# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021140
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

 1 Dipartimento di Matematica, Universita' degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy 2 Dipartimento di Matematica, Universita' degli Studi di Pavia and IMATI-C.N.R., Via Ferrata 5, 27100 Pavia, Italy

* Corresponding author

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: Tania Biswas would like to thank Department of Mathematics, University of Pavia for providing financial support and stimulating environment for the research and Prof. Elisabetta Rocca, Prof. Pierluigi Colli for fruitful discussions. This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics "F. Casorati", University of Pavia. In addition, this research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR "Matematica d'Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l'ATtRattività dell'ateneo pavese". The present paper also benefits from the support of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica)

We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [2]. It is comprised of phase-field equation to describe tumor growth, which is coupled to a reaction-diffusion type equation for generic nutrient for the tumor. An additional equation couples the concentration of prostate-specific antigen (PSA) in the prostatic tissue and it obeys a linear reaction-diffusion equation. The system completes with homogeneous Dirichlet boundary conditions for the tumor variable and Neumann boundary condition for the nutrient and the concentration of PSA. Here we investigate the long time dynamics of the model. We first prove that the initial-boundary value problem generates a strongly continuous semigroup on a suitable phase space that admits the global attractor in a proper phase space. Moreover, we also discuss the convergence of a solution to a single stationary state and obtain a convergence rate estimate under some conditions on the coefficients.

Citation: Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021140
##### References:
 [1] C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Applied Mathematics & Optimization, 83 (2021), 739-787.  doi: 10.1007/s00245-019-09562-5.  Google Scholar [2] P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali and E. Rocca, Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Mathematical Models and Methods in Applied Sciences, 30 (2020), 1253-1295.  doi: 10.1142/S0218202520500220.  Google Scholar [3] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Applied Mathematics & Optimization, 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar [4] P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discrete & Continuous Dynamical Systems, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.  Google Scholar [5] 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 [6] P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth, Discrete & Continuous Dynamical Systems, 35 (2015), 2423-2442.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar [7] P. Colli, G. Gilardi, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete & Continuous Dynamical Systems, 25 (2019), 63-81.  doi: 10.3934/dcds.2009.25.63.  Google Scholar [8] M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. Schonbek, Analysis of a diffuse interface model for multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.  doi: 10.1088/1361-6544/aa6063.  Google Scholar [9] M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, Journal of Differential Equations, 266 (2019), 5998-6036.  doi: 10.1016/j.jde.2018.10.045.  Google Scholar [10] M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calculus of Variations, 58 (2019), Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar [11] E. Feireisl, F. Issard-Roch and H. Petzeltova, Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete & Continuous Dynamical Systems, 10 (2004), 239-252.  doi: 10.3934/dcds.2004.10.239.  Google Scholar [12] S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumour growth, European Journal of Applied Mathematics, 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.  Google Scholar [13] 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 [14] H. Garcke and K. F. Lam, Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis, Discrete & Continuous Dynamical Systems, 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.  Google Scholar [15] H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Applied Mathematics & Optimization, 78 (2018), 495-544.  doi: 10.1007/s00245-017-9414-4.  Google Scholar [16] H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar [17] H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences, 28 (2018), 525-577.  doi: 10.1142/S0218202518500148.  Google Scholar [18] W. Hao, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic–hyperbolic phase-field system with Neumann boundary conditions, Mathematical Models and Methods in Applied Sciences, 17 (2007), 125-153.  doi: 10.1142/S0218202507001851.  Google Scholar [19] J. Jiang, H. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth, Journal of Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.  Google Scholar [20] P. Laurençot, Long-time behaviour for a model of phase-field type, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 126 (1996), 167-185.  doi: 10.1017/S0308210500030663.  Google Scholar [21] G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), E7663–E7671. doi: 10.1073/pnas.1615791113.  Google Scholar [22] J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European Journal of Applied Mathematics, 24 (2013) 691–734. doi: 10.1017/S0956792513000144.  Google Scholar [23] A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, Journal of Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.  Google Scholar [24] E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Physica D: Nonlinear Phenomena, 192 (2004), 279-307.  doi: 10.1016/j.physd.2004.01.024.  Google Scholar [25] A. Sergiu, E. Feireisl and F. Issard–Roch, Long–time convergence of solutions to a phase–field system, Mathematical methods in the applied sciences, 24 (2001), 277-287.  doi: 10.1002/mma.215.  Google Scholar [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [27] J. Xu, G. Vilanova and H. Gomez, A mathematical model coupling tumor growth and angiogenesis, PLoS ONE, 11 (2016), e0149422. doi: 10.1371/journal.pone.0149422.  Google Scholar

