@ -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 . 2 cm }
\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 . 2 cm }
\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 . 2 cm }
\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 } % Lef t 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}
\end { frame}
\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}
@ -364,105 +414,104 @@ 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: unders amp ling}
\frametitle { Results for channel: alias ing 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=0.6\textwidth] { images/histo_ channel.png}
\caption { \footnotesize Histograms of different undersampling rates for the channel}
\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: unders amp ling}
\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=0.6\textwidth] { images/undersampling_ press.png}
\caption { \footnotesize Different undersampling rates for the channel}
\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 { Numerical tests: aorta}
\begin { columns} [c]
\column { .6\textwidth } % Left column and width
\frametitle { Results for channel: aliasing and noise}
\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}
For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\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}
\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 { columns}
\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}
\end { frame}
\begin { frame}
\frametitle { Results for aorta: aliasing and noise}
\frametitle { Results for channel : 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 }
\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 aorta: undersampling}
\frametitle { Results for channel : 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 }
\includegraphics [height=0.6\textwidth] { images/histo_ channel .png}
\caption { \footnotesize Histograms of different undersampling rates for the channel }
\end { center}
\end { figure}
@ -471,23 +520,96 @@ Also perturbations were added into the measurements:
\begin { frame}
\frametitle { Results for aorta : undersampling}
\frametitle { Results for channel : 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 }
\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}
% \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}
%
%
\section [4D flow data] { Experiments using real 4D flow data }
@ -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}
