Gray matter concentration and effective connectivity changes in Alzheimer’s disease: a longitudinal structural MRI study

Abstract

Introduction

Understanding disease progression in Alzheimer’s disease (AD) awaits the resolution of three fundamental questions: first, can we identify the location of “seed” regions where neuropathology is first present? Some studies have suggested the medial temporal lobe while others have suggested the hippocampus. Second, are there similar atrophy rates within affected regions in AD? Third, is there evidence of causality relationships between different affected regions in AD progression?

Methods

To address these questions, we conducted a longitudinal MRI study to investigate the gray matter (GM) changes in AD progression. Abnormal brain regions were localized by a standard voxel-based morphometry method, and the absolute atrophy rate in these regions was calculated using a robust regression method. Primary foci of atrophy were identified in the hippocampus and middle temporal gyrus (MTG). A model based upon the Granger causality approach was developed to investigate the cause–effect relationship over time between these regions based on GM concentration.

Results

Results show that in the earlier stages of AD, primary pathological foci are in the hippocampus and entorhinal cortex. Subsequently, atrophy appears to subsume the MTG.

Conclusion

The causality results show that there is in fact little difference between AD and age-matched healthy control in terms of hippocampus atrophy, but there are larger differences in MTG, suggesting that local pathology in MTG is the predominant progressive abnormality during intermediate stages of AD development.

Appendices

Appendix 1: AR model within subjects for effective connectivity study

For the subjects who have at least three longitudinal scans, we assume the brain GM concentration is an autoregressive (AR) function, i.e., the GM change at a later time point is related to the GM at a previous time point. This is a reasonable assumption [16], and based on this assumption, for each affected region i, we have:

$$ {y_i}(t) = {a_i}{y_i}(t - 1) + {e_i}(t) $$

(A1)

where t is the time, y _i (t) is current GM concentration value; y _i (t − 1) is the previous GM concentration value; a _i is the AR coefficient; and e _i (t) is the model error (Gaussian noise). For two-connected regions, we can consider the mutual interaction between these regions (for example hippocampus and MTG as shown in Fig. 2); we thus have a GCM as follows [19, 22]:

$$ {y_2}(t) = {b_1}{y_1}(t - 1) + {b_2}{y_2}(t - 1) + {e_2}(t) $$

(A2)

where y ₁(t) and y ₂(t) are the current averaged GM concentration in hippocampus and MTG, respectively, and y ₁ (t − 1), y ₂(t − 1) are the corresponding previous average GM concentration in hippocampus and MTG regions as shown in Fig. 3b. If a subject has been scanned only twice over time, we cannot estimate the coefficients based on individual subjects. Equation A2 has two parameters, but two “visits” can only produce one equation. Assuming for each subject, left and right hemispheres have the same model within each subject (Fig. 5), we can build a general linear model for each subject and combine the left and right hemisphere regional GM concentration within this model, i.e. first-level analysis for Eq. A2 [39]:

$$ Y = {X_1}{b_1} + {X_2}{b_2} + e(t) $$

(A3)

where Y = [y _2,1(t), y _2,2(t), …, y _2,n(t)]′, $n = (V - 1) \times 2$, where V is the total number of visits (we multiply by 2 because we combine the left and right hemisphere GM concentrations within the model); X ₁ = [y _1,1(t − 1), y _1,2(t − 1), …, y _1,n(t − 1)]′, X ₂ = [y _2,1(t − 1), y _2,2(t − 1), …, y _2,n(t − 1)]′. X ₂ represents the AR term of y ₂(t), and X ₁ denotes the influence from the other connected region and $e(t) = [{e_{{2,1}}}(t),{e_{{2,2}}}(t), \cdots, {e_{{2,n}}}(t)]{}^{\prime}\sim N(0,\sigma {}^2)$. The estimated GM concentration response is:

$$ \widehat{Y} = X\widehat{\beta } + e(t) $$

(A4)

where $\widehat{\beta }$ can be estimated by:

$$ \widehat{\beta } = X{}^{+} Y $$

(A5)

and X = [X ₁, X ₂], and X ⁺ is the Moore–Penrose pseudoinverse of the matrix. To study the influence from MTG to hippocampus, we swap the y ₂(t) and y ₁(t) in Eq. A2.

