# Fast reconstruction of 3D blood flows from Doppler ultrasound images and reduced models

## TL;DR

This paper introduces improved reduced models for fast 3D blood flow reconstruction from Doppler ultrasound, enhancing accuracy over traditional linear methods through task-adapted, potentially nonlinear mappings.

## Contribution

It proposes new reduced space construction methods tailored to blood flow reconstruction, outperforming classical linear PBDW approaches.

## Key findings

- Task-adapted reduced models improve reconstruction accuracy.
- Nonlinear mappings can outperform linear approaches.
- Numerical experiments validate the proposed methods.

## Abstract

This paper deals with the problem of building fast and reliable 3D reconstruction methods for blood flows for which partial information is given by Doppler ultrasound measurements. This task is of interest in medicine since it could enrich the available information used in the diagnosis of certain diseases which is currently based essentially on the measurements coming from ultrasound devices. The fast reconstruction of the full flow can be performed with state estimation methods that have been introduced in recent years and that involve reduced order models. One simple and efficient strategy is the so-called Parametrized Background Data-Weak approach (PBDW). It is a linear mapping that consists in a least squares fit between the measurement data and a linear reduced model to which a certain correction term is added. However, in the original approach, the reduced model is built a priori and independently of the reconstruction task (typically with a proper orthogonal decomposition or a greedy algorithm). In this paper, we investigate the construction of other reduced spaces which are built to be better adapted to the reconstruction task and which result in mappings that are sometimes nonlinear. We compare the performance of the different algorithms on numerical experiments involving synthetic Doppler measurements. The results illustrate the superiority of the proposed alternatives to the classical linear PBDW approach.