diff --git a/presentation/images/4dflow.png b/presentation/images/4dflow.png index 0214908..3ec6fbb 100644 Binary files a/presentation/images/4dflow.png and b/presentation/images/4dflow.png differ diff --git a/presentation/images/channel_aliasing.png b/presentation/images/channel_aliasing.png new file mode 100644 index 0000000..4067d6e Binary files /dev/null and b/presentation/images/channel_aliasing.png differ diff --git a/presentation/images/channel_curves_SNR10.png b/presentation/images/channel_curves_SNR10.png new file mode 100644 index 0000000..6063765 Binary files /dev/null and b/presentation/images/channel_curves_SNR10.png differ diff --git a/presentation/images/channel_curves_SNRinf.png b/presentation/images/channel_curves_SNRinf.png new file mode 100644 index 0000000..fdfd46f Binary files /dev/null and b/presentation/images/channel_curves_SNRinf.png differ diff --git a/presentation/images/channel_noise.png b/presentation/images/channel_noise.png new file mode 100644 index 0000000..52ca65b Binary files /dev/null and b/presentation/images/channel_noise.png differ diff --git a/presentation/images/channel_ppt_1.png b/presentation/images/channel_ppt_1.png new file mode 100644 index 0000000..5297536 Binary files /dev/null and b/presentation/images/channel_ppt_1.png differ diff --git a/presentation/images/channel_ppt_2.png b/presentation/images/channel_ppt_2.png new file mode 100644 index 0000000..ea86d21 Binary files /dev/null and b/presentation/images/channel_ppt_2.png differ diff --git a/presentation/images/channel_ppt_3.png b/presentation/images/channel_ppt_3.png new file mode 100644 index 0000000..3aed2dd Binary files /dev/null and b/presentation/images/channel_ppt_3.png differ diff --git a/presentation/images/channel_ppt_4.png b/presentation/images/channel_ppt_4.png new file mode 100644 index 0000000..f3512f5 Binary files /dev/null and b/presentation/images/channel_ppt_4.png differ diff --git a/presentation/images/channel_under.png b/presentation/images/channel_under.png new file mode 100644 index 0000000..34df21a Binary files /dev/null and b/presentation/images/channel_under.png differ diff --git a/presentation/pres03.tex b/presentation/pres03.tex index ed4b648..8dca76c 100755 --- a/presentation/pres03.tex +++ b/presentation/pres03.tex @@ -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\\ 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] \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] \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] \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] \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 $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$. \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 $ \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 \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} + +\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} @@ -364,8 +414,72 @@ 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: 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=0.45\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: 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=0.45\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{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=0.45\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{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} @@ -373,6 +487,23 @@ Also perturbations were added into the measurements: \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_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 channel: undersampling} \footnotesize @@ -406,86 +537,77 @@ Also perturbations were added into the measurements: -\begin{frame} - \frametitle{Numerical tests: aorta} +%\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} -\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} +%\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} +% +% @@ -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 \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 \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}