A Bayesian Hierarchical Approach to Jointly Model Cortical Thickness and Covariance Networks

Abstract

Estimation of structural biomarkers and covariance networks from MRI have provided valuable insight into the morphological processes and organisation of the human brain. State-of-the-art analyses such as linear mixed effects (LME) models and pairwise descriptive correlation networks are usually performed independently, providing an incomplete picture of the relationships between the biomarkers and network organisation. Furthermore, descriptive network analyses do not generalise to the population level. In this work, we develop a Bayesian generative model based on wombling that allows joint statistical inference on biomarkers and connectivity covariance structure. The parameters of the wombling model were estimated via Markov chain Monte Carlo methods, which allow for simultaneous inference of the brain connectivity matrix and the association of participants’ biomarker covariates. To demonstrate the utility of wombling on real data, the method was used to characterise intrahemispheric cortical thickness and networks in a study cohort of subjects with Alzheimer’s disease (AD), mild-cognitive impairment and healthy ageing. The method was also compared with state-of-the-art alternatives. Our Bayesian modelling approach provided posterior probabilities for the connectivity matrix of the wombling model, accounting for the uncertainty for each connection. This provided superior inference in comparison with descriptive networks. On the study cohort, there was a loss of connectivity across diagnosis levels from healthy to Alzheimer’s disease for all network connections (posterior probability ≥ 0.7). In addition, we found that wombling and LME model approaches estimated that cortical thickness progressively decreased along the dementia pathway. The major advantage of the wombling approach was that spatial covariance among the regions and global cortical thickness estimates could be estimated. Joint modelling of biomarkers and covariance networks using our novel wombling approach allowed accurate identification of probabilistic networks and estimated biomarker changes that took into account spatial covariance. The wombling model provides a novel tool to address multiple brain features, such as morphological and connectivity changes facilitating a better understanding of disease pathology.

Appendices

Appendix: Methods and Applications

Additional material to supplement simulation study results, posterior diagnostic checks, wombling ROI cortical thickness estimates at the population and participant levels, independent Bayesian mixed effect model results, WAIC values and Pearson correlation networks can be found in this Appendix. R code to implement the wombling model can be found at the following GitHub repository https://github.com/MarcelaCespedes/Brain_wombling.

Simulation Study

The simulation study described in Sect. 7.2.3 provided a thorough assessment of the Bayesian brain wombling algorithm. The four scenarios in the simulation study are; contiguous balanced (each person had an equal number of replicates) and unbalanced (the number of replicates varied per person), and a structured balanced and unbalanced designs. The results for fixed effect parameters β and residual variance σ ² are shown in Table 7.3. While the results for the structured configuration show a slightly lower recovery of fixed effect parameters, they do not represent a potential biological configuration. Hence performance of the wombling algorithm is better assessed on the contiguous configuration, whose performance of the recovery of the parameters is approximately 95%.

Table 7.3 Percentage (%) of fixed effect and residual variance parameter recovery for four scenarios, each with 50 simulations

As discussed in Sect. 7.2.3.1, spatial scale variance $\sigma ^2_s$ is a biased estimate and was not recovered in our simulation study.

Figure 7.9 shows the histograms on the percentage of the recovered random effects for each scenario. The simulation study comprised of 50 independently simulated data sets for each scenario, each data set consisted of I = 100 simulated participants, each with K = 35 ROI resulting in 3500 random effects per simulated data set to estimate. Overall we can see that there is approximately 95% recovery of the random effects for each scenario.

As ρ in Model (7.1) is a fixed value, we investigated the effect recovering the parameters in the structured scenario for ρ values [0.85, 0.9, 0.95, 0.99]. Table 7.4 summarises the results.

Contact the author for additional simulation study results such as MCMC convergence checks, estimation of credible intervals, and posterior predictive plots.

Table 7.4 Parameter values for ρ set to 0.85, 0.9, 0.95, 0.99 values

Posterior Diagnostic Checks for AIBL Data Set

Posterior predictive plots for each AIBL group analysed were used to assess goodness-of-fit for each wombled model. The plots in Figs. 7.10, 7.11 and 7.12 show the expected mean of the data was recovered well, however there is a slight overestimation of the variance, as the proportion of predicted values inside the 95% credible intervals is slightly over 0.95. However these results show our models adequately captured the uncertainty in the data.

