A statistical approach to data assimilation for hemodynamics

Cardiovascular applications recently gave a strong impulse to numerical methods for fluid
dynamics. Furthermore, thanks to new precise measurement devices and efficient image processing techniques, medicine is experiencing a tremendous increment of available data,
inevitably affected by noise. Beyond validation, these data can be combined with numerical
simulations in order to develop mathematical tools of clinical impact. Techniques for merging
data and mathematical models are known as data assimilation (DA) methods. In the context
of hemodynamics the reliability of assimilated solutions is crucial and it is mandatory to
quantify the uncertainty of numerical results.

In this talk we discuss a DA technique for hemodynamics based on a Bayesian approach to
inverse problems for the estimation of statistical properties of the blood velocity and related variables. This method is formulated as a control problem where a weighted misfit between
data and velocity is minimized under the constraint of the Navier-Stokes equations. The
optimization problem is solved with a discretize-then-optimize approach relying on the finite
element method. The result of the inversion is the probability density function of the normal
stress at the inflow section of the vessel; using such distribution we derive statistical estimators (namely the maximum a posteriori and the maximum likelihood estimators) and
confidence intervals for the velocity.

We present numerical results on 2-dimensional and 3-dimensional axisymmetric geometries approximating blood vessels, we compare statistical and deterministic estimators and we
draw confidence regions for quantities of interest.