Whole brain voxel-wise analysis of single-subject serial DTI by permutation testing
Introduction
In recent years, diffusion tensor magnetic resonance imaging (DTI) (Basser et al., 1994a, Basser et al., 1994b) has been widely used in studying normal development and aging, and diverse pathological conditions of the human brain due to its unique ability to identify microstructural abnormalities. Since DTI is a relatively new technique and is fundamentally different from other imaging techniques in that each voxel contains not a single value but a 3 × 3 matrix with 6 unique elements called the diffusion tensor, the statistical analysis of DTI is still under development. Multiple approaches exist for defining the target regions for statistical comparisons, including manually traced region of interest (ROI), fiber tracking-defined ROI, whole brain histogram, and whole brain voxel-wise analysis by statistical parametric mapping SPM (Friston et al., 1995) or by tract-based spatial statistics (Smith et al., 2006). There are also diverse DTI-derived parameters to compare, such as mean diffusivity (Dav), parallel/transverse diffusivity, fractional anisotropy (FA) and the primary eigenvector (Schwartzman et al., 2005).
Most DTI studies with statistical testing performed in one or more of the aforementioned ways have one thing in common; they are all multiple-subject group comparison studies, usually one group of experimental subjects compared with a matched group of healthy control subjects. Group comparison studies are possible when the effects of interest are located in stereotypical anatomic structures, such as the same white matter tracts, across subjects. In certain conditions where the effects are expected to be focal (or multi-focal) with spatial distributions that are highly specific for individual subjects, finding and grouping subjects that share effects in similar anatomic locations may be difficult. Furthermore, group analyses demand that significant effects be larger than group variability and therefore may suffer from decreased sensitivity. A statistical analysis that can be performed in individual subjects is thus required.
Due to the non-Gaussianity, DTI parameters such as anisotropy indices (Pajevic and Basser, 2003, Skare et al., 2000) makes parametric testing less optimal. Non-parametric resampling techniques have great potential in the statistical testing of these DTI data since the recent trend of oversampling DTI data makes it suitable for resampling. One of these techniques called bootstrap has been shown to be very useful in DTI (Chung et al., 2006a, Heim et al., 2004, Jones, 2003, Lazar and Alexander, 2005, Pajevic and Basser, 2003, Whitcher et al., 2007). These works have shown the ability of bootstrap to estimate the uncertainties of DTI parameters (descriptive statistics) but have not addressed the issue of statistical significance testing (inferential statistics). For statistical testing when comparing two groups, permutation testing (Edgington, 1995, Good, 2005, Manly, 2007), another non-parametric resampling technique, can be useful as well.
Permutation testing provides statistical significance testing of differences between groups, with the unique ability of directly estimating the null distribution of the statistic describing the difference, rather than assuming a null distribution of analytically known form (such as T-distribution). Multiple groups are required for resampling and permutation testing provides exact (or almost exact) p-values but does not estimate uncertainties of sample statistics (such as the standard error of mean). Bootstrap is mainly used to estimate the accuracy of sample statistics by resampling from one group (multiple groups are not required). Though it can be used not only in descriptive statistics (i.e., standard errors) but also in inferential statistics (i.e., significance testing), bootstrap testing is not able to estimate a data-driven null distribution and bootstrap-estimated p-value is only approximate.
We focus on permutation testing in this work. Permutation testing is well established in the field of neuroimaging, especially functional MRI where numerous works advocate the strength of this approach in the last decade (for a review refer to Nichols and Holmes, 2002). Permutation based method 1) makes minimal assumptions and thus can be applied even in situations where the assumptions of parametric approaches are not met or cannot be verified, 2) is conceptually simple and provides intuitive solutions to the multiple comparison problem, and 3) is easily applicable to any test statistics allowing the researchers to freely choose the statistic best suited for their studies. Unlike bootstrap, permutation testing can elegantly incorporate the whole statistical procedure of voxel-wise comparison and multiple-comparison correction in a completely non-parametric way due to the ability to estimate data-specific null distribution as described above, possibly with more accurate p-values.
In this study, we describe how permutation testing can be properly implemented in voxel-wise analysis of single-subject serial DTI studies. We will pay special attention to not violate exchangeability assumption for permutation testing that if groups are not different then any re-grouping of the samples (permutation) are equally likely as the original grouping (observation). Then, we present a novel statistical analysis framework called PERVADE (permutation voxel-wise analysis of diffusion estimates) that is designed to localize subtle and local microstructural changes over time in the whole brain of a single subject without any prior hypothesis. PERVADE includes 1) non-linear registration between two time-points to account for any morphological changes over time, 2) voxel-wise calculation of p-values by permutation testing, and 3) suprathreshold cluster analysis with permutation testing to deal with the multiple comparison problem. Our preliminary results of local microstructural changes detected outside as well as inside the focal lesions of patients in the earliest stage of multiple sclerosis show the potential of this technique to provide additional information about microstructural white matter injury.
Section snippets
Permutation testing in group comparison
Before presenting the proposed voxel-wise permutation testing in a single-subject serial DTI study, we first examine how it would be done in a multiple-subject group comparison study (i.e., comparison of group-averaged metric between two cohorts), a simple and typical scenario of permutation testing. To test whether the observed difference in group-averaged FA, i.e., , is statistically significant, permutation testing can be performed as follows. All subjects are randomly assigned
Monte Carlo simulation
Comparison of the distribution of observed p-values from the simulation of permutation testing and the expected p-values are shown (Fig. 3). Fig. 3a shows that when the diffusion directions were rotated by 20° each around x-, y- and z-axes for one time point, and if the permutation was carried out ignoring the diffusion gradient directions, the observed p-values were deviated from the ideal distribution, with the trend of over-estimating p-values. This is not surprising since increased
Discussion
In this paper, we have proposed a novel non-parametric statistical framework for detecting subtle and local diffusion MRI changes over time in a single subject. To our knowledge, this is the first study to analyze single-subject serial DTI data at the voxel/cluster level. This was possible with the ability of DTI permutation testing to calculate voxel-wise statistic p-values and to correct for multiple comparisons. We have demonstrated that we can ensure exchangeability of the DWIs which
Acknowledgments
We are grateful to Daniel Handwerker and Pratik Mukherjee for helpful discussions. This study was supported by RG3240A1 from the National Multiple Sclerosis Society.
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