\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
@ -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.\\
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).
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
There exists a unique solution of Problem \ref{eq:Corrector_discrete} under condition: $$\rho/\tau+ C_\Omega^{-2}\mu/2-\rho3\|\nabla\vec u_{meas}^k\|_\infty > 0$$ 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
\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
\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