Friday, November 17, 2017

William Bialek, Winner of the 2018 Max Delbruck Prize in Biological Physcis

William Bialek is this year’s winner of the American Physical Society’s 2018 Max Delbruck Prize in Biological Physics “for the application of general theoretical principles of physics and information theory to help understand and predict how biological systems function across a variety of scales, form molecules and cells, to brains and animal collectives.”

Bialek is author of the textbook Biophysics: Searching for Principles. When preparing this blog post, I checked out this book from the Oakland University Library and glanced through it. It is different from Intermediate Physics for Medicine and Biology in many ways. For instance, it is a graduate text, aimed at grad students in physics, whereas IPMB is targeted at undergraduates who have had a year of introductory physics and a year of calculus. Biophysics emphasizes events at the molecular scale, while in the preface to IPMB Russ Hobbie and I write "molecular biophysics has been almost completely ignored: excellent texts already exist, and this is not our area of expertise." One of those excellent texts would be Bialek's Biophysics.

I particularly enjoyed Bialek's introduction, which is a readable and personal account of his experiences at the intersection of physics and biology. At one point he lists a series of questions that anyone interested in applying physics to biology should think about:
  • "Where is the boundary between physics and biology?
  • Is biophysics really physics or just the application of methods from physics to the problems of biology?
  • My biologist friends tell me that 'theoretical biology' is nonsense [Yikes!], so what would theoretical physicists be doing if they got interested in this field?
  • In the interaction between physics and biology, what happens to chemistry?
  • How much biology do I need to know to make progress?
  • Why do physicists and biologists seem to be speaking such different languages?
  • Can I be interested in biological problems and still be a physicist, or do I have to become a biologist?"
The entire introduction (indeed, the entire book) is an attempt to answer these questions. If you don't have access to the book, you could read his similarly themed article available on the physics archive: Perspectives on Theory at the Interface of Physics and Biology.

Like IPMB, Bialek's book has many homework problems. Readers of this blog know that I enjoy reducing biological and medical physics principles down to homework problems, so this footnote from Bialek's introduction resonated with me:
"In some sections I found it difficult to formulate manageable problems. I worry that this reflects poorly on my understanding."
For those of you who prefer video over text, below is a three-part interview with Bialek from the International Centre for Theoretical Physics

Also, here is a video of Bialek giving the 2017 Buhl Lecture "The Physics of Life: How Much Can We Calculate?"


Friday, November 10, 2017


The Intermediate Physics for Medicine and Biology Facebook Group has now reached 150 members.

Yes, IPMB has a Facebook group. I use it to circulate blog posts every Friday morning, but I occasionally share other posts of interest to readers of IPMB. The group photo is my Ideal Bookshelf picture highlighting books about physics applied to medicine and biology.

Group members include my family (including my dog Suki Roth, who has her own Facebook Page) and former students. But members I don't know come from countries all over the world, including:
In particular, many members are from India and Pakistan.

I am amazed and delighted to have members from all over the world. I don't know if universities teach classes based on IPMB in all these places, or if people just stumble upon the group.

The IPMB Facebook group welcomes everyone interested in physics applied to medicine and biology. I am delighted to have you. And for those who are not yet members, just go to Facebook, search for "Intermediate Physics for Medicine and Biology," and click "Join Group." Let's push for 200 members!

Friday, November 3, 2017

Countercurrent Transport of Oxygen in the Gills of a Fish

In Section 5.8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss countercurrent transport.
“The countercurrent principle is found in the renal tubules (Hall 2011, p. 309; Patton et al. 1989, p. 1081), in the villi of the small intestine (Patton et al., 1989, p. 915), and in the lamellae of fish gills (Schmidt-Nielsen 1971, p. 45). The principle is also used to conserve heat in the extremities—such as people’s arms and legs, whale flippers, or the leg of a duck. If a vein returning from an extremity runs closely parallel to the artery feeding the extremity, the blood in the artery will be cooled and the blood in the vein warmed. As a result, the temperature of the extremity will be lower and the heat loss to the surroundings will be reduced.”
In a homework problem, Russ and I ask the student to analyze a countercurrent heat exchanger. In this blog post, I present a new exercise studying countercurrent oxygen exchange in fish gills.
Problem 19 ½. Fish use countercurrent transport to increase uptake of oxygen in their gills. Consider a capillary extending from x = 0 to x = L, with blood flowing in the positive x direction. Blood entering the capillary has a low oxygen concentration that we take as zero. Seawater flowing outside the capillary has an oxygen concentration of [O2] where it enters the gill. Consider the case when |ain|=|aout|=a in Eq. 5.24. The goal is to calculate the oxygen concentration in the blood, Cin, and the oxygen concentration in the seawater, Cout, as functions of x, and in particular to determine the blood oxygen concentration at the end of the gill where it reenters the fish's body, Cin(L).