show all references

##### References:
 [1] C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Applied Mathematics & Optimization, 83 (2021), 739-787.  doi: 10.1007/s00245-019-09562-5.  Google Scholar [2] P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali and E. Rocca, Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Mathematical Models and Methods in Applied Sciences, 30 (2020), 1253-1295.  doi: 10.1142/S0218202520500220.  Google Scholar [3] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Applied Mathematics & Optimization, 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar [4] P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discrete & Continuous Dynamical Systems, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.  Google Scholar [5] 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 [6] P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth, Discrete & Continuous Dynamical Systems, 35 (2015), 2423-2442.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar [7] P. Colli, G. Gilardi, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete & Continuous Dynamical Systems, 25 (2019), 63-81.  doi: 10.3934/dcds.2009.25.63.  Google Scholar [8] M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. Schonbek, Analysis of a diffuse interface model for multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.  doi: 10.1088/1361-6544/aa6063.  Google Scholar [9] M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, Journal of Differential Equations, 266 (2019), 5998-6036.  doi: 10.1016/j.jde.2018.10.045.  Google Scholar [10] M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calculus of Variations, 58 (2019), Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar [11] E. Feireisl, F. Issard-Roch and H. Petzeltova, Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete & Continuous Dynamical Systems, 10 (2004), 239-252.  doi: 10.3934/dcds.2004.10.239.  Google Scholar [12] S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumour growth, European Journal of Applied Mathematics, 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.  Google Scholar [13] 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 [14] H. Garcke and K. F. Lam, Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis, Discrete & Continuous Dynamical Systems, 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.  Google Scholar [15] H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Applied Mathematics & Optimization, 78 (2018), 495-544.  doi: 10.1007/s00245-017-9414-4.  Google Scholar [16] H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar [17] H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences, 28 (2018), 525-577.  doi: 10.1142/S0218202518500148.  Google Scholar [18] W. Hao, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic–hyperbolic phase-field system with Neumann boundary conditions, Mathematical Models and Methods in Applied Sciences, 17 (2007), 125-153.  doi: 10.1142/S0218202507001851.  Google Scholar [19] J. Jiang, H. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth, Journal of Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.  Google Scholar [20] P. Laurençot, Long-time behaviour for a model of phase-field type, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 126 (1996), 167-185.  doi: 10.1017/S0308210500030663.  Google Scholar [21] G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), E7663–E7671. doi: 10.1073/pnas.1615791113.  Google Scholar [22] J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European Journal of Applied Mathematics, 24 (2013) 691–734. doi: 10.1017/S0956792513000144.  Google Scholar [23] A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, Journal of Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.  Google Scholar [24] E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Physica D: Nonlinear Phenomena, 192 (2004), 279-307.  doi: 10.1016/j.physd.2004.01.024.  Google Scholar [25] A. Sergiu, E. Feireisl and F. Issard–Roch, Long–time convergence of solutions to a phase–field system, Mathematical methods in the applied sciences, 24 (2001), 277-287.  doi: 10.1002/mma.215.  Google Scholar [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [27] J. Xu, G. Vilanova and H. Gomez, A mathematical model coupling tumor growth and angiogenesis, PLoS ONE, 11 (2016), e0149422. doi: 10.1371/journal.pone.0149422.  Google Scholar
 [1] Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1 [2] Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873 [3] Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375 [4] Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193 [5] Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367 [6] Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949 [7] Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63 [8] Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077 [9] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [10] Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683 [11] Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824 [12] Eduard Feireisl, Françoise Issard-Roch, Hana Petzeltová. Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 239-252. doi: 10.3934/dcds.2004.10.239 [13] Erica M. Rutter, Yang Kuang. Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1001-1021. doi: 10.3934/dcdsb.2017050 [14] Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018 [15] Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098 [16] Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041 [17] Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272 [18] Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021140 [19] Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058 [20] A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373

2020 Impact Factor: 1.327

Article outline

[Back to Top]