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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} |
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%\author[Jeremías Garay Labra] |
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%{Jeremías Garay Labra} |
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\institute[University of Groningen] |
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{ |
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Bernoulli Institute\\ |
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Faculty of Sciences and Engineering\\ |
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University of Groningen\\[0.5cm] |
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%\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf} |
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% \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm] |
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\texttt{Jeremías Garay Labra \\ \ j.e.garay.labra@rug.nl} |
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} |
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\date{\today} |
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\begin{document} |
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\frame{\titlepage} |
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\begin{frame} |
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\frametitle{Index} |
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\tableofcontents |
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\end{frame} |
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\section{4D flow MRI} |
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\begin{frame} |
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\frametitle{4D flow MRI} |
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\begin{columns}[c] |
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\column{.55\textwidth} % Left column and width |
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\footnotesize |
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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. |
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\column{.5\textwidth} % Right column and width |
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\end{columns} |
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\end{frame} |
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\begin{frame} |
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\frametitle{4D flow MRI} |
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\footnotesize |
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Main limitation for its clinical applicability is the long scan times involved. Therefore, multiple strategies emerged in order to make acquisition faster> |
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\begin{itemize} |
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\item Navigator gating |
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\item modest spatial resolutions $2.5 \times 2.5 \times 2.5 \ mm3$ |
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\item partial data coverage |
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\end{itemize} |
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Typical quality estimators are> SNR, VNR, peak flows/velocities, mass conservation (zero divergence |
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We want to introduce a novel measure for quantify the quality of the 4D flow measurements, using the conservation of momentum of the flow. |
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\end{frame} |
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\section{The corrector field} |
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\begin{frame} |
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\frametitle{The corrector field} |
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\begin{columns}[c] |
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\column{.6\textwidth} % Left column and width |
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\footnotesize |
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\onslide<1-> We assume a perfect velocity \begin{eqnarray*} |
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\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} |
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\end{eqnarray*} |
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\onslide<2-> And a corrector field which |
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\begin{align} |
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\vec{u} & \approx \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\ |
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\nabla \cdot \vec w & = 0 \quad \text{in} \quad \Omega \label{eq:correctorDiv} \\ |
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\vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC} |
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\end{align} |
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\onslide<3-> asd |
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\begin{itemize} |
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\footnotesize |
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\item[]<4-> $u = u_{in} \quad \text{in} \quad \Gamma_{inlet}$ |
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\end{itemize} |
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\column{.5\textwidth} % Right column and width |
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\begin{figure}[!hbtp] |
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\onslide<1-> |
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\begin{center} |
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\includegraphics[height=\textwidth]{images/aorta_blender.png} |
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\caption{Aortic mesh } |
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\end{center} |
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\end{figure} |
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\end{columns} |
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\end{frame} |
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\begin{frame} |
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\frametitle{The corrector field} |
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\footnotesize |
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To study the corrector in several scenarios> synthetic data, experimental phantom and healthy volunteers. |
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\end{frame} |
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\begin{frame} |
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\frametitle{The corrector field} |
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\footnotesize |
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different data treatments> aliasing and noise. Undersampling |
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\end{frame} |
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\section{Results} |
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\begin{frame} |
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\frametitle{Results} |
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\footnotesize |
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results for the synthetic data. Comparison againts a perfect correction case with du. |
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\end{frame} |
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\begin{frame} |
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\frametitle{Results} |
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\footnotesize |
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results for experimental phantom |
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\end{frame} |
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\begin{frame} |
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\frametitle{Results} |
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\footnotesize |
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results in healthy volunteers |
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\end{frame} |
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\section{Conclusions} |
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\begin{frame} |
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\frametitle{Results} |
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\footnotesize |
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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 |
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\end{frame} |
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\begin{frame} |
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\begin{center} |
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\huge{Thank you for your time!} |
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\end{center} |
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\end{frame} |
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%\includegraphics<1>[height=4.5cm]{images/pat1.png} |
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%\includegraphics<2>[height=4.5cm]{images/pat2.png} |
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\end{document} |
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