Cardiovascular stents as PDE nets: 1D vs. 3D

This manuscript gives an overview of a new framework for modelling multi-component, mesh-like structures such as cardiovascular (endovascular) stents. By using dimension reduction and multi-scale approaches, a metallic, multi-component, mesh-like structure is described by a mathematical model which is in the form of a hyperbolic net problem. This contrasts classical approaches in bioengineering in which a stent is modelled as a single, 3D elastic body, using commercial software packages based on 3D finite element method structure approximations. Simulating slender stent struts using 3D approaches is computationally very expensive, typically producing simulation results with poor accuracy due to the insufficient mesh refinement imposed by the large memory requirements associated with the use of 3D meshes to approximate the slender stent struts. The approach presented in this manuscript is based on approximating the slender stent struts as 1D curved rods modelled by a system of 1D hyperbolic conservation laws. The slender curved rods are combined into a 3D hyperbolic stent net via certain geometric conditions, and via the contact conditions describing the physics of the interaction between the rods. Numerical simulation of the resulting stent net is simpler and substantially cheaper, providing a manageable and simple framework for the studies of stents mechanical properties and for the studies of optimal stent design. In this manuscript we begin by formulating the time-dependent, evolution problem for a hyperbolic stent-net, and then consider a linearized, static problem which is solved numerically. The main new result of this manuscript is a comparison between the numerical solutions of the full 3D model and the solution of the novel, linearized, static 1D hyperbolic net model. Our results show that the 3D solutions approach, with 3D mesh refinement, the solution of the 1D-based stent net model.