Murray’s Law

Homework Problem 33 in Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology is about Murray’s law, a relationship describing the radii of branching vessels.

A parent vessel of radius R_p branches into two daughter vessels of radii R_d1 and R_d2. Find a relationship between the radii such that the shear stress on the vessel wall is the same in each vessel. (Hint: Use conservation of the volume flow.) This relationship is called ‘Murray’s Law’. Organisms may use shear stress to determine the appropriate size of vessels for fluid transport [LaBarbera (1990)].

The reference is to

LaBarbera, M. (1990) “Principles of Design of Fluid Transport Systems in Zoology.” Science, Volume 249, Pages 992–1000.

Vital Circuits,
by Steven Vogel.

In his book Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems, Steven Vogel provides a clear and engaging discussion of Murray’s law.

Our problem of figuring the cheapest arrangement of pipes turns out to involve nothing more nor less than calculating the relative dimensions of pipes so that the steepness of the speed gradient at all walls is the same. This calculation was done by Cecil D. Murray, of Bryn Mawr College, back in 1926, and is spoken of, when (uncommonly) it’s mentioned, as “Murray’s law.”

Murray’s law isn’t especially complicated, and anyone with a hand calculator can play around with it (but you can ignore the specifics without missing the present message). The rule is that the cube of the radius of the parental vessel equals the sum of the cubes of the radii of the daughter vessels. If a pipe with a radius of two units splits into a pair of pipes, each of the pair ought to have a radius of about 1.6 units. (To check, cube 1.6 and then double the result—you get about 2 cubed.) The daughters are smaller, but only a little (Figure 5.6). Still, if the parental one eventually divides into a hundred progeny, the progeny do come out substantially smaller, each about a fifth of the radius of the parent. (Their aggregate cross-section area is, of course, greater than the parental one—to be specific, four fold greater.)

The relationship predicts the relative sizes of both our arteries and our veins quite well. It only fails for the very smallest arterioles and capillaries….

It would be indefensibly anthropocentric to suppose that we’re the only creatures to follow Mr. Murray. My friend, Michael LaBarbera (who introduced me to the whole issue) has tested the law on several systems that are very unlike us structurally and functionally, and very distant from us evolutionarily…Murray’s law again proves applicable…

The mechanism … is becoming clear. Without getting into the details, it looks as if the cells lining the blood vessels can quite literally sense changes in the speed gradient next to them. An increase in the speed of flow through a vessel increases the speed gradient at its walls. An increase in gradient stimulates cell division, which would increase vessel diameter as appropriate to offset the faster flow. Neither change in blood pressure nor cutting the nerve supply makes any difference—this is apparently a direct effect of the gradient on synthesis of some chemical signal by the cells. Perhaps the neatest feature of the scheme is that a cell needn’t know anything about the size of the vessel of which it’s a part. As a consequence of Murray’s Law, it can be given the same specific instruction wherever it might be located, a command telling it to divide when the speed gradient exceeds a specific value.

Vogel is a faculty member in the Biology Department at Duke University. He has published several fine books, including Vital Circuits quoted above and the delightful Life in Moving Fluids (Princeton University Press, 1994), both cited in Intermediate Physics for Medicine and Biology.

A Useful Website

While I have many goals when writing this blog (with the top being to sell textbooks!), sometimes I simply like to point out useful websites relevant to readers of the 4th edition of Intermediate Physics for Medicine and Biology. One example is the website of Rob MacLeod, a professor of bioengineering at the University of Utah. MacLeod’s research, like mine, centers on the numerical simulation of cardiac electrophysiology, so we find many of the same topics interesting.

I particularly enjoy his list of Background Links for Rob’s Courses. You will find many books listed, some of which Russ Hobbie and I cite in Intermediate Physics for Medicine and Biology, and some that we don’t cite but should. For example, MacLeod speaks highly of the book Mathematical Physiology by Keener and Sneyd, but somehow Russ and I never reference it. I didn’t know Malmivuo and Plonsey’s book Bioelectromagnetism (which we do cite) is now available online and free of charge. The Welcome Trust Heart Atlas is beautiful, as is the Virtual Heart website. MacLeod’s list of books about “Cardiology and Medicine” look fascinating, with a heavy emphasis on the relevant history and biography. If I start running out of topics for these blog posts, I could probably find a year of material by exploring the sources listed on this page.

If you visit MacLeod's website (and I hope you do), make sure to click on the link “Information on Writing.” I am an admirer of good writing, especially in nonfiction, and am frustrated when presented with a poorly written scientific book or paper. (I review a lot of papers for journals, and often find myself venting and fuming.) My advice to a young scientist is: Learn To Write. Throughout your scientific career you will be judged primarily on your papers and your grant proposals, which are both written documents. Maybe your science is so good that it can overcome poor writing and still impress the reader, but I doubt it. Learn to write.

Bethesda

A couple months ago I went to Bethesda, Maryland to review grant proposals for the National Institutes of Health. They swear us to secrecy, so I can’t divulge any details about the specific research. But I will share a few general observations.

