Wavelet-based cluster analysis: data-driven grouping of voxel time courses with application to perfusion-weighted and pharmacological MRI of the rat brain
Introduction
The current functional MRI (fMRI) literature encompasses a considerable range of experimental protocols, involving both hypothesis- and data-driven analysis techniques. The design and analysis of fMRI experiments are rapidly developing with a variety of paradigms from which to choose. The most popular analysis of fMRI data involves fitting a general linear model (GLM) whose functional form is based on the a priori experimental procedure, usually convolved with a hemodynamic response function, and testing the fit of this model on a voxel by voxel basis (Lange, 1999, Petersson et al., 1999, Rabe-Hesketh et al., 1997). Recently, data-driven or exploratory approaches have also begun to be employed in brain imaging in order to identify and separate time courses of interest, along with their associated spatial patterns. Possible approaches include independent component analysis (ICA) (McKeown et al., 1998, Calhoun et al., 2001, Calhoun et al., 2002, Kiviniemi et al., 2003, Netsiri et al., 2003), and methods that are sensitive to measures of “structure” in the data from the distribution of values in the time courses, or from spatial or temporal (auto-)correlation (Baumgartner et al., 2000, Friman et al., 2002, Somorjai et al., 2002). In between these two approaches are methods designed to determine from the data, and validate, an optimum model based on the experimental protocol for subsequent hypothesis-based testing (Kherif et al., 2002, Luo and Nichols, 2003).
The modeling of pharmacological MRI (phMRI) data is relatively new in the literature and poses some different problems (Chen et al., 1997, Houston et al., 2001, Preece et al., 2001, Reese et al., 2000, Xi et al., 2002). Here we consider phMRI experiments associated with a single event (injection of a compound) producing a transient in the voxel time series. The identification of “active” voxels in such experiments is possible using a parametric model, similar to the standard GLM-based approach for fMRI data, for the time curves; for example, a compartmental model (Bloom et al., 1999, Ritschel and Kearns, 1999, Xi et al., 2002). However, the level of association between the voxel time series and any parametric model is suspect and may lead to the loss of truly active voxels based on model misspecification. Another class of experiment with similar temporal characteristics is perfusion-weighted imaging (PWI) based on an analysis of signal changes during the first pass circulation of a rapid bolus of contrast agent. Here, a common parametric model is the gamma-variate function (Rosen et al., 1990). In both of these cases, the models are nonlinear and possess all the difficulties with fitting nonlinear models to noisy data.
In this paper, we investigate grouping (clustering) voxels by their temporal characteristics. Various types of cluster analysis have been used on fMRI data in the past. Moser et al. (1999) used fuzzy clustering on the raw voxel time series. Filzmoser et al. (1999) used k-means and principal component analysis (PCA) to analyze fMRI time series. Baune et al. (1999) implemented dynamic cluster analysis using k-means clustering (adaptive clustering). Baudelet and Gallez (2003) performed k-means clustering on a subset of the raw voxel time series, where white noise voxels were discarded by testing a smoothed estimate of their spectral density function. Goutte et al. (1999) performed hierarchical and k-means clustering on the cross-correlation sequence (truncated to ± lags) between the design function and each voxel time series.
Instead of the cross-correlation function, we decompose each voxel time series using the discrete wavelet transform (DWT). The DWT produces a vector of wavelet coefficients associated with temporal scales and specific locations in time. The DWT is well known to track transient events in time series and thus presents itself as a reasonable candidate to decompose voxel time series in imaging experiments such as perfusion-weighted imaging and phMRI. The DWT isolates a deterministic signal into a few large coefficients while the background noise is spread across most, if not all, of the wavelet coefficients. Testing a subset of wavelet coefficients against a threshold is the most common way to separate the signal and noise, commonly known in the literature as wavelet shrinkage or de-noising (Donoho and Johnstone, 1994, Donoho and Johnstone, 1995). Wavelet shrinkage per se is not implemented here, instead entire scales of wavelet coefficients are removed from the decomposition—usually from higher frequency bands. The experimental design (in this case a single stimulus), and thus the underlying physical process involved, dictates which scales should be removed. The observed data are not used to determine which wavelet coefficients survive. This dimension reduction procedure avoids any unwarranted assumptions on the noise structure of the data.
By grouping the data according to their own activation pattern, wavelet-based cluster analysis (WCA) uses DWT to discriminate distinct cerebral response patterns. WCA is an exploratory procedure, but with efficient implementation over the brain volume it identifies brain regions that follow similar activation patterns.
We have targeted our methodology in this paper toward two classes of MRI experiments in the rat brain. The first involves tracking a bolus of contrast agent to map cerebral perfusion in an animal model of stroke. The second comprises a phMRI experiment involving an acute pharmacological challenge following intravascular administration of cocaine that typically produces first-order pharmacokinetic signal changes. In addition, we compare the WCA results on a “phantom” data set with spatial and temporal ICA to help determine how well WCA performs with respect to other blind analysis techniques.
Section snippets
MRI experiments
All animal experiments were ethically peer reviewed and approved. Experiments in the United Kingdom were carried out according to the Animals (Scientific Procedures) Act (1986). Experiments in Italy were carried out in accordance with Italian regulations governing animal welfare and protection. Both the perfusion-weighted and pharmacological in vivo experiments were performed using Bruker 4.7T BioSpec MRI systems (Bruker, Ettlingen, Germany).
Phantom data
WCA was performed on the phantom data using K = 2,…,9 groups. Identification of active regions was generally poor for K = 2,3, but produced good voxel groupings for K = 4,5. Allowing the number of clusters to increase past K = 5 did not improve the signal detection or differentiation, it only partitioned the background noise into several groups. The WCA results for the phantom data with K = 6 clusters are shown in Fig. 2. Average cluster time courses are shown in Fig. 2b. The active regions are
Discussion
Clustering temporal changes in MRI images is extremely helpful in localizing, without a priori assumptions, brain regions—not necessarily spatially adjacent—that are affected in a similar way by drug administration where the input model is unknown or progressive. The associated time courses may be used to inform the construction of appropriate models for subsequent hypothesis-based analysis. Automatic analyses such as wavelet-based cluster analysis (WCA) have the advantage of removing
Conclusions
We have proposed a methodology called wavelet-based cluster analysis (WCA) that efficiently partitions repetitive MRI spatiotemporal data containing nonperiodic transient temporal features into K distinct groups. The discrete wavelet transform is used in WCA to compress the data by reducing the temporal dimension and retaining only key transient features. With a much larger collection of basis functions available, this method is applicable beyond the range of wavelet coefficients selected for
Acknowledgments
The authors would like to thank Torsten Reese, Alessandro Gozzi, and Angelo Bifone (Psychiatry CEDD, GlaxoSmithKline) for constructive comments on this work. The authors would also like to thank two anonymous reviewers whose comments greatly improved the quality of this publication.
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