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An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law
Cahn–Hilliard–Brinkman systems for tumour growth
1. | Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany |
2. | Department of Mathematics, University of Trento, Trento, Italy |
A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.
References:
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H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013.
doi: 10.1142/S0218202511500138. |
[2] |
A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani,
A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.
doi: 10.1002/mma.4548. |
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D. Ambrosi and L. Preziosi,
On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sci., 12 (2002), 737-754.
doi: 10.1142/S0218202502001878. |
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P. R. Amestoy, T. A. Davis and I. S. Duff,
Algorithm 837: AMD, an approximate minimum degree ordering algorithm, ACM Trans. Math. Software, 30 (2004), 381-388.
doi: 10.1145/1024074.1024081. |
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L. Baňas and R. Nürnberg,
Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202-232.
doi: 10.1007/s10915-008-9203-y. |
[6] |
J. W. Barrett, H. Garcke and R. Nürnberg, Chapter 4 - Parametric finite element approximations of curvature-driven interface evolutions, in Geometric Partial Differential Equations - Part I, Handbook of Numerical Analysis, 21, Elsevier (2020), 275–423.
doi: 10.1016/bs.hna.2019.05.002. |
[7] |
J. W. Barrett, H. Garcke and R. Nürnberg,
Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., 34 (2014), 1289-1327.
doi: 10.1093/imanum/drt044. |
[8] |
J. W. Barrett, R. Nürnberg and V. Styles,
Finite element approximation of a phase field model for void electromigration, SIAM J. Numer. Anal., 42 (2004), 738-772.
doi: 10.1137/S0036142902413421. |
[9] |
N. Bellomo, N. K. Li and P. K. Maini,
On the foundations of cancer modelling: selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
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H. Byrne and M. A. J. Chaplain,
Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-658.
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P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn–Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
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V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, Selected Topics in Cancer Modeling, Birkhäuser Boston, (2008), 113–181. |
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V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise,
Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763.
doi: 10.1007/s00285-008-0215-x. |
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Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
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T. A. Davis,
Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), 196-199.
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T. A. Davis,
Algorithm 849: A concise sparse Cholesky factorization package, ACM Trans. Math. Software, 31 (2005), 587-591.
doi: 10.1145/1114268.1114277. |
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M. Ebenbeck, Cahn–Hilliard–Brinkman Models for Tumour Growth: Modelling, Analysis and Optimal Control, Ph.D thesis, University Regensburg, 2020. Google Scholar |
[21] |
M. Ebenbeck and H. Garcke,
Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998-6036.
doi: 10.1016/j.jde.2018.10.045. |
[22] |
M. Ebenbeck and H. Garcke,
On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM J. Math. Anal., 51 (2019), 1868-1912.
doi: 10.1137/18M1228104. |
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C. Eck, H. Garcke and P. Knabner, Mathematical Modeling, Springer, Cham, 2017.
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C. M. Elliott and H. Garcke,
On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
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J. Eyles, J. R. King and V. Styles,
A tractable mathematical model for tissue growth, Interfaces Free Bound., 21 (2019), 463-493.
doi: 10.4171/IFB/428. |
[26] |
S. J. Franks and J. R. King,
Interactions between a uniformly proliferating tumour and its surroundings: Stability analysis for variable material properties, Internat. J. Engrg. Sci., 47 (2009), 1182-1192.
doi: 10.1016/j.ijengsci.2009.07.004. |
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S. J. Franks and J. R. King,
Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 47-89.
doi: 10.1093/imammb/20.1.47. |
[28] |
H. B. Frieboes, J. S. Lowengrub, S. Wise, X. Zheng, P. Macklin, E. L. Bearer and V. Cristini,
Computer simulation of glioma growth and morphology, NeuroImage, 37 (2007), 59-70.
doi: 10.1016/j.neuroimage.2007.03.008. |
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A. Friedman,
A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound., 8 (2006), 247-261.
doi: 10.4171/IFB/142. |
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A. Friedman, Free boundary problems associated with multiscale tumor models, Math. Model. Nat. Phenom., 4 (2009), 134-155.
doi: 10.1051/mmnp/20094306. |
[31] |
A. Friedman,
Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[32] |
A. Friedmann and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
[33] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
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S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (Springer INdAM ser.), 22 Springer Cham, (2017), 217–254. |
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M. Fritz, E. A. B. F. Lima, J. T. Oden and B. Wohlmuth,
On the unsteady Darcy–Forchheimer–Brinkman equation in local and nonlocal tumor growth models, Math. Models Methods Appl. Sci., 29 (2019), 1691-1731.
doi: 10.1142/S0218202519500325. |
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H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Math., 1 (2016), 318-360.
doi: 10.3934/Math.2016.3.318. |
[37] |
H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, Trends in Applications of Mathematics to Mechanics, Springer, Cham 27 (2018), 243–264. |
[38] |
H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka,
A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525-577.
doi: 10.1142/S0218202518500148. |
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H. Garcke, K. F. Lam and A. Signori, On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects, Nonlinear Anal. Real World Appl., 57 (2021), 103192.
doi: 10.1016/j.nonrwa.2020.103192. |
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H. Garcke, K. F. Lam, E. Sitka and V. Styles,
A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.
doi: 10.1142/S0218202516500263. |
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Second order phase field asymptotics for multi-component systems, Interfaces Free Bound., 8 (2006), 131-157.
