Biomedical Engineering Reference
In-Depth Information
processing step for computing the quantity to be compared with the data. “Regular-
ization” stands for some possible Tikhonov-like regularizing term with the role of
making the mathematical and numerical problem more tractable (see e.g. [12, 15]).
Control problems with constraints represented by partial differential equations have
been studied since a long time ([3, 12, 17, 26, 29, 42, 43, 44]). In computational
hemodynamics, these problems have been considered, for example, for the prescrip-
tion of defective boundary conditions [45, 46, 47].
There are several issues when solving this kind of problems. Particularly relevant
for our applications are:
•
the
existence
of an admissible CV that attains the minimal distance between data
and results. This can depend on the location (in space and time) of the available
data and can be forced by a proper regularization term;
•
the
noise
that invariably affects the data to be assimilated; this has a major impact
on the reliability of the entire data assimilation process.
In the examples presented below we will partially address these issues, pointing
out available results and open problems for each application. We will split each ex-
ample in three sections after the presentation of the specific problem and its medical
motivations, namely (i) the formalization of the problem in mathematical and nu-
merical terms - with a specific link to the feedback loop above - (ii) the discussion of
some preliminary numerical results and (iii) of the associated prospective research.
Far from being a conclusive review of methods and applications, the present work
pinpoints several open challenging problems in the adoption of variational meth-
ods for DA in computational hemodynamics. These are anticipated to become an
important tool for pursuing more reliability of numerical simulations in the general
perspective of
data driven simulations
[48] and
inverse cardiovascular mathematics
.
Accuracy and reliability of scientific computing are in fact an increasingly critical is-
sue for the progressive inclusion of numerical simulations in the validation protocol
of medical devices/drugs as well as in the decision making of medical doctors [49].
12.2 Variational assimilation of velocity data for the
incompressible Navier-Stokes equations
Bicuspid aortic valve (BAV) is the most common congenital heart defect, occurring
in about 1 % of the population [14]. At a mean age of 17.8 years 52 % of males with
normally functioning BAV already have aortic dilatation [9] which may eventually
lead to aortic regurgitation or dissection or aortic aneurysms. Medical doctors are in-
terested in developing a better understanding of the hemodynamics contributing to
aortic dilatation not only in patients with BAV but also in other forms of congenital
heart disease in which aortic dilatation is common [50]. Such an understanding may
allow early risk stratification, possibly leading to guidelines for earlier intervention
in high-risk groups, with an anticipated resultant reduction in morbidity and mor-
tality for these patients. Some studies suggest that BAV morphology results in ab-
normal flow patterns in the ascending aorta, anticipating that valves with significant
