American Institute of Mathematical Sciences

Advanced
2013, 10(3): 649-665. doi: 10.3934/mbe.2013.10.649

Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms

Alberto Gambaruto 1, , João Janela 2, , Alexandra Moura 1, and  Adélia Sequeira 1,

1. 

Univ Tecn Lisboa, Inst Super Tecn, Dept Matemat and CEMAT, P-1049-001, Lisbon, Portugal, Portugal, Portugal
2. 

Univ Tecn Lisboa, ISEG, Dept Matemat and CEMAPRE, P-1200-781 Lisbon, Portugal

Received  September 2012 Revised  September 2012 Published  April 2013

Two different generalized Newtonian mathematical models for blood flow, derived for the same experimental data, are compared, together with the Newtonian model, in three different anatomically realistic geometries of saccular cerebral aneurysms obtained from rotational CTA. The geometries differ in size of the aneurysm and the existence or not of side branches within the aneurysm. Results show that the differences between the two generalized Newtonian mathematical models are smaller than the differences between these and the Newtonian solution, in both steady and unsteady simulations.
Keywords: patient-specific geometry, cerebral aneurysm, Blood flow modeling, non-Newtonian blood models..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Alberto Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Mathematical Biosciences & Engineering, 2013, 10 (3) : 649-665. doi: 10.3934/mbe.2013.10.649
References:
[1]

H. Baek, M. V. Jayaraman, P. D. Richardson and G. E. Karniadakis, Flow instability and wall shear stress variation in intracranial aneurysms, Journal of The Royal Society Interface, 7 (2010), 967-988. doi: 10.1098/rsif.2009.0476.  Google Scholar

[2]

M. Bessis, "Living Blood Cells and Their Ultrastructure," Springer-Verlag, 1973. Google Scholar

[3]

A. C. Burleson, C. M. Strother and V. T. Turitto, Computer modeling of intracranial saccular and lateral aneurysms for the study of their haemodynamics, Neurosurgery, 37 (1995), 774-784. Google Scholar

[4]

C. G. Caro, T. J. Pedley, R. C. Schroter and W. A. Seed, "The Mechanics of the Circulation," Oxford University Press, 1978. Google Scholar

[5]

J. R. Cebral, M. A. Castro, S. Appanaboyina, C. M. Putman, D. Millan and A. F. Frangi, Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm haemodynamics: Technique and sensitivity, IEEE Transactions on Medical Imaging, 24 (2005), 457-467. doi: 10.1109/TMI.2005.844159.  Google Scholar

[6]

J. R. Cebral, F. Mut, M. Raschi, E. Scrivano, P. Lylyk and C. M. Putman, Aneurysm rupture following treatment with flow diverting stents: Computational hemodynamics analysis of treatment, AJNR Am. J. Neuroradiol, 32 (2011), 27-33. doi: 10.3174/ajnr.A2398.  Google Scholar

[7]

J. R. Cebral, F. Mut, J. Weir and C. M. Putman, Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms, AJNR Am. J. Neuroradiol, 32 (2011), 145-152. doi: 10.3174/ajnr.A2419.  Google Scholar

[8]

M. D. Ford, S. W. Lee, S. P. Lownie, D. W. Holdsworth and D. A. Steinman, On the effect of parent-aneurysm angle on flow patterns in basilar tip aneurysms: Towards a surrogate geometric marker of intra-aneurismal haemodynamics, Journal of Biomechanics, 41 (2008), 241-248. doi: 10.1016/j.jbiomech.2007.09.032.  Google Scholar

[9]

A. Gambaruto, J. Janela, A. Moura and A. Sequeira, Sensitivity of haemodynamics in patient specific cerebral aneurysms to vascular geometry and blood rheology, Mathematical Biosciences and Engineering, 8 (2011), 409-423. doi: 10.3934/mbe.2011.8.409.  Google Scholar

[10]

A. Gambaruto, A. Moura and A. Sequeira, Topological flow structures and stir mixing for steady flow in a peripheral bypass graft with uncertainty, Int. J. Numer. Meth. Biomed. Engng., 26 (2010), 926-953. doi: 10.1002/cnm.1393.  Google Scholar

[11]

