March  2019, 8(1): 221-230. doi: 10.3934/eect.2019012

Shock wave formation in compliant arteries

1. 

Division of Imaging Sciences and Biomedical Engineering, King's College London, St. Thomas' Hospital, SE1 7EH London, United Kingdom

2. 

Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 València, Spain

3. 

CoMMLab, Departament d'Informàtica, Universitat de València, E-46100 Burjassot, València, Spain

* Corresponding author: J. Alberto Conejero

Received  March 2018 Revised  July 2018 Published  January 2019

We focus on the problem of shock wave formation in a model of blood flow along an elastic artery. We analyze the conditions under which this phenomenon can appear and we provide an estimation of the instant of shock formation. Numerical simulations of the model have been conducted using the Discontinuous Galerkin Finite Element Method. The results are consistent with certain phenomena observed by practitioners in patients with arteriopathies, and they could predict the possible formation of a shock wave in the aorta.

Citation: Cristóbal Rodero, J. Alberto Conejero, Ignacio García-Fernández. Shock wave formation in compliant arteries. Evolution Equations & Control Theory, 2019, 8 (1) : 221-230. doi: 10.3934/eect.2019012
References:
[1]

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S. ČanićJ. TambačaG. GuidoboniA. MikelićC. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006), 164-193.  doi: 10.1137/060651562.  Google Scholar

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[5]

I. C. ChristovV. CognetT. C. Shidhore and H. A. Stone, Flow rate-pressure drop relation for deformable shallow microfluidic channels, Journal of Fluid Mechanics, 841 (2018), 267-286.  doi: 10.1017/jfm.2018.30.  Google Scholar

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L. CozijnsenR. L. BraamR. A. WaalewijnM. A. SchepensB. L. LoeysM. F. van OosterhoutD. Q. Barge-Schaapveld and B. J. Mulder, What is new in dilatation of the ascending aorta?, Circulation, 123 (2011), 924-928.  doi: 10.1161/CIRCULATIONAHA.110.949131.  Google Scholar

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T. A. Crowley and V. Pizziconi, Isolation of plasma from whole blood using planar microfilters for lab-on-a-chip applications, Lab on a Chip, 5 (2005), 922-929.  doi: 10.1039/b502930a.  Google Scholar

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C. T. DotterD. J. Roberts and I. Steinberg, Aortic length: Angiocardiographic measurements, Circulation, 2 (1950), 915-920.  doi: 10.1161/01.CIR.2.6.915.  Google Scholar

[10]

A. Elgarayhi, E. El-Shewy, A. A. Mahmoud and A. A. Elhakem, Propagation of nonlinear pressure waves in blood, ISRN Computational Biology, 2013 (2013), Article ID 436267, 5 pages. doi: 10.1155/2013/436267.  Google Scholar

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R. Erbel and H. Eggebrecht, Aortic dimensions and the risk of dissection, Heart, 92 (2006), 137-142.  doi: 10.1136/hrt.2004.055111.  Google Scholar

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Y. C. Fung, Biomechanics: Circulation, Springer Science & Business Media, 2013. Google Scholar

[13]

J. E. Hall, Guyton and Hall Textbook of Medical Physiology E-Book, Elsevier Health Sciences, 13 edition, 2015. Google Scholar

[14]

P. Hunter, Numerical Simulation of Arterial Blood Flow, PhD thesis, ResearchSpace@ Auckland, 1972. Google Scholar

[15]

J. Keener and J. Sneyd, Mathematical Physiology, volume 8 of Interdisciplinary Applied Mathematics, Springer: New York, 1998.  Google Scholar

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P. S. Laplace, Traité de Mécanique Céleste, Courcier, 1805. Google Scholar

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P. Lax, Hyperbolic systems of conservation laws ii, Communications on Pure and Applied Mathematics, 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

