New equations for the dose under pulsative/periodic conditions in the design of coated stents.

The notion of dose that comes from the biologists has been introduced by Delfour et al. (2005 SIAM J. Appl. Math. 65(3):858-881) in the context of coated stents to control restenosis. Assuming a stationary velocity profile of the blood flow in the lumen, it leads to a time-independent equation for the dose that considerably simplifies the analysis and the design problem. Under stable conditions the blood flow is pulsative, that is the velocity field can be assumed to be periodic. So it is necessary to justify the replacement of the periodic field by its time average over the pulsation period. In this paper, firstly we introduce the new unfolded dose and its equations without a priori constraint on the size of the period. So it can be used in biochemical problems where the period is large compared to the time constants of the system. Secondly, we show that, as the period goes to zero, the velocity field can be replaced by its average over the period. Numerical tests on a one-dimensional example are included to illustrate the theory.

General patterns and asymptotic dose in the design of coated stents.

Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent to prevent early reclosure due to the proliferation of smooth muscle cells. It is recognised that it is the dose of the agent that effectively controls the growth. This paper focusses on the asymptotic behaviour of the dose for general families of coated stents under a fixed ratio between the coated region of the stent and the targeted region of the vessel and set therapeutic bounds on the dose. It generalises the results of Delfour, Garon and Longo for stents made of a sequence of thin equally spaced rings to stents with an arbitrary pattern. It gives the equation of the asymptotic dose for a normal tiling of the target region using the theory of tilings, patterns and motifs on a cylinder.