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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

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.
[1]

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

[2]

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

[3]

Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35

[4]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255

[5]

Wouter Huberts, E. Marielle H. Bosboom, Frans N. van de Vosse. A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function. Mathematical Biosciences & Engineering, 2009, 6 (1) : 27-40. doi: 10.3934/mbe.2009.6.27

[6]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[7]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[8]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[9]

Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015

[10]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[11]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1

[12]

Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279

[13]

Rafael Vázquez, Emmanuel Trélat, Jean-Michel Coron. Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 925-956. doi: 10.3934/dcdsb.2008.10.925

[14]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073

[15]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[16]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

[17]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[18]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[19]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[20]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

2017 Impact Factor: 1.049

Article outline

Figures and Tables

[Back to Top]