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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}


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

Applying the decomposition $\vec{u}   \approx   \vec{u}_{meas} + \vec{w}$ 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 
\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*}

or in simple terms:
\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

In the Discrete, we can write the problem as follows:

\begin{equation}
A_{k}(\vec w,p;\vec v ,q ) + S^{conv}_{k}(\vec w;\vec v) + S^{press}_{k}(\vec w,p;\vec v ,q)  =  \mathcal{L}_j (\vec v)
\end{equation}

With $ S^{conv}_{k}(\vec w;\vec v)$ and $ S^{press}_{k}(\vec w,p;\vec v ,q)$ terms for the stabilization of the convection  and pressure respectively.


\begin{itemize} 
\small
\item $
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 $
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 $
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 $ \mathcal{L}_j (\vec v) := \int_{\Omega} \frac{\rho}{\tau} \vec{w}^{k-1} \cdot \vec{v} + \mathcal{\ell}_j (\vec v,q)   $
\end{itemize}

\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
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 for channel: 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 for channel: undersampling}
\footnotesize

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

other results concerning undersampling....

\column{.5\textwidth} % Right column and width
\begin{figure}[!hbtp]
 \begin{center}
  \includegraphics[height=1.2\textwidth]{images/undersampling_final.png}
\caption{ \footnotesize Different undersampling rates for the channel}
 \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=0.7\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 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
\item 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 A stenosis of $11 \ mm$ of diameter was added in the descending aorta 
\item  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{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}

