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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=1.5cm]{Imagenes/escudoU2014.pdf}   
  %  \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
  \texttt{Jeremías Garay Labra \\ \ j.e.garay.labra@rug.nl}
}
\date{\today}


\begin{document}
\frame{\titlepage}


% \onslide<1->




\begin{frame}
  \frametitle{Index}
  \tableofcontents
\end{frame}


\section[4D flow MRI]{4D flow MRI}
\begin{frame}
  \frametitle{4D flow MRI}  
 \begin{columns}[c] 
\column{.55\textwidth} % Left column and width
\footnotesize

4D flow MRI has been shown potential in the assesment of blood flow dynamics in the heart and also large arteries, allowing wide variety of options for visualization and quantification. 

Some advantages respect others techniques:
\begin{itemize}
\item Full 3D coverage of the region of interest
\item Retrospective plane positioning
\item Rich post-proccesing: derived parameters 
\end{itemize}

\column{.5\textwidth} % Right column and width

\end{columns}
\end{frame}


\begin{frame}
  \frametitle{4D flow MRI}
\footnotesize
Main limitation for its clinical applicability is the long scan times involved. Therefore, multiple strategies emerged in order to make acquisition faster, such as:
\begin{itemize}
\item Navigator gating
\item modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
\item partial data coverage  
\end{itemize}

Typical quality estimators: 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 (Navier-Stokes compatibility).

\end{frame}


\section[]{The corrector field}

\begin{frame}
 \frametitle{The corrector field}
\footnotesize

We assume a perfect physical velocity field $\vec{u}$ 
\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}
\end{eqnarray*}

And a corrector field $\vec{w}$ which satisfies:
\begin{align}
\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}
\end{align}

The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.

\end{frame}



\section[Synthetic data]{Experiments using synthetic data }


\begin{frame}
 \frametitle{Numerical tests}

\footnotesize
We tested the corrector using CFD simulations as a measurements, in the following testcases:

\begin{itemize}
\item Womersley flow in a cilinder
\item Navier-Stokes simulations in an aortic mesh
\end{itemize}

Also perturbations were added into the measurements: 
\begin{itemize}
\item  velocity aliasing (varying the $venc$ parameter)
\item  additive noise (setting SNR in decibels)
\item  simulated k-space undersampling (compressed sensing for the reconstruction)
\end{itemize}

All simulations were done using a stabilized finite element method implemented in FEniCS. Afterwards, all numerical simulations were interpolated into a voxel-type structured mesh

\end{frame}


\begin{frame}
 \frametitle{Numerical tests: details}
\begin{columns}[c] 
\column{.6\textwidth} % Left column and width
\footnotesize
\textbf{Channel:}
\begin{itemize}
\item Convective term was neglected
\item Non-slip condition at walls
\item Oscilatory pressure at $\Gamma_{inlet}$
\end{itemize}


\column{.5\textwidth} % Right column and width
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.3\textwidth]{images/cilinder_2.png}
 \end{center}
 \end{figure}
\end{columns}


\begin{columns}[c] 
\column{.6\textwidth} % Left column and width
\footnotesize
\textbf{Aorta}
\begin{itemize}
\item a mild coartation was added in the descending aorta
\item $u_{inlet}$ simulates a cardiac cycle
\item 3-element Windkessel for the outlets
\item Non-slip condition at walls
\end{itemize}


\column{.5\textwidth} % Right column and width
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.7\textwidth]{images/aorta_blender.png}
\caption{\tiny{Channel mesh}}
 \end{center}
 \end{figure}
\end{columns}


\end{frame}


\begin{frame}
 \frametitle{Results: aliasing and noise}
\footnotesize

For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$

\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
\caption{Different perturbation scenarios}
 \end{center}
 \end{figure}


\end{frame}


\begin{frame}
 \frametitle{Results: undersampling}
\footnotesize



\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.6\textwidth]{images/undersampling_final.png}
\caption{Different perturbation scenarios}
 \end{center}
 \end{figure}


\end{frame}



\section[4D flow data]{Experiments using real 4D flow data }



\begin{frame}
 \frametitle{Experiments}
\footnotesize
\begin{itemize}
\item We performed 4D flow measurements in a silicon aortic phantom
\item 4 healthy volunteers were scanned using a clinical standard 4D flow protocol.
\end{itemize}

\end{frame}





\begin{frame}
 \frametitle{Results}
\footnotesize

results for experimental phantom

\end{frame}





\section{Conclusions}


\begin{frame}
 \frametitle{Conclusions and future}
\footnotesize

potential of the new quality parameter:
\begin{itemize}
\item analize real data
\item use the specificity for label zones with strong disagreedment
\item  Use the field for create new inverse problems which can be used for further accelerations
\end{itemize}

\end{frame}





\begin{frame}
\begin{center}
\huge{Thank you for your time!}
\end{center}
\end{frame}







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\end{document}