https://orcid.org/0000-0002-2550-6847
Abstract
Topological features of time-dependent, three-dimensional (3D) vector flow fields, such as wall shear stress (WSS) fixed points, are considered surrogates of pathological blood flow dynamics in cardiovascular diseases. Fixed-point visualizations are typically constrained to two-dimensional (2D) spaces, yet they aim to display complex spatiotemporal (four-dimensional (4D)) dynamics. There is a need for visualization strategies to reduce occlusion and reliance on animations to allow the detection of holistic flow patterns. Using intracranial aneurysms as a use case, we present the fixed-point carousel, a novel approach to visually depicting the “4D” nature of WSS fixed points via (1) topographic mapping of the 3D aneurysm sac to overcome occlusion while preserving fixed-point distances and sac morphological features; and (2) arranging these into a carousel model to present with temporal dimension holistically. Examples are presented for image-based computational fluid dynamic (CFD) models of intracranial aneurysms, illuminating the intricate and distinct fixed-point trajectories and interactions, a necessary step toward understanding the volumetric flow manifolds that drive them for this and other cardiovascular—and potentially nonbiomedical—fluid dynamics applications.
References
1.
Wootton
, D. M.
, and Ku
, D. N.
, 1999
, “Fluid Mechanics of Vascular Systems, Diseases, and Thrombosis
,” Annu. Rev. Biomed. Eng.
, 1
(1
), pp. 299
–329
.10.1146/annurev.bioeng.1.1.2992.
Dai
, G.
, Kaazempur-Mofrad
, M. R.
, Natarajan
, S.
, Zhang
, Y.
, Vaughn
, S.
, Blackman
, B. R.
, Kamm
, R. D.
, García-Cardeña
, G.
, and Gimbrone
, M. A.
, Jr., 2004
, “Distinct Endothelial Phenotypes Evoked by Arterial Waveforms Derived From Atherosclerosis-Susceptible and -Resistant Regions of Human Vasculature
,” Proc. Natl. Acad. Sci. U.S.A.
, 101
(41
), pp. 14871
–14876
.10.1073/pnas.04060731013.
Arzani
, A.
, and Shadden
, S. C.
, 2018
, “Wall Shear Stress Fixed Points in Cardiovascular Fluid Mechanics
,” J. Biomech.
, 73
, pp. 145
–152
.10.1016/j.jbiomech.2018.03.0344.
Tobak
, M.
, and Peake
, D. J.
, 1982
, “Topology of Three-Dimensional Separated Flows
,” Annu. Rev. Fluid Mech.
, 14
(1
), pp. 61
–85
.10.1146/annurev.fl.14.010182.0004255.
Joshi
, A.
, Caban
, J.
, Rheingans
, P.
, and Sparling
, L.
, 2009
, “Case Study on Visualizing Hurricanes Using Illustration-Inspired Techniques
,” IEEE Trans. Visualization Comput. Graphics
, 15
(5
), pp. 709
–718
.10.1109/TVCG.2008.1056.
Julian
, M.
, 2015
, Illustration-Inspired Visualization of Blood Flow Dynamics
, University of Toronto
, Toronto, ON, Canada
.7.
Farghadan
, A.
, and Arzani
, A.
, 2019
, “The Combined Effect of Wall Shear Stress Topology and Magnitude on Cardiovascular Mass Transport
,” Int. J. Heat Mass Transfer
, 131
, pp. 252
–260
.10.1016/j.ijheatmasstransfer.2018.11.0518.
Mazzi
, V.
, Morbiducci
, U.
, Calò
, K.
, De Nisco
, G.
, Lodi Rizzini
, M.
, Torta
, E.
, Caridi
, G. C. A.
, Chiastra
, C.
, and Gallo
, D.
, 2021
, “Wall Shear Stress Topological Skeleton Analysis in Cardiovascular Flows: Methods and Applications
,” Mathematics
, 9
(7
), p. 720
.10.3390/math90707209.
Mazzi
, V.
, De Nisco
, G.
, Hoogendoorn
, A.
, Calò
, K.
, Chiastra
, C.
, Gallo
, D.
, Steinman
, D. A.
, Wentzel
, J. J.
, and Morbiducci
, U.
, 2021
, “Early Atherosclerotic Changes in Coronary Arteries Are Associated With Endothelium Shear Stress Contraction/Expansion Variability
,” Ann. Biomed. Eng.
