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. 2010 Apr;29(4):1039-49.
doi: 10.1109/TMI.2010.2040625. Epub 2010 Mar 22.

FRATS: Functional Regression Analysis of DTI Tract Statistics

Hongtu Zhu  1 , Martin Styner, Niansheng Tang, Zhexing Liu, Weili Lin, John H Gilmore
Affiliations

FRATS: Functional Regression Analysis of DTI Tract Statistics

Hongtu Zhu et al. IEEE Trans Med Imaging. 2010 Apr.

Abstract

Diffusion tensor imaging (DTI) provides important information on the structure of white matter fiber bundles as well as detailed tissue properties along these fiber bundles in vivo. This paper presents a functional regression framework, called FRATS, for the analysis of multiple diffusion properties along fiber bundle as functions in an infinite dimensional space and their association with a set of covariates of interest, such as age, diagnostic status and gender, in real applications. The functional regression framework consists of four integrated components: the local polynomial kernel method for smoothing multiple diffusion properties along individual fiber bundles, a functional linear model for characterizing the association between fiber bundle diffusion properties and a set of covariates, a global test statistic for testing hypotheses of interest, and a resampling method for approximating the p-value of the global test statistic. The proposed methodology is applied to characterizing the development of five diffusion properties including fractional anisotropy, mean diffusivity, and the three eigenvalues of diffusion tensor along the splenium of the corpus callosum tract and the right internal capsule tract in a clinical study of neurodevelopment. Significant age and gestational age effects on the five diffusion properties were found in both tracts. The resulting analysis pipeline can be used for understanding normal brain development, the neural bases of neuropsychiatric disorders, and the joint effects of environmental and genetic factors on white matter fiber bundles.

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Figures

Fig. 1
Fig. 1
A schematic overview of FRATS: a nonparametric model for regularizing individual tracts, a functional linear model, a global test statistic for hypothesis testing, and a resampling method for estimating the p-value of the global test statistic.
Fig. 2
Fig. 2
Splenium tract and diffusion properties along the splenium tract: (a) the splenium tract extracted from the tensor atlas with color representing mean FA value; (b) FA; (c) MD; (d) λ1; (e) λ2; (f) λ3. The diffusion properties in panels (b)–(f) are sampled along the atlas-normalized arc length for all 128 subjects.
Fig. 3
Fig. 3
Simulation study: Type I and Type II error rates. Rejection rates of Sn based on the resampling method are calculated at five different values of c for sample sizes of 64, 128 subjects at the 5% (green) and 1% (red) significance levels: (a) n = 128; (b) n = 64.
Fig. 4
Fig. 4
Results from the analysis of FA and MD on the splenium tract: reconstructed curves f^i(s)sec for FA in panel (a) and MD in panel (b); (c) estimated correlation between FA and MD along the tract; estimated covariance matrices Γ^(s,t) for FA in panel (d) and MD in panel (e); (f) estimated regression coefficient functions for FA: β^11(s) for intercept (blue), β^12(s) for gender (red), β^13(s) for gestational age (green), and β^14(s) for age (black).
Fig. 5
Fig. 5
Results from the analysis of FA and MD on the splenium tract: the −log10(p) values of test statistics Sn(sj) for testing gender effect in panel (a), gestational age effect in panel (b), and age effect in panel (c) on FA; the −log10(p) values of test statistics Sn(sj) for testing gender effect in panel (d), gestational age effect in panel (e), and age effect in panel (f) on MD.
Fig. 6
Fig. 6
Results from the analysis of the three eigenvalues of diffusion tensor on the splenium tract: the −log10(p) values of test statistics Sn(sj) for testing gender effect in panel (a), gestational age effect in panel (b), and age effect in panel (c) on λ1 (red), λ2 (blue) and λ3 (green); correlations among λ1, λ2 and λ3 in panel (d); scatter plots of λ1 (red), λ2 (blue) and λ3 (green) against gestational age and age in panels (e) and (f), respectively.
Fig. 7
Fig. 7
Right internal capsule tract and diffusion properties along the tract: (a) the right internal capsule tract extracted from the tensor atlas with color representing mean FA value; (b) FA; (c) MD; (d) λ1; (e) λ2; (f) λ3. The diffusion properties in panels (b)–(f) are sampled along the atlas-normalized arc length for all 128 subjects.
Fig. 8
Fig. 8
Results obtained from the analysis of FA and MD on the right internal capsule tract: the estimated mean function and covariance function for FA in panels (a) and (b), respectively; the estimated mean function and covariance function for MD in panels (d) and (e), respectively; (g) the correlation between FA and MD; the −log10(p) values of test statistics Sn(sj) for testing gestational age and age effect on FA (green), MD (blue), and (FA, MD) (red) in panels (c) and (f), respectively; scatter plots of MD and FA measures from a selected grid point against standardized gestational age, abbreviated as S-gestation age, in panels (h) and (i), respectively.
Fig. 9
Fig. 9
Results from the analysis of the three eigenvalues of diffusion tensor on the right internal capsule tract: (a) the estimated mean functions for λ1 (red), λ2 (blue), and λ3 (blue); (b) correlations among λ1, λ2 and λ3; (c) scatter plot of λ1 (red), λ2 (blue) and λ3 (green) from a selected grid point against gestational age; the −log10(p) values of test statistics Sn(sj) for testing gender effect in panel (d), gestational age effect in panel (e), and age effect in panel (f) on λ1 (red), λ2 (blue) and λ3 (green).
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