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P. Lax, Development of singularities of solution of nonlinear hyperbolic partial differential equations, Journal of Mathematical Physics, 5 (1964), 611-613.  doi: 10.1063/1.1704154.  Google Scholar

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S. S. MaoN. AhmadiB. ShahD. BeckmannA. ChenL. NgoF. R. FloresY. L. Gao and M. J. Budoff, Normal thoracic aorta diameter on cardiac computed tomography in healthy asymptomatic adults: impact of age and gender, Academic Radiology, 15 (2008), 827-834.   Google Scholar

[20]

D. MowatN. Haites and J. Rawles, Aortic blood velocity measurement in healthy adult using a simple ultrasound technique, Cardiovascular Research, 17 (1983), 75-80.  doi: 10.1093/cvr/17.2.75.  Google Scholar

[21]

P. R. Painter, The velocity of the arterial pulse wave: A viscous-fluid shock wave in an elastic tube, Theoretical Biology and Medical Modelling, 5 (2008), p15. doi: 10.1186/1742-4682-5-15.  Google Scholar

[22]

P. Perdikaris and G. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models, Ann. Biomed. Eng., 42 (2014), 1012-1023.  doi: 10.1007/s10439-014-0970-3.  Google Scholar

[23]

K. PerktoldM. Resch and H. Florian, Pulsatile non-newtonian flow characteristics in a three-dimensional human carotid bifurcation model, Journal of biomechanical engineering, 113 (1991), 464-475.  doi: 10.1115/1.2895428.  Google Scholar

[24]

G. PorentaD. Young and T. Rogge, A finite-element model of blood flow in arteries including taper, branches, and obstructions, J. Biomech. Eng., 108 (1986), 161-167.  doi: 10.1115/1.3138596.  Google Scholar

[25]

J. K. RainesM. Y. Jaffrin and A. H. Shapiro, A computer simulation of arterial dynamics in the human leg, J. Biomech., 7 (1974), 77-91.  doi: 10.1016/0021-9290(74)90072-4.  Google Scholar

[26]

C. Rodero, Analysis of blood flow in one dimensional elastic artery using Navier-Stokes conservation laws, Master's thesis, Universitat de València / Universitat Politècnica de València, 2017. Google Scholar

[27]

S. SherwinV. FrankeJ. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.  doi: 10.1023/B:ENGI.0000007979.32871.e2.  Google Scholar

[28]

S. J. SherwinL. FormaggiaJ. 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, Int. J. Numer. Methods Fluids, 43 (2003), 673-700.  doi: 10.1002/fld.543.  Google Scholar

[29]

R. Shoucri and M. Shoucri, Application of the method of characteristics for the study of shock waves in models of blood flow in the aorta, Cardiovascular Engineering, 7 (2007), 1-6.   Google Scholar

[30]

T. Sochi, Flow of Navier-Stokes fluids in cylindrical elastic tubes, J. Appl. Fluid Mech., 8 (2015), 181-188.   Google Scholar

[31]

E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: A practical introduction, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[32]

E. F. Toro, Brain venous haemodynamics, neurological diseases and mathematical modelling. a review, Appl. Math. Comput., 272 (2016), 542-579.  doi: 10.1016/j.amc.2015.06.066.  Google Scholar

[33]

N. Westerhof and A. Noordergraaf, Arterial viscoelasticity: A generalized model, J. Biomech., 3 (1970), 357-379.  doi: 10.1016/0021-9290(70)90036-9.  Google Scholar

[34]

R. J. WhittakerM. HeilO. E. Jensen and S. L. Waters, A rational derivation of a tube law from shell theory, Applied Mathematics, 63 (2010), 465-496.  doi: 10.1093/qjmam/hbq020.  Google Scholar

[35]

T. Young, Ⅲ. An essay on the cohesion of fluids, Philosophical Transactions of the Royal Society of London, 95 (1805), 65-87.   Google Scholar

show all references

References:
[1]

