## Saturday, June 9, 2012

### Law of Laplace

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I often introduce topics in the homework problems that we don’t have room to discuss fully in the text. For instance, Problem 18 in Chapter 1 asks the reader to derive the Law of Laplace, f = p R, a relationship between the pressure p inside a cylinder, its radius R, and its wall tension f.

 Vital Circuits, by Steven Vogel.
In his book Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems, Steven Vogel explains the physiological significance of the Law of Laplace, particularly for blood vessels.
The wall of the aorta of a dog is about 0.6 millimeters thick, while the wall of the artery leading to the head is only half that. Pressure differences across the wall are the same…, but the aorta has an internal diameter three times as great. The bigger vessel simply needs a thicker wall. An arteriole is 100 times skinnier yet; it wall is fifteen times thinner than that of the artery going to the head. A capillary is eight times still skinnier, with walls another twenty times thinner… A general rule that wall thickness is proportional to vessel diameter is clearly evident, just the relationship expected from Laplace’s law.
In our Homework Problem 18, Russ and I write that “Sometimes a patient will have an aneurysm in which a portion of an artery will balloon out and possibly rupture. Comment on this phenomenon in light of the R dependence of the force per unit length.” The answer (spoiler alert) is explained by Vogel. He first examines what happens when inflating a balloon.
About the same pressure is needed throughout the inflation, except for an extra bit to get started and (if you persist) another extra bit just before the final explosion… Pressure gets more effective in generating tension—stretch—as the balloon gets bigger [p = f/R], automatically providing the extra force needed as the rubber is expanded.
What is the implication for an aneurysm?
Pressure, we noted, is more effective in generating tension in the walls of a big cylinder than in those of a small cylinder. Blow into a cylindrical balloon, and one part of the balloon will inflate almost fully before the remainder expands. Pressure inside at any instant is the same everywhere, but the responsive stretching is curiously irregular…any part partially inflated is easier to inflate further than any part not yet inflated at all.
Thus, the law of Laplace provides insight into aneurysms: just think of a bulge that develops when inflating a cylindrical balloon. Now the question can be turned on its head: why don’t all cylindrical vessels immediately develop aneurysms as soon as pressure is applied? In other words, why are we not all dead, killed by the Law of Laplace? Vogel addresses this point too. “The primary question isn’t why aneurysms sometimes occur, but why they don’t normally happen…Why an arterial wall...behaves in a much friendlier manner.”

Vogel’s answer is that arteries “should surely stretch strangely” (I love the alliteration). The walls of an artery as designed such that
a disproportionate force is needed for each incremental bit of stretch—the thing gets stiffer as it stretches further…As the vessels expand, pressure inside is increasingly effective at generating tension in their walls—that’s the unavoidable consequence of Laplace’s law. But that tension, the stress in the walls, is decreasingly effective in causing the walls to stretch. It all comes down to that curved, J-shaped line on the stress-strain graph, which means no aneurysm.
Thank goodness nature found a way to avoid the aneurysms predicted by the Law of Laplace!

There are many more biomedical applications of the Law of Laplace. In a review article, Jeffrey Basford describes the “Law of Laplace and Its Relevance to Contemporary Medicine and Rehabilitation” (Archives of Physical Medidine and Rehabilitation, Volume 83, Pages 1165–1170, 2002). Basford considers many examples, including bladder function, compressive wraps to treat peripheral edema, the choice of where in the uterus to perform a Cesarean section. That is a lot of insight from one simple law relating pressure, radius and wall tension.