INTEGRATIVE BAYESIAN ANALYSIS OF NEUROIMAGING-GENETIC DATA THROUGH HIERARCHICAL DIMENSION REDUCTION

Abstract

Advances in neuromedicine have emerged from endeavors to elucidate the distinct genetic factors that influence the changes in brain structure that underlie various neurological conditions. We present a framework for examining the extent to which genetic factors impact imaging phenotypes described by voxel-wise measurements organized into collections of functionally relevant regions of interest (ROIs) that span the entire brain. Statistically, the integration of neuroimaging and genetic data is challenging. Because genetic variants are expected to impact different regions of the brain, an appropriate method of inference must simultaneously account for spatial dependence and model uncertainty. Our proposed framework combines feature extraction using generalized principal component analysis to account for inherent short- and long-range structural dependencies with Bayesian model averaging to effectuate variable selection in the presence of multiple genetic variants. The methods are demonstrated on a cocaine dependence study to identify ROIs associated with genetic factors that impact diffusion parameters.

1. INTRODUCTION

Chronic cocaine consumption leads to psychological and physical problems. Several studies have suggested that genetics contribute to addiction. More specifically, it has been asserted that certain DNA polymorphisms may enhance vulnerability to drug abuse and may be associated with an individual’s response to treatment [1]. By facilitating the interrogation of neuropathological markers of cognitive disorders, non-invasive imaging technologies such as functional magnetic resonance imaging (fMRI) and diffusion tensor imaging have provided new insights into the underlying mechanism of addiction. In particular, chronic cocaine users have been shown to exhibit subtle abnormalities in particular brain regions. Current multidisciplinary efforts endeavor to elucidate the precise environmental, genetic, and psychological mechanisms underlying the alteration of brain imaging parameters in chronic cocaine users [2, 3].

Developing analytic approaches for integrating imaging and genetic data is challenging because the statistical and computational frameworks must account for (i) high-dimensionality of the imaging and/or genetic data; (ii) existence of complex correlation structures such as short- and long-range dependence (e.g., spatial/serial correlation) among the imaging features; and (iii) spatial heterogeneity in variable selection in the presence of a large set of genetic variants that might impact different parts of the brain.

To overcome these limitations to statistical inference, we introduce a hierarchical statistical framework called integrative Bayesian analysis of neuroimaging-genetic data through hierarchical dimensional reduction (iBANG-HD). Briefly, we leverage both biological knowledge (in defining regions of interest, ROIs) and principled dimension reduction techniques (e.g., generalized principal component analyses) hierarchically, as suitable projections for the high-dimensional imaging features. Subsequently, we account for model (genetic) uncertainty by using Bayesian model averaging (BMA) procedures in the reduced space. This results in a fast, scalable method that can accommodate up to hundreds of thousands of voxellevel observables while facilitating coherent probabilistic inference to account for the multiple sources of inherent variation.

Our methods are motivated by a recent cocaine addiction study that used DTI to identify brain regions that emit strong evidence of differential diffusion patterns, demonstrated by fractional anisotropy (FA) values, among candidate genetic variants, cocaine users, and demographic features. The results suggested that chronic cocaine users exhibit subtle abnormalities in the anterior and posterior corpus callosum and in tracts in the frontal and parietal regions of the brain. Our results suggested that cocaine consumption is associated with diminished FA in most brain regions. Additionally, gene polymorphisms associated with GABAergic neurotransmitters and receptors exhibited evidence of association with FA. This finding has potential implications in the development and progression of addiction.

2. HIERARCHICAL STATISTICAL FRAMEWORK

In this section, we introduce the hierarchical statistical framework for iBANG-HD, which proceeds via the following steps.

2.1. Model formulation

Suppose the entire brain region can be mapped to r = 1,…, R ROIs defined using an appropriate brain atlas. In each ROI, the (nested) imaging features (e.g., FA values) at the voxel ν^(r) = 1,…, V^(r) for subject i = 1,…, n are represented by $y_i^{(r)}$ (ν), and the corresponding (m-dimensional) genetic and demographic variables are represented by {x_i1,…, x_im}. Assume Y ^(r) (ν) is a column-wise matrix of imaging features $\{y_1^{(r)}(\nu ),\dots ,y_n^{(r)}(\nu )\}$, X is an n × m matrix, and β^(r)(ν) is the full m-dimensional vector of the regression coefficients. A linear regression model for the νth voxel in the rth ROI can be expressed as

$${\mathit{Y}}^{(r)}(\nu )={\alpha }^{(r)}+\mathit{X}{\mathit{\beta }}^{(r)}(\nu )+{\sigma }^{(r)}{\epsilon }^{(r)},$$ (1)

where α^(r) is an intercept, σ^(r) ε ℝ₊ is a scale parameter, and ε ^(r) is random noise (accounting for unknown sources of variation), which follows an n-dimensional normal distribution with zero mean and the identity covariance matrix.

