March  2016, 11(1): 69-87. doi: 10.3934/nhm.2016.11.69

A shallow water with variable pressure model for blood flow simulation

1. 

Laboratoire J.A. Dieudonné, UMR CNRS 7351 & Polytech Nice Sophia, Université de Nice - Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France

2. 

Sorbonne Universités, CNRS and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert - 4, place Jussieu, Boîte 162, 75005 Paris, France, France, France

Received  April 2015 Revised  September 2015 Published  January 2016

We performed numerical simulations of blood flow in arteries with a variable stiffness and cross-section at rest using a finite volume method coupled with a hydrostatic reconstruction of the variables at the interface of each mesh cell. The method was then validated on examples taken from the literature. Asymptotic solutions were computed to highlight the effect of the viscous and viscoelastic source terms. Finally, the blood flow was computed in an artery where the cross-section at rest and the stiffness were varying. In each test case, the hydrostatic reconstruction showed good results where other simpler schemes did not, generating spurious oscillations and nonphysical velocities.
Citation: Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks and Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69
References:
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E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090.

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A. Bermúdez, A. Dervieux, J.-A. Desideri and M. E. Vázquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Computer Methods in Applied Mechanics and Engineering, 155 (1998), 49-72. doi: 10.1016/S0045-7825(97)85625-3.

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A. Bermúdez and M. E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms, Computers & Fluids, 23 (1994), 1049-1071. doi: 10.1016/0045-7930(94)90004-3.

[4]

C. Berthon and F. Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, Journal of Computational Physics, 231 (2012), 4993-5015. doi: 10.1016/j.jcp.2012.02.031.

[5]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, volume 2/2004, Birkhäuser Basel, 2004. doi: 10.1007/b93802.

[6]

F. Bouchut and T. Morales De Luna, A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal., 48 (2010), 1733-1758. doi: 10.1137/090758416.

[7]

M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes, Technical Report 4282, INRIA, 2001.

[8]

M. J. Castro, A. Pardo and C. Parès, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Mathematical Models and Methods in Applied Sciences, 17 (2007), 2055-2113. doi: 10.1142/S021820250700256X.

[9]

N. Cavallini, V. Caleffi and V. Coscia, Finite volume and WENO scheme in one-dimensional vascular system modeling, Computers and Mathematics with Applications, 56 (2008), 2382-2397. doi: 10.1016/j.camwa.2008.05.039.

[10]

N. Cavallini and V. Coscia, One-dimensional modeling of venous pathologies: Finite volume and WENO schemes, in Advances in Mathematical Fluid Mechanics (eds, R. Rannacher and A. Sequeira), Springer Berlin Heidelberg, (2010), 147-170. doi: 10.1007/978-3-642-04068-9_9.

[11]

T. Chacón Rebollo, A. Domínguez Delgado and E. D. Fernández Nieto, Asymptotically balanced schemes for non-homogeneous hyperbolic systems-application to the shallow water equations, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 85-90. doi: 10.1016/j.crma.2003.11.008.

[12]

O. Delestre, Simulation du Ruissellement D'eau de Pluie sur des Surfaces Agricoles/Rain Water Overland Flow on Agricultural Fields Simulation, Ph.D thesis, Université d'Orléans in Orléans, 2010.

[13]

O. Delestre and P.-Y. Lagrée, A "well-balanced" finite volume scheme for blood flow simulation, International Journal for Numerical Methods in Fluids, 72 (2013), 177-205. doi: 10.1002/fld.3736.

[14]

L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1d arterial networks coupled with a lumped parameters description of the heart, Computer Methods in Biomechanics and Biomedical Engineering, 9 (2006), 273-288. doi: 10.1080/10255840600857767.

[15]

J.-M. Fullana and S. Zaleski, A branched one-dimensional model of vessel networks, J. Fluid. Mech., 621 (2009), 183-204. doi: 10.1017/S0022112008004771.

[16]

T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Computers & Fluids, 32 (2003), 479-513. doi: 10.1016/S0045-7930(02)00011-7.

[17]

D. L. George, Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation, Journal of Computational Physics, 227 (2008), 3089-3113. doi: 10.1016/j.jcp.2007.10.027.

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[19]

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[21]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35-61. doi: 10.1137/1025002.

[22]

J. Hou, F. Simons, Q. Liang and R. Hinkelmann, An improved hydrostatic reconstruction method for shallow water model, Journal of Hydraulic Research, 52 (2014), 432-439. doi: 10.1080/00221686.2013.858648.

[23]

T. J. R. Hughes and J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels, Mathematical Biosciences, 18 (1973), 161-170. doi: 10.1016/0025-5564(73)90027-8.

[24]

S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN, 35 (2001), 631-645. doi: 10.1051/m2an:2001130.

