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presentation/pres03.tex

@ -97,7 +97,7 @@ @@ -97,7 +97,7 @@
\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}
\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]
@ -107,8 +107,7 @@ Faculty of Sciences and Engineering\\ @@ -107,8 +107,7 @@ 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.}
}
Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
\date{\today}
@ -134,15 +133,13 @@ University of Groningen\\[0.5cm] @@ -134,15 +133,13 @@ University of Groningen\\[0.5cm]
\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
\item<2-> Full 3D coverage of the region of interest
\item<3-> Rich post-proccesing: derived parameters
\end{itemize}
\column{.54\textwidth} % Right column and width
\onslide<1->
\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=0.9\textwidth]{images/4dflow.png}
@ -156,28 +153,69 @@ University of Groningen\\[0.5cm] @@ -156,28 +153,69 @@ University of Groningen\\[0.5cm]
\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:
\onslide<1-> Disadvantages:
\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
\item<2-> Long scan time
\item<3-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
\item<4-> partial data coverage
\end{itemize}
\vspace{0.5cm}
\onslide<6-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
\begin{columns}[c]
\column{.4\textwidth} % Right column and width
\onslide<5->
\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<6->
\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<7->
\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{0.3cm}
\onslide<8-> 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).
\onslide<9-> This work $\longrightarrow$ 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
@ -189,7 +227,7 @@ University of Groningen\\[0.5cm] @@ -189,7 +227,7 @@ University of Groningen\\[0.5cm]
\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} \\
\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}
@ -203,8 +241,8 @@ University of Groningen\\[0.5cm] @@ -203,8 +241,8 @@ University of Groningen\\[0.5cm]
\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<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*}
@ -233,7 +271,7 @@ for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$. @@ -233,7 +271,7 @@ for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
\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)}
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}
@ -244,12 +282,14 @@ A_{k}(\vec w,p;\vec v ,q ) := \int_{\Omega} \frac{\rho}{\tau} \vec{w} \cdot \vec @@ -244,12 +282,14 @@ A_{k}(\vec w,p;\vec v ,q ) := \int_{\Omega} \frac{\rho}{\tau} \vec{w} \cdot \vec
$ \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}$
\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}
@ -304,55 +344,65 @@ Experiments using synthetic data @@ -304,55 +344,65 @@ Experiments using synthetic data
\begin{frame}
\frametitle{Numerical tests}
\onslide<1->
\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
\column{.4\textwidth} % Right 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
Simulated channel flow as measurements (Stokes flow)
\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}
\includegraphics[height=0.35\textwidth]{images/cilinder_2.png}\\
(b) Channel mesh
%\caption{Aliasing}
\end{center}
\end{figure}
\end{columns}
\end{frame}
\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}
@ -364,105 +414,104 @@ Also perturbations were added into the measurements: @@ -364,105 +414,104 @@ Also perturbations were added into the measurements:
\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: $\vec{w} \times 4$ }
\includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
\caption{\small Fields for the channel in terms of (SNR,$venc$)}
\end{center}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Results for channel: undersampling}
\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=0.6\textwidth]{images/histo_channel.png}
\caption{ \footnotesize Histograms of different undersampling rates for the channel}
\includegraphics[height=0.45\textwidth]{images/channel_ppt_2.png}
\caption{\small Fields for the channel in terms of (SNR,$venc$)}
%\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}
\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=0.6\textwidth]{images/undersampling_press.png}
\caption{ \footnotesize Different undersampling rates for the channel}
\includegraphics[height=0.45\textwidth]{images/channel_ppt_3.png}
\caption{\small Fields for the channel in terms of (SNR,$venc$)}
%\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{Numerical tests: aorta}
\begin{columns}[c]
\column{.6\textwidth} % Left column and width
\frametitle{Results for channel: aliasing and noise}
\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}
For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
\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}
\includegraphics[height=0.45\textwidth]{images/channel_ppt_4.png}
\caption{\small Fields for the channel in terms of (SNR,$venc$)}
%\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{columns}
\end{frame}
\begin{frame}
\frametitle{Results for channel: 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{Results for aorta: aliasing and noise}
\frametitle{Results for channel: 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}
\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{Results for aorta: undersampling}
\frametitle{Results for channel: 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}
\includegraphics[height=0.6\textwidth]{images/histo_channel.png}
\caption{ \footnotesize Histograms of different undersampling rates for the channel}
\end{center}
\end{figure}
@ -471,23 +520,96 @@ Also perturbations were added into the measurements: @@ -471,23 +520,96 @@ Also perturbations were added into the measurements:
\begin{frame}
\frametitle{Results for aorta: undersampling}
\frametitle{Results for channel: 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}
\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 }
@ -512,7 +634,7 @@ Experiments using real 4D flow data @@ -512,7 +634,7 @@ Experiments using real 4D flow data
\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<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}
@ -551,7 +673,7 @@ Experiments using real 4D flow data @@ -551,7 +673,7 @@ Experiments using real 4D flow data
\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}
\caption{At peak systole: a) measurements b) corrector field c) corrected measurements: $\vec u_{meas} + \vec w$}
\end{center}
\end{figure}
@ -578,11 +700,18 @@ Conclusions @@ -578,11 +700,18 @@ Conclusions
\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
\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}

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