, 49
(9
), pp. 2606
–2621
.10.1007/s10439-021-02829-510.
Morbiducci
, U.
, Mazzi
, V.
, Domanin
, M.
, De Nisco
, G.
, Vergara
, C.
, Steinman
, D. A.
, and Gallo
, D.
, 2020
, “Wall Shear Stress Topological Skeleton Independently Predicts Long-Term Restenosis After Carotid Bifurcation Endarterectomy
,” Ann. Biomed. Eng.
, 48
(12
), pp. 2936
–2949
.10.1007/s10439-020-02607-911.
Candreva
, A.
, Pagnoni
, M.
, Rizzini
, M. L.
, Mizukami
, T.
, Gallinoro
, E.
, Mazzi
, V.
, Gallo
, D.
, et al., 2022
, “Risk of Myocardial Infarction Based on Endothelial Shear Stress Analysis Using Coronary Angiography
,” Atherosclerosis
, 342
, pp. 28
–35
.10.1016/j.atherosclerosis.2021.11.01012.
Zhang
, Y.
, Takao
, H.
, Murayama
, Y.
, and Qian
, Y.
, 2013
, “Propose a Wall Shear Stress Divergence to Estimate the Risks of Intracranial Aneurysm Rupture
,” Sci. World J.
, 2013
(1
), p. 508131
.10.1155/2013/50813113.
Goodarzi Ardakani
, V.
, Tu
, X.
, Gambaruto
, A. M.
, Velho
, I.
, Tiago
, J.
, Sequeira
, A.
, and Pereira
, R.
, 2019
, “Near-Wall Flow in Cerebral Aneurysms
,” Fluids
, 4
(2
), p. 89
.10.3390/fluids402008914.
Brisman
, J. L.
, Song
, J. K.
, and Newell
, D. W.
, 2006
, “Cerebral Aneurysms
,” New Engl. J. Med.
, 355
(9
), pp. 928
–939
.10.1056/NEJMra05276015.
Korja
, M.
, Lehto
, H.
, and Juvela
, S.
, 2014
, “Lifelong Rupture Risk of Intracranial Aneurysms Depends on Risk Factors: A Prospective Finnish Cohort Study
,” Stroke
, 45
(7
), pp. 1958
–1963
.10.1161/STROKEAHA.114.00531816.
Hilditch
, C. A.
, Brinjikji
, W.
, Tsang
, A.
, Nicholson
, P.
, Kostynskyy
, A.
, Tymianski
, M.
, Krings
, T.
, Radovanovic
, I.
, and Pereira
, V.
, 2021
, “Application of PHASES and ELAPSS Scores to Ruptured Cerebral Aneurysms: How Many Would Have Been Conservatively Managed?
,” J. Neurosurg. Sci.
, 65
(1
), pp. 33
–37
.10.23736/S0390-5616.18.04498-317.
Murayama
, Y.
, Fujimura
, S.
, Suzuki
, T.
, and Takao
, H.
, 2019
, “Computational Fluid Dynamics as a Risk Assessment Tool for Aneurysm Rupture
,” Neurosurg. Focus
, 47
(1
), p. E12
.10.3171/2019.4.FOCUS1918918.
Chung
, B. J.
, and Cebral
, J. R.
, 2015
, “CFD for Evaluation and Treatment Planning of Aneurysms: Review of Proposed Clinical Uses and Their Challenges
,” Ann. Biomed. Eng.
, 43
(1
), pp. 122
–138
.10.1007/s10439-014-1093-619.
Meng
, H.
, Tutino
, V. M.
, Xiang
, J.
, and Siddiqui
, A.
, 2014
, “High WSS or Low WSS? Complex Interactions of Hemodynamics With Intracranial Aneurysm Initiation, Growth, and Rupture: Toward a Unifying Hypothesis
,” Am. J. Neuroradiol.
, 35
(7
), pp. 1254
–1262
.10.3174/ajnr.A355820.
Chung
, B. J.
, Mut
, F.
, Putman
, C. M.
, Hamzei-Sichani
, F.
, Brinjikji
, W.
, Kallmes
, D.
, Jimenez
, C. M.
, and Cebral
, J. R.
, 2018
, “Identification of Hostile Hemodynamics and Geometries of Cerebral Aneurysms: A Case-Control Study
,” Am. J. Neuroradiol.
, 39
(10
), pp. 1860
–1866
.10.3174/ajnr.A576421.
Salimi Ashkezari
, S. F.
