Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation

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Abstract

In this paper, we study a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable and the chemical potential, Robin boundary conditions for the nutrient and a “no-friction” boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the variables of the state equation during the evolution and the distribution of tumor cells at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish first-order necessary optimality conditions for the optimal control problem.

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References

  1. 1.

    Abels, H., Terasawa, Y.: On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Agosti, A., Cattaneo, C., Giverso, C., Ambrosi, D., Ciarletta, P.: A computational framework for the personalized clinical treatment of glioblastoma multiforme. ZAMM J. Appl. Math. Mech. 98, 2307–2327 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Amann, H.: Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory. Birkhäuser, Basel (1995)

    Google Scholar 

  4. 4.

    Astanin, S., Preziosi, L.: Multiphase models of tumour growth. In: Angelis, E., Chaplain, M.A.J., Bellomo, N. (eds.) Selected topics in cancer modeling, Modeling and Simulation in Science, Engineering and Technology, pp. 223–253. Birkhäuser Boston, Boston (2008)

    Google Scholar 

  5. 5.

    Bearer, E.L., Lowengrub, J.S., Frieboes, H.B., Chuang, Y.L., Jin, F., Wise, S.M., Ferrari, M., Agus, V., David, B., Cristini, V.: Multiparameter computational modeling of tumor invasion. Cancer Res. 69(10), 4493–4501 (2009)

    Article  Google Scholar 

  6. 6.

    Benosman, C., Aïnseba, B., Ducrot, A.: Optimization of cytostatic leukemia therapy in an advection–reaction–diffusion model. J. Optim. Theory Appl. 167(1), 296–325 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Biswas, T., Dharmatti, S., Mohan, M.T.: Pontryagin’s maximum principle for optimal control of the nonlocal Cahn–Hilliard–Navier-Stokes systems in two dimensions (2018). ArXiv e-prints: arXiv:1802.08413

  8. 8.

    Cavaterra, C., Rocca, E., Wu, H.: Long-time dynamics and optimal control of a diffuse interface model for tumor growth. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09562-5

  9. 9.

    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company Inc, New York (1955)

    Google Scholar 

  10. 10.

    Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53(4), 2696–2721 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Colli, P., Gilardi, G., Hilhorst, D.: On a Cahn–Hilliard type phase field system related to tumor growth. Discrete Contin. Dyn. Syst. 35(6), 2423–2442 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30(6), 2518–2546 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions. SIAM J. Control Optim. 56(3), 1665–1691 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Cristini, V., Li, X., Lowengrub, J.S., Wise, S.M.: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58(4–5), 723–763 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Ebenbeck, M., Garcke, H.: On a Cahn-Hilliard-Brinkman model for tumour growth and its singular limits. SIAM. J. Math. Anal. 51(3), 1868–1912 (2018)

    MATH  Google Scholar 

  16. 16.

    Ebenbeck, M., Garcke, H.: Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis. J. Differ. Equ. 266(9), 5998–6036 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Frieboes, H.B., Lowengrub, J., Wise, S., Zheng, X., Macklin, P., Bearer, E., Cristini, V.: Computer simulation of glioma growth and morphology. NeuroImage 37(Suppl 1), 59–70 (2007). 02

    Article  Google Scholar 

  18. 18.

    Frigeri, S., Grasselli, M., Rocca, E.: On a diffuse interface model of tumour growth. Eur J. Appl. Math. 26(2), 215–243 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Galdi, G.P.: An introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer Monographs in Mathematics. Steady-State Problems. Springer, New York (2011)

    Google Scholar 

  20. 20.

    Garcke, H., Lam, K.F.: Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Math. 1(Math–01–00318), 318–360 (2016)

    Article  MATH  Google Scholar 

  21. 21.

    Garcke, H., Lam, K.F.: Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28(2), 284–316 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Garcke, H., Lam, K.F.: On a Cahn–Hilliard-Darcy system for tumour growth with solution dependent source terms. In: Trends in Applications of Mathematics to Mechanics, Volume 27 of Springer INdAM Series, pp. 243–264. Springer, Cham (2018)

    Google Scholar 

  23. 23.

