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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}
%\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]
  Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
\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{.5\textwidth} % Left column and width
\footnotesize

\begin{itemize}
\item<2-> Full 3D coverage of the region of interest
\item<3-> Rich post-proccesing: derived parameters 
\end{itemize}

\onslide<4-> Disadvantages:
\begin{itemize}
\item<5-> Long scan time
\end{itemize}




\column{.54\textwidth} % Right column and width
\onslide<1-> 
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.9\textwidth]{images/4dflow.png}
  \caption{\footnotesize 4D flow MRI of a human thorax}
 \end{center}
 \end{figure}
\end{columns}
\end{frame}


\begin{frame}
  \frametitle{4D flow MRI}
\footnotesize
\onslide<1-> Strategies:
\begin{itemize}
\item<2-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
\item<3-> partial data coverage  
\end{itemize}


\begin{columns}[c] 
\column{.4\textwidth} % Right column and width
\onslide<4->
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.25\textwidth]{images/channel_noise.png} \\
  (a) Noise
  %\caption{Noise}
 \end{center}
 \end{figure}
 \column{.4\textwidth} % Right column and width
\onslide<5->
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.25\textwidth]{images/channel_aliasing.png}\\
  (b) Aliasing
  %\caption{Aliasing}
 \end{center}
 \end{figure}
  \column{.4\textwidth} % Right column and width
\onslide<6->
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.25\textwidth]{images/channel_under.png}\\
  (c) Undersampling
  %\caption{Aliasing}
 \end{center}
 \end{figure}
\end{columns}

\vspace{0.5cm}

\onslide<7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 

\vspace{0.5cm}

\onslide<8-> This work $\longrightarrow$  \textbf{conservation of linear momentum} (Navier-Stokes compatibility).

\end{frame}


\section[]{The corrector field}

\begin{frame}
 \frametitle{The corrector field}
\begin{center}
Methodology
\end{center}
\end{frame}



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

\onslide<1-> We assume a perfect physical velocity field $\vec{u}$ 
\onslide<2-> \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*}

\onslide<3-> And a corrector field $\vec{w}$ which satisfies:
\onslide<4-> \begin{align}
 \vec{u}  & =   \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}

\onslide<5-> The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.

\end{frame}


\begin{frame}
 \frametitle{The corrector field: Continuum problem}
\footnotesize

\onslide<1-> Applying the decomposition $\vec{u}   =  \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have:\\[0.2cm]
Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that: 
\onslide<2-> \begin{equation*} 
\int_{\Omega} \rho \frac{\partial \vec{w}}{\partial t} \cdot \vec{v} +  \rho \big ( ( \vec{u}_{meas} + \vec w)  \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w}  \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v}  + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w} \notag 
\end{equation*}
\begin{equation*}
= - \int_{\Omega}  \rho \frac{\partial \vec{u}_{meas}}{\partial t} \cdot \vec{v} + \rho \big ( \vec{u}_{meas}  \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v} + \mu \nabla \vec{u}_{meas} : \nabla \vec{v} + q \nabla \cdot \vec{u}_{meas} 
\end{equation*}

\vspace{0.2cm}

\onslide<3-> or in simple terms:
\onslide<4-> \begin{equation*}
A(\vec w,p;\vec v ,q )  =  \mathcal{L} (\vec v)
\end{equation*}


for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.

\end{frame}



\begin{frame}
 \frametitle{The corrector field: Discrete problem}
\footnotesize

\onslide<1-> In the Discrete, we can write the problem as follows:

\onslide<2-> \begin{equation}
A_{k}(\vec w,p;\vec v ,q ) + \color{blue}{S^{press}_{k}(\vec w,p;\vec v ,q)} + \color{red}{S^{conv}_{k}(\vec w;\vec v)} \color{black}{ =  \mathcal{L}_j (\vec v)}
\label{eq:Corrector_discrete}
\end{equation}