Fig. 7.18

Global posterior cortical thickness means for (red) APOE ε4 carriers and (green) APOE ε4 non-carriers

Table 7.5 shows the Gelman-Rubin diagnostic, upper 95% credible interval for convergence checks of the four chains for β ₀, β ₁, σ ² and $\sigma ^2_s$.

Table 7.5 Gelman-Rubin diagnostic upper confidence limit values for each group in AIBL study

Wombling Cortical Thickness Estimates at the ROI Level

As discussed in Sect. 7.4.2.1, we investigated an adaptation to the wombling model to account for ROI means via fixed effect parameters. The extended model is of the form

$$\displaystyle \begin{aligned} y_{irk}| b_{ik},\boldsymbol{\beta}, \sigma^2 &\sim N(\beta_0 + \beta_1R_2 + \beta_2R_3 + \ldots + \beta_{34}R_{35} + b_{ik}, \sigma^2) \\ {\mathbf{b}}_i &\sim MVN(\mathbf{0}, \sigma_s^2Q) \\ Q^{-1} &= \rho(D_w - W) + (1-\rho)\mathbb{I}. \end{aligned} $$

(7.9)

Where the response (y _irk), spatial random effects (b _ik), residual (σ ²) and spatial scale variance $(\sigma ^2_s)$ terms are the same as those presented in Sect. 7.2.2. The precentral gyrus is the baseline ROI whose cortical thickness (in mm) is estimated by β ₀. The fixed effect parameter β _k−1 estimates the deviation of ROI k away from β ₀ when the binary indicator variable R _k is equal to one. Estimation of β is attained by the same conditional distribution described in Sect. 7.2.2, with minor modifications to account for the design matrix R rather than X. Figures 7.17 and 7.18 show participant specific cortical thickness estimates as caterpillar plots (β _k + b _ik) colour coded for diagnosis and age groups respectively. Figure 7.13 shows the posterior means of W for HC (top), MCI (middle) and AD (bottom) groups. While the posterior mean for the HC group is similar that in Fig. 7.4, with the same 36 links present in both networks and 468 absent connections in common, the matrices for MCI and AD group show the probability of each link is close to 0.5. We believe that the reason for this is because the HC group has a substantially larger sample size (120 individuals) compared to the MCI and AD groups (with 21 and 26 individuals respectively). Hence, the more complex model in Eq. (7.9) requires data with larger sample sizes, compared to the original wombling model, in order to derive meaningful W estimates.

Fig. 7.19

Posterior mean and 95% credible interval on participants estimated cortical thickness (random effects b _i), for nine key regions associated with AD cortical signature. Colour coded as follows; (red) APOE ε4 carriers and (green) APOE ε4 non-carriers

Figure 7.14 shows the marginal posterior densities for the ROI means for 35 regions. These results resemble the independent Bayesian LME ROI estimates in Fig. 7.20, particularly for ROIs associated with early onset of AD such as the inferior, middle and superior temporal gyrus, posterior cingulate gyrus.

Fig. 7.20

Marginal posterior distributions of total cortical thickness estimates for each ROI. AD, MCI and HC diagnosis are dentoed by red, yellow and green posterior densities

Wombling Cortical Thickness Estimates at the Participant Level

As described in Sect. 7.2.4.4 and discussed in Sect. 7.3.2.4, the wombling model derived participant specific estimates on all ROIs. Figures 7.15 and 7.16 shows the posterior means and 95% credible intervals (as error bars) for each participant.

