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\title{\huge Diffusion Tensor Imaging: \\ An Introduction \\ \small Seminar "Processing of Matrix-Valued Images"}
\author{Jan Hendrik Dithmar}
\institute{\texttt{jadi5003@stud.uni-saarland.de}}
\date{May 07, 2008}

\begin{document}

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\section{Motivation}
\begin{frame}[t]
	\frametitle{Motivation \\}
	
	\begin{itemize}
		\item We want to help patients who suffer from a disease like a stroke, epilepsy, multiple sclerosis, dementia and other white-matter diseases.
		\item Optimally, the measurements should be non-invasive.
		\item We want to discuss the following techniques:
		%\item Diffusion Tensor MRI is a good choice to do a non-invasive measurement of the brain and other parts in the (human) body.
	\end{itemize}
	\begin{center}
		\includegraphics[width=0.4\textwidth, height=0.4\textheight]{pictures/mri-diagram.eps}
	\end{center}
\end{frame}

\section{Diffusion MRI}
\subsection{Magnetic Resonance Imaging}
\begin{frame}[t]
	\frametitle{Magnetic Resonance Imaging}
	
	\begin{center}
		\includegraphics[width=0.8\textwidth, height=0.8\textheight]{pictures/Modern_3T_MRI.eps} \\
		{\scriptsize Modern 3 tesla clinical MRI scanner. Source: \textit{Wikipedia}.}
	\end{center}
\end{frame}

\begin{frame}[fragile, t]
	\frametitle{Magnetic Resonance Imaging (2)}
	
	\begin{center}
		\includegraphics[width=0.7\textwidth, height=0.7\textheight]{pictures/mri-scanner.eps} \\
	\end{center}
	{\scriptsize Schematic of a MRI scanner. Source: \verb!http://www.magnet.fsu.edu/education/tutorials/magnetacademy/mri/!}
\end{frame}

\begin{frame}[t]
	\frametitle{Magnetic Resonance Imaging (3) \\}
	
	\begin{itemize}
		\item MRI (Kernspintomographie) is a medical imaging technique used to visualize the structure and the function of the body.
		\item Uses a \important{strong magnetic field} instead of radiation.
		\item Measures \important{displacements of water molecules} using gradients in different directions. Usually six (or even four) directions are used.
		\item Better soft tissue contrast than for example for CT.
	\end{itemize}
	\begin{center}
		\includegraphics[width=0.35\textwidth, height=0.35\textheight]{pictures/ct-image-with-eyes.eps} \quad
		\includegraphics[width=0.35\textwidth, height=0.35\textheight]{pictures/mri-image-with-eyes.eps}
	\end{center}
	{\scriptsize \textbf{Left:} CT image of the head. Eyes can be clearly seen. \textbf{Right:} Approximately the same image, but measured with MRI. Source: \textit{Wikipedia}.}
\end{frame}

\subsection{Diffusion Magnetic Resonance Imaging}
\begin{frame}[t]
	\frametitle{Diffusion Magnetic Resonance Imaging \\}
	
	\begin{itemize}
		\item Diffusion Magnetic Resonance Imaging measures the \important{diffusion} of water molecules.
%		\item Diffusion is usually described by a \important{diffusion equation}. \\
%			Example: linear isotropic diffusion in 1-D
%			$$\partial_t u = \partial_{xx} u$$
%		\item \important{Brownian motion} is the \important{random movement of particles} suspended in a liquid or gas or the mathematical model used to describe such random movements.
		\item In the case of an \important{isotropic medium}, water molecules move randomly according to Brownian motion.
		\item In biological tissues, diffusion can be \important{anisotropic}.
	\end{itemize}
	\ \\[0.5cm]
	Reminder:
	\begin{itemize}
		\item Anisotropy: the property of being directionally dependent.
		\item Isotropy: the opposite of anisotropy.
	\end{itemize}
\end{frame}

