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Sampling theory and automated simulations for vertical sections, applied to human brain



    Corresponding author
    1. Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Santander, Spain
    • Correspondence to: Professor Luis M. Cruz-Orive, Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, E–39005 Santander, Spain. Tel.: +34 942 20 14 24; fax: +34 942 20 14 02; e-mail:

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    1. School of Informatics, University of Edinburgh, Edinburgh, U.K.
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    1. Clinical Research Imaging Centre (CRIC), The Queen's Medical Research Institute (QMRI), School of Clinical Sciences, College of Medicine and Veterinary Medicine, University of Edinburgh, Edinburgh, U.K.
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In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images – although convenient manual application is always an option – and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the ‘lambda method’) is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable – one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.

Lay Description

The so-called ‘vertical sections’ method of design stereology, first proposed in 1986, provides an unbiased and often highly efficient approach to estimate surface area. This paper explores the vertical sections design in the following aspects. (a) The theory is completed. (b) A new estimation method is developed for automatic image analysis from digitized vertical sections. (c) Whenever manual, or semiautomatic, image analysis is imperative, a new predictor is proposed for the error variance contribution of intersection counting with cycloids. Finally, (d) the performance of the variance splitting predictors due to orientation, Cavalieri sectioning, and intersection counting with cycloids, is assessed in detail, and practical recommendations are given.

The numerical analyses were performed on a healthy human brain. High resolution MR images were used to triangulate the pial and the subcortical surfaces of the brain in three dimensions by means of the recently available FreeSurfer software. Vertical sections of such triangulations yielded polygonal trace approximations of the real surface traces which were computed analytically. This allowed us to perform automatic Monte Carlo simulations on a real object for the first time in a stereological context. In particular, the aforementioned new method (b) could be implemented. The variance predictor for intersection counting quoted in (c) could also be checked by Monte Carlo simulations by means of a new software which determines automatically the number of intersections between a bounded polygonal trace curve and a test grid of cycloids. The same software allowed us to perform the task (d) automatically by means of further Monte Carlo resampling. The triangulations were therefore used only as a working bench with theoretical and methodological purposes in mind - the accuracy of the surface area approximations yielded by such triangulations was not questioned.