Elsevier

NeuroImage

Volume 44, Issue 2, 15 January 2009, Pages 373-384
NeuroImage

Multiple-subjects connectivity-based parcellation using hierarchical Dirichlet process mixture models

https://doi.org/10.1016/j.neuroimage.2008.08.044Get rights and content

Abstract

We propose a hierarchical infinite mixture model approach to address two issues in connectivity-based parcellations: (i) choosing the number of clusters, and (ii) combining data from different subjects. In a Bayesian setting, we model voxel-wise anatomical connectivity profiles as an infinite mixture of multivariate Gaussian distributions, with a Dirichlet process prior on the cluster parameters. This type of prior allows us to conveniently model the number of clusters and estimate its posterior distribution directly from the data. An important benefit of using Bayesian modelling is the extension to multiple subjects clustering via a hierarchical mixture of Dirichlet processes. Data from different subjects are used to infer on class parameters and the number of classes at individual and group level. Such a method accounts for inter-subject variability, while still benefiting from combining different subjects data to yield more robust estimates of the individual clusterings.

Introduction

In-vivo segregation of brain regions into distinct functional sub-units is an ambitious yet important task in neuroscience. Sulcal/Gyral landmarks have long been used systematically to parcellate individual cortices for functional or structural studies (Fischl et al., 2004). Such approaches may be limited by a variability in correspondence between anatomical landmarks and functional or cytoarchitectonic borders (Amunts et al., 1999). Atlas-based approaches to localise brain function are limited by the high degree of inter-subject variability (Fischl et al., 2007). Functional sub-units in the cortex can be imaged directly using functional brain imaging techniques, but such studies are limited to examining functional divisions that are implicated in a particular task under investigation. This not only imposes practical limitations on achieving a functional division of the cortex due to the number of different task contrasts necessary, but also requires that the correct functional dissociations are known prior to the design of the experiment.

Diffusion-based tractography has been used recently to subdivide cortical and sub-cortical regions non-invasively on the basis of their connections (Behrens et al., 2003, Johansen-Berg et al., 2004, Klein et al., 2007, Rushworth et al., 2006, Tomassini et al., 2007, Anwander et al., 2007). The rationale behind such an approach is that the function of a brain region is constrained by its connections (Passingham et al., 2002). Indeed, independent fMRI data have shown a close correspondence between connectionally and functionally defined sub-units in the Thalamus (Johansen-Berg et al., 2005), Supplementary Motor Area (Johansen-Berg et al., 2004) and the lateral premotor cortex (Tomassini et al., 2007).

Early attempts to segment brain regions on the basis of their connection patterns have fallen into two general categories. They have either relied on strong neuro-anatomical hypotheses for the connection patterns of different functional sub-units (Behrens et al., 2003, Johansen-Berg et al., 2004, Zarei et al., 2007, Lehericy et al., 2004, Newton et al., 2006, Leh et al., 2007) or they have used machine learning techniques to perform “blind classification” of voxel-wise connectivity patterns with no deference to knowledge of the anatomy(Johansen-Berg et al., 2004, Anwander et al., 2007). Each of these approaches has clear advantages. The ability to impose and test explicit anatomical hypotheses in the data allows connectional differences to be interpreted clearly with respect to their predicted functional consequences, and lends weight to the resultant parcellations through comparison with connection patterns known from invasive techniques. Conversely, the ability to classify connectivity patterns without such prior knowledge allows for exploratory analyses in brain regions where the anatomy is not well known, either through lack of invasive data or because of expected differences between species. Neither of these approaches, however, are able to infer the number of functional sub-units that exist within a brain region. They both rely on direct input from the experimenter. This limitiation is especially important in blind parcellation approaches that aim to detect functional sub-units without strong prior hypotheses. Lastly, such blind classification approaches have not been able to establish a correspondence between functional sub-units across individuals. Such an inferred correspondence enhances the interpretability and robustness of the parcellation, but moreover it is a crucial step in solving the difficult problem of inferring the number of sub-units that best describe the data. In situations where the evidence for or against a new sub-unit is not compelling, formal comparison across different individuals offers an invaluable source of information.

In this article, we take an alternative approach that lies between “hypothesis-driven” and “blind” connectivity-based classifications. Voxel-wise connection patterns are modelled using a Bayesian generative model that represents the data as samples from a mixture of probability distributions. The data here are the connectivity patterns to anatomical brain regions, which allows us to impose explicit anatomical hypotheses whenever they are available. On the other hand, having a generative model allows us to have an explicit representation of the number of sub-units that constitute a given brain region. Moreover, it gives us a rigourous mathematical framework within which formal comparison across different subjects becomes a trivial extension of the single subject case.

