Bayesian Predictive Modeling Based on Multidimensional Connectivity Profiling

Abstract

Dysfunction of brain structural and functional connectivity is increasingly being recognized as playing an important role in many brain disorders. Diffusion tensor imaging (DTI) and functional magnetic resonance (fMR) imaging are widely used to infer structural and functional connectivity, respectively. How to combine structural and functional connectivity patterns for predictive modeling is an important, yet open, problem. We propose a new method, called Bayesian prediction based on multidimensional connectivity profiling (BMCP), to distinguish subjects at the individual level based on structural and functional connectivity patterns. BMCP combines finite mixture modeling and Bayesian network classification. We demonstrate its use in distinguishing young and elderly adults based on DTI and resting-state fMR data.

Introduction

Predictive modeling has great potential in clinical applications such as diagnosis and prognosis as well as monitoring functional status during and after treatment, and generating an individualized treatment plan.¹ Magnetic resonance (MR) imaging is a noninvasive imaging technique to examine brain structure and function. MR-based predictive modeling centers on discriminating subjects at the individual level based on their MR imaging data.² To the extent that diseases affect brain structure and function, MR imaging should be able to characterize these manifestations, and perhaps lead to early diagnosis and accurate prognosis, or indicate promising therapeutic avenues.

The brain is a complex network of structurally and functionally connected neural modules. The dysfunction of brain networks is recognized as playing an important role in many brain disorders.³ Diffusion tensor imaging (DTI) and functional magnetic resonance (fMR) are widely used for inferring structural and functional connectivity, respectively.⁴ Several studies have centered on identifying connectivity patterns predictive of clinical status.^2,5–7 For example, Dosenbach et al. used support vector machine based multivariate pattern analysis to analyze functional connectivity patterns of 238 subjects, and made predictions about individuals’ brain maturity across development.² They found that binary support vector machine classification of individuals as either children or adults was 91% accurate.

It is well recognized that combining information from several imaging modalities may improve prediction accuracy. How to combine structural and functional connectivity patterns is an open problem. We propose a new method, called Bayesian prediction based on multidimensional connectivity profiling (BMCP), to distinguish subjects at the individual level based on structural and functional connectivity patterns. In BMCP, the connection between two brain regions p and q, E_p-q, is the variable of interest. For a connection E, the joint probability distribution between its structural and functional connectivity score is modeled as a finite mixture model. We call this mixture model the multidimensional connection signature of E, and we refer to the signatures of a set of connections as their multidimensional connectivity profile, which provides an overall picture of structure-function associations. Let C denote the group-membership variable (case or control, young or elderly). Our goal is to predict C based on a multidimensional connectivity profile. We use Bayesian network (BN) classifiers for this task.

Methods

The overall approach of BMCP is depicted in Figure 1. The input to BMCP includes DTI and resting-state fMR (rs-fMR) data, and the group-membership variable C. BMCP consists of three modules. The first module performs data preprocessing, to generate structural and functional connectivity scores. The second module performs multidimensional connectivity profiling (MCP), in which, we model the joint probability density of structural and functional connectivity scores, based on finite mixture modeling. The third module generates a classifier based on the profile. We base this module on a Bayesian-network representation with inverse-tree structure, to identify a set of connections that are predictive of C; this module then generates a naïve Bayes classifier from these connection variables.

Figure 1.

The architecture of BMCP.

Data preprocessing

The data preprocessing module operates on T1-weighted, DTI, and rs-fMR image data. Algorithms in this pipeline have been widely used in the neuroimaging research community.

The T1-weighted MR image-analysis procedure includes skull stripping, segmentation, and registration. First, the brain extraction tool⁸ is used to exclude non-brain tissues. Then, the FAST algorithm⁹ is used for segmentation into gray matter, white matter, and cerebrospinal fluid. Then we normalize the image to the Montreal Neurological Institute (MNI) stereotaxic space using the nonlinear registration algorithm FNIRT in the FMRIB Software Library (FSL).⁹ Based on the deformation field generated by FNIRT, we parcellate each individual brain into 90 structures defined in the Automated Anatomical Labeling (AAL) template.¹⁰

We also use an FSL-based pipeline to process DTI data.¹¹ First, we correct raw DTI data for head movement and eddy-current distortion. Then we fit a diffusion-tensor model for each voxel. For each subject, the labeled brain regions obtained by T1-weighted MR parcellation are registered to the subject’s diffusion space, using the transformation matrix from structural to diffusion space generated by FSL FDT registration; this step generates a brain parcellation defined in the diffusion space. Then we use PROBTRACTX,¹² a probabilistic fiber tracking algorithm, to calculate the connectivity score between two brain regions. Based on the PROBTRACTX result, we calculate the number of samples starting with voxels in brain region p and passing through brain region q, which represents the connection strength between regions p and q; we normalize this quantity by the total number of streamline samples. Noting that the density of fiber tracts between two regions, as calculated from DTI data, is often skewed, we apply a log transformation in order to increase normality.¹³ This transformed value is the structural connectivity score.

