777 lines
19 KiB
TeX
Executable File
777 lines
19 KiB
TeX
Executable File
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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}
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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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Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
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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{.5\textwidth} % Left column and width
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\footnotesize
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\begin{itemize}
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\item<2-> Full 3D coverage of the region of interest
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\item<3-> Rich post-proccesing: derived parameters
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\end{itemize}
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\onslide<4-> Disadvantages:
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\begin{itemize}
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\item<5-> Long scan time
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\end{itemize}
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\column{.54\textwidth} % Right column and width
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\onslide<1->
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.9\textwidth]{images/4dflow.png}
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\caption{\footnotesize 4D flow MRI of a human thorax}
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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{4D flow MRI}
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\footnotesize
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\onslide<1-> Strategies:
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\begin{itemize}
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\item<2-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
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\item<3-> partial data coverage
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\end{itemize}
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\begin{columns}[c]
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\column{.4\textwidth} % Right column and width
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\onslide<4->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_noise.png} \\
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(a) Noise
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%\caption{Noise}
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\end{center}
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\end{figure}
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\column{.4\textwidth} % Right column and width
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\onslide<5->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_aliasing.png}\\
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(b) Aliasing
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%\caption{Aliasing}
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\end{center}
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\end{figure}
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\column{.4\textwidth} % Right column and width
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\onslide<6->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_under.png}\\
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(c) Undersampling
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%\caption{Aliasing}
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\end{center}
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\end{figure}
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\end{columns}
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\vspace{0.5cm}
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\onslide<7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
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\vspace{0.5cm}
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\onslide<8-> This work $\longrightarrow$ \textbf{conservation of linear momentum} (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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\begin{center}
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Methodology
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\end{center}
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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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\onslide<1-> We assume a perfect physical velocity field $\vec{u}$
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\onslide<2-> \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<3-> And a corrector field $\vec{w}$ which satisfies:
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\onslide<4-> \begin{align}
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\vec{u} & = \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<5-> 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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\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]
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Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that:
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\onslide<2-> \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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\vspace{0.2cm}
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\onslide<3-> or in simple terms:
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\onslide<4-> \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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\onslide<1-> In the Discrete, we can write the problem as follows:
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\onslide<2-> \begin{equation}
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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)}
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\label{eq:Corrector_discrete}
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\end{equation}
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\begin{itemize}
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\small
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\item<3-> $
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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<3-> $ \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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\vspace{0.2cm}
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\item<4-> \color{blue}$
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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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$
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\vspace{0.2cm}
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\item<5-> \color{red}$
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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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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Well-posedness}
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\footnotesize
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\onslide<1->
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\begin{theorem}
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There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
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\end{theorem}
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\onslide<2->
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We can furthermore prove the following energy balance:
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\onslide<3->
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\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}), with $\ell_j(\vec v,q)=0$ it holds
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\begin{equation*}\label{eq:energy}
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\| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)}
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\end{equation*}
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under the condition
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\begin{equation*}\label{eq:condstab}
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\mu \geq C_\Omega^2 \rho \| \nabla \vec u_{meas}^k\|_\infty
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\end{equation*}
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\end{theorem}
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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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\onslide<1->
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\footnotesize
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\begin{columns}[c]
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\column{.4\textwidth} % Right column and width
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\footnotesize
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Simulated channel flow as measurements (Stokes flow)
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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=0.35\textwidth]{images/cilinder_2.png}\\
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(b) Channel mesh
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%\caption{Aliasing}
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\end{center}
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\end{figure}
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\end{columns}
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\vspace{0.2cm}
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%\onslide<1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
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%\onslide<2->
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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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\onslide<2-> Afterwards, perturbations were added:
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\begin{itemize}
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\item<3-> velocity aliasing (varying the $venc$ parameter)
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\item<4-> additive noise (setting SNR in decibels)
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\item<5-> simulated k-space undersampling (compressed sensing for the reconstruction)
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\end{itemize}