a) Consider the case when the seawater flows in the same direction as the blood. Draw a picture illustrating this case. Derive expressions for Cin(L) and Cout(x) in terms of [O2], a, and L. Plot qualitatively Cin(x) and Cout(x) versus x when aL is greater than 1. 

b) Now consider the case of countercurrent transport, when the seawater flows in the opposite direction as the blood. Draw a picture, derive expressions, and make plots.

c) Which case results in the highest oxygen concentration in the blood when it leaves the gill and enters the fish’s body?

d) Explain why countercurrent transport is so effective using words instead of equations.
The solution is given at the bottom of this blog post. You can probably guess that countercurrent transport is more efficient for absorbing oxygen. In one of my favorite books, How Animals Work, Knut Schmidt-Nielsen describes countercurrent transport in the fish gill. His description would be an excellent answer to part d) of the new homework problem.
“In the lamellae of the fish gill, water and blood flow in opposite directions (figure 27). As a consequence, the blood, just as it is about to leave the gill, encounters the incoming water which still has all its oxygen; that is, the oxygen tension of the blood will approach that of the water before any oxygen has been removed. At the other end of the lamellae, the water that is about to exit encounters venous blood, so that, even though much oxygen has already been removed from the water, more can still be taken from it by the blood. As a result of this arrangement, fish may extract as much as 80 to 90% of the oxygen in the water, an efficiency which could not easily be achieved without a countercurrent flow.”

Friday, October 27, 2017


A few months ago I started an account on Twitter. It is not a personal account, but instead is for the Oakland University Center for Biomedical Research, which I direct. If you are interested, the Twitter handle is @OaklandUniv_CBR. Most of my tweets are useful for faculty and students at OU: announcing seminars, congratulating principal investigators for their new grants, highlighting accomplishments of students, and that sort of thing. However, I follow a lot of accounts that are related to Intermediate Physics for Medicine and Biology. If you are on Twitter, you might like these:
If you use Twitter and know some accounts that readers of IPMB should follow, mention them in the comments.


Friday, October 20, 2017

Galvani’s Spark: The Story of the Nerve Impulse

I recently finished a wonderful history of neurophysiology. Galvani’s Spark: The Story of the Nerve Impulse, by Alan McComas, covers several topics that Russ Hobbie and I discuss in Intermediate Physics for Medicine and Biology, such as the nerve action potential, patch clamping, and the structure of the potassium channel. While I enjoyed these parts of the book, I particularly liked the earlier history about scientists like Charles Sherrington and Edgar Adrian.

One of my favorite chapters is about Keith Lucas, an English physiologist who worked at Cambridge. Lucas showed that when he increased the strength of an electrical stimulus to a muscle, the response increased in discrete steps. From this, he deduced that each fiber responded in an all-or-none way, and the increase in response with stimulus strength resulted from recruiting more fibers. Lucas had the skills of an engineer as well as a biologist, and would make his own equipment to record action potentials. He probably would have made many more discoveries, except that during World War I he left academic research to work for the military. McComas describes the work well.
"Living in a small wooden hut and rising a 4 in the morning, Lucas grappled with a number of problems that beset the pilots of the early flying machines. One was to improve a bombsight, and another to eliminate the unreliability of the pilot’s compass as the plane was made to turn. Once again, just as it had in the Physiological Laboratory in Cambridge, Lucas’s flair for analysis and design, and for constructing equipment himself, served him in good stead, and the problems were solved. To gain first-hand experience of a particular problem, and to see if his solution was effective, Lucas would fly himself, initially as a passenger and then as a trained pilot. For this, he transferred to the Central Flying School at Upavon in Wiltshire."
Tragically, in October 1916 he was killed in a midair collision between two planes.