Resting-state fMR data preprocessing includes: motion correction, B0 field unwarping, slice-timing correction, skull-stripping, and regression against the motion parameter time courses. We register the base fMR volume to the subject’s T1-weighted MR images. Using the deformation field generated in the registration step, we generate an fMR-space AAL parcellation. After fMR space parcellation, for each AAL structure, we compute the average time series for each subject. Then we calculate Pearson’s correlation coefficient between pairs of time series. We transform each correlation coefficient into a Z-value using Fisher's Z transformation, in order to increase normality.¹³ This Z-value is the functional connectivity score.

Multidimensional connectivity profiling

The goal of multidimensional connectivity profiling (MCP) is to model the joint probability density of structural and functional connectivity scores, based on finite mixture modeling.^14,15 For a connection E, let Y_fmr and Y_dti represent the functional and structural connectivity scores, respectively. Y = (Y_fmr, Y_dti). f(Y) is the joint probability density of these two connectivity scores. Finite mixture modeling is an extremely flexible method of modeling unknown distribution shapes, based on a semiparametric framework. In finite mixture modeling, f(Y) is written in parametric form as

$$f(\mathbf{y};\Psi )=\sum _{i=1}^g\pi ifi(\mathbf{y};\theta i),$$ (1)

where g is the number of components in the model, θ_i is parameters for the ith component, π_i (∑π_i = 1) are called the mixing proportions, and Ψ{π₁, … , π_g , θ₁ , … , θ_g} contains all parameters. This mixture model represents the raw data set in a compact form, and can be considered a fingerprint or signature of the raw data set.

Figure 2 is a hypothetical study illustrating the conceptual framework of MCP. In this study, our analysis involves three brain regions: the right anterior cingulum (Cinglum_Ant_R), the right hippocampus (Hippocampus_R), and the left parahippocampal gyrus (ParaHippocampal_L). There are three connections: Cinglum_Ant_R - Hippocampus_R, Cinglum_Ant_R - ParaHippocampal_L, and Hippocampus_R-ParaHippocampal_L. (A-C) show the finite mixture models generated by MCP. We call the finite mixture model for a connection its signature, and we call the collection of these signatures a subject’s connectivity profile. The generated model is descriptive. For example, in (A), there are two clusters for the connection Cinglum_Ant_R - Hippocampus_R: the blue and green points represent high and low anatomical connectivity, respectively. In this example, there is significant variability in functional connectivity in both clusters.

Figure 2.

The conceptual framework of multidimensional connection profiling (MCP). In this example, our analysis involves a fully connected network (the upper portion) with three brain regions: the right anterior cingulum, the right hippocampus, and the left parahippocampal gyrus. There are three connections (thick black lines) in this network. A) The signature of the right anterior cingulum - right hippocampus. B) The signature of the right hippocampus - left parahippocampal gyrus. C) The signature of the right anterior cingulum - left parahippocampal gyrus. Each signature is represented as a finite mixture model. We call the collection of these signatures the connectivity profile.

The most widely used parametric family for f_i(y; θ_i) is the normal family, which leads to a multivariate normal mixture. The component i is the multivariate normal density with mean μ_i and covariance matrix Σ_i. We parameterize the covariance matrix using an eigenvalue-decomposition method; Σ_i is represented as

$$\Sigma i=\lambda iDiAi{D}_i^T$$ (2)

where D_i is the orthogonal matrix of eigenvectors, A_i is a diagonal matrix whose elements are proportional to the eigenvalues of Σ_i, and λ_i is an eigenvalue. The shape, orientation, and volume of the components are determined by A_i, D_i, and λ_i.

Mixture-model parameters are estimated using the expectation-maximization (EM) algorithm.¹⁶ In the E-step, the probability that an instance belongs to a cluster is estimated, conditional on the current parameter estimates. In the M-step, the parameters are updated given the current clustering result. The EM algorithm for mixture modeling is initialized with a model-based hierarchical clustering step.¹⁴ Model selection, including the specification of the covariance matrices and the number of clusters, is based on the Bayesian information criterion.¹⁷

MCP is a feature-extraction process: for each connection E, MCP generates a vector representing the group to which an instance belongs. In this manner, MCP partitions the instance space. We refer to this vector as the coloring for E. The colorings for all connections are the input to our next step.

BN classifier generation

In this step, our task is to build a predictive model of C based the colorings for all connections.