\begin{itemize} 
\small
\item<3-> $
A_{k}(\vec w,p;\vec v ,q ) := \int_{\Omega} \frac{\rho}{\tau} \vec{w} \cdot \vec{v} +  \rho \big ( ( \vec{u}_{meas}^k + \vec{w}^{k-1} )  \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w}  \cdot \nabla \big) \vec{u}_{meas}^k \cdot \vec{v}  + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w}
$ \vspace{0.2cm}
\item<3-> $ \mathcal{L}_j (\vec v) := \int_{\Omega} \frac{\rho}{\tau} \vec{w}^{k-1} \cdot \vec{v} + \mathcal{\ell}_j (\vec v,q)   $
 \vspace{0.2cm}
\item<4-> \color{blue}$
S^{press}_{k}(\vec w,p;\vec v ,q) := \delta  \sum_{K \in \Omega}\int_{K} \frac{h_j^2}{\mu} \bigg ( \rho \big ( (\vec u^k_{meas} + \vec w^{k-1})  \cdot \nabla \big)  \vec{w}  +   \rho \big ( \vec{w}  \cdot \nabla \big)  \vec{u}_{meas}^k  + \nabla p  \bigg) \cdot \notag  \bigg ( \rho \big ( (\vec u^k_{meas} + \vec w^{k-1})  \cdot \nabla \big)  \vec{v}  + \rho \big ( \vec{v}  \cdot \nabla \big)  \vec{u}_{meas}^k   + \nabla  q     \bigg ) 
$
 \vspace{0.2cm}
\item<5-> \color{red}$
S^{conv}_{k}(\vec w;\vec v) := \int_{\Omega} \frac{\rho}{2} \ \big(  \nabla \cdot  (\vec u^k_{meas} + \vec w^{k-1}) \big) \  \vec{w} \cdot \vec{v}
$ \vspace{0.2cm}

\end{itemize}

\end{frame}




\begin{frame}
 \frametitle{The corrector field: Well-posedness}
\footnotesize
\onslide<1->
\begin{theorem} 
There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
\end{theorem}
\onslide<2->
We can furthermore prove the following energy balance:
\onslide<3->
\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}),  with $\ell_j(\vec v,q)=0$ it holds
\begin{equation*}\label{eq:energy}
   \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} 
\end{equation*}
under the condition
\begin{equation*}\label{eq:condstab}
\mu \geq C_\Omega^2 \rho \| \nabla \vec u_{meas}^k\|_\infty
\end{equation*}
\end{theorem}


\end{frame}









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

\begin{frame}
 \frametitle{Experiments}
\begin{center}
Experiments using synthetic data
\end{center}
\end{frame}






\begin{frame}
 \frametitle{Numerical tests}
 
\onslide<1-> 
\footnotesize
\begin{columns}[c] 
\column{.4\textwidth} % Right column and width
\footnotesize
 Simulated channel flow as measurements (Stokes flow)
 \column{.5\textwidth} % Right column and width
\footnotesize
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.35\textwidth]{images/cilinder_2.png}\\
  (b) Channel mesh
  %\caption{Aliasing}
 \end{center}
 \end{figure}
\end{columns}


\vspace{0.2cm}

%\onslide<1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
%\onslide<2->
%\begin{itemize}
%\item Womersley flow in a cilinder
%\item Navier-Stokes simulations in an aortic mesh
%\end{itemize}
\onslide<2-> Afterwards, perturbations were added: 
\begin{itemize}
\item<3->  velocity aliasing (varying the $venc$ parameter)
\item<4->  additive noise (setting SNR in decibels)
\item<5->  simulated k-space undersampling (compressed sensing for the reconstruction)
\end{itemize}
%\onslide<7-> 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: channel}
%\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=1.0\textwidth]{images/cilinder.png}
%  \caption{3D channel mesh}
% \end{center}
% \end{figure}
%\end{columns}
%\end{frame}
%

\begin{frame}
 \frametitle{Numerical tests}
\begin{center}
Results
\end{center}
\end{frame}




\begin{frame}
 \frametitle{Aliasing and noise}
\footnotesize

\onslide<1-> For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$

\onslide<2->
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
\caption{\small Fields for the channel: $(SNR,venc) = (\infty,120\%)$. $\vec{w} \times 200$}
 \end{center}
 \end{figure}

\end{frame}

\begin{frame}
 \frametitle{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=0.45\textwidth]{images/channel_ppt_2.png}
  \caption{\small Fields for the channel: $(SNR,venc) = (\infty,80\%)$. $\vec{w} \times 4$ }
%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
 \end{center}
 \end{figure}

\end{frame}


\begin{frame}
 \frametitle{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=0.45\textwidth]{images/channel_ppt_3.png}
  \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,120\%)$. $\delta \vec{u}, \vec{w} \times 4$}
%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
 \end{center}
 \end{figure}

\end{frame}


\begin{frame}
 \frametitle{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=0.45\textwidth]{images/channel_ppt_4.png}
  \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,80\%)$.  $\vec{w} \times 4$}
%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
 \end{center}
 \end{figure}

