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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 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

\onslide<1-> 4D flow MRI has been shown potential in the assesment of blood flow dynamics in the heart and also large arteries.\\[0.2cm]
\onslide<2-> Some advantages:
\begin{itemize}
\item<3-> Full 3D coverage of the region of interest
\item<4-> Retrospective plane positioning
\item<5-> Rich post-proccesing: derived parameters 
\end{itemize}

\column{.54\textwidth} % Right column and width
\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->  Main limitation $\longrightarrow$ long scan times involved.\\
\vspace{0.2cm}
\onslide<2-> In order to mitigate:
\begin{itemize}
\item<3-> Navigator gating
\item<4-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
\item<5-> partial data coverage  
\end{itemize}

\vspace{0.5cm}

\onslide<6-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 

\vspace{0.5cm}

\onslide<7-> This work $\longrightarrow$  conservation of linear momentum (Navier-Stokes compatibility).

\end{frame}


\section[]{The corrector field}

\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 the following:\\ 
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{red}{S^{conv}_{k}(\vec w;\vec v)} + \color{blue}{S^{press}_{k}(\vec w,p;\vec v ,q)} \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{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}
\item<5-> \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 ) 
$
\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}

\footnotesize

\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<3->
Also perturbations were added into the measurements: 
\begin{itemize}
\item<4->  velocity aliasing (varying the $venc$ parameter)
\item<5->  additive noise (setting SNR in decibels)
\item<6->  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{Results for channel: 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=0.5\textwidth]{images/perturbation_pres.png}
\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{Results for channel: 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 injects to the system a blood mimicking fluid and allows the control of: 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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\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}
 \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-> The detect zones with strong disagreedment
\item<3-> To better recognize common acquisition artifacts
\item<4-> 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}







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\end{document}

