Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties

Abstract

In this paper we derive a set of equations which can be used to study wall deformations and transmural pressure at the anchoring sites of endovascular prostheses. The equations are the jump conditions associated with the underlying model equations. The model equations are derived from the Navier–Stokes equations to describe the blood flow through compliant axi-symmetric vessels after endovascular repair. They are in the form of a quasilinear hyperbolic system of partial differential equations with discontinuous coefficients. Since the weak form of the equations contains the product of the Dirac delta distribution with the Heaviside function, the jump conditions and the weak form cannot be obtained using the standard distribution theory. Driven by the undelying application in mind, we present a preliminary analysis leading to the jump conditions by interpreting the ambiguous products as a mean value with respect to the measure obtained in the limit of the “regularizing kernels” [17]. We show that the numerical solution obtained by using the Richtmyer two-step Lax–Wendroff method satisfies the weak form of the equations associated with a symmetric regularizing kernel in which case the weak form is independent of the particular choice of the kernel. We give an example (treatement of aortic abdominal aneurysm using multiple overlapping stents) where the conditions obtained in this paper can be used in the optimal design of an endovascular procedure.