1. Winston Churchill said that “Democracy is the worst form of government except all the others that have been tried.” That sums up my opinion of the NIH review process. There are all sorts of problems with the way we select the best research to fund, but I can’t think of a better way than that used by NIH. Each time I participate, I come away with a great respect for the process. Of course, from the outside the review process can resemble a casino, but I don’t see how you can eliminate some randomness while at the same time keeping the process fair, with wide input, and a focus on the significance and impact of the research.
2. If you are a young biomedical researcher, or hope to be one someday, then you should take advantage of any opportunity to review grant proposals. It is like going to grant writing school. No book, no website, no video, no workshop is more useful for learning how to prepare a proposal. It is a lot of work, but you will gain much, especially the first time or two you do it. However, if you simply are not able to participate in a review panel, then at least watch this video (see below), which is a fairly accurate description of what goes on.
3. After reviewing grant proposals, I am optimistic about the future of the scientific enterprise in the United States, because of all the fascinating and important research being proposed. I am also pessimistic about my chances for winning additional funding, because the competition is so fierce. But, we must soldier on. To quote Churchill again, “Never give in, never give in, never, never, never, never.” So I’ll keep trying.
4. Research is becoming more and more interdisciplinary, and many proposals now come from multidisciplinary teams. Each individual researcher cannot know everything, but they must know enough to understand each other, and to talk to each other intelligently. I believe this is one of the virtues of the 4th edition of Intermediate Physics for Medicine and Biology. It helps bridge the gap between physicists and engineers on the one side, and biologists and medical doctors on the other. The book won’t turn a physicist into a biologist, but it may help a physicist talk to and better appreciate a biologist. This is crucial for performing modern collaborative research, and for obtaining funding to pay for that research. After reviewing all those proposals, I came away proud of our textbook.

We finished our review session a couple hours earlier than anticipated, so I used the time to visit the new Martin Luther King Memorial in Washington, DC. It is just across the tidal basin from the Jefferson Memorial, and the statues of King and Jefferson stare at each other across the water. If you happen to be going to DC soon, prepare yourself for a shock. The beautiful reflecting pool between the Washington Monument and the Lincoln Memorial is now a dried-up, plowed-up mud flat. Apparently they are renovating it. But the other attractions are as beautiful as ever, including the Vietnam Veterans Memorial, the Korean War Veterans Memorial, the National World War II Memorial, and the Franklin Delano Roosevelt Memorial. I even saw one I had somehow missed in previous visits: the George Mason Memorial, near the Jefferson Memorial. All this site seeing was a little bonus after reviewing all those grants (packed into two frantic hours between leaving the review session and reaching the airport).

NIH Peer Review Revealed.

https://www.youtube.com/watch?v=fBDxI6l4dOA&

The Mathematics of Diffusion

The Mathematics of Diffusion,
by John Crank.

Diffusion is one of those topics that is rarely covered in an introductory physics class, but is essential for understanding biology. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss diffusion and its biomedical applications. One of the books we cite is The Mathematics of Diffusion by John Crank. Hard-core mathematical physicists who are interested in biology and medicine will find Crank’s book to be a good fit. Physiologists who want to avoid as much mathematical analysis as possible may prefer to learn their diffusion from Random Walks in Biology, by Howard Berg.
Crank died five years ago this week. Like Wilson Greatbatch, who I discussed in my last blog entry, Crank was one of those scientists who came of age serving in the military during World War Two (Tom Brokaw would call them members the “Greatest Generation”). Crank’s 2006 obituary in the British newspaper The Telegraph states:

John Crank was born on February 6 1916 at Hindley, Lancashire, the only son of a carpenter’s pattern-maker. He studied at Manchester University, where he gained his BSc and MSc. At Manchester he was a student of the physicist Lawrence Bragg, the youngest-ever winner of a Nobel prize, and of Douglas Hartree, a leading numerical analyst.

Crank was seconded to war work during the Second World War, in his case to work on ballistics. This was followed by employment as a mathematical physicist at Courtaulds Fundamental Research Laboratory from 1945 to 1957. He was then, from 1957 to 1981, professor of mathematics at Brunel University (initially Brunel College in Acton).

Crank published only a few research papers, but they were seminal. Even more influential were his books. His work at Courtaulds led him to write The Mathematics of Diffusion, a much-cited text that is still an inspiration for researchers who strive to understand how heat and mass can be transferred in crystalline and polymeric material. He subsequently produced Free and Moving Boundary Problems, which encompassed the analysis and numerical solution of a class of mathematical models that are fundamental to industrial processes such as crystal growth and food refrigeration.

Crank is best known for a numerical technique to solve equations like the diffusion equation, developed with Phyllis Nicolson and known as the Crank-Nicolson method. The algorithm has the advantage that it is numerically stable, which can be shown using von Neuman stability analysis. They published this method in a 1947 paper in the Proceedings of the Cambridge Philosophical Society

Crank, J., and P. Nicolson (1947) “A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type,” Proc. Camb. Phil. Soc., Volume 43, Pages 50–67.

Rather than describe the Crank-Nicolson method, I will let the reader explore it in a new homework problem.

Section 4.8

Problem 24 ½ The numerical approximation for the diffusion equation, derived as part of Problem 24, has a key limitation: it is unstable if the time step is too large. This problem can be avoided using the Crank-Nicolson method. Replace the first time derivative in the diffusion equation with a finite difference, as was done in Problem 24. Next, replace the second space derivative with the finite difference approximation from Problem 24, but instead of evaluating the second derivative at time t, use the average of the second derivative evaluated at times t and t+Δt.
(a) Write down this numerical approximation to the diffusion equation, analogous to Eq. 4 in Problem 24.
(b) Explain why this expression is more difficult to compute than the expression given in the first two lines of Eq. 4. Hint: consider how you determine C(t+Δt) once you know C(t).
The difficulty you discover in part (b) is offset by the advantage that the Crank-Nicolson method is stable for any time step. For more information about the Crank-Nicolson method, stability, and other numerical issues, see Press et al. (1992).

The citation is to my favorite book on computational methods: Numerical Recipes (of course, the link is to the FORTRAN 77 version, which is the edition that sits on my shelf).