diff --git a/presentation/images/histo_blender.png b/presentation/images/histo_blender.png new file mode 100644 index 0000000..01ed7f0 Binary files /dev/null and b/presentation/images/histo_blender.png differ diff --git a/presentation/images/histo_channel.png b/presentation/images/histo_channel.png new file mode 100644 index 0000000..aed07a2 Binary files /dev/null and b/presentation/images/histo_channel.png differ diff --git a/presentation/images/norms_blender.png b/presentation/images/norms_blender.png new file mode 100644 index 0000000..5498cbf Binary files /dev/null and b/presentation/images/norms_blender.png differ diff --git a/presentation/images/norms_channel.png b/presentation/images/norms_channel.png new file mode 100644 index 0000000..ddc9b2a Binary files /dev/null and b/presentation/images/norms_channel.png differ diff --git a/presentation/images/undersampling_press.png b/presentation/images/undersampling_press.png new file mode 100644 index 0000000..2cdecfc Binary files /dev/null and b/presentation/images/undersampling_press.png differ diff --git a/presentation/pres03.tex b/presentation/pres03.tex index 54747a4..2cb30b6 100755 --- a/presentation/pres03.tex +++ b/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 @@ -284,16 +333,75 @@ 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{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=0.5\textwidth]{images/perturbation_pres.png} +\caption{Different perturbation scenarios} + \end{center} + \end{figure} + + +\end{frame} + + +\begin{frame} + \frametitle{Results for channel: undersampling} +\footnotesize + +\begin{figure}[!hbtp] + \begin{center} + \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 @@ -311,8 +419,8 @@ All simulations were done using a stabilized finite element method implemented i \footnotesize \begin{figure}[!hbtp] \begin{center} - \includegraphics[height=0.7\textwidth]{images/aorta_blender.png} -\caption{\tiny{Channel mesh}} + \includegraphics[height=1.0\textwidth]{images/aorta_blender.png} +\caption{Aortic mesh} \end{center} \end{figure} \end{columns} @@ -321,43 +429,10 @@ All simulations were done using a stabilized finite element method implemented i \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} @@ -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}