Biomedical Engineering Reference
In-Depth Information
Stochastic methods.
These are based on the extension to nonlinear problems of the
Kalman filter
, which is a statistical approach for prediction of linear systems affected
by uncertainty [8, 18, 35], relying upon a Bayesian maximum likelihood argument.
On the contrary, there are relatively few studies devoted to a mathematically
sound assimilation of data in hemodynamics, probably because the availability of
more and more accurate measurements is the result of truly recent advancements.
In particular, we mention [36, 37, 38, 39, 40], essentially based on Kalman filtering
techniques, and [41] for the variational approach.
In this chapter we consider three possible applications of DA based on
variational
methods
. In particular we present possible techniques for:
•
Merging velocity data into the numerical solution of the incompressible Navier-
Stokes equations, so to eventually retrieve non primitive variables like the
Wall
Shear Stress
(WSS);
•
Including images into the simulation of blood flow in a moving domain, so to
perform the fluid dynamics simulations including the measured movement of the
vessel;
•
Estimating physiological parameters of clinical interest by matching numerical
simulations and available data.
In all these examples we face a common structure that can be depicted as a clas-
sical feedback loop illustrated in the scheme below.
Measures (noisy)
v
=
FW
(Input
,
CV)
Data
Input
Results
v
FORWARD
PROBLEM
POST
PROCESSING
CV
f
(
v
)
CONTROL
J ≡
dist
(
f
(
v
)
,
Data)
Control
(+ Regularization)
Variable
At an abstract level, all these applications actually lead to solve a problem in the
form: Find the Control Variable CV (belonging to a suitable functional space) such
that it minimizes the distance
J ≡
dist
(
f
(
v
,
Data))
(+
Regularization
)
,
(12.1)
where
is the set of (noisy) measures,
v
the solution of the Forward Problem
FW, which depends on some
Input
variables and
CV
. Finally,
f
Data
(
·
)
represents a post
