Latent growth curve (LGC) models estimate change over time in a cohort's serially obtained measurements. We have applied LGC techniques to a spatial distribution of Alzheimer's disease (AD) pathology using autopsy data from 435 participants in the Honolulu-Asia Aging Study. Neurofibrillary tangle (NFT) and neuritic plaques (NP) were distributed across differently ordered sets of anatomical regions. The gradient of spatial change in neuritic plaque (dNP), was significantly associated with that of neurofibrillary tangle (dNFT), but weakly and inversely (r = −0.12; p < 0.001). Both dNFT and dNP correlated significantly and inversely with Braak stage. Sixty-one percent of the variance in Braak stage was explained by dNFT independent of covariates. Only dNFT was significantly associated with longitudinal change in cognition. Only dNP was associated with apolipoprotein (APOE) e4 burden. This is the first application of LGC models to spatially-ordered data. The result is a quantification of the interindividual variation in the interregional vulnerability to Alzheimer's disease lesions.
Latent “class” or growth curve (LGC) and growth mixture models (GMM) represent the state of the art in longitudinal data analysis. LGC estimate the trajectory of change over time in a cohort's serially-obtained measurements (Willet and Sayer, 1994). GMM can be used to define subsets among a cohort with homogenous trajectory parameters. Through them, it is possible to use intra- and interindividual change over time as outcome variables or as predictors, e.g., in structural equation models (SEM) (McArdle and Epstein, 1987 and Willet and Sayer, 1994). This allows one to assess mediating/moderating effects on longitudinal outcomes. Another valuable feature of LGC models is that measurement error is explicitly assessed, and removed from the latent construct. This can strengthen statistical power and improve model fit.
However, it may also be possible to extend the application of these techniques beyond temporally ordinal datasets, i.e., to measures repeated across spatial dimensions. Particularly useful applications might be in the analysis of neuropathological or neuroimaging data (Royall, 2007). For example, Braak and others have suggested that neurofibrillary tangles (NFT) and Lewy body lesions propagate transynaptically within neuronal networks (Braak and Braak, 1991, Braak et al., 2006, Pearson and Powell, 1989 and Saper et al., 1985). The interconnections of those networks may thus determine the exquisite regional, and even laminar vulnerability of neuronal populations to NFT (Armstrong et al., 2001 and Arnold et al., 1991). These networks can be conceived as an ordinally arranged sequence of anatomical regions along a hierarchical gradient of interregional vulnerability. As such, the 3-dimensional propagation of neuropathology through the network may be amenable to modeling with LGC and GMM techniques.
If applied to NFT counts, this approach would result in 2 latent parameters, the network's “intercept” (e.g., the estimated mean NFT count within the first in a hierarchically arranged sequence of anatomical regions that together define the Braak neuropathological hierarchy) and its “slope” (e.g., the mean change in NFT counts across regions, from the most to the least vulnerable in the sequence). Both parameters would have associated estimates of variability about their means, and both linear and nonlinear gradients in NFT counts across the network could be independently estimated. These “slope” parameters can be interpreted as representing the network's “vulnerability” to, or alternatively, its resistance against, penetration by the Alzheimer's disease (AD) process. Biomarkers, genes, or other variables could then be tested as determinants of this vulnerability.
If there is significant variability about the estimated mean change in NFT counts across the network, then GMM could be employed to identify homogeneous subgroups within the cohort with significantly different network intercepts and vulnerability gradients. These could be interpreted as subpopulations within the cohort with differing risks of AD pathology. Thus, the variables responsible for those differences could also be identified, in regression models of trajectory class membership. Associating these gradients with longitudinal change in cognitive measures, i.e., in SEM, may eventually allow the direct quantification of cognitive reserve and its related biomarkers. In this report, we demonstrate the feasibility of this approach using autopsy and clinical data from the Honolulu-Asia Aging Study (HAAS).
