Computational modelling of multiscale, multiphase fluid mixtures with application to tumour growth. (English) Zbl 1439.76003

Summary: In this work we consider the discretization of our recently formulated [Eur. J. Appl. Math. 28, No. 3, 499–534 (2017; Zbl 1375.92030)] multiscale model for drug- and nutrient-limited tumour growth. The key contribution of this work is the proposal of a discontinuous Galerkin finite element scheme which incorporates a non-standard coupling across a singular surface, and the presentation of full details of a suitable discretization for the coupled flow and transport systems, such as that arising in [loc. cit.] and other similar works. We demonstrate the application of the proposed discretizations via representative numerical experiments; furthermore, we present a short numerical study of convergence for the proposed microscale scheme, in which we observe optimal rates of convergence for sufficiently smooth data.


76-10 Mathematical modeling or simulation for problems pertaining to fluid mechanics
74L15 Biomechanical solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76T99 Multiphase and multicomponent flows
76Z05 Physiological flows
92C42 Systems biology, networks


Zbl 1375.92030


Triangle; MUMPS
Full Text: DOI Link


[1] Davit, Y.; Bell, C. G.; Byrne, H. M.; Chapman, L. A.C.; Kimpton, L. S.; Lang, G. E.; Leonard, K. H.L.; Oliver, J. M.; Pearson, N. C.; Shipley, R. J.; Waters, S. L.; Whiteley, J. P.; Wood, B. D.; Quintard, M., Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?, Adv. Water Resour., 62, 178-206 (2013)
[2] Brezzi, F., Numerical Analysis, 69-82 (2000), CRC, Chapter Interacting with the Subgrid World
[3] Efendiev, Y.; Hou, T. Y., Multiscale Finite Element Methods: Theory and Applications (2009), Springer: Springer New York · Zbl 1163.65080
[4] Farmer, C., Upscaling: A review, Internat. J. Numer. Methods Fluids, 40, 63-78 (2002) · Zbl 1058.76574
[5] Keller, J. B., Darcy’s law for flow in porous media and the two-scale method, (Sternberg, R. L.; Kalinowski, A. J.; Papadakis, J. S., Nonlinear PDE in Engineering and Applied Sciences (1980), Marcel Dekker) · Zbl 0439.76017
[6] Rubinstein, J.; Torquato, S., Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds, J. Fluid Mech., 206, 25-46 (1989) · Zbl 0681.76098
[7] O’Dea, R. D.; Nelson, M. R.; El Haj, A. J.; Waters, S. L.; Byrne, H. M., A multiscale analysis of nutrient transport and biological tissue growth in vitro, Math. Med. Biol. (2014)
[8] Ptashnyk, M.; Roose, T., Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory, SIAM J. Appl. Math., 70, 7, 2097-2118 (2010) · Zbl 1230.35014
[9] Shipley, R. J., Multiscale modelling of fluid and drug transport in vascular tumours (2008), University of Oxford, (PhD thesis) · Zbl 1198.92028
[10] Araujo, R. P.; McElwain, D. L.S., A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66, 1039-1091 (2004) · Zbl 1334.92187
[11] Alarcón, T.; Byrne, H. M.; Maini, P. K., A cellular automaton model for tumour growth in inhomogeneous environment, J. Theoret. Biol., 225, 257-274 (2003)
[12] Alarcón, T.; Byrne, H. M.; Maini, P. K., Towards whole-organ modelling of tumour growth, Prog. Biophys. & Mol. Biol., 85, 451-472 (2004)
[13] Alarcón, T.; Byrne, H. M.; Maini, P. K., A multiple scale model for tumor growth, Multiscale Model. Simul., 3, 2, 440-475 (2005) · Zbl 1107.92019
[14] Alarcón, T.; Owen, M. R.; Byrne, H. M.; Maini, P. K., Multiscale modelling of tumour growth and therapy: The influence of vessel normalisation on chemotherapy, Comput. Math. Methods Med., 7, 2-3, 85-119 (2006) · Zbl 1111.92023
[15] Macklin, P.; McDougall, S.; Anderson, A. R.A.; Chaplain, M. A.J.; Cristini, V.; Lowengrub, J., Multiscale modelling and nonlinear simulation of vascular tumour growth, J. Math. Biol., 58, 4-5, 765-798 (2009) · Zbl 1311.92040
[16] Owen, M. R.; Alarcón, T.; Maini, P. K.; Byrne, H. M., Angiogenesis and vascular remodelling in normal and cancerous tissues, Journal of Mathematical Biology, 58, 4-5, 689-721 (2009) · Zbl 1311.92034
[17] Owen, M. R.; Stamper, I. J.; Muthana, M.; Dobson, G. W.; Lewis, C. E.; Byrne, H. M., Mathematical modeling predicts synergistic antitumor effects of combining a macrophage-based, hypoxia-targeted gene therapy with chemotherapy, Cancer Res., 71, 8, 2826-2837 (2011)
