Biomedical Engineering Reference
In-Depth Information
for detecting possible anomalies or performing individual-based numerical simu-
lations. Here, numerical differential models play the role of a bridge between the
measurable data and the unknown parameters. There are many critical issues at the
computational level to be addressed. Even if practical applications in general de-
mand for less accuracy than the one usually considered acceptable from the numer-
ical viewpoint, the noise is supposed to play a relevant role on the reliability of the
entire approach. Moreover, the frequency of sampling of images, currently driven
by technological limits, has probably a major impact on the accuracy of the results.
From the computational viewpoint, in these preliminary applications we resorted
to standard numerical tools like the BFGS method. Extension of this approach to
real 3D cases rises new issues on the computational effectiveness of the methods.
An extensive comparison among different possible options (in particular for the se-
quence of optimization and discretization steps) and different possible algorithms is
required for a massive use of these methods in practice.
A long term follow up of the present research is the extension of this optimization
procedure to more complex sets of CV, such as the configuration and geometrical
features of the cardiac fibres. In fact, we mention here that one of the open challeng-
ing problems in heart imaging and modelling is the estimation of the orientation of
the fibres driving the mechanical contraction and the electrical potential propagation
in the cardiac tissue.
12.5 Conclusions
A mathematically sound adoption of numerical models for investigating the vas-
cular blood dynamics originates from pioneering works in the late 80s (among the
others, see e.g. [30, 84, 85]). At that time, numerical simulations were carried out in
idealized domains, moving from basic geometrical primitives to realistic shapes of
regions of interest. Simulations were intended to provide an insight to physiological
and pathological dynamics for a better understanding of the most relevant diseases.
The impact of these simulations was mostly at a qualitative level, since data and ge-
ometries were realistic but not patient-specific. Successively, in the '90s, the advent
of new imaging technologies and corresponding numerical methods allowed the in-
troduction of “patient-specific” simulations. The geometry of the single patient at a
given instant was reconstructed from digital subtraction angiographies or computed
tomographies and used as the computational domain, possibly together with individ-
ual measures of data for the boundary conditions. This “sequential” merging of data
and simulations (i.e.,
first
the data,
then
the simulations fed by the data) led to a more
quantitative relevance of numerical models, closer to the clinical activity. Reliabil-
ity of numerical models have been progressively increased by removing many of the
simplifying assumptions postulated in the first simulations, e.g. rigid geometries or
Newtonian rheology (see e.g. [16]).
The development of more sophisticated mathematical and numerical models has
been corresponded by the development of more sophisticated measurements and
imaging tools. Nowadays, these instruments provide more data and more images, so
