Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 393

Data Assimilation (DA) is a process for integrating the knowledge provided by
numerical models and measurements with the purpose of improving the reliability
of quantitative analysis. This approach has been developed since the mid of the
twentieth century having as preferential application the weather forecasting. The
rationale is that the predictions provided by numerical models, that we may
call a background knowledge , being partially based on universal physical and
constitutive laws, are affected by uncertainties in real-world problems. These are
the consequence of simplifying assumptions as well as of an incomplete knowledge
of parameters usually needed by the constitutive laws forming a mathematical
model. For instance, referring to biomedical applications, blood viscosity (that in
a Newton constitutive law is supposed to be constant) or compliance of an artery
(that in a Hookean material is supposed to be represented by a parameter, the Young
modulus) is available as estimated on samples, but when dealing with a specific
patient they are in general not known, being impossible or inconvenient to measure.
The integration with available measures, that we may call a foreground knowledge ,
since they are specific of the case, is expected to be beneficial to the quantitative
analysis, reducing the uncertainty in the mathematical models. On the other hand,
background models improve the knowledge extracted from the data, providing a
way for filtering noise. In particular, this is important for at least three purposes,
1. estimate the state of a dynamical system (e.g., the velocity, the pressure) or its
derivated quantities for which noisy data and mathematical models are available,
2. predict the state of a dynamical system for which data are available in the past,
3. identify one or more parameters involved in the mathematical model, adjusting
their values on the basis of available data.
In the global picture—that we have represented in Fig. 6.1 —DA reduces possibil-
ities of failure in estimating, predicting, and identifying by merging background
and foreground in a unique quantitative analysis. The necessity of this process in
the traditional development of numerical models in cardiovascular mathematics
becomes progressively more urgent with the increment of available data and, more
importantly, of patients that may benefit from quantitative analysis.
In this chapter we want to provide an introduction to some topics brought in by
DA in Cardiovascular Mathematics, with a particular emphasis to FSI problems.
It is important to stress that, as such, this introduction cannot be complete. First,
there are several ways for approaching DA and it is basically impossible to provide
an exhaustive global picture of the possible methodologies. We refer to [ 8 ]asa
more general introduction. Second, DA in cardiovascular modeling is a relatively
recent topic and many questions and challenges are still open, so it is hard to draw
conclusive statements about the adequacy of a methodology for a specific problem.
Our perspective is to provide some examples that have been recently considered
in the literature and a self-contained introduction to the methods used there. In
particular, we have selected examples tackled with different approaches, providing
different perspectives for solving the same problem. This is intended to give not only
the idea of the complexity of the problems but also of the variety of approaches, the