Our primary construct for inference is the effect surfaces β^(r)(ν) across the ROIs, which capture the associations between the imaging features and each of the m genetic covariates across the brain. However, this requires estimation of V^(r) × m number of parameters over all ROIs, which in our case is (14 × 10⁴) × 24 ≈ 33 × 10⁴ parameters, and presents considerable analytical and computational challenges. To circumvent this, we decouple the model fitting and inference using a three-step component-wise analysis pipeline:

• Step I: Apply hierarchical dimension reduction to each ROI via generalized principal component analysis that accounts for both short- and long-range spatial dependencies (Section 2.2).

• Step II: Estimate the association between genetic and demographic variables via Bayesian model averaging on the reduced dimensional space of each ROI (Section 2.3).

• Step III: Use reverse projections to obtain posterior inferences across the entire brain region (Section 2.4).

2.2. Generalized principal component analysis

For notational simplicity, we drop the superscript r from the ensuing discussions, noting that model fitting is performed for each ROI independently and in parallel. Using a model based on principal component analysis (PCA), we project the imaging features, Y (ν), as Y (ν) = M+UDV^T +E, where M denotes the mean matrix, D is the singular values, U and V are the left and right (eigen-) factors, respectively, and E is the error matrix. Assuming Z(ν) denotes the centered data, Z(ν) = Y(ν) − M. The matrix errors can be written as vec(E) ∼ (0, P⁻¹ ⊗ Q⁻¹).

If we assume the error matrix values are independent and identical i.e., P = I and Q = I, the above model reduces to the standard PCA model with independent errors. While this accounts for long-range dependencies, it does not account for short-range “local” spatial dependencies. This limitation can be addressed by considering unequal weighting of matrix errors (E) according to the data structure, using a recently introduced generalized PCA (GPCA) model [4]. This assumes the noise covariance is smooth with respect to the structured imaging data, as in fMRI and DTI data [5], through suitable specification of the smoothing operators: P,Q. In our context, they are defined as follows: for P_ν×ν, we consider a Laplacian graphical operator to account for the spatial structure of the data based on the Euclidean distance for each ROI [6]. This accounts for the local spatial architecture by allowing nearby voxels to have similar loadings. Assume G is a graph that denotes the grid structures based on the Euclidean distance between the voxels in each ROI. We define P_ν×ν based on the Laplacian matrix as P_ν×ν ≡ L(G) = {D(G) − A(G)}, where A(G) is the adjacency matrix, and D(G) represents the diagonal matrix of the vertex degree, suitably defined. Q_n×n is defined as the identity matrix since the patients are considered to be independent.

The loss function of the transportable quadratic norm under unequal weighting of the matrix error terms can be expressed as

$$\begin{array}{l}{\Vert \mathit{Z}-\mathit{U}\mathit{D}{\mathit{V}}^T\Vert }_{\mathit{Q},\mathit{P}}^2={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{\nu }{\displaystyle \sum _{i^{\prime }=1}^n{\displaystyle \sum _{j^{\prime }=1}^{\nu }{\mathit{Q}}_{i{i}^{\prime }}{\mathit{P}}_{j{j}^{\prime }}}}}}\\ \times (Z_{ij}-{\mathit{u}}_i\mathit{D}{\mathit{v}}_j^T)(Z_{i^{\prime }{j}^{\prime }}-{\mathit{u}}_{i^{\prime }}\mathit{D}{\mathit{v}}_{j^{\prime }}^T),\end{array}$$ (2)

where u_i is the i^th column of U, and v_j is the j^th column of V. This can be expressed as a generalized least-square matrix decomposition (GMD) optimization problem to find the best rank-K approximation of the data with respect to Q, P –norm by minimize_U,D,V ${\Vert \mathit{Z}-\mathit{U}\mathit{D}{\mathit{V}}^{\mathit{T}}\Vert }_{\mathit{Q},\mathit{P}}^2$, subject to U^TQU = I_(k), V^TRV = I_(k), and diag(D) ≥ 0, where Q, P are the left and right quadratic operators, respectively. We use the proposed GMD algorithm, which is feasible for the massive data sets commonly encountered in neuroimaging [4].

