On the effect of sharp rises in blood pressure in the Shah-Humphrey model for intracranial saccular aneurysms.

We consider the model originally proposed by Shah and Humphrey (J Biomech 32:593-599, 1999) for a class of intracranial saccular aneurysms and show that for constant pressure the addition of the viscoelastic term corresponding to the presence of cerebral spinal fluid outside the membrane, no matter how small, does ensure convergence to an equilibrium. Our arguments apply to a general equation of this type, and thus also hold for variations of this model such as that proposed by David and Humphrey (J Biomech 36:1143-1150, 2003). On the other hand, it is known that the presence of damping may destabilize periodic orbits of periodically forced systems or even prevent them from existing altogether. We present numerical simulations showing that for some forcing terms the high-frequency oscillations do not die out with time, and a much more complex behaviour may emerge as a discontinuous forcing term is approached. The key point for this situation to occur is related to the derivative of the forcing term, supporting the hypothesis that sharper rises (or falls) in blood pressure may increase the risk of aneurysm rupture.