How does Bernoulli’s Principle apply to the cardiovascular system?

Question

Below are graphs which illustrate the cross-sectional area, velocity, and fluid pressure through each vascular segment of the cardiovascular system.

It makes sense that velocity and cross-sectional area should be inversely related graphically (by the equation of continuity, ($v_1A_1=v_2A_2$). However, Bernoulli’s principle states that where velocity is high, pressure is low, and vice versa. However, in the graph above, clearly pressure is decreasing as you move away from the aorta, regardless of velocity.

I can understand how the pressure in capillaries will be higher than those in venules, since a higher pressure in capillaries would accelerate the blood as it moves into the venules with a lesser total cross-sectional area. This agrees graphically, since, past the purple stripe, the pressure curve decreases as the velocity curve rises. Yet the mean arterial pressure in the aorta and arteries seems to violate the relationship between velocity and pressure stated in Bernoulli’s principle. What is there to make of this?

(Image taken from Campbell Biology, 10th edition)

Answer 1

Bernoulli's principle can be a little tricky when applied to the cardiovascular system, but it still holds true across the entire system. You mention a good point that the relationship doesn't seem quite right at the aorta or arteries, because of the constant fluctuation of pressure between systolic and diastolic without a significant change in diameter of the vessel. Remember that this is due to the pulsatile flow of blood from the heart, and has nothing to do with Bernoulli's principle. If the flow was constant (i.e. not pulsatile) then the pressure would be constant as the diameter of the vessels is constant. As the diameter of the vessels begins to decrease the velocity would increase to maintain a constant value.

As you mention, Bernoulli's principle describes how the product of area and velocity of flow must be constant across a system. When the total surface area of the system increases, the pressure decreases as well as flow. It is important to remember that even though the capillaries are so small and individually high resistance (which you would think according to Bernoulli's principle should increase the velocity), you are effectively adding an innumerable number of these very small resistors in parallel (not in series) and the resistance overall (and subsequently pressure + flow) decreases greatly.

When adding resistance in series the total resistance is additive and it adds up quickly! $$R_{Total} = R_1 + R_2 + R_3$$ While resistance in parallel adds the reciprocal of the resistor, meaning that as more resistors (or capillary pathways) are added the total resistance continues to decrease because of the following relationship: $$\frac{1}{R_{Total}} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$$

The parallel resistance stuff was difficult for me to grasp when I was a student, because it never seemed to make sense in my mind that the resistance significantly decreased when you added all these small capillaries. One of my physiology professors described it well when he said, "Think of adding resistance in parallel, in the way that capillary beds and other vessels are in the cardiovascular system, like giving the blood another place to go." If the blood has more places that it can go, then it's easier to see why resistance falls so much when you add a bunch of extra places for the blood to flow.

Answer 2

I think there are several things that keep Bernoulli's principle from being straightforwardly applied.

First, Bernoulli's principle would help us calculate the pressure exerted by the blood on the walls of the vessels, but that's not what blood pressure measures. Blood pressure measures the amount of force that has to be applied to stop the blood from flowing forward. There's some relationship between the two, but it's not clean.

Second, the branching makes the surface area calculations different from a single pipe. The arteries have more cross-sectional area than the aorta, but they have have much, much more surface area-- way more of a difference than a single pipe with those cross-sectional characteristics would have.

So Bernoulli's principle does mean that blood is exerting more pressure on the surface of the arteries and arterioles than it is in the aorta. However, pressure is force per unit area, and there's much more area in the smaller vessels, so the pressure in any single vessel is less.

Answer 3

I love this question, and there is a number of things to unpack.

Exactly where we are measuring a pressure in that circut matters. We usually measure a persons BP in the brachial artery, so relatvely on the left side of your picture.

Here, we can consider the pressure to be part of a simple circut, where Pressure = Resistance * Flow (ohms law). Vasoconstriction of artieroles further downstream could be considered to be increasing resistance. This results in an increase in pressure at the level of the arteries and specifically, where we measure it at the brachial artery.

(As a side note: it doesnt overly matter at this level that the distal system is in parallel. For simplicity, vasoconstriction will still increase the overall resistance. In a parallel system, increasing all or most of the resistors still increases the overall circuit resistance)

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If we were to look at what is happening in the microvasculature, its a very very different kettle of fish, and much more akin to what you are describing with Bernoulli's principle. In areas subject to vasoconstriction, there is indeed a decrease in pressure. This means there is less perfusion at that area (due to a decrease in hydrostatic pressure, see starling forces). In areas not subject to vasoconstriction, there is a relative increase in pressure and therefore an increase in perfusion.

This about why this matters - We want to keep perfusion high in places that need blood (the brain, the kidneys), so they will stay relatively dilated. Conversly, if we dont need to perfuse our muscles, we can direct blood away from them and perfuse other organs preferentially by constricting their blood vessels.

Its probably worth noting as well - the resistance happens PRIOR to the capillaries. Capillaries dont change size. The artieroles and meta-arterioles do but the pressure before and after them does change and the pressure will be lower in areas that have undergone constriction.

To make matters more complicated however (read on at your peril); If we also apply Ohms law at the level of the microvasculature and we consider this to be a parallel circuit, consider that increasing resistance will also lead to a decrease in flow. This is "shunting", where blood will flow preferentially to areas of less resistance. This extra flow 'would' (with an ideal fluid and non-compliant vessel), increase flow and decrease pressure. However, blood does not behave like an ideal fluid, and its viscosity prevents its speed from increasing as you'd expect. Moreover, the blood vessels tend to expand with the increased blood volume, increasing the radius of the vessel which significnatly decreases speed and overall actually probably increases its pressure (since radius is SUPER important in determining speed).

NB: This is STILL a huge oversimplification of a complex physics system - we havnt even touched on the effect of gravity and columns of water (is the perfusion pressure in our feet HUGE??!)

Happy learning!