Another interesting chapter was about three American neurophysiologists—Joseph Erlanger, Herbert Gasser, and George Bishop—who pioneered the use of an oscilloscope for recording action potentials. Gasser is portrayed as saintly, but Erlanger doesn’t come across as an attractive figure. At one point, Bishop published a paper without passing it by Erlanger first, and Erlanger threw a fit.
"Erlanger’s violent temper was well known to this family, but at work it had usually been controlled. Now, however, it was unleashed in its full fury. Bishop was sent for, an accusatory letter written, and then came expulsion from the physiology department."
In 1944, Erlanger and Gasser were awarded the Nobel Prize in Physiology or Medicine, but Bishop was not included. McComas disagrees with this decision, describing Bishop as “the man who should have shared the Nobel Prize with Gasser and Erlanger.”

Another interesting story is of the debate between Henry Dale and John Eccles about the nature of the nerve-muscle synapse. Dale favored a chemical synapse, with acetylcholine as the neurotransmitter. Eccles championed a synapse having a direct electrical connection. Apparently they engaged in a heated battle at the 1935 Cambridge meeting of the Physiological Society. Dale won this battle, and shared the 1936 Nobel Prize with Otto Loewi for their “discoveries relating to chemical transmission of nerve impulses”. Eccles, after a difficult time finding his scientific home, eventually made landmark discoveries about neural transmission in the central nervous system, and won his own Nobel Prize.

American Kenneth Cole is a complex character. On the one hand, Cole was generous in sharing his ideas with the young Alan Hodgkin when Hodgkin visited his Woods Hole laboratory in 1938. Yet, McComas writes
"Respected and admired as a pioneer in the study of the nerve impulse, the recipient of medals and honorary degrees, Kenneth Cole was not content. This kind and unassuming man continued to resent the fact that his preparation and his voltage clamp had been used by Hodgkin without due acknowledgement."
He adds this interesting insight: “Unlike Cole, perpetually bedeviled by problems of one kind or another, success always seemed to follow Hodgkin.”

Another scientist depicted almost tragically is the Spaniard Rafael Lorente de No. McComas says
"The publication of the Hodgkin-Huxley papers had been a bitter blow. Having labored for 10 years on his monumental study of peripheral nerve, Lorente now found that it was largely irrelevant, or, even worse, wrong in its main conclusions….Yet he refused to capitulate, let alone to walk away from a battle that only he wished to right. He would appear at international meetings, rejecting the general applicability of the Hodgkin-Huxley findings, and referring dismissively to the ‘so-called sodium hypothesis.’ It was a sad end to a career that had been so full of promise."
The climax of the book is the story of Alan Hodgkin and Andrew Huxley developing their model of the squid giant axon, a model described in Chapter 6 of IPMB. Here is my favorite passage:
"It was the intention of Hodgkin and Huxley to use the Cambridge University computer—the only computer in the entire university—to carry out the formidable amount of calculation involved, but the machine was undergoing major modifications at the time and would not be available for six months. Huxley then suggested to Hodgkin that he, Huxley, attempt to solve them himself, with the aid of his hand-operated Brunsviga calculating machine. It was an extraordinarily ambitious undertaking. The calculator, rather like an old-fashioned cash register, required that the data were entered by moving small levers in slots to appropriate positions beside numerically inscribed wheels. A handle at the side of the machine would then be turned so many times in one direction or another, and the results read off on the numbered wheels. These results would then have to be written down on paper, before proceeding to the next stage of the calculation. And these steps had to be repeated over and over again. The reconstruction of the action potential required numerical integration, and a complete set of data had to be produced for each small time interval. To calculate a complete ‘run’ required 8 hours of intense mental and physical activity. It has been said that, in all the calculations, more than a million separate steps were involved. It is doubtful if anyone other than Huxley could have brought it off."
If you are looking for a history of the early years of neuroscience, I highly recommend Galvani’s Spark. To tell you the truth, when I started the book I didn’t think it would be this good. Enjoy!

Friday, October 13, 2017

John Clark, Biomedical Engineer (1936-2017)

John W. Clark passed away on August 6, in Houston, Texas. He was a professor of Engineering at Rice University for 49 years.

