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

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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.

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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.

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Correspondence to Ahmed Darwish.

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Associate Editor Igor Efimov oversaw the review of this article.

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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

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