Computational models are bringing novel perspectives into the investigation of cardiovascular issues. Together with imaging improvements, computational tools are increasingly adopted toward the understanding of patient-specific pathological situations, in the development of surgical planning, and in the support of interventional medical decisions. However, the complexity of patient-specific problems calls for an adequate and knowledgeable use of computational tools to obtain valuable results.

Accordingly, after a discussion on the role and development of computational methods, the present chapter provides basic concepts and examples complemented by important references to the recent literature. In particular, the first part of the chapter is devoted to the numerical simulation of tissues and structures. It focuses on structural simulations of endovascular treatments and discusses the main steps and issues inherent with such simulations, with an emphasis on carotid artery stenting and transcatheter aortic valve implantation. The discussion is mainly devoted to classical approaches, such as finite element analysis, with an overview on a promising extension known as isogeometric analysis.

The second part of the chapter introduces numerical modeling of human hemodynamics, starting from the basic concepts for the numerical modeling of blood as an incompressible fluid in three-dimensional domains to a discussion on specific methods to simulate the complexity that may arise when considering fluid-structure interaction problems occurring in computational hemodynamics, such as the interaction between blood and the vascular walls, and heart valve dynamics. Moreover, as the computational cost of the numerical approximations of the problems is generally high, their complexity may need to be conveniently reduced, calling for specific model reduction techniques based on the online/offline paradigm.

Therefore, the final part of this chapter will be a primer on reduced-order model methods and especially those that look particularly promising to fulfill clinical timelines.