One of the figures was featured on the cover of the March, 2000 issue the journal. Lindblom made some lovely movies to go along with these figures, but they’re now lost in antiquity. I later discovered that a simple cellular automata model could reproduce many of these results (see my previous post for details).

Two pages from Lindblom et al. (2000),
showing some of the color figures.

The editor asked Art Winfree to write an editorial to go along with our article (see my previous post about Winfree). I especially like his closing remarks.

This is clearly a landmark event in cardiac electrophysiology at the end of our century. It is sure to have major implications for clinical electrophysiologic work and for defibrillator design.

In retrospect, he was overly optimistic; the paper was an incremental contribution, not a landmark event of the 20^th century. But I appreciated his kind words.