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show all references
References:
[1] |
H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013.
doi: 10.1142/S0218202511500138. |
[2] |
A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani,
A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.
doi: 10.1002/mma.4548. |
[3] |
D. Ambrosi and L. Preziosi,
On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sci., 12 (2002), 737-754.
doi: 10.1142/S0218202502001878. |
[4] |
P. R. Amestoy, T. A. Davis and I. S. Duff,
Algorithm 837: AMD, an approximate minimum degree ordering algorithm, ACM Trans. Math. Software, 30 (2004), 381-388.
doi: 10.1145/1024074.1024081. |
[5] |
L. Baňas and R. Nürnberg,
Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202-232.
doi: 10.1007/s10915-008-9203-y. |
[6] |
J. W. Barrett, H. Garcke and R. Nürnberg, Chapter 4 - Parametric finite element approximations of curvature-driven interface evolutions, in Geometric Partial Differential Equations - Part I, Handbook of Numerical Analysis, 21, Elsevier (2020), 275–423.
doi: 10.1016/bs.hna.2019.05.002. |
[7] |
J. W. Barrett, H. Garcke and R. Nürnberg,
Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., 34 (2014), 1289-1327.
doi: 10.1093/imanum/drt044. |
[8] |
J. W. Barrett, R. Nürnberg and V. Styles,
Finite element approximation of a phase field model for void electromigration, SIAM J. Numer. Anal., 42 (2004), 738-772.
doi: 10.1137/S0036142902413421. |
[9] |
N. Bellomo, N. K. Li and P. K. Maini,
On the foundations of cancer modelling: selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
[10] |
H. Byrne and M. A. J. Chaplain,
Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-658.
doi: 10.1017/S0956792597003264. |
[11] |
P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, North-Holland Publishing Co., Amsterdam, 1988. |
[12] |
P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn–Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
[13] |
V. Cristini, H. B. Frieboes, R. Gatenby, S. Caserta, M. Ferrari and J. Sinek,
Morphologic instability and cancer invasion, Clin. Cancer Res., 11 (2005), 6772-6779.
doi: 10.1158/1078-0432.CCR-05-0852. |
[14] |
V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, Selected Topics in Cancer Modeling, Birkhäuser Boston, (2008), 113–181. |
[15] |
V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise,
Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763.
doi: 10.1007/s00285-008-0215-x. |
[16] |
V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511781452.![]() |
[17] |
V. Cristini, J. Lowengrub and Q. Nie,
Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
doi: 10.1007/s00285-002-0174-6. |
[18] |
T. A. Davis,
Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), 196-199.
doi: 10.1145/992200.992206. |
[19] |
T. A. Davis,
Algorithm 849: A concise sparse Cholesky factorization package, ACM Trans. Math. Software, 31 (2005), 587-591.
doi: 10.1145/1114268.1114277. |
[20] |
M. Ebenbeck, Cahn–Hilliard–Brinkman Models for Tumour Growth: Modelling, Analysis and Optimal Control, Ph.D thesis, University Regensburg, 2020. Google Scholar |
[21] |
M. Ebenbeck and H. Garcke,
Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998-6036.
doi: 10.1016/j.jde.2018.10.045. |
[22] |
M. Ebenbeck and H. Garcke,
On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM J. Math. Anal., 51 (2019), 1868-1912.
doi: 10.1137/18M1228104. |
[23] |
C. Eck, H. Garcke and P. Knabner, Mathematical Modeling, Springer, Cham, 2017.
doi: 10.1007/978-3-319-55161-6. |
[24] |
C. M. Elliott and H. Garcke,
On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
[25] |
J. Eyles, J. R. King and V. Styles,
A tractable mathematical model for tissue growth, Interfaces Free Bound., 21 (2019), 463-493.
doi: 10.4171/IFB/428. |
[26] |
S. J. Franks and J. R. King,
Interactions between a uniformly proliferating tumour and its surroundings: Stability analysis for variable material properties, Internat. J. Engrg. Sci., 47 (2009), 1182-1192.
doi: 10.1016/j.ijengsci.2009.07.004. |
[27] |
S. J. Franks and J. R. King,
Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 47-89.
doi: 10.1093/imammb/20.1.47. |
[28] |
H. B. Frieboes, J. S. Lowengrub, S. Wise, X. Zheng, P. Macklin, E. L. Bearer and V. Cristini,
Computer simulation of glioma growth and morphology, NeuroImage, 37 (2007), 59-70.
doi: 10.1016/j.neuroimage.2007.03.008. |
[29] |
A. Friedman,
A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound., 8 (2006), 247-261.
doi: 10.4171/IFB/142. |
[30] |
A. Friedman, Free boundary problems associated with multiscale tumor models, Math. Model. Nat. Phenom., 4 (2009), 134-155.
doi: 10.1051/mmnp/20094306. |
[31] |
A. Friedman,
Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[32] |
A. Friedmann and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
[33] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[34] |
S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (Springer INdAM ser.), 22 Springer Cham, (2017), 217–254. |
[35] |
M. Fritz, E. A. B. F. Lima, J. T. Oden and B. Wohlmuth,
On the unsteady Darcy–Forchheimer–Brinkman equation in local and nonlocal tumor growth models, Math. Models Methods Appl. Sci., 29 (2019), 1691-1731.
doi: 10.1142/S0218202519500325. |
[36] |
H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Math., 1 (2016), 318-360.
doi: 10.3934/Math.2016.3.318. |
[37] |
H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, Trends in Applications of Mathematics to Mechanics, Springer, Cham 27 (2018), 243–264. |
[38] |
H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka,
A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525-577.
doi: 10.1142/S0218202518500148. |
[39] |
H. Garcke, K. F. Lam and A. Signori, On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects, Nonlinear Anal. Real World Appl., 57 (2021), 103192.
doi: 10.1016/j.nonrwa.2020.103192. |
[40] |
H. Garcke, K. F. Lam, E. Sitka and V. Styles,
A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.
doi: 10.1142/S0218202516500263. |
[41] |
H. Garcke and B. Stinner,
Second order phase field asymptotics for multi-component systems, Interfaces Free Bound., 8 (2006), 131-157.
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