A Mathematical Approach to Factors Affecting Blood Flow

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OpenStax Anatomy and Physiology

Jean Louis Marie Poiseuille was a French physician and physiologist who devised a mathematical equation describing blood flow and its relationship to known parameters. The same equation also applies to engineering studies of the flow of fluids. Although understanding the math behind the relationships among the factors affecting blood flow is not necessary to understand blood flow, it can help solidify an understanding of their relationships. Please note that even if the equation looks intimidating, breaking it down into its components and following the relationships will make these relationships clearer, even if you are weak in math. Focus on the three critical variables: radius (r), vessel length (λ), and viscosity (η).

Poiseuille’s equation:

Source: OpenStax Anatomy and Physiology

• π is the Greek letter pi, used to represent the mathematical constant that is the ratio of a circle’s circumference to its diameter. It may commonly be represented as 3.14, although the actual number extends to infinity.

• ΔP represents the difference in pressure.

• r 4 is the radius (one-half of the diameter) of the vessel to the fourth power.

• η is the Greek letter eta and represents the viscosity of the blood.

• λ is the Greek letter lambda and represents the length of a blood vessel.

One of several things this equation allows us to do is calculate the resistance in the vascular system. Normally this value is extremely difficult to measure, but it can be calculated from this known relationship:

Source: OpenStax Anatomy and Physiology

If we rearrange this slightly,

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Then by substituting Pouseille’s equation for blood flow:

Source: OpenStax Anatomy and Physiology

By examining this equation, you can see that there are only three variables: viscosity, vessel length, and radius, since 8 and π are both constants. The important thing to remember is this: Two of these variables, viscosity and vessel length, will change slowly in the body. Only one of these factors, the radius, can be changed rapidly by vasoconstriction and vasodilation, thus dramatically impacting resistance and flow. Further, small changes in the radius will greatly affect flow, since it is raised to the fourth power in the equation.

We have briefly considered how cardiac output and blood volume impact blood flow and pressure; the next step is to see how the other variables (contraction, vessel length, and viscosity) articulate with Pouseille’s equation and what they can teach us about the impact on blood flow.


Betts, J. G., Young, K. A., Wise, J. A., Johnson, E., Poe, B., Kruse, D. H., … DeSaix, P. (n.d.). Anatomy and Physiology. Houston, Texas: OpenStax. Access for free at: https://openstax.org/details/books/anatomy-and-physiology