Use of mathematical models in the study of local hemodynamics at the vascular wall

The behaviour of the vascular endothelium is ruled by several, complex interacting factors. One of them is known to be the local flow field, i.e. the local distribution of velocity and pressure. In particular, the shear stress at the wall, generated by the local velocity gradient (shear rate), is believed to be a potential cause of endothelial damage. In vivo evaluation of wall shear stress is heavily restricted by the technology available for accurate measurements of local fluid dynamic quantities (such as velocity and shear rate) and it turns out to be practically impossible. It is important to say that this kind of variables is the typical output from a class of mathematical models of the flow field, well-known in the engineering practice, based on the Finite Element Method (FEM). These models are generally based on highly detailed descriptions (either two- or three-dimensional) of the fluid domain of interest and offer two major benefits: the ability to control and isolate relevant variables as well as the ability to obtain quantitative data. The determination of proper boundary conditions can be achieved by means of a different class of mathematical models, able to describe the hydraulic behaviour of the global circulation, but discarding the dependence of the fluid dynamic quantities from the local geometry. This means that the generic fluid dynamic variable ν is now a function of only one spatial co-ordinate (which gives the location across the hydraulic network) as well as of time. These models are typically arranged as electrical networks including resistance, compliance and inertance components. A major issue is the level of detail to be adopted for the network model of the circulation: the more detailed it is, the more correct boundary conditions for the FEM model are obtained. On the other hand, a high detail in the network model could require the knowledge of too many of its parameters. Provided a correct balance is achieved, the FEM model representing the flow field of interest at the endothelial wall can be thought as a built-in component of the global network model of the circulation. Using this approach simulations of the aorto coronary bypass fluid dynamics have been performed in steady and unsteady state and for both two- and three-dimensional models. The results are discussed for comparison.