, Mut
, F.
, Chung
, B. J.
, Yu
, A. K.
, Stapleton
, C. J.
, See
, A. P.
, Amin-Hanjani
, S.
, et al., 2021
, “Hemodynamics in Aneurysm Blebs With Different Wall Characteristics
,” J. Neurointerventional Surg.
, 13
(7
), pp. 642
–646
.10.1136/neurintsurg-2020-01660122.
Valen-Sendstad
, K.
, and Steinman
, D. A.
, 2014
, “Mind the Gap: Impact of Computational Fluid Dynamics Solution Strategy on Prediction of Intracranial Aneurysm Hemodynamics and Rupture Status Indicators
,” Am. J. Neuroradiol.
, 35
(3
), pp. 536
–543
.10.3174/ajnr.A379323.
Khan
, M. O.
, Arana
, V. T.
, Najafi
, M.
, MacDonald
, D. E.
, Natarajan
, T.
, Valen-Sendstad
, K.
, and Steinman
, D. A.
, 2021
, “On the Prevalence of Flow Instabilities From High-Fidelity Computational Fluid Dynamics of Intracranial Bifurcation Aneurysms
,” J. Biomech.
, 127
, p. 110683
.10.1016/j.jbiomech.2021.11068324.
MacDonald
, D. E.
, Najafi
, M.
, Temor
, L.
, and Steinman
, D. A.
, 2022
, “Spectral Bandedness in High-Fidelity Computational Fluid Dynamics Predicts Rupture Status in Intracranial Aneurysms
,” ASME J. Biomech. Eng.
, 144
(6
), p. 061004
.10.1115/1.405340325.
Goubergrits
, L.
, Thamsen
, B.
, Berthe
, A.
, Poethke
, J.
, Kertzscher
, U.
, Affeld
, K.
, Petz
, C.
, Hege
, H. C.
, Hoch
, H.
, and Spuler
, A.
, 2010
, “In Vitro Study of Near-Wall Flow in a Cerebral Aneurysm Model With and Without Coils
,” Am. J. Neuroradiol.
, 31
(8
), pp. 1521
–1528
.10.3174/ajnr.A212126.
Reininghaus
, J.
, and Hotz
, I.
, 2010
, “Combinatorial 2D Vector Field Topology Extraction and Simplification
,” Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications
, Springer
, Berlin, Germany
, pp. 103
–114
.10.1007/978-3-642-15014-2_927.
Petz
, C.
, Pöthkow
, K.
, and Hege
, H. C.
, 2012
, “Probabilistic Local Features in Uncertain Vector Fields With Spatial Correlation
,” Comput. Graphics Forum
, 31
(3pt2
), pp. 1045
–1054
.10.1111/j.1467-8659.2012.03097.x28.
Sujudi
, D.
, and Haimes
, R.
, 1995
, “Identification of Swirling Flow in 3D Vector Fields
,” AIAA
Paper No. 1995-1715.10.2514/6.1995-171529.
Rossl
, C.
, and Theisel
, H.
, 2012
, “Streamline Embedding for 3D Vector Field Exploration
,” IEEE Trans. Visualization Comput. Graphics
, 18
(3
), pp. 407
–420
.10.1109/TVCG.2011.7830.
Mazzi
, V.
, Gallo
, D.
, Calò
, K.
, Najafi
, M.
, Khan
, M. O.
, De Nisco
, G.
, Steinman
, D. A.
, and Morbiducci
, U.
, 2020
, “A Eulerian Method to Analyze Wall Shear Stress Fixed Points and Manifolds in Cardiovascular Flows
,” Biomech. Model. Mechanobiol.
, 19
(5
), pp. 1403
–1423
.10.1007/s10237-019-01278-331.
Van der Stigchel
, S.
, 2020
, “An Embodied Account of Visual Working Memory
,” Visual Cognit.
, 28
(5–8
), pp. 414
–419
.10.1080/13506285.2020.174282732.
Coppin
, P.
, Harvey
, J.
, Valen-Sendstad
, K.
, Steinman
, D.
, and Steinman
, D. A.
, 2014
, “Illustration-Inspired Visualization of Blood Flow Dynamics
,” 2014 18th International Conference on Information Visualisation
, Paris, France
, July 16–18, pp. 333
–335
.10.1109/IV.2014.1933.
Valen-Sendstad
, K.
, Mardal
, K. A.
, and Steinman
, D. A.
