Previous |  Up |  Next


Title: A phase-field method applied to interface tracking for blood clot formation (English)
Author: Čapek, Marek
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 65
Issue: 4
Year: 2020
Pages: 447-481
Summary lang: English
Category: math
Summary: The high shear rate thrombus formation was only recently recognized as another way of thrombosis. Models proposed in Weller (2008), (2010) take into account this type of thrombosis. This work uses the phase-field method to model these evolving interface problems. A loosely coupled iterative procedure is introduced to solve the coupled system of equations. Convergence behavior on two levels of refinement of perfusion chamber geometry and cylinder geometry is then studied. The perfusion chamber simulations show good agreement with the original results of Weller. The code is implemented in FEM-library deal.ii Alzeta et al.\ (2018), which enables distribution of computations to large number of processing units. A scalability and numerical performance study of the loosely coupled iterative procedure is performed, combined with several preconditioners for the linear subproblems. (English)
Keyword: thrombus growth
Keyword: free boundary problem
Keyword: fluid dynamics
Keyword: phase field method
Keyword: finite element method
Keyword: scalability
Keyword: high shear rate thrombosis
MSC: 76D05
MSC: 76M10
MSC: 76T99
idZBL: 07250671
idMR: MR4134143
DOI: 10.21136/AM.2020.0056-19
Date available: 2020-09-07T09:47:35Z
Last updated: 2020-11-18
Stable URL:
Reference: [1] Aarts, P. A., Broek, S. A. van den, Prins, G. W., Kuiken, G. D., Sixma, J. J., Heethaar, R. M.: Blood platelets are concentrated near the wall and red blood cells, in the center in flowing blood.Arteriosclerosis 8 (1988), 819-824. 10.1161/01.atv.8.6.819
Reference: [2] Affeld, K., Reininger, A. J., Gadischke, J., Grunert, K., Schmidt, S., Thiele, F.: Fluid mechanics of the stagnation point flow chamber and its platelet deposition.Artif. Organs. 19 (1995), 597-602. 10.1111/j.1525-1594.1995.tb02387.x
Reference: [3] G. Alzetta, D. Arndt, W. Bangerth, V. Boddu, B. Brands, D. Davydov, R. Gassmöller, T. Heister, L. Heltai, K. Kormann, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells: The deal.II library, version 9.0.J. Numer. Math. 26 (2018), 173-183. Zbl 1410.65363, MR 3893339, 10.1515/jnma-2018-0054
Reference: [4] Anand, M., Rajagopal, K., Rajagopal, K. R.: A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood.J. Theor. Med. 5 (2003), 183-218. Zbl 1078.92017, MR 2158290, 10.1080/10273660412331317415
Reference: [5] D. Arndt, W. Bangerth, T. C. Clevenger, D. Davydov, M. Fehling, D. Garcia-Sanchez, G. Harper, T. Heister, L. Heltai, M. Kronbichler, R. M. Kynch, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells: The deal.II library, version 9.1.J. Numer. Math. 27 (2019), 203-213. Zbl 07181764, MR 4078181, 10.1515/jnma-2019-0064
Reference: [6] Balay, S, Gropp, W. D., McInnes, L. C., Smith, B. F.: PETSc: Portable, Extensible Toolkit for Scientific Computation---Toolkit for Advanced Computation.Available at (2018).
Reference: [7] Bangerth, W., Hartmann, R., Kanschat, G.: deal.ii---a general-purpose object-oriented finite element library.ACM Trans. Math. Softw. 33 (2007), Article ID 24, 27 pages. Zbl 1365.65248, MR 2404402, 10.1145/1268776.1268779
Reference: [8] Barlas, G.: Multicore and GPU Programming: An Integrated Approach.Morgan Kaufmann Publishers, San Francisco (2015).
Reference: [9] Bodnár, T., Fasano, A., Sequeira, A.: Mathematical models for blood coagulation.Fluid-Structure Interaction and Biomedical Applications T. Bodnár et al. Advances in Mathematical Fluid Mechanics. Birkhäuser/Springer, Basel (2014), 483-569. Zbl 1351.76320, MR 3329023, 10.1007/978-3-0348-0822-4_7
Reference: [10] Burstedde, C., Wilcox, L. C., Ghattas, O.: p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees.SIAM J. Sci. Comput. 33 (2011), 1103-1133. Zbl 1230.65106, MR 2800566, 10.1137/100791634
Reference: [11] Casa, L. D. C., Deaton, D. H., Ku, D. N.: Role of high shear rate in thrombosis.J. Vasc. Surg. 61 (2015), 1068-1080 (2015). 10.1016/j.jvs.2014.12.050
Reference: [12] Casa, L. D. C., Ku, D N.: Thrombus formation at high shear rates.Ann. Rev. Biomed. Eng. 19 (2017), 415-433. 10.1146/annurev-bioeng-071516-044539
Reference: [13] Colman, R. W., Marder, V. J., Clowes, A. W., George, J. N., Goldhaber, S. Z.: Hemostasis and Thrombosis: Basic Principles and Clinical Practice.Lippincott Williams & Wilkins, Philadelphia (2005).
