# Law of Laplace

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A 35-year-old man presents for evaluation of chest pain and dyspnea that worsens with exertion. Medical history is notable for untreated Streptococcal pharyngitis as a child. An echocardiogram is performed and reveals aortic stenosis. The patient is found to have a left ventricular wall thickness of T and a radius of R. The intraventricular pressure during systole is P. The patient is lost to follow-up and, upon returning a year later, is found to have a wall thickness of 1.2T and radius of 0.6R. Intraventricular pressure remains unchanged. Which of the following best characterizes decreased left ventricular wall stress over the past year?

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#### Contributors

The law of Laplace, named in honor of French scholar Pierre Simon Laplace, is a law in physics that states that the tension in the walls of a hollow sphere or cylinder is dependent on the pressure of its contents and its radius.

The concept was then later applied to medicine since there are many hollow spherical and cylindrical shaped organs in our bodies that deal with pressures.

Important examples include the blood vessels and the chambers of the heart.

Okay, so according to the law of Laplace, wall tension is proportional to pressure (P) times radius (r).

Now, let's break it down.

The wall tension is the force in the container’s walls that resists the force trying to expand it.

So if we’re blowing up a balloon, we can think of the wall tension as the force created by the elastic rubber wall that resists the outward force applied by the pressure inside the balloon.

Now if we break the wall tension into components, we have a vertical vector of force that’s counteracting the expansion of the balloon and a horizontal vector of force that’s stretching and tearing the balloon’s wall.

So, for pressure, if we were to blow more air into a balloon, we would expect the pressure inside to build up and the wall tension of the balloon would increase as the walls push back against the expansion.

If the pressure trying to expand the balloon is greater than the wall tension, the balloon will expand... or pop!

Now another factor is the radius.

A smaller radius means more pressure is needed to overcome the wall tension in order for the container to expand.

This is why it’s harder to blow up a small, deflated balloon than it is to blow up a half inflated balloon.

An example of this can be seen in the alveoli in the lungs of a newborn.

Let’s plug in some easy, imaginary numbers and forego units to make this concept easier to understand!

Normally, an unused alveolus in a newborn is collapsed, so let’s say it has a radius of 2, and the wall tension is 8.

The baby starts crying and inhales.

The pressure of the inhaled air in the alveolus is 4.

So our equation is 4 * 2 which gives us 8, and since this is the same as the wall pressure, the alveolus doesn’t expand.

### Summary

The law of Laplace is a law in physics that states that the wall tension of a hollow sphere or cylinder is proportional to both the pressure of its contents and its radius. Wall stress is the wall tension divided by 2 times the wall thickness. Now, when applied to hollow spherical objects like the left ventricle of the heart, the following formula is used: wall stress = P x r / 2w, where P is pressure, r is total radius, and w is wall thickness. Put simply, the law of Laplace states that wall tension is directly proportional to pressure and radius; and wall stress is proportional to the wall tension but inversely proportional to two times the wall thickness.

### Sources

1. "Medical Physiology" Elsevier (2016)
2. "Physiology" Elsevier (2017)
3. "Human Anatomy & Physiology" Pearson (2017)
4. "Principles of Anatomy and Physiology" Wiley (2014)
5. "Microcirculation: Mechanics of Blood Flow in Capillaries" Annual Review of Fluid Mechanics (1971)
6. "Measuring Wall Shear Stress Using Velocity-Encoded MRI" Current Cardiovascular Imaging Reports (2014)

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