\subsection{The Diffusion Tensor}
\begin{frame}[t]	
	\frametitle{The Diffusion Tensor \\}
	
	\begin{itemize}
		\item In an \important{isotropic} medium, the diffusion can be fully described by a single (scalar) parameter. The \important{attenuation} $A$ is described as follows:
			$$A = \exp(-bD)$$
			where
			\begin{itemize}
				\item $b$: "b-factor" which characterizes the magnetic field gradient pulses (timing, amplitude, shape),
				\item $D$: diffusion coefficient.
			\end{itemize}
	\end{itemize}
\end{frame}

\begin{frame}[t]
	\frametitle{The Diffusion Tensor (2) \\}
	
	\begin{itemize}
		\item In an \important{anisotropic} medium, one requires a tensor instead of a single coefficient. The tensor is {\color{blue}{symmetric}} and looks like this:
			$$\textbf{D} = \begin{array}{ccc}\color{black}{D_{xx}} & \color{blue}{D_{xy}} & \color{blue}{D_{xz}} \\ \color{blue}{D_{xy}} & \color{black}{D_{yy}} & \color{blue}{D_{yz}} \\ \color{blue}{D_{xz}} & \color{blue}{D_{yz}} & \color{black}{D_{zz}}\end{array}$$
			where $D_{xx}$ is the diffusivity in $x$, $D_{yy}$ is the diffusivity in $y$ and $D_{zz}$ is the diffusivity in $z$. $\textbf{D}$ is a positive definite 3 $\times$ 3 matrix.
		\item The \important{attenuation} $A$ is described as follows:
			$$A = \exp\left(- \sum_{i=x,y,z} \sum_{j=x,y,z} b_{i,j} D_{i,j}\right)$$
			where $\textbf{b}$ is now a matrix that replaces the "b-factor".
	\end{itemize}
\end{frame}

\begin{frame}[t]
	\frametitle{The Diffusion Tensor (3)}
	
	\begin{center}
		\includegraphics[height=0.7\textheight]{pictures/corpus-callosum.eps}
	\end{center}
	{\scriptsize The diffusion tensor, axial slices. Diffusivity along $x$, $y$ and $z$, shown in $D_{xx}$ (a), $D_{yy}$ (d) and $D_{zz}$ (f) images. Authors: Le Bihan et al.}
\end{frame}

\begin{frame}[t]
	\frametitle{The Diffusion Tensor (4)}
	
	\begin{center}
		\includegraphics[height=0.63\textheight]{pictures/different-shapes.eps}
	\end{center}
	{\scriptsize Four microstructures found in the brain. Black lines are barriers to the movement of water molecules. Red shapes indicate expected diffusion. \textbf{(a)} fluid-filled region, \textbf{(b)} isotropic grey matter, \textbf{(c)} and \textbf{(d)} white matter with one and two dominant fiber orientations. Author: Daniel C. Alexander.}
\end{frame}

\subsection{Diffusion Tensor MRI}
\begin{frame}[t]
	\frametitle{Diffusion Tensor MRI \\}
	
	\begin{itemize}
		\item Let $p$ be a probability density function of particle displacements $\textbf{x}$ over a fixed time $t$.
		\item Diffusion Tensor MRI computes the diffusion tensor on the assumption that $p$ is a zero-mean trivariate Gaussian distribution:
			$$G(\textbf{x}; \textbf{D}, t) = ((4 \pi t)^3 \det(\textbf{D}))^{-\frac{1}{2}} \exp \left(-\frac{\textbf{x}^T \textbf{D}^{-1} \textbf{x}}{4t}\right).$$
		\item This indicates the probability density for a water molecule to move from $\textbf{x} = 0$ at time $t = 0$ to the point $\textbf{x}$ in time $t$.
		\item We get the \important{Stejskal-Tanner formula} which gives the following:
			$$S(\textbf{q}) = S_0 \cdot \exp\left(-t \textbf{q}^T \textbf{D} \textbf{q}\right)$$
			where $S_0$ is the signal strength of an unweighted measurement and $S(\textbf{q})$ is the signal strength under a gradient field in direction $\textbf{q}$.
	\end{itemize}
\end{frame}