Appendix 2: Granger model for the effective connectivity

Granger causality analysis [61, 62] is derived based on F statistics. For Eq. A2, the test for determining Granger causality (GC) is: (1) y ₁(t) is GC of y ₂(t) [61], if b ₁ = 0 in Eq. A2 is not true. Given the data, we reach this conclusion if b ₁ = 0 is rejected. (2) Similarly, y ₂(t) is GC of y ₁(t) can be investigated by reversing the input–output roles of the two series. F statistics (see Appendix 3) are developed to detect significant relations within subject, and t statistics are developed for between-subjects analysis (Appendix 4).

Appendix 3: F test (within subject)

After the covariates and their coefficients b ₁ and b ₂ in Eqs. A3/A4 have been estimated by the least squares method in Appendix 1 from the GM concentration response, the F test [63] is applied to test the inference of the connectivity between different regions. Accordingly from Eq. A4, we partitioned the coefficients $\widehat{\beta }$ as: $\widehat{\beta } = ({\widehat{\beta }_1}:{\widehat{\beta }_2})$ and X = (X ₁ : X ₂), we can write this test as:

$$ {H_0}{\text{:}}{\widehat{\beta }_2} = 0\,{\text{versus}}\,{H_1}:{\widehat{\beta }_2} \ne 0; $$

For one GM concentration response, the F test on the causality is given by [64]:

$$ \frac{{R{}^2}}{{1 - R{}^2}}\frac{{n - k - s}}{s}\sim F(s,n - k - s) $$

(A6)

where s is the column of X ₂; k is the column of X ₁; n is the total number of visits minus one (multiply by 2 if combining two hemispheres within the brain). $R{}^2 = \frac{{{\text{RS}}{{\text{S}}_0} - {\text{RSS}}}}{{{\hbox{RS}}{{\hbox{S}}_0}}}$, where RSS₀ (original system when β ₂ = 0, without interaction terms for the two-connection network (Fig. 2)) and RSS are the residual sums of squares (when both the AR and feedback terms exists in the system), ${\text{RS}}{{\text{S}}_0} = (Y - \widehat{Y}){}^\prime \times (Y - \widehat{Y})$(under H ₀); ${\text{RSS}} = (Y - \widehat{Y}){}^\prime \times (Y - \widehat{Y})$. Then, we apply Eq. A6 to test the influences between regions.

Appendix 4: Mixed effect model (first level: within subject)

Apart from the F test conducted within subjects, we can also inference an effect by defining a contrast matrix c and using the t statistics, we start from estimation of an effect:

$$ {E_j} = c\widehat{\beta } $$

(A7)

j = 1,…, n, where n is the number of visits minus one (×2 if combine two hemispheres within the model in A3/A4). For example, we can select c = [0,1] at first level contrast to study the influence from MTG to hippocampus.

$$ {S_j} = \left\| {cX{}^{+} } \right\|\widehat{\sigma } $$

(A8)

where $\widehat{\sigma } = \sqrt {{r\prime r/v}}$, $r = Y - X\widehat{\beta }$, $v = m - {\hbox{rank}}(X)$is the degree of freedom (df); $m = 2(n - 1)$is the total number of effect in Eq. A7 for two hemispheres.

Appendix 4: Mixed effect model (second level: between subjects)

For the second-level analysis, a general linear mixed model [65] was adopted, i.e.:

$$ E = Z \cdot \gamma + \eta $$

(A9)

where E = (E ₁, …, E _n* )′, S = (S ₁, …, S _n ), and η is normally distributed with zero mean and variance $S + \sigma_{\rm{random}}^2$ independently for j = 1, …, n. We want to compare the effects in VBM using covariates (Z is the design matrix for comparison in the general linear model):

$$ Z = \left[ {\begin{array}{*{20}{c}} {{I_{{n_1^{ * }}}}} & {{O_1}} \\{{O_2}} & {{I_{{n_2^{ * }}}}} \\\end{array} } \right] $$

(A10)

If we are interested in the difference between two groups, we can set ${I_{{{n_1}}}} = \left[ {1, \cdots, 1} \right]_{{1 \times n_1^{ * }}}^{\prime;}$,${I_{{{n_2}}}} = \left[ {1, \cdots, 1} \right]_{{1 \times n_2^{ * }}}^{\prime;}$,${O_2} = \left[ {0, \cdots, 0} \right]_{{1 \times n_2^{ * }}}^{\prime;}$, and ${O_1} = \left[ {0, \cdots, 0} \right]_{{1 \times n_1^{ * }}}^{\prime;}$. In this study, $n_1^{ * } = 34$(number of controls), $n_2^{ * } = 20$ (number of AD subjects).

