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Biomedical Applications of Geometric Functional Data Analysis

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

In this chapter, we describe several biomedical applications of geometric functional data analysis methods for modeling probability density functions, amplitude and phase components in functional data, and shapes of curves and surfaces. We begin by reviewing parameterization-invariant Riemannian metrics and corresponding simplifying square-root transforms for each case. These tools allow for computationally efficient implementations of statistical procedures on the appropriate representation spaces, including computation of the Karcher mean and exploration of variability via principal component analysis. We then showcase applications of these tools in multiple biomedical case studies based on various datasets including Glioblastoma Multiforme tumors, Diffusion Tensor Magnetic Resonance Image-based white matter tracts and fractional anisotropy functions, electrocardiogram signals, endometrial tissue surfaces and subcortical surfaces in the brain.

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References

  1. Bakas, S., Zeng, K., Sotiras, A., Rathore, S., Akbari, H., Gaonkar, B., Rozycki, M., Pati, S., Davatzikos, C.: GLISTRboost: combining multimodal MRI segmentation, registration, and biophysical tumor growth modeling with gradient boosting machines for glioma segmentation. In: BrainLes, pp. 144–155 (2015)

    Google Scholar 

  2. Bauer, M., Bruveris, M., Harms, P., Møller-Andersen, J.: A numerical framework for Sobolev metrics on the space of curves. SIAM J. Imag. Sci. 10(1), 47–73 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bharath, K., Kurtek, S., Rao, A., Baladandayuthapani, V.: Radiologic image-based statistical shape analysis of brain tumours. J. R. Stat. Soc. Ser. C 67(5), 1357–1378 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bhattacharyya, A.: On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109 (1943)

    MathSciNet  MATH  Google Scholar 

  5. Bousseljot, R., Kreiseler, D., Schnabel, A.: Nutzung der EKG-Signaldatenbank CARDIODAT der PTB uber das internet. Biomedizinische Technik 40(1), S317–S318 (1995)

    Google Scholar 

  6. Brosens, I., Puttemans, P., Campo, R., Gordts, S., Kinkel, K.: Diagnosis of endometriosis: Pelvic endoscopy and imaging techniques. Best Pract. Res. Clin. Obstet. Gynaecol. 18(2), 285–303 (2004)

    Article  Google Scholar 

  7. Čencov, N.N.: Statistical Decision Rules and Optimal Inference. American Mathematical Society, Providence (1982)

    Google Scholar 

  8. Clifford, G.D., Azuaje, F., McSharry, P.: Advanced Methods And Tools for ECG Data Analysis. Artech House, Inc., Boston (2006)

    Google Scholar 

  9. De Sousa, F., Melo, E., Vermeulen, L., Fessler, E., Medema, J.P.: Cancer heterogeneity—a multifaceted view. EMBO Rep. 14(8), 686–695 (2013)

    Article  Google Scholar 

  10. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, New York (1998)

    MATH  Google Scholar 

  11. Garg, A., Appel-Cresswell, S., Popuri, K., McKeown, M.J., Beg, M.F.: Morphological alterations in the caudate, putamen, pallidum, and thalamus in Parkinson’s disease. Front. Neurosci. 9, 101 (2015)

    Article  Google Scholar 

  12. Goldberger, A., Amaral, L., Glass, L., Hausdorff, J., Mark, R., Mietus, J., Moody, G., Peng, C., Stanley, H.: PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000)

    Article  Google Scholar 

  13. Goldsmith, J., Crainiceanu, C.M., Caffo, B.S., Reich, D.S.: Penalized functional regression analysis of white-matter tract profiles in multiple sclerosis. NeuroImage 57(2), 431–439 (2011)

    Article  Google Scholar 

  14. Goldsmith, J., Scheipl, F., Huang, L., Wrobel, J., Gellar, J., Harezlak, J., Mclean, M., Swihart, B., Crainiceanu, L.X.C., Reiss, P., Chen, Y., Greven, S., Huo, L., Kunda, M.G., Park, S.Y., Miller, D., Staicu, A.M.: Refund: Regression with Functional Data (2018). https://cran.r-project.org/web/packages/refund/

  15. Holland, E.C.: Glioblastoma multiforme: the terminator. Proc. Natl. Acad. Sci. 97(12), 6242–6244 (2000)

    Article  Google Scholar 

  16. Ismail, M., Keynton, R., Mostapha, M., Eltanboly, A., Casanova, M., Gimel’farb, G., El-Baz, A.: Studying autism spectrum disorder with structural and diffusion magnetic resonance imaging: A survey. Front. Hum. Neurosci. 10 (2016)

    Google Scholar 

  17. Jacobson, S., Jacobson, J., Sokol, R., Martier, S., Chiodo, L.: New evidence for neurobehavioral effects of in utero cocaine exposure. J. Pediatr. 129(4) pp. 581–590 (1996)

    Article  Google Scholar 

  18. Jacobson, S., Jacobson, J., Sokol, R., Chiodo, L., Corobana, R.: Maternal age, alcohol abuse history, and quality of parenting as moderators of the effects of prenatal alcohol exposure on 7.5-year intellectual function. Alcohol. Clin. Exp. Res. 28, 1732–1745 (2004)

    Google Scholar 

  19. Jermyn, I.H., Kurtek, S., Klassen, E., Srivastava, A.: Elastic shape matching of parameterized surfaces using square root normal fields. In: European Conference on Computer Vision, pp. 804–817 (2012)

    Google Scholar 

  20. Jermyn, I.H., Kurtek, S., Laga, H., Srivastava, A.: Elastic Shape Analysis of Three-Dimensional Objects. Morgan & Claypool Publishers, San Rafael (2017)

    Book  Google Scholar 

  21. Joshi, S., Klassen, E., Srivastava, A., Jermyn, I.: Removing shape-preserving transformations in square-root elastic (SRE) framework for shape analysis of curves. In: EMMCVPR, pp. 387–398 (2007)

    Google Scholar 

  22. Joshi, S., Xie, Q., Kurtek, S., Srivastava, A., Laga, H.: Surface shape morphometry for hippocampal modeling in Alzheimer’s disease. In: International Conference on Digital Image Computing: Techniques and Applications (DICTA), pp. 1–8 (2016)

    Google Scholar 

  23. Just, N.: Improving tumour heterogeneity MRI assessment with histograms. Br. J. Cancer 111(12), 2205 (2014)

    Article  Google Scholar 

  24. Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)

    Article  MathSciNet  Google Scholar 

  25. Kass, R.E., Vos, P.W.: Geometrical Foundations of Asymptotic Inference, vol. 908. Wiley, New York (2011)

    MATH  Google Scholar 

  26. Kendall, D.G.: Shape manifolds, procrustean metrics and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)

    Article  MathSciNet  Google Scholar 

  27. Kurtek, S.: A geometric approach to pairwise Bayesian alignment of functional data using importance sampling. Electron. J. Stat. 11(1), 502–531 (2017)

    Article  MathSciNet  Google Scholar 

  28. Kurtek, S., Needham, T.: Simplifying transforms for general elastic metrics on the space of plane curves. ArXiv:1803.10894v1 (2018)

    Google Scholar 

  29. Kurtek, S., Klassen, E., Ding, Z., Jacobson, S., Jacobson, J., Avison, M., Srivastava, A.: Parameterization-invariant shape comparisons of anatomical surfaces. IEEE Trans. Med. Imaging 30(3), 849–858 (2011)

    Article  Google Scholar 

  30. Kurtek, S., Srivastava, A., Klassen, E., Ding, Z.: Statistical modeling of curves using shapes and related features. J. Am. Stat. Assoc. 107(499), 1152–1165 (2012)

    Article  MathSciNet  Google Scholar 

  31. Kurtek, S., Su, J., Grimm, C., Vaughan, M., Sowell, R., Srivastava, A.: Statistical analysis of manual segmentations of structures in medical images. Comput. Vis. Image Underst. 117, 1036–1050 (2013)

    Article  Google Scholar 

  32. Kurtek, S., Wu, W., Christensen, G., Srivastava, A.: Segmentation, alignment and statistical analysis of biosignals with application to disease classification. J. Appl. Stat. 40(6), 1270–1288 (2013)

    Article  MathSciNet  Google Scholar 

  33. Kurtek, S., Samir, C., Ouchchane, L.: Statistical shape model of elastic endometrial tissue surfaces. In: International Conference on Pattern Recognition Applications and Methods (2014)

    Google Scholar 

  34. Kurtek, S., Xie, Q., Samir, C., Canis, M.: Statistical model for simulation of deformable elastic endometrial tissue shapes. Neurocomputing 173(P1), 36–41 (2016)

    Article  Google Scholar 

  35. Laga, H., Xie, Q., Jermyn, I.H., Srivastava, A.: Numerical inversion of SRNF maps for elastic shape analysis of genus-zero surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 39(12), 2451–2464 (2017)

    Article  Google Scholar 

  36. Le, H.: Locating Frechet means with application to shape spaces. Adv. Appl. Probab. 33(2), 324–338 (2001)

    Article  MathSciNet  Google Scholar 

  37. Maier-Hein, K.H., Neher, P.F., Houde, J.C., Côté, M.A., Garyfallidis, E., Zhong, J., Chamberland, M., Yeh, F.C., Lin, Y.C., Ji, Q., Reddick, W.E., Glass, J.O., Chen, D.Q., Feng, Y., Gao, C., Wu, Y., Ma, J., Renjie, H., Li, Q., Westin, C.F., Deslauriers-Gauthier, S., González, J.O.O., Paquette, M., St-Jean, S., Girard, G., Rheault, F., Sidhu, J., Tax, C.M.W., Guo, F., Mesri, H.Y., Dávid, S., Froeling, M., Heemskerk, A.M., Leemans, A., Boré, A., Pinsard, B., Bedetti, C., Desrosiers, M., Brambati, S., Doyon, J., Sarica, A., Vasta, R., Cerasa, A., Quattrone, A., Yeatman, J., Khan, A.R., Hodges, W., Alexander, S., Romascano, D., Barakovic, M., Auría, A., Esteban, O., Lemkaddem, A., Thiran, J.P., Cetingul, H.E., Odry, B.L., Mailhe, B., Nadar, M.S., Pizzagalli, F., Prasad, G., Villalon-Reina, J.E., Galvis, J., Thompson, P.M., Requejo, F.D.S., Laguna, P.L., Lacerda, L.M., Barrett, R., Dell’Acqua, F., Catani, M., Petit, L., Caruyer, E., Daducci, A., Dyrby, T.B., Holland-Letz, T., Hilgetag, C.C., Stieltjes, B., Descoteaux, M.: The challenge of mapping the human connectome based on diffusion tractography. Nat. Commun. 8(1), 1349 (2017)

    Google Scholar 

  38. Marron, J., Ramsay, J.O., Sangalli, L.M., Srivastava, A.: Functional data analysis of amplitude and phase variation. Stat. Sci. 30(4), 468–484 (2015)

    Article  MathSciNet  Google Scholar 

  39. Oishi, K., Mielke, M., Albert, M., Lyketsos, C., Mori, S.: DTI analyses and clinical applications in Alzheimer’s disease. In: Advances in Alzheimer’s Disease, vol. 2, pp. 525–534 (2011)

    Google Scholar 

  40. Pennec, X.: Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)

    Article  MathSciNet  Google Scholar 

  41. Ramsay, J., Silverman, B.: Functional Data Analysis. Springer, Berlin (2005)

    Book  Google Scholar 

  42. Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. In: Breakthroughs in Statistics, pp. 235–247. Springer, Berlin (1992)

    Google Scholar 

  43. Robinson, D.T.: Functional Data Analysis and Partial Shape Matching in the Square Root Velocity Framework. Ph.D. thesis, Florida State University, Tallahassee (2012)

    Google Scholar 

  44. Saha, A., Banerjee, S., Kurtek, S., Narang, S., Lee, J., Rao, G., Martinez, J., Bharath, K., Rao, A., Baladandayuthapani, V.: DEMARCATE: Density-based magnetic resonance image clustering for assessing tumor heterogeneity in cancer. NeuroImage Clin. 12, 132–143 (2016)

    Article  Google Scholar 

  45. Samir, C., Kurtek, S., Srivastava, A., Canis, M.: Elastic shape analysis of cylindrical surfaces for 3D/2D registration in endometrial tissue characterization. IEEE Trans. Med. Imaging 33(5), 1035–1043 (2014)

    Article  Google Scholar 

  46. Srivastava, A., Jermyn, I.H., Joshi, S.H.: Riemannian analysis of probability density functions with applications in vision. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2007)

    Google Scholar 

  47. Srivastava, A., Klassen, E., Joshi, S.H., Jermyn, I.H.: Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1415–1428 (2011)

    Article  Google Scholar 

  48. Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J.S.: Registration of functional data using Fisher-Rao metric. ArXiv:1103.3817v2 (2011)

    Google Scholar 

  49. Srivastave, A., Klassen, E.P.: Functional and Shape Data Analysis. Springer, Berlin (2016)

    Book  Google Scholar 

  50. Wang, L., Beg, M.F., Ratnanather, J.T., Ceritoglu, C., Younes, L., Morris, J.C., Csernansky, J.G., Miller, M.I.: Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE Trans. Med. Imaging 26(4), 462–470 (2007)

    Article  Google Scholar 

  51. Xie, Q., Kurtek, S., Le, H., Srivastava, A.: Parallel transport of deformations in shape space of elastic surfaces. In: International Conference on Computer Vision (2013)

    Google Scholar 

  52. Yavariabdi, A., Samir, C., Bartoli, A., Ines, D.D., Bourdel, N.: Contour-based TVUS-MR image registration for mapping small endometrial implants. In: Abdominal Imaging, vol. 8198, pp. 145–154 (2013)

    Google Scholar 

  53. Younes, L.: Computable elastic distance between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998)

    Article  MathSciNet  Google Scholar 

  54. Younes, L.: Elastic distance between curves under the metamorphosis viewpoint. ArXiv:1804.10155 (2018)

    Google Scholar 

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Acknowledgements

We thank Arvind Rao for sharing the GBM tumor dataset, and acknowledge Joonsang Lee, Juan Martinez, Shivali Narang and Ganesh Rao for their roles in processing the MRIs used to produce the tumor outlines. We thank Zhaohua Ding for providing the DT-MRI tract dataset used in Sect. 24.4.3 and the dataset of subcortical structures for ADHD classification. Finally, we thank Chafik Samir for providing the endometrial tissue surfaces. Shariq Mohammed would like to acknowledge Institutional Research Support from The University of Michigan. Additionally, this work was supported in part by grants NSF DMS-1613054, NSF CCF-1740761, NSF CCF-1839252 and NIH R37-CA214955.

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Authors and Affiliations

  1. Department of Statistics, The Ohio State University, Columbus, OH, USA

    James Matuk & Sebastian Kurtek

  2. Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA

    Shariq Mohammed

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Karthik Bharath

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  1. James Matuk
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  2. Shariq Mohammed
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  3. Sebastian Kurtek
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  4. Karthik Bharath
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Correspondence to Sebastian Kurtek .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Matuk, J., Mohammed, S., Kurtek, S., Bharath, K. (2020). Biomedical Applications of Geometric Functional Data Analysis. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_24

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