Skip to main content
SpringerLink
Log in

Spectral-Clustering of Lagrangian Trajectory Graphs: Application to Abdominal Aortic Aneurysms

  • Short Communication
  • Published:
Cardiovascular Engineering and Technology Aims and scope Submit manuscript

Abstract

Purpose

Identification of coherent structures in cardiovascular flows is crucial to describe the transport and mixing of blood. Coherent structures can highlight locations where minimal blood mixing takes place, thus, potential thrombus formation can be expected thither. Graph-based approaches have recently been introduced in order to describe fluid transport and mixing between multiple Lagrangian trajectories, where each trajectory serves as a node that can be connected to another trajectory based on their relative distance during the course of time.

Methods

In this study, we compute the Lagrangian trajectories from in vitro planar instantaneous velocity fields in two models of abdominal aortic aneurysms, (AAA) namely single bulge and bi-lobed. Then, we construct unweighted and undirected graphs based on the pairwise distance of Lagrangian trajectories. We report local measures of the graph namely the degree and the clustering coefficient. We also perform spectral clustering of the graph Laplacian to extract the flow coherent sets.

Results

Local graph measures reveal fluid regions of high mixing such as vortex boundaries. Through spectral clustering, the fluid is partitioned into a reduced number of coherent sets where within each set, inner mixing of fluid is maximized while the fluid mixing between different coherent sets is minimized. The approach reveals multiple coherent sets adjacent to the AAA bulge that have sustained this adjacency to the wall through their coherent motion during one cardiac cycle.

Conclusion

Identifying coherent sets enables tracking their transport during the cardiac cycle and identify their role in the flow dynamics. Moreover, the size and the transport of the long residing coherent sets inside the AAA bulges can be deduced which may aid in predicting thrombus formation at such location.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4

References

  1. Allshouse, M. R., and T. Peacock. Lagrangian based methods for coherent structure detection. Chaos 25(9):097617, 2015. https://doi.org/10.1063/1.4922968.

  2. Allshouse, M. R., and J.-L. Thiffeault. Detecting coherent structures using braids. Physica D 241(2):95-105, 2012. https://doi.org/10.1016/j.physd.2011.10.002.

  3. Arzani, A., and S. C. Shadden. Characterization of the transport topology in patient-specific abdominal aortic aneurysm models. Phys. Fluids 24(8):081901, 2012. https://doi.org/10.1063/1.4744984.

  4. Badas, M. G., F. Domenichini, and G. Querzoli. Quantification of the blood mixing in the left ventricle using Finite Time Lyapunov Exponents. Meccanica 52(3):529–544, 2016. https://doi.org/10.1007/s11012-016-0364-8.

  5. Balasuriya, S., N. T. Ouellette, and I. I. Rypina. Generalized Lagrangian coherent structures. Physica D 372:31–51, 2018. https://doi.org/10.1016/j.physd.2018.01.011.

  6. Banisch, R., P. Koltai, and K. Padberg-Gehle. Network measures of mixing. Chaos 29(6):063125, 2019. https://doi.org/10.1063/1.5087632.

  7. Biasetti, J., F. Hussain, and T. C. Gasser. Blood flow and coherent vortices in the normal and aneurysmatic aortas: a fluid dynamical approach to intra-luminal thrombus formation. J. R. Soc. Interface 8(63):1449–1461, 2011. https://doi.org/10.1098/rsif.2011.0041.

  8. Bluestein, D., L. Niu, R. T. Schoephoerster, et al. Steady flow in an aneurysm model: correlation between fluid dynamics and blood platelet deposition. J. Biomech. Eng. 118(3):280-286, 1996. https://doi.org/10.1115/1.2796008.

  9. Chung, F. Spectral Graph Theory. Providence: American Mathematical Society, 1996. https://doi.org/10.1090/cbms/092.

  10. Darwish, A., G. Di Labbio, W. Saleh, et al. In vitro characterization of Lagrangian fluid transport downstream of a dysfunctional bileaflet mechanical aortic valve. AIP Adv. 10(9):095319, 2020. https://doi.org/10.1063/5.0021372.

  11. Di Labbio, G., J. Vétel, and L. Kadem. Material transport in the left ventricle with aortic valve regurgitation. Phys. Rev. Fluids 3(11):113101, 2018. https://doi.org/10.1103/physrevfluids.3.113101.

  12. Di Labbio, G., J.-L. Thiffeault, and L. Kadem. Deducing global mixing information in the heart from sparse particle trajectory data. APS Division of Fluid Dynamics Meeting Abstracts. APS Meeting Abstracts, January 2020.

  13. Donner, R. V., Y. Zou, J. F. Donges, et al. Ambiguities in recurrence-based complex network representations of time series. Phys. Rev. E 81(1):015101, 2010. https://doi.org/10.1103/physreve.81.015101.

  14. Froyland, G., and K. Padberg-Gehle. A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data. Chaos 25(8):087406, 2015. https://doi.org/10.1063/1.4926372.

  15. Hadjighasem, A., D. Karrasch, H. Teramoto, et al. Spectral-clustering approach to Lagrangian vortex detection. Phys. Rev. E 93(6):063107, 2016. https://doi.org/10.1103/physreve.93.063107.

  16. Joly, F., G. Soulez, D. Garcia, et al. Flow stagnation volume and abdominal aortic aneurysm growth: insights from patient-specific computational flow dynamics of Lagrangian-coherent structures. Comput. Biol. Med. 92:98–109, 2018. https://doi.org/10.1016/j.compbiomed.2017.10.033.

  17. Lloyd, S. Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2):129-137, 1982. https://doi.org/10.1109/tit.1982.1056489.

  18. Meschi, S. S., A. Farghadan, and A. Arzani. Flow topology and targeted drug delivery in cardiovascular disease. J. Biomech. 119:110307, 2021. https://doi.org/10.1016/j.jbiomech.2021.110307.

  19. Newman, M. Networks. Oxford: Oxford University Press, 2010. https://doi.org/10.1093/acprof:oso/9780199206650.001.0001.

  20. Norouzi, S. Flow Characteristics in Abdominal Aortic Aneurysms: An in vitro Study. MA thesis. Concordia University, 2020.

  21. Padberg-Gehle, K., and C. Schneide. Network-based study of Lagrangian transport and mixing. Nonlinear Processes Geophys. 24(4):661–671, 2017. https://doi.org/10.5194/npg-24-661-2017.

  22. Ser-Giacomi, E., V. Rossi, C. López, et al. Flow networks: a characterization of geophysical fluid transport. Chaos 25(3):036404, 2015. https://doi.org/10.1063/1.4908231.

  23. Shadden, S. C., and A. Arzani. Lagrangian postprocessing of computational hemodynamics. Ann. Biomed. Eng. 43(1):41–58, 2014. https://doi.org/10.1007/s10439-014-1070-0.

  24. Shadden, S. C., and S. Hendabadi. Potential fluid mechanic pathways of platelet activation. Biomech. Model. Mechanobiol. 12(3):467–474, 2012. https://doi.org/10.1007/s10237-012-0417-4.

  25. Shi, J., and J. Malik. Normalized cuts and image segmentation. In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society. https://doi.org/10.1109/cvpr.1997.609407.

  26. Taebi, A., C. T. Vu, and E. Roncali. Multiscale computational fluid dynamics modeling for personalized liver cancer radioembolization dosimetry. J. Biomech. Eng. 143(1):011002, 2020. https://doi.org/10.1115/1.4047656.

Download references

Author Contributions

AD and LK conceptualized the study. SN designed and performed the experiments to provide the flow fields. AD developed the post-processing codes and performed the analysis. AD and LK wrote the original draft while all authors revised and edited the final manuscript. LK supervised the project and acquired the funding.

Funding

This work is supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) (Grant No. 343164-07).

Data Availability

The codes used in this manuscript are available at https://github.com/AhmedDarwish466/TrajectoryBasedNetworks.

Conflict of Interest

Ahmed Darwish, Shahrzad Norouzi and Lyes Kadem declare that they have no conflict of interest.

Author information

Author notes

  1. Ahmed Darwish and Shahrzad Norouzi contributed equally to this work.

Authors and Affiliations

  1. Laboratory of Cardiovascular Fluid Dynamics, Concordia University, Montréal, QC, H3G 1M8, Canada

    Ahmed Darwish, Shahrzad Norouzi & Lyes Kadem

  2. Mechanical Engineering Department, Assiut University, 71515, Assiut, Egypt

    Ahmed Darwish

Authors
  1. Ahmed Darwish
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shahrzad Norouzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lyes Kadem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Darwish.

Additional information

Associate Editor Igor Efimov oversaw the review of this article.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Below is the link to the electronic supplementary material.

Supplementary file1 (DOCX 308 KB)

Supplementary Video 1 (MP4 10363 KB)

Supplementary Video 1 (MP4 30840 KB)

Supplementary Video 1 (MP4 13803 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Darwish, A., Norouzi, S. & Kadem, L. Spectral-Clustering of Lagrangian Trajectory Graphs: Application to Abdominal Aortic Aneurysms. Cardiovasc Eng Tech 13, 504–513 (2022). https://doi.org/10.1007/s13239-021-00590-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13239-021-00590-3

Keywords

Search

Navigation