In essence, the above GPCA model defines a projection of the original ν−dimensional image matrix Y (ν) to a lower K-orthogonal dimensional Z(K) matrix, using the following (conformable) projection matrix ${\mathit{W}}_{\mathit{K}\times {\mathit{\nu }}^{(r)}}={\mathit{D}}_{\mathit{K}\times \mathit{K}}^{1/2}{\mathit{V}}_{\mathit{K}\times {\mathit{\nu }}^{(r)}}^T(\mathrm{i}.\mathrm{e}.,Z(K)\equiv \mathit{W}\mathit{Y}(\nu ))$. These projections have two important properties: they not only take into account the spatial structure, but also are (nearly) loss-less transformations since they capture most of the modes of variation in the data. More importantly, these projects serve as substantial dimension reduction devices. For example, for the middle cerebellar peduncle ROI with 5280 voxels, using GPCA, we can explain 95% of the variation in this ROI using only K = 50 principal components. Overall, the average number of voxels in each ROI is around 2882.10 and the mean number of principal components needed to explain 95% of the variability is around 42.22, which indicates that the GPCA is capable of almost 68.25-fold dimension reduction.

2.3. Bayesian model averaging

The lower dimensional orthogonal projections Z(K) for each ROI serve as responses to construct the effect surfaces β^(r)(ν) that capture the associations between the imaging features and each of the m genetic covariates across the brain. However, this necessitates estimation of K × m number of parameters, which in our case for a given ROI (e.g., the middle cerebellar peduncle ROI) is 50 × 24 ≈ 1200 parameters without accounting for model uncertainty. In other words, we do not expect the same set of genetic covariates to have the same impact across all brain regions; hence, the need to incorporate covariate (model) selection into our modeling strategy. However, the number of models increases exponentially to 50 × 2²⁴ when accounting for model uncertainty over all possible configurations of models, which represents substantial analytical and computational challenges. To overcome this challenge, we utilize BMA procedures, which account for model uncertainty by shrinking the influence of insignificant covariates (to zero) through appropriate model weights, and provides a unified method of inference for all voxels [7], as detailed below.

Let $\mathcal{M}=\{M_j:j=1,\dots ,2^m\}$ define the model space. Suppose that each ROI can be mapped to K components. For Z^(r)(K) obtained from GPCA, a specific model M_j has a subset of X_j clinical, demographic and genetic variables, leading to the following equation:

$$Z^{(r)}(K)={\alpha }^{\ast (r)}+{\mathit{X}}_j{\mathit{\beta }}_j^{\ast (r)}(K)+{\sigma }^{\ast (r)}{\epsilon }^{\ast },$$ (3)

where the superscript * is used to define the parameters in the reduced space, and ${\mathit{\beta }}_j^{\ast (r)}\in {ℝ}^{m_j}$ (0 ≤ m_j ≤ m) is the reduced subset of covariates with the elimination of ${\mathit{\beta }}_{m-m_j}^{\ast (r)}(K)$. Priors: Priors are defined for (i) model space $\mathcal{M}$ and (ii) corresponding parameters ${\mathit{\beta }}_j^{\ast }(K)$, α* and σ*. On the model space, to elucidate no a priori model preference in the absence of prior knowledge, we select a uniform distribution. For the regression parameters, we assume improper non-informative priors for α* and σ* such that p(α*, σ*) ∝ σ⁻¹ to achieve maximal learning from the data. The residual error variance, σ*, explains unknown sources of variability. Also, we assume a g-prior structure for ${\mathit{\beta }}_j^{\ast }(K)$ whereby $p\{{\mathit{\beta }}_j^{\ast }(K)|{\alpha }^{\ast },{\sigma }^{\ast },M_j\}$ is modeled as an m_j-dimensional normal distribution with mean zero and covariance matrix of ${\sigma }^{\ast 2}{(g{{\mathit{X}}^{\mathit{\prime }}}_j{\mathit{X}}_j)}^{-1}$, where g = 1/max {n, m²}, which guarantees asymptotic consistency for selecting the correct model [8].

