Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 397

Fig. 6.2 Pipe where an
incompressible fluid flows
and velocity measures are
available in the points P i :
how is it possible to reliably
compute the wall shear stress
at the wall?

u ;p

P 5

P 2

P 1 P 4

P 3

solution. In such case, the accuracy of the estimated stress will also depend on
the numerical approximation.
3. Data assimilation procedure : suppose that the assumptions behind the Poiseuille
solution are acceptable but our knowledge is incomplete, for instance the
viscosity and the pressure gradient G P are unknown; in this case, we may
take advantage of the velocity measures to fill the gap and eventually to compute
the WSS by formulating the following problem. Find and G P such that u
minimizes the mismatch

N

X

u m .P i / u p .P i / 2

J D

i D 1

where u m .P i / is the measured velocity and u p is the Poiseuille solution ( 6.2 ). In
this way, we are fitting the physical parameters and G P so that the background
model is matching the foreground knowledge. Once and G P are computed, the
WSS (both as an estimate or as a prediction) is quantified. Contextually, the noise
affecting the data is filtered by our background knowledge of fluid mechanics in
the physically driven least squares procedure. Notice that when quantifying the
viscosity we are solving an identification problem .
In the more realistic case that the Poiseuille solution cannot be applied, we
replace u p with the (numerical) solution of the Navier-Stokes equations. In this
case, the minimization procedure requiring to find the stationary points of J

regarded as a function of and G P is clearly nontrivial (as we will see in the
next sections).

This simple example (that will be developed in Sect. 6.4 ) shows the relevance
of DA in biomedical applications, especially related to the clinical practice. As
a matter of fact, patient-specific knowledge of parameters that form a mathe-
matical/numerical model is always incomplete. As for the boundary conditions,
this has led to extensive investigation of the so-called defective boundary data
problems (see, for instance, [ 24 , 26 ]). Concerning parameter identification, we
mention elastography as a method for detecting the rigidity of soft tissues by solving
inverse elasticity problems [ 4 , 5 ]. In this case, parameter identification is not only
functional to the computation of a specific variable of interest, but it is by itself an
important procedure for diagnostic purposes (e.g., breast cancer).