The excess internal pressure (above the external pressure) tries to open the bubble while the surface tension tries to close it. The result is the Laplace bubble equation that relates the surface tension γ (gamma — unit of force/length) to the pressure difference, Δp, to the radius of the bubble, R:

Δp = 2γ/R .

A.1 If we are modeling the heart as a spherical bubble, estimate the radius of a typical heart and how much that radius changes during a heartbeat. Discuss how you found the values you chose and why you chose them.

A.2 The excess pressure of the heart in a healthy human varies from 120 mmHg (systolic) to 80 mmHg (diastolic). Using your value for the relaxed heart and the diastolic pressure, get and estimate for the surface tension in the relaxed heart muscle. (Please convert to SI units of Pascals.)

A.3 From your estimate of how much the radius changes during a heartbeat and the systolic pressure. Can you estimate the change in the tension of the heart muscle during a heartbeat? Is it reasonable to consider that the tension does not change?

Did the tension increase or decrease during the heartbeat that drove the pressure up? We have a certain expectation (what is it?) but our model is so simplified that it might not yield the expected result. Discuss what you found, what you expected, and whether the result is plausible.

A.4 In this simple model, there is a simple relationship among our three variables: pressure difference, radius, and tension in the muscle (really surface tension – force/length, not force). From this relationship, what would you expect for a larger heart? Is it easier or harder to provide the same pressure difference as in a smaller heart? This has implications both for the medical condition of an enlarged heart and for a comparison of the hearts of smaller vs larger animals. Discuss the implications of the Laplace Bubble equation for these two situations.

B. Improved model

As we can see from the reasonably realistic image at the top of the page (a CG image – not real), the thickness of the heart wall is not small compared to the radius. Let’s explore some aspects of an improved model in which we take this into account.

Since the tension is provided by muscle fibers, what is more likely to be a good parameter describing the force in the heart is surface tension per unit thickness of the heart, or circumferential stress, σ (sigma). We’ll use the Greek “tau” – τ – to represent thickness. (In handwriting it looks like a “t” with the bar moved up to the top and the curve at the bottom turned to the left instead of the right.) Let’s model the tension in the heart as

σ = γ/τ

Our model says that as the thickness changes, so does the surface tension and the surface tension that can be provided is directly proportional to the thickness.

B.1 Derive the equation for the relation of the pressure difference provided by the heart muscle and the parameters of the heart, σ, τ, and R.

B.2 Now suppose that the inner radius of the heart is R and the outer radius is R + τ. Still assuming a (now thick) spherical bubble, find equations for the volume of heart muscle, Vheart, and the volume of fluid contained in the sphere, Vblood.

B.3 Suppose that the heart is enlarged by 20 percent (the radius increases by 20%) but the volume of muscle does not change. Find an equation for the new thickness of the heart wall. (If it’s too messy, simplify by assuming that τ/R is small and can be ignored compared to a constant factor, and that (τ/R )2 can likewise be ignored.)

B.4 Continuing with the 20% enlarged heart, find the ratio , the surface tension in the enlarged heart over that in the heart before being enlarged. (Assuming γ is constant.)

C. Next steps

Even the improved model is not very good. It doesn’t include the multiple chambers of the heart, for example. How useful do you think this analysis is? Might it only be useful for simpler animals such as worms that have an “aortic arch” rather than a full heart? Or perhaps for animals with two chambers hearts like fish? Or does it have value for thinking about the human heart?