Posterior computations: The posterior distribution of ${\mathit{\beta }}_j^{\ast }(K)$ can be derived as

$$\mathcal{P}\{{\mathit{\beta }}_j^{\ast }(K)|\mathit{Z}(K)\}={\displaystyle \sum _{j=1}^{2^m}\mathcal{P}\{{\mathit{\beta }}_j^{\ast }(K)|\mathit{Z}(K),M_j\}\mathcal{P}\{M_j|\mathit{Z}(K)\}},$$ (4)

where the posterior model-specific probability $\mathcal{P}\{M_j|\mathit{Y}(K)\}$ is calculated as

$$\mathcal{P}\{M_j|\mathit{Z}(K)\}=\frac{\{{\mathcal{L}}_{(K)}(M_j)\}P(M_j)}{{\displaystyle {\sum }_{j=1}^{2^m}\{{\mathcal{L}}_{(K)}(M_h)\}P(M_h)}},$$ (5)

and the marginal likelihood of model M_j which we denote by ℒ_(K)(M_j), is

$$\begin{array}{r}{\mathcal{L}}_{(K)}(M_j)={\displaystyle \int p\{\mathit{Z}(K)|{\alpha }^{\ast },{\mathit{\beta }}_j^{\ast }(K),{\sigma }^{\ast },M_j\}}p({\alpha }^{\ast },{\sigma }^{\ast })\\ p\{{\mathit{\beta }}_j^{\ast }(K)|{\alpha }^{\ast },{\sigma }^{\ast },M_j\}d{\alpha }^{\ast }d{\mathit{\beta }}_j^{\ast }(K)d{\sigma }^{\ast },\end{array}$$ (6)

where $p\{\mathit{Z}(K)|{\alpha }^{\ast },{\mathit{\beta }}_j^{\ast }(K),{\sigma }^{\ast },M_j\}$ which represents the sampling model, (3), p(α*, σ*) and $p\{{\mathit{\beta }}_j^{\ast }(K)|{\alpha }^{\ast },{\sigma }^{\ast },M_j\}$ are the prior distributions for the intercept, scale, and regression coefficients, respectively. We use Markov chain Monte Carlo (MCMC)-based methods to estimate all model parameters and posterior probabilities, specifically the Metropolis-based sampling schemes described in [9].

2.4. Posterior inference via reverse projections

The hierarchical implementation of GPCA and BMA provides the posterior means and standard deviations of the effect surfaces for each genetic covariate-ROI combination in the reduced space. For coherent biological and scientific interpretations, we need inferences at the original voxel/brain level, however. In our framework, this can be achieved by using reverse projections that map the estimates in the reduced (Z) space back to the data (Y) space as follows. The coefficient surfaces, β^*(r)(ν), in the reduced space for the rth ROI can be projected to the entire (brain-wide) ROI through the following procedure. Let $\mathit{Z}(K)={\mathit{U}}_{n\times K}{\mathit{D}}_{K\times K}^{1/2}\times {\mathit{D}}_{K\times K}^{1/2}{\mathit{V}}_{K\times {\nu }^{\tau }}^T$, and the projection matrix of ${\mathit{W}}_{\mathit{K}\times {\mathit{\nu }}^{(r)}}={\mathit{D}}_{\mathit{K}\times \mathit{K}}^{1/2}{\mathit{V}}_{\mathit{K}\times {\mathit{\nu }}^{(r)}}^T$. Since a BMA-based MCMC procedure was conducted for each Z(k), considering genetic, clinical and demographic features, we use the MCMC samples of the regression coefficients of the reduced space ${\mathit{\beta }}_j^{\ast (r)}(K)$ to reconstruct the coefficient surface on the entire ROI space using the following reverse projections: ${\mathit{\beta }}_j^{(r)}(\nu )={\mathit{\beta }}_j^{\ast }(K)\times {\mathit{W}}^{-1}\equiv \stackrel{\mathit{\sim }}{\mathit{\beta }}$. Note that since these are linear projections, we can assess significance using the 95% upper and lower credible intervals for each $\stackrel{\mathit{\sim }}{\mathit{\beta }}$ on the original data space to evaluate the true effect of each genetic covariate on the imaging values.

3. APPLICATION TO COCAINE STUDY

The iBANG-HD statistical framework described in Section 2 was used to identify ROIs that exhibit strong evidence of differential patterns among the candidate genetic variants and demographic variables. Before model fitting, we defined the ROIs using the Johns Hopkins University white matter atlas, which includes 48 ROIs [10]. We focused our analysis on voxels with FA values that exceed 0.2 to capture white matter regions of the brain. This resulted in a total of 138,667 voxels for downstream analysis. We examined 21 candidate genetic variants in 17 genes that we hypothesized might play a role in addiction vulnerability and which have been associated with addiction vulnerability, psychiatric morbidities, or neurotransmitter pathways [11]. This includes several polymorphisms in the dopamine and serotonin transporters and in the norepinephrine postsynaptic receptor.