    Garcke, H., Lam, K.F., Rocca, E.: Optimal control of treatment time in a diffuse interface model of tumor growth. Appl. Math. Optim. 78(3), 495–544 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Garcke, H., Lam, K.F.: Analysis of a Cahn–Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete Contin. Dyn. Syst. 37(8), 42–77 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Garcke, H., Lam, K.F., Nürnberg, R., Sitka, E.: A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Models Methods Appl. Sci. 28(3), 525–577 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Garcke, H., Lam, K.F., Sitka, E., Styles, V.: A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26(6), 1095–1148 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Gilardi, G., Sprekels, J.: Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions. Nonlinear Anal. 178, 1–31 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Hawkins-Daarud, A., van der Zee, K.G., Oden, J.T.: Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Methods Biomed. Eng. 28(1), 3–24 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Hilhorst, D., Kampmann, J., Nguyen, T.N., Van Der Zee, K.G.: Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25(6), 1011–1043 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388–418 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259(7), 3032–3077 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Kahle, C., Lam, K.F.: Parameter identification via optimal control for a Cahn-Hilliard-chemotaxis system with a variable mobility. Appl. Math. Optim. (2018). https://doi.org/10.1007/s00245-018-9491-z

    Article  Google Scholar 

  33. 33.

    Ledzewicz, U., Schättler, H.: Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. Optim. Theory Appl. 153(1), 195–224 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Oden, J.T., Hawkins, A., Prudhomme, S.: General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20(3), 477–517 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Oke, S.I., Matadi, M.B., Xulu, S.S.: Optimal control analysis of a mathematical model for breast cancer. Math. Comput. Appl. 23(2), 21 (2018)

    MathSciNet  Google Scholar 

  36. 36.

    Preziosi, L., Tosin, A.: Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58(4), 625 (2008)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Schättler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies, Volume 42 of Interdisciplinary Applied Mathematics. An Application of Geometric Methods. Springer, New York (2015)

    Google Scholar 

  38. 38.

    Shibata, Y., Shimizu, S.: On the Stokes equation with Neumann boundary condition. In: Regularity and Other Aspects of the Navier–Stokes Equations, Volume 70 of Banach Center Publication, pp. 239–250. Polish Academy of Sciences Institute of Mathematics, Warsaw (2005)

  39. 39.

    Signori, A.: Optimal distributed control of an extended model of tumor growth with logarithmic potential. Appl. Math. Optim. (2018). https://doi.org/10.1007/s00245-018-9538-1

  40. 40.

    Signori, A.: Vanishing parameter for an optimal control problem modeling tumor growth (2019). ArXiv e-prints: arXiv:1903.04930

  41. 41.

    Simon, J.: Compact sets in the space \(L^p(0, T, B)\). Ann. Mat. Pura Appl. 146(1), 65–96 (1986)

    MATH  Article  Google Scholar 

  42. 42.

    Sprekels, J., Wu, H.: Optimal distributed control of a Cahn–Hilliard–Darcy system with mass sources. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09555-4

  43. 43.

    Swan, G.W.: Role of optimal control theory in cancer chemotherapy. Math. Biosci. 101(2), 237–284 (1990)

    MATH  Article  Google Scholar 

  44. 44.

    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam (1978)

    Google Scholar 

  45. 45.

    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)

    Google Scholar 

  46. 46.

    Zhao, X., Liu, C.: Optimal control of the convective Cahn–Hilliard equation. Appl. Anal. 92(5), 1028–1045 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Zhao, X., Liu, C.: Optimal control for the convective Cahn–Hilliard equation in 2D case. Appl. Math. Optim. 70(1), 61–82 (2014)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

Matthias Ebenbeck was supported by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). The support is gratefully acknowledged.

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Ebenbeck, M., Knopf, P. Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation. Calc. Var. 58, 131 (2019). https://doi.org/10.1007/s00526-019-1579-z

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Mathematics Subject Classification

  • 35K61
  • 76D07
  • 49J20
  • 49K20
  • 92C50