\begin{frame}[t]
	\frametitle{Diffusion Tensor MRI (2) \\}
	
	\begin{itemize}
		\item It is difficult to interpret raw tensor data.
		\item Any positive definite 3 $\times$ 3 matrix can be represented by an ellipsoid. DT-MRI depicts the diffusion tensor $\textbf{D}$ by ellipsoids.
	\end{itemize}
%	\begin{columns}
%		\column{0.5\textwidth}
%			\begin{center}
%				\includegraphics[height=0.64\textheight]{pictures/different-shapes.eps}
%			\end{center}
%		\column{0.45\textwidth}
%			{\scriptsize \ \\ \ \\ \ \\ \ \\ \ \\ \ \\ \ \\ Four microstructures found in the brain. Black lines are barriers to the movement of water molecules. Red shapes indicate expected diffusion. \textbf{(a)} fluid-filled region, \textbf{(b)} isotropic grey matter, \textbf{(c)} and \textbf{(d)} white matter with one and two dominant fiber orientations. Author: Daniel C. Alexander.}
%	\end{columns}
\end{frame}

\subsection{How to interpret ellipsoids?}
\begin{frame}[t]
	\frametitle{How to interpret ellipsoids? \\}
	
	\begin{itemize}
		\item An ellipsoid's principal axes point into the directions of the eigenvectors of $\textbf{D}$ and have the eigenvalues of $\textbf{D}$ as their length.
		\item So it is a \important{3-D representation} of the \important{diffusion distance} in a given \important{diffusion time} $T_d$.
		\item The \important{eccentricity} (Exzentrizit\"{a}t) of the ellipsoid provides information about the \important{degree of anisotropy} and its \important{symmetry}. Isotropic diffusion would be seen as a sphere.
	\end{itemize}
	\begin{columns}
		\column[b]{3cm}
			\includegraphics[height=0.4\textheight]{pictures/simple-example.eps}
		\column[b]{6.3cm}
			{\scriptsize Two ellipsoids. \\ \textit{(Visualisation software written by Stephan Didas, Luis Pizarro, and Stephan Zimmer.)}}
	\end{columns}
\end{frame}

\subsection{Example}
\begin{frame}[t]
	\frametitle{Example}
	
	\begin{center}
		\includegraphics[height=0.42\textheight]{pictures/adultslice-whole_brain.eps} \,
		\includegraphics[height=0.42\textheight]{pictures/adultslice-cc.eps} \,
		\includegraphics[height=0.42\textheight]{pictures/adultslice-cc-big_scaling.eps}
	\end{center}
	{\scriptsize Diffusion Tensors visualized by ellipsoids. \textbf{Left:} One brain slice of a human head on top view. \textbf{Middle:} The part where the corpus callosum is located. \textbf{Right:} Same view as before, but with higher scaling of the ellipsoids. This gives a better visual impression of what is isotropic and anisotropic diffusion. Images measured by the University of Eindhoven. \textit{(Visualisation software written by Stephan Didas, Luis Pizarro, and Stephan Zimmer.)}}
\end{frame}

\subsection{Another visualisation}
\begin{frame}[t]
	\frametitle{Another visualisation}
	
	\begin{center}
		\includegraphics[height=0.7\textheight]{pictures/3d-dt.eps}
	\end{center}
	{\scriptsize 3-D visualisation of the diffusion tensor. Main eigen-values are shown as cylinders, the length of which is scaled with the degree of anisotropy. Corpus callosum fibers are displayed around ventricles, thalami, putamen, and caudate nuclei. Authors: Le Bihan et al.}
\end{frame}

