  Arterial deformations arise in blood flow when surrounding tissue invades the
space available for a blood vessel to maintain its circular cross section, the
most immediate effects being a reduction in blood flow and redistribution of
shear stress. Here we consider deformations from circular to elliptic cross
sections. Solution of this problem in steady flow is fairly straightforward.
The focus in the present paper is on pulsatile flow where the change from
circular to elliptic cross sections is associated with a transition in the
character of the equations governing the flow from Bessel to Mathieu equations.
The study of this problem has been hampered in the past because of difficulties
involved in the solution of the governing equations. In the present study we
describe methods we have used to overcome some of these difficulties and
present a comprehensive set of results based on these methods. In particular,
vessel deformation is examined under two different conditions relevant to blood
flow regulation: (i) keeping cross sectional area constant and (ii) keeping
cross sectional circumference constant. The results provide an important
context for the mechanism of neurovascular control of blood flow under the
pathological conditions of vessel deformation.
