Figure P2.10 shows a blood vessel of radius R1 branching into “daughter” vessels of radii R₂ and R₃. The respective flow rates are Q₁, Q₂, and Q₃. Anatomical studies of the mammalian circulation indicate that the radii in such a bifurcation tend to be related as

independently of the vessel lengths and branching angle. This is called Murray's law, after an American biologist who proposed the explanation that is outlined below.

Murray (1926) postulated that the circulation evolved to minimize the metabolic energy needed for its operation and maintenance. The energy expenditure for a vessel of radius R and length L was viewed as having two parts: (1) the pumping power needed for a given blood flow rate Q, and (2) the energy per unit time required to renew the blood in the vessel and maintain the surrounding cells. The maintenance energy was assumed to be proportional to the blood volume. The total rate of energy usage E_t was expressed as

where the proportionality constant b was assumed to be independent of vessel size. Blood flow in the circulation is almost entirely laminar. Thus, Poiseuille's law was used for Q, even though the flow is pulsatile (especially in large arteries) and not precisely Newtonian (especially in capillaries).

(a) Assuming that Q and L for a given vessel are fixed (by the need to supply oxygen and nutrients to a tissue), find the value of R that minimizes E_t for that vessel.

(b) Show that Murray's law is obtained by applying the result from part (a) to each vessel at a bifurcation.

(c) Each set of branchings in a network creates a new “generation” of vessels. Let R_ij be the radius of vessel i in generation j and let N_j be the total number of vessels in that generation. If the result from part (a) applies to all blood vessels, show that

where generations j and k are not necessarily adjacent. That is, the energyminimization principle predicts that the sum of the cubes will remain constant as the network is traversed. That is approximately what is found (Sherman, 1981).

Transcribed Image Text:

R₁ = R₁ + R² (P2.10-1)