The input to our algorithm is a training data set D. D = {EC₁, … , EC_p, C}, where C assumes a value in {+, −}, and EC_i is the coloring for connection E_i. The output is a classifier.

Our algorithm is based on BN classification; our rationale for employing BN classifiers is as follows. First, the coloring of a connection is a discrete variable; BN modeling is a powerful approach for describing interactions among discrete variables, and has a formal mathematical foundation for generating models under uncertainty.¹⁸ Second, connection-based studies contain more variables than the number of subjects, leading to undersampling. In this case, the resulting model is prone to overfitting (poor generalizability).¹⁹ BN classifiers have a variety of variable-selection mechanisms to solve this problem.

We limit our algorithm to BN classifiers with inverse-tree structure (BNCIT) for joint classification and variable selection.^20–22 In BNCIT, the group-membership variable C is a leaf node, and predictive variables are parents of C. BNCIT is a computationally efficient method for detecting variables that are jointly predictive of C for high-dimensional data. In BNCIT, we use the Bayesian Dirichlet (BDe) score^23,24 to measure the goodness-of-fit of a candidate BN structure to the observed data. Then we search biomarkers of C based on

$$X^{*}=\arg \hspace{1em}maxX\hspace{1em}\mathrm{BDe}(C,X),$$ (3)

where X is the parent set of C.

However, due to undersampling, the model generated by BNCIT may not be stable to small perturbations of the data D. We employ ensemble learning to improve BNCIT stability. Ensemble learning stabilizes the model-generation process²⁵ and improves classification performance.^26–28 The procedure of BNCIT with ensemble learning is as follows. We employ bootstrap resampling to resample D K times, resulting in kth data set D^k. From D^k, we generate a BN model B^k using BNCIT. Based on B^k, we obtain the feature set, X^k(C), which is the parent set of C. We thereby obtain a feature ensemble {X¹(C), … , X^K(C)}. In the feature-aggregation step, we calculate the frequency of connection, freq(E). That is,

$$\mathrm{freq}(E)=\sum _{k=1}^{K}I[E\in X^k(C)],$$ (4)

where I_[condition] = 1 if condition is true; otherwise, I_[condition] = 0. We rank connections based on freq(E), and choose the top-ranked features as the aggregated feature set. The number of highly ranked features to be included in the classifier can be determined by internal cross-validation or a threshold. We denote this ensemble-derived aggregated feature set by X_e.

Given X_e, we can construct a naïve Bayes classifier to classify C. We use a naïve Bayes classifier because, given a set of predictors that are informative of C, naïve Bayes manifests excellent prediction performance in empirical studies.²⁹ Frequently, X_e includes more than ten variables; in this case, parameter estimation for a BN with inverse tree structure would be unreliable for undersampled data. Therefore, we adopt the naïve Bayes architecture.

We use leave-one-out cross-validation (LOOCV) to evaluate the generated classifier. In LOOCV, the data set D with K samples is partitioned into K subsets. For each subset, we generate a predictive model based on subjects in this subset, and assess the model based on the remaining subject. The overall performance is the average across all K subsets. To evaluate the classifier, we compute prediction accuracy, sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV).

Experimental Results

We applied BMCP to data from a DTI and rs-fMR study of normal aging.³⁰ This study included 70 participants with normal cognitive function. Exclusion criteria were previous brain surgery, major CNS trauma, or prior documented history of stroke (not including transient ischemic attacks) that resulted in lasting sequelae, active neurological dysfunction, or the use of antipsychotic and/or antiepileptic medications with known neurological side effects. This data set included two groups: young and old. The young adult group had 23 participants, aged 18–38 years, 11 of which were male. The old adult group included 47 participants, ages 65–90 years, 22 of which were male. All subjects completed and signed a consent form, which was approved by the IRBs at Virgina Tech and the Wake Forest University School of Medicine. MR data were collected on a GE 1.5 T scanner with an eight-channel head coil. T1-weighted, rs-fMR, and DTI images were collected for each subject.

Several studies on healthy aging reported the default mode network and the salient network change with age.³¹ Therefore, we investigated whether the multidimensional signatures of connections involving these two networks can discriminate between the young and old adult subjects at the individual level. There were 350 connections involving the bilateral medial prefrontal gyrus (MFG), which is in the default mode network, and the bilateral anterior cingulate cortex (ACC), which is in the salient network. Our analysis focuses on these connections.

Using BMCP, we generated classifiers to classify C based on multidimensional signatures.

Table 1 lists the classification results. In this table, the label BMCP-s indicates that the top s ranked variables were used to construct the BMCP classifier. We found that BMCP achieved high accuracy for a wide range of s (s = 10, 20, 30, 40, 50). The mean [standard deviation (SD)] of accuracy, sensitivity, specificity, PPV, NPV were 0.900 (0.023), 0.927 (0.024), 0.843 (0.024), 0.924 (0.012), 0.852 (0.045). BMCP’s classification performance was stable across different values of s.