The model that we use here is an infinite mixture model. In a Bayesian setting, we model the connectivity profiles of a brain region as an infinite mixture of multivariate Gaussian distributions, with a Dirichlet Process prior on the cluster parameters (Escobar and West, 1995). The model falls into the class of non-parametric Bayes, in which the number of parameters in the model (which depends on the number of classes) can adaptively change depending on the data. With this type of priors, one can therefore estimate a posterior distribution for the number of clusters directly from the data. An important benefit of using Bayesian modelling is the extension to multiple subjects clustering via a hierarchical mixture of Dirichlet processes (Teh et al., 2006). Data from different subjects are then used to infer on class parameters and number of classes at individual and group level, accounting for inter-subject variability, but still benefiting from combining data from different subjects to yield more robust estimates of the individual clusterings.

Mixture models with Dirichlet process priors have already been used in the context of brain segmentation. In fMRI, Thirion et al. (2007) have formalised brain activations using spatial Dirichlet process mixture (DPM) models at an individual level, and then used a Bayesian Network to combine information across different subjects. On the other hand, Kim and Smyth (2006) have used hierarchical DPM to solve the same problem, by combining different subjects in the same mixture model, and allowing for inter-subject variability by including random effects on the class parameters. Da Silva (2007) used DPM to classify structural images into different tissue types based on T1-weighted structural MR images. In the present article, DPMs are used to segment regions on the basis of their connectivity. Although structural, functional and diffusion data pertain to different types of information, it is clear however that they share underlying spatial information. Combining different sources of structural data has proven to give promising results for clustering grey matter (Yovel and Assaf, 2007). It is conceivable that merging structural and diffusion data in a common spatial model might, in the future, serve to improve the sensitivity and specificity of the clustering.

Section snippets

Methods

Our clustering strategy is based on mixture modelling. Usually, this requires fixing the number of clusters a priori. We will use recent advances in non-parametric Bayesian statistics to estimate the number of clusters from the data via an infinite mixture model. A hierarchical extension of this model will allow us to combine data from different subjects. This literature being relatively new, we begin by introducing some of the basic concepts that are relevant for our application.

Results

We will focus on two parcellation experiments: the segregation of the thalamus and that of the supplementary motor area (SMA), although we will also show some results on simulated data to illustrate how the number of clusters is estimated, depending on the between-cluster variance, the dimensionality of the data and the initialisation of the Gibbs sampler. In both experiments (thalamus and SMA), we show the effect of the spatial priors and the estimation of the number of clusters. We also show

Discussion

In this article we have described a Bayesian approach to clustering brain regions on the basis of their connections. Tractography data results are injected into a non-parametric Bayesian model, and used to classify brain voxels with no prior knowledge on the number of clusters. Using Dirichlet process priors allows us to estimate the number of clusters from the data. The use of a hierarchical extension of the DPM makes combining data from different subjects straightforward, and allows groups of

Acknowledgments

We thank Brian Patenaude for his patient help on 3D figure rendering. The authors would like to acknowledge funding from the Dr. Hadwen Trust for Humane Research (SJ), the UK Engineering and Physical Sciences Research Council (MWW) and the UK Medical Research Council (TEJB).

References (35)

  • BleiD.M. et al.

    Variational methods for Dirichlet process mixtures

    Bayesian Analysis

    (2006)
  • da SilvaA.R.F.

    A dirichlet process mixture model for brain mri tissue classification

    Med. Image Anal.

    (2007)
  • DeoniS.C.L. et al.

    Visualization of thalamic nuclei on high resolution, multi-averaged t1 and t2 maps acquired at 1.5 t

    Hum. Brain Mapp.

    (2005)
  • EscobarM.D. et al.

    Bayesian density estimation and inference using mixtures

    J. Am. Stat. Assoc.

    (1995)
  • FergussonT.S.

    A Bayesian analysis of some nonparametric problems

    Ann. Stat.

    (1973)
  • FischlB. et al.

    Automatically parcellating the human cerebral cortex

    Cereb. Cortex

    (2004)
  • FischlB. et al.

    Cortical folding patterns and predicting cytoarchitecture

    Cereb. Cortex

    (2008)
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