On the number of clusters and the fuzziness index for unsupervised FCA application to BOLD fMRI time series
Introduction
Functional MRI is a recent technique for determining the neural correlates of cognitive processes. It can be sensitised to changes of physiological parameters (CBF, CBV, blood oxygenation) during cognitive tasks (Ogawa et al., 1991, Kwong et al., 1992). The most convenient method uses blood as an endogenous contrast agent allowing non-invasive examination of focal signal intensity changes resulting from hemodynamics. The activation-induced increase in blood oxygenation decreases intravascular deoxyhemoglobin and, therefore, decreases susceptibility-induced intravoxel dephasing. Therefore, spin coherence increases, resulting in a relative signal enhancement (Thurlborn, 1982, Ogawa et al., 1991, Bandettini et al., 1992), an effect known as the BOLD contrast. Brain activation is therefore observed as a weak localized signal enhancement in time series images obtained using sequences sensitive to small changes in T2* and T2. Many difficulties must be addressed when processing fMRI data such as the weakness of the signal enhancement and the multiple sources of artifacts. These limitations are all the more strict for post-processing strategies since the size of population of the analysed voxels is large.
In order to extract functional information and detect activated regions using fMRI, the most widely adopted procedures are generally based on signal detection theory and are strongly paradigm dependent (Bandettini et al., 1993, Friston et al., 1994, Bullmore and Brammer, 1996, Lange, 1996, Ruan et al., 1996, Kuppussumy et al., 1997). In contrast with these approaches, other authors have applied different methods such as rotated principal component analysis (PCA) and fuzzy clustering analysis (FCA) which have the benefit of being exploratory and model-free (Scarth and McIntyre, 1995, Backfrieder, 1996). These techniques have the capability of separating different types of responses (e.g. BOLD from inflow effects), without any knowledge and hypothesis about the paradigm or the hemodynamic response function. FCA is a potent way to investigate fMRI data that has been used successfully, but it also presents several drawbacks. Indeed, the a priori definition of the number of clusters has not been discussed in previous works (Scarth and McIntyre, 1995, Baugmartner et al., 1998, Coutte et al., 1999) and remains an open problem called the ‘cluster validity’ problem. In addition, the data size remains an important issue since the activated regions represent a small proportion of the brain and can be embedded in the large amount of voxels that are not activated. This is the so-called ill-balanced data problem in the classification literature (Bezdek, 1981, Coutte et al., 1999).
In this paper, the fuzzy c-means (FCM) algorithm is briefly presented and adapted to the fMRI case. The clustering is applied to the data in the temporal domain. The cluster validity problem is then addressed and solved using heuristics (Fadili et al., 1998). An original cluster validity measure is introduced and validated using simulations. The effect of the fuzziness index choice on the detection power is studied using simulations with real fMRI noise (data acquired from subjects under rest condition). These effects are quantified using the area under the ROC as a single index of merit.
The format of the paper is as follows. In Section 2.1, the algorithm is presented in detail. Then, the influence of the Fuzzy Clustering Analysis parameters (e.g. number of a priori clusters, fuzziness index) is studied and discussed in order to optimise the proposed strategy. The emphasis is put on the potential of the method for use in fMRI time series analysis.
Section snippets
Overview of the fuzzy c-means (FCM) algorithm
Fuzzy cluster analysis (Zadeh, 1977) presents an alternative to hard clustering. It attempts to find a partition of a data set X of n feature vectors , by producing, for a pre-selected number of clusters c, c vectors in a feature space , called cluster centroids or prototypes. They represent points around which the data are concentrated. The FCA also produces, for each datum, a membership vector whose components are real numbers between 0 and 1, which measures the
Imaging
The data were acquired using a single shot EPI sequence on a 1.5T Signa scanner (72 blocks of 26 axial interleaved slices, TR=5 s, TE=60 ms, α=90°, 64×64 matrix, voxels of 3.75×3.75 mm2 in the plane and 5 mm thickness with no gap). For the motor activation study, four healthy right-handed young subjects were scanned. The paradigm began with four task volumes and then consisted of eight task-rest cycles, each cycle containing eight volumes (four under each condition). The task was a self-paced
Simulation description
All the presented measures may not lead to the same partition, e.g. the criteria using only membership degrees are usually too restrictive in the case of overlapping non-spherical clusters. The CS and S criteria are more efficient because of their dependence on data structure and shape. However, CS and S can become unpredictable and sensitive to the fuzziness index. Thus, we carried out simulations in order to study the joint influence of c and m on each of the three validity measures SCF, CS
Conclusion
This paper has presented a strategy for analysis of fMRI time series using a partitional fuzzy clustering method based on the FCM algorithm. We have described the algorithm, addressed its parameters and limitations. We have suggested some original solutions to the choice of the parameters, and validated these solutions.
The influence of the fuzziness index was studied in detail. Its impact on the partition quality was analysed using the MFs. It was then quantified by using the ROC methodology.
Acknowledgements
We would like to thank the anonymous reviewers for very helpful comments and suggestions concerning the first version of this paper. A part of this work is presented in the 6th international conference of Functional Mapping of the Human Brain San Antonio’2000.
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