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\begin{document}


\title{\small Information Processing in Medical Imaging 2005\\
\LARGE \bf Unified Statistical Approach to Cortical Thickness
Analysis}
\author{Moo K. Chung$^{1,2,3}$, Steve Robbins$^4$, Alan C. Evans$^4$\\
$^1$Department of Statistics, $^2$ Department of Biostatistics and
Medical Informatics\\
$^3$Waisman Laboratory for Brain Imaging and Behavior\\
University of Wisconsin-Madison\\
%,Madison, WI 53706\\
$^4$Montreal Neurological Institute, McGill University, Canada\\
\tt{mchung@stat.wisc.edu, stever, alan@bic.mni.mcgill.ca}}


\date{July 13, 2005}
 \maketitle \thispagestyle{empty}

\begin{multicols}{3}
\section*{Introduction} We present a unified image processing and
analysis framework for cortical thickness in characterizing a
clinical population. Due to the convoluted non-Euclidean surface
geometry, data smoothing and analysis on the cortex are inherently
difficult. When measurements lie on a curved surface, it is natural
to assign kernel smoothing weights based on the geodesic distance
along the surface rather than the Euclidean distance. We present a
new data smoothing framework that addresses this problem implicitly
without computing the geodesic distance.

\section*{Cortical thickness}
The two surfaces that bound the gray matter in the human brain are
extracted from magnetic resonance images (MRI). The distance between
the two surfaces is usually referred as the {\em cortical thickness}
(Figure 1). The details of extracting the cortical surfaces can be
found in [2, 3, 4].

\begin{figure}
%\centering
\small
\includegraphics[width=182mm]{twosurface.eps}
\caption{Left: part of the cortex showing the outer and inner
surface that bound gray matter. Right: enlargement of the boxed
region. The cortical thickness measures the distance between outer
and inner surfaces.} \label{fig:thickness}
\end{figure}

\section*{Surface registration}

\begin{figure}
\centering \small
\includegraphics[width=182mm]{sulcus.eps}
\caption{\label{fig:spam-cachia-labels} Demonstration of surface
registration. Automatically generated traces of the central and
superior temporal sulcal fundi are normalized on the unit sphere
[2]. The first column shows the traces generated for the template
surface. The second column shows the probability of sulcal matching
based on 149 normal subjects before any surface normalization. The
third column shows the probability after surface normalization.}
\end{figure}

Spatial normalization of the surfaces is necessary to facilitate
vertex-by-vertex inter-subject thickness comparison. The surface
deformation field from one surface to the template surface is
obtained by minimizing an objective function that measures the
global fit of two surfaces, while maximizing the smoothness of the
deformation in such a way that the pattern of gyral ridges are
matched smoothly. The detail can be found in [2].




\section*{Heat kernel smoothing}

\begin{figure}
\centering \small
\includegraphics[width=182mm]{simulation.eps} \caption{Top: Heat kernel
 smoothing of cortical thickness with $\sigma = 1$ and $k=20, 100, 200$ iterations.  Bottom: Heat kernel
smoothing on simulated data with $\sigma=1$ and $k=20, 200, 5000$
iterations.} \label{fig:hk}
\end{figure}

The cortical thickness measurements are always contaminated with
noise. In order to increase the signal-to-noise ratio, new
surface-based smoothing called {\em heat kernel smoothing} is
developed [2]. Observations $Y$ measured on the cortex $\partial
\Omega$ is assumed to follow the additive model of true signal
$\theta$ plus noise $\epsilon$: $$Y(p) = \theta(p) + \epsilon(p), p
\in \partial \Omega \label{eq:model1}.$$ Then we estimate $\theta$
using {\em heat kernel smoothing}: $$ \widehat{\theta}(p) =
K_{\sigma} *Y (p)=\int_{\partial
\Omega}K_{\sigma}(p,q)Y(q)\;d\mu(q),$$ where $\mu(q)$ is the surface
measure and $K_{\sigma}$ is the heat kernel with bandwidth $\sigma$
[2]. This is a more efficient technique than {\em diffusion
smoothing} [1, 3, 4]. In the diffusion smoothing formulation, the
isotropic diffusion equation
$$\frac{\partial f}{\partial t} = \nabla f, f(p,0)=Y(p)$$
is solved via the finite element method on the cortex [3, 4].
$K_{\sigma}*Y$ is the solution of the diffusion equation after time
$t=\sigma^2/2$.


For small $\sigma$ we approximate $K_{\sigma}$ by the {\em
parametrix expansion} [5]: \bq K_{\sigma}(p,q) &=&
\frac{1}{(2\pi\sigma)^{1/2}}e^{-\frac{d^2(p,q)}{2\sigma^2}}\big[u_0(p,q)
+ O(\sigma^2)\big],\eq where $d(p,q)$ is the geodesic distance
between $x$ and $y$ and $u_0(p,q) \to 1$ as $p \to q$.

For large $\sigma$, the smoothing is performed by iteratively with a
smaller bandwidth: $$K_{\sqrt k \sigma}
* Y = \underbrace{K_{\sigma} *\cdots * K_{\sigma}}_{k \mbox{
\small{times} }}*Y.$$ The {\tt MATLAB} implementation can be found
in {\tt
http://www.stat.wisc.edu/}\\
{\tt $\sim$mchung/hk/hk.html.}



\section*{Statistical inference on manifolds}

As an illustration, we have applied the method in detecting the
regions of abnormal cortical thickness in 16 high functioning
autistic children. We set up a general linear model (GLM) on
cortical thickness $Y_j$ for subject $j$:
 \bq K_{\sigma}*Y_j(p) = \lambda_1(p) + \lambda_2(p)\cdot {\tt{age}}_j\\
+\lambda_3(p) \cdot {\tt{volume}}_j + \beta(p) \cdot {\tt{group}}_j
+ \epsilon_j,\label{eq:ge-age-volume}\eq where $\tt{group}_j$ is 1
for an autistic subject and $0$ for a normal subject. ${\tt
volume}_j$ is the total gray matter volume and ${\tt age}_j$ is the
age. Then we test the group difference by performing a hypothesis
testing:
$$H_0: \beta(p) =0 \mbox{ for all } p \in \partial \Omega$$
v.s.
$$H_1: \beta(p) \neq 0 \mbox{ for some } p
\in \partial \Omega.$$ Based on $F$ random field $F(p)$, the test
statistic $\sup_{p \in
\partial \Omega} F(p)$ is used to compute the $p$-value
corresponding to the above multiple hypotheses.

\begin{figure}
\centering \small
\includegraphics[width=182mm]{autism-fstat3.eps}
\caption{Corrected $p$ value maps of $F$-test removing the effect of
age and relative gray matter volume difference projected onto the
average outer (top) and inner surfaces (bottom). It shows relatively
asymmetric thickness difference between two groups.
\label{fig:autism-tstat}}
\end{figure}







%Chung, M.K., Worsley, K.J., Robbins, S., Paus, P., Taylor, J.,
%Giedd, J.N., Rapoport, J.L., Evans, A.C. 2003. Deformation-Based
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%{\em NeuroImage}. {\bf 18}:198-213.

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\end{multicols}
\end{document}
