THE STUDY OF HUMAN BRAIN SYMMETRY
Lei Wang
Introduction
While the human brain exhibits marked symmetry across the sagittal plane,
the left and right hemispheres are not exact mirror images. Asymmetries
exist between left and right cortical structures as well as substructures
such as the ventricle and the hippocampi.
Studies have suggested that inter-hemispheric variation due to asymmetry
within the same individual is much less than variation seen across subjects
~\cite{galaburda:87, galaburda:91}.
Studies are under way to investigate any correlation
between brain sub-structure symmetries and certain brain functionalities
and diseases. In our application, it is a hypothesis that
schizophrenia may involve the internsification or reversal of asymmetries
found in normal individuals, due to defects in neurodevelopment, the
ability to assess inter-hemispheric symmetry in the human brain with
precision is an critical need.
Assessment of brain symmetry can be made quantitatively precise
using mathematical representations known as the symmetry groups. This
symmetry group action will be performed with respect to a sagittal plane,
which is to be defined using mathematical terms. The
notion of left and right sided brain structures, as well as variation in
the measurement of such structures, can then be quantified.
We begin by embedding the three dimensional MR data set in the coordinate
frame of a unit cube,
Let two structures under study be ${\mathcal X}=\{x|x\in\Omega\}$
on one side of the brain and one, and ${\mathcal Y}=\{y|y\in\Omega\}$ on
the other side. We decompose
the transformation $h({\mathcal X}) (h:\Omega\rightarrow\Omega)$
into a symmetry part and an asymmetry part:
\begin{equation}
h = h_{\mbox{symm}} \circ h_{\mbox{asym}}\,.
\end{equation}
The symmetry map consists of a reflection group action with respect to a
sagittal plane which is to be defined. A perfect symmetry will have
$h_{\mbox{asym}}=0$; anything away from zero will give a
measure of asymmetry.
For symmetry across the sagittal plane,
consider standard mathematical representations known as symmetry groups.
Most straightforwardly, let us examine the dihedral group ${\mathcal D}$,
which expresses axis-flipping symmetry across the sagittal plane. We
choose the convention that the sagittal plane through the center of the
longitudinal fissure is the set of all points
$\{x=(x_1,x_2,x_3)\in\Omega|x_3=0\}$. The dihedral group,
${\mathcal D}=\{I,R\}$, consists of the identity transformation $I$, the
$3\times 3$ identity matrix, and the reflection matrix $R$
which reflects points across the sagittal plane,
$R:(x_1,x_2,x_3)\longrightarrow (x_1,x_2,-x_3)$:
We then re-define the sagittal plane across which the symmtry of the
structures ${\mathcal X}$ and ${\mathcal Y}$ is studied as follows.
The symmetry map captures the global rotation, reflection, and translation,
from one rigid body to another. Using the rigid motion to characterize
lines and planes of symmetry, we define a rigid flip as
$(O, R, t): x\mapsto y=ORx+t$, where $O$ is an orthonormal rotation matrix,
$R$ a reflection matrix with respect to the $z$-axis (the dihedral
group), and $t$ a translation vector.
The symmetry map can be improved by further selecting points in
${\mathcal X}$ and ${\mathcal Y}$ as {\it landmarks}.
Treating the landmarks as 0-dimensional manifolds and
embedding the problem in a Bayesian framework, the {\it flip} group is
defined as
where $x$ are the landmark points, $n$ is the number of landmarks, and
$K(x,x_i)$ is the covariance matrix of the landmark transformation.
The sagittal plane $P_{\mbox{sag}}$ is defined as the plane of symmetry
whose points are invariant to the above transformation. I.e., the plane of
symmetry is defined as the set of points:
.
Once this sagittal plane is defined,
then the question of left- and right-sidedness, and the quantitative
representation of interhemispheric differences, becomes straightforward.
We take ${\mathcal X}$ as the
template and ${\mathcal Y}$ the target, and quantitatively study the
left-right asymmetry. Namely, we constrain the transformation from
${\mathcal X}$ to ${\mathcal Y}$ to be of the form~\cite{gec:thesis:94}
\begin{equation}
x\mapsto Fx-u(Fx)\,,
\end{equation}
with $F$ being the flip transformation.
The transformation from $\mathcal X$ to $\mathcal Y$ can be summarized as
follows:
If the left and right sides of the
identified brain structure (e.g., hippocampus) were perfectly symmetric,
then $u(\cdot)$ would be identically zero. The variation of $u(\cdot)$
away from zero then becomes a measure of asymmetry. Examining the average
size and locality of variation of the $u(\cdot)$ fields when comparing the
left and right sides of the brain then becomes a quantitative measure of
asymmetry within and across populations.
Experiments
Shown in Figure
\ref{lei-symmetry-figure}
are an example of the symmetries in the human brain between the left
and right hippocampus.
The left column
shows a section through an MRI of the template hippocampus (top row)
and a 3-D rendering of the template (bottom row).
The middle column shows the RHS target that the template has been
mapped into.
The right column
shows the result of flipping via
the rigid axis-flipping group,
and then mapping the RHS to the LHS.
\caption{
Left column shows the
template section of MRI (top row) and 3-D surface rendering (bottom row).
The middle and right panels show the template mapped to the RHS
(middle column)
and the RHS mapped to the LHS (right column).}
\label{lei-symmetry-figure}
\end{figure}
%%% \psfull
Lei Wang
([email protected]);
page last updated on Wed Nov 6 01:02:07 1985.
