Page R. Painter, Patrik Edén and Hans-Uno Bengtsson

**Pulsatile blood flow, shear force, energy dissipation and Murray's Law**

*Theoretical Biology and Medical Modelling*

**3**, 31 (2006)

**Abstract:**

**Background.** Murray's Law states that, when a parent
blood vessel branches into daughter vessels, the cube of the
radius of the parent vessel is equal to the sum of the cubes
of the radii of daughter blood vessels. Murray derived this
law by defining a cost function that is the sum of the energy
cost of the blood in a vessel and the energy cost of pumping
blood through the vessel. The cost is minimized when vessel
radii are consistent with Murray's Law. This law has also
been derived from the hypothesis that the shear force of
moving blood on the inner walls of vessels is constant
throughout the vascular system. However, this derivation,
like Murray's earlier derivation, is based on the assumption
of constant blood flow.

**Methods.** To determine the implications of the
constant shear force hypothesis and to extend Murray's energy
cost minimization to the pulsatile arterial system, a model
of pulsatile flow in an elastic tube is analyzed. A new and
exact solution for flow velocity, blood flow rate and shear
force is derived.

**Results.** For medium and small arteries with
pulsatile flow, Murray's energy minimization leads to
Murray's Law. Furthermore, the hypothesis that the maximum
shear force during the cycle of pulsatile flow is constant
throughout the arterial system implies that Murray's Law is
approximately true. The approximation is good for all but the
largest vessels (aorta and its major branches) of the
arterial system.

**Conclusion.** A cellular mechanism that senses shear
force at the inner wall of a blood vessel and triggers
remodeling that increases the circumference of the wall when
a shear force threshold is exceeded would result in the
observed scaling of vessel radii described by Murray's
Law.

LU TP 06-31