When I was a graduate student at Vanderbilt University in the 1980s, I was influenced by the papers of Robert Plonsey and his graduate student Clark. They calculated the extracellular electrical potential outside a nerve axon from the transmembrane action potential by expressing the transmembrane potential in terms of its Fourier transform, and then using Bessel functions to calculate the Fourier transform of the extracellular potential. Russ Hobbie and I outline this technique in Problem 30 of Chapter 7 in Intermediate Physics for Medicine and Biology. James Woolsey, my PhD advisor John Wikswo, and I used a similar method—inspired by Clark and Plonsey’s work—to calculate the magnetic field of a nerve axon (see Problem 16 of Chapter 8 in IPMB). Moreover, my first work on the bidomain model of the heart was analyzing cylindrical strands of cardiac tissue using methods that were an extension of Clark and Plonsey’s work. If I were to list the articles that had the biggest impact on my own work, near the top of that list would be Clark and Plonsey’s 1968 paper in the Biophysical Journal (Volume 8, Pages 842-864).

Clark graduated from Case Western Reserve University at about the time this Biophysical Journal  paper was published, and joined the faculty at Rice. Rarely do you see a professor's career span half a century at one institution. He was a Life Fellow of the Institute of Electrical and Electronics Engineers (IEEE) "for contributions to modeling in electrophysiology, and cardiopulmonary systems." He played a role in establishing the field of biomedical engineering, and served as President of the IEEE Engineering in Medicine and Biology Society.

To learn more about Clark and his contributions, see obituaries here, here and here.

Friday, October 6, 2017

Implantable Biocompatible Lasers!

Russ Hobbie and I discuss lasers several times in Intermediate Physics for Medicine and Biology. For instance, Homework Problem 6 in Chapter 14 is:
Problem 6. The left side of Fig. 14.1 shows the emission of a photon during a transition from an initial state with energy Ei to a final one with energy Ef . Usually the Boltzmann factor ensures that the population of the initial state is less than the final state. In some cases however, when the initial state is metastable, one can create a population inversion. Photons with energy corresponding to the energy difference EiEf can produce stimulated emission of other photons with the same energy, a type of positive feedback. Lasers work on this principle. Suppose a laser is made using two states having an energy difference of 1.79 eV. What is the wavelength of the output light? What color does this correspond to? Lasers have many uses in medicine (Peng et al.2008).
I thought I was familiar with most biomedical applications of lasers, until I read the recent article by Tami Freeman in
Sep 27, 2017
Implantable biolasers line up for therapy, monitoring
Biolasers -- miniature implantable lasers made of biocompatible materials -- are the subject of increasing research interest. Such lasers, which offer narrow emission linewidth, high coherence and high intensity, could enable novel imaging, diagnostic and therapeutic applications, as well as real-time physiological monitoring of temperature or glucose levels...
Implantable lasers made of biocompatible materials? Wow! The article concludes
...The researchers concluded that the availability of biocompatible and biodegradable microlasers made from materials approved for medical use or substances already present in the human body may open new opportunities for light-based diagnostics and therapies, as well as basic research.

"One of the first applications could be sensing and diagnostics," [Marjaž] Humar [from the Jožef Stefan Institute] told medicalphysicsweb. "For example, the biolasers could be functionalized to be sensitive to glucose. A person having these lasers implanted into the skin would simply measure their glucose level by reading the laser output with a small optical reader."
Not only is this article fascinating, but also it reminds me: have you been keeping up with medicalphysicsweb? Anyone interested in medical physics should read it regularly. Medicalphysicsweb is a community website from IOP [Institute of Physics] publishing. The English IOP is similar to the USA’s American Physical Society, supporting physics education, research, and industry. Tami Freeman does a superb job editing medicalphysicsweb. To hear more about her story, see

Friday, September 29, 2017

James Mattiello, Medical Physicist (1958-2017)

James Mattiello passed away on March 19, 2017, at the age of 59, in Utica, Michigan. Jim was a friend of mine from when we both worked at the National Institutes of Health, where he contributed to the development of a magnetic resonance imaging technique called Diffusion Tensor Imaging. He was the first graduate of the Oakland University Medical Physics PhD Program, which I now direct. When I was at NIH, I had never heard of Oakland University until Jim mentioned it as his alma mater. Little did I know that I would have a 20-year career at OU, teaching and doing research.

