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presentation/pres03.tex

@ -1,4 +1,5 @@
\documentclass[xcolor=dvipsnames]{beamer}
\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
\usepackage{amsmath}
%\documentclass{beamer}
\usepackage[english]{babel}
%\usepackage[latin1]{inputenc}
@ -13,11 +14,20 @@
%\usepackage{graphicx}
%\usepackage{movie15}
%\usepackage{media9}[2013/11/04]
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{multimedia}
\usepackage{media9}
\usepackage{listings,xcolor,caption, mathtools, wrapfig}
\usepackage{amsfonts}
\usepackage{amssymb,graphicx,enumerate}
\usepackage{hyperref}
\usepackage[normalem]{ulem} % for strike out command \sout
@ -97,7 +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 \\ \ j.e.garay.labra@rug.nl}
\texttt{Jeremías Garay Labra join with Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
}
\date{\today}
@ -124,13 +134,12 @@ University of Groningen\\[0.5cm]
\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:
\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 Full 3D coverage of the region of interest
\item Retrospective plane positioning
\item Rich post-proccesing: derived parameters
\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{.5\textwidth} % Right column and width
@ -142,16 +151,22 @@ Some advantages respect others techniques:
\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:
\onslide<1-> Main limitation $\longrightarrow$ long scan times involved.\\
\vspace{0.2cm}
\onslide<2-> In order to mitigate:
\begin{itemize}
\item Navigator gating
\item modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
\item partial data coverage
\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}
Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
\vspace{0.5cm}
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).
\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}
@ -162,19 +177,19 @@ We want to introduce a novel measure for quantify the quality of the 4D flow mea
\frametitle{The corrector field}
\footnotesize
We assume a perfect physical velocity field $\vec{u}$
\begin{eqnarray*}
\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*}
And a corrector field $\vec{w}$ which satisfies:
\begin{align}
\onslide<3-> And a corrector field $\vec{w}$ which satisfies:
\onslide<4-> \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.
\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}
@ -183,16 +198,18 @@ The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow me
\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*}
\onslide<1-> 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
\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*}
or in simple terms:
\begin{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*}
@ -207,27 +224,26 @@ for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
\frametitle{The corrector field: Discrete problem}
\footnotesize
In the Discrete, we can write the problem as follows:
\onslide<1-> 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)
\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}
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 $
\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 $
\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 $
\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 )
$ \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}
@ -235,6 +251,37 @@ $ \vspace{0.2cm}
\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 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}
@ -253,27 +300,29 @@ Experiments using synthetic data
\frametitle{Numerical tests}
\footnotesize
We tested the corrector using CFD simulations as a measurements, in the following testcases:
\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 velocity aliasing (varying the $venc$ parameter)
\item additive noise (setting SNR in decibels)
\item simulated k-space undersampling (compressed sensing for the reconstruction)
\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}
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
%\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: details}
\frametitle{Numerical tests: channel}
\begin{columns}[c]
\column{.6\textwidth} % Left column and width
\footnotesize
@ -284,82 +333,108 @@ All simulations were done using a stabilized finite element method implemented i
\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}
\includegraphics[height=1.0\textwidth]{images/cilinder.png}
\caption{3D channel mesh}
\end{center}
\end{figure}
\end{columns}
\end{frame}
\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
\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=0.7\textwidth]{images/aorta_blender.png}
\caption{\tiny{Channel mesh}}
\includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
\caption{Different perturbation scenarios}
\end{center}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Results for channel: aliasing and noise}
\frametitle{Results for channel: undersampling}
\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}
\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}
other results concerning undersampling....
\column{.5\textwidth} % Right column and width
\footnotesize
\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=1.2\textwidth]{images/undersampling_final.png}
\caption{ \footnotesize Different undersampling rates for the channel}
\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
@ -374,6 +449,23 @@ other results concerning undersampling....
\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
@ -413,10 +505,10 @@ Experiments using real 4D flow data
\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)
\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}
@ -426,7 +518,7 @@ Experiments using real 4D flow data
\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)}
\caption{\footnotesize{Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)}}
\end{center}
\end{figure}
@ -467,14 +559,15 @@ Experiments using real 4D flow data
\begin{frame}
\frametitle{Conclusions and future}
\frametitle{Conclusions and future work}
\footnotesize
potential of the new quality parameter:
\onslide<1-> 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
\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}

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