The results for corpus callosum ROIs are summarized in Figure 1, where each row depicts the dominant magnitude and direction of the effect of cocaine consumption (top row) and gene GAD1^a (bottom row) on FA alteration in the corpus callosum. Precedent preclinical and clinical studies have identified relationships between cocaine use and GAD1. Enoch et al. [12] showed that GAD1 expression levels in postmortem brains were related to cocaine use. In addition, our findings support reduction in white matter FA in cocaine users within corpus callosum ROIs, which is consistent with the findings of previous studies [13, 14, 15]. In terms of the computation time for the entire procedure, steps I-III took approximately 75 seconds for fitting each ROI using a standard multi-core computing server.

Fig. 1.

Average magnitude of FA alteration among ROIs that comprise the corpus callosum. Averages of the estimated regression coefficients are provided calculated among significant voxels with positive (left) and negative (right) posterior means. Color represents the extent of FA alteration: green indicates enhancement; red diminishment.

4. MODEL PERFORMANCE

We evaluated the performance of our BMA-based model fitting compared to that of a full Bayesian model (Full) with no model averaging (i.e., including all covariates in the model). We computed two model selection metrics as follows.

Approximate deviance information criterion (aDIC)

We used a variant of the deviance information criterion (DIC), which is a hierarchical modeling generalization of the Akaike information criterion (AIC) and Bayesian information criterion (BIC) [16]. In BMA, we consider the marginal likelihoods of models, so DIC_j(K) in model space $\mathcal{M}$ is calculated as DIC_j(K) = 2 {m_j (K) + 1} −2 {ℒ_j (K)}, where m_j (K) represents the total number of regressors in model M_j, and ℒ_j (K) is the marginal likelihood of model M_j. Using the proportion of MCMC frequencies as weights, we can extend DIC to BMA settings as a weighted average of the model-specific DICs. Thus, aDIC for component K, aDIC_j (K), in model space $\mathcal{M}$ is calculated as $\mathit{aDIC}(K)={\displaystyle {\sum }_{j=1}DI{C}_j(K)}\times w_j$, where w_j is a weight determined by the MCMC sampling frequency of model j. To formally compare the DIC of BMA versus that of the full model, we used paired t–tests of the null hypothesis H₀: DIC_BMA − DIC_Full = 0 versus a (one-sided) alternative hypothesis of the H_a: DIC_BMA < DIC_Full. The p-values are very close to zero (p < 2.2 × e⁻¹⁶) for all 48 ROIs, suggesting that BMA effectuates estimators with a significantly enhanced model fit versus complexity trade-offs.

Bayesian information criterion (BIC)

BIC is another model section criterion that is closely related to AIC. Similarly, for component K, BIC_j(K) of the models in model space $\mathcal{M}$ is derived by BIC_j(ν) = −2(ℒ_j(K)) + (m_j (K) + 1) × log(n), where ℒ_j(K) is the marginal likelihood of model M_j, and m_j(K) is the total number of regressors in model M_j. In BMA, BIC(K) is calculated by $\mathit{BIC}(K)={\displaystyle {\sum }_{j=1}BI{C}_j(K)\times w_j}$, where w_j is a weight computed from the MCMC sampling frequency of model j. Because the Full model involves a single likelihood function, BIC (and DIC) can be calculated without considering these weights. Again, p-values for comparing the methods for goodness-of-fit are very close to zero (p < 2.2 × e⁻¹⁶) for all 48 ROIs, demonstrating enhanced performance for BMA.

Figure 2 compares the goodness-of-fit measures between BMA and Full models across all ROIs. The resultant values of both DIC and BIC are substantially lower for BMA when compared to the Full model, which shows enhanced goodness-of-fit for BMA was evident among all 48 ROIs. We highlight (in red) the corpus callosum ROIs that were depicted in Figure 1.

Fig. 2.

Left (right) panels depict the mean DIC (BIC) for full Bayesian (x-axis) model versus Bayesian model averaging (y-axis) for all 48 ROIs. Red cross: genu of corpus callosum; red circle: body of corpus callosum: red triangle: splenium of corpus callosum.