Reference: [14] Fasano, A., Sequeira, A.: Hemorheology and hemodynamics.Hemomath Modeling, Simulation and Applications 18. Springer, Cham (2017), 1-77. MR 3727113, 10.1007/978-3-319-60513-5_1
Reference: [15] Gross, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows.Springer Series in Computational Mathematics 40. Springer, Berlin (2011). Zbl 1222.76002, MR 2799400, 10.1007/978-3-642-19686-7
Reference: [16] Guermond, J. L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows.Comput. Methods Appl. Mech. Eng. 195 (2006), 6011-6045. Zbl 1122.76072, MR 2250931, 10.1016/j.cma.2005.10.010
Reference: [17] M. A. Heroux, R. A. Bartlett, V. E. Howle, R. J. Hoekstra, J. J. Hu, T. G. Kolda, R. B. Lehoucq, K. R. Long, R. P. Pawlowski, E. T. Phipps, A. G. Salinger, H. Thornquist, R. S. Tuminaro, J. M. Willenbring, A. Williams, K. S. Stanley: An overview of the Trilinos project.ACM Trans. Math. Softw. 31 (2005), 397-423. Zbl 1136.65354, MR 2266800, 10.1145/1089014.1089021
Reference: [18] al., V. Huck et: A2 - Research. Available at
Reference: [19] Karypis, G., Kumar, V.: MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0.Available at {metis}, 2009.
Reference: [20] Key, N. S., Makris, M., Lillicrap, D., eds.: Practical Hemostasis and Thrombosis.John Wiley, Chichester (2017). 10.1002/9781118344729
Reference: [21] Kuzmin, D.: Introduction to Computational Fluid Dynamics.Available at {kuzmin/cfdintro/lecture8.pdf}, 2010.
Reference: [22] Li, X., Lowengrub, J., Rätz, A., Voigt, A.: Solving PDEs in complex geometries: a diffuse domain approach.Commun. Math. Sci. 7 (2009), 81-107. Zbl 1178.35027, MR 2512834, 10.4310/CMS.2009.v7.n1.a4
Reference: [23] Mohan, A.: Modeling the Growth and Dissolution of Clots in Flowing Blood.PhD Thesis, Texas A&M University, College Station (2005).
Reference: [24] Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces.Applied Mathematical Sciences 153. Springer, New York (2003). Zbl 1026.76001, MR 1939127, 10.1007/b98879
Reference: [25] Reinders, J.: Intel Threading Building Blocks: Outfitting C++ for Multi-Core Processor Parallelism.O'Reilly, Sebastopol (2007).
Reference: [26] Sakariassen, K. S., Orning, L., Turitto, V. T.: The impact of blood shear rate on arterial thrombus formation.Future Sci. OA 1 (2015), Article ID FSO30. 10.4155/fso.15.28
Reference: [27] Sethian, J. A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science.Cambridge Monographs on Applied and Computational Mathematics 3. Cambridge University Press, Cambridge (1999). Zbl 0973.76003, MR 1700751
Reference: [28] Tokarev, A. A., Butylin, A. A., Ataullakhanov, F. I.: Platelet adhesion from shear blood flow is controlled by near-wall rebounding collisions with erythrocytes.Biophys. J. 100 (2011), 799-808. 10.1016/j.bpj.2010.12.3740
Reference: [29] Tokarev, A., Sirakov, I., Panasenko, G., Volpert, V., Shnol, E., Butylin, A., Ataullakhanov, F.: Continuous mathematical model of platelet thrombus formation in blood flow.Russ. J. Numer. Anal. Math. Model. 27 (2012), 191-212. Zbl 06032989, MR 2910582, 10.1515/rnam-2012-0011
Reference: [30] Turek, S.: Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach.Lecture Notes in Computational Science and Engineering 6. Springer, Berlin (1999). Zbl 0930.76002, MR 1691839, 10.1007/978-3-642-58393-3
Reference: [31] Weller, F.: Modeling, Analysis, and Simulation of Thrombosis and Hemostasis: PhD Thesis.Ruprecht-Karls-Universität, Heidelberg (2008). 10.11588/heidok.00008558
Reference: [32] Weller, F. F.: Platelet deposition in non-parallel flow: influence of shear stress and changes in surface reactivity.J. Math. Biol. 57 (2008), 333-359. Zbl 1149.35332, MR 2411224, 10.1007/s00285-008-0163-5
Reference: [33] Weller, F. F.: A free boundary problem modeling thrombus growth. Model development and numerical simulation using the level set method.J. Math. Biol. 61 (2010), 805-818. Zbl 1205.92010, MR 2726451, 10.1007/s00285-009-0324-1
Reference: [34] Xu, Z., Chen, N., Shadden, S. C., Marsden, J. E., Kamocka, M. M., Rosen, E. D., Alber, M.: Study of blood flow impact on growth of thrombi using a multiscale model.Soft Matter 5 (2009), 769-779. 10.1039/B812429A
Reference: [35] Zlobina, K. E., Guria, G. Th.: Platelet activation risk index as a prognostic thrombosis indicator.Scientific Reports 6 (2016), Article ID 30508. 10.1038/srep30508

Fulltext not available (moving wall 24 months)

Partner of
EuDML logo