# American Institute of Mathematical Sciences

May  2022, 27(5): 2455-2469. doi: 10.3934/dcdsb.2021140

## 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 Published  May 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (5) : 2455-2469. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [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.
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