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doi: 10.3934/dcdss.2022003
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Optimal control of an Allen-Cahn model for tumor growth through supply of cytotoxic drugs

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

2. 

Lebanese University, Khawarizmi Laboratory for Mathematics and Applications, Beirut, Lebanon

3. 

Lebanese International University, Department of Mathematics and Physics, Lebanon

4. 

Xiamen University, School of Mathematical Sciences, Xiamen, Fujian, China

5. 

Université de Poitiers, Laboratoire I3M and Laboratoire de Mathématiques et Applications, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

*Corresponding author: Alain Miranville

Received  August 2021 Revised  November 2021 Early access January 2022

Our aim in this paper is to study an optimal control problem for a tumor growth model. The state system couples an Allen-Cahn equation and a reaction diffusion equation that models the evolution of tumor in the presence of nutrient supply. Elimination of cancer cells via cytotoxic drug is considered and the concentration of the cytotoxic drug is represented as a control variable.

Citation: Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Optimal control of an Allen-Cahn model for tumor growth through supply of cytotoxic drugs. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022003
References:
[1]

H. Alsayed, H. Fakih, A. Miranville and A. Wehbe, On an optimal control problem describing lactate production inhibition, Applicable Analysis, (2021), 1–21. doi: 10.1080/00036811.2021.1999418.

[2]

R. Araujo, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[3]

N. BellomoN. 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.

[4]

H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.

[5]

H. Byrne, The role of mathematics in solid tumour growth, Math. Today, (Southend-on-Sea), 35 (1999), 48–53.

[6]

H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Mathematical Biosciences, 130 (1995), 151-181.  doi: 10.1016/0025-5564(94)00117-3.

[7]

H. Byrne and S. Gourley, The role of growth factors in avascular tumour growth, Math. Comput. Modelling, 26 (1997), 35-55.  doi: 10.1016/S0895-7177(97)00143-X.

[8]

J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in fe-al alloy domain growth kinetics, J. Phys. Colloques, 38 (1977), 51-54.  doi: 10.1051/jphyscol:1977709.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[10]

C. CavaterraE. Rocca and H. Wu, Long-Time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim, 83 (2021), 739-787.  doi: 10.1007/s00245-019-09562-5.

[11]

R. V. J. Chari, Targeted cancer therapy: Conferring specificity to cytotoxic drugs, Acc. Chem. Res., 41 (2008), 98-107.  doi: 10.1021/ar700108g.

[12]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.

[13]

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.

[14]

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.

[15]

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.

[16]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[17]

P. ColliH. GomezG. LorenzoG. MarinoschiA. Reali and E. Rocca, Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth, Math. Models Methods Appl. Sci., 31 (2021), 1419-1468.  doi: 10.1142/S0218202521500299.

[18]

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.

[19]

R. Dautray and J. Lions, Analyse Mathématique Et Calcul Numérique Pour Les Sciences Et Les Techniques, Masson, Paris, 1988.

[20]

T. S. DeisboeckZ. WangP. Macklin and V. Cristini, Multiscale cancer modeling, Annual Review of Biomedical Engineering, 13 (2011), 127-155. 

[21]

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

[22]

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.

[23]

L. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[24]

H. Fakih, A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.

[25]

H. FakihR. Mghames and N. Nasreddine, On the Cahn-Hilliard equation with mass source for biological applications, Commun. Pure Appl. Anal., 20 (2021), 495-510.  doi: 10.3934/cpaa.2020277.

[26]

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

[27]

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.

[28]

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.

[29]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418.  doi: 10.1137/110824152.

[30]

J. JiangH. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.

[31] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[32]

A. Miranville, The Cahn-Hilliard equation with a nonlinear source term, J. Differential Equations, 294 (2021), 88-117.  doi: 10.1016/j.jde.2021.05.045.

[33]

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.

[34]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680.  doi: 10.1137/140964308.

[35]

F. Tröltzsch and J. Sprekels, 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.

show all references

References:
[1]

H. Alsayed, H. Fakih, A. Miranville and A. Wehbe, On an optimal control problem describing lactate production inhibition, Applicable Analysis, (2021), 1–21. doi: 10.1080/00036811.2021.1999418.

[2]

R. Araujo, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[3]

N. BellomoN. 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.

[4]

H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.

[5]

H. Byrne, The role of mathematics in solid tumour growth, Math. Today, (Southend-on-Sea), 35 (1999), 48–53.

[6]

H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Mathematical Biosciences, 130 (1995), 151-181.  doi: 10.1016/0025-5564(94)00117-3.

[7]

H. Byrne and S. Gourley, The role of growth factors in avascular tumour growth, Math. Comput. Modelling, 26 (1997), 35-55.  doi: 10.1016/S0895-7177(97)00143-X.

[8]

J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in fe-al alloy domain growth kinetics, J. Phys. Colloques, 38 (1977), 51-54.  doi: 10.1051/jphyscol:1977709.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[10]

C. CavaterraE. Rocca and H. Wu, Long-Time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim, 83 (2021), 739-787.  doi: 10.1007/s00245-019-09562-5.

[11]

R. V. J. Chari, Targeted cancer therapy: Conferring specificity to cytotoxic drugs, Acc. Chem. Res., 41 (2008), 98-107.  doi: 10.1021/ar700108g.

[12]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.

[13]

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.

[14]

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.

[15]

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.

[16]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[17]

P. ColliH. GomezG. LorenzoG. MarinoschiA. Reali and E. Rocca, Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth, Math. Models Methods Appl. Sci., 31 (2021), 1419-1468.  doi: 10.1142/S0218202521500299.

[18]

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.

[19]

R. Dautray and J. Lions, Analyse Mathématique Et Calcul Numérique Pour Les Sciences Et Les Techniques, Masson, Paris, 1988.

[20]

T. S. DeisboeckZ. WangP. Macklin and V. Cristini, Multiscale cancer modeling, Annual Review of Biomedical Engineering, 13 (2011), 127-155. 

[21]

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

[22]

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.

[23]

L. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[24]

H. Fakih, A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.

[25]

H. FakihR. Mghames and N. Nasreddine, On the Cahn-Hilliard equation with mass source for biological applications, Commun. Pure Appl. Anal., 20 (2021), 495-510.  doi: 10.3934/cpaa.2020277.

[26]

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

[27]

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.

[28]

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.

[29]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418.  doi: 10.1137/110824152.

[30]

J. JiangH. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.

[31] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[32]

A. Miranville, The Cahn-Hilliard equation with a nonlinear source term, J. Differential Equations, 294 (2021), 88-117.  doi: 10.1016/j.jde.2021.05.045.

[33]

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.

[34]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680.  doi: 10.1137/140964308.

[35]

F. Tröltzsch and J. Sprekels, 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.

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