Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

Signori Andrea

doi:10.3934/mcrf.2019040

Article Contents

2020, Volume 10, Issue 2: 305-331. Doi: 10.3934/mcrf.2019040

This issue Previous Article On the exact controllability and the stabilization for the Benney-Luke equation Next Article Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

Signori Andrea

Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125 Milano, Italy

Received: January 31, 2019

Revised: April 30, 2019

Early access: August 2019

Published: April 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long-time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, we assume some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide the existence of optimal controls and a necessary condition for optimality is addressed.

Keywords:

Mathematics Subject Classification: Primary: 35K61, 35Q92, 49J20, 49K20; Secondary: 35K86, 92C50.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	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.
[2]	V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597. doi: 10.1016/0022-247X(81)90125-6.
[3]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, , Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leyden, 1976.
[4]	H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973.
[5]	C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., (2019), 1–49. doi: 10.1007/s00245-019-09562-5.
[6]	P. Colli, M. H. Farshbaf-Shaker, G. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721. doi: 10.1137/140984749.
[7]	P. Colli, M. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24. doi: 10.1007/s00245-014-9250-8.
[8]	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.
[9]	P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112. doi: 10.3934/mcrf.2016.6.95.
[10]	P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116. doi: 10.3934/eect.2018006.
[11]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108. doi: 10.1016/j.nonrwa.2015.05.002.
[12]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546. doi: 10.1088/1361-6544/aa6e5f.
[13]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modeling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54. doi: 10.3934/dcdss.2017002.
[14]	P. Colli, G. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994. doi: 10.1016/j.jmaa.2014.05.008.
[15]	P. Colli, G. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Opt., 73 (2016), 195-225. doi: 10.1007/s00245-015-9299-z.
[16]	P. Colli, G. Gilardi and J. Sprekels, Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition, Nonlinear Anal., 170 (2018), 171-196. doi: 10.1016/j.na.2018.01.003.
[17]	P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691. doi: 10.1137/17M1146786.
[18]	P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A 'deep quench' approach, J. Convex Anal., 26 (2019), 485–514. Available from: http://www.heldermann.de/JCA/JCA26/JCA262/jca26024.htm.
[19]	P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234. doi: 10.1137/120902422.
[20]	V. Cristini, X. R. 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.
[21]	V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Leiden, 2010. doi: 10.1017/CBO9780511781452.
[22]	M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658. doi: 10.1088/1361-6544/aa6063.
[23]	M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, preprint, arXiv: 1903.00333.
[24]	M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019). doi: 10.1007/s00526-019-1579-z.
[25]	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.
[26]	S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243. doi: 10.1017/S0956792514000436.
[27]	S. Frigeri, K. F. Lam, E. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821-856. doi: 10.4310/CMS.2018.v16.n3.a11.
[28]	S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 217–254. doi: 10.1007/978-3-319-64489-9_9.
[29]	H. Garcke and K. F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European. J. Appl. Math., 28 (2017), 284-316. doi: 10.1017/S0956792516000292.
[30]	H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277-4308. doi: 10.3934/dcds.2017183.
[31]	H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. doi: 10.3934/Math.2016.3.318.
[32]	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 INdAM Ser., Springer, Cham, 27 (2018), 243–264. doi: 10.1007/978-3-319-75940-1_12.
[33]	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.
[34]	H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495-544. doi: 10.1007/s00245-017-9414-4.
[35]	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.
[36]	G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.
[37]	A. Hawkins-Daarud, S. Prudhomme, K. G. van der Zee and J. T. Oden, Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, J. Math. Biol., 67 (2013), 1457-1485. doi: 10.1007/s00285-012-0595-9.
[38]	A. Hawkins-Daruud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng., 28 (2012), 3-24. doi: 10.1002/cnm.1467.
[39]	D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011-1043. doi: 10.1142/S0218202515500268.
[40]	S. Kurima, Asymptotic analysis for Cahn-Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci., 42 (2019), 2431-2454. doi: 10.1002/mma.5520.
[41]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Mathematical Monographs, 23. American mathematical society, Providence, R. I. 1968.
[42]	J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Grundlehren der Mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
[43]	A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544. doi: 10.3934/Math.2017.2.479.
[44]	A. Miranville, E. Rocca and and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642. doi: 10.1016/j.jde.2019.03.028.
[45]	A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.
[46]	J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517. doi: 10.1142/S0218202510004313.
[47]	A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, preprint, arXiv: 1903.04930.
[48]	A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, preprint, arXiv: 1811.08626.
[49]	A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9538-1.
[50]	J. Simon, Compact sets in the space L^p(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65–96. Available from: https://doi.org/10.1007/BF01762360.
[51]	J. Sprekels and H. Wu, Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources, Appl. Math. Optim., (2019), 1–42. doi: 10.1007/s00245-019-09555-4.
[52]	F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.
[53]	S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method, J. Theor. Biol., 253 (2008), 524-543. doi: 10.1016/j.jtbi.2008.03.027.
[54]	X. Wu, G. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahnf-Hilliard models with applications to diffuse–interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180-203. doi: 10.1002/cnm.2597.