The Gompertz Mortality Function

In Section 2.4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss exponential decay with a variable rate. If the rate is constant, the fraction of a population remaining after a time t decays exponentially. This is not a good way to estimate the lifespan of humans, because as we age the likelihood of death increases. A simple model is to assume that the mortality rate increases exponentially, leading to the Gompertz mortality function. IPMB explores this behavior in a homework problem.

Problem 15. When we are dealing with death or component failure, we often write Eq. 2.17 in the form y(t) = y₀ exp[-∫₀^t m(t') dt'] and call m(t) the mortality function. Various forms for the mortality function can represent failure of computer components, batteries in pacemakers, or the death of organisms. (This is not the most general possible mortality model. For example, it ignores any interaction between organisms, so it cannot account for effects such as overcrowding or a limited supply of nutrients.)

(a) For human populations, the mortality function is often written as m(t) = m₁e ^−b₁t + m₂ + m₃e ^+b₃t . What sort of processes does each of these terms represent?

(b) Assume that m₁ and m₂ are zero. Then m(t) is called the Gompertz mortality function. Obtain an expression for y(t) with the Gompertz mortality function. Time t_max is sometimes defined to be the time when y(t) = 1. It depends on y₀. Obtain an expression for t_max.

I won’t solve this problem for you (after all, it’s your homework problem). Instead, I’ll examine this behavior in a different way. First, let’s recast the governing differential equation in terms of dimensionless variables. Let p(t) = y(t)/y₀ be the fraction surviving after time t, where y₀ is the initial number at t = 0. Also, define a dimensionless time scale as T = m₃t, and a dimensionless ratio of rates as X = b₃/m₃. The differential equation governing p(T) is then

dp/dT = - exp(XT) p

where p = 1 at T = 0. This form of the equation shows that, aside from scale factors, the behavior depends only on X.

The homework problem asks you to find an analytical expression for p(T). This is a valuable exercise, but you can also learn about the behavior by solving for p(T) numerically. The figure below shows p(T) for several values of X, calculated using Euler’s method. If the increase in mortality is slow compared to the decay of p (that is, X is much less than 1), the decay is approximately exponential (the red X=0 curve). However, if X is large the decay starts exponentially (for T less than about 0.1 the curves in the figure are all nearly equal) but then accelerates as the rate grows.

The surviving fraction p as a function of time T, for different values of X.

An exponential decay of mortality was first analyzed by Benjamin Gompertz (1779-1865), an English mathematician and actuary. His 1825 article “On the Nature of the Function Expressive of the Law of Human Mortality” helped establish two fields of study: actuarial science and the biology of aging. Thomas Kirkwood’s 2015 paper describes Gompertz’s life and work. The title and abstract are below.

Deciphering death: a commentary on Gompertz (1825) ‘On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies’

In 1825, the actuary Benjamin Gompertz read a paper, “On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies,” to the Royal Society in which he showed that over much of the adult human lifespan, age-specific mortality rates increased in an exponential manner. Gompertz’s work played an important role in shaping the emerging statistical science that underpins the pricing of life insurance and annuities. Latterly, as the subject of ageing itself became the focus of scientific study, the Gompertz model provided a powerful stimulus to examine the patterns of death across the life course not only in humans but also in a wide range of other organisms. The idea that the Gompertz model might constitute a fundamental ‘law of mortality’ has given way to the recognition that other patterns exist, not only across the species range but also in advanced old age. Nevertheless, Gompertz’s way of representing the function expressive of the pattern of much of adult mortality retains considerable relevance for studying the factors that influence the intrinsic biology of ageing.

The Goiania Accident

Thirty years ago this week (September 13, 1987) a cesium-137 radiotherapy unit was taken from a abandoned hospital in Goiania Brazil, triggering a tragic radiological accident. Below I reproduce part of the executive summary of a report about this accident published in 1988 by the International Atomic Energy Agency.

