Sunday, October 28, 2012

Hemodynamics - Pouiselle's Law

Although a very simplified version of blood flow through a vessel -- Pouiselle's model for flow in a cylindrical tube is useful for arranging the relationships between variables in the hemodynamic circulation. In particular, the incredible importance of blood flow on the radius of the vessel (directly proportional to the fourth power), which is conceptually applicable in all sorts of clinical situations (vasodilators/vasocontrictors/atherosclerosis/etc.)

What complexities are left out of the Pouiselle model?
- The model applies to laminar flow. Turbulence and pulsatility is not accounted for.
- The model applies to tubes of constant width, whereas most arteries taper.
- The model applies to a rigid tube, whereas blood vessels have some elasticity. Similarly, the model applies to steady flow, rather than pulsatile flow.
- The model applies to flow in an ideal, noncompressible liquid.

Pouiselle's formula (above) can be re-written to express the laminar resistance in a tube. The relationship is similar to that of an electrical circuit (V = IR)

change in pressure =  (flow in the tube) (laminar resistance in a tube)

And from here it follows that the laminar resistance of a vessel is:

laminar resistance in a tube = (change in pressure) / (flow in the tube)

laminar resistance in a tube= (8 η L) / (π r^4 )

This rearrangement re-emphasizes the dependence of vascular resistance on the size of the vessels (esp. arterioles). In normal circumstances, the length and viscosity of a vascular bed do not vary much. It also expresses the minimum possible resistance, because as soon as turbulence occurs (arterial branch points, atherosclerosis),  resistance will increase.

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1. "Circulatory Physiology: The Essentials" Smith and Kampine. 3rd ed. (1990)
2. "Biomechanics: Circulation" Fung YC. 2nd ed (1997)