A. M. Gambaruto, J. Peiró, D. J. Doorly and A. G. Radaelli, Reconstruction of shape and its effect on flow in arterial conduits, Int. J. Numer. Meth. Fluids, 57 (2008), 495-517. doi: 10.1002/fld.1642.  Google Scholar

[12]

T. Hassan, E. V. Timofeev, T. Saito, H. Shimizu, M. Ezura, Y. Matsumoto, K. Takayama, T. Tominaga and A. Takahashi, A proposed parent vessel geometry-based categorization of saccular intracranial aneurysms: computational flow dynamics analysis of the risk factors for lesion rupture, Journal of Neurosurgery, 103 (2005), 662-680. doi: 10.3171/jns.2005.103.4.0662.  Google Scholar

[13]

S. G. Imbesi and C. W. Kerber, Analysis of slipstream flow in a wide-necked basilar artery aneurysm: Evaluation of potential treatment regimens, American Journal of Neuroradiology, 22 (2001), 721-724. Google Scholar

[14]

J. Janela, A. Moura and A. Sequeira, Towards a geometrical multiscale approach to non-Newtonian blood flow simulations, in "Advances in Mathematical Fluid Mechanics", Springer, Berlin and Heidelberg, (2010), 295-310. doi: 10.1007/978-3-642-04068-9_18.  Google Scholar

[15]

M. Kim, D. B. Taulbee, M. Tremmel and H. Meng, Comparison of two stents in modifying cerebral aneurysm haemodynamics, Annals of Biomedical Engineering, 36 (2008), 726-741. doi: 10.1007/s10439-008-9449-4.  Google Scholar

[16]

D. Krex, H. K. Schackert and G. Schackert, Genesis of cerebral aneurysms - an update, Acta Neurochirurgica, 143 (2001), 429-449. Google Scholar

[17]

S. W. Lee and D. A. Steinman, On the relative importance of rheology for image-based CFD models of the carotid bifurcation, J. Biomech. Eng., 129 (2007), 273-278. Google Scholar

[18]

G. D. O. Lowe, "Clinical Blood Rheology," CRC Press, Boca Raton, FL, 1988. Google Scholar

[19]

H. Meng, Z. Wang, Y. Hoi, L. Gao, E. Metaxa, D. D. Swartz and J. Kolega, Complex haemodynamics at the apex of an arterial bifurcation induces vascular remodeling resembling cerebral aneurysm initiation, Stroke, 38 (2007), 1924-1931. doi: 10.1161/STROKEAHA.106.481234.  Google Scholar

[20]

T. Passerini, L. M. Sangalli, S. Vantini, M. Piccinelli, S. Bacigaluppi, L. Antiga, E. Boccardi, P. Secchi and A. Veneziani, An integrated statistical investigation of internal carotid arteries of patients affected by cerebral aneurysms, Cardiovascular Engineering and Technology, 3 (2012), 22-40. doi: 10.1007/s13239-011-0079-x.  Google Scholar

[21]

B. Pincombe and J. Mazdumar, The effects of post-stenotic dilatations on the flow of blood analogue through stenosed coronary arteries, Math. Comput. Modelling, 25 (1997), 57-70. doi: 10.1016/S0895-7177(97)00039-3.  Google Scholar

[22]

S. Ramalho, A. Moura, A. M. Gambaruto and A. Sequeira, Sensitivity to outflow boundary conditions and level of geometry description for a cerebral aneurysm, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 697-713. doi: 10.1002/cnm.2461.  Google Scholar

[23]

S. Ramalho, A. Moura, A. M. Gambaruto and A. Sequeira, Influence of fluid mathematical model and outflow conditions in numerical simulations of cerebral aneurysms, in "Mathematical Methods and Models in Biomedicine" (Eds. U. Ledzewicz, H. Schättler, A. Friedman and E. Kashdan), Springer, (2013), 149-175. Google Scholar

[24]

V. L. Rayz, L. Boussel, M. T. Lawton, G. Acevedo-Bolton, L. Ge, W. L. Young, R. T. Higashida and D. Saloner, Numerical modeling of the flow in intracranial aneurysms: Prediction of regions prone to thrombus formation, Ann. Biomed. Eng., 36 (2008), 1793-1804. doi: 10.1007/s10439-008-9561-5.  Google Scholar

[25]

A. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, in "Hemodynamical Flows", Modeling, Analysis and Simulation, Oberwolfach Seminars, 37 (2008), 63-120, Birkhäuser. doi: 10.1007/978-3-7643-7806-6_2.  Google Scholar

[26]

M. Shojima, M. Oshima, K. Takagi, et al., Role of the bloodstream impacting force and the local pressure elevation in the rupture of cerebral aneurysms, Stroke, 36 (2005), 193338. doi: 10.1161/01.STR.0000177877.88925.06.  Google Scholar

[27]

D. Steinman, J. Milner, C. Norley, S. Lownie and D. Holdsworth, Image-based computational simulation of flow dynamics in a giant intracranial aneurysm, American Journal of Neuroradiology, 24 (2003), 55966. Google Scholar

[28]

P. Venugopal, D. Valentino, H. Schmitt, J. P. Villablanca, F. Viñuela and G. Duckwiler, Sensitivity of patient-specific numerical simulation of cerebral aneurysm haemodynamics to inflow boundary conditions, Journal of Neurosurgery, 106 (2007), 1051-1060. Google Scholar

[29]

K. K. Yeleswarapu, M. V. Kameneva, K. R. Rajagopal and J. F. Antaki, The flow of blood in tubes: Theory and experiment, Mech. Res. Commun., 25 (1998), 257-262. doi: 10.1016/S0093-6413(98)00036-6.  Google Scholar

[30]

Z. Zeng, F. K. David, M. Durika, Y. Ding, D. A. Lewis, R. Kadirvel and A. M. Robertson, Sensitivity of CFD based hemodynamic results in rabbit aneurysm models to idealizations in surrounding vasculature, Journal of Biomechanical Engineering, 132 (2010), 091009. doi: 10.1115/1.4001311.  Google Scholar

show all references

References:
[1]

H. Baek, M. V. Jayaraman, P. D. Richardson and G. E. Karniadakis, Flow instability and wall shear stress variation in intracranial aneurysms, Journal of The Royal Society Interface, 7 (2010), 967-988. doi: 10.1098/rsif.2009.0476.  Google Scholar

[2]

M. Bessis, "Living Blood Cells and Their Ultrastructure," Springer-Verlag, 1973. Google Scholar

[3]

A. C. Burleson, C. M. Strother and V. T. Turitto, Computer modeling of intracranial saccular and lateral aneurysms for the study of their haemodynamics, Neurosurgery, 37 (1995), 774-784. Google Scholar

[4]

C. G. Caro, T. J. Pedley, R. C. Schroter and W. A. Seed, "The Mechanics of the Circulation," Oxford University Press, 1978. Google Scholar

[5]

J. R. Cebral, M. A. Castro, S. Appanaboyina, C. M. Putman, D. Millan and A. F. Frangi, Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm haemodynamics: Technique and sensitivity, IEEE Transactions on Medical Imaging, 24 (2005), 457-467. doi: 10.1109/TMI.2005.844159.  Google Scholar

[6]

J. R. Cebral, F. Mut, M. Raschi, E. Scrivano, P. Lylyk and C. M. Putman, Aneurysm rupture following treatment with flow diverting stents: Computational hemodynamics analysis of treatment, AJNR Am. J. Neuroradiol, 32 (2011), 27-33. doi: 10.3174/ajnr.A2398.  Google Scholar

[7]

J. R. Cebral, F. Mut, J. Weir and C. M. Putman, Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms, AJNR Am. J. Neuroradiol, 32 (2011), 145-152. doi: 10.3174/ajnr.A2419.  Google Scholar

[8]

M. D. Ford, S. W. Lee, S. P. Lownie, D. W. Holdsworth and D. A. Steinman, On the effect of parent-aneurysm angle on flow patterns in basilar tip aneurysms: Towards a surrogate geometric marker of intra-aneurismal haemodynamics, Journal of Biomechanics, 41 (2008), 241-248. doi: 10.1016/j.jbiomech.2007.09.032.  Google Scholar

[9]

A. Gambaruto, J. Janela, A. Moura and A. Sequeira, Sensitivity of haemodynamics in patient specific cerebral aneurysms to vascular geometry and blood rheology, Mathematical Biosciences and Engineering, 8 (2011), 409-423. doi: 10.3934/mbe.2011.8.409.  Google Scholar

[10]