It is now known that at about the end of 1985 a private radiotherapy institute, the Institute Goiano de Radioterapia in Goiania, Brazil, moved to new premises, taking with it a cobalt-60 teletherapy unit and leaving in place a caesium-137 teletherapy unit without notifying the licensing authority as required under the terms of the institute's licence. The former premises were subsequently partly demolished. As a result, the caesium-137 teletherapy unit became totally insecure. Two people entered the premises and, not knowing what the unit was but thinking it might have some scrap value, removed the source assembly from the radiation head of the machine. This they took home and tried to dismantle.

In the attempt the source capsule was ruptured. The radioactive source was in the form of caesium chloride salt, which is highly soluble and readily dispersible. Contamination of the environment ensued, with one result being the external irradiation and internal contamination of several persons. Thus began one of the most serious radiological accidents ever to have occurred.

After the source capsule was ruptured, the remnants of the source assembly were sold for scrap to a junkyard owner. He noticed that the source material glowed blue in the dark. Several persons were fascinated by this and over a period of days friends and relatives came and saw the phenomenon. Fragments of the source the size of rice grains were distributed to several families. This proceeded for five days, by which time a number of people were showing gastrointestinal symptoms arising from their exposure to radiation from the source.

The symptoms were not initially recognized as being due to irradiation. However, one of the persons irradiated connected the illnesses with the source capsule and took the remnants to the public health department in the city. This action began a chain of events which led to the discovery of the accident. A local physicist was the first to assess, by monitoring, the scale of the accident and took actions on his own initiative to evacuate two areas. At the same time the authorities were informed, upon which the speed and the scale of the response were impressive. Several other sites of significant contamination were quickly identified and residents evacuated.

The report then addresses the health consequences of the radiation exposure.

Shortly after it had been recognized that a serious radiological accident had occurred, specialists — including physicists and physicians — were dispatched from Rio de Janeiro and Sao Paulo to Goiania. On arrival they found that a stadium had been designated as a temporary holding area where contaminated and/or injured persons could be identified. Medical triage was carried out, from which 20 persons were identified as needing hospital treatment.

Fourteen of these people were subsequently admitted to the Marciho Dias Naval Hospital in Rio de Janeiro. The remaining six patients were cared for in the Goiania General Hospital. Here a whole body counter was set up to assist in the bioassay programme and to monitor the efficacy of the drug Prussian Blue, which was given to patients in both hospitals to promote the decorporation of caesium. Cytogenetic analysis was very helpful in distinguishing the severely irradiated persons from those less exposed who did not require intensive medical care…

Four of the casualties died within four weeks of their admission to hospital. The post-mortem examinations showed haemorrhagic and septic complications associated with the acute radiation syndrome. The best independent estimates of the total body radiation doses of these four people, by cytogenetic analysis, ranged from 4.5 Gy to over 6 Gy. Two patients with similar estimated doses survived…. 

Cesium-137 is a notorious radioactive isotope that has been released in many nuclear accidents. It undergoes beta decay to metastable barium-137m, with an average beta energy of 512 keV and a half-life of about 30 years. ^137mBa has a half-life of 153 seconds and decays to ¹³⁷Ba by emitting a 662 keV gamma ray.

In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of radiation exposure. Typical background exposures are a few mSv per year (the unit of a sievert, Sv, is related to a gray, Gy, by multiplying by a dimensionless factor called the relative biological effectiveness; for ¹³⁷Cs the decays are all beta and gamma, this factor is about one, and we can take the sievert and gray to be the same). Typically about 5 Sv is a fatal dose.

For those of you who would prefer to learn visually, below is a video about the Goiania accident.

Anode Break Excitation

Problem 57 in Chapter 6 of Intermediate Physics for Medicine and Biology analyzes anode break excitation.

