Law of Laplace

Definitions & Key takeaways

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.

Chapters:

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Introduction0:00–0:30

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).

Law of Laplace0:30–1:08

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.

Pressure1:08–1:27

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

Radius1:27–1:43

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 is can be seen in the alveoli in the lungs of a newborn.

Case Study1:43–2:43

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.
In other words, the baby will need to breathe in enough air to generate greater than 4 units of pressure to inflate it. Now let’s say that the alveolus has expanded after the baby took a few breaths, and the radius is now 4, and the wall tension remains the same.
So this time, the baby only needs to breathe in enough air to generate greater than 2 units of pressure to expand the alveolus, so it takes a lot less work!
Now, a concept related to wall tension is wall stress, which uses a modified version of Laplace’s law that factors in the thickness of the wall.

Wall Stress2:43–4:00

The equation for wall stress is pressure times radius over 2 times the wall thickness. Based on this equation, wall stress is wall tension divided by 2 times the wall thickness, so the thicker the wall, the less wall stress.
To see a real life example, let’s look at the heart, more specifically the chamber of the left ventricle, which resembles a sturdy, round cup.
Let’s say we have a person with aortic stenosis, which means the aortic valve is narrowed and it makes it harder for blood to flow through.
In order to overcome this, the heart will pump harder, causing an increase in the pressure inside the heart, and as a result, we increase the wall tension.
Now in order to push more blood out, the left ventricle hypertrophies, meaning the cardiac cells grow thicker by adding more sarcomeres, the contractile units of cardiac cells.
As a result, there's an increase in the wall thickness and a decrease in the radius of the chamber, and according to our equation, both will reduce wall stress.
Thanks to this, the heart is less likely to tear open when under pressure! Alright, as a quick recap, 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.

Review4:00–4:47

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.
In short, 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.