Automated voxel-based 3D cortical thickness measurement in a combined Lagrangian–Eulerian PDE approach using partial volume maps

Abstract

Accurate cortical thickness estimation is important for the study of many neurodegenerative diseases. Many approaches have been previously proposed, which can be broadly categorised as mesh-based and voxel-based. While the mesh-based approaches can potentially achieve subvoxel resolution, they usually lack the computational efficiency needed for clinical applications and large database studies. In contrast, voxel-based approaches, are computationally efficient, but lack accuracy. The aim of this paper is to propose a novel voxel-based method based upon the Laplacian definition of thickness that is both accurate and computationally efficient. A framework was developed to estimate and integrate the partial volume information within the thickness estimation process. Firstly, in a Lagrangian step, the boundaries are initialized using the partial volume information. Subsequently, in an Eulerian step, a pair of partial differential equations are solved on the remaining voxels to finally compute the thickness. Using partial volume information significantly improved the accuracy of the thickness estimation on synthetic phantoms, and improved reproducibility on real data. Significant differences in the hippocampus and temporal lobe between healthy controls (NC), mild cognitive impaired (MCI) and Alzheimer's disease (AD) patients were found on clinical data from the ADNI database. We compared our method in terms of precision, computational speed and statistical power against the Eulerian approach. With a slight increase in computation time, accuracy and precision were greatly improved. Power analysis demonstrated the ability of our method to yield statistically significant results when comparing AD and NC. Overall, with our method the number of samples is reduced by 25% to find significant differences between the two groups.

A comparison of voxel and surface based cortical thickness estimation methods

“…In contrast, voxel based methods (Lohmann et al, 2003;Hutton et al, 2008;Acosta et al, 2009;Aganj et al, 2009;Das et al, 2009;Cardoso et al, 2011;Scott et al, 2009) work directly on the voxel grid and are computationally very efficient. However, they are considered to be less accurate due to the limited resolution of the voxel grid, less robust to noise and mis-segmentation and significantly affected by partial volume (PV) effects at the boundaries of convoluted structures such as deep sulci (Acosta et al, 2009).…”

“…However, they are considered to be less accurate due to the limited resolution of the voxel grid, less robust to noise and mis-segmentation and significantly affected by partial volume (PV) effects at the boundaries of convoluted structures such as deep sulci (Acosta et al, 2009). …”

“…Laplacian approaches (Hutton et al, 2008;Acosta et al, 2009;Cardoso et al, 2011), solve the Laplace equation (Jones et al, 2000) using boundary value relaxation (Press et al, 1991) or matrix methods (Haidar et al, 2005), calculate thickness by integrating the tangent to the Laplacian scalar field (Jones et al, 2000), summing the Euclidean distance from neighbouring voxels on the same streamline, or using a partial differential equation (Yezzi and Prince, 2003) with boundaries set to zero (Yezzi and Prince, 2003), half the mean voxel dimension (Diep et al, 2007) or using Lagrangian initialisation (Bourgeat et al, 2008;Acosta et al, 2009). In contrast, the registration based approach of Das et al (2009) uses a greedy diffeomorphic registration algorithm to warp the WM segment to match the GM+WM segment.…”

“…1 http://www.loni.ucla.edu/Software/BrainSuite 2 http://brainvisa.info/ 3 http://surfer.nmr.mgh.harvard.edu/ Recently there has been significant interest in the development of voxel based methods (Hutton et al, 2008;Scott et al, 2009;Acosta et al, 2009;Aganj et al, 2009;Cardoso et al, 2011;Das et al, 2009). In addition, voxel based methods have featured in a comparison with voxel based morphometry (Hutton et al, 2009), been used to correlate changes of cortical thickness with diffusion measures using sparse canonical correlation analysis (Avants et al, 2010) and used in clinical studies (Querbes et al, 2009).…”

“…We chose FreeSurfer as it is the most widely used of the surface based methods (Nakamura et al, 2010). Of the voxel based methods, we chose a Laplacian method similar to Acosta et al (2009) as many of the references above are Laplacian based, and a registration method similar to Das et al (2009) as there is current interest in diffeomorphic registration algorithms, many of which could be applied to this application. The methods are compared in terms of reproducibility, disease differentiation and the ability to detect changes of cortical thickness in longitudinal imaging studies.…”