Table 1.

The classification performance for the aging study.

	Accuracy	Sensitivity	Specificity	PPV	NPV
BMCP-10	0.871	0.894	0.826	0.913	0.791
BMCP-20	0.886	0.914	0.826	0.915	0.826
BMCP-30	0.900	0.936	0.826	0.917	0.864
BMCP-40	0.928	0.957	0.870	0.938	0.909
BMCP-50	0.914	0.936	0.870	0.936	0.870
Naïve Bayes	0.857	0.783	0.894	0.783	0.894
Gaussian Process	0.786	0.783	0.787	0.643	0.881
SVM	0.829	0.696	0.894	0.762	0.857

We computed the one-sample statistic for a binomial proportion to assess the statistical significance of our results. Random guessing achieved an accuracy of 0.67 (47/70). For our model, we tested the null hypothesis: accuracy = 0.67, versus the alternative hypothesis accuracy ≠ 0.67. P-values were <0.001 for s = 10, 20, 30, 40, 50; we found that our model’s performance was significantly different from that of random guessing.

We compared BMCP to three widely used classification methods: naïve Bayes,³² linear Gaussian process classifier,³³ and linear support vector machines (SVMs).³⁴ These classifiers do not have an embedded variable-selection mechanism. We used the filtering method¹⁹ for variable selection; in the filtering process, we use Fisher’s exact test to identify variables associated with C, and set the p-value cutoff to 0.01. Thus, if the p-value of a variable is below 0.01, we consider this variable to be predictive of C, and include it in the classifier. Given the selected predictors of C, the next step is to construct a model to predict C. Classifier parameters are tuned using internal fivefold cross-validation. Table 1 lists the classification results of these three classifiers. We found that BMCP achieved superior classification results relative to the other classifiers. The mean classification accuracy for these three classifiers was 0.824, whereas the mean accuracy of BMCP across s was 0.900.

Since BMCP’s classification performance was stable across s, we examined the detected biomarkers based on s = 30. During cross-validation, different connections were selected for different iterations. Among them, 11 were selected for all iterations. The regions involved in these selected connections included MFG, ACC, insula, the superior and middle temporal gyri, and the inferior frontal gyrus; Figure 3 shows these connections. For a connection, its association with C is measured by the p-value based on Fisher’s exact test. In this figure, the intensity of a connection is proportional to –log(p-value).

Figure 3.

Connections characterizing normal aging.

Discussion

We have proposed a method, BMCP, to analyze structural and functional connectivity patterns and use them to predict disease/cognitive states. BMCP has great potential in generating multimodal neuroimaging biomarkers that characterize a disease, clinical state, or cognitive process. We have shown that BMCP is applicable to integrated DTI/rs-fMR data analysis with good results, and that BMCP generates interpretable results.

BMCP combines finite mixture modeling and BN classification. Finite mixture modeling provides a framework to model unknown distribution shapes; and is widely used for clustering and density estimation. Finite mixture modeling has been successfully applied to problems in bioinformatics and medical imaging. One advantage of finite mixture modeling is computational efficiency. For the aging study, generating multidimensional connection signatures required approximately one minute for the algorithm to converge on a workstation with Intel Core 7 2.4 G CPU and 8 G memory. In this paper, we used the EM algorithm to estimate parameters of the mixture model; we used a model-based hierarchical clustering algorithm for initialization of EM parameters, which we found to produce good results.

BN classification has been used to generate classifiers based on a broad range of data, including neuroimaging data.^35–37 BNCIT is a type of BN classifier particularly suited to high-dimensional data. In this paper, we incorporated ensemble learning to stabilize BNCIT model generation. Our evaluation demonstrated that BNCIT with ensemble learning can accurately distinguish between the young and old adult subjects at the individual level, based on multidimensional connectivity profiles.

One future extension of this work is to expand BMCP to handle high-dimensional neuroimaging connectivity scores. In the current version of BMCP, we focus on integrated analysis of DTI/rs-fMR data, because DTI and rs-fMR are among the most widely used MR techniques to study brain connectivity. Magnetoencephalography (MEG) is an alternative means of measuring functional connectivity.³⁸ We will extend BMCP to accommodate DTI, rs-fMR, and MEG data. In that case, each connection would have three attributes: a functional connectivity score based on rs-fMR, a functional connectivity score based on MEG, and a structural connectivity score based on DTI. The generated mixture model would represent interactions among these connectivity measures.

Funding

This work is supported by the University of Maryland's Center for Health Informatics and Bioimaging, and the State of Maryland MPower initiative.

Conflict of interest

The authors declare no conflict of interest.