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.