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\usepackage{multicol} % indice en 2 columnas
%\usepackage{ccfonts} % Font family: Concrete Math
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\title[A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI data]{ A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI data}
%\author[Jeremías Garay Labra]
%{Jeremías Garay Labra}
\institute[University of Groningen]
Bernoulli Institute\\
Faculty of Sciences and Engineering\\
University of Groningen\\[0.5cm]
% \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
\texttt{Jeremías Garay Labra \\ \}
\section{4D flow MRI}
\frametitle{4D flow MRI}
\column{.55\textwidth} % Left column and width
4D flow MRI has been shown potential in the assesment of blood flow dynamics in heart and large arteries, allowing wide variety of options for visualization and quantification.
\column{.5\textwidth} % Right column and width
\frametitle{4D flow MRI}
Main limitation for its clinical applicability is the long scan times involved. Therefore, multiple strategies emerged in order to make acquisition faster>
\item Navigator gating
\item modest spatial resolutions $2.5 \times 2.5 \times 2.5 \ mm3$
\item partial data coverage
Typical quality estimators are> SNR, VNR, peak flows/velocities, mass conservation (zero divergence
We want to introduce a novel measure for quantify the quality of the 4D flow measurements, using the conservation of momentum of the flow.
\section{The corrector field}
\frametitle{The corrector field}
\column{.6\textwidth} % Left column and width
\onslide<1-> We assume a perfect velocity \begin{eqnarray*}
\rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \quad \text{in} \quad \Omega \label{eq:NSmom}
\onslide<2-> And a corrector field which
\vec{u} & \approx \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\
\nabla \cdot \vec w & = 0 \quad \text{in} \quad \Omega \label{eq:correctorDiv} \\
\vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC}
\onslide<3-> asd
\item[]<4-> $u = u_{in} \quad \text{in} \quad \Gamma_{inlet}$
\column{.5\textwidth} % Right column and width
\caption{Aortic mesh }
\frametitle{The corrector field}
To study the corrector in several scenarios> synthetic data, experimental phantom and healthy volunteers.
\frametitle{The corrector field}
different data treatments> aliasing and noise. Undersampling
results for the synthetic data. Comparison againts a perfect correction case with du.
results for experimental phantom
results in healthy volunteers
potential of the new quality parameter> analize real data. use the specificity for label zones with strong disagreedment. Use the field for create new inverse problems which can be used for further accelerations
\huge{Thank you for your time!}