\subsection{How many scanning directions?}
\begin{frame}[t]
	\frametitle{How many scanning directions?}
	
	\begin{center}
		\includegraphics[height=0.7\textheight]{pictures/number-of-directions.eps}
	\end{center}
	{\scriptsize Schemes for directional sampling. Initial sampling schemes consisted of six (or even four) directions of measurement. Progress is made in scanning space more uniformly along many directions. Authors: Le Bihan et al.}
\end{frame}

\subsection{Applications}
\begin{frame}[t]
	\frametitle{Applications \\}
	
	\begin{itemize}
		\item \important{Fibers in the brain} can be detected because water molecules prefereably move along them. This allows to diagnose
			\begin{itemize}
		{\color{blue}{
				\item brain ischemia (Blutleere)
				\item strokes (Schlaganf\"{a}lle)
				\item multiple sclerosis
				\item schizophrenia
				\item epilepsy
				\item dementia
		}}
			\end{itemize}
			and to study {\color{blue}{brain connectivity}}.
		\item One can also have a look at \important{fibers in the body}, for example muscle fibers. Typical organs would be the tongue or the heart. However, in this case the images suffer from strong respiratory motion artifacts.
	\end{itemize}	
\end{frame}

\comments{
\section{Outlook}
\begin{frame}[t]
	\frametitle{Outlook: Diffusion-Weighted MRI \\}
	
	\begin{itemize}
		\item Aquires measurements that are sensitive to the motion of nuclei possessing a net spin.
		\item Spin-echo sequences can detect displacements of spins during a measurement.
		\item One can do this by using one radio-frequency (RF) pulse which tips the spins and apply another RF pulse which negates the phase of the spins.
		\item Stationary spins do not experience any net phase shift.
		\item Diffusion measurement are only possible in one direction. To measure diffusion in more than one direction one has to repeat the measurement with respect to the new direction.
	\end{itemize}
\end{frame}
}

\section{Summary}
\subsection{Summary}
\begin{frame}[t]
	\frametitle{Summary \\}
	
	\begin{itemize}
		\item Diffusion Tensor MRI is now a routine clinical technique. It can detect anisotropies.
		\item It is a non-invasive, harmless measurement technique to get images of the brain.
		\item The image depicts the diffusion of water molecules, which is important for many applications, namely fiber tracking or brain connectivity.
		\item Diffusion tensors can be visualized by ellipsoids.
	\end{itemize}
\end{frame}

\subsection{References}
\begin{frame}[fragile, t]
	\frametitle{References \\}
	
	\begin{itemize}
		\item Denis Le Bihan et al., \textit{Diffusion Tensor Imaging: Concepts and Applications}. Journal of Magnetic Resonance Imaging 13, p. 534-546, 2001.
		\item Daniel C. Alexander, \textit{An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond}. Department of Computer Science, University College London, p. 77-100, 2006.
		\item {\small \verb!http://de.wikipedia.org/wiki/Diffusions-Tensor-Bildgebung!} \\
			\textit{\small (Wikipedia article on DT-MRI (in german). Very good article, better than the English one.)}
		\item {\small \verb!http://en.wikipedia.org/wiki/Magnetic_resonance_imaging!} \\
			\textit{\small (Wikipedia article on Magnetic Resonance Imaging.)}
		\item {\small \verb!http://de.wikipedia.org/wiki/Magnetresonanztomographie!} \\
			\textit{\small (Wikipedia article on Magnetic Resonance Imaging (in german).)}
%		\item {\small \verb!http://en.wikipedia.org/wiki/Brownian_motion!} \\
%			\textit{\small (Wikipedia article on Brownian motion.)}
	\end{itemize}
\end{frame}

\section{Thank you}
\begin{frame}
	\begin{center}
		\ \\ \ \\
		\Huge{\color{blue}{Thanks for your attention!}} \\
		\Large{\color{blue}{Feel free to ask questions.}}
	\end{center}
\end{frame}

\end{document}