[25]

T. Katsaounis, B. Perthame and C. Simeoni, Upwinding sources at interfaces in conservation laws, Applied Mathematics Letters, 17 (2004), 309-316. doi: 10.1016/S0893-9659(04)90068-7.

[26]

R. Kirkman, T. Moore and C. Adlard, The Walking Dead, Image Comics, Berkeley, 2003.

[27]

A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system, Mathematical Modelling and Numerical Analysis, 36 (2002), 397-425. doi: 10.1051/m2an:2002019.

[28]

R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in mathematics ETH Zurich, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.

[29]

R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, Journal of Computational Physics, 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058.

[30]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[31]

Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms, Advances in Water Resources, 32 (2009), 873-884. doi: 10.1016/j.advwatres.2009.02.010.

[32]

J. Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1978.

[33]

V. Martin, F. Clément, A. Decoene and J.-F. Gerbeau, Parameter identification for a one-dimensional blood flow model, in ESAIM: PROCEEDINGS (eds, E. Cancès and . J.-F. Gerbeau), EDP Sciences, 14 (2005), 174-200.

[34]

V. Melicher and V. Gajdosík, A numerical solution of a one-dimensional blood flow model-moving grid approach, Journal of Computational and Applied Mathematics, 215 (2008), 512-520. doi: 10.1016/j.cam.2006.03.065.

[35]

P. Munz, I. Hudea, J. Imad and R. J. Smith, When zombies attack!: Mathematical modelling of an outbreak of zombie infection, Infectious Disease Modelling Research Progress, (2009), 133-150.

[36]

S. Noelle, N. Pankratz, G. Puppo and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, Journal of Computational Physics, 213 (2006), 474-499. doi: 10.1016/j.jcp.2005.08.019.

[37]

S. Noelle, Y. Xing and C. W. Shu, High-order well-balanced finite volume weno schemes for shallow water equation with moving water, Journal of Computational Physics, 226 (2007), 29-58. doi: 10.1016/j.jcp.2007.03.031.

[38]

M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), 1281-1299. doi: 10.1114/1.1326031.

[39]

B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231. doi: 10.1007/s10092-001-8181-3.

[40]

M. Saito, Y. Ikenaga, M. Matsukawa, Y. Watanabe, T. Asada and P.-Y. Lagrée, One-dimensional model for propagation of a pressure wave in a model of the human arterial network: Comparison of theoretical and experimental, Journal of Biomechanical Engineering, 133 (2011), 121005. doi: 10.1115/1.4005472.

[41]

S. J. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, International Journal for Numerical Methods in Fluids, 43 (2003), 673-700. doi: 10.1002/fld.543.

[42]

N. Stergiopulos, D. F. Young and T. R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomechanics, 25 (1992), 1477-1488. doi: 10.1016/0021-9290(92)90060-E.

[43]

J. C. Stettler, P. Niederer and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations - part i: Mathematical model and prediction of normal pulse patterns, Annals of Biomedical Engineering, 9 (1981), 145-164. doi: 10.1007/BF02363533.

[44]

M. D. Thanh, M. Fazlul Karim and A. I. M. Ismail, Well-balanced scheme for shallow water equations with arbitrary topography, Int. J. Dynamical Systems and Differential Equations, 1 (2008), 196-204. doi: 10.1504/IJDSDE.2008.019681.

[45]

E. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows, John Wiley and Sons Ltd., England, 2001.

[46]

X. Wang, O. Delestre, J.-M. Fullana, M. Saito, Y. Ikenaga, M. Matsukawa and P.-Y. Lagrée, Comparing different numerical methods for solving arterial 1d flows in networks, Computer Methods in Biomechanics and Biomedical Engineering, 15 (2012), 61-62. doi: 10.1080/10255842.2012.713677.

[47]

X. Wang, J.-M. Fullana and P.-Y. Lagrée, Verification and comparison of four numerical schemes for a 1d viscoelastic blood flow model, Computer Methods in Biomechanics and Biomedical Engineering, 18 (2015), 1704-1725. doi: 10.1080/10255842.2014.948428.

[48]

M. Willemet, V. Lacroix and E. Marchandise, Inlet boundary conditions for blood flow simulations in truncated arterial networks, Journal of Biomechanics, 44 (2011), 897-903. doi: 10.1016/j.jbiomech.2010.11.036.

[49]

D. Xiu and S. J. Sherwin, Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network, Journal of Computational Physics, 226 (2007), 1385-1407. doi: 10.1016/j.jcp.2007.05.020.

[50]

M. Zagzoule, J. Khalid-Naciri and J. Mauss, Unsteady wall shear stress in a distensible tube, J. Biomechanics, 24 (1991), 435-439. doi: 10.1016/0021-9290(91)90031-H.

