Flow is proportional to the fourth power of the radius
Mnemonic
“Radius to the fourth power” is the whole of the law:
Flow = (pi x pressure difference x r^4) / (8 x viscosity x length)
- Halving the radius reduces flow 16-fold
- Doubling the radius increases flow 16-fold
Everything else is linear; only radius is to the fourth, which is why arteriolar tone dominates peripheral resistance and why a small change in bronchial calibre transforms airway resistance.
The practical applications follow directly:
- For rapid infusion, use a short, wide cannula: a grey 16G in the antecubital fossa beats a long central line
- Bronchoconstriction and croup cause disproportionate obstruction because the airway is already narrow, and children start with smaller airways
- Coronary and renal artery stenoses become haemodynamically significant abruptly
The law assumes laminar flow of a Newtonian fluid in a rigid tube, none of which is strictly true of blood, but it holds well enough to be useful.
Expansion
Flow = (pi x pressure difference x radius to the fourth power) / (8 x viscosity x length)
So resistance is proportional to viscosity x length / radius to the fourth power.
The dominant term is radius, raised to the fourth power. Consequences:
- Arterioles are the main resistance vessels, because they can change radius most
- A wide, short cannula gives far more flow than a long narrow one; a 14 gauge peripheral cannula outperforms a long central line for rapid transfusion
- Polycythaemia raises viscosity and therefore resistance, while anaemia lowers it
- A coronary stenosis becomes haemodynamically significant only once it narrows the lumen substantially, because of this steep relationship
Note the assumptions: steady, laminar flow of a Newtonian fluid in a rigid tube. Blood is not Newtonian and vessels are not rigid, so the law is a guide to relationships rather than an exact calculation.