, 2013
, “High-Resolution CFD Detects High-Frequency Velocity Fluctuations in Bifurcation, but Not Sidewall, Aneurysms
,” J. Biomech.
, 46
(2
), pp. 402
–407
.10.1016/j.jbiomech.2012.10.04234.
Cancelliere
, N. M.
, Najafi
, M.
, Brina
, O.
, Bouillot
, P.
, Vargas
, M. I.
, Lovblad
, K. O.
, Krings
, T.
, Pereira
, V. M.
, and Steinman
, D. A.
, 2020
, “4D-CT Angiography Versus 3D-Rotational Angiography as the Imaging Modality for Computational Fluid Dynamics of Cerebral Aneurysms
,” J. Neurointerventional Surg.
, 12
(6
), pp. 626
–630
.10.1136/neurintsurg-2019-01538935.
Hoi
, Y.
, Wasserman
, B. A.
, Xie
, Y. J.
, Najjar
, S. S.
, Ferruci
, L.
, Lakatta
, E. G.
, Gerstenblith
, G.
, and Steinman
, D. A.
, 2010
, “Characterization of Volumetric Flow Rate Waveforms at the Carotid Bifurcations of Older Adults
,” Physiol. Meas.
, 31
(3
), pp. 291
–302
.10.1088/0967-3334/31/3/00236.
Mortensen
, M.
, and Valen-Sendstad
, K.
, 2015
, “Oasis: A High-Level/High-Performance Open Source Navier–Stokes Solver
,” Comput. Phys. Commun.
, 188
, pp. 177
–188
.10.1016/j.cpc.2014.10.02637.
Khan
, M. O.
, Valen-Sendstad
, K.
, and Steinman
, D. A.
, 2015
, “Narrowing the Expertise Gap for Predicting Intracranial Aneurysm Hemodynamics: Impact of Solver Numerics Versus Mesh and Time-Step Resolution
,” Am. J. Neuroradiol.
, 36
(7
), pp. 1310
–1316
.10.3174/ajnr.A426338.
Eulzer
, P.
, Engelhardt
, S.
, Lichtenberg
, N.
, De Simone
, R.
, and Lawonn
, K.
, 2019
, “Temporal Views of Flattened Mitral Valve Geometries
,” IEEE Trans. Visualization Comput. Graphics
, 26
(1
), pp. 971
–980
.10.1109/TVCG.2019.293433739.
Meuschke
, M.
, Voß
, S.
, Preim
, B.
, and Lawonn
, K.
, 2018
, “Exploration of Blood Flow Patterns in Cerebral Aneurysms During the Cardiac Cycle
,” Comput. Graphics
, 72
, pp. 12
–25
.10.1016/j.cag.2018.01.01240.
Natarajan
, T.
, Singh-Gryzbon
, S.
, Chen
, H.
, Sadri
, V.
, Ruile
, P.
, Neumann
, F. J.
, Yoganathan
, A. P.
, and Dasi
, L. P.
, 2024
, “Sensitivity of Post-TAVR Hemodynamics to the Distal Aortic Arch Anatomy: A High-Fidelity CFD Study
,” Cardiovasc. Eng. Technol.
, 15
(4
), pp. 463
–480
.10.1007/s13239-024-00728-z41.
Hong
, W.
, Gu
, X.
, Qiu
, F.
, Jin
, M.
, and Kaufman
, A.
, 2006
, “Conformal Virtual Colon Flattening
,” Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling
, Wales, UK, June 6–8, pp. 85
–93
.10.1145/1128888.112890142.
Kreiser
, J.
, Meuschke
, M.
, Mistelbauer
, G.
, Preim
, B.
, and Ropinski
, T.
, 2018
, “A Survey of Flattening-Based Medical Visualization Techniques
,” Comput. Graphics Forum
, 37
(3
), pp. 597
–624
.10.1111/cgf.1344543.
Kurihara
, N.
, 2011
, “MR Imaging of Cerebral Aneurysms
,” Neurovascular Imaging: MRI and Microangiography
, Springer-Verlag, London, UK, pp. 345
–372
.10.1007/978-1-84882-134-744.
Lévy
, B.
, Petitjean
, S.
, Ray
, N.
, and Maillot
, J.
, 2002, “Least Squares Conformal Maps for Automatic Texture Atlas Generation
,” ACM Trans. Graphics
, 21
(3
), pp. 362
–371
.10.1145/566654.56659045.
Rabinovich
, M.
