Longitudinal deformation models, spatial regularizations and learning strategies to quantify Alzheimer's disease progression

Abstract

In the context of Alzheimer's disease, two challenging issues are (1) the characterization of local hippocampal shape changes specific to disease progression and (2) the identification of mild-cognitive impairment patients likely to convert. In the literature, (1) is usually solved first to detect areas potentially related to the disease. These areas are then considered as an input to solve (2). As an alternative to this sequential strategy, we investigate the use of a classification model using logistic regression to address both issues (1) and (2) simultaneously. The classification of the patients therefore does not require any a priori definition of the most representative hippocampal areas potentially related to the disease, as they are automatically detected. We first quantify deformations of patients' hippocampi between two time points using the large deformations by diffeomorphisms framework and transport these deformations to a common template. Since the deformations are expected to be spatially structured, we perform classification combining logistic loss and spatial regularization techniques, which have not been explored so far in this context, as far as we know. The main contribution of this paper is the comparison of regularization techniques enforcing the coefficient maps to be spatially smooth (Sobolev), piecewise constant (total variation) or sparse (fused LASSO) with standard regularization techniques which do not take into account the spatial structure (LASSO, ridge and ElasticNet). On a dataset of 103 patients out of ADNI, the techniques using spatial regularizations lead to the best classification rates. They also find coherent areas related to the disease progression.

Keywords: Alzheimer's disease; Brain imaging; Coefficient map; Deformation model; Disease progression; Karcher mean; LDDMM; Logistic regression; Spatial regularization; Transport.

Fig. 1

Four steps are needed to classify patient evolutions using local descriptors of shape deformations: (1) the local descriptors are computed for each patient independently, (2) a population template is computed, (3) all local shape deformation descriptors are transported towards this template, and (4) classification is performed.

Fig. 2

For each patient, the initial momentum encoding the hippocampus evolution is computed in a two-step process.

Fig. 3

Local descriptors of hippocampus evolutions are transported to the template in a two-step process. First the deformation field from the patient space to the population template. Second, this deformation field is used to transport the local descriptors.

Fig. 4

Empirical measures of convergence of the Karcher template algorithm. On this dataset, we notice that (1) the convergence speeds are coherent with the ones presented in Vialard et al. (2011) and Vialard et al. (2012b), i.e. only a few Karcher iterations are required for convergence, and (2) the alternate minimization over T and {R_i}_{1 ≤ i ≤ n} provides a faster convergence than the one over T with the {R_i} fixed.

Fig. 5

The region of interest Ω_S (visualized with transparency) is designed to select voxels close to the boundary (i.e. close to the surface) of T. It is obtained via standard morphological operations, and in this study Ω_S contains 12,531 voxels.

Fig. 6

Effects of various regularizations on the solution $\widehat{\mathbf{w}}$ of the optimization problem. Each small image represents the coefficients of one 2D slice of $\widehat{\mathbf{w}}$, which is a 3D volume. Zero coefficients are displayed in light green, higher values are going red and lower values are going blue. On each row, the regularization is increasing from left to right, and the 10th and 90th percentiles of the coefficients (resp. P₁₀ and P₉₀) correspond to the saturation limits of the colorbar. Panels a, b and c show standard regularizations whereas Panels d, e and f show spatial regularizations. Spatial regularizations provide more structured coefficients.