[51]

M. Zagzoule and J.-P. Marc-Vergnes, A global mathematical model of the cerebral circulation in man, J. Biomechanics, 19 (1986), 1015-1022. doi: 10.1016/0021-9290(86)90118-1.

show all references

References:
[1]

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090.

[2]

A. Bermúdez, A. Dervieux, J.-A. Desideri and M. E. Vázquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Computer Methods in Applied Mechanics and Engineering, 155 (1998), 49-72. doi: 10.1016/S0045-7825(97)85625-3.

[3]

A. Bermúdez and M. E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms, Computers & Fluids, 23 (1994), 1049-1071. doi: 10.1016/0045-7930(94)90004-3.

[4]

C. Berthon and F. Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, Journal of Computational Physics, 231 (2012), 4993-5015. doi: 10.1016/j.jcp.2012.02.031.

[5]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, volume 2/2004, Birkhäuser Basel, 2004. doi: 10.1007/b93802.

[6]

F. Bouchut and T. Morales De Luna, A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal., 48 (2010), 1733-1758. doi: 10.1137/090758416.

[7]

M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes, Technical Report 4282, INRIA, 2001.

[8]

M. J. Castro, A. Pardo and C. Parès, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Mathematical Models and Methods in Applied Sciences, 17 (2007), 2055-2113. doi: 10.1142/S021820250700256X.

[9]

N. Cavallini, V. Caleffi and V. Coscia, Finite volume and WENO scheme in one-dimensional vascular system modeling, Computers and Mathematics with Applications, 56 (2008), 2382-2397. doi: 10.1016/j.camwa.2008.05.039.

[10]

N. Cavallini and V. Coscia, One-dimensional modeling of venous pathologies: Finite volume and WENO schemes, in Advances in Mathematical Fluid Mechanics (eds, R. Rannacher and A. Sequeira), Springer Berlin Heidelberg, (2010), 147-170. doi: 10.1007/978-3-642-04068-9_9.

[11]

T. Chacón Rebollo, A. Domínguez Delgado and E. D. Fernández Nieto, Asymptotically balanced schemes for non-homogeneous hyperbolic systems-application to the shallow water equations, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 85-90. doi: 10.1016/j.crma.2003.11.008.

[12]

O. Delestre, Simulation du Ruissellement D'eau de Pluie sur des Surfaces Agricoles/Rain Water Overland Flow on Agricultural Fields Simulation, Ph.D thesis, Université d'Orléans in Orléans, 2010.

[13]

O. Delestre and P.-Y. Lagrée, A "well-balanced" finite volume scheme for blood flow simulation, International Journal for Numerical Methods in Fluids, 72 (2013), 177-205. doi: 10.1002/fld.3736.

[14]

L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1d arterial networks coupled with a lumped parameters description of the heart, Computer Methods in Biomechanics and Biomedical Engineering, 9 (2006), 273-288. doi: 10.1080/10255840600857767.

[15]

J.-M. Fullana and S. Zaleski, A branched one-dimensional model of vessel networks, J. Fluid. Mech., 621 (2009), 183-204. doi: 10.1017/S0022112008004771.

[16]

T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Computers & Fluids, 32 (2003), 479-513. doi: 10.1016/S0045-7930(02)00011-7.

[17]

D. L. George, Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation, Journal of Computational Physics, 227 (2008), 3089-3113. doi: 10.1016/j.jcp.2007.10.027.

[18]

E. Godlewski and P.-A. Raviart, Numerical Approximations of Hyperbolic Systems of Conservation Laws, volume Applied Mathematical Sciences 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9.

[19]

J. M. Greenberg and A.-Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equation, SIAM Journal on Numerical Analysis, 33 (1996), 1-16. doi: 10.1137/0733001.

[20]

L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-balanced and Asymptotic-Preserving, SIMAI Springer Series 2, Springer, Milano, 2013. doi: 10.1007/978-88-470-2892-0.

[21]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35-61. doi: 10.1137/1025002.

[22]

J. Hou, F. Simons, Q. Liang and R. Hinkelmann, An improved hydrostatic reconstruction method for shallow water model, Journal of Hydraulic Research, 52 (2014), 432-439. doi: 10.1080/00221686.2013.858648.

[23]

T. J. R. Hughes and J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels, Mathematical Biosciences, 18 (1973), 161-170. doi: 10.1016/0025-5564(73)90027-8.

[24]

S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN, 35 (2001), 631-645. doi: 10.1051/m2an:2001130.

[25]

T. Katsaounis, B. Perthame and C. Simeoni, Upwinding sources at interfaces in conservation laws, Applied Mathematics Letters, 17 (2004), 309-316. doi: 10.1016/S0893-9659(04)90068-7.