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%\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
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\end{frame}
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%
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%\begin{frame}
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% \frametitle{Numerical tests: channel}
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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=1.0\textwidth]{images/cilinder.png}
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% \caption{3D 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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%
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\begin{frame}
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\frametitle{Numerical tests}
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\begin{center}
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Results
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Aliasing and noise}
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\footnotesize
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\onslide<1-> For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
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\onslide<2->
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
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\caption{\small Fields for the channel: $(SNR,venc) = (\infty,120\%)$. $\vec{w} \times 200$}
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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{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=0.45\textwidth]{images/channel_ppt_2.png}
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\caption{\small Fields for the channel: $(SNR,venc) = (\infty,80\%)$. $\vec{w} \times 4$ }
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%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
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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{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=0.45\textwidth]{images/channel_ppt_3.png}
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\caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,120\%)$. $\delta \vec{u}, \vec{w} \times 4$}
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%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
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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{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=0.45\textwidth]{images/channel_ppt_4.png}
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\caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,80\%)$. $\vec{w} \times 4$}
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%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
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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{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=0.5\textwidth]{images/channel_curves_SNRinf.png}
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\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
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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{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=0.5\textwidth]{images/channel_curves_SNR10.png}
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\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
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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{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=0.6\textwidth]{images/histo_channel.png}
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\caption{ \footnotesize Histograms of different undersampling rates for the channel}
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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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%
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%\begin{figure}[!hbtp]
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% \begin{center}
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% \includegraphics[height=0.6\textwidth]{images/undersampling_press.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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%
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%
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%\end{frame}
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%
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%\begin{frame}
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% \frametitle{Numerical tests: aorta}
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%
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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=1.0\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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%
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%
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%\end{frame}
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%
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%
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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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%
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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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%
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%\end{frame}
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%
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%
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%\begin{frame}
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% \frametitle{Results for aorta: undersampling}
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%\footnotesize
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%
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%\begin{figure}[!hbtp]
|
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% \begin{center}
|
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% \includegraphics[height=0.6\textwidth]{images/histo_blender.png}
|
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%\caption{ \footnotesize Histograms of different undersampling rates for the aortic mesh}
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% \end{center}
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% \end{figure}
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%
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%\end{frame}
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%
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%\begin{frame}
|
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% \frametitle{Results for aorta: undersampling}
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%\footnotesize
|
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%
|
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%\begin{figure}[!hbtp]
|
|
% \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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%
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%\end{frame}
|
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%
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%
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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}
|
|
Experiments using real 4D flow data
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\end{center}
|
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\end{frame}
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\begin{frame}
|
|
\frametitle{Experiments}
|
|
\footnotesize
|
|
|
|
\begin{columns}[c]
|
|
\column{.6\textwidth} % Left column and width
|
|
|
|
\begin{itemize}
|
|
\item<1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
|
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\item<2-> A controled pump (heart rate, peak flow, stroke volume and flow waveform)
|
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\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}
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|
|
\column{.5\textwidth} % Right column and width
|
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|
|
\begin{figure}[!hbtp]
|
|
\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)}}
|
|
\end{center}
|
|
\end{figure}
|
|
|
|
\end{columns}
|
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|
|
%\includemedia[width=0.6\linewidth,height=0.6\linewidth,activate=pageopen,
|
|
%passcontext,
|
|
%transparent,
|
|
%addresource=images/phantom.mp4,
|
|
%flashvars={source=images/phantom.mp4}
|
|
%]{\includegraphics[width=0.6\linewidth]{images/phantom.jpg}}{VPlayer.swf}
|
|
%
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|
\end{frame}
|
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\begin{frame}
|
|
\frametitle{Results}
|
|
\footnotesize
|
|
|
|
\begin{figure}
|
|
\begin{subfigure}{.31\textwidth}
|
|
\centering
|
|
\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png}
|
|
\caption*{(a) $\vec{u}_{meas}$}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.01\textwidth}
|
|
\hfill
|
|
\end{subfigure}
|
|
\begin{subfigure}{.31\textwidth}
|
|
\centering
|
|
\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png}
|
|
\caption*{(b) $\vec{w}$}
|
|
\end{subfigure}
|
|
\begin{subfigure}{.01\textwidth}
|
|
\hfill
|
|
\end{subfigure}
|
|
\begin{subfigure}{.31\textwidth}
|
|
\centering
|
|
\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png}
|
|
\caption*{(c) $\vec{u}_{meas}+\vec{w}$}
|
|
\end{subfigure}
|
|
\caption{Measurements, corrector fields and corrected velocities for all the cases.}
|
|
\label{fig:phantom_resolution}
|
|
\end{figure}
|
|
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
|
|
\section{Conclusions}
|
|
|
|
\begin{frame}
|
|
\frametitle{Experiments}
|
|
\begin{center}
|
|
Conclusions
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Conclusions and future work}
|
|
\footnotesize
|
|
|
|
\onslide<1-> Potential of the new quality parameter:
|
|
|
|
\begin{itemize}
|
|
\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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|
|
|
|
|
|
\begin{frame}
|
|
\begin{center}
|
|
\huge{Thank you for your time!}
|
|
\end{center}
|
|
\end{frame}
|
|
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|
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|
|
%\includegraphics<1>[height=4.5cm]{images/pat1.png}
|
|
%\includegraphics<2>[height=4.5cm]{images/pat2.png}
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\end{document}
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