[18] Perfahl, H.; Byrne, H. M.; Chen, T.; Estrella, V.; Alarcón, T.; Lapin, A.; Gatenby, R. A.; Gillies, R. J.; Lloyd, M. C.; Maini, P. K.; Reuss, M.; Owen, M. R., Multiscale modelling of vascular tumour growth in 3d: the roles of domain size and boundary conditions, PLoS One, 6, 4, e14790 (2011)
[19] Frieboes, H. B.; Lowengrub, J. S.; Wise, S. M.; Zheng, X.; Macklin, P.; Bearer, E. L.; Cristini, V., Computer simulation of glioma growth and morphology, Neuroimage, 37, S58-S70 (2007)
[20] Shipley, R. J.; Chapman, S. J.; Jawad, R., Multiscale modeling of fluid transport in tumors, Bull. Math. Biol., 70, 8, 2334-2357 (2010) · Zbl 1169.92307
[21] Powathil, G. G.; Swat, M.; Chaplain, M. A.J., Systems oncology: Towards patient-specific treatment regimes informed by multiscale mathematical modelling, Sem. Cancer Biol., 30, 13-20 (2015)
[23] Hubbard, M. E.; Byrne, H. M., Multiphase modelling of vascular tumour growth in two spatial dimensions, J. Theoret. Biol., 316, 70-89 (2013)
[24] Breward, C. J.W.; Byrne, H. M.; Lewis, C. E., A multiphase model describing vascular tumour growth, Bull. Math. Biol., 1, 1-28 (2004)
[25] Shipley, R. J.; Chapman, S. J., Multiscale modelling of fluid and drug transport in vascular tumours, Bull. Math. Biol., 72, 6, 1464-1491 (2010) · Zbl 1198.92028
[26] Bresch, D.; Koko, J., Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids, Internat. J. Appl. Math. Comput. Sci., 16, 4, 419-429 (2006) · Zbl 1119.35346
[27] O’Dea, R. D.; Waters, S. L.; Byrne, H. M., A two-fluid model for tissue growth within a dynamic flow environment, European J. Appl. Math., 19, 607-634 (2008) · Zbl 1256.92015
[28] Franks, S. J.; King, J. R., Interactions between a uniformly proliferating tumour and its surroundings, Math. Med. Biol., 20, 47-89 (2003) · Zbl 1044.92032
[29] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002
[30] Toselli, A., hp-finite element discontinuous Galerkin approximations for the Stokes problem, M3AS, 12, 1565-1616 (2002) · Zbl 1041.76045
[31] Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., The development of discontinuous Galerkin methods, (Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., Discontinuous Galerkin Finite Element Methods. Discontinuous Galerkin Finite Element Methods, Lect. Notes Comput. Sci. Eng., vol. 11 (2000), Springer-Verlag: Springer-Verlag Berlin), 3-50 · Zbl 0989.76045
[33] Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of the Finite Element Method. Mathematical Aspects of the Finite Element Method, Lectures Notes in Math., vol. 606 (1977), Springer-Verlag: Springer-Verlag New York)
[34] Nedelec, J. C., Mixed finite elments in \(R^3\), Numer. Math., 35, 315-341 (1980)
[35] Houston, P.; Süli, E., hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems, SIAM J. Sci. Comput., 23, 4, 1226-1252 (2001) · Zbl 1029.65130
[36] Discacciati, M.; Gervasio, P.; Quarteroni, A., Interface control domain decomposition methods for heterogeneous problems, Internat. J. Numer. Methods Fluids, 76, 8, 471-496 (2014)
[37] Discacciati, M.; Gervasio, P.; Giacomini, A.; Quarteroni, A., The interface control domain decomposition method for Stokes-Darcy coupling, SIAM J. Numer. Anal., 54, 2, 1039-1068 (2016) · Zbl 1337.49056
[38] Brezzi, F.; Douglas, J.; Marini, L. D., Recent results on mixed finite element methods for second order elliptic problems, (Balakrishanan; Dorodnitsyn; Lions, Vistas in Applied Math., Numerical Analysis, Atmospheric Sciences, Immunology (1986), Optimization Software Publications: Optimization Software Publications New York) · Zbl 0611.65071
[39] Brezzi, F.; Douglas, J.; Duran, R.; Fortin, M., Mixed finite element methods for second order elliptic problems in three variables, Numer. Math., 51, 237-250 (1987) · Zbl 0631.65107
[40] Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, Math. Model. Numer. Anal., 21, 581-604 (1987) · Zbl 0689.65065
[42] Amestoy, P. R.; Duff, I. S.; L’Excellent, J.-Y.; Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018
[43] Amestoy, P. R.; Guermouche, A.; L’Excellent, J.-Y.; Pralet, S., Hybrid scheduling for the parallel solution of linear systems, Parallel Comput., 32, 2, 136-156 (2006)
[44] Shewchuk, J. R., Triangle: Engineering a 2D quality mesh generator and delaunay triangulator, (Lin, Ming C.; Manocha, Dinesh, Applied Computational Geometry: Towards Geometric Engineering. Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148 (1996), Springer-Verlag), 203-222, From the First ACM Workshop on Applied Computational Geometry
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.