Biomedical Engineering Reference
In-Depth Information
Unsteady problems.
When solving unsteady problems, following again a DO ap-
proach, we first discretize in time and at each instants solve the optimization problem.
In this case, the extension of the method devised for the steady case is pretty imme-
diate. However, possible computational concerns arise from the nesting of the time
and the optimization loops. Selection of appropriate effective preconditioners is in
order. Another issue refers to the initial conditions that in general are not known. In
meteorological applications, these are included in the set of CV and used for driving
the assimilation procedure. In cardiovascular applications an alternative approach
consists of forcing periodicity of the solution. This approach will be investigated
elsewhere.
12.3 Image assimilation in a moving domain simulation
Rigid-wall models for blood motion in arteries are often accurate enough for a quan-
titative analysis of hemodynamics (see e.g. [63]). However, there are situations in
which the magnitude of the mechanical forces involved and the deformation expe-
rienced by the vessels cannot be neglected and their effects should be appropriately
considered while modelling the coupled system.
The standard strategy to simulate the blood flow in a compliant vessel is to write
the models for both the blood (the incompressible Navier-Stokes equations) and the
wall (see e.g. [64]) together with appropriate matching conditions at the interface
between the two domains (Fluid-Structure Interaction - FSI). At the numerical level,
the coupled model is then solved either with a monolithic approach or by segregated
solvers managing iteratively the sequence of fluid and solid problems (see e.g. [25]).
This strategy allows the accurate computation of both fluid and solid mechanics and
is challenging from both the modelling and the numerical point of view. In fact, the
constitutive laws for modelling the arterial wall still deserve extensive investigations
especially in the presence of vascular pathologies (see e.g. [65]), not to mention
the difficulty to obtain
in vivo
measurements that can accurately estimate the model
parameters for an individual patient (see Sect. 12.4). Moreover, vessels are subject
to external loads due to the presence of the surrounding tissues, which are in general
unknown or not easy to model. We mention for example the effects of cardiac motion
on the aortic arch. From the numerical point of view, the strongly heterogeneous
nature of the problem raises issues concerning numerical stability and efficiency of
FSI algorithms (see e.g. [66, 67]).
Here we consider an alternative approach based on a DA procedure, that exploits
the technological development experienced in the last decade by medical imaging
techniques. The advent of high resolution imaging devices allows the fast acquisition
of 4D (space + time) images. From those images it is possible to reconstruct anatom-
ical structures not just in one specific instant, but in multiple ones over the cardiac
cycle. Following this approach, the vessel motion, instead of being computed, is re-
trieved from images and plugged into the Navier-Stokes solver. The main advantage
of this approach is the direct inclusion into the simulations of patient-specific data,
i.e. the motion of the vessel (depending on its mechanical characteristics and those of
the surrounding organs). This is done through the use of medical images at a limited
