Biomedical Engineering Reference
In-Depth Information
sign of new therapies and, more in general, in the decision making process of medical
doctors.
12.1 Introduction
In the last 20 years mathematical and numerical models have been progressively
used as a tool for supporting medical research in the cardiovascular science.
In sil-
ico
experiments can provide remarkable insights into a physio-pathological process
completing more traditional
in vitro
and
in vivo
investigations. Numerical models
have been playing the role of “individual based” simulators, able to furnish a dy-
namical representation of the biology of a specific patient as a support to the prog-
nostic activity. At the same time, the need for quantitative responses for diagnostic
purposes has strongly stimulated the design of new methods and instruments for
measurements and imaging. On the one hand, we can simulate in 3D large portions
of the cardiovascular system of a real patient properly including simplified models
for the peripheral sites (see e.g. [16, 21, 31, 32, 33]). On the other hand, thanks to new
instruments, images and measures nowadays provide doctors and bioengineers with
a huge amount of data. These data offer obviously new possible benchmarks for the
numerical simulations (see e.g. [34]). However, beyond the validation, it is possible
to merge simulations and measures by means of more sophisticated numerical tech-
niques. This procedure is called
Data Assimilation
(DA) (see e.g. [22]). With this
name we mean the ensemble of methods for merging observed (generally sparse and
noisy) information into a numerical model based on the approximation of physical
and constitutive laws. The merging improves the quality of the information brought
both by numerical results and by measurements:
•
numerical simulations are improved by the merging of data that allow to include
effects otherwise difficult to model (at the qualitative or quantitative level), such
as the presence of tissues surrounding an artery or the motion of heart affecting
the aortic dynamics;
•
measures are in general affected by noise, so that assimilation of results based
on physical and constitutive laws introduces a sophisticated filter, forcing the
consistency with basic principles.
In some fields, these techniques are quite mature and tested, in particular in geo-
physics and meteorology (see the excellent review of methods in [22]). There are
basically two classes of methods for performing DA, both with pros and cons.
Variational methods.
DA is performed by minimizing a functional, estimating the
discrepancy between numerical results and measures. The optimization problem is
solved by using the mathematical model as a constraint, upon the identification of a
proper set of control variables. In environmental studies this is often the initial state
of the system of interest. In some cases (
Nudging
or
Dynamic Relaxation Methods
)
the functional to be minimized is properly “altered” so to include the data to be
assimilated directly in the equations of the model.
