Elsevier

Medical Image Analysis

Volume 11, Issue 2, April 2007, Pages 169-182
Medical Image Analysis

A Dirichlet process mixture model for brain MRI tissue classification

https://doi.org/10.1016/j.media.2006.12.002Get rights and content

Abstract

Accurate classification of magnetic resonance images according to tissue type or region of interest has become a critical requirement in diagnosis, treatment planning, and cognitive neuroscience. Several authors have shown that finite mixture models give excellent results in the automated segmentation of MR images of the human normal brain. However, performance and robustness of finite mixture models deteriorate when the models have to deal with a variety of anatomical structures. In this paper, we propose a nonparametric Bayesian model for tissue classification of MR images of the brain. The model, known as Dirichlet process mixture model, uses Dirichlet process priors to overcome the limitations of current parametric finite mixture models. To validate the accuracy and robustness of our method we present the results of experiments carried out on simulated MR brain scans, as well as on real MR image data. The results are compared with similar results from other well-known MRI segmentation methods.

Introduction

Magnetic resonance imaging (MRI) is an effective diagnostic tool in the study of the human brain. Changes in the composition of brain tissues can be used to identify physiological processes and characterize disease severity (Heindel et al., 1994). For instance, changes in sulcal cerebrospinal fluid volume have been related to the neurodegeneration hypothesis in schizophrenia (Molina et al., 2002). Measures of the amounts of gray matter (GM), white matter (WM), cerebrospinal fluid (CSF), as well as their spatial distribution, have been used to support diagnosis of degenerative brain illnesses such as Alzheimer’s disease (DeCarli et al., 1992). For detecting tissue abnormalities such as cancers and injuries, regions of interest (ROIs) need to be examined in detail. In functional neuroimaging there is the need to correlate brain structure and function. To understand the relationship between activity in certain brain areas and specific mental functions, the brain can be partitioned into several anatomical regions (e.g., brainstem, cerebellum, diencephalon, and cerebrum) and structures which are associated with high-level mechanisms such as sensation, motor control, and cognition (Solo et al., 2001, Parry and Matthews, 2002). Low-level classification of the brain improves the localization of signal in magnetoencephalography and electroencephalography data (Dale and Sereno, 1993). Accurate classification of magnetic resonance (MR) images according to tissue type or region of interest has become a critical requirement in diagnosis, treatment planning, and cognitive neuroscience.

Manual segmentation methods are impractical for large amounts of data, and also are highly subjective and non-reproducible. In view of its practical importance, automated MRI segmentation has been a highly researched area in recent years. The myriad of different segmentation methods that have been proposed and implemented can be perceived by browsing the ITK Software Guide (Ibáñez and Schroeder, 2003). Statistically-based classification methods, especially parametric ones, are popular in brain MRI research. In particular, the expectation-maximization algorithm (Dempster et al., 1977), and the frequentist mixture model clustering algorithm (Hartigan, 1975) have been adapted to MRI intensity estimation and classification by many researchers and practitioners (Santago and Gage, 1993, Wells III et al., 1996, Guillemaud and Brady, 1997, Leemput et al., 1999, Ashburner and Friston, 2000, Ruan et al., 2000, Zhang et al., 2001).

In conventional statistical classifiers, the ideal brain MR image is assumed to have a piecewise constant mean intensity. Clustering methods can be used to partition MR images into different tissue types, and estimate the defining parameters for each tissue class. Typically, healthy brain tissue is classified into three broad tissue classes: gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF). Since each class is assumed to be modeled by a single Gaussian distribution, with mean and variance as parameters, tissue classification amounts to the estimation of a three component Gaussian mixture. There are several aspects that contribute to rendering the task of MR image segmentation difficult. Due to limitations of the image acquisition process, image intensity values are typically corrupted by inhomogeneities and noise. As a consequence of the finite resolution of the imaging process and the complexity of tissue boundaries, there is a partial volume effect which causes many voxels in MRI images to contain a mixture of more than one tissue type. The partial volume effect causes the distributions of the intensities to deviate from normal (Frackowiak et al., 2003). Similarly, any clusters representing the background are not well modeled by a single Gaussian (Cocosco et al., 2003). In addition, image noise produces speckled regions and ambiguous tissue boundaries, with direct impact on the accuracy of intensity-based classification procedures.

To address the abovementioned concerns, more complex Gaussian mixture models have been proposed for the segmentation of MR images of the human normal brain. For instance, the statistical model proposed in Grabowski et al. (2000), as well as the model in Ruan et al. (2000) are based on a five Gaussian mixture model. The authors assume that the brain can be segmented into three basic constituent classes (CSF, GM, and WM) and two mixed classes CG and GW, where CG denotes a mixture of CSF and GM, and GW denotes a mixture of GM and WM. However, several studies indicate that a five component mixture model is still insufficient for modeling levels of intensity inhomogeneity and variability in tissues as a result of specific biological processes. As an example, intensity histograms of abnormal brains show large deviations from the intensity histograms of the normal population of brains which are used in most automatic segmentation algorithms (Frackowiak et al., 2003). Another example is the automatic segmentation of MR images of the newborn brain. Automatic segmentation methods typically fail in segmenting the different structures of developing tissues apparent in newborn brain MRI (Prastawa et al., 2005).

In this paper, we propose a nonparametric Bayesian model for tissue classification of MR images of the brain. The model, known as Dirichlet process (DP) mixture model, uses Dirichlet process priors to overcome some of the limitations of current parametric finite mixture models. The DP mixture model is a Bayesian nonparametric methodology that relies on Markov chain Monte Carlo simulations for exploring mixture models with an unknown number of components. An important feature of the method is that it is weakly informative about a priori assumptions of the parameters. This feature supports data-adaptive estimations. We exemplify the application of the DP mixture model to the case of conjugate models with normal structure. The basic DP mixture model is extended with a Markov random field (MRF) model to incorporate spatial coherence assumptions. To validate the accuracy and robustness of our method, we present the results of experiments carried out on simulated MR brain scans and real MR image data. The results are compared with some other well-established MRI segmentation methods.

The paper is organized as follows. In Section 2, we recall some of the main characteristics of finite mixture models, as a preliminary step for the introduction of DP mixture models. Section 3 presents the DP mixture model used in this work. In Section 4, we detail a DP mixture specification with normal structure. In Section 5, we outline the main characteristics of the MRF model. Section 6 describes the experimental validation analysis. Finally, Section 7 discusses some of the advantages and limitations of the proposed approach.

Section snippets

Finite mixture models

We consider the case where we wish to model data y = (y1,  , yn) as independent observations from a mixture with a specified number of components k,f(yi)=j=1kωjfj(yi),i=1,,n,where the distributions fj(1  j  k) are known up to a parameter and the mixing proportions 0 < ωj < 1 satisfy j=1kωj=1. When the fj’s are from a given parametric family, with unknown parameter θj, we obtain the parametric mixture modelyij=1kωjf(yi|θj).It is common to assume that the mixture coefficients ω = (ω1  , ωk) are all from

The Dirichlet process mixture model

Traditional parametric models constrain inference to a specific parametric form indexed by a finite dimensional parameter, such as the mean and variance in normal models. However, constraining inference to a specific parametric family may limit the scope and type of inferences that can be drawn from such models.

One conventional nonparametric method of density estimation that is often used is the kernel-density estimator (Venables and Ripley, 1997). Suppose y1,  , yn is a sample from f(.), where f

The conjugate normal–normal DP mixture model

In this section, we exemplify the application of the DP mixture model described in Section 3 to the case of conjugate models with normal structure. Following (7), we consider a DP mixture modeled as a convolution with a normal kernel in the formyiN(yi|θi,σ2)dG(θ),i=1,,nwith DP prior on the mixing measure G  DP(αG0) and a standard normal base measure G0 = N(0,1). For the precision σ−2 we use the gamma prior σ−2  Γ(a,b) with parameters a and b. The hierarchical mixture model may then be specified

Markov random field modeling

Noise in the magnetic fields used during MR image acquisition prevent characterization of voxel tissue content based solely on image intensity. Standard finite mixture models work with voxel intensities and do not take into account spatial information. Such limitation causes the finite mixture model to work well only with moderate levels of noise. To compensate for noise disturbances, several studies have extended the basic finite mixture model by taking into account the spatial information

Experimental validation

The segmentation method proposed in this paper was tested on simulated data for which the classification ground truth is known. Several image models were used to test the ability of the DP mixture model to assess the number of classes in noisy images. Different noise conditions as well as spatially smoothly varying intensity inhomogeneities or bias fields were tested to simulate real imaging conditions. In addition, tests were also performed with real MR images provided by the Internet Brain

Discussion and conclusion

Several authors have shown that finite mixture models give excellent results in the automated segmentation of MR images of the human normal brain. In particular, finite mixture models based on a five Gaussian mixture model have recently gained wide acceptance (see Cuadra et al. (2005) for a comparative study of classification methods in T1-weighted MR brain images). In a Bayesian setting, these models rely on the specification of parametric priors for well-defined tissue types. However,

Acknowledgments

The author thanks the anonymous reviewers for their constructive comments and valuable recommendations.

References (54)

  • D. Blackwell et al.

    Ferguson distributions via Polya urn schemes

    The Annals of Statistics

    (1973)
  • C.A. Bush et al.

    A semiparametric Bayesian model for randomised block designs

    Biometrika

    (1996)
  • Casella, G., Robert, C.P., Wells, M.T., 2000. Mixture models, latent variables and partitioned importance sampling....
  • D.L. Collins et al.

    Design and construction of a realistic digital brain phantom

    IEEE Transactions on Medical Imaging

    (1998)
  • M.B. Cuadra et al.

    Comparison and validation of tissue modelization and statistical classification methods in T1-weighted MR brain images

    IEEE Transactions on Medical Imaging

    (2005)
  • A.M. Dale et al.

    Improved localization of cortical activity by combining EEG and MEG with MRI cortical surface reconstruction: a linear approach

    Journal of Cognitive Neuroscience

    (1993)
  • C. DeCarli et al.

    Method of quantification of brain, ventricular, and sub-arachnoid csf volumes for MRI images

    JCT

    (1992)
  • A. Dempster et al.

    Maximum likelihood from incomplete data via the EM algorithm (with discussion)

    Journal of the Royal Statistical Society B

    (1977)
  • J. Diebolt et al.

    Estimation of finite mixture distributions through Bayesian sampling

    Journal of the Royal Statistical Society B

    (1994)
  • M.D. Escobar

    Estimating normal means with a Dirichlet process prior

    Journal of the American Statistical Association

    (1994)
  • M.D. Escobar et al.

    Bayesian density estimation and inference using mixtures

    Journal of the American Statistical Association

    (1995)
  • T.S. Ferguson

    A Bayesian analysis of some nonparametric problems

    The Annals of Statistics

    (1973)
  • R.S.J. Frackowiak et al.

    Human Brain Function

    (2003)
  • A. Gelman et al.

    Bayesian Data Analysis

    (2004)
  • S. Geman et al.

    Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images

    IEEE Transactions on Pattern Analysis and Machine Intelligence

    (1984)
  • R. Guillemaud et al.

    Estimating the bias field of MR images

    IEEE Transactions on Medical Imaging

    (1997)
  • J.A. Hartigan

    Clustering Algorithms

    (1975)
  • Cited by (0)

    View full text