Biomedical Engineering Reference
In-Depth Information
different levels of integration in the CVS will be the focus of developments in com-
putational modelling in the forthcoming years. As a matter of fact, this kind of inte-
grative models will be capable of providing a better understanding of the processes
involved in the onset and development of vascular diseases [12, 20, 32, 41, 51, 59]
such as stenotic plaques, aneurism growth and rupture, elevated arterial pressure,
atherosclerosis, among others. Moreover, the capabilities of such models will go
beyond these applications, thus providing guidelines to assist and plan surgical pro-
cedures.
Regarding the fluid dynamics aspects of the CVS we identify what we call lev-
els of integration: (i) the overall systemic behaviour, (ii) the hemodynamics of large
arteries and (iii) the local circulation in specific districts. Several models have been
proposed to take into account the relevant phenomena at each level of integration.
For instance, specific vessels can be modelled using a full 3D representation, whereas
the systemic arteries can be accounted for using simplified 1D representations and
peripheral beds may be seen as lumped vascular entities, that is 0D models. In this
sense, it is possible to select the level of complexity of each component based on the
type of data available about a certain vascular entity and the information to be re-
trieved from the model. Particularly, modelling the blood flow in deformable vessels
comprises several challenging issues. Indeed, the propagatory nature of the pulse
wave, which is related to the compliant properties of the arterial walls, poses the
problem of defining boundary conditions once a specific district has been artificially
isolated from the rest of the system.
Thus, the simulation of blood flow in specific vessels can be carried out through
several approaches. The simplest one is based just on the imposition of boundary
conditions obtained from patient records (see for instance [47, 48]). A more com-
plex approach, presented in [28, 62, 63], is based on accounting for the phenomena
occurring downstream the 3D region through the computation of the downstream
vascular impedance. Finally, the most sophisticated approach consists in coupling
dimensionally-heterogeneous models (3D-1D-0D models). With this approach a high
level of detail can be attained in specific zones (3D models), while the physical
phenomena corresponding to the remaining part of the system is modelled using
dimensionally-reduced (1D-0D) models. This kind of formulation ensures proper
systemic functioning from which physiological regimes for 3D simulations can be
readily achieved. This line of research was pioneered in [17] and continued in [6, 8,
9, 18, 19, 34, 43, 60].
In [17, 18, 60] the authors make use of
a priori
assumptions which are directly
incorporated into the set of partial differential equations with the aim of coupling full
3D model based on the Navier-Stokes equations with simplified distributed and/or
lumped parameter equations. Specifically, this is accomplished by prescribing the
continuity of some of the involved quantities (for instance flow rate and mean pres-
sure) at the so-called coupling interfaces (points to which the heterogeneous models
converge). In turn, in [19, 23] the concept of defective boundary conditions is intro-
duced so as to write down a well-posed problem by means of further assumptions, in
view of the lack of information available at the communicating interfaces between
a 3D model and the rest of the system.
