A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks

doi:10.1007/s42967-021-00126-5

Abstract

Abstract: This paper formulates an efficient numerical method for solving the convection diffusion solute transport equations coupled to blood flow equations in vessel networks. The reduced coupled model describes the variations of vessel cross-sectional area, radially averaged blood momentum and solute concentration in large vessel networks. For the discretization of the reduced transport equation, we combine an interior penalty discontinuous Galerkin method in space with a novel locally implicit time stepping scheme. The stability and the convergence are proved. Numerical results show the impact of the choice for the steady-state axial velocity profile on the numerical solutions in a fifty-five vessel network with physiological boundary data.

Key words: Reduced models, Blood flow, Solute transport, Coriolis coefficient, Vessel networks, Junction conditions, Locally implicit

CLC Number:

Rami Masri, Charles Puelz, Beatrice Riviere. A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 500-529.

TrendMD

References

1.Alastruey, J., Parker, K.H., Sherwin, S.J.:Arterial pulse wave haemodynamics.In:11th International Conference on Pressure Surges, vol.30, pp.401-443.Virtual PiE Led t/a BHR Group Lisbon, Portugal (2012)
2.Azer, K.:Taylor diffusion in time dependent flow.Int.J.Heat Mass Transf.48(13), 2735-2740 (2005)
3.Barnard, A., Hunt, W., Timlake, W., Varley, E.:A theory of fluid flow in compliant tubes.Biophys.J.6(6), 717-724 (1966)
4.Boileau, E., Nithiarasu, P., Blanco, P.J., Müller, L.O., Fossan, F.E., Hellevik, L.R., Donders, W.P., Huberts, W., Willemet, M., Alastruey, J.:A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling.Int.J.Numer.Methods Biomed.Eng.31(10), e02732 (2015)
5.Calamante, F., Gadian, D.G., Connelly, A.:Delay and dispersion effects in dynamic susceptibility contrast MRI:simulations using singular value decomposition.Magn.Reson.Med.44(3), 466-473 (2000)
6.Čanić, S., Kim, E.H.:Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels.Math.Methods Appl.Sci.26(14), 1161-1186 (2003)
7.Cheng, Y., Shu, C.W.:A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives.Math.Comput.77(262), 699-730 (2008)
8.Cockburn, B., Shu, C.W.:TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws.II.General framework.Math.Comput.52(186), 411-435 (1989)
9.D'Angelo, C.:Multiscale modelling of metabolism and transport phenomena in living tissues.Tech.rep., EPFL (2007)
10.D'Angelo, C., Quarteroni, A.:On the coupling of 1D and 3D diffusion-reaction equations:application to tissue perfusion problems.Math.Models Methods Appl.Sci.18(08), 1481-1504 (2008)
11.Dolejší, V., Feistauer, M., Hozman, J.:Analysis of semi-implicit DGFEM for nonlinear convectiondiffusion problems on nonconforming meshes.Comput.Methods Appl.Mech.Eng.196(29/30), 2813-2827 (2007)
12.Formaggia, L., Lamponi, D., Quarteroni, A.:One dimensional models for blood flow in arteries.J.Eng.Math.47(3/4), 251-276 (2003)
13.Köppl, T., Schneider, M., Pohl, U., Wohlmuth, B.:The influence of an unilateral carotid artery stenosis on brain oxygenation.Med.Eng.Phys.36(7), 905-914 (2014)
14.Köppl, T., Wohlmuth, B., Helmig, R.:Reduced one-dimensional modelling and numerical simulation for mass transport in fluids.Int.J.Numer.Methods Fluids 72(2), 135-156 (2013)
15.Marbach, S., Alim, K.:Active control of dispersion within a channel with flow and pulsating walls.Phys.Rev.Fluids 4(11), 114202 (2019)
16.Marbach, S., Alim, K., Andrew, N., Pringle, A., Brenner, M.P.:Pruning to increase Taylor dispersion in physarum polycephalum networks.Phys.Rev.Lett.117(17), 178103 (2016)
17.Masri, R., Puelz, C., Riviere, B.:A reduced model for solute transport in compliant blood vessels with arbitrary axial velocity profile.International Journal of Heat and Mass Transfer 176, 121379 (2021)
18.Mynard, J.P., Smolich, J.J.:One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation.Ann.Biomed.Eng.43(6), 1443-1460 (2015)
19.Ozisik, S., Riviere, B., Warburton, T.:On the constants in inverse inequalities in L2.Tech.rep.(2010)
20.Puelz, C., Čanić, S., Riviere, B., Rusin, C.G.:Comparison of reduced models for blood flow using Runge Kutta discontinuous Galerkin methods.Appl.Numer.Math.115, 114-141 (2017)
21.Reichold, J., Stampanoni, M., Keller, A.L., Buck, A., Jenny, P., Weber, B.:Vascular graph model to simulate the cerebral blood flow in realistic vascular networks.J.Cereb.Blood Flow Metab.29(8), 1429-1443 (2009)
22.Riviere, B.:Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations:Theory and Implementation.SIAM, Philadelphia (2008)
23.Sherwin, S., Formaggia, L., Peiro, J., Franke, V.: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(6-7), 673-700 (2003)
24.Taylor, G.I.:Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion.Proc.R.Soc.Lond.A 225(1163), 473-477 (1954)
25.Wang, H., Shu, C.W., Zhang, Q.:Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems.Appl.Math.Comput.272, 237-258 (2016)
26.Warburton, T., Hesthaven, J.S.:On the constants in hp-finite element trace inverse inequalities.Comput.Methods Appl.Mech.Eng.192(25), 2765-2773 (2003)
27.Zhang, Q., Shu, C.W.:Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws.SIAM J.Numer.Anal.42(2), 641-666 (2004)
28.Zhang, Q., Shu, C.W.:Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws.SIAM J.Numer.Anal.44(4), 1703-1720 (2006)

[1]	Xiaozhou Li. How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 759-782.
[2]	Hendrik Ranocha, Gregor J. Gassner. Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 880-903.
[3]	Poorvi Shukla, J. J. W. van der Vegt. A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 904-944.
[4]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1011-1056.
[5]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part II: Time Convergence, Error Analysis and Numerical Results [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1057-1104.
[6]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[7]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[8]	Erik Burman, Omar Duran, Alexandre Ern. Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 597-633.
[9]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[10]	Qi Tao, Yan Xu, Xiaozhou Li. Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 250-270.
[11]	Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.
[12]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[13]	Jie Du, Eric Chung, Yang Yang. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 353-379.
[14]	Junhong Tian, Hengfei Ding. A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 571-584.
[15]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.