\end{frame}



\begin{frame}
 \frametitle{Aliasing and noise}
\footnotesize

\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.5\textwidth]{images/channel_curves_SNRinf.png}
\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
 \end{center}
 \end{figure}


\end{frame}


\begin{frame}
 \frametitle{Aliasing and noise}
\footnotesize

\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.5\textwidth]{images/channel_curves_SNR10.png}
\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
 \end{center}
 \end{figure}


\end{frame}




\begin{frame}
 \frametitle{Undersampling}
\footnotesize

\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.6\textwidth]{images/histo_channel.png}
\caption{ \footnotesize Histograms of different undersampling rates for the channel}
 \end{center}
 \end{figure}

\end{frame}




%\begin{frame}
% \frametitle{Results for channel: undersampling}
%\footnotesize
%
%\begin{figure}[!hbtp]
% \begin{center}
%  \includegraphics[height=0.6\textwidth]{images/undersampling_press.png}
%\caption{ \footnotesize Different undersampling rates for the channel}
% \end{center}
% \end{figure}
%
%
%\end{frame}
%



%\begin{frame}
% \frametitle{Numerical tests: aorta}
%
%\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=1.0\textwidth]{images/aorta_blender.png}
%\caption{Aortic mesh}
% \end{center}
% \end{figure}
%\end{columns}
%
%
%\end{frame}
%
%


%\begin{frame}
% \frametitle{Results for aorta: aliasing and noise}
%\footnotesize
%
%\begin{figure}[!hbtp]
% \begin{center}
%  \includegraphics[height=0.7\textwidth]{images/aorta_perturbation.png}
%\caption{Different perturbation scenarios for the aortic mesh}
% \end{center}
% \end{figure}
%
%\end{frame}
%
%
%\begin{frame}
% \frametitle{Results for aorta: undersampling}
%\footnotesize
%
%\begin{figure}[!hbtp]
% \begin{center}
%  \includegraphics[height=0.6\textwidth]{images/histo_blender.png}
%\caption{ \footnotesize Histograms of different undersampling rates for the aortic mesh}
% \end{center}
% \end{figure}
%
%\end{frame}
%
%\begin{frame}
% \frametitle{Results for aorta: undersampling}
%\footnotesize
%
%\begin{figure}[!hbtp]
% \begin{center}
%  \includegraphics[height=0.7\textwidth]{images/undersampling_blender.png}
%\caption{ \footnotesize Different undersampling rates for the aortic mesh}
% \end{center}
% \end{figure}
%
%\end{frame}
%
%



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



\begin{frame}
 \frametitle{Experiments}
\begin{center}
Experiments using real 4D flow data
\end{center}
\end{frame}




\begin{frame}
 \frametitle{Experiments}
\footnotesize

\begin{columns}[c] 
\column{.6\textwidth} % Left column and width

\begin{itemize}
\item<1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
\item<2-> A controled pump (heart rate, peak flow, stroke volume and flow waveform)
\item<3-> A stenosis of $11 \ mm$ of diameter was added in the descending aorta 
\item<4->  The phantom was scanned using a clinical $1.5 \ T$ MR scanner (Philips Achieva, Best, The Netherlands)
\end{itemize}


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

\begin{figure}[!hbtp]
 \begin{center}
 \footnotesize
  \includegraphics[height=\textwidth]{images/phantom.jpg}
\caption{\footnotesize{Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)}}
 \end{center}
 \end{figure}
 
\end{columns}

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%addresource=images/phantom.mp4,
%flashvars={source=images/phantom.mp4}
%]{\includegraphics[width=0.6\linewidth]{images/phantom.jpg}}{VPlayer.swf}
%

\end{frame}





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

\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=0.5\textwidth]{images/phantom_cib.png}
\caption{At peak systole: a) measurements b) corrector field c) corrected measurements: $\vec u_{meas} + \vec w$}
 \end{center}
 \end{figure}

\end{frame}





\section{Conclusions}

\begin{frame}
 \frametitle{Experiments}
\begin{center}
Conclusions
\end{center}
\end{frame}


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

\onslide<1->  Potential of the new quality parameter:

\begin{itemize}
\item<2-> Vector fields has more details
\item<3-> Artifacts recognition
\end{itemize}


\onslide<4->  Future:
\begin{itemize}
\item<5-> The use of 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}







%\includegraphics<1>[height=4.5cm]{images/pat1.png}
%\includegraphics<2>[height=4.5cm]{images/pat2.png}


 
\end{document}