|Abstract:|| This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can identify in 1997) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.
Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem,
its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives.
We then address specific methods for stand-alone models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called ``defective problems'' naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process.
We also describe specific issues related to th boundary treatment of reduced models, particularly relevant to the multiscale coupling.
Next, we detail the most popular numerical
algorithms for the solution of these problems.
Finally, we review some of the most representative works - from different research groups - which addressed the geometric multiscale approach in the past years.
A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled.|