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Varying coefficient model for modeling diffusion tensors along white matter tracts. (English) Zbl 1454.62425
Summary: Diffusion tensor imaging provides important information on tissue structure and orientation of fiber tracts in brain white matter in vivo. It results in diffusion tensors, which are \(3\times 3\) symmetric positive definite (SPD) matrices, along fiber bundles. This paper develops a functional data analysis framework to model diffusion tensors along fiber tracts as functional data in a Riemannian manifold with a set of covariates of interest, such as age and gender. We propose a statistical model with varying coefficient functions to characterize the dynamic association between functional SPD matrix-valued responses and covariates. We calculate weighted least squares estimators of the varying coefficient functions for the log-Euclidean metric in the space of SPD matrices. We also develop a global test statistic to test specific hypotheses about these coefficient functions and construct their simultaneous confidence bands. Simulated data are further used to examine the finite sample performance of the estimated varying coefficient functions. We apply our model to study potential gender differences and find a statistically significant aspect of the development of diffusion tensors along the right internal capsule tract in a clinical study of neurodevelopment.

MSC:
62P10 Applications of statistics to biology and medical sciences; meta analysis
62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62H35 Image analysis in multivariate analysis
62R10 Functional data analysis
Software:
fda (R); KernSmooth
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References:
[1] Anderson, A. W. (2001). Theoretical analysis of the effects of noise on diffusion tensor imaging. Magn. Reson. Med. 46 1174-1188.
[2] Arsigny, V. (2006). Processing data in lie groups: An algebraic approach. Application to non-linear registration and diffusion tensor MRI. Ph.D. thesis, Ecole Polytechnique.
[3] Assemlal, H.-E., Tschumperlé, D., Brun, L. and Siddiqi, K. (2011). Recent advances in diffusion MRI modeling: Angular and radial reconstruction. Med. Image Anal. 15 369-396.
[4] Barmpoutis, A., Vemuri, B. C., Shepherd, T. M. and Forder, J. R. (2007). Tensor splines for interpolation and approximation of DT-MRI with applications to segmentation of isolated rat hippocampi. IEEE Trans. Med. Imaging 26 1537-1546.
[5] Basser, P. J., Mattiello, J. and LeBihan, D. (1994a). Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance Ser. B 103 247-254.
[6] Basser, P. J., Mattiello, J. and LeBihan, D. (1994b). MR diffusion tensor spectroscopy and imaging. Biophys. J. 66 259-267.
[7] Basser, P. J., Pajevic, S., Pierpaoli, C., Duda, J. and Aldroubi, A. (2000). In vivo fiber tractography using DT-MRI data. Magn. Reson. Med. 44 625-632.
[8] Davis, B. C., Bullitt, E., Fletcher, P. T. and Joshi, S. (2010). Population shape regression from random design data. Int. J. Comput. Vis. 90 255-266.
[9] Dryden, I. L., Koloydenko, A. and Zhou, D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3 1102-1123. · Zbl 1196.62063
[10] Fan, J. and Gijbels, I. (1992). Variable bandwidth and local linear regression smoothers. Ann. Statist. 20 2008-2036. · Zbl 0765.62040
[11] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66 . Chapman & Hall, London. · Zbl 0873.62037
[12] Fan, J., Yao, Q. and Cai, Z. (2003). Adaptive varying-coefficient linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 57-80. · Zbl 1063.62054
[13] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491-1518. · Zbl 0977.62039
[14] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand. J. Stat. 27 715-731. · Zbl 0962.62032
[15] Fan, J. and Zhang, W. (2008). Statistical methods with varying coefficient models. Stat. Interface 1 179-195. · Zbl 1230.62031
[16] Goldsmith, A. J., Crainiceanu, C. M., Caffo, B. S. and Reich, D. (2011). Penalized functional regression analysis of white-matter tract profiles in multiple sclerosis. NeuroImage 57 431-439.
[17] Goodlett, C. B., Fletcher, P. T., Gilmore, J. H. and Gerig, G. (2009). Group analysis of DTI fiber tract statistics with application to neurodevelopment. NeuroImage 45 S133-S142.
[18] Kim, P. T. and Richards, D. S. (2011). Deconvolution density estimation on spaces of positive definite symmetric matrices. In Nonparametric Statistics and Mixture Models : A Festschrift in Honor of Thomas P. Hettmansperger 147-168. World Scientific Press, Singapore.
