1/Rt = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn
For a vascular circuit like the one above, this means that flow through all resistances (Rt) is greater than flow through any one resistance. This makes intuitive sense.
The resistance of any one tube can be estimated by the Pouiselle law (see post from 10/28/12), so:
laminar resistance in a tube (R) = (8 η L) / (π r^4 )
Remembering that resistance through a Poiseuille tube is highly dependent on the radius term (r^4), large shifts in resistance through any tube can be acheived with only small changes in caliber size. The blood then redirects through the lower resistance circuit.
This dynamic results in two fundamental features of the circulatory system: collaterals and physiologic shunting.
Collateral formation is simple to apply to this situation. If you take two vessels in parallel (common iliac vein and lumbar vein), one is lower resistance (common iliac vein (larger r, shorter l) and receives the most flow.... but if this larger flow becomes narrowed and less favorable to flow (R1 above though atherosclerosis, compression, etc.), then other pathways in parallel becomes more attractive as a route of flow (R2, R3, and R4 above, dilatation of the lumbar vein in May-Thurner syndrome, for instance).
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| The anatomy of a collateral: stem arteries, midzone collaterals, and re-entry arteries. Flow continues into the peripheral vascular bed. |
The other application of the concept is nonpathologic. If the body needs to shunt more blood to a certain area of the body (for instance, to the legs if running away from a tiger) then it can constrict the arterioles of non-leg arterial systems (say, the mesenteric arterial system), thereby redirecting flow to the area it needs (lower extremities).
A pressure gradient and a higher velocity of flow through the midzone arteries (above) is theorized to result in collateral artery enlargement, but the pathway is still not totally understood.
Another way to apply these principles is in the somewhat paradoxical rise in pressures as main arteries branch all the way out to capillaries... the pressure should fall dramatically as resistance is reduced by so many parallel circuits in series... but it doesn't... why? because the vessels become so small that the (r^4) term in resistance dominates. Resistance actually increases down to the capillary level..
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1. "Circulatory Physiology: The
Essentials" Smith and Kampine. 3rd ed. (1990)
2. "Biomechanics: Circulation" Fung YC. 2nd
ed (1997)
3. "Rutherford's Vascular Surgery" Cronenwett and Johnston. 7th ed (2010)
3. "Rutherford's Vascular Surgery" Cronenwett and Johnston. 7th ed (2010)
