Substituting equations (5.4.1) and (5.4.2) into (5.4.6)’ we obtain:
which gives an expression relating flow to the optimal vessel radius. We
see that the flow is proportional to the cube of the vessel radius:
This relation is the well-known cube law.
It is sometimes known as
Murray’s law. It states that in order to achieve a minimum amount of the
rate of energy, the blood flow required to perfuse a blood vessel must be
proportional to the cubic power of the radius. Controversy arises in the
application of Murray’s law. This stems from the fact that most of the
resistance to blood flow are presented by small peripheral vessels
(equation (5.4.1)), but that flow dominates
large vessels, such as the
aorta. The applicability of the Murray’s law therefore relies on where in
the vasculature it is applied to.
cylindrical blood vessel with radius
Murray’s (1926) minimum energy and Rosen’s (1967) “Optimality
Principles in Biology” have influenced earlier analysis of optimum
branching. They both considered the use of a “cost function” which is
commonly used in control systems engineering.
The cost function
considered is the sum of the rate of work done on the perfusing blood
and the rate at which energy is utilized. This results in a cost function in
terms of power associated with the flow-vessel interaction: