American Institute of Mathematical Sciences

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doi: 10.3934/dcds.2020373

Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential

Andrea Signori

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

Received  May 2020 Revised  September 2020 Published  November 2020

A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn–Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimised possesses some objective terms and it also penalises long treatments time, which may affect harm to the patients, and big aggregations of tumourous cells. Hence, the optimisation problem leads to the optimal strategy which reduces the time exposure of the patient to the medication and at the same time allows the doctors to achieve suitable clinical goals.

Keywords: Optimal control, free terminal time, phase field, tumour growth, Cahn–Hilliard equation, adjoint system, necessary optimality conditions.
Mathematics Subject Classification: Primary: 35Q92, 49J20, 49K20; Secondary: 35K86, 92C50.
Citation: Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020373
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.  Google Scholar

[2]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10] 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.  Google Scholar
[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 71, 38 pp. doi: 10.1051/cocv/2019059.  Google Scholar

[14]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar

[15]

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.  Google Scholar

[16]

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, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Cham, 22 (2017), 217–254. doi: 10.1007/978-3-319-64489-9_9.  Google Scholar

[17]

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.   Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, in Trends on Applications of Mathematics to Mechanics, E. Rocca, U. Stefanelli, L. Truskinovski, A. Visintin (ed.), Springer INdAM Series, Springer, Cham, 27 (2018), 243–264. doi: 10.1007/978-3-319-75940-1_12.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

J.-L. Lions, Contrôle Optimal de Systèmes Gouverneś par des Equations aux Dérivées Partielles, Dunod, Paris, 1968.  Google Scholar

[29]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.  Google Scholar

[30]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.   Google Scholar

[31]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV(eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[32]

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.  Google Scholar

[33]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43–66. doi: 10.3233/ASY-191546.  Google Scholar

[34]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305–331. doi: 10.3934/mcrf.2019040.  Google Scholar

[35]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193–217. doi: 10.3934/eect.2020003.  Google Scholar

[36]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[37]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[38]

J. Sprekels and H. Wu, Optimal distributed control of a Cahn–Hilliard–Darcy system with Mass sources, Appl. Math. Optim., (2019), 1–42. Google Scholar

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Grad. Stud. in Math., 112, AMS, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[40]

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.  Google Scholar

[41]

X. Wu, G. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahn–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.  Google Scholar

show all references

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.  Google Scholar

[2]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10] 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.  Google Scholar
[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 71, 38 pp. doi: 10.1051/cocv/2019059.  Google Scholar

[14]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar

[15]

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.  Google Scholar

[16]

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, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Cham, 22 (2017), 217–254. doi: 10.1007/978-3-319-64489-9_9.  Google Scholar

[17]

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.   Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, in Trends on Applications of Mathematics to Mechanics, E. Rocca, U. Stefanelli, L. Truskinovski, A. Visintin (ed.), Springer INdAM Series, Springer, Cham, 27 (2018), 243–264. doi: 10.1007/978-3-319-75940-1_12.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

J.-L. Lions, Contrôle Optimal de Systèmes Gouverneś par des Equations aux Dérivées Partielles, Dunod, Paris, 1968.  Google Scholar

[29]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.  Google Scholar

[30]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.   Google Scholar

[31]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV(eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[32]

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.  Google Scholar

[33]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43–66. doi: 10.3233/ASY-191546.  Google Scholar

[34]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305–331. doi: 10.3934/mcrf.2019040.  Google Scholar

[35]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193–217. doi: 10.3934/eect.2020003.  Google Scholar

[36]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[37]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[38]

J. Sprekels and H. Wu, Optimal distributed control of a Cahn–Hilliard–Darcy system with Mass sources, Appl. Math. Optim., (2019), 1–42. Google Scholar

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Grad. Stud. in Math., 112, AMS, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[40]

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.  Google Scholar

[41]

X. Wu, G. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahn–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.  Google Scholar

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