Mixed-Effects Shape Models for Estimating Longitudinal Changes in Anatomy

Abstract

In this paper, we propose a new method for longitudinal shape analysis that fits a linear mixed-effects model, while simultaneously optimizing correspondences on a set of anatomical shapes. Shape changes are modeled in a hierarchical fashion, with the global population trend as a fixed effect and individual trends as random effects. The statistical significance of the estimated trends are evaluated using specifically designed permutation tests. We also develop a permutation test based on the Hotelling T² statistic to compare the average shapes trends between two populations. We demonstrate the benefits of our method on a synthetic example of longitudinal tori and data from a developmental neuroimaging study.

Fig. 1

Example of randomly-generated, synthetic longitudinal data. Data points for each “individual” are displayed with different symbols. The estimated mixed-effects model is shown with both the fixed effects, i.e., group trend (solid black line), and random effects, i.e., individual trends (dashed lines). The estimated linear regression model, which ignores correlations within individuals, is shown as a red line.

Fig. 2

Visualizing fixed effects on the synthetic data

Fig. 3

Visualizing the trends in the two groups created to test group differences: constant trend (top), trend with increasing r (bottom)

Fig. 4

Visualizing the fixed-effects of brain structures (blue denotes expansion, and yellow denotes contraction)

Fig. 5

Visualizing the random-effects of brain structures (red denotes high variance regions)

Fig. 6

Variation in group trends for the HR+ (top) and LR− (bottom) groups (yellow denotes contraction, blue denotes expansion)