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505 lines
13 KiB
505 lines
13 KiB
\documentclass[xcolor=dvipsnames]{beamer}
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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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% \onslide<1->
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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]{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 the heart and also large arteries, allowing wide variety of options for visualization and quantification.
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Some advantages respect others techniques:
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\begin{itemize}
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\item Full 3D coverage of the region of interest
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\item Retrospective plane positioning
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\item Rich post-proccesing: derived parameters
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\end{itemize}
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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, such as:
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\begin{itemize}
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\item Navigator gating
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\item modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
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\item partial data coverage
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\end{itemize}
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Typical quality estimators: 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 (Navier-Stokes compatibility).
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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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\footnotesize
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We assume a perfect physical velocity field $\vec{u}$
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\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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And a corrector field $\vec{w}$ which satisfies:
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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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The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Continuum problem}
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\footnotesize
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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
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\begin{equation*}
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\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
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\end{equation*}
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\begin{equation*}
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= - \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}
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\end{equation*}
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or in simple terms:
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\begin{equation*}
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A(\vec w,p;\vec v ,q ) = \mathcal{L} (\vec v)
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\end{equation*}
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for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Discrete problem}
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\footnotesize
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In the Discrete, we can write the problem as follows:
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\begin{equation}
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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)
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\end{equation}
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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.
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\begin{itemize}
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\small
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\item $
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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}
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$ \vspace{0.2cm}
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\item $
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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}
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$ \vspace{0.2cm}
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\item $
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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 )
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$ \vspace{0.2cm}
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\item $ \mathcal{L}_j (\vec v) := \int_{\Omega} \frac{\rho}{\tau} \vec{w}^{k-1} \cdot \vec{v} + \mathcal{\ell}_j (\vec v,q) $
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\end{itemize}
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\end{frame}
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\section[Synthetic data]{Experiments using synthetic data }
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\begin{frame}
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\frametitle{Experiments}
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\begin{center}
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Experiments using synthetic data
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Numerical tests}
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\footnotesize
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We tested the corrector using CFD simulations as a measurements, in the following testcases:
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\begin{itemize}
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\item Womersley flow in a cilinder
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\item Navier-Stokes simulations in an aortic mesh
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\end{itemize}
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Also perturbations were added into the measurements:
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\begin{itemize}
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\item velocity aliasing (varying the $venc$ parameter)
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\item additive noise (setting SNR in decibels)
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\item simulated k-space undersampling (compressed sensing for the reconstruction)
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\end{itemize}
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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
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\end{frame}
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\begin{frame}
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\frametitle{Numerical tests: details}
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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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\textbf{Channel:}
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\begin{itemize}
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\item Convective term was neglected
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\item Non-slip condition at walls
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\item Oscilatory pressure at $\Gamma_{inlet}$
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\end{itemize}
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\column{.5\textwidth} % Right column and width
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.3\textwidth]{images/cilinder_2.png}
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\end{center}
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\end{figure}
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\end{columns}
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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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\textbf{Aorta}
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\begin{itemize}
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\item a mild coartation was added in the descending aorta
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\item $u_{inlet}$ simulates a cardiac cycle
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\item 3-element Windkessel for the outlets
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\item Non-slip condition at walls
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\end{itemize}
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\column{.5\textwidth} % Right column and width
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.7\textwidth]{images/aorta_blender.png}
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\caption{\tiny{Channel 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{Results for channel: aliasing and noise}
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\footnotesize
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For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
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\caption{Different perturbation scenarios}
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\end{center}
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\end{figure}
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\end{frame}
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\begin{frame}
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\frametitle{Results for channel: undersampling}
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\footnotesize
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\begin{columns}[c]
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\column{.6\textwidth} % Left column and width
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other results concerning undersampling....
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\column{.5\textwidth} % Right column and width
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=1.2\textwidth]{images/undersampling_final.png}
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\caption{ \footnotesize Different undersampling rates for the channel}
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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{Results for aorta: aliasing and noise}
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.7\textwidth]{images/aorta_perturbation.png}
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\caption{Different perturbation scenarios for the aortic mesh}
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\end{center}
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\end{figure}
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\end{frame}
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\begin{frame}
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\frametitle{Results for aorta: undersampling}
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.7\textwidth]{images/undersampling_blender.png}
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\caption{ \footnotesize Different undersampling rates for the aortic mesh}
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\end{center}
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\end{figure}
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\end{frame}
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\section[4D flow data]{Experiments using real 4D flow data }
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\begin{frame}
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\frametitle{Experiments}
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\begin{center}
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Experiments using real 4D flow data
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Experiments}
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\footnotesize
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\begin{columns}[c]
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\column{.6\textwidth} % Left column and width
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\begin{itemize}
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\item 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
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\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
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\item A stenosis of $11 \ mm$ of diameter was added in the descending aorta
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\item The phantom was scanned using a clinical $1.5 \ T$ MR scanner (Philips Achieva, Best, The Netherlands)
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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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\begin{center}
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\footnotesize
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\includegraphics[height=\textwidth]{images/phantom.jpg}
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\caption{\footnotesize Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)}
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\end{center}
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\end{figure}
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\end{columns}
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%\includemedia[width=0.6\linewidth,height=0.6\linewidth,activate=pageopen,
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%passcontext,
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%transparent,
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%addresource=images/phantom.mp4,
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%flashvars={source=images/phantom.mp4}
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%]{\includegraphics[width=0.6\linewidth]{images/phantom.jpg}}{VPlayer.swf}
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%
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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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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.5\textwidth]{images/phantom_cib.png}
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\caption{At peak systole: a) measurements b) corrector field c) corrected measurements}
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\end{center}
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\end{figure}
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\end{frame}
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\section{Conclusions}
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\begin{frame}
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\frametitle{Conclusions and future}
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\footnotesize
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potential of the new quality parameter:
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\begin{itemize}
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\item analize real data
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\item use the specificity for label zones with strong disagreedment
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\item Use the field for create new inverse problems which can be used for further accelerations
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\end{itemize}
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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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