[26]

R. Kirkman, T. Moore and C. Adlard, The Walking Dead, Image Comics, Berkeley, 2003.

[27]

A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system, Mathematical Modelling and Numerical Analysis, 36 (2002), 397-425. doi: 10.1051/m2an:2002019.

[28]

R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in mathematics ETH Zurich, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.

[29]

R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, Journal of Computational Physics, 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058.

[30]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[31]

Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms, Advances in Water Resources, 32 (2009), 873-884. doi: 10.1016/j.advwatres.2009.02.010.

[32]

J. Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1978.

[33]

V. Martin, F. Clément, A. Decoene and J.-F. Gerbeau, Parameter identification for a one-dimensional blood flow model, in ESAIM: PROCEEDINGS (eds, E. Cancès and . J.-F. Gerbeau), EDP Sciences, 14 (2005), 174-200.

[34]

V. Melicher and V. Gajdosík, A numerical solution of a one-dimensional blood flow model-moving grid approach, Journal of Computational and Applied Mathematics, 215 (2008), 512-520. doi: 10.1016/j.cam.2006.03.065.

[35]

P. Munz, I. Hudea, J. Imad and R. J. Smith, When zombies attack!: Mathematical modelling of an outbreak of zombie infection, Infectious Disease Modelling Research Progress, (2009), 133-150.

[36]

S. Noelle, N. Pankratz, G. Puppo and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, Journal of Computational Physics, 213 (2006), 474-499. doi: 10.1016/j.jcp.2005.08.019.

[37]

S. Noelle, Y. Xing and C. W. Shu, High-order well-balanced finite volume weno schemes for shallow water equation with moving water, Journal of Computational Physics, 226 (2007), 29-58. doi: 10.1016/j.jcp.2007.03.031.

[38]

M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), 1281-1299. doi: 10.1114/1.1326031.

[39]

B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231. doi: 10.1007/s10092-001-8181-3.

[40]

M. Saito, Y. Ikenaga, M. Matsukawa, Y. Watanabe, T. Asada and P.-Y. Lagrée, One-dimensional model for propagation of a pressure wave in a model of the human arterial network: Comparison of theoretical and experimental, Journal of Biomechanical Engineering, 133 (2011), 121005. doi: 10.1115/1.4005472.

[41]

S. J. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, International Journal for Numerical Methods in Fluids, 43 (2003), 673-700. doi: 10.1002/fld.543.

[42]

N. Stergiopulos, D. F. Young and T. R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomechanics, 25 (1992), 1477-1488. doi: 10.1016/0021-9290(92)90060-E.

[43]

J. C. Stettler, P. Niederer and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations - part i: Mathematical model and prediction of normal pulse patterns, Annals of Biomedical Engineering, 9 (1981), 145-164. doi: 10.1007/BF02363533.

[44]

M. D. Thanh, M. Fazlul Karim and A. I. M. Ismail, Well-balanced scheme for shallow water equations with arbitrary topography, Int. J. Dynamical Systems and Differential Equations, 1 (2008), 196-204. doi: 10.1504/IJDSDE.2008.019681.

[45]

E. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows, John Wiley and Sons Ltd., England, 2001.

[46]

X. Wang, O. Delestre, J.-M. Fullana, M. Saito, Y. Ikenaga, M. Matsukawa and P.-Y. Lagrée, Comparing different numerical methods for solving arterial 1d flows in networks, Computer Methods in Biomechanics and Biomedical Engineering, 15 (2012), 61-62. doi: 10.1080/10255842.2012.713677.

[47]

X. Wang, J.-M. Fullana and P.-Y. Lagrée, Verification and comparison of four numerical schemes for a 1d viscoelastic blood flow model, Computer Methods in Biomechanics and Biomedical Engineering, 18 (2015), 1704-1725. doi: 10.1080/10255842.2014.948428.

[48]

M. Willemet, V. Lacroix and E. Marchandise, Inlet boundary conditions for blood flow simulations in truncated arterial networks, Journal of Biomechanics, 44 (2011), 897-903. doi: 10.1016/j.jbiomech.2010.11.036.

[49]

D. Xiu and S. J. Sherwin, Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network, Journal of Computational Physics, 226 (2007), 1385-1407. doi: 10.1016/j.jcp.2007.05.020.

[50]

M. Zagzoule, J. Khalid-Naciri and J. Mauss, Unsteady wall shear stress in a distensible tube, J. Biomechanics, 24 (1991), 435-439. doi: 10.1016/0021-9290(91)90031-H.

[51]

M. Zagzoule and J.-P. Marc-Vergnes, A global mathematical model of the cerebral circulation in man, J. Biomechanics, 19 (1986), 1015-1022. doi: 10.1016/0021-9290(86)90118-1.

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