A. Gambaruto, A. Moura and A. Sequeira, Topological flow structures and stir mixing for steady flow in a peripheral bypass graft with uncertainty, Int. J. Numer. Meth. Biomed. Engng., 26 (2010), 926-953. doi: 10.1002/cnm.1393.  Google Scholar

[11]

A. M. Gambaruto, J. Peiró, D. J. Doorly and A. G. Radaelli, Reconstruction of shape and its effect on flow in arterial conduits, Int. J. Numer. Meth. Fluids, 57 (2008), 495-517. doi: 10.1002/fld.1642.  Google Scholar

[12]

T. Hassan, E. V. Timofeev, T. Saito, H. Shimizu, M. Ezura, Y. Matsumoto, K. Takayama, T. Tominaga and A. Takahashi, A proposed parent vessel geometry-based categorization of saccular intracranial aneurysms: computational flow dynamics analysis of the risk factors for lesion rupture, Journal of Neurosurgery, 103 (2005), 662-680. doi: 10.3171/jns.2005.103.4.0662.  Google Scholar

[13]

S. G. Imbesi and C. W. Kerber, Analysis of slipstream flow in a wide-necked basilar artery aneurysm: Evaluation of potential treatment regimens, American Journal of Neuroradiology, 22 (2001), 721-724. Google Scholar

[14]

J. Janela, A. Moura and A. Sequeira, Towards a geometrical multiscale approach to non-Newtonian blood flow simulations, in "Advances in Mathematical Fluid Mechanics", Springer, Berlin and Heidelberg, (2010), 295-310. doi: 10.1007/978-3-642-04068-9_18.  Google Scholar

[15]

M. Kim, D. B. Taulbee, M. Tremmel and H. Meng, Comparison of two stents in modifying cerebral aneurysm haemodynamics, Annals of Biomedical Engineering, 36 (2008), 726-741. doi: 10.1007/s10439-008-9449-4.  Google Scholar

[16]

D. Krex, H. K. Schackert and G. Schackert, Genesis of cerebral aneurysms - an update, Acta Neurochirurgica, 143 (2001), 429-449. Google Scholar

[17]

S. W. Lee and D. A. Steinman, On the relative importance of rheology for image-based CFD models of the carotid bifurcation, J. Biomech. Eng., 129 (2007), 273-278. Google Scholar

[18]

G. D. O. Lowe, "Clinical Blood Rheology," CRC Press, Boca Raton, FL, 1988. Google Scholar

[19]

H. Meng, Z. Wang, Y. Hoi, L. Gao, E. Metaxa, D. D. Swartz and J. Kolega, Complex haemodynamics at the apex of an arterial bifurcation induces vascular remodeling resembling cerebral aneurysm initiation, Stroke, 38 (2007), 1924-1931. doi: 10.1161/STROKEAHA.106.481234.  Google Scholar

[20]

T. Passerini, L. M. Sangalli, S. Vantini, M. Piccinelli, S. Bacigaluppi, L. Antiga, E. Boccardi, P. Secchi and A. Veneziani, An integrated statistical investigation of internal carotid arteries of patients affected by cerebral aneurysms, Cardiovascular Engineering and Technology, 3 (2012), 22-40. doi: 10.1007/s13239-011-0079-x.  Google Scholar

[21]

B. Pincombe and J. Mazdumar, The effects of post-stenotic dilatations on the flow of blood analogue through stenosed coronary arteries, Math. Comput. Modelling, 25 (1997), 57-70. doi: 10.1016/S0895-7177(97)00039-3.  Google Scholar

[22]

S. Ramalho, A. Moura, A. M. Gambaruto and A. Sequeira, Sensitivity to outflow boundary conditions and level of geometry description for a cerebral aneurysm, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 697-713. doi: 10.1002/cnm.2461.  Google Scholar

[23]

S. Ramalho, A. Moura, A. M. Gambaruto and A. Sequeira, Influence of fluid mathematical model and outflow conditions in numerical simulations of cerebral aneurysms, in "Mathematical Methods and Models in Biomedicine" (Eds. U. Ledzewicz, H. Schättler, A. Friedman and E. Kashdan), Springer, (2013), 149-175. Google Scholar

[24]