Jim performed his PhD research with Prof. Fred Hetzel, and graduated with his PhD in 1987. His dissertation described an in vivo experimental investigation on the interaction between photodynamic therapy and hyperthermia. A copy of his dissertation sits in our Physics Department office, and I often show it to prospective students because it is the thickest dissertation on the shelf, over 480 pages. Hetzel, Norm Tepley, Michael Chopp, and Abe Liboff formed the dissertation committee (I didn't arrive at OU until ten years later). Three journal articles resulting from this work are:
Mattiello J, Hetzel FW (1986) Hematoporphyrin-derivative optical-fluorescence-detection instrument for localization of bladder and bronchous carcinoma in situ. Review of Scientific Instruments 57:2339-2342.
Mattiello J, Hetzel F, Vandenheede L (1987) Intratumor temperature measurements during photodynamic theorapy. Photochemistry and Photobiology 46:873-879.
Mattiello J, Evelhoch JL, Brown E, Schaap AP, Hetzel FW (1990) Effect of photodynamic therapy on RIF-1 tumor metabolism and blood flow examined by 31P and 2H NMR spectroscopy. NMR in Biomedicine 3:64-70.
A news article about Jim's research appeared in the Spring 1984 issue of The Oakland University Magazine (above left).

After graduation, Jim obtained a fellowship to work at the intramural program of the National Institutes of Health in Bethesda, Maryland, where I first met him. Below I quote from an NIH oral history interview with Peter Basser, which describes how Basser, Denis LeBihan, and Jim developed Diffusion Tensor Imaging in the early 1990s.
“Well actually this was an amazing story too, because there’s so many people involved and activities that had to be done in order to bring this from bench to bedside. So the first thing is Denis and I started corresponding, and Jim Mattiello then, who was working with Denis and who was also working in our program [Biomedical Engineering and Instrumentation Program], was a little frustrated with some of the projects he was working on and decided that he wanted to start working with us. So I was excited about that because Jim had a technical background in MRI, he had been working in the area for a few–maybe a year and a half at that point, and he would provide a lot of experimental help which I really couldn’t provide because my knowledge at that point of the NMRI [Nuclear Magnetic Resonance Imaging] hardware and sequences and things was almost nonexistent. And so we started doing diffusion experiments with water. The first thing that we – in pork loin – the first thing that we started doing was – Denis got us some magnetic time down at the NMRI center and we started to – since we had this mathematical framework that related the signal that we measured to the diffusion tensor the first thing that you want to do is show that the diffusion tensor in water is an isotropic tensor, which means that if you look at the diffusion process along any direction that it appears the same and that has a characteristic – a special form when you write it as a tensor and it’s something that if you can’t do that you can’t look at other materials that are more complex."
I can remember the morning when Peter came in to NIH carrying a pork loin from a local grocery store. I asked him why he brought a chunk of raw meat to work, and he told me that he and Jim were going to use it that day in their first DTI experiment on muscle. Later in the oral history interview, Basser describes this experiment.
"we wrote our first abstract describing it [Anisotropic Diffusion Tensor Imaging] at the ISMRM [International Society for Magnetic Resonance in Medicine Conference] I think which we presented in Berlin in 1992, we looked at a sample of pork loin and we showed that we first measured the diffusion tensor for a large region of that pork loin specimen, and then we actually physically rotated that – Jim Mattiello actually physically rotated the pork loin specimen in the magnet. We repeated the experiments, calculated the tensor and we were able to show that the directions that we calculated for the pork loin muscles followed the direction of the rotation that he had applied physically on that sample, so that we were measuring something intrinsic to the tissue. These principle directions that we were able to extract from the diffusion tensor were fundamental to the tissue architecture and were independent of the coordinate system that we made the measurement in, which was really, I think, a very important demonstration then."
Jim is a coauthor on two classic papers about DTI that are widely cited in the medical literature.
Basser PJ, Mattiello J, LeBihan D (1994) MR Diffusion Tensor Spectroscopy and Imaging. Biophysical Journal 66:259-267. (4495 citations in Google Scholar as of 9-23-2017)

Basser PJ, Mattiello J, LeBihan D (1994) Estimation of the Effective Self-Diffusion Tensor From the NMR Spin Echo. Journal of Magnetic Resonance B 103:247-254. (3261 citations)
I know many scientists who have had long and successful careers, but few of them can claim they contributed to a paper with over 4000 citations, a significant achievement (that averages to one citation every other day for over two decades). My most cited article, published about the same time, has only 500 citations, and I consider myself to be a successful scientist. Jim was also the lead author on two related papers.
Mattiello J, Basser PJ, LeBihan D (1994) Analytical Expressions for the B Matrix in NMR Diffusion Imaging and Spectroscopy. Journal of Magnetic Resonance A, 108:131-141. (224 citations)

Mattiello J, Basser PJ, LeBihan D (1997) The B Matrix in Diffusion Tensor Echo-Planar Imaging. Magnetic Resonance in Medicine 37:292-300. (227 citations)
In addition, Jim is listed as an inventor on a key patent for DTI.
Basser PJ, Mattiello JH, LeBihan D. Method and System for Measuring the Diffusion Tensor and for Diffusion Tensor Imaging. US Patent 5,539,310.
Russ Hobbie and I cite the 1994 Biophysical Journal paper and the 1994 Journal of Magnetic Resonance A paper in Intermediate Physics for Medicine and Biology. Our Figure 18.40 is based in part on the pulse sequence he helped developed for DTI. Nowadays Diffusion Tensor Imaging is used to make beautiful maps of fiber tracts in the brain.

Jim spent the later part of his career teaching physics at St. Clair County Community College in Port Huron, Michigan. I last saw him when he returned to Oakland University in 2002 to give a physics colloquium about DTI.

James Mattiello's contributions to magnetic resonance imaging, and specifically to diffusion tensor imaging, have had a lasting impact on the field of medical physics. He will be missed.

Friday, September 22, 2017

The Beautiful Brain: The Drawings of Santiago Ramon y Cajal

An art exhibit titled "The Beautiful Brain: The Drawings of Santiago Ramón y Cajal," which is traveling through museums in the United States and Canada, relates to topics covered in Intermediate Physics for Medicine and Biology. Unfortunately it won't pass through Detroit, but I was able to enjoy the wonderful book that accompanies the exhibit (thank you Oakland University Interlibrary Loan Department!). The introduction begins
Santiago Ramón y Cajal has rightly been credited as the father of modern neuroscience, the study of the structure and function of the brain. Cajal, who lived from 1852 to 1934, was a neuroanatomist who, over the course of five decades, produced more than twenty-nine hundred drawings that reveal the nervous system as we know it today. He studied many aspects of the brain, from the structure of individual neurons…and the connections between them, to the changes that occur in the brain during early life and following injury. He did this by examining thin slices of the brain under a microscope. He treated these slices with chemical stains to highlight different types of brain cells and structures within these cells. Most notably, he used a stain developed by the Italian biologist Camillo Golgi, which colors brain cells a deep, rich black. Cajal improved upon the original formulation of the Golgi stain to obtain exquisite images of neurons.
The introduction then summarizes the contents of the book.
“This book presents eighty of Cajal's original drawings of the brain... Some of these drawings are well known, while others have not been published previously except in Cajal's original scientific papers. Captions accompanying the drawings describe their subject matter and their scientific importance. Two essays focus respectively on Cajal's life and scientific achievements, and his mastery of the art of drawings. A third essay brings us up-to-date, describing modern neuroscience imaging methods that Cajal, undoubtedly, would have appreciated. We hope you enjoy Cajal's vision of the beautiful brain.”
As a teaser, below I present some of Cajal's drawings.

The structure of the retina

Cells of the cerebellum

Purkinje neurons from the cerebellum

A pyramidal neuron in the cerebral cortex

I especially like the last drawing, because it is the one Sheldon was supposed to give to Amy Farrah Fowler for Valentine's Day, but he decided to keep it for himself instead!

Cajal shared the 1906 Nobel Prize in Physiology or Medicine with Golgi,"in recognition of their work on the structure of the nervous system." Below is a photo of Cajal sitting at his microscope. He was a pioneer in photography as well as drawing.

Santiago Ramón y Cajal

Here is the schedule for the exhibit, in case you are lucky enough to have it visit your town.
You can listen to a National Public Radio broadcast about the exhibit here, and read a review of the exhibit here.


Friday, September 15, 2017

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[-∫0t 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 tmax is sometimes defined to be the time when y(t) = 1. It depends on y0. Obtain an expression for tmax.
I won’t solve this problem for you (after all, it's your homework problem). Instead, I will 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.

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.