Problem 57. When a squid nerve axon is hyperpolarized by a stimulus (the transmembrane potential is more negative than resting potential) for a long time and then released, the transmembrane potential drifts back towards resting potential, overshoots v_r and becomes more positive than v_r, and eventually reaches threshold and fires an action potential. This process is called anode-break excitation: anode because the membrane is hyperpolarized, and break because the excitation does not occur until after the stimulus ends. Modify the program in Figure 6.38 [to solve the Hodgkin-Huxley equations], so that the stimulus lasts 3 ms, and the stimulus strength is −0.15 A m⁻². Show that the program predicts anode break stimulation. Determine the mechanism responsible for anode break stimulation. Hint: pay particular attention of the sodium inactivation gate (the h gate). You may want to plot h versus time to see how it behaves.

Anode break is interesting because it is an unexpected, peculiar behavior. I first learned about anode break in Hodgkin and Huxley’s Nobel Prize-winning paper “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve” (Journal of Physiology, 117:500–544, 1952). They write:

Anode break excitation. Our [squid] axons with the long electrode in place often gave anode break responses at the end of a period during which current was made to flow inward through the membrane. The corresponding response of our theoretical model was calculated for the case in which a current sufficient to bring the membrane potential to 30 mV above [Hodgkin and Huxley used an unusual sign convention, in which a transmembrane potential “above” rest means hyperpolarization] the resting potential was suddenly stopped after passing for a time long compared with all the time constants of the membrane. To do this, eqn. (26)

was solved with I = 0 and the initial conditions that V = + 30 mV, and m, n and h [gates opening and closing the sodium and potassium channels] have their steady state values for V = + 30 mV, when t = 0. The calculation was made for a temperature of 6 3° C. A spike resulted, and the time course of membrane potential is plotted in Fig. 22A. A tracing of an experimental anode break response is shown in Fig. 22B; the temperature is 18-50 C, no record near 6° being available. It will be seen that there is good general agreement. (The oscillations after the positive phase in Fig. 22B are exceptionally large; the response of this axon to a small constant current was also unusually oscillatory as shown in Fig. 23.)

The basis of the anode break excitation is that anodal polarization decreases the potassium conductance and removes inactivation [of the sodium channel]. These effects persist for an appreciable time so that the membrane potential reaches its resting value with a reduced outward potassium current and an increased inward sodium current. The total ionic current is therefore inward at V = 0 and the membrane undergoes a depolarization which rapidly becomes regenerative.

Russ Hobbie and I have prepared a solution manual for IPMB that we distribute to instructors. Below is a sample from the solution manual for Problem 57 about anode break excitation. We introduce each homework question by a sentence or two explaining why the problem is important. If you are an instructor—Russ and I will ask you to verify this—and would like a copy of the solution manual, contact us by email.

6.57* Sometimes the true power of a mathematical model becomes evident when it correctly predicts unexpected, odd behavior. In this example, students use numerical computations to show that the Hodgkin-Huxley model predicts anode break excitation.

The plot shows the transmembrane potential as a function of time for anode break stimulation. A stimulus of −0.15 A m⁻² lasts from 0.5 to 3.5 ms. The action potential fires about 6 ms after the end of the stimulus.

Anode break excitation.

The mechanism for anode break stimulation can be understood from the plots of the gate variables. During the hyperpolarizing stimulus, the h-gate opens to a value of about 0.8, which is higher than its resting value of about 0.6. After the stimulus ends, the h-gate decreases, but very slowly. Once the transmembrane potential returns to rest (about t = 8 ms), the sodium current is larger than at rest because of the still large value of h. This causes the membrane to further depolarize, until it reaches threshold and fires an action potential. The closing of the n-gate during the hyperpolarizing stimulus also contributes to the anode break mechanism, but because the n-gate is slightly faster than the h-gate, the h-gate provides the main effect. Note that the stimulus must be long enough so the h-gate has time to open. Brief stimuli will not work well.

Hodgkin and Huxley observed anode break excitation in their 1952 paper.

I’m not surprised that the Hodgkin-Huxley model correctly describes voltage clamp data from the squid axon; it was designed to do that and the model parameters were fit to the voltage clamp data. Moreover, I’m not too surprised that the model correctly predicts the action potential; the purpose of Hodgkin and Huxley’s research was to understand nerve excitation and conduction. But I am surprised that the model is so good that it can reproduce oddball behavior such as anode break excitation. That’s impressive!