Fig. 7.21

Participant specific estimates of each ROI ranked according to posterior mean with 90% credible interval error bar. Colour coded according to the diagnosis levels HC, MCI and AD (green, yellow and red respectively)

Fig. 7.22

Marginal posterior distributions of total cortical thickness estimates for each ROI. Age groups A, B and C denoted by green, yellow and green posterior densities

Fig. 7.23

Participant specific estimates of each ROI ranked according to posterior mean with 90% credible interval error bar. Colour coded according to age groups A, B and C (colours: green, yellow and red respectively)

Fig. 7.24

Marginal posterior distributions of total cortical thickness estimates for each ROI. APOE ε4 carrier (red) and non-carrier (green) groups

APOE Wombling Results

Carriers of the Apolipoprotein (APOE) ε4 gene have known to be at higher risk of developing AD compared to non-carriers, hence in neuroimaging studies, it is a key biomarker to investigate. For exploration purposes, we applied the wombling model on AIBL data divided into APOE ε4 carrier and non-carrier groups. Figures 7.17, 7.18 and 7.19 show the cortical networks, global estimates across all ROI and participant specific rankings for key AD regions as described in Sect. 7.2.4.3.

Fig. 7.25

Participant specific estimates of each ROI ranked according to posterior mean with 90% credible interval error bar. Colour coded according to APOE ε4 carrier (red) and non-carrier (green) groups

Fig. 7.26

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on HC diagnosis. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

Fig. 7.27

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on MCI diagnosis. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

Fig. 7.28

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on AD diagnosis. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

Fig. 7.29

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on age group A. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

There were no strong differences APOE ε4 carrier and non-carrier groups in any of the ROI. We believe the reason for this is due to APOE ε carrier and non-carrier groups comprising of participants across the entire spectrum (HC, MCI and AD), large variety of ages and many other AD biomarkers, making it difficult to assess the deterioration differences associated with the APOE ε gene. Unfortunately due to our low sample size, we did not have sufficient data to investigate more meaningful biomarker groups such as APOE ε4 carrier and non-carrier groups that were clinically diagnosed as HC or AD.

Bayesian Linear Mixed Effect Models on Each ROI

As described in Sect. 7.2.4.4 and discussed in Sect. 7.3.2.4, Bayesian linear mixed effect models were independently applied to each ROI on groups; diagnosis levels HC, MCI and AD and age groups A, B and C. For exploration purposes we also investigated APOE ε4 allele carriers and non-carriers. All models were of the form

$$\displaystyle \begin{aligned} y_{ij} | \sigma^2, \beta_1, \mu_{0i} \sim& N(\mu_{0i} + \beta_1x_i, \sigma^2) \\ \mu_{0i}| \mu_0, \sigma^2_0 \sim& N(\mu_0, \sigma^2_0). \end{aligned} $$

(7.10)

In order to make the models comparable with the wombling approach, covariate x _i is gender as described in Sect. 7.2.3.1, with x _i = 1 for male and 0 otherwise. The residual variance prior for σ ² and the random effects prior, $\sigma ^2_0$, is the same as discussed in Sect. 7.2.3.1. Similarly, the prior for the intercept effect μ ₀ is also relatively vague with a N(0, 10) distribution.

Figures 7.20, 7.21, 7.22, 7.23, 7.24 and 7.25 show the marginal posterior mean population distributions and participants ranked according to posterior means with 95% credible interval.

Fig. 7.30

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on age group B. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

Fig. 7.31

Pearson pairwise correlation plots for baseline (left top and bottom) and repeated measures (right top and bottom) on age group C. Top: networks binarised according to threshold of τ = 0.7 applied on the absolute value of each element on correlation matrices above

WAIC Results

As described in Sect. 7.2.4.4, we applied the WAIC criterion on the wombled and independent Bayesian LME models to assess model choice. Table 7.6 shows the results of the WAIC for the wombling model applied to each group, and the combined WAIC criterion for the independent Bayesian LME analyses for each region.

Table 7.6 WAIC values for diagnosis groups

Pearson Correlation Networks for Each Group

Cortical networks derived by Pearson’s pairwise correlation networks for each group are shown in Figs. 7.26, 7.27, 7.28, 7.29, 7.30, and 7.31. As Pearson’s pairwise networks do not accommodate the repeated measure structure of the data, we derived networks at both baseline (independent and identically distributed (IID) observations) as well as on the whole data, with repeated measures treated as IID to investigate any potential differences.