To estimate γ in Eq. A9, we first use the restricted maximum likelihood [66, 67] algorithm to estimate $\widehat{\sigma }_{\text{random}}^2$. In the expectation maximization algorithm [67–69], let S = diag(S ₁, …, S _n* ) and I be the n ^*  × n ^* identity matrix ($n{}^{ * } = n_1^{ * } + n_2^{ * } = 54$ in this study). From (A9), we have the variance matrix of the effects vector E = (E ₁ , …, E _n )′:

$$ {\sum { = {{\rm\mathbf{S}}^2} + {\rm\mathbf{I}}} {{\sigma_{{random}}^2}}} $$

(A11)

Define the weighted residual matrix:

$$ {\text{R}}_{{{\sum {} }}} = {\sum {^{{ - 1}} } } - {\sum {^{{ - 1}} } }{\mathbf{Z}}{\left( {{\mathbf{Z}}\prime {\sum {^{{ - 1}} {\mathbf{Z}}} }} \right)} + {\mathbf{Z}}\prime {\sum {^{{ - 1}} } } $$

(A12)

Starting with an initial value of $\sigma_{{random}}^2 = {\mathbf{E}^{{\prime;}}}{\mathbf{R}_{\rm{I}}}\mathbf{E/}{v^{*}}$ based on assuming that the fixed effects variances are zero. The updated estimate is:

$$ \widehat{\sigma }^{2}_{{random}} = {\left( {\sigma ^{2}_{{random}} {\left( {p* + {\text{tr}}{\left( {{\mathbf{S}}^{2} {\mathbf{R}}{\sum {} }} \right)}} \right)} + \sigma ^{4}_{{random}} {\mathbf{E}}\prime {\mathbf{R}}\begin{array}{*{20}c} {2} \\ {{{\sum {} }}} \\ \end{array} {\mathbf{E}}} \right)}/n* $$

(A13)

where p ^* = rank(Z). Replace $\sigma_{\rm{random}}^{{2}}$ with $\widehat{\sigma }_{\text{random}}^2$ in (A11) and iterate (A11–A13) to convergence. In practice, 10 iterations appear to be enough [69]. After convergence, step (A11) is repeated with $\sigma_{\rm{random}}^{{2}}$ by $\widehat{\sigma }^{2}_{{{\text{random}}}}$, then the estimate of γ is:

$$ \widehat{\gamma } = {\left( {Z\prime {\sum^{ - 1}}Z} \right)^{+} }{Z^\prime }{\sum^{ - 1}}E $$

(A14)

And its estimated variance matrix is:

$$ V\widehat{a}r(\widehat{\gamma } = {\left( {{Z^\prime }\sum {^{ - 1}} Z} \right)^ + } $$

(A15)

In the case when the variances of E are not homogeneous across the level 2 unit (for example, different scanner), Eq. A15 should be replaced by other terms [40, 70]:

$$ V\widehat{a}r(\widehat{\gamma }) = {\left( {{Z^\prime }\sum {^{ - 1}} Z} \right)^ + }{Z^\prime }\sum {^{ - 1}} \left( {E - Z\widehat{\gamma }} \right){\left( {E - Z\widehat{\gamma }} \right)^\prime }\sum {^{ - 1}} Z{\left( {{Z^\prime }\sum {^{ - 1}} Z} \right)^ + } $$

(A16)

Finally, effects defined by a contrasts matrix (second level, b = [1 −1] in this study for control compared to AD) b in γ can be estimated by $E^{ * } = b\widehat{\gamma }$ with standard deviation:

$$ {S^*} = \sqrt {{bV\widehat{a}r\left( {\widehat{\gamma }} \right){b^\prime }}} $$

(A17)

The T statistic is:

$$ T{}^{ * } = E{}^{ * }/S{}^{ * } $$

(A18)

with a nominal ν ^* df ($v^{ * } = n{}^{*} - {\hbox{rank}}(Z)$) can then be used to detect such an effect.