K. AndoT. SanadaK. InabaJ. DamazoJ. ShepherdT. Colonius and C. Brennen, Shock propagation through a bubbly liquid in a deformable tube, Journal of Fluid Mechanics, 671 (2011), 339-363.  doi: 10.1017/S0022112010005707.  Google Scholar

[2]

S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Method Appl. Sci., 26 (2003), 1161-1186.  doi: 10.1002/mma.407.  Google Scholar

[3]

S. ČanićJ. TambačaG. GuidoboniA. MikelićC. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006), 164-193.  doi: 10.1137/060651562.  Google Scholar

[4]

I. ChristovP. Jordan and C. Christov, Nonlinear acoustic propagation in homentropic perfect gases: A numerical study, Physics Letters A, 353 (2006), 273-280.  doi: 10.1016/j.physleta.2005.12.101.  Google Scholar

[5]

I. C. ChristovV. CognetT. C. Shidhore and H. A. Stone, Flow rate-pressure drop relation for deformable shallow microfluidic channels, Journal of Fluid Mechanics, 841 (2018), 267-286.  doi: 10.1017/jfm.2018.30.  Google Scholar

[6]

L. CozijnsenR. L. BraamR. A. WaalewijnM. A. SchepensB. L. LoeysM. F. van OosterhoutD. Q. Barge-Schaapveld and B. J. Mulder, What is new in dilatation of the ascending aorta?, Circulation, 123 (2011), 924-928.  doi: 10.1161/CIRCULATIONAHA.110.949131.  Google Scholar

[7]

T. A. Crowley and V. Pizziconi, Isolation of plasma from whole blood using planar microfilters for lab-on-a-chip applications, Lab on a Chip, 5 (2005), 922-929.  doi: 10.1039/b502930a.  Google Scholar

[8]

V. Dolejš and M. Feistaner, Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow, Springer, Cham, 2015. doi: 10.1007/978-3-319-19267-3.  Google Scholar

[9]

C. T. DotterD. J. Roberts and I. Steinberg, Aortic length: Angiocardiographic measurements, Circulation, 2 (1950), 915-920.  doi: 10.1161/01.CIR.2.6.915.  Google Scholar

[10]

A. Elgarayhi, E. El-Shewy, A. A. Mahmoud and A. A. Elhakem, Propagation of nonlinear pressure waves in blood, ISRN Computational Biology, 2013 (2013), Article ID 436267, 5 pages. doi: 10.1155/2013/436267.  Google Scholar

[11]

R. Erbel and H. Eggebrecht, Aortic dimensions and the risk of dissection, Heart, 92 (2006), 137-142.  doi: 10.1136/hrt.2004.055111.  Google Scholar

[12]

Y. C. Fung, Biomechanics: Circulation, Springer Science & Business Media, 2013. Google Scholar

[13]

J. E. Hall, Guyton and Hall Textbook of Medical Physiology E-Book, Elsevier Health Sciences, 13 edition, 2015. Google Scholar

[14]

P. Hunter, Numerical Simulation of Arterial Blood Flow, PhD thesis, ResearchSpace@ Auckland, 1972. Google Scholar

[15]

J. Keener and J. Sneyd, Mathematical Physiology, volume 8 of Interdisciplinary Applied Mathematics, Springer: New York, 1998.  Google Scholar

[16]

P. S. Laplace, Traité de Mécanique Céleste, Courcier, 1805. Google Scholar

[17]

P. Lax, Hyperbolic systems of conservation laws ii, Communications on Pure and Applied Mathematics, 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

[18]

P. Lax, Development of singularities of solution of nonlinear hyperbolic partial differential equations, Journal of Mathematical Physics, 5 (1964), 611-613.  doi: 10.1063/1.1704154.  Google Scholar

[19]

S. S. MaoN. AhmadiB. ShahD. BeckmannA. ChenL. NgoF. R. FloresY. L. Gao and M. J. Budoff, Normal thoracic aorta diameter on cardiac computed tomography in healthy asymptomatic adults: impact of age and gender, Academic Radiology, 15 (2008), 827-834.   Google Scholar