V. L. Rayz, L. Boussel, M. T. Lawton, G. Acevedo-Bolton, L. Ge, W. L. Young, R. T. Higashida and D. Saloner, Numerical modeling of the flow in intracranial aneurysms: Prediction of regions prone to thrombus formation, Ann. Biomed. Eng., 36 (2008), 1793-1804. doi: 10.1007/s10439-008-9561-5.  Google Scholar

[25]

A. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, in "Hemodynamical Flows", Modeling, Analysis and Simulation, Oberwolfach Seminars, 37 (2008), 63-120, Birkhäuser. doi: 10.1007/978-3-7643-7806-6_2.  Google Scholar

[26]

M. Shojima, M. Oshima, K. Takagi, et al., Role of the bloodstream impacting force and the local pressure elevation in the rupture of cerebral aneurysms, Stroke, 36 (2005), 193338. doi: 10.1161/01.STR.0000177877.88925.06.  Google Scholar

[27]

D. Steinman, J. Milner, C. Norley, S. Lownie and D. Holdsworth, Image-based computational simulation of flow dynamics in a giant intracranial aneurysm, American Journal of Neuroradiology, 24 (2003), 55966. Google Scholar

[28]

P. Venugopal, D. Valentino, H. Schmitt, J. P. Villablanca, F. Viñuela and G. Duckwiler, Sensitivity of patient-specific numerical simulation of cerebral aneurysm haemodynamics to inflow boundary conditions, Journal of Neurosurgery, 106 (2007), 1051-1060. Google Scholar

[29]

K. K. Yeleswarapu, M. V. Kameneva, K. R. Rajagopal and J. F. Antaki, The flow of blood in tubes: Theory and experiment, Mech. Res. Commun., 25 (1998), 257-262. doi: 10.1016/S0093-6413(98)00036-6.  Google Scholar

[30]

Z. Zeng, F. K. David, M. Durika, Y. Ding, D. A. Lewis, R. Kadirvel and A. M. Robertson, Sensitivity of CFD based hemodynamic results in rabbit aneurysm models to idealizations in surrounding vasculature, Journal of Biomechanical Engineering, 132 (2010), 091009. doi: 10.1115/1.4001311.  Google Scholar

[1]

Alberto M. Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Mathematical Biosciences & Engineering, 2011, 8 (2) : 409-423. doi: 10.3934/mbe.2011.8.409
[2]

Changli Yuan, Mojdeh Delshad, Mary F. Wheeler. Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework. Networks & Heterogeneous Media, 2010, 5 (3) : 583-602. doi: 10.3934/nhm.2010.5.583
[3]

Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333
[4]

Mette S. Olufsen, Ali Nadim. On deriving lumped models for blood flow and pressure in the systemic arteries. Mathematical Biosciences & Engineering, 2004, 1 (1) : 61-80. doi: 10.3934/mbe.2004.1.61
[5]

Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic & Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361
[6]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[7]

Yukun Song, Yang Chen, Jun Yan, Shuai Chen. The existence of solutions for a shear thinning compressible non-Newtonian models. Electronic Research Archive, 2020, 28 (1) : 47-66. doi: 10.3934/era.2020004
[8]

Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure & Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839
[9]

Adélia Sequeira, Rafael F. Santos, Tomáš Bodnár. Blood coagulation dynamics: mathematical modeling and stability results. Mathematical Biosciences & Engineering, 2011, 8 (2) : 425-443. doi: 10.3934/mbe.2011.8.425
[10]

Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks & Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527
[11]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521
[12]

Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037
[13]

Benchawan Wiwatanapataphee, Yong Hong Wu, Thanongchai Siriapisith, Buraskorn Nuntadilok. Effect of branchings on blood flow in the system of human coronary arteries. Mathematical Biosciences & Engineering, 2012, 9 (1) : 199-214. doi: 10.3934/mbe.2012.9.199
[14]

Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69
[15]

B. Wiwatanapataphee, D. Poltem, Yong Hong Wu, Y. Lenbury. Simulation of Pulsatile Flow of Blood in Stenosed Coronary Artery Bypass with Graft. Mathematical Biosciences & Engineering, 2006, 3 (2) : 371-383. doi: 10.3934/mbe.2006.3.371
[16]

Tony Lyons. The 2-component dispersionless Burgers equation arising in the modelling of blood flow. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1563-1576. doi: 10.3934/cpaa.2012.11.1563
[17]

Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419
[18]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255
[19]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[20]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy