State Estimation with Model Reduction and Shape Variability. Application to biomedical problems

Abstract

We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces precomputed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.

Inverse problems in blood flow modeling: A review

“…Moreover, an inequality constraint on the basis coefficients was included, avoiding penalization parameters in the optimization functional. The approach was extended in Galarce et al 182 to account for domain shape uncertainties when the domain's geometry is not easily parametrizable.…”