Finally, anode break excitation in nerves is very different from anode break excitation in cardiac tissue. That is another story.

David Goodsell

The Machinery of Life,
by David Goodsell.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I recommend the book The Machinery of Life by David Goodsell. I have mentioned Goodsell several times in this blog (see, for example, here and here). Today, I’ll tell you more about him, and show you some of his artwork (at his website, he has a few illustrations available for use on the internet). For instance, Russ and I discuss the bacterium E. coli several times in IPMB. Below is Goodsell’s illustration of it.

E. coli, by David Goodsell.

Atomic Evidence,
by David Goodsell.

Last year, Goodsell published Atomic Evidence: Seeing the Molecular Basis of Life. In the introduction, he writes

In this book, I will take an evidence-based approach to current knowledge about the structure of biomolecules and their place in our lives, inviting us to explore how we know what we know and how current gaps in knowledge may influence our individual approach to the information. The book is separated into a series of short essays that present some of the foundational concepts of biomolecular science, with many examples of the molecules that perform the basic functions of life.

In particular, I recommend his pictures of insulin in action (his Fig. 16.1), of a nerve synapse (Fig. 19.10), and of a poliovirus neutralized by antibodies (Fig. 21.1). His series of illustrations of human immunodeficiency virus are stunning. Below is a picture of HIV (boooo!) in blood; the red y-shaped things attacking its surface are antibodies (yay!!!).

HIV attacked by antibodies, by David Goodsell.

Often IPMB mentions red blood cells. Below is Goodsell's illustration of part of a red blood cell (bottom left, red) in blood. There’s a lot more stuff floating in the blood than I expected.

A red blood cell, by David Goodsell.

If you want to learn more about David Goodsell, I recommend these two videos, where you can hear him describe how he creates his lovely artwork.

Tenth Anniversary of this Blog About Intermediate Physics for Medicine and Biology

Intermediate Physics for
Medicine and Biology.

This week marks the tenth anniversary of this blog dedicated to the textbook Intermediate Physics for Medicine and Biology. I posted the first entry on Tuesday, August 21, 2007. Soon, I started posting weekly on Friday mornings, and I have been doing so now for ten years.

The blog began shortly after the publication of the 4th edition of IPMB, and continued through the 5th edition. Although the initial posts were brief, they soon become longer essays. If you look at the blog website under “labels” you will find several generic types of posts, such as book reviews, obituaries, and new homework problems. My personal favorites are called…er…“personal favorites.” These include Trivial Pursuit IPMB (a great game for a hot August night with nothing to do), Strat-O-Matic Baseball (because I love to write about myself), Physics of Phoxhounds (I’m a dog lover), The Amazing World of Auger Electrons (I think my cannon-ball/double-canister artillery analogy is clever), My Ideal Bookshelf (which provided the cover picture for the IPMB’s Facebook page), Aliasing (containing a lame joke based on The Man Who Shot Liberty Valance), IPMB Tourist (to help with your vacation plans), The leibniz (a quixotic attempt by John Wikswo and me to introduce a new unit equal to a mole of differential equations), The Rest of the Story (Paul Harvey!), and Myopia (because I love that quote from Mornings on Horseback).

I want this blog to be useful to instructors and students using IPMB in their classes. Although I sometimes drift off topic, they all are my target audience. If you look at posts labeled “Useful for Instructors” you’ll find tips about teaching at the intersection of physics and biology. Instructors should also visit the book’s website, which includes useful information such as the errata and downloadable game cards for Trivial Pursuit IPMB. Instructors can email Russ Hobbie or me about getting a copy of the IPMB solution manual (sorry students; we send it to instructors only).

How much longer will I keep writing the blog? I don’t know, but I don’t expect to stop any time soon. I enjoy it, and I suspect the blog is helpful for instructors and students. I know the blog has only a handful of readers, but their quality more than makes up for the quantity.

Enjoy!