[19] Kosorok, M. R. (2003). Bootstraps of sums of independent but not identically distributed stochastic processes. J. Multivariate Anal. 84 299-318. · Zbl 1016.62063
[20] O’Donnell, L. J., Westin, C.-F. and Golby, A. J. (2009). Tract-based morphometry for white matter group analysis. NeuroImage 45 832-844.
[21] Pierpaoli, C. and Basser, P. J. (1996). Toward a quantitative assessment of diffusion anisotropy. Magn. Reson. Med. 36 893-906.
[22] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[23] Rose, J., Mirmiran, M., Butler, E., Lin, C., Barnes, P. D., Kermoian, R. and Stevenson, D. K. (2007). Neonatal microstructural development of the internal capsule on diffusion tensor imaging correlates with severity of gait and motor deficits. Developmental Medicine and Child Neurology 49 745-750.
[24] Schwartzman, A., Mascarenhas, W. F. and Taylor, J. E. (2008). Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices. Ann. Statist. 36 2886-2919. · Zbl 1196.62067
[25] Smith, S. M., Jenkinson, M., Johansen-Berg, H., Rueckert, D., Nichols, T. E., Mackay, C. E., Watkins, K. E., Ciccarelli, O., Cader, M. Z., Matthews, P. M. and Behrens, T. E. J. (2006). Tract-based spatial statistics: Voxelwise analysis of multi-subject diffusion data. NeuroImage 31 1487-1505.
[26] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Monographs on Statistics and Applied Probability 60 . Chapman & Hall, London. · Zbl 0854.62043
[27] Wang, L., Li, H. and Huang, J. Z. (2008). Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. J. Amer. Statist. Assoc. 103 1556-1569. · Zbl 1286.62034
[28] Welsh, A. H. and Yee, T. W. (2006). Local regression for vector responses. J. Statist. Plann. Inference 136 3007-3031. · Zbl 1094.62053
[29] Wu, C. O. and Chiang, C.-T. (2000). Kernel smoothing on varying coefficient models with longitudinal dependent variable. Statist. Sinica 10 433-456. · Zbl 0945.62047
[30] Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis . Wiley-Interscience, Hoboken, NJ. · Zbl 1127.62041
[31] Yuan, Y., Zhu, H., Ibrahim, J. G., Lin, W. and Peterson, B. S. (2008). A note on the validity of statistical bootstrapping for estimating the uncertainty of tensor parameters in diffusion tensor images. IEEE Trans. Med. Imaging 27 1506-1514.
[32] Yuan, Y., Zhu, H., Lin, W. and Marron, J. S. (2012). Local polynomial regression for symmetric positive definite matrices. J. R. Stat. Soc. Ser. B Stat. Methodol. 74 697-719.
[33] Yushkevich, P. A., Zhang, H., Simon, T. J. and Gee, J. C. (2008). Structure-specific statistical mapping of white matter tracts. NeuroImage 41 448-461. · Zbl 1171.68823
[34] Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data. Ann. Statist. 35 1052-1079. · Zbl 1129.62029
[35] Zhu, H., Li, R. and Kong, L. (2010). Multivariate varying coefficient models for functional responses. Technical report, Univ. North Carolina at Chapel Hill. · Zbl 1373.62169
[36] Zhu, H., Ibrahim, J. G., Tang, N., Rowe, D. B., Hao, X., Bansal, R. and Peterson, B. S. (2007a). A statistical analysis of brain morphology using wild bootstrapping. IEEE Trans. Med. Imaging 26 954-966.
[37] Zhu, H., Zhang, H., Ibrahim, J. G. and Peterson, B. S. (2007b). Statistical analysis of diffusion tensors in diffusion-weighted magnetic resonance imaging data. J. Amer. Statist. Assoc. 102 1085-1102. · Zbl 1332.62222
[38] Zhu, H., Chen, Y., Ibrahim, J. G., Li, Y., Hall, C. and Lin, W. (2009). Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging. J. Amer. Statist. Assoc. 104 1203-1212. · Zbl 1388.62198
[39] Zhu, H., Styner, M., Tang, N., Liu, Z., Lin, W. and Gilmore, J. H. (2010). FRATS: Functional regression analysis of DTI tract statistics. IEEE Trans. Med. Imaging 29 1039-1049.
[40] Zhu, H., Kong, L., Li, R., Styner, M., Gerig, G., Lin, W. and Gilmore, J. H. (2011). FADTTS: Functional analysis of diffusion tensor tract statistics. NeuroImage 56 1412-1425.
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