Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study

Authors Andrew Aukerman , Mathieu Carrière, Chao Chen , Kevin Gardner, Raúl Rabadán , Rami Vanguri



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Andrew Aukerman
  • Department of Pathology & Cell Biology, Columbia University, New York, NY, United States
Mathieu Carrière
  • Department of Systems Biology, Columbia University, New York, NY, United States
Chao Chen
  • Department of Biomedical Informatics, Stony Brook University, NY, United States
Kevin Gardner
  • Department of Pathology & Cell Biology, Columbia University, New York, NY, United States
Raúl Rabadán
  • Department of Systems Biology, Columbia University, New York, NY, United States
Rami Vanguri
  • Department of Pathology & Cell Biology, Columbia University, New York, NY, United States

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Andrew Aukerman, Mathieu Carrière, Chao Chen, Kevin Gardner, Raúl Rabadán, and Rami Vanguri. Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.11

Abstract

Persistent homology is a common tool of topological data analysis, whose main descriptor, the persistence diagram, aims at computing and encoding the geometry and topology of given datasets. In this article, we present a novel application of persistent homology to characterize the spatial arrangement of immune and epithelial (tumor) cells within the breast cancer immune microenvironment. More specifically, quantitative and robust characterizations are built by computing persistence diagrams out of a staining technique (quantitative multiplex immunofluorescence) which allows us to obtain spatial coordinates and stain intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers of cancer subtype and prognostic biomarker of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. This provides new insights and possibilities on the general problem of building (topology-based) biomarkers that are characteristic and predictive of cancer subtype, overall survival and response to therapy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Topological data analysis
  • persistence diagrams

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References

  1. Shahira Abousamra, Danielle Fassler, Le Hou, Yuwei Zhang, Rajarsi Gupta, Tahsin Kurc, Luisa F. Escobar-Hoyos, Dimitris Samaras, Beatrice Knudson, Kenneth Shroyer, Joel Saltz, and Chao Chen. Weakly-supervised deep stain decomposition for multiplex ihc images. In IEEE International Symposium on Biomedical Imaging (ISBI), 2019. Google Scholar
  2. Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8), 2017. Google Scholar
  3. Magnus Botnan and William Crawley-Boevey. Decomposition of persistence modules. arXiv, November 2018. URL: http://arxiv.org/abs/1811.08946.
  4. Magnus Botnan and Michael Lesnick. Algebraic stability of zigzag persistence modules. Algebraic and Geometric Topology, 18(6):3133-3204, October 2018. Google Scholar
  5. Rickard Brüel-Gabrielsson, Bradley Nelson, Anjan Dwaraknath, Primoz Skraba, Leonidas Guibas, and Gunnar Carlsson. A topology layer for machine learning. arXiv, May 2019. URL: http://arxiv.org/abs/1905.12200.
  6. Peter Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(77):77-102, 2015. Google Scholar
  7. Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016. Google Scholar
  8. Samantha Burugu, Karama Asleh-Aburaya, and Torsten O Nielsen. Immune infiltrates in the breast cancer microenvironment: detection, characterization and clinical implication. Breast Cancer, 24(1):3-15, 2017. Google Scholar
  9. Jung Byun, Sandeep Singhal, Samson Park, IK Dae, Ambar Caban, Nasreen Vohra, Eliseo Perez-Stable, Anna Napoles, and Kevin Gardner. Transcription regulatory networks associated with luminal master regulator expression and breast cancer survival, 2019. Google Scholar
  10. Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete and Computational Geometry, 42(1):71-93, July 2009. Google Scholar
  11. Mathieu Carrière, Marco Cuturi, and Steve Oudot. Sliced Wasserstein kernel for persistence diagrams. In International Conference on Machine Learning, volume 70, pages 664-673, July 2017. Google Scholar
  12. Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan. Topology of viral evolution. Proceedings of the National Academy of Sciences, 110(46):18566-18571, 2013. Google Scholar
  13. Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas Guibas, and Steve Oudot. Proximity of persistence modules and their diagrams. In International Symposium on Computational Geometry, page 237, 2009. Google Scholar
  14. Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. Springer International Publishing, 2016. Google Scholar
  15. Chao Chen, Xiuyan Ni, Qinxun Bai, and Yusu Wang. A topological regularizer for classifiers via persistent homology. In International Conference on Artificial Intelligence and Statistics, pages 2573-2582, 2019. URL: http://proceedings.mlr.press/v89/chen19g.html.
  16. Jérémy Cochoy and Steve Oudot. Decomposition of exact pfd persistence bimodules. arXiv, May 2016. URL: http://arxiv.org/abs/1605.09726.
  17. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, January 2007. Google Scholar
  18. David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko. Lipschitz functions have l p-stable persistence. Foundations of computational mathematics, 10(2):127-139, 2010. Google Scholar
  19. René Corbet, Ulderico Fugacci, Michael Kerber, Claudia Landi, and Bei Wang. A kernel for multi-parameter persistent homology. Computers & Graphics: X, 2:100005, December 2019. Google Scholar
  20. Xiaofeng Dai, Liangjian Xiang, Ting Li, and Zhonghu Bai. Cancer hallmarks, biomarkers and breast cancer molecular subtypes. Journal of Cancer, 7(10):1281, 2016. Google Scholar
  21. Herbert Edelsbrunner and John Harer. Persistent homology-a survey. Contemporary mathematics, 453:257-282, 2008. Google Scholar
  22. Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010. Google Scholar
  23. Danielle J Fassler, Shahira Abousamra, Rajarsi Gupta, Chao Chen, Maozheng Zhao, David Paredes-Merino, Syeda Areeha Batool, Beatrice Knudsen, Luisa Escobar-Hoyos, Kenneth R Shroyer, Dimitris Samaras, Tahsin Kurc, and Joel Saltz. Deep learning-based image analysis methods for brightfield-acquired multiplex immunohistochemistry images, 2019. under review. Google Scholar
  24. Mingchen Gao, Chao Chen, Shaoting Zhang, Zhen Qian, Dimitris Metaxas, and Leon Axel. Segmenting the papillary muscles and the trabeculae from high resolution cardiac ct through restoration of topological handles. In International Conference on Information Processing in Medical Imaging, pages 184-195. Springer, 2013. Google Scholar
  25. Robyn Gartrell, Douglas Marks, Thomas Hart, Gen Li, Danielle Davari, Alan Wu, Zoe Blake, Yan Lu, Kayleigh Askin, Anthea Monod, et al. Quantitative analysis of immune infiltrates in primary melanoma. Cancer immunology research, 6(4):481-493, 2018. Google Scholar
  26. Arthur Gretton, Karsten Borgwardt, Malte Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13:723-773, 2012. Google Scholar
  27. Arthur Gretton, Ralf Herbrich, Alexander Smola, Olivier Bousquet, and Bernhard Schölkopf. Kernel methods for measuring independence. Journal of Machine Learning Research, 6:2075-2129, 2005. Google Scholar
  28. Heather Harrington, Nina Otter, Hal Schenck, and Ulrike Tillmann. Stratifying multiparameter persistent homology. SIAM Journal on Applied Algebra and Geometry, 3(3):439-471, January 2019. Google Scholar
  29. Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements of statistical learning. Springer-Verlag, 2003. Google Scholar
  30. Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures. In Advances in Neural Information Processing Systems, pages 1634-1644, 2017. Google Scholar
  31. Xiaoling Hu, Li Fuxin, Dimitris Samaras, and Chao Chen. Topology-preserving deep image segmentation. In the Thirty-third Conference on Neural Information Processing Systems (NeurIPS), 2019. Google Scholar
  32. Jessica Kalra and Jennifer Baker. Multiplex immunohistochemistry for mapping the tumor microenvironment. In Signal Transduction Immunohistochemistry, pages 237-251. Springer, 2017. Google Scholar
  33. Lida Kanari, Paweł Dłotko, Martina Scolamiero, Ran Levi, Julian Shillcock, Kathryn Hess, and Henry Markram. A topological representation of branching neuronal morphologies. Neuroinformatics, 16(1):3-13, 2018. Google Scholar
  34. Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. Persistence weighted Gaussian kernel for topological data analysis. In International Conference on Machine Learning, volume 48, pages 2004-2013, June 2016. Google Scholar
  35. Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, and Ulrich Bauer. Statistical topological data analysis - a kernel perspective. In Advances in Neural Information Processing Systems, pages 3070-3078, 2015. Google Scholar
  36. Théo Lacombe, Marco Cuturi, and Steve Oudot. Large scale computation of means and clusters for persistence diagrams using optimal transport. In Advances in Neural Information Processing Systems, pages 9770-9780, 2018. Google Scholar
  37. Peter Lawson, Andrew B Sholl, J Quincy Brown, Brittany Terese Fasy, and Carola Wenk. persistent homology for the quantitative evaluation of architectural features in prostate cancer histology. Scientific reports, 9, 2019. Google Scholar
  38. Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998. Google Scholar
  39. Hyekyoung Lee, Hyejin Kang, Moo K Chung, Bung-Nyun Kim, and Dong Soo Lee. Persistent brain network homology from the perspective of dendrogram. IEEE transactions on medical imaging, 31(12):2267-2277, 2012. Google Scholar
  40. Michael Lesnick and Matthew Wright. Interactive visualization of 2D persistence modules. arXiv, December 2015. URL: http://arxiv.org/abs/1512.00180.
  41. Yanjie Li, Dingkang Wang, Giorgio A Ascoli, Partha Mitra, and Yusu Wang. Metrics for comparing neuronal tree shapes based on persistent homology. PloS one, 12(8):e0182184, 2017. Google Scholar
  42. Zhixian Liu, Mengyuan Li, Zehang Jiang, and Xiaosheng Wang. A comprehensive immunologic portrait of triple-negative breast cancer. Translational oncology, 11(2):311-329, 2018. Google Scholar
  43. Yuriy Mileyko, Sayan Mukherjee, and John Harer. Probability measures on the space of persistence diagrams. Inverse Problems, 27(12):124007, December 2011. Google Scholar
  44. Monica Nicolau, Arnold J Levine, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences, 108(17):7265-7270, 2011. Google Scholar
  45. Steve Oudot. Persistence theory: from quiver representations to data analysis. American Mathematical Society, 2015. Google Scholar
  46. Deepti Pachauri, Chris Hinrichs, Moo K Chung, Sterling C Johnson, and Vikas Singh. Topology-based kernels with application to inference problems in alzheimer’s disease. IEEE transactions on medical imaging, 30(10):1760-1770, 2011. Google Scholar
  47. Edwin Roger Parra, Alejandro Francisco-Cruz, and Ignacio Ivan Wistuba. State-of-the-art of profiling immune contexture in the era of multiplexed staining and digital analysis to study paraffin tumor tissues. Cancers, 11(2):247, 2019. Google Scholar
  48. Adrien Poulenard, Primoz Skraba, and Maks Ovsjanikov. Topological function optimization for continuous shape matching. In Computer Graphics Forum, volume 37, pages 13-25. Wiley Online Library, 2018. Google Scholar
  49. Lajos Pusztai, Thomas Karn, Anton Safonov, Maysa M Abu-Khalaf, and Giampaolo Bianchini. New strategies in breast cancer: immunotherapy. Clinical Cancer Research, 22(9):2105-2110, 2016. Google Scholar
  50. Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In IEEE Conference on Computer Vision and Pattern Recognition, 2015. Google Scholar
  51. Abbas Rizvi, Pablo Cámara, Elena Kandror, Thomas Roberts, Ira Schieren, Tom Maniatis, and Raul Rabadan. Single-cell topological RNA-seq analysis reveals insights into cellular differentiation and development. Nature Biotechnology, 35(6):551-560, May 2017. Google Scholar
  52. Andrew Robinson and Katharine Turner. Hypothesis testing for topological data analysis. Journal of Applied and Computational Topology, 1(2):241-261, December 2017. Google Scholar
  53. Rebecca L Siegel, Kimberly D Miller, and Ahmedin Jemal. Cancer statistics, 2019. CA: a cancer journal for clinicians, 69(1):7-34, 2019. Google Scholar
  54. Alexandra Signoriello, Marcus Bosenberg, Mark Shattuck, and Corey O'Hern. Modeling the spatiotemporal evolution of the melanoma tumor microenvironment. In APS Meeting Abstracts, 2016. Google Scholar
  55. Carl-Johann Simon-Gabriel and Bernhard Schölkopf. Kernel distribution embeddings: universal kernels, characteristic kernels and kernel metrics on distributions. Journal of Machine Learning Research, 19(44):1-29, 2018. Google Scholar
  56. Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson. Topological methods for the analysis of high dimensional data sets and 3D object recognition. In Eurographics Symposium on Point-Based Graphics, pages 91-100, 2007. Google Scholar
  57. Bharath K Sriperumbudur, Kenji Fukumizu, and Gert RG Lanckriet. Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research, 12(Jul):2389-2410, 2011. Google Scholar
  58. Mikael Vejdemo-Johansson and Sayan Mukherjee. Multiple testing with persistent homology. arXiv, December 2018. URL: http://arxiv.org/abs/1812.06491.
  59. Pengxiang Wu, Chao Chen, Yusu Wang, Shaoting Zhang, Changhe Yuan, Zhen Qian, Dimitris Metaxas, and Leon Axel. Optimal topological cycles and their application in cardiac trabeculae restoration. In International Conference on Information Processing in Medical Imaging, pages 80-92. Springer, 2017. Google Scholar
  60. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249-274, 2005. Google Scholar
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