June  2021, 41(6): 2519-2542. 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

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

Received  May 2020 Revised  September 2020 Published  June 2021 Early access  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.

Citation: Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373
References:
[1]

A. AgostiP. F. AntoniettiP. CiarlettaM. 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]

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.

[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.

[4]

P. ColliG. 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.

[5]

P. ColliG. GilardiG. 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.

[6]

P. ColliG. GilardiE. 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.

[7]

P. ColliG. GilardiE. 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.

[8]

P. ColliG. GilardiE. 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.

[9]

V. CristiniX. LiJ. 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.

[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.
[11]

M. DaiE. FeireislE. RoccaG. 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.

[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.

[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.

[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.

[15]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.

[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.

[17]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821-856. 

[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.

[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.

[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.

[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.

[22]

H. GarckeK. F. LamR. 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.

[23]

H. GarckeK. 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.

[24]

H. GarckeK. F. LamE. 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.

[25]

A. Hawkins-DaarudS. PrudhommeK. 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.

[26]

A. Hawkins-DaruudK. 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.

[27]

D. HilhorstJ. KampmannT. 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.

[28]

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

[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.

[30]

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

[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.

[32]

J. T. OdenA. 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.

[33]

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

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

S. M. WiseJ. S. LowengrubH. 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.

[41]

X. WuG. 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.

show all references

References:
[1]

A. AgostiP. F. AntoniettiP. CiarlettaM. 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]

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.

[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.

[4]

P. ColliG. 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.

[5]

P. ColliG. GilardiG. 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.

[6]

P. ColliG. GilardiE. 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.

[7]

P. ColliG. GilardiE. 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.

[8]

P. ColliG. GilardiE. 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.

[9]

V. CristiniX. LiJ. 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.

[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.
[11]

M. DaiE. FeireislE. RoccaG. 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.

[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.

[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.

[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.

[15]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.

[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.

[17]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821-856. 

[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.

[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.

[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.

[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.

[22]

H. GarckeK. F. LamR. 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.

[23]

H. GarckeK. 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.

[24]

H. GarckeK. F. LamE. 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.

[25]

A. Hawkins-DaarudS. PrudhommeK. 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.

[26]

A. Hawkins-DaruudK. 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.

[27]

D. HilhorstJ. KampmannT. 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.

[28]

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

[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.

[30]

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

[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.

[32]

J. T. OdenA. 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.

[33]

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

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

S. M. WiseJ. S. LowengrubH. 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.

[41]

X. WuG. 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.

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