Monday, November 5, 2012

Hemodynamics -- Turbulence & flow separation

If only blood flow were as simple as concentric fluid layers sliding over each other in an ideal tube. The truth is much more turbulent.

Pouiselle's law (see the 10/28/2012 post) acts as if blood were a laminar fluid. In this situation, resistance is linear with respect to pressure and flow. But human blood vessels don't usually fufill the criteria necessay for the development of the nice, orderly parabolic flow of Pouiselle's law. Tortuosity and multiple branch points end up causing shear to the blood flow, and complex helical blood flow is often the result.

Furthermore, at a certain velocity -- the critical velocity -- the laminar model breaks down and the fluid takes on a life of its own.  Eddies, vortices, and random velocity vectors form in the flow and become a separate, significant factor for resistance to the flow.  When this occurs, flow is roughly proportional to √ (∆P).

So what affects how quickly a fluid becomes turbulent?  Reynolds found that the fluid's  

density (ρ) g/cm3
velocity (ν) cm/sec
diameter of the vessel (D) (cm), and
viscosity (μ) g/sec/cm2

were all factors, and related by:

Reynold's original 1883 apparatus from which he formulated the relationship of Reynold's number.

Many of these factors make intuitive sense.  In the circulatory system, the variables that are most applicable are the velocity and diameter terms, implying that there is more turbulence in larger, faster vessels.  In anemia, the viscosity decreases, causing a relative decrease in turbulence.  The magic number seems to be around Re = 2000. With Re < 2000 (most peripheral arteries) local disturbances are dampened out by the viscous forces, and turbulence is unlikely to occur.

The turbulence of blood flow contributes to the normal physiologic sounds that were used to, such as arterial bruits or abnormal cardiac sounds from valvular damage.

A specialized type of turbulent flow occuring at arterial branch points is flow separation. When blood flows through a vessel, the outermost layer is called the boundary layer, and this layer experiences both the frictional force of the vessel wall as well as viscous forces from the fluid streams more centrally. When there is an abrupt change in the shape of the tube (as at a bifurcation or a stenosis), small pressure gradients are created and the boundary layer flow can stop or reverse direction (below). This local microenvironment of different flow is known as flow separation. This slow or reversed flow occurs in an area of low shear stress, that is the force requires to move one layer of flow by another (shear).

Using the carotid bifurcation as a model, flow separation is thought to play a role in the development of atherosclerosis. It was noticed that carotid bifurcation atherosclerosis occurs primarily on the outer wall, which experiences less shear stress than the inner wall, as well as helical flow patterns and flow reversal.  Distal to this point, more laminar flow returns.  But whether this flow separation in the region of atherosclerosis is causative is not entirely certain.

1. "Circulatory Physiology - The Essentials" Smith JJ and Kampine JP. 3rd ed (1990)
2. "Rutherford's Vascular Surgery" Cronenwett and Johnston. 7th ed (2010).