Study of the Genetic Encoding of the Central Sulcus Shape Using Principal Component Analysis

G Le Goualher$^\dagger$, A.M. Argenti $^\dagger$$^\ddagger$ and A.C. Evans$^\dagger$

$^\dagger$McConnell Brain Imaging Centre, Montréal Neurological Institute,
McGill University, Montréal, Canada ; $^\ddagger$ Unité de Recherche d'épidemiologie génétique, INSERM,Unité 155, Université Paris VII, Jussieu, Paris

Introduction

Principal Component Analysis (PCA) allows the description of shape variability with a restricted number of parameters. These parameters (or modes) can be used to quantify the difference between two shapes through the computation of a modal distance. A statistical test can then be applied to this set of measurements in order to detect a significant difference between two groups of subjects. We have applied this methodology to highlight the genetic encoding of the shape of an anatomical structure. To underline genetic constraint, we have studied if central sulci were more similar within pairs of monozygotic twins than between independent subjects. This comparison was performed using a Mantel Permutation Test.

Methods

We had at our disposal 20 tri-dimensional MRI volumes corresponding to 10 pairs of monozygotic twins (average age : 34.6 $\pm$ 9.85; 8 males, 12 females). We have also included in our analysis 20 additional tri-dimensional MRIs corresponding to 20 subjects (average age : 29.6 $\pm$6.22; 8 males, 12 females) selected from a database of 150 volunteers corresponding to normal young subjects acquired as part of the International Consortium for Brain Mapping (ICBM) project.
We have previously developed a method (referred to as an Active Ribbon) which models the median surface of a sulcus by a parametric surface [1]. Principal Component Analysis (PCA) can be applied to a collection of such numerical representation of shapes. This eigenanalysis technique quantifies shape in term of deformation modes which represent unique and natural coordinates for shape comparisons. In the modal space, modal distance quantifies the observed difference between two shapes. In the eigen vector space, the modal distance between two samples is defined as :

$$d(S_i,S_j)= \Vert \vec{b_j} - \vec{b_i} \Vert$$ (1)

The closer this norm is to zero, the more similar the two samples are.

If a genetic encoding exists, one can argue that the minimum sulcal shape distance d_min(S_i,S_j) will systematically occur within a pair of twins : $d(S_a,S_b) \leq d(S_a,S_j)$, $\forall S_j$, if S_a and S_b is a pair of twins. However, this assumption may not be true even if a genetic encoding exists simply by chance. A statistical test is therefore needed to show that the intra-pair modal distances are statistically inferior to inter-pair modal distances.

The statistical we have used is the Mantel Permutation Test [2]. The Mantel Permutation Test requires no knowledge of population distribution (by opposition the classical student t-test is applied for the comparison of two populations each of them following a normal distribution; moreover to apply a student t-test the variance of the two populations must be similar) and makes meaningful use of all available data [3].

Conclusion

Our study highlights genetic constraints on the shape of the central sulcus. We found from 10 pairs of monozygotic twins that the intra-pair modal distances of the central sulcus were significantly inferior from inter-pair modal distances for the left central sulcus (z = -3.3179 ; p < 0.0005 ) as well as for the right central sulcus (z = -2.5030; p < 0.006). However, since the magnitude of the z-values is not large, it appears that environmental factors play a non-negligible role in the definition of the central sulcus shape.

Bibliography

1

G. LE GOUALHER, C. BARILLOT, and Y. BIZAIS.
Modeling cortical sulci using active ribbons.
International Journal of Pattern Recognition and Artificial Intelligence, 11(8):1295-1315, 1997.

2

N. MANTEL.
The detection of disease clustering and a generalized regression approach.
Cancer Research, 27:209-220, 1967.

3

K.J. WORSLEY, A.C. EVANS, S.C. STROTHER, and J.L. TYLER.
A Linear Spatial Correlation Model, with Applications to Positron Emission Tomography.
Journal of the American Statistical Association, 86(413):55-67, March 1991.

$^\ast$Supported by U.S. Human Brain Map Project and the International Consortium for Brain Mapping. The authors are grateful to Michel Duyme of Unité de Recherche d'épidemiologie génétique, INSERM,Unité 155, Université Paris VII, Jussieu, Paris for providing them with the twins data.

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