[20]

D. MowatN. Haites and J. Rawles, Aortic blood velocity measurement in healthy adult using a simple ultrasound technique, Cardiovascular Research, 17 (1983), 75-80.  doi: 10.1093/cvr/17.2.75.  Google Scholar

[21]

P. R. Painter, The velocity of the arterial pulse wave: A viscous-fluid shock wave in an elastic tube, Theoretical Biology and Medical Modelling, 5 (2008), p15. doi: 10.1186/1742-4682-5-15.  Google Scholar

[22]

P. Perdikaris and G. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models, Ann. Biomed. Eng., 42 (2014), 1012-1023.  doi: 10.1007/s10439-014-0970-3.  Google Scholar

[23]

K. PerktoldM. Resch and H. Florian, Pulsatile non-newtonian flow characteristics in a three-dimensional human carotid bifurcation model, Journal of biomechanical engineering, 113 (1991), 464-475.  doi: 10.1115/1.2895428.  Google Scholar

[24]

G. PorentaD. Young and T. Rogge, A finite-element model of blood flow in arteries including taper, branches, and obstructions, J. Biomech. Eng., 108 (1986), 161-167.  doi: 10.1115/1.3138596.  Google Scholar

[25]

J. K. RainesM. Y. Jaffrin and A. H. Shapiro, A computer simulation of arterial dynamics in the human leg, J. Biomech., 7 (1974), 77-91.  doi: 10.1016/0021-9290(74)90072-4.  Google Scholar

[26]

C. Rodero, Analysis of blood flow in one dimensional elastic artery using Navier-Stokes conservation laws, Master's thesis, Universitat de València / Universitat Politècnica de València, 2017. Google Scholar

[27]

S. SherwinV. FrankeJ. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.  doi: 10.1023/B:ENGI.0000007979.32871.e2.  Google Scholar

[28]

S. J. SherwinL. FormaggiaJ. 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, Int. J. Numer. Methods Fluids, 43 (2003), 673-700.  doi: 10.1002/fld.543.  Google Scholar

[29]

R. Shoucri and M. Shoucri, Application of the method of characteristics for the study of shock waves in models of blood flow in the aorta, Cardiovascular Engineering, 7 (2007), 1-6.   Google Scholar

[30]

T. Sochi, Flow of Navier-Stokes fluids in cylindrical elastic tubes, J. Appl. Fluid Mech., 8 (2015), 181-188.   Google Scholar

[31]

E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: A practical introduction, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[32]

E. F. Toro, Brain venous haemodynamics, neurological diseases and mathematical modelling. a review, Appl. Math. Comput., 272 (2016), 542-579.  doi: 10.1016/j.amc.2015.06.066.  Google Scholar

[33]

N. Westerhof and A. Noordergraaf, Arterial viscoelasticity: A generalized model, J. Biomech., 3 (1970), 357-379.  doi: 10.1016/0021-9290(70)90036-9.  Google Scholar

[34]

R. J. WhittakerM. HeilO. E. Jensen and S. L. Waters, A rational derivation of a tube law from shell theory, Applied Mathematics, 63 (2010), 465-496.  doi: 10.1093/qjmam/hbq020.  Google Scholar

[35]

T. Young, Ⅲ. An essay on the cohesion of fluids, Philosophical Transactions of the Royal Society of London, 95 (1805), 65-87.   Google Scholar

Figure 1.  An artery as a compliant tube, where variable x denotes the spatial coordinate and t the temporal one.
Figure 2.  Decomposition of the domain $ D $.
Figure 3.  Several beat like boundary conditions (23) for $u(0, 0) = 0$.
Figure 4.  Formation of a shock wave with a beat-like boundary condition. The discontinuity at $x = 20$ is due to the nature of the DG method, which provides two values in the frontier between elements.
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