, Poranne
, R.
, Panozzo
, D.
, and Sorkine-Hornung
, O.
, 2017
, “Scalable Locally Injective Mappings
,” ACM Trans. Graphics
, 36
(2
), pp. 1
–16
.10.1145/298362146.
Liu
, L.
, Zhang
, L.
, Xu
, Y.
, Gotsman
, C.
, and Gortler
, S. J.
, 2008
, “A Local/Global Approach to Mesh Parameterization
,” Comput. Graphics Forum
, 27
(5
), pp. 1495
–1504
.10.1111/j.1467-8659.2008.01290.x47.
Jacobson
, A.
, Panozzo
, D.
, Schüller
, C.
, Diamanti
, O.
, Zhou
, Q.
, and Pietroni
, N.
, 2018
, “libigl: A Simple C++ Geometry Processing Library
,” Online, accessed Aug. 2024, https://libigl.github.io/48.
Salimi Ashkezari
, S. F.
, Detmer
, F. J.
, Mut
, F.
, Chung
, B. J.
, Yu
, A. K.
, Stapleton
, C. J.
, See
, A. P.
, et al., 2021
, “Blebs in Intracranial Aneurysms: Prevalence and General Characteristics
,” J. Neurointerventional Surg.
, 13
(3
), pp. 226
–230
.10.1136/neurintsurg-2020-01627449.
Piccinelli
, M.
, Steinman
, D. A.
, Hoi
, Y.
, Tong
, F.
, Veneziani
, A.
, and Antiga
, L.
, 2012
, “Automatic Neck Plane Detection and 3D Geometric Characterization of Aneurysmal Sacs
,” Ann. Biomed. Eng.
, 40
(10
), pp. 2188
–2211
.10.1007/s10439-012-0577-550.
Steinman
, D. A.
, Coppin
, P. W.
, and Steinman
, D. A.
, 2021
, “Narcissus and Echo: Reflections on an Art-Science Collaboration
,” Leonardo
, 54
(5
), pp. 552
–557
.10.1162/leon_a_0200951.
McCloud
, S.
, and Martin
, M.
, 1993
, Understanding Comics: The Invisible Art
, Vol. 106, Kitchen Sink Press
, Northampton, MA
.52.
Ramachandran
, V. S.
, and Hirstein
, W.
, 1999
, “The Science of Art: A Neurological Theory of Aesthetic Experience
,” J. Conscious. Stud.
, 6
(6–7
), pp. 15
–51
.https://www.researchgate.net/publication/233556531_The_Science_of_Art_A_Neurological_Theory_of_Aesthetic_Experience53.
Hunt
, J. C.
, Abell
, C. J.
, Peterka
, J. A.
, and Woo
, H.
, 1978
, “Kinematical Studies of the Flows Around Free or Surface-Mounted Obstacles; Applying Topology to Flow Visualization
,” J. Fluid Mech.
, 86
(1
), pp. 179
–200
.10.1017/S002211207800106854.
Gerrits
, T.
, Rössl
, C.
, and Theisel
, H.
, 2017
, “Glyphs for General Second-Order 2D and 3D Tensors
,” IEEE Trans. Visualization Comput. Graphics
, 23
(1
), pp. 980
–989
.10.1109/TVCG.2016.259899855.
Wilde
, T.
, Rossl
, C.
, and Theisel
, H.
, 2019
, “Recirculation Surfaces for Flow Visualization
,” IEEE Trans. Visualization Comput. Graphics
, 25
(1
), pp. 946
–955
.10.1109/TVCG.2018.286481356.
Oeltze-Jafra
, S.
, Cebral
, J. R.
, Janiga
, G.
, and Preim
, B.
, 2016
, “Cluster Analysis of Vortical Flow in Simulations of Cerebral Aneurysm Hemodynamics
,” IEEE Trans. Visualization Comput. Graphics
, 22
(1
), pp. 757
–766
.10.1109/TVCG.2015.246720357.
Oeltze
, S.
, Lehmann
, D. J.
, Kuhn
, A.
, Janiga
, G.
, Theisel
, H.
, and Preim
, B.
, 2014
, “Blood Flow Clustering and Applications in Virtual Stenting of Intracranial Aneurysms
,” IEEE Trans. Visualization Comput. Graphics
, 20
(5
), pp. 686
–701
.10.1109/TVCG.2013.2297914Copyright © 2025 by ASME
You do not currently have access to this content.
