Network Neurodegeneration in Alzheimer’s Disease via MRI Based Shape Diffeomorphometry and High-Field Atlasing

This paper examines MRI analysis of neurodegeneration in Alzheimer’s Disease (AD) in a network of structures within the medial temporal lobe using diffeomorphometry methods coupled with high-field atlasing in which the entorhinal cortex is partitioned into eight subareas. The morphometry markers for three groups of subjects (controls, preclinical AD, and symptomatic AD) are indexed to template coordinates measured with respect to these eight subareas. The location and timing of changes are examined within the subareas as it pertains to the classic Braak and Braak staging by comparing the three groups. We demonstrate that the earliest preclinical changes in the population occur in the lateral most sulcal extent in the entorhinal cortex (alluded to as transentorhinal cortex by Braak and Braak), and then proceeds medially which is consistent with the Braak and Braak staging. We use high-field 11T atlasing to demonstrate that the network changes are occurring at the junctures of the substructures in this medial temporal lobe network. Temporal progression of the disease through the network is also examined via changepoint analysis, demonstrating earliest changes in entorhinal cortex. The differential expression of rate of atrophy with progression signaling the changepoint time across the network is demonstrated to be signaling in the intermediate caudal subarea of the entorhinal cortex, which has been noted to be proximal to the hippocampus. This coupled to the findings of the nearby basolateral involvement in amygdala demonstrates the selectivity of neurodegeneration in early AD.

The disparity between surfaces or normed error between , ′ is given by where is the smoothing window over which the integral is computed, and 〈⋅,⋅ 〉 is the inner-product between normal vectors.
Since the vector space ∈ of vector fields is spatially smooth it has a reproducing kernel defined as implying that the variational minimizers of Eqn. (1) will involve the kernel (see below). The variational problem of Equation 1 is solved by representing the deforming surfaces as a dynamical system, with state 0 , ∈ , = ( ), 0 = . Denoting the 3 × 3 Jacobian matrix as ( ) = ( ), ( ) matrix transpose, the solution satisfies ∈ [0,1], subject to = ∇ ( , ), = 1, … with ∇ ( , ) denoting the 3 × 1 gradient of the matching cost with respect to the state. The target surfaces enter through boundary conditions involving the state transforming the template.

Appendix 2: Linear mixed-effects Modelling for Group Comparisons: Control versus Preclinical and Control versus Symptomatic
Calculation of the MLE parameters for the linear mixed-effects model: The parameters , ′ , , ′ , 2 are estimated by maximum likelihood for all dimensions v for each of the two hypotheses. Evaluating the log-likelihood in each case at the MLEs of the parameters gives loglikelihood essentially determined by the mixed sums of squares.
We describe the maximum likelihood estimation algorithm, focusing on the general hypothesis 1 . The null hypothesis 0 is handled the same way, without the parameters , ′ , or = ′ = 0. We also work with fixed , since the models across shape coordinates do not share any parameter and can be estimated independently from each other. The model parameters for 1 are then = ( , ′ , , ′ ), 2 and ; for 0 then = ( , ′ , 0, 0), 2 .
We now describe the estimation procedure. Let denote the -dimensional vector with all coordinates equal to 1. The covariance matrix of ( ) is the × matrix ( ) = 2 ( + ).
This implies that the log-likelihood of the sample is (up to an additive constant) The procedure loops over the following two steps until convergence (which usually requires a small number of iterations) Step 1: Least square estimation, updating all parameters except . Then, a direct computation shows that the least square estimator of is given by To estimate the variance, define the residual ( ) = ( ) = ( )̂. For a given , let ̅ ( ) = ( ) = ∑ ( ) =1 , then, one must take Step 2: Update with all other parameters fixed. Focusing on the part of the likelihood that depends on , we see that ̂ minimizes the function This minimization problem has no closed-form solution and must be performed numerically.
Note that the computations in steps 1 and 2 are made independently across shape coordinates.
The maximization in Δ is made, in both cases, by computing ( ) (or ( , )) for all over a discrete time interval.
Tests for Significance: The test statistic is the log-likelihood difference between the null hypothesis 0 : ′ = 0 and the general hypothesis 1 , namely = 1 − 0 .
The global statistic is then defined by * = max . P-values are computed using permutation sampling run until a 10% accuracy is reached with high probability. Permutations affect the value of the onset time by expressing as = 1 ( ) + , where 1 ( ) is the age of subject at the beginning of the study (first scan), and permuting the values of , so that, for a permutation π, the permuted times are = 1( ) + ( ) .
A global p-value is obtained as the fraction of permutations for which the resulting statistic, say * , is larger than the observed one * . When using the heterogeneous onset model, variables for